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ALE Testing - Article.ipynb
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{
"cells": [
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"scrolled": true
},
"outputs": [],
"source": [
"from main import testing_ALE, testing_defs_ALE, testing_defs_ALE_CT\n",
"#testing_defs_ALE(solver_to_use='LU')\n",
"#testing_defs_ALE_CT(solver_to_use='LU')\n",
"dict_mt, dict_ct = testing_ALE(solver_to_use='LU') # 'LU', 'Krylov'"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"import matplotlib.pyplot as plt\n",
"def testing_comparative_plots(dict_fields_mt: dict, dict_fields_ct: dict, *args) -> None:\n",
" \"\"\"\n",
" Computes error figures between the schemes with Jacobian evaluation at times {n, n+1}, i.e.\n",
" with explicit and implicit treatment of the time-derivative term.\n",
" ...\n",
" \n",
" Input Parameters\n",
" ----------------\n",
" dict_fields_mt : tuple -> (dict, dict, dict)\n",
" tuple containing dictionaries of fields (def, vel, pre) from the iNSE ALE Monolithic solver.\n",
" \n",
" dict_fields_ct : tupel -> (dict, dict, dict)\n",
" tuple containing dictionaries of fields (def, vel, pre) from the iNSE ALE CT solver. \n",
" \"\"\"\n",
" print(\"\\nComputing error figures for both schemes...\\n\")\n",
" # Creates dictionaries for \"expanding\" and \"contracting\" functions\n",
" # Time dictionary\n",
" time_list = {\n",
" 'MT': {jac_name: [] for jac_name in dict_fields_mt.keys()},\n",
" 'CT': {jac_name: [] for jac_name in dict_fields_ct.keys()}\n",
" }\n",
" # Numerical dissipation dictionary --> int (| J^{n+1} u^{n+1} - J^{n} u^{n} |)\n",
" dis_value_exp = {\n",
" 'MT': {jac_name: [] for jac_name in dict_fields_mt.keys()},\n",
" 'CT': {jac_name: [] for jac_name in dict_fields_ct.keys()}\n",
" }\n",
" dis_value_con = {\n",
" 'MT': {jac_name: [] for jac_name in dict_fields_mt.keys()},\n",
" 'CT': {jac_name: [] for jac_name in dict_fields_ct.keys()}\n",
" }\n",
" # Physical dissipation dictionary --> int ( 2*mu* | eps(u) |^2 )\n",
" eps_value_exp = {\n",
" 'MT': {jac_name: [] for jac_name in dict_fields_mt.keys()},\n",
" 'CT': {jac_name: [] for jac_name in dict_fields_ct.keys()}\n",
" }\n",
" eps_value_con = {\n",
" 'MT': {jac_name: [] for jac_name in dict_fields_mt.keys()},\n",
" 'CT': {jac_name: [] for jac_name in dict_fields_ct.keys()}\n",
" }\n",
" # 'delta' values dictionary\n",
" delta_value_exp = {\n",
" 'MT': {jac_name: [] for jac_name in dict_fields_mt.keys()},\n",
" 'CT': {jac_name: [] for jac_name in dict_fields_ct.keys()}\n",
" }\n",
" delta_value_con = {\n",
" 'MT': {jac_name: [] for jac_name in dict_fields_mt.keys()},\n",
" 'CT': {jac_name: [] for jac_name in dict_fields_ct.keys()}\n",
" }\n",
" # Normalized 'delta' values dictionary (w.r.t physical dissipation)\n",
" delta_hat_value_exp = {\n",
" 'MT': {jac_name: [] for jac_name in dict_fields_mt.keys()},\n",
" 'CT': {jac_name: [] for jac_name in dict_fields_ct.keys()}\n",
" }\n",
" delta_hat_value_con = {\n",
" 'MT': {jac_name: [] for jac_name in dict_fields_mt.keys()},\n",
" 'CT': {jac_name: [] for jac_name in dict_fields_ct.keys()}\n",
" }\n",
" # Normalized 'delta' values dictionary (w.r.t numerical dissipation)\n",
" delta_hat_dis_exp = {\n",
" 'MT': {jac_name: [] for jac_name in dict_fields_mt.keys()},\n",
" 'CT': {jac_name: [] for jac_name in dict_fields_ct.keys()}\n",
" }\n",
" delta_hat_dis_con = {\n",
" 'MT': {jac_name: [] for jac_name in dict_fields_mt.keys()},\n",
" 'CT': {jac_name: [] for jac_name in dict_fields_ct.keys()}\n",
" }\n",
" # Iterate over the different cases in MT (Monolithic) and CT (Chorin-Temam) schemes\n",
" for scheme in ['MT', 'CT']: # 'MT', 'CT'\n",
" if scheme == 'MT':\n",
" dict_fields = dict_fields_mt\n",
" else:\n",
" dict_fields = dict_fields_ct\n",
" # Iterate over different cases 'j_implicit', 'j_explicit'\n",
" for jac_name in dict_fields.keys():\n",
" dict_vel = dict_fields[jac_name][1]\n",
" for i in range(len(dict_vel['High_Osc_exp'])):\n",
" # Saving time steps\n",
" time_list[scheme][jac_name].append(dict_vel['High_Osc_exp'][i][1])\n",
" # Add 'delta' values\n",
" delta_value_exp[scheme][jac_name].append(dict_vel['High_Osc_exp'][i][2])\n",
" #delta_value_con[scheme][jac_name].append(dict_vel['High_Osc_con'][i][2])\n",
" # Add physical dissipation\n",
" eps_value_exp[scheme][jac_name].append(dict_vel['High_Osc_exp'][i][3])\n",
" #eps_value_con[scheme][jac_name].append(dict_vel['High_Osc_con'][i][3])\n",
" # Add numerical dissipation\n",
" dis_value_exp[scheme][jac_name].append(dict_vel['High_Osc_exp'][i][4])\n",
" #dis_value_con[scheme][jac_name].append(dict_vel['High_Osc_con'][i][4])\n",
" # Compute normalized 'delta' values w.r.t the physical dissipation\n",
" delta_hat_value_exp[scheme][jac_name].append(dict_vel['High_Osc_exp'][i][2]/dict_vel['High_Osc_exp'][i][3])\n",
" #delta_hat_value_con[scheme][jac_name].append(dict_vel['High_Osc_con'][i][2]/dict_vel['High_Osc_con'][i][3])\n",
" # Compute normalized 'delta' values w.r.t the numerical dissipation\n",
" delta_hat_dis_exp[scheme][jac_name].append(dict_vel['High_Osc_exp'][i][2]/dict_vel['High_Osc_exp'][i][4])\n",
" #delta_hat_dis_con[scheme][jac_name].append(dict_vel['High_Osc_con'][i][2]/dict_vel['High_Osc_con'][i][4])\n",
"\n",
" # Auxiliary indexes for plotting\n",
" # TODO: Test this line and automatize the \"100\"!\n",
" init = 0\n",
" out = 101 #len(time_list['CT']['j_implicit']) # at tau = 0.01, T = 2 --> 200 time steps\n",
"\n",
" # ---------------------- #\n",
" # DELTA VALUE CASE\n",
" # ---------------------- #\n",
" # CASE CT\n",
" plt.subplots(figsize=(10, 4))\n",
" # Top \n",
" lf_ax = plt.subplot(1, 2, 2)\n",
" lf_j_exp, = lf_ax.plot(time_list['CT']['j_explicit'][init:out], delta_value_exp['CT']['j_explicit'][init:out], color='blue', ls='--')\n",
" lf_j_imp, = lf_ax.plot(time_list['CT']['j_implicit'][init:out], delta_value_exp['CT']['j_implicit'][init:out], color='blue')\n",
" lf_ax.set_xlabel(r\"Simulation time [t]\")\n",
" lf_ax.set_ylabel(r\"$\\delta_{CT}$\", color='blue')\n",
" lf_ax.set_title(r\"Residual Energy $\\delta_{CT}$\")\n",
" lf_ax.tick_params(axis='y', labelcolor='blue')\n",
" lf_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" #rh_ax = lf_ax.twinx()\n",
" #rh_j_exp, = rh_ax.plot(time_list['CT']['j_explicit'][init:out], delta_value_con['CT']['j_explicit'][init:out], color='red', ls='--')\n",
" #rh_j_imp, = rh_ax.plot(time_list['CT']['j_implicit'][init:out], delta_value_con['CT']['j_implicit'][init:out], color='red')\n",
" #rh_ax.set_ylabel(r\"$\\delta_{CT}$ def. II\", color='red')\n",
" #rh_ax.tick_params(axis='y', labelcolor='red')\n",
" #rh_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" plt.legend((lf_j_exp, lf_j_imp), (r'CT $\\bigstar\\bigstar = n$', r'CT $\\bigstar\\bigstar = n+1$'), loc='center right')\n",
" plt.grid(True)\n",
" # CASE M\n",
" # Bottom\n",
" lf_ax = plt.subplot(1, 2, 1)\n",
" lf_j_exp, = lf_ax.plot(time_list['MT']['j_explicit'][init:out], delta_value_exp['MT']['j_explicit'][init:out], color='blue', ls='--')\n",
" lf_j_imp, = lf_ax.plot(time_list['MT']['j_implicit'][init:out], delta_value_exp['MT']['j_implicit'][init:out], color='blue')\n",
" lf_ax.set_xlabel(r\"Simulation time [t]\")\n",
" lf_ax.set_ylabel(r\"$\\delta_{M}$\", color='blue')\n",
" lf_ax.set_title(r\"Residual Energy $\\delta_{M}$\")\n",
" lf_ax.tick_params(axis='y', labelcolor='blue')\n",
" lf_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" #rh_ax = lf_ax.twinx()\n",
" #rh_j_exp, = rh_ax.plot(time_list['MT']['j_explicit'][init:out], delta_value_con['MT']['j_explicit'][init:out], color='red', ls='--')\n",
" #rh_j_imp, = rh_ax.plot(time_list['MT']['j_implicit'][init:out], delta_value_con['MT']['j_implicit'][init:out], color='red')\n",
" #rh_ax.set_ylabel(r\"$\\delta_{M}$ def. II\", color='red')\n",
" #rh_ax.tick_params(axis='y', labelcolor='red')\n",
" #rh_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" plt.legend((lf_j_exp, lf_j_imp), (r'M $\\bigstar\\bigstar = n$', r'M $\\bigstar\\bigstar = n+1$'), loc='center right')\n",
" plt.grid(True)\n",
" plt.subplots_adjust(hspace=0.8)\n",
" plt.tight_layout()\n",
" plt.savefig('Comparison_Delta_Value_GCL_{}_Solver_{}.png'.format(args[0], args[1]), dpi=300)\n",
" plt.close()\n",
"\n",
" # ---------------------- #\n",
" # NORMALIZED DELTA CASE (w.r.t physical dissipation)\n",
" # ---------------------- #\n",
" plt.subplots(figsize=(10, 4))\n",
" # CASE CT\n",
" # Top\n",
" lf_ax = plt.subplot(1, 2, 2)\n",
" lf_j_exp, = lf_ax.plot(time_list['CT']['j_explicit'][init:out], delta_hat_value_exp['CT']['j_explicit'][init:out], color='blue', ls='--')\n",
" lf_j_imp, = lf_ax.plot(time_list['CT']['j_implicit'][init:out], delta_hat_value_exp['CT']['j_implicit'][init:out], color='blue')\n",
" lf_ax.set_xlabel(r\"Simulation time [t]\")\n",
" lf_ax.set_ylabel(r\"$\\hat{\\delta}_{CT}$\", color='blue')\n",
" lf_ax.set_title(r\"Normalized value $\\hat{\\delta}_{CT}$\")\n",
" lf_ax.tick_params(axis='y', labelcolor='blue')\n",
" lf_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" #rh_ax = lf_ax.twinx()\n",
" #rh_j_exp, = rh_ax.plot(time_list['CT']['j_explicit'][init:out], delta_hat_value_con['CT']['j_explicit'][init:out], color='red', ls='--')\n",
" #rh_j_imp, = rh_ax.plot(time_list['CT']['j_implicit'][init:out], delta_hat_value_con['CT']['j_implicit'][init:out], color='red')\n",
" #rh_ax.set_ylabel(r\"$\\hat{\\delta}_{CT}$ def. II\", color='red')\n",
" #rh_ax.tick_params(axis='y', labelcolor='red')\n",
" #rh_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" plt.legend((lf_j_exp, lf_j_imp), (r'CT $\\bigstar\\bigstar = n$', r'CT $\\bigstar\\bigstar = n+1$'), loc='lower right')\n",
" plt.grid(True)\n",
" # CASE M\n",
" # Bottom\n",
" lf_ax = plt.subplot(1, 2, 1)\n",
" lf_j_exp, = lf_ax.plot(time_list['MT']['j_explicit'][init:out], delta_hat_value_exp['MT']['j_explicit'][init:out], color='blue', ls='--')\n",
" lf_j_imp, = lf_ax.plot(time_list['MT']['j_implicit'][init:out], delta_hat_value_exp['MT']['j_implicit'][init:out], color='blue')\n",
" lf_ax.set_xlabel(r\"Simulation time [t]\")\n",
" lf_ax.set_ylabel(r\"$\\hat{\\delta}_{M}$\", color='blue')\n",
" lf_ax.set_title(r\"Normalized value $\\hat{\\delta}_{M}$\")\n",
" lf_ax.tick_params(axis='y', labelcolor='blue')\n",
" lf_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" #rh_ax = lf_ax.twinx()\n",
" #rh_j_exp, = rh_ax.plot(time_list['MT']['j_explicit'][init:out], delta_hat_value_con['MT']['j_explicit'][init:out], color='red', ls='--')\n",
" #rh_j_imp, = rh_ax.plot(time_list['MT']['j_implicit'][init:out], delta_hat_value_con['MT']['j_implicit'][init:out], color='red')\n",
" #rh_ax.set_ylabel(r\"$\\hat{\\delta}_{M}$ def. II\", color='red')\n",
" #rh_ax.tick_params(axis='y', labelcolor='red')\n",
" #rh_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" plt.legend((lf_j_exp, lf_j_imp), (r'M $\\bigstar\\bigstar = n$', r'M $\\bigstar\\bigstar = n+1$'), loc='upper right')\n",
" plt.grid(True)\n",
" plt.subplots_adjust(hspace=0.8)\n",
" plt.tight_layout()\n",
" plt.savefig('Comparison_Delta_Hat_Value_GCL_{}_solver_{}.png'.format(args[0], args[1]), dpi=300)\n",
" plt.close()\n",
"\n",
" # ---------------------- #\n",
" # NORMALIZED DELTA CASE (w.r.t numerical dissipation)\n",
" # ---------------------- #\n",
" plt.subplots(figsize=(10, 4))\n",
" # CASE CT\n",
" # Top\n",
" lf_ax = plt.subplot(1, 2, 2)\n",
" lf_j_exp, = lf_ax.plot(time_list['CT']['j_explicit'][init:out], delta_hat_dis_exp['CT']['j_explicit'][init:out], color='blue', ls='--')\n",
" lf_j_imp, = lf_ax.plot(time_list['CT']['j_implicit'][init:out], delta_hat_dis_exp['CT']['j_implicit'][init:out], color='blue')\n",
" lf_ax.set_xlabel(r\"Simulation time [t]\")\n",
" lf_ax.set_ylabel(r\"$\\hat{\\delta}_{CT}$ dis. \", color='blue')\n",
" lf_ax.set_title(r\"Normalized value $\\hat{\\delta}_{CT}$ (dis)\")\n",
" lf_ax.tick_params(axis='y', labelcolor='blue')\n",
" lf_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" #rh_ax = lf_ax.twinx()\n",
" #rh_j_exp, = rh_ax.plot(time_list['CT']['j_explicit'][init:out], delta_hat_dis_con['CT']['j_explicit'][init:out], color='red', ls='--')\n",
" #rh_j_imp, = rh_ax.plot(time_list['CT']['j_implicit'][init:out], delta_hat_dis_con['CT']['j_implicit'][init:out], color='red')\n",
" #rh_ax.set_ylabel(r\"$\\hat{\\delta}_{CT}$ def. II\", color='red')\n",
" #rh_ax.tick_params(axis='y', labelcolor='red')\n",
" #rh_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" plt.legend((lf_j_exp, lf_j_imp), (r'CT $\\bigstar\\bigstar = n$', r'CT $\\bigstar\\bigstar = n+1$'), loc='lower right')\n",
" plt.grid(True)\n",
" # CASE M\n",
" # Bottom\n",
" lf_ax = plt.subplot(1, 2, 1)\n",
" lf_j_exp, = lf_ax.plot(time_list['MT']['j_explicit'][init:out], delta_hat_dis_exp['MT']['j_explicit'][init:out], color='blue', ls='--')\n",
" lf_j_imp, = lf_ax.plot(time_list['MT']['j_implicit'][init:out], delta_hat_dis_exp['MT']['j_implicit'][init:out], color='blue')\n",
" lf_ax.set_xlabel(r\"Simulation time [t]\")\n",
" lf_ax.set_ylabel(r\"$\\hat{\\delta}_{M}$ dis.\", color='blue')\n",
" lf_ax.set_title(r\"Normalized value $\\hat{\\delta}_{M}$ (dis)\")\n",
" lf_ax.tick_params(axis='y', labelcolor='blue')\n",
" lf_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" #rh_ax = lf_ax.twinx()\n",
" #rh_j_exp, = rh_ax.plot(time_list['MT']['j_explicit'][init:out], delta_hat_dis_con['MT']['j_explicit'][init:out], color='red', ls='--')\n",
" #rh_j_imp, = rh_ax.plot(time_list['MT']['j_implicit'][init:out], delta_hat_dis_con['MT']['j_implicit'][init:out], color='red')\n",
" #rh_ax.set_ylabel(r\"$\\hat{\\delta}_{M}$ def. II\", color='red')\n",
" #rh_ax.tick_params(axis='y', labelcolor='red')\n",
" #rh_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" plt.legend((lf_j_exp, lf_j_imp), (r'M $\\bigstar\\bigstar = n$', r'M $\\bigstar\\bigstar = n+1$'), loc='lower right')\n",
" plt.grid(True)\n",
" plt.subplots_adjust(hspace=0.8)\n",
" plt.tight_layout()\n",
" plt.savefig('Comparison_Delta_Hat_Dis_Value_GCL_{}_solver_{}.png'.format(args[0], args[1]), dpi=300)\n",
" plt.close()\n",
"\n",
" # ---------------------- #\n",
" # PHYSICAL DISSIPATION PLOT\n",
" # ---------------------- #\n",
" plt.subplots(figsize=(10, 4))\n",
" # CASE CT\n",
" # Top\n",
" lf_ax = plt.subplot(1, 2, 2)\n",
" lf_j_exp, = lf_ax.plot(time_list['CT']['j_explicit'][init:out], eps_value_exp['CT']['j_explicit'][init:out], color='blue', ls='--')\n",
" lf_j_imp, = lf_ax.plot(time_list['CT']['j_implicit'][init:out], eps_value_exp['CT']['j_implicit'][init:out], color='blue')\n",
" lf_ax.set_xlabel(r\"Simulation time [t]\")\n",
" lf_ax.set_ylabel(r\"CT\", color='blue')\n",
" lf_ax.set_title(r\"Physical Dissipation (CT)\")\n",
" lf_ax.tick_params(axis='y', labelcolor='blue')\n",
" lf_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" #rh_ax = lf_ax.twinx()\n",
" #rh_j_exp, = rh_ax.plot(time_list['CT']['j_explicit'][init:out], eps_value_con['CT']['j_explicit'][init:out], color='red', ls='--')\n",
" #rh_j_imp, = rh_ax.plot(time_list['CT']['j_implicit'][init:out], eps_value_con['CT']['j_implicit'][init:out], color='red')\n",
" #rh_ax.set_ylabel(r\"On def. II\", color='red')\n",
" #rh_ax.tick_params(axis='y', labelcolor='red')\n",
" #rh_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" plt.legend((lf_j_exp, lf_j_imp), \\\n",
" (r'CT $\\star\\star = n$', r'CT $\\bigstar\\bigstar = n+1$'), loc='lower right')\n",
" plt.grid(True)\n",
" # CASE M\n",
" # Bottom\n",
" lf_ax = plt.subplot(1, 2, 1)\n",
" lf_j_exp, = lf_ax.plot(time_list['MT']['j_explicit'][init:out], eps_value_exp['MT']['j_explicit'][init:out], color='blue', ls='--')\n",
" lf_j_imp, = lf_ax.plot(time_list['MT']['j_implicit'][init:out], eps_value_exp['MT']['j_implicit'][init:out], color='blue')\n",
" lf_ax.set_xlabel(r\"Simulation time [t]\")\n",
" lf_ax.set_ylabel(r\"M\", color='blue')\n",
" lf_ax.set_title(r\"Physical Dissipation (M)\")\n",
" lf_ax.tick_params(axis='y', labelcolor='blue')\n",
" lf_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" #rh_ax = lf_ax.twinx()\n",
" #rh_j_exp, = rh_ax.plot(time_list['MT']['j_explicit'][init:out], eps_value_con['MT']['j_explicit'][init:out], color='red', ls='--')\n",
" #rh_j_imp, = rh_ax.plot(time_list['MT']['j_implicit'][init:out], eps_value_con['MT']['j_implicit'][init:out], color='red')\n",
" #rh_ax.set_ylabel(r\"On def. II\", color='red')\n",
" #rh_ax.tick_params(axis='y', labelcolor='red')\n",
" #rh_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" plt.legend((lf_j_exp, lf_j_imp), (r'M $\\bigstar\\bigstar = n$', r'M $\\bigstar\\bigstar = n+1$'), loc='lower right')\n",
" plt.grid(True)\n",
" plt.subplots_adjust(hspace=0.8)\n",
" plt.tight_layout()\n",
" plt.savefig('Comparison_Eps_Value_GCL_{}_solver_{}.png'.format(args[0], args[1]), dpi=300)\n",
" plt.close()\n",
"\n",
" # ---------------------- #\n",
" # NUMERICAL DISSIPATION PLOT\n",
" # ---------------------- #\n",
" plt.subplots(figsize=(10, 4))\n",
" # CASE CT\n",
" # Top\n",
" lf_ax = plt.subplot(1, 2, 2)\n",
" lf_j_exp, = lf_ax.plot(time_list['CT']['j_explicit'][init:out], dis_value_exp['CT']['j_explicit'][init:out], color='blue', ls='--')\n",
" lf_j_imp, = lf_ax.plot(time_list['CT']['j_implicit'][init:out], dis_value_exp['CT']['j_implicit'][init:out], color='blue')\n",
" lf_ax.set_xlabel(r\"Simulation time [t]\")\n",
" lf_ax.set_ylabel(r\"CT\", color='blue')\n",
" lf_ax.set_title(r\"Numerical Dissipation (CT)\")\n",
" lf_ax.tick_params(axis='y', labelcolor='blue')\n",
" lf_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" #rh_ax = lf_ax.twinx()\n",
" #rh_j_exp, = rh_ax.plot(time_list['CT']['j_explicit'][init:out], dis_value_con['CT']['j_explicit'][init:out], color='red', ls='--')\n",
" #rh_j_imp, = rh_ax.plot(time_list['CT']['j_implicit'][init:out], dis_value_con['CT']['j_implicit'][init:out], color='red')\n",
" #rh_ax.set_ylabel(r\"On def. II\", color='red')\n",
" #rh_ax.tick_params(axis='y', labelcolor='red')\n",
" #rh_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" plt.legend((lf_j_exp, lf_j_imp), (r'CT $\\bigstar\\bigstar = n$', r'CT $\\bigstar\\bigstar = n+1$'), loc='lower right')\n",
" plt.grid(True)\n",
" # CASE M\n",
" # Bottom\n",
" lf_ax = plt.subplot(1, 2, 1)\n",
" lf_j_exp, = lf_ax.plot(time_list['MT']['j_explicit'][init:out], dis_value_exp['MT']['j_explicit'][init:out], color='blue', ls='--')\n",
" lf_j_imp, = lf_ax.plot(time_list['MT']['j_implicit'][init:out], dis_value_exp['MT']['j_implicit'][init:out], color='blue')\n",
" lf_ax.set_xlabel(r\"Simulation time [t]\")\n",
" lf_ax.set_ylabel(r\"M\", color='blue')\n",
" lf_ax.set_title(r\"Numerical Dissipation (M)\")\n",
" lf_ax.tick_params(axis='y', labelcolor='blue')\n",
" lf_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" #rh_ax = lf_ax.twinx()\n",
" #rh_j_exp, = rh_ax.plot(time_list['MT']['j_explicit'][init:out], dis_value_con['MT']['j_explicit'][init:out], color='red', ls='--')\n",
" #rh_j_imp, = rh_ax.plot(time_list['MT']['j_implicit'][init:out], dis_value_con['MT']['j_implicit'][init:out], color='red')\n",
" #rh_ax.set_ylabel(r\"On def. II\", color='red')\n",
" #rh_ax.tick_params(axis='y', labelcolor='red')\n",
" #rh_ax.ticklabel_format(style='sci', axis='y', scilimits=(0,0))\n",
" plt.legend((lf_j_exp, lf_j_imp), (r'M $\\bigstar\\bigstar = n$', r'M $\\bigstar\\bigstar = n+1$'), loc='lower right')\n",
" plt.grid(True)\n",
" plt.subplots_adjust(hspace=0.8)\n",
" plt.tight_layout()\n",
" plt.savefig('Comparison_Num_Dis_Value_GCL_{}_solver_{}.png'.format(args[0], args[1]), dpi=300)\n",
" plt.close()\n",
" print(\"\\nAll computations done and figures saved...\\n\")\n",
" print(\"\\n------------------------------------------\\n\")\n",
" return None\n",
"\n",
"testing_comparative_plots(dict_mt, dict_ct, True, 'LU')"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.6.9"
}
},
"nbformat": 4,
"nbformat_minor": 2
}

10
README.rst
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@ -5,8 +5,8 @@ This repository implements via finite element solvers for incompressible
Navier-Stokes (iNSE) equations in Arbitrary Lagrangian-Eulerian (ALE) formalism the schemes
proposed on [RA20]_.
In particular, it aims to replicate the energy results shown in the article [RA20]_
for both Monolithic and Chorin-Temam solvers.
It aims to replicate the energy results shown in the article [RA20]_
for both Monolithic and Chorin-Temam solvers. TeX submissions are added for reference.
Solvers
-------
@ -15,7 +15,7 @@ Both monolithic solver and a fractional step solver are implemented.
* Monolithic solver for the iNSE-ALE problem with linearized convective term and Taylor-Hood (P2/P1) stable finite element space.
* Fractional step solver for the iNSE-ALE problem with linealized convective term and P1/P1 finite element space. The Chorin-Temam schemes proposed is described in [RA20]_.
* Fractional step solver for the iNSE-ALE problem with linealized convective term and P1/P1 finite element space. Chorin-Temam schemes proposed here are described in [RA20]_.
Flow model
----------
@ -32,9 +32,7 @@ no configuration files where implemented to further customize the problem.
Nevertheless, the solvers are easily modified since its implementation is done via
FEniCS [LO12]_.
An ipynb file is included to reproduce the results and recommended to use.
If desired to run the simulations only, execute::
To run the simulations and generate the figures depicted in our article, execute::
python main.py

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*.aux
*.log
*.pdf
*.synctex.gz
*.sh
# Submission 1
1_submission/*.synctex.gz
1_submission/*.log
1_submission/*.aux
1_submission/*.pdf
# Submission 2
2_submission/*.synctex.gz
2_submission/*.log
2_submission/*.aux
2_submission/*.pdf
# HAL Submission
hal_submission/*.synctex.gz
hal_submission/*.log
hal_submission/*.aux
hal_submission/*.pdf

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tex/1_submission/README Normal file
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@ -0,0 +1,5 @@
Latest modifications done by: R.A. Arostica
We are using Elsevier's journal article template.
The main tex file containing the article under writing in paper.tex
I've chosen the single column, but we can easily change it for a double column if necessary.

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%%
%% Copyright 2019-2020 Elsevier Ltd
%%
%% This file is part of the 'CAS Bundle'.
%% --------------------------------------
%%
%% It may be distributed under the conditions of the LaTeX Project Public
%% License, either version 1.2 of this license or (at your option) any
%% later version. The latest version of this license is in
%% http://www.latex-project.org/lppl.txt
%% and version 1.2 or later is part of all distributions of LaTeX
%% version 1999/12/01 or later.
%%
%% The list of all files belonging to the 'CAS Bundle' is
%% given in the file `manifest.txt'.
%%
%% Template article for cas-dc documentclass for
%% double column output.
%\documentclass[a4paper,fleqn,longmktitle]{cas-dc}
\documentclass[a4paper,fleqn]{cas-dc}
%\usepackage[authoryear,longnamesfirst]{natbib}
%\usepackage[authoryear]{natbib}
\usepackage[numbers]{natbib}
%%%Author definitions
\def\tsc#1{\csdef{#1}{\textsc{\lowercase{#1}}\xspace}}
\tsc{WGM}
\tsc{QE}
\tsc{EP}
\tsc{PMS}
\tsc{BEC}
\tsc{DE}
%%%
\begin{document}
\let\WriteBookmarks\relax
\def\floatpagepagefraction{1}
\def\textpagefraction{.001}
\shorttitle{Leveraging social media news}
\shortauthors{CV Radhakrishnan et~al.}
\title [mode = title]{This is a specimen $a_b$ title}
\tnotemark[1,2]
\tnotetext[1]{This document is the results of the research
project funded by the National Science Foundation.}
\tnotetext[2]{The second title footnote which is a longer text matter
to fill through the whole text width and overflow into
another line in the footnotes area of the first page.}
\author[1,3]{CV Radhakrishnan}[type=editor,
auid=000,bioid=1,
prefix=Sir,
role=Researcher,
orcid=0000-0001-7511-2910]
\cormark[1]
\fnmark[1]
\ead{cvr_1@tug.org.in}
\ead[url]{www.cvr.cc, cvr@sayahna.org}
\credit{Conceptualization of this study, Methodology, Software}
\address[1]{Elsevier B.V., Radarweg 29, 1043 NX Amsterdam, The Netherlands}
\author[2,4]{Han Theh Thanh}[style=chinese]
\author[2,3]{CV Rajagopal}[%
role=Co-ordinator,
suffix=Jr,
]
\fnmark[2]
\ead{cvr3@sayahna.org}
\ead[URL]{www.sayahna.org}
\credit{Data curation, Writing - Original draft preparation}
\address[2]{Sayahna Foundation, Jagathy, Trivandrum 695014, India}
\author%
[1,3]
{Rishi T.}
\cormark[2]
\fnmark[1,3]
\ead{rishi@stmdocs.in}
\ead[URL]{www.stmdocs.in}
\address[3]{STM Document Engineering Pvt Ltd., Mepukada,
Malayinkil, Trivandrum 695571, India}
\cortext[cor1]{Corresponding author}
\cortext[cor2]{Principal corresponding author}
\fntext[fn1]{This is the first author footnote. but is common to third
author as well.}
\fntext[fn2]{Another author footnote, this is a very long footnote and
it should be a really long footnote. But this footnote is not yet
sufficiently long enough to make two lines of footnote text.}
\nonumnote{This note has no numbers. In this work we demonstrate $a_b$
the formation Y\_1 of a new type of polariton on the interface
between a cuprous oxide slab and a polystyrene micro-sphere placed
on the slab.
}
\begin{abstract}
This template helps you to create a properly formatted \LaTeX\ manuscript.
\noindent\texttt{\textbackslash begin{abstract}} \dots
\texttt{\textbackslash end{abstract}} and
\verb+\begin{keyword}+ \verb+...+ \verb+\end{keyword}+
which
contain the abstract and keywords respectively.
\noindent Each keyword shall be separated by a \verb+\sep+ command.
\end{abstract}
\begin{graphicalabstract}
\includegraphics{figs/grabs.pdf}
\end{graphicalabstract}
\begin{highlights}
\item Research highlights item 1
\item Research highlights item 2
\item Research highlights item 3
\end{highlights}
\begin{keywords}
quadrupole exciton \sep polariton \sep \WGM \sep \BEC
\end{keywords}
\maketitle
\section{Introduction}
The Elsevier cas-dc class is based on the
standard article class and supports almost all of the functionality of
that class. In addition, it features commands and options to format the
\begin{itemize} \item document style \item baselineskip \item front
matter \item keywords and MSC codes \item theorems, definitions and
proofs \item lables of enumerations \item citation style and labeling.
\end{itemize}
This class depends on the following packages
for its proper functioning:
\begin{enumerate}
\itemsep=0pt
\item {natbib.sty} for citation processing;
\item {geometry.sty} for margin settings;
\item {fleqn.clo} for left aligned equations;
\item {graphicx.sty} for graphics inclusion;
\item {hyperref.sty} optional packages if hyperlinking is
required in the document;
\end{enumerate}
All the above packages are part of any
standard \LaTeX{} installation.
Therefore, the users need not be
bothered about downloading any extra packages.
\section{Installation}
The package is available at author resources page at Elsevier
(\url{http://www.elsevier.com/locate/latex}).
The class may be moved or copied to a place, usually,\linebreak
\verb+$TEXMF/tex/latex/elsevier/+, %$%%%%%%%%%%%%%%%%%%%%%%%%%%%%
or a folder which will be read
by \LaTeX{} during document compilation. The \TeX{} file
database needs updation after moving/copying class file. Usually,
we use commands like \verb+mktexlsr+ or \verb+texhash+ depending
upon the distribution and operating system.
\section{Front matter}
The author names and affiliations could be formatted in two ways:
\begin{enumerate}[(1)]
\item Group the authors per affiliation.
\item Use footnotes to indicate the affiliations.
\end{enumerate}
See the front matter of this document for examples.
You are recommended to conform your choice to the journal you
are submitting to.
\section{Bibliography styles}
There are various bibliography styles available. You can select the
style of your choice in the preamble of this document. These styles are
Elsevier styles based on standard styles like Harvard and Vancouver.
Please use Bib\TeX\ to generate your bibliography and include DOIs
whenever available.
Here are two sample references:
\cite{Fortunato2010}
\cite{Fortunato2010,NewmanGirvan2004}
\cite{Fortunato2010,Vehlowetal2013}
\section{Floats}
{Figures} may be included using the command,\linebreak
\verb+\includegraphics+ in
combination with or without its several options to further control
graphic. \verb+\includegraphics+ is provided by {graphic[s,x].sty}
which is part of any standard \LaTeX{} distribution.
{graphicx.sty} is loaded by default. \LaTeX{} accepts figures in
the postscript format while pdf\LaTeX{} accepts {*.pdf},
{*.mps} (metapost), {*.jpg} and {*.png} formats.
pdf\LaTeX{} does not accept graphic files in the postscript format.
\begin{figure}
\centering
\includegraphics[scale=.75]{figs/Fig1.pdf}
\caption{The evanescent light - $1S$ quadrupole coupling
($g_{1,l}$) scaled to the bulk exciton-photon coupling
($g_{1,2}$). The size parameter $kr_{0}$ is denoted as $x$ and
the \PMS is placed directly on the cuprous oxide sample ($\delta
r=0$, See also Table \protect\ref{tbl1}).}
\label{FIG:1}
\end{figure}
The \verb+table+ environment is handy for marking up tabular
material. If users want to use {multirow.sty},
{array.sty}, etc., to fine control/enhance the tables, they
are welcome to load any package of their choice and
{cas-dc.cls} will work in combination with all loaded
packages.
\begin{table}[width=.9\linewidth,cols=4,pos=h]
\caption{This is a test caption. This is a test caption. This is a test
caption. This is a test caption.}\label{tbl1}
\begin{tabular*}{\tblwidth}{@{} LLLL@{} }
\toprule
Col 1 & Col 2 & Col 3 & Col4\\
\midrule
12345 & 12345 & 123 & 12345 \\
12345 & 12345 & 123 & 12345 \\
12345 & 12345 & 123 & 12345 \\
12345 & 12345 & 123 & 12345 \\
12345 & 12345 & 123 & 12345 \\
\bottomrule
\end{tabular*}
\end{table}
\section[Theorem and ...]{Theorem and theorem like environments}
{cas-dc.cls} provides a few shortcuts to format theorems and
theorem-like environments with ease. In all commands the options that
are used with the \verb+\newtheorem+ command will work exactly in the same
manner. {cas-dc.cls} provides three commands to format theorem or
theorem-like environments:
\begin{verbatim}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newdefinition{rmk}{Remark}
\newproof{pf}{Proof}
\newproof{pot}{Proof of Theorem \ref{thm2}}
\end{verbatim}
The \verb+\newtheorem+ command formats a
theorem in \LaTeX's default style with italicized font, bold font
for theorem heading and theorem number at the right hand side of the
theorem heading. It also optionally accepts an argument which
will be printed as an extra heading in parentheses.
\begin{verbatim}
\begin{theorem}
For system (8), consensus can be achieved with
$\|T_{\omega z}$ ...
\begin{eqnarray}\label{10}
....
\end{eqnarray}
\end{theorem}
\end{verbatim}
\newtheorem{theorem}{Theorem}
\begin{theorem}
For system (8), consensus can be achieved with
$\|T_{\omega z}$ ...
\begin{eqnarray}\label{10}
....
\end{eqnarray}
\end{theorem}
The \verb+\newdefinition+ command is the same in
all respects as its \verb+\newtheorem+ counterpart except that
the font shape is roman instead of italic. Both
\verb+\newdefinition+ and \verb+\newtheorem+ commands
automatically define counters for the environments defined.
The \verb+\newproof+ command defines proof environments with
upright font shape. No counters are defined.
\section[Enumerated ...]{Enumerated and Itemized Lists}
{cas-dc.cls} provides an extended list processing macros
which makes the usage a bit more user friendly than the default
\LaTeX{} list macros. With an optional argument to the
\verb+\begin{enumerate}+ command, you can change the list counter
type and its attributes.
\begin{verbatim}
\begin{enumerate}[1.]
\item The enumerate environment starts with an optional
argument `1.', so that the item counter will be suffixed
by a period.
\item You can use `a)' for alphabetical counter and '(i)'
for roman counter.
\begin{enumerate}[a)]
\item Another level of list with alphabetical counter.
\item One more item before we start another.
\item One more item before we start another.
\item One more item before we start another.
\item One more item before we start another.
\end{verbatim}
Further, the enhanced list environment allows one to prefix a
string like `step' to all the item numbers.
\begin{verbatim}
\begin{enumerate}[Step 1.]
\item This is the first step of the example list.
\item Obviously this is the second step.
\item The final step to wind up this example.
\end{enumerate}
\end{verbatim}
\section{Cross-references}
In electronic publications, articles may be internally
hyperlinked. Hyperlinks are generated from proper
cross-references in the article. For example, the words
\textcolor{black!80}{Fig.~1} will never be more than simple text,
whereas the proper cross-reference \verb+\ref{tiger}+ may be
turned into a hyperlink to the figure itself:
\textcolor{blue}{Fig.~1}. In the same way,
the words \textcolor{blue}{Ref.~[1]} will fail to turn into a
hyperlink; the proper cross-reference is \verb+\cite{Knuth96}+.
Cross-referencing is possible in \LaTeX{} for sections,
subsections, formulae, figures, tables, and literature
references.
\section{Bibliography}
Two bibliographic style files (\verb+*.bst+) are provided ---
{model1-num-names.bst} and {model2-names.bst} --- the first one can be
used for the numbered scheme. This can also be used for the numbered
with new options of {natbib.sty}. The second one is for the author year
scheme. When you use model2-names.bst, the citation commands will be
like \verb+\citep+, \verb+\citet+, \verb+\citealt+ etc. However when
you use model1-num-names.bst, you may use only \verb+\cite+ command.
\verb+thebibliography+ environment. Each reference is a\linebreak
\verb+\bibitem+ and each \verb+\bibitem+ is identified by a label,
by which it can be cited in the text:
\noindent In connection with cross-referencing and
possible future hyperlinking it is not a good idea to collect
more that one literature item in one \verb+\bibitem+. The
so-called Harvard or author-year style of referencing is enabled
by the \LaTeX{} package {natbib}. With this package the
literature can be cited as follows:
\begin{enumerate}[\textbullet]
\item Parenthetical: \verb+\citep{WB96}+ produces (Wettig \& Brown, 1996).
\item Textual: \verb+\citet{ESG96}+ produces Elson et al. (1996).
\item An affix and part of a reference:\break
\verb+\citep[e.g.][Ch. 2]{Gea97}+ produces (e.g. Governato et
al., 1997, Ch. 2).
\end{enumerate}
In the numbered scheme of citation, \verb+\cite{<label>}+ is used,
since \verb+\citep+ or \verb+\citet+ has no relevance in the numbered
scheme. {natbib} package is loaded by {cas-dc} with
\verb+numbers+ as default option. You can change this to author-year
or harvard scheme by adding option \verb+authoryear+ in the class
loading command. If you want to use more options of the {natbib}
package, you can do so with the \verb+\biboptions+ command. For
details of various options of the {natbib} package, please take a
look at the {natbib} documentation, which is part of any standard
\LaTeX{} installation.
\appendix
\section{My Appendix}
Appendix sections are coded under \verb+\appendix+.
\verb+\printcredits+ command is used after appendix sections to list
author credit taxonomy contribution roles tagged using \verb+\credit+
in frontmatter.
\printcredits
%% Loading bibliography style file
%\bibliographystyle{model1-num-names}
\bibliographystyle{cas-model2-names}
% Loading bibliography database
\bibliography{cas-refs}
%\vskip3pt
\bio{}
Author biography without author photo.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
\endbio
\bio{figs/pic1}
Author biography with author photo.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
\endbio
\bio{figs/pic1}
Author biography with author photo.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
\endbio
\end{document}

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%%
%% This is file `cas-sc.cls'.
%%
%% This file is part of the 'CAS Bundle'.
%% ......................................
%%
%% It may be distributed under the conditions of the LaTeX Project Public
%% License, either version 1.2 of this license or (at your option) any
%% later version. The latest version of this license is in
%% http://www.latex-project.org/lppl.txt
%% and version 1.2 or later is part of all distributions of LaTeX
%% version 1999/12/01 or later.
%%
%% The list of all files belonging to the 'CAS Bundle' is
%% given in the file `manifest.txt'.
%%
%% $Id: cas-dc.cls 49 2020-03-14 09:05:10Z rishi $
\def\RCSfile{cas-dc}%
\def\RCSversion{2.1}%
\def\RCSdate{2020/03/14}%
\NeedsTeXFormat{LaTeX2e}[1995/12/01]
\ProvidesClass{\RCSfile}[\RCSdate, \RCSversion: Formatting class
for CAS double column articles]
%
\def\ABD{\AtBeginDocument}
%
% switches
%
\newif\iflongmktitle \longmktitlefalse
\newif\ifdc \global\dctrue
\newif\ifsc \global\scfalse
\newif\ifcasreviewlayout \global\casreviewlayoutfalse
\newif\ifcasfinallayout \global\casfinallayoutfalse
\newcounter{blind}
\setcounter{blind}{0}
\def\blstr#1{\gdef\@blstr{#1}}
\def\@blstr{1}
\newdimen\@bls
\@bls=\baselineskip
\DeclareOption{singleblind}{\setcounter{blind}{1}}
\DeclareOption{doubleblind}{\setcounter{blind}{2}}
\DeclareOption{longmktitle}{\global\longmktitletrue}
\DeclareOption{final}{\global\casfinallayouttrue}
\DeclareOption{review}{\global\casreviewlayouttrue}
\ExecuteOptions{a4paper,10pt,oneside,fleqn,review}
\DeclareOption*{\PassOptionsToClass{\CurrentOption}{article}}
\ProcessOptions
\LoadClass{article}
\RequirePackage{graphicx}
\RequirePackage{amsmath,amsfonts,amssymb}
\allowdisplaybreaks
\RequirePackage{expl3,xparse}
\@ifundefined{regex_match:nnTF}{\RequirePackage{l3regex}}{}
\RequirePackage{etoolbox,balance}
\RequirePackage{booktabs,makecell,multirow,array,colortbl,dcolumn,stfloats}
\RequirePackage{xspace,xstring,footmisc}
\RequirePackage[svgnames,dvipsnames]{xcolor}
\RequirePackage[colorlinks]{hyperref}
\colorlet{scolor}{black}
\colorlet{hscolor}{DarkSlateGrey}
\hypersetup{%
pdftitle={\csuse{__short_title:}},
pdfauthor={\csuse{__short_authors:}},
pdfcreator={LaTeX3; cas-sc.cls; hyperref.sty},
pdfproducer={pdfTeX;},
linkcolor={hscolor},
urlcolor={hscolor},
citecolor={hscolor},
filecolor={hscolor},
menucolor={hscolor},
}
\let\comma\@empty
\let\tnotesep\@empty
\let\@title\@empty
%
% Load Common items
%
\RequirePackage{cas-common}
%
% Specific to Single Column
%
\ExplSyntaxOn
\RenewDocumentCommand \maketitle { }
{
\ifbool { usecasgrabsbox }
{
\setcounter{page}{0}
\thispagestyle{empty}
\unvbox\casgrabsbox
} { }
\pagebreak
\ifbool { usecashlsbox }
{
\setcounter{page}{0}
\thispagestyle{empty}
\unvbox\casauhlbox
} { }
\pagebreak
\thispagestyle{first}
\ifbool{longmktitle}
{
\LongMaketitleBox
\ProcessLongTitleBox
}
{
\twocolumn[\MaketitleBox]
\printFirstPageNotes
}
\setcounter{footnote}{\int_use:N \g_stm_fnote_int}
\renewcommand\thefootnote{\arabic{footnote}}
\gdef\@pdfauthor{\infoauthors}
\gdef\@pdfsubject{Complex ~STM ~Content}
}
%
% Fonts
%
\RequirePackage[T1]{fontenc}
\file_if_exist:nTF { stix.sty }
{
\file_if_exist:nTF { charis.sty }
{
\RequirePackage[notext]{stix}
\RequirePackage{charis}
}
{ \RequirePackage{stix} }
}
{
\iow_term:x { *********************************************************** }
\iow_term:x { ~Stix ~ and ~ Charis~ fonts ~ are ~ not ~ available ~ }
\iow_term:x { ~ in ~TeX~system.~Hence~CMR~ fonts~ are ~ used. }
\iow_term:x { *********************************************************** }
}
\file_if_exist:nTF { inconsolata.sty }
{ \RequirePackage[scaled=.85]{inconsolata} }
{ \tex_gdef:D \ttdefault { cmtt } }
\ExplSyntaxOff
%
% Page geometry
%
\usepackage[%
paperwidth=210mm,
paperheight=280mm,
vmargin={19.5mm,18.2mm},
hmargin={18.1mm,18.1mm},
headsep=12pt,
footskip=12pt,
columnsep=18pt
]{geometry}
\endinput
%
% End of class 'cas-sc'
%

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@article{FEniCS2015,
author = {Aln\ae{}s, Martin and Blechta, Jan and Hake, Johan and Johansson, August and Kehlet, Benjamin and Logg, Anders and Richardson, Chris and Ring, Johannes and Rognes, Marie E and Wells, Garth N},
language = {eng},
title = {The FEniCS Project Version 1.5},
journal = {<p>Archive of Numerical Software},
volume = {Vol 3},
pages = {<strong>Starting Point and Frequency: </strong>Year: 2013</p>--},
publisher = {University Library Heidelberg},
year = {2015},
copyright = {Authors who publish with this journal agree to the following terms: Authors retain copyright and grant the journal right of first publication with the descriptive part of the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal. The code part of the work is licensed under a suitable OSI approved Open Source license. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.}
}
@book{Quarteroni2009Cardiovascular,
year = {2009},
publisher = {Springer Milan},
editor = {Luca Formaggia and Alfio Quarteroni and Alessandro Veneziani},
title = {Cardiovascular Mathematics}
}
@book{Richter2017,
year = {2017},
publisher = {Springer International Publishing},
author = {Thomas Richter},
title = {Fluid-structure Interactions}
}
@article{bertoglio2018benchmark,
title={Benchmark problems for numerical treatment of backflow at open boundaries},
author={Bertoglio, Crist{\'o}bal and Caiazzo, Alfonso and Bazilevs, Yuri and Braack, Malte and Esmaily, Mahdi and Gravemeier, Volker and L. Marsden, Alison and Pironneau, Olivier and E. Vignon-Clementel, Irene and A. Wall, Wolfgang},
journal={International journal for numerical methods in biomedical engineering},
volume={34},
number={2},
pages={e2918},
year={2018},
publisher={Wiley Online Library}
}
@article{Bert2018,
author = {Bertoglio, C. and Nu{\~n}ez, R. and Galarce, F. and Nordsletten, D. and Osses, A.},
title = {Relative pressure estimation from velocity measurements in blood flows: State-of-the-art and new approaches},
journal = {International Journal for Numerical Methods in Biomedical Engineering},
volume = {2018;34:e2925. https://doi.org/10.1002/cnm.2925},
year = {2018}
}
@article{Lozovskiy2018,
year = {2018},
month = may,
publisher = {Elsevier {BV}},
volume = {333},
pages = {55--73},
author = {Alexander Lozovskiy and Maxim A. Olshanskii and Yuri V. Vassilevski},
title = {A quasi-Lagrangian finite element method for the Navier{\textendash}Stokes equations in a time-dependent domain},
journal = {Computer Methods in Applied Mechanics and Engineering}
}
@article{Liu2018,
year = {2018},
month = aug,
publisher = {Elsevier {BV}},
volume = {337},
pages = {549--597},
author = {Ju Liu and Alison L. Marsden},
title = {A unified continuum and variational multiscale formulation for fluids, solids, and fluid{\textendash}structure interaction},
journal = {Computer Methods in Applied Mechanics and Engineering}
}
@article{Nordsletten2008,
year = {2008},
publisher = {Wiley},
volume = {56},
number = {8},
pages = {1457--1463},
author = {D. A. Nordsletten and P. J. Hunter and N. P. Smith},
title = {Conservative and non-conservative arbitrary Lagrangian{\textendash}Eulerian forms for ventricular flows},
journal = {International Journal for Numerical Methods in Fluids}
}
@article{Landajuela2016,
year = {2016},
month = jul,
publisher = {Wiley},
volume = {33},
number = {4},
pages = {e2813},
author = {Mikel Landajuela and Marina Vidrascu and Dominique Chapelle and Miguel A. Fern{\'{a}}ndez},
title = {Coupling schemes for the {FSI} forward prediction challenge: Comparative study and validation},
journal = {International Journal for Numerical Methods in Biomedical Engineering}
}
@article{Burtschell2017,
year = {2017},
month = apr,
publisher = {Elsevier {BV}},
volume = {182},
pages = {313--324},
author = {Bruno Burtschell and Dominique Chapelle and Philippe Moireau},
title = {Effective and energy-preserving time discretization for a general nonlinear poromechanical formulation},
journal = {Computers {\&} Structures}
}
@article{Basting2017,
year = {2017},
month = feb,
publisher = {Elsevier {BV}},
volume = {331},
pages = {312--336},
author = {Steffen Basting and Annalisa Quaini and Sun{\v{c}}ica {\v{C}}ani{\'{c}} and Roland Glowinski},
title = {Extended {ALE} Method for fluid{\textendash}structure interaction problems with large structural displacements},
journal = {Journal of Computational Physics}
}
@article{Deparis2016,
year = {2016},
month = dec,
publisher = {Elsevier {BV}},
volume = {327},
pages = {700--718},
author = {Simone Deparis and Davide Forti and Gwenol Grandperrin and Alfio Quarteroni},
title = {{FaCSI}: A block parallel preconditioner for fluid{\textendash}structure interaction in hemodynamics},
journal = {Journal of Computational Physics}
}
@incollection{Colciago2017,
year = {2017},
month = nov,
publisher = {De Gruyter},
author = {Claudia M. Colciago and Simone Deparis and Davide Forti},
editor = {Stefan Frei and B\"{a}rbel Holm and Thomas Richter and Thomas Wick and Huidong Yang},
title = {7. Fluid-structure interaction for vascular flows: From supercomputers to laptops},
booktitle = {Fluid-Structure Interaction}
}
@article{LeTallec2001,
year = {2001},
month = mar,
publisher = {Elsevier {BV}},
volume = {190},
number = {24-25},
pages = {3039--3067},
author = {P. Le Tallec and J. Mouro},
title = {Fluid structure interaction with large structural displacements},
journal = {Computer Methods in Applied Mechanics and Engineering}
}
@phdthesis{smaldone2014,
TITLE = {{Numerical analysis and simulations of coupled problems for the cariovascular system}},
AUTHOR = {Smaldone, Saverio},
SCHOOL = {{L'UNIVERSIT{\'E} PIERRE ET MARIE CURIE - Paris VI }},
YEAR = {2014},
MONTH = Oct,
TYPE = {Theses},
HAL_ID = {tel-01287506},
HAL_VERSION = {v1},
}
@article{Balzani2015,
year = {2015},
month = dec,
publisher = {Wiley},
volume = {32},
number = {10},
pages = {e02756},
author = {Daniel Balzani and Simone Deparis and Simon Fausten and Davide Forti and Alexander Heinlein and Axel Klawonn and Alfio Quarteroni and Oliver Rheinbach and Joerg Schr\"{o}der},
title = {Numerical modeling of fluid-structure interaction in arteries with anisotropic polyconvex hyperelastic and anisotropic viscoelastic material models at finite strains},
journal = {International Journal for Numerical Methods in Biomedical Engineering}
}
@misc{langer2014numerical,
title={Numerical Simulation of Fluid-Structure Interaction Problems with Hyperelastic Models: A Monolithic Approach},
author={Ulrich Langer and Huidong Yang},
year={2014},
eprint={1408.3737},
archivePrefix={arXiv},
primaryClass={math.NA}
}
@misc{failer2020impact,
title={On the Impact of Fluid Structure Interaction in Blood Flow Simulations: Stenotic Coronary Artery Benchmark},
author={Lukas Failer and Piotr Minakowski and Thomas Richter},
year={2020},
eprint={2003.05214},
archivePrefix={arXiv},
primaryClass={physics.comp-ph}
}
@article{Langer2016,
year = {2016},
month = feb,
publisher = {Wiley},
volume = {108},
number = {4},
pages = {303--325},
author = {Ulrich Langer and Huidong Yang},
title = {Robust and efficient monolithic fluid-structure-interaction solvers},
journal = {International Journal for Numerical Methods in Engineering}
}
@article{Boffi2004,
year = {2004},
month = oct,
publisher = {Elsevier {BV}},
volume = {193},
number = {42-44},
pages = {4717--4739},
author = {Daniele Boffi and Lucia Gastaldi},
title = {Stability and geometric conservation laws for {ALE} formulations},
journal = {Computer Methods in Applied Mechanics and Engineering}
}
@article{Murea2016,
year = {2016},
month = jun,
publisher = {Wiley},
volume = {109},
number = {8},
pages = {1067--1084},
author = {Cornel Marius Murea and Soyibou Sy},
title = {Updated Lagrangian/Arbitrary Lagrangian-Eulerian framework for interaction between a compressible neo-Hookean structure and an incompressible fluid},
journal = {International Journal for Numerical Methods in Engineering}
}
@article{Hessenthaler2017,
year = {2017},
month = feb,
publisher = {Wiley},
volume = {33},
number = {8},
pages = {e2845},
author = {Andreas Hessenthaler and Oliver R\"{o}hrle and David Nordsletten},
title = {Validation of a non-conforming monolithic fluid-structure interaction method using phase-contrast {MRI}},
journal = {International Journal for Numerical Methods in Biomedical Engineering}
}
@Inbook{Wall2009,
author="Wall, W. A.
and Gerstenberger, A.
and Mayer, U. M.",
editor="Eberhardsteiner, Josef
and Hellmich, Christian
and Mang, Herbert A.
and P{\'e}riaux, Jacques",
title="Advances in Fixed-Grid Fluid Structure Interaction",
bookTitle="ECCOMAS Multidisciplinary Jubilee Symposium: New Computational Challenges in Materials, Structures, and Fluids",
year="2009",
publisher="Springer Netherlands",
address="Dordrecht",
pages="235--249",
isbn="978-1-4020-9231-2",
doi="10.1007/978-1-4020-9231-2_16",
url="https://doi.org/10.1007/978-1-4020-9231-2_16"
}
@inproceedings{Tallec2003,
author = {Tallec, Le and Hauret, Patrice},
year = {2003},
month = {01},
pages = {},
title = {Energy conservation in fluid-structure interactions}
}
@misc{Wang2020,
title={An energy stable one-field monolithic arbitrary Lagrangian-Eulerian formulation for fluid-structure interaction},
author={Yongxing Wang and Peter K. Jimack and Mark A. Walkley and Olivier Pironneau},
year={2020},
eprint={2003.03819},
archivePrefix={arXiv},
primaryClass={cs.CE}
}

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%%
%% This is file `cas-dc.cls'.
%%
%% This file is part of the 'CAS Bundle'.
%% ......................................
%%
%% It may be distributed under the conditions of the LaTeX Project Public
%% License, either version 1.2 of this license or (at your option) any
%% later version. The latest version of this license is in
%% http://www.latex-project.org/lppl.txt
%% and version 1.2 or later is part of all distributions of LaTeX
%% version 1999/12/01 or later.
%%
%% The list of all files belonging to the 'CAS Bundle' is
%% given in the file `manifest.txt'.
%%
%% $Id: cas-sc.cls 49 2020-03-14 09:05:10Z rishi $
\def\RCSfile{cas-sc}%
\def\RCSversion{2.1}%
\def\RCSdate{2020/03/14}%
\NeedsTeXFormat{LaTeX2e}[1995/12/01]
\ProvidesClass{\RCSfile}[\RCSdate, \RCSversion: Formatting class
for CAS single column articles]
%
\def\ABD{\AtBeginDocument}
%
% switches
%
\newif\iflongmktitle \longmktitlefalse
\newif\ifdc \global\dcfalse
\newif\ifsc \global\sctrue
\newif\ifcasreviewlayout \global\casreviewlayoutfalse
\newif\ifcasfinallayout \global\casfinallayoutfalse
\newcounter{blind}
\setcounter{blind}{0}
\def\blstr#1{\gdef\@blstr{#1}}
\def\@blstr{1}
\newdimen\@bls
\@bls=\baselineskip
\DeclareOption{singleblind}{\setcounter{blind}{1}}
\DeclareOption{doubleblind}{\setcounter{blind}{2}}
\DeclareOption{longmktitle}{\global\longmktitletrue}
\DeclareOption{final}{\global\casfinallayouttrue}
\DeclareOption{review}{\global\casreviewlayouttrue}
\ExecuteOptions{a4paper,10pt,oneside,fleqn,review}
\DeclareOption*{\PassOptionsToClass{\CurrentOption}{article}}
\ProcessOptions
\LoadClass{article}
\RequirePackage{graphicx}
\RequirePackage{amsmath,amsfonts,amssymb}
\allowdisplaybreaks
\RequirePackage{expl3,xparse}
\@ifundefined{regex_match:nnTF}{\RequirePackage{l3regex}}{}
\RequirePackage{etoolbox}
\RequirePackage{booktabs,makecell,multirow,array,colortbl,dcolumn,stfloats}
\RequirePackage{xspace,xstring,footmisc}
\RequirePackage[svgnames,dvipsnames]{xcolor}
\RequirePackage[colorlinks]{hyperref}
\colorlet{scolor}{black}
\colorlet{hscolor}{DarkSlateGrey}
\hypersetup{%
pdfcreator={LaTeX3; cas-sc.cls; hyperref.sty},
pdfproducer={pdfTeX;},
linkcolor={hscolor},
urlcolor={hscolor},
citecolor={hscolor},
filecolor={hscolor},
menucolor={hscolor},
}
% \AtEndDocument{\hypersetup
% {pdftitle={\csuse{__short_title:}},
% pdfauthor={\csuse{__short_authors:}}}}
\let\comma\@empty
\let\tnotesep\@empty
\let\@title\@empty
%
% Load Common items
%
\RequirePackage{cas-common}
%
% Specific to Single Column
%
\ExplSyntaxOn
\RenewDocumentCommand \maketitle {}
{
\ifbool { usecasgrabsbox }
{
\setcounter{page}{0}
\thispagestyle{empty}
\unvbox\casgrabsbox
} { }
\pagebreak
\ifbool { usecashlsbox }
{
\setcounter{page}{0}
\thispagestyle{empty}
\unvbox\casauhlbox
} { }
\pagebreak
\thispagestyle{first}
\ifbool{longmktitle}
{
\LongMaketitleBox
\ProcessLongTitleBox
}
{
\MaketitleBox
\printFirstPageNotes
}
\normalcolor \normalfont
\setcounter{footnote}{\int_use:N \g_stm_fnote_int}
\renewcommand\thefootnote{\arabic{footnote}}
\gdef\@pdfauthor{\infoauthors}
\gdef\@pdfsubject{Complex ~STM ~Content}
}
%
% Fonts
%
\RequirePackage[T1]{fontenc}
\file_if_exist:nTF { stix.sty }
{
\file_if_exist:nTF { charis.sty }
{
\RequirePackage[notext]{stix}
\RequirePackage{charis}
}
{ \RequirePackage{stix} }
}
{
\iow_term:x { *********************************************************** }
\iow_term:x { ~Stix ~ and ~ Charis~ fonts ~ are ~ not ~ available ~ }
\iow_term:x { ~ in ~TeX~system.~Hence~CMR~ fonts~ are ~ used. }
\iow_term:x { *********************************************************** }
}
\file_if_exist:nTF { inconsolata }
{ \RequirePackage[scaled=.85]{inconsolata} }
{ \tex_gdef:D \ttdefault { cmtt } }
\ExplSyntaxOff
%
% Page geometry
%
\usepackage[%
paperwidth=192mm,
paperheight=262mm,
% vmargin={12.4mm,11.5mm},
vmargin={19mm,19mm},
hmargin={13.7mm,13.7mm},
headsep=12pt,
footskip=12pt,
]{geometry}
\endinput
%
% End of class 'cas-sc'
%

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%%
%% Copyright 2019-2020 Elsevier Ltd
%%
%% This file is part of the 'CAS Bundle'.
%% ---------------------------------------------
%%
%% It may be distributed under the conditions of the LaTeX Project Public
%% License, either version 1.2 of this license or (at your option) any
%% later version. The latest version of this license is in
%% http://www.latex-project.org/lppl.txt
%% and version 1.2 or later is part of all distributions of LaTeX
%% version 1999/12/01 or later.
%%
%% The list of all files belonging to the 'CAS Bundle' is
%% given in the file `manifest.txt'.
%%
%% $Id: elsdoc-cas.tex 35 2020-02-25 09:04:59Z rishi $
%%
\documentclass[a4paper,12pt]{article}
\usepackage[xcolor,qtwo]{rvdtx}
\usepackage{multicol}
\usepackage{color}
\usepackage{xspace}
\usepackage{pdfwidgets}
\usepackage{enumerate}
\def\ttdefault{cmtt}
\headsep4pc
\makeatletter
\def\bs{\expandafter\@gobble\string\\}
\def\lb{\expandafter\@gobble\string\{}
\def\rb{\expandafter\@gobble\string\}}
\def\@pdfauthor{C.V.Radhakrishnan}
\def\@pdftitle{CAS templates: A documentation}
\def\@pdfsubject{Document formatting with CAS template}
\def\@pdfkeywords{LaTeX, Elsevier Ltd, document class}
\def\file#1{\textsf{#1}\xspace}
%\def\LastPage{19}
\DeclareRobustCommand{\LaTeX}{L\kern-.26em%
{\sbox\z@ T%
\vbox to\ht\z@{\hbox{\check@mathfonts
\fontsize\sf@size\z@
\math@fontsfalse\selectfont
A\,}%
\vss}%
}%
\kern-.15em%
\TeX}
\makeatother
\def\figurename{Clip}
\setcounter{tocdepth}{1}
\AtBeginDocument{
\setcounter{topnumber}{2}
\setcounter{bottomnumber}{2}
\setcounter{totalnumber}{4}
\renewcommand{\topfraction}{0.85}
\renewcommand{\bottomfraction}{0.85}
\renewcommand{\textfraction}{0.15}
\renewcommand{\floatpagefraction}{0.7}
}
\begin{document}
\def\testa{This is a specimen document. }
\def\testc{\testa\testa\testa\testa}
\def\testb{\testc\testc\testc\testc\testc}
\long\def\test{\testb\par\testb\par\testb\par}
\pinclude{\copy\contbox\printSq{\LastPage}}
\title{Documentation for Elsevier's Complex Article Service (CAS)
\LaTeX\ template}
\author{Elsevier Ltd}
\contact{elsarticle@stmdocs.in}
\version{1.0}
\date{\today}
\maketitle
\section{Introduction}
Two classfiles namely \file{cas-sc.cls} and \file{cas-dc.cls} were
written for typesetting articles submitted in journals of Elsevier's
Complex Article Service (CAS) workflow.
\subsection{Usage}
\begin{enumerate}
\item \file{cas-sc.cls} for single column journals.
\begin{vquote}
\documentclass[<options>]{cas-sc}
\end{vquote}
\item \file{cas-dc.cls} for single column journals.
\begin{vquote}
\documentclass[<options>]{cas-dc}
\end{vquote}
\end{enumerate}
and have an option longmktitle to handle long front matter.
\section{Front matter}
\begin{vquote}
\title [mode = title]{This is a specimen $a_b$ title}
\tnotemark[1,2]
\tnotetext[1]{This document is the results of the research
project funded by the National Science Foundation.}
\tnotetext[2]{The second title footnote which is a longer text
matter to fill through the whole text width and overflow into
another line in the footnotes area of the first page.}
\author[1,3]{CV Radhakrishnan}[type=editor,
auid=000,bioid=1,
prefix=Sir,
role=Researcher,
orcid=0000-0001-7511-2910]
\cormark[1]
\fnmark[1]
\ead{cvr_1@tug.org.in}
\ead[url]{www.cvr.cc, cvr@sayahna.org}
\end{vquote}
\begin{vquote}
\credit{Conceptualization of this study, Methodology,
Software}
\address[1]{Elsevier B.V., Radarweg 29, 1043 NX Amsterdam,
The Netherlands}
\author[2,4]{Han Theh Thanh}[style=chinese]
\author[2,3]{CV Rajagopal}[%
role=Co-ordinator,
suffix=Jr,
]
\fnmark[2]
\ead{cvr3@sayahna.org}
\ead[URL]{www.sayahna.org}
\credit{Data curation, Writing - Original draft preparation}
\address[2]{Sayahna Foundation, Jagathy, Trivandrum 695014,
India}
\author[1,3]{Rishi T.}
\cormark[2]
\fnmark[1,3]
\ead{rishi@stmdocs.in}
\ead[URL]{www.stmdocs.in}
\address[3]{STM Document Engineering Pvt Ltd., Mepukada,
Malayinkil, Trivandrum 695571, India}
\cortext[cor1]{Corresponding author}
\cortext[cor2]{Principal corresponding author}
\fntext[fn1]{This is the first author footnote. but is common
to third author as well.}
\fntext[fn2]{Another author footnote, this is a very long
footnote and it should be a really long footnote. But this
footnote is not yet sufficiently long enough to make two lines
of footnote text.}
\end{vquote}
\begin{vquote}
\nonumnote{This note has no numbers. In this work we
demonstrate $a_b$ the formation Y\_1 of a new type of
polariton on the interface between a cuprous oxide slab
and a polystyrene micro-sphere placed on the slab.
}
\begin{abstract}[S U M M A R Y]
This template helps you to create a properly formatted
\LaTeX\ manuscript.
\noindent\texttt{\textbackslash begin{abstract}} \dots
\texttt{\textbackslash end{abstract}} and
\verb+\begin{keyword}+ \verb+...+ \verb+\end{keyword}+
which contain the abstract and keywords respectively.
Each keyword shall be separated by a \verb+\sep+ command.
\end{abstract}
\begin{keywords}
quadrupole exciton \sep polariton \sep \WGM \sep \BEC
\end{keywords}
\maketitle
\end{vquote}
\begin{figure}
\includegraphics[width=\textwidth]{sc-sample.pdf}
\caption{Single column output (classfile: cas-sc.cls).}
\end{figure}
\begin{figure}
\includegraphics[width=\textwidth]{dc-sample.pdf}
\caption{Double column output (classfile: cas-dc.cls).}
\end{figure}
\subsection{Title}
\verb+\title+ command have the below options:
\begin{enumerate}
\item \verb+title:+ Document title
\item \verb+alt:+ Alternate title
\item \verb+sub:+ Sub title
\item \verb+trans:+ Translated title
\item \verb+transsub:+ Translated sub title
\end{enumerate}
\begin{vquote}
\title[mode=title]{This is a title}
\title[mode=alt]{This is a alternate title}
\title[mode=sub]{This is a sub title}
\title[mode=trans]{This is a translated title}
\title[mode=transsub]{This is a translated sub title}
\end{vquote}
\subsection{Author}
\verb+\author+ command have the below options:
\begin{enumerate}
\item \verb+auid:+ Author id
\item \verb+bioid:+ Biography id
\item \verb+alt:+ Alternate author
\item \verb+style:+ Style of author name chinese
\item \verb+prefix:+ Prefix Sir
\item \verb+suffix:+ Suffix
\item \verb+degree:+ Degree
\item \verb+role:+ Role
\item \verb+orcid:+ ORCID
\item \verb+collab:+ Collaboration
\item \verb+anon:+ Anonymous author
\item \verb+deceased:+ Deceased author
\item \verb+twitter:+ Twitter account
\item \verb+facebook:+ Facebook account
\item \verb+linkedin:+ LinkedIn account
\item \verb+plus:+ Google plus account
\item \verb+gplus:+ Google plus account
\end{enumerate}
\begin{vquote}
\author[1,3]{Author Name}[type=editor,
auid=000,bioid=1,
prefix=Sir,
role=Researcher,
orcid=0000-0001-7511-2910,
facebook=<facebook id>,
twitter=<twitter id>,
linkedin=<linkedin id>,
gplus=<gplus id>]
\end{vquote}
\subsection{Various Marks in the Front Matter}
The front matter becomes complicated due to various kinds
of notes and marks to the title and author names. Marks in
the title will be denoted by a star ($\star$) mark;
footnotes are denoted by super scripted Arabic numerals,
corresponding author by of an Conformal asterisk (*) mark.
\subsubsection{Title marks}
Title mark can be entered by the command, \verb+\tnotemark[<num>]+
and the corresponding text can be entered with the command
\verb+\tnotetext[<num>]+ \verb+{<text>}+. An example will be:
\begin{vquote}
\title[mode=title]{Leveraging social media news to predict
stock index movement using RNN-boost}
\tnotemark[1,2]
\tnotetext[1]{This document is the results of the research
project funded by the National Science Foundation.}
\tnotetext[2]{The second title footnote which is a longer
text matter to fill through the whole text width and
overflow into another line in the footnotes area of
the first page.}
\end{vquote}
\verb+\tnotetext+ and \verb+\tnotemark+ can be anywhere in
the front matter, but shall be before \verb+\maketitle+ command.
\subsubsection{Author marks}
Author names can have many kinds of marks and notes:
\begin{vquote}
footnote mark : \fnmark[<num>]
footnote text : \fntext[<num>]{<text>}
affiliation mark : \author[<num>]
email : \ead{<emailid>}
url : \ead[url]{<url>}
corresponding author mark : \cormark[<num>]
corresponding author text : \cortext[<num>]{<text>}
\end{vquote}
\subsubsection{Other marks}
At times, authors want footnotes which leave no marks in
the author names. The note text shall be listed as part of
the front matter notes. Class files provides
\verb+\nonumnote+ for this purpose. The usage
\begin{vquote}
\nonumnote{<text>}
\end{vquote}
\noindent and should be entered anywhere before the \verb+\maketitle+
command for this to take effect.
\subsection{Abstract and Keywords}
Abstract shall be entered in an environment that starts
with \verb+\begin{abstract}+ and ends with
\verb+\end{abstract}+. Longer abstracts spanning more than
one page is also possible in Class file even in double
column mode. We need to invoke longmktitle option in the
class loading line for this to happen smoothly.
The key words are enclosed in a \verb+{keyword}+
environment.
\begin{vquote}
\begin{abstract}
This is a abstract. \lipsum[3]
\end{abstract}
\begin{keywords}
First keyword \sep Second keyword \sep Third
keyword \sep Fourth keyword
\end{keywords}
\end{vquote}
\section{Main Matter}
\subsection{Tables}
\subsubsection{Normal tables}
\begin{vquote}
\begin{table}
\caption{This is a test caption.}
\begin{tabular*}{\tblwidth}{@{} LLLL@{} }
\toprule
Col 1 & Col 2\\
\midrule
12345 & 12345\\
12345 & 12345\\
12345 & 12345\\
\bottomrule
\end{tabular*}
\end{table}
\end{vquote}
\subsubsection{Span tables}
\begin{vquote}
\begin{table*}[width=.9\textwidth,cols=4,pos=h]
\caption{This is a test caption.}
\begin{tabular*}{\tblwidth}{@{} LLLLLL@{} }
\toprule
Col 1 & Col 2 & Col 3 & Col4 & Col5 & Col6 & Col7\\
\midrule
12345 & 12345 & 123 & 12345 & 123 & 12345 & 123 \\
12345 & 12345 & 123 & 12345 & 123 & 12345 & 123 \\
12345 & 12345 & 123 & 12345 & 123 & 12345 & 123 \\
\bottomrule
\end{tabular*}
\end{table*}
\end{vquote}
\subsection{Figures}
\subsubsection{Normal figures}
\begin{vquote}
\begin{figure}
\centering
\includegraphics[scale=.75]{Fig1.pdf}
\caption{The evanescent light - $1S$ quadrupole coupling
($g_{1,l}$) scaled to the bulk exciton-photon coupling
($g_{1,2}$). The size parameter $kr_{0}$ is denoted as $x$ and
the \PMS is placed directly on the cuprous oxide sample ($\delta
r=0$, See also Fig. \protect\ref{FIG:2}).}
\label{FIG:1}
\end{figure}
\end{vquote}
\subsubsection{Span figures}
\begin{vquote}
\begin{figure*}
\centering
\includegraphics[width=\textwidth,height=2in]{Fig2.pdf}
\caption{Schematic of formation of the evanescent polariton on
linear chain of \PMS. The actual dispersion is determined by
the ratio of two coupling parameters such as exciton-\WGM
coupling and \WGM-\WGM coupling between the microspheres.}
\label{FIG:2}
\end{figure*}\end{vquote}
\subsection{Theorem and theorem like environments}
CAS class file provides a few hooks to format theorems and
theorem like environments with ease. All commands the
options that are used with \verb+\newtheorem+ command will work
exactly in the same manner. Class file provides three
commands to format theorem or theorem like environments:
\begin{enumerate}
\item \verb+\newtheorem+ command formats a theorem in
\LaTeX's default style with italicized font for theorem
statement, bold weight for theorem heading and theorem
number typeset at the right of theorem heading. It also
optionally accepts an argument which will be printed as an
extra heading in parentheses. Here is an example coding and
output:
\begin{vquote}
\newtheorem{theorem}{Theorem}
\begin{theorem}\label{thm}
The \WGM evanescent field penetration depth into the
cuprous oxide adjacent crystal is much larger than the
\QE radius:
\begin{equation*}
\lambda_{1S}/2 \pi \left({\epsilon_{Cu2O}-1}
\right)^{1/2} = 414 \mbox{ \AA} \gg a_B = 4.6
\mbox{ \AA}
\end{equation*}
\end{theorem}
\end{vquote}
\item \verb+\newdefinition+ command does exactly the same
thing as with except that the body font is up-shape instead
of italic. See the example below:
\begin{vquote}
\newdefinition{definition}{Definition}
\begin{definition}
The bulk and evanescent polaritons in cuprous oxide
are formed through the quadrupole part of the light-matter
interaction:
\begin{equation*}
H_{int} = \frac{i e }{m \omega_{1S}} {\bf E}_{i,s}
\cdot {\bf p}
\end{equation*}
\end{definition}
\end{vquote}
\item \verb+\newproof+ command helps to define proof and
custom proof environments without counters as provided in
the example code. Given below is an example of proof of
theorem kind.
\begin{vquote}
\newproof{pot}{Proof of Theorem \ref{thm}}
\begin{pot}
The photon part of the polariton trapped inside the \PMS
moves as it would move in a micro-cavity of the effective
modal volume $V \ll 4 \pi r_{0}^{3} /3$. Consequently, it
can escape through the evanescent field. This evanescent
field essentially has a quantum origin and is due to
tunneling through the potential caused by dielectric
mismatch on the \PMS surface. Therefore, we define the
\emph{evanescent} polariton (\EP) as an evanescent light -
\QE coherent superposition.
\end{pot}
\end{vquote}
\end{enumerate}
\subsection{Enumerated and Itemized Lists}
CAS class files provides an extended list processing macros
which makes the usage a bit more user friendly than the
default LaTeX list macros. With an optional argument to the
\verb+\begin{enumerate}+ command, you can change the list
counter type and its attributes. You can see the coding and
typeset copy.
\begin{vquote}
\begin{enumerate}[1.]
\item The enumerate environment starts with an optional
argument `1.' so that the item counter will be suffixed
by a period as in the optional argument.
\item If you provide a closing parenthesis to the number in the
optional argument, the output will have closing
parenthesis for all the item counters.
\item You can use `(a)' for alphabetical counter and `(i)' for
roman counter.
\begin{enumerate}[a)]
\item Another level of list with alphabetical counter.
\item One more item before we start another.
\begin{enumerate}[(i)]
\item This item has roman numeral counter.
\end{vquote}
\begin{vquote}
\item Another one before we close the third level.
\end{enumerate}
\item Third item in second level.
\end{enumerate}
\item All list items conclude with this step.
\end{enumerate}
\section{Biography}
\verb+\bio+ command have the below options:
\begin{enumerate}
\item \verb+width:+ Width of the author photo (default is 1in).
\item \verb+pos:+ Position of author photo.
\end{enumerate}
\begin{vquote}
\bio[width=10mm,pos=l]{tuglogo.jpg}
\textbf{Another Biography:}
Recent experimental \cite{HARA:2005} and theoretical
\cite{DEYCH:2006} studies have shown that the \WGM can travel
along the chain as "heavy photons". Therefore the \WGM
acquires the spatial dispersion, and the evanescent
quadrupole polariton has the form (See Fig.\ref{FIG:3}):
\endbio
\end{vquote}
\section[CRediT...]{CRediT authorship contribution statement}
Give the authorship contribution after each author as
\begin{vquote}
\credit{Conceptualization of this study, Methodology,
Software}
\end{vquote}
To print the details use \verb+\printcredits+
\begin{vquote}
\author[1,3]{V. {{\=A}}nand Rawat}[auid=000,
bioid=1,
prefix=Sir,
role=Researcher,
orcid=0000-0001-7511-2910]
\end{vquote}
\begin{vquote}
\cormark[1]
\fnmark[1]
\ead{cvr_1@tug.org.in}
\ead[url]{www.cvr.cc, www.tug.org.in}
\credit{Conceptualization of this study, Methodology,
Software}
\address[1]{Indian \TeX{} Users Group, Trivandrum 695014,
India}
\author[2,4]{Han Theh Thanh}[style=chinese]
\author[2,3]{T. Rishi Nair}[role=Co-ordinator,
suffix=Jr]
\fnmark[2]
\ead{rishi@sayahna.org}
\ead[URL]{www.sayahna.org}
\credit{Data curation, Writing - Original draft preparation}
. . .
. . .
. . .
\printcredits
\end{vquote}
\section{Bibliography}
For CAS categories, two reference models are recommended.
They are \file{model1-num-names.bst} and \file{model2-names.bst}.
Former will format the reference list and their citations according to
numbered scheme whereas the latter will format according name-date or
author-year style. Authors are requested to choose any one of these
according to the journal style. You may download these from
The above bsts are available in the following location for you to
download:
\url{https://support.stmdocs.in/wiki/index.php?title=Model-wise_bibliographic_style_files}
\hfill $\Box$
\end{document}

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tex/1_submission/doc/makefile Normal file
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# $Id: makefile 37 2020-02-25 09:06:02Z rishi $
file=elsdoc-cas
all: pdf out
make pdf
make pdf
out:
if [ -f $(file).out ] ; then cp $(file).out tmp.out; fi ;
sed 's/BOOKMARK/dtxmark/g;' tmp.out > x.out; mv x.out tmp.out ;
pdf:
pdflatex $(file).tex
index:
makeindex -s gind.ist -o $(file).ind $(file).idx
changes:
makeindex -s gglo.ist -o $(file).gls $(file).glo
xview:
xpdf -z 200 $(file).pdf &>/dev/null
view:
open -a 'Adobe Reader.app' $(file).pdf
ins:
latex $(file).ins
diff:
diff $(file).sty ../$(file).sty |less
copy:
cp $(file).sty ../

384
tex/1_submission/doc/pdfwidgets.sty Normal file
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%%
%% pdfwidgets.sty
%%
%% $Id: pdfwidgets.sty,v 1.2 2007-10-22 09:45:17 cvr Exp $
%%
%% (c) C. V. Radhakrishnan <cvr@river-valley.org>
%%
%% This package may be distributed under the terms of the LaTeX Project
%% Public License, as described in lppl.txt in the base LaTeX distribution.
%% Either version 1.0 or, at your option, any later version.
%%
%\RequirePackage[oldstyle]{minion}
%\RequirePackage[scaled=.8]{prima}
%\RequirePackage[scaled=.9]{lfr}
\usepackage[dvipsnames,svgnames]{xcolor}
\RequirePackage{graphicx}
\RequirePackage{tikz}
\usetikzlibrary{backgrounds}
%\def\thesection{\ifnum\c@section<10
% \protect\phantom{0}\fi\arabic{section}}
\newdimen\lmrgn
\def\rulecolor{orange}
\def\rulewidth{1pt}
\pgfdeclareshape{filledbox}{%
\inheritsavedanchors[from=rectangle] % this is nearly a rectangle
\inheritanchorborder[from=rectangle]
\inheritanchor[from=rectangle]{center}
\inheritanchor[from=rectangle]{north}
\inheritanchor[from=rectangle]{south}
\inheritanchor[from=rectangle]{west}
\inheritanchor[from=rectangle]{east}
% ... and possibly more
\backgroundpath{% this is new
% store lower right in xa/ya and upper right in xb/yb
\southwest \pgf@xa=\pgf@x \pgf@ya=\pgf@y
\northeast \pgf@xb=\pgf@x \pgf@yb=\pgf@y
% compute corner of ``flipped page''
\pgf@xc=\pgf@xb \advance\pgf@xc by-5pt % this should be a parameter
\pgf@yc=\pgf@yb \advance\pgf@yc by-5pt
% construct main path
\pgfsetlinewidth{\rulewidth}
\pgfsetstrokecolor{\rulecolor}
\pgfpathmoveto{\pgfpoint{\pgf@xa}{\pgf@ya}}
\pgfsetcornersarced{\pgfpoint{9pt}{9pt}}
\pgfpathlineto{\pgfpoint{\pgf@xa}{\pgf@yb}}
% \pgfsetcornersarced{\pgforigin}
\pgfsetcornersarced{\pgfpoint{9pt}{9pt}}
\pgfpathlineto{\pgfpoint{\pgf@xb}{\pgf@yb}}
\pgfsetcornersarced{\pgfpoint{9pt}{9pt}}
\pgfpathlineto{\pgfpoint{\pgf@xb}{\pgf@ya}}
\pgfsetcornersarced{\pgforigin}
\pgfpathclose ;
% \draw(\pgf@xa,\pgf@ya) -- (\pgf@xa,\pgf@yb) ;
}%
}
\pgfdeclareshape{roundedbox}{%
\inheritsavedanchors[from=rectangle] % this is nearly a rectangle
\inheritanchorborder[from=rectangle]
\inheritanchor[from=rectangle]{center}
\inheritanchor[from=rectangle]{north}
\inheritanchor[from=rectangle]{south}
\inheritanchor[from=rectangle]{west}
\inheritanchor[from=rectangle]{east}
% ... and possibly more
\backgroundpath{% this is new
% store lower right in xa/ya and upper right in xb/yb
\southwest \pgf@xa=\pgf@x \pgf@ya=\pgf@y
\northeast \pgf@xb=\pgf@x \pgf@yb=\pgf@y
% compute corner of ``flipped page''
\pgf@xc=\pgf@xb \advance\pgf@xc by-5pt % this should be a parameter
\pgf@yc=\pgf@yb \advance\pgf@yc by-5pt
% construct main path
\pgfsetlinewidth{\rulewidth}
\pgfsetstrokecolor{\rulecolor}
\pgfpathmoveto{\pgfpoint{\pgf@xa}{\pgf@ya}}
\pgfsetcornersarced{\pgfpoint{4pt}{4pt}}
\pgfpathlineto{\pgfpoint{\pgf@xa}{\pgf@yb}}
% \pgfsetcornersarced{\pgforigin}
\pgfsetcornersarced{\pgfpoint{4pt}{4pt}}
\pgfpathlineto{\pgfpoint{\pgf@xb}{\pgf@yb}}
\pgfsetcornersarced{\pgfpoint{4pt}{4pt}}
\pgfpathlineto{\pgfpoint{\pgf@xb}{\pgf@ya}}
% \pgfsetcornersarced{\pgforigin}
\pgfsetcornersarced{\pgfpoint{4pt}{4pt}}
\pgfpathclose ;
% \draw(\pgf@xa,\pgf@ya) -- (\pgf@xa,\pgf@yb) ;
}%
}
\pgfdeclareshape{buttonbox}{%
\inheritsavedanchors[from=rectangle] % this is nearly a rectangle
\inheritanchorborder[from=rectangle]
\inheritanchor[from=rectangle]{center}
\inheritanchor[from=rectangle]{north}
\inheritanchor[from=rectangle]{south}
\inheritanchor[from=rectangle]{west}
\inheritanchor[from=rectangle]{east}
% ... and possibly more
\backgroundpath{% this is new
% store lower right in xa/ya and upper right in xb/yb
\southwest \pgf@xa=\pgf@x \pgf@ya=\pgf@y
\northeast \pgf@xb=\pgf@x \pgf@yb=\pgf@y
% compute corner of ``flipped page''
\pgf@xc=\pgf@xb \advance\pgf@xc by-5pt % this should be a parameter
\pgf@yc=\pgf@yb \advance\pgf@yc by-5pt
% construct main path
\pgfsetlinewidth{1pt}
\pgfsetstrokecolor{blue!10}
\pgfpathmoveto{\pgfpoint{\pgf@xa}{\pgf@ya}}
\pgfsetcornersarced{\pgfpoint{4pt}{4pt}}
\pgfpathlineto{\pgfpoint{\pgf@xa}{\pgf@yb}}
% \pgfsetcornersarced{\pgforigin}
\pgfsetcornersarced{\pgfpoint{4pt}{4pt}}
\pgfpathlineto{\pgfpoint{\pgf@xb}{\pgf@yb}}
\pgfsetcornersarced{\pgforigin}
% \pgfsetcornersarced{\pgfpoint{9pt}{9pt}}
\pgfpathlineto{\pgfpoint{\pgf@xb}{\pgf@ya}}
\pgfsetcornersarced{\pgforigin}
\pgfpathclose ;
% \draw(\pgf@xa,\pgf@ya) -- (\pgf@xa,\pgf@yb) ;
}%
}
\pgfdeclareshape{quotedbox}{%
\inheritsavedanchors[from=rectangle] % this is nearly a rectangle
\inheritanchorborder[from=rectangle]
\inheritanchor[from=rectangle]{center}
\inheritanchor[from=rectangle]{north}
\inheritanchor[from=rectangle]{south}
\inheritanchor[from=rectangle]{west}
\inheritanchor[from=rectangle]{east}
% ... and possibly more
\backgroundpath{% this is new
% store lower right in xa/ya and upper right in xb/yb
\southwest \pgf@xa=\pgf@x \pgf@ya=\pgf@y
\northeast \pgf@xb=\pgf@x \pgf@yb=\pgf@y
% compute corner of ``flipped page''
\pgf@xc=\pgf@xb \advance\pgf@xc by-5pt % this should be a parameter
\pgf@yc=\pgf@yb \advance\pgf@yc by-5pt
% construct main path
\pgfsetlinewidth{\rulewidth}
\pgfsetstrokecolor{\rulecolor}
\pgfpathmoveto{\pgfpoint{\pgf@xa}{\pgf@ya}}
\pgfsetcornersarced{\pgfpoint{9pt}{9pt}}
\pgfpathlineto{\pgfpoint{\pgf@xa}{\pgf@yb}}
\pgfsetcornersarced{\pgforigin}
% \pgfsetcornersarced{\pgfpoint{4pt}{4pt}}
\pgfpathlineto{\pgfpoint{\pgf@xb}{\pgf@yb}}
\pgfsetcornersarced{\pgforigin}
% \pgfsetcornersarced{\pgfpoint{9pt}{9pt}}
\pgfpathlineto{\pgfpoint{\pgf@xb}{\pgf@ya}}
\pgfsetcornersarced{\pgforigin}
\pgfpathclose ;
% \draw(\pgf@xa,\pgf@ya) -- (\pgf@xa,\pgf@yb) ;
}%
}
\newcounter{clip}
\newdimen\mywidth
\mywidth=\linewidth
\def\src#1{\gdef\@src{#1}}\let\@src\@empty
\def\includeclip{\@ifnextchar[{\@includeclip}{\@includeclip[]}}
\def\@includeclip[#1]#2#3#4{\par
% \vskip.75\baselineskip plus 3pt minus 1pt
\computeLinewidth{\mywidth}%
\begingroup\color{white}%
\noindent%
\begin{tikzpicture}
%\node[fill=black!10,draw,shape=filledbox,
\node[fill=black!10,%
draw,
shade,%
top color=blue!10,
bottom color=cyan!5,
shape=filledbox,
inner sep=\Sep,
text width=\Linewidth] (x)
{\parbox{\Linewidth}
{\ifx\@src\@empty\else\refstepcounter{clip}\label{clip\theclip}%
{\par\vskip6pt\color{orange}\sffamily\small
~Clip \theclip:\space\@src.}%
\par\vskip3pt\fi\normalcolor
\includegraphics[width=\Linewidth,page={#2},%
viewport={#3},clip=true,#1]{#4}}
\hspace*{-10pt}};
\end{tikzpicture}
\endgroup
% \par\vskip.5\baselineskip
% plus 3pt minus 1pt
}
%%
%% include clippings from a pdf document:
%% #1 => Optional argument for \includegraphics
%% #2 => page number
%% #3 => co-ordinates
%% #4 => file name
\newenvironment{quoted}{%\bigskip
\computeLinewidth{.95\linewidth}%
\global\setbox0=\hbox\bgroup
\begin{minipage}{.95\linewidth}\color{brown}%
\footnotesize\ttfamily\obeyspaces\obeylines}
{\end{minipage}\egroup
\vskip12pt plus 3pt minus 3pt\noindent\begin{tikzpicture}
\node[fill=blue!10,draw,shade,top color=orange!10,
bottom color=white,shape=filledbox,
inner sep=8pt,text width=\Linewidth] (x) {\box0} ;
\end{tikzpicture}%
\vskip12pt plus 3pt minus 3pt}
\newdimen\Linewidth
\newdimen\Sep
\def\computeLinewidth#1{\global\setlength\Linewidth{#1}%
\global\addtolength{\Linewidth}{-2\Sep}}
\newdimen\npskip
\npskip=0mm
\long\def\NavigationPanel{%
\global\setbox0=\hbox\bgroup
\begin{minipage}[t][.8125\panelheight][t]{.9\panelwidth}\color{brown}%
%\centering
\ifx\@pinclude\empty\relax\par\vfill\else
\@pinclude\fi
%River Valley Technologies
\end{minipage}\egroup
\Sep=.5cm
\@tempdima=\panelwidth
\advance\@tempdima-1cm
\computeLinewidth{\@tempdima}%
\def\rulewidth{.2pt}%
\noindent\begin{tikzpicture}
\node[fill=blue!10,draw,shade,bottom color=brown!30,
top color=white,shape=filledbox,
inner sep=\the\Sep,text width=\Linewidth] (x)
{\hspace*{\npskip}\box0} ;
\end{tikzpicture}%
\vspace*{.0125\panelheight}
}
\long\def\pinclude#1{\gdef\@pinclude{#1}}
\let\@pinclude\empty
\def\Strut{\vrule depth 2pt height 10pt width 0pt}
\def\pdfButton#1#2{\begin{tikzpicture}
\node[fill=blue!10,draw,shade,top color=blue!50,
bottom color=white,shape=buttonbox,
inner sep=2pt,text width=#1](x)
{\parbox{#1}{\centering\Strut#2}}; \end{tikzpicture}}
\def\vpanel{\def\@linkcolor{blue}%
\def\@urlcolor{blue}%
\def\@menucolor{blue}%
\begin{minipage}[t][\vpanelheight][c]{\paperwidth}%
\normalsfcodes%
\hspace*{.25cm}
\begin{minipage}[c][\vpanelheight][c]{17cm}
\parbox[c][27mm][b]{15mm}%
% {\includegraphics[width=15mm]{logo4.pdf}}\hfill%\hspace{1cm}
{\def\rulecolor{Goldenrod}%
\def\rulewidth{1pt}%
\begin{tikzpicture}%
%\node[fill=black!10,draw,shape=filledbox,
\node[fill=white!10,%
draw,
% shade,%
% top color=blue!10,
% bottom color=white,
shape=roundedbox,
inner sep=2mm,
text width=13mm] (x)
{\includegraphics[width=13mm]{els-logo.pdf}};
\end{tikzpicture}}\hfill
%
\parbox[c][24mm][b]{145mm}%
{{\fontsize{30}{30}\selectfont\textsf{\color{white}elsarticle.cls}}
\quad{\fontsize{14}{14}\selectfont\sffamily\color{blue!50}
A better way to format your submission}}
\end{minipage}
\hfill
\begin{minipage}[c][\vpanelheight][b]{7.9cm}
\sffamily\footnotesize
\pdfButton{2cm}{\href{mailto:elsarticle@river-valley.com}{BUGS}}
\pdfButton{2cm}{\href{http://support.river-valley.com}{SUPPORT}}
\pdfButton{2cm}%
{\href{http://www.elsevier.com/locate/latex}%
{RESOURCES}}
% \pdfButton{2cm}{\Acrobatmenu{GoToPage}{GoTo}}
\end{minipage}\\
\rule{\paperwidth}{0.1pt}
\end{minipage}%
}
\@ifundefined{backgroundcolor}%
{\def\backgroundcolor#1{\gdef\@backgroundcolor{#1}}}{}
\colorlet{panelbackground}{orange!10}
\backgroundcolor{orange!10}
\def\@urlcolor{brown}
\def\@linkcolor{brown}
\def\@menucolor{brown}
\RequirePackage{moreverb}
\newenvironment{vquote}%
{\medskip
\verbatimwrite{tmp.tex}}
{\endverbatimwrite
\aftergroup\printBox}
\def\printBox{\bgroup\def\rulecolor{orange}%
\def\rulewidth{.2pt}%
\noindent\begin{tikzpicture}
\node[fill=blue!10,draw,shade,top color=white!10,
bottom color=cyan!5,shape=quotedbox,
inner sep=8pt,text width=.95\linewidth]
{\color{orange}\vspace*{-1pc}%
\verbatiminput{tmp.tex}%
\vspace*{-\baselineskip}%
} ;
\end{tikzpicture}%
\egroup
\medskip
}
\def\red{\color{Sepia}}
\def\verbatim@font{\red\normalfont\ttfamily}
\def\verbatimcontinuewrite{%
\@bsphack
% \verbatim@out=#1
\let\do\@makeother\dospecials
\obeyspaces\catcode`\^^M\active \catcode`\^^I=12
\def\verbatim@processline{%
\immediate\write\verbatim@out
{\the\verbatim@line}}%
\verbatim@start}
\def\@@@lbr{\expandafter\@gobble\string\{}
\def\@@@rbr{\expandafter\@gobble\string\}}
\def\@@@pcr{\expandafter\@gobble\string\%}
%\immediate\write18{touch mytool.tex
% ^^J rm mytool.tex ^^J touch mytool.tex}
\newenvironment{toolwrite}[1]%
{\@tempdima=#1
\verbatimwrite{xx}}
{\endverbatimwrite
\immediate\write18{echo
"\string\Clear\@@@lbr\the\@tempdima\@@@rbr\@@@lbr\@@@pcr">>mytool.tex^^J
cat xx.tex >> mytool.tex ^^J
echo "\@@@rbr" >> mytool.tex}}
\tikzstyle{place}=[scale=.39,rectangle,draw=blue!90,fill=blue!30,thin,%
minimum height=1mm,minimum width=13mm]
\tikzstyle{trans}=[scale=.39,rectangle,draw=Olive,fill=Olive!20,thin,%
minimum height=1mm,minimum width=13mm]
\tikzstyle{past}=[scale=.39,rectangle,draw=Olive,fill=Olive!60,thin,%
minimum height=1mm,minimum width=13mm]
\def\printSq#1{\parbox{107mm}{\@tempcnta=1
\let\printfill\@empty
\loop\ifnum\@tempcnta<#1
{\printfill\ifnum\c@page=\@tempcnta
\tikz\node at(0,0) [place]{};\else
\ifnum\c@page<\@tempcnta
\hyperlink{page.\the\@tempcnta}{\tikz\node at(0,0)
[trans]{};}%
\else
\hyperlink{page.\the\@tempcnta}{\tikz\node at(0,0)
[past]{};}%
\fi\fi}%
\advance\@tempcnta 1 \let\printfill\,\repeat}}
\endinput

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% Copyright 2019-2020 Elsevier Ltd
%
% This file is part of the 'CAS Bundle'.
% --------------------------------------
%
% It may be distributed and/or modified under the
% conditions of the LaTeX Project Public License, either version 1.2
% of this license or (at your option) any later version.
% The latest version of this license is in
% http://www.latex-project.org/lppl.txt
% and version 1.2 or later is part of all distributions of LaTeX
% version 1999/12/01 or later.
%
% The list of all files belonging to the LaTeX 'CAS Bundle' is
% given in the file `manifest.txt'.
%
% CONTENTS OF THE CAS BUNDLE
% ==========================
Directory elsevier-cas-template/
cas-sc.cls
Classfile to be used for single column format
cas-dc.cls
Classfile to be used for double column format
model2-names.bst
BibTeX style file
cas-sc-template.tex
TeX template
cas-sc-template.pdf
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cas-dc-template.tex
TeX template
cas-dc-template.pdf
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manifest.txt
this file
README
small readme documentation
Directory doc/
The following files are graphic files needed for creating pdf output
of the documentation from elsdoc.tex:
dc-sample.pdf
sc-sample.pdf
elsdoc-cas.tex -- LaTeX source file of documentation
elsdoc-cas.pdf -- documentation for elsarticle.cls
Directory thumbnails/
Contains thumbnail images which will be included in the
typeset PDF.
cas-email.jpeg
cas-facebook.jpeg
cas-gplus.jpeg
cas-linkedin.jpeg
cas-twitter.jpeg
cas-url.jpeg
Directory figs/
Dummy figures used in the template files.
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Fig2.pdf
Fig3.pdf
grabs.pdf
pic1.pdf
The following files are files written out every time elsdoc.tex is
compiled:
elsdoc-cas.aux
elsdoc-cas.log
elsdoc-cas.out
tmp-cas.tex
Auxiliary packages needed to generate pdf output from elsdoc.tex:
rvdtx.sty
pdfwidgets.sty

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%%
%% Copyright 2019-2020 Elsevier Ltd
%%
%% This file is part of the 'CAS Bundle'.
%% --------------------------------------
%%
%% It may be distributed under the conditions of the LaTeX Project Public
%% License, either version 1.2 of this license or (at your option) any
%% later version. The latest version of this license is in
%% http://www.latex-project.org/lppl.txt
%% and version 1.2 or later is part of all distributions of LaTeX
%% version 1999/12/01 or later.
%%
%% The list of all files belonging to the 'CAS Bundle' is
%% given in the file `manifest.txt'.
%%
%% Template article for cas-sc documentclass for
%% single column output.
%\documentclass[a4paper,fleqn,longmktitle]{cas-sc}
\documentclass[a4paper,fleqn]{cas-sc}
\usepackage[numbers]{natbib}
%\usepackage[authoryear]{natbib}
%\usepackage[authoryear,longnamesfirst]{natbib}
\usepackage[normalem]{ulem}
%%%Author macros
\def\tsc#1{\csdef{#1}{\textsc{\lowercase{#1}}\xspace}}
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%%%
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\begin{document}
\let\WriteBookmarks\relax
\def\floatpagepagefraction{1}
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\shorttitle{Incompressible flows in moving domains}
\shortauthors{Ar\'ostica and Bertoglio}
%\begin{frontmatter}
\title[mode = title]{On monolithic and Chorin-Temam schemes for incompressible flows in moving domains}
\tnotetext[1]{This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 852544 - CardioZoom).}
\author[1]{Reidmen Ar\'{o}stica}
\ead{r.a.arostica.barrera@rug.nl}
%\ead[url]{https://sites.google.com/view/cvmath/people/ar\'{o}stica?authuser=0}
%\credit{Conceptualization, Analysis, Implementation}
\author[1]{Crist\'{o}bal Bertoglio}
\ead{c.a.bertoglio@rug.nl}
%\ead[url]{https://sites.google.com/view/cvmath/people/bertoglio?authuser=0}
%\credit{Conceptualization, Methodology, Guidance}
\address[1]{Bernoulli Institute, University of Groningen, Groningen, 9747AG, The Netherlands}
%\cortext[cor1]{Corresponding author}
%\cortext[cor2]{Principal corresponding author}
\begin{abstract}
%When working with numerical schemes for the incompressible Navier-Stokes equations (iNSE), several schemes can be proposed and diverse techniques be used, from monolithic approaches to Chorin-Temam methods. We study a standard but general monolithic scheme that characterizes several approaches used in literature, where unconditional energy stability will holds under constrains with the addition of consistent terms. With such finding, we derive a Chorin-Temam scheme and show the unconditional stability of it.
Several time discretization schemes for the incompressible Navier-Stokes equations (iNSE) in moving domains have been proposed. Here we introduce them in a unified fashion, allowing a common well possedness and time stability analysis. It can be therefore shown that only a particular choice of the numerical scheme ensures such properties. The analysis is performed for monolithic and Chorin-Temam schemes. Results are supported by numerical experiments.
\end{abstract}
%\begin{graphicalabstract}
%%\includegraphics{figs/grabs.pdf}
%\end{graphicalabstract}
%\begin{highlights}
%\item Research highlights item 1
%\item Research highlights item 2
%\item Research highlights item 3
%\end{highlights}
\begin{keywords}
incompressible flows \sep moving domains \sep monolithic coupling \sep Chorin-Temam \sep stability analysis %\\
\end{keywords}
% EDITING COMMANDS
\newcommand{\del}[1]{\sout{#1}}
\renewcommand{\mod}[2]{\sout{#1}$\mapsto$#2}
\newcommand{\RA}[1]{{\color{magenta}RA: #1}}
\newcommand{\CB}[1]{{\color{blue}CB: #1}}
\maketitle
\section{Introduction}
%When working with flows from a simulation point of view, several schemes are proposed depending on its applications, suitable for specific requirement e.g. high spatio-temporal resolution, fidelity, stability or fast implementation/simulation times.
%The literature is vast, but to our knowledge there is not a overview trying to summarize their approaches in a single scheme.
Several works have been reported dealing with the numerical solution of the iNSE in moving domains within an Arbitrary Lagrangian Eulerian formulation (ALE), primarily in the context of fluid-solid coupling.
In particular different choices of time discretization have been reported on \cite{Basting2017, Murea2016, Landajuela2016,Lozovskiy2018,smaldone2014,langer2014numerical,LeTallec2001,Liu2018,failer2020impact,Hessenthaler2017}.
% In \cite{Basting2017, Murea2016, Landajuela2016} the FSI within Arbitrary Lagrangian-Eulerian (ALE) formalism in a linearized approach is studied, and \cite{Lozovskiy2018,smaldone2014} also include mass conservation terms. \cite{langer2014numerical,LeTallec2001,Liu2018} propose fully non-linear FSI schemes, and extensions using Crank-Nicholson are used by \cite{failer2020impact,Hessenthaler2017}. \CB{No entiendo la clasificacion de los papers, me parece que le falta un poco de coherencia. No deberian tambien haber mas papers, o estos son todos? }
%Thougthful
To the best of the authors knowledge, only a few monolithic schemes have been thoroughly analyzed, e.g. in \cite{Lozovskiy2018, Burtschell2017, LeTallec2001, smaldone2014}, while no analysis has been reported for Chorin-Temam (CT) methods.
The goal of this work is therefore to assess well-posedness and unconditional energy balance of the iNSE-ALE for all reported monolithic and CT discretization schemes within a single formulation.
% Maybe we need to add the works of \cite{Boffi2004} for general surveys of ALE schemes.
%State-of-the art: monolithic (what is proven), CT (not proven, just "used") \\
%, which as it will be seen later on, holds under some restrictions with help of consistent stabilization terms.
%The findings for the monolithic case are then used to introduce a Chorin-Temam scheme, for which unconditional energy stability holds.
The reminder of this paper is structured as follows:
Section \ref{sec:continuous_problem} provides the continuous problem that will be studied. Section \ref{sec:monolithic_schemes} introduces a general monolithic scheme that characterizes several approaches used in literature, well-posedness and energy stability are studied and discussed. Section \ref{sec:chorin_temam_schemes} introduces the Chorin-Temam schemes where time stability is analyzed. Finally, Section \ref{sec:numerical_examples} provides numerical examples testing our results.
\section{The continuous problem}
\label{sec:continuous_problem}
%\CB{propongo sacar el $$, en el paper no tiene mucho sentido, solo tiene sentido en FSI. Ademas asi se va a achicar el texto}
In the following, let us consider a domain $\Omega_0 \subset \mathbb{R}^d$ with $d = 2,3$ and a deformation mapping $\mathcal{X}: \mathbb{R}^d\times \mathbb{R}_{+}\mapsto\mathbb{R}^d$ that defines the time evolving domain $\Omega_t := \mathcal{X}(\Omega_0, t)$. We assume $\mathcal{X}$ a continuous mapping, 1-to-1 with continuous inverse. We denote $\mathbf{X} \in \mathbb{R}^d$ the cartesian coordinate system in $\Omega_0$ and $\mathbf{x}_t := \mathcal{X}(\mathbf{X}, t)$ the one in $\Omega_t$, by $F_t := \frac{\partial \mathbf{x}_t}{\partial \mathbf{X}}$ the deformation gradient, $F^{-1}_t$ its inverse and $J_t := det(F_t)$ its jacobian.
Similarly, $Grad(\mathbf{\mathbf{g}}) := \frac{\partial \mathbf{g}}{\partial \mathbf{X}}$, $Div(\mathbf{g}) := \frac{\partial}{\partial \mathbf{X}} \cdot \mathbf{g}$ denote the gradient and divergence operators, respectively, and $\epsilon^t(\mathbf{g}) := \frac{1}{2}( Grad(\mathbf{g})F_t^{-1} + F_t^{-T} Grad(\mathbf{g})^{T})$ the symmetric gradient, for $\mathbf{g}$ a well-defined vector function. We consider the weak form of the iNSE in ALE form \cite[Ch. 5]{Richter2017}: Find $(\mathbf{u}(t), p(t)) \in \mathbf{H}^{1}_0(\Omega_0) \times L^2_0(\Omega_0)$ with $\mathbf{u}(0) = \mathbf{u}_{init}$ s.t.
\begin{equation}
\label{eq:continuous_formulation}
\begin{aligned}
% \int_{\Omega_0} \rho J^t \partial_{t} \mathbf{u} \cdot \mathbf{v} + \int_{\Omega_0} \rho J^t Grad(\mathbf{u}) F_t^{-1} (\mathbf{u} - \mathbf{w}) \cdot \mathbf{v} + \int_{\Omega_0} J^t 2\mu \, \epsilon^t(\mathbf{u}):\epsilon^t(\mathbf{v}) - \int_{\Omega_0} Div(J^t F_t^{-1} \mathbf{v}) p & \\
% \textcolor{red}{ + \alpha \int_{\Omega_0} \frac{\rho}{2} \left(\partial_{t} J^t - Div\left(J^t F_t^{-1} \mathbf{w} \right) \right) \mathbf{u} \cdot \mathbf{v} + \beta \int_{\Omega_0} \frac{\rho}{2} Div\left(J^t F_t^{-1} \mathbf{u}\right) \mathbf{u} \cdot \mathbf{v} }
% + \int_{\Omega_0} Div(J^t F_t^{-1} \mathbf{u})q
% & = 0 \\
\int_{\Omega_0} \rho J^t \partial_{t} \mathbf{u} \cdot \mathbf{v} + \rho J^t Grad(\mathbf{u}) F_t^{-1} (\mathbf{u} - \mathbf{w}) \cdot \mathbf{v} + J^t 2\mu \, \epsilon^t(\mathbf{u}):\epsilon^t(\mathbf{v}) - Div(J^t F_t^{-1} \mathbf{v}) p + Div(J^t F_t^{-1} \mathbf{u})q = 0%& \\
\end{aligned}
\end{equation}
for all $(\mathbf{v}, q) \in \mathbf{H}^1_0(\Omega_0) \times L^2_0(\Omega_0),\, t > 0$, $\mathbf{u}_{init}, \mathbf{w}(t) \in \mathbf{H}^1_0(\Omega_0)$ given initial and domain velocities, where $\mathbf{H}^1_0(\Omega_0)$ is the standard Sobolev space of vector fields $\mathbf{u}$ defined on $\Omega_0$ with values in $\mathbb{R}^d$ s.t. $\mathbf{u} = \mathbf{0}$ on $\partial \Omega_0$, $L^2_0(\Omega_0)$ the standard square integrable space of functions $p$ s.t. $p(\mathbf{0}) = 0$.
Notice that the flow at time $t$ is given by $\mathbf{u} \circ \mathcal{X}^{-1}(\cdot, t)$.
%
%\begin{remark}
%Formulation \eqref{eq:continuous_formulation} includes strongly consistent \textit{Geometric Conservation Law} (GCL) and \textit{Temam} stabilization terms multiplied by scalars $\alpha$ and $\beta$ respectively. This allows for a unified analysis for all reported methods.
%\end{remark}
\begin{remark}
Although Dirichlet boundary conditions are used throughout this work, it can be extended straightforwardly to the Neumann case by including the so called \textit{backflow stabilizations}, see e.g. \cite{bertoglio2018benchmark}. Moreover, in the discrete case the extension of well-posedness results to the case with non-zero boundary conditions follow from trace theorem.
\end{remark}
%An energy estimate can be deduced from \eqref{eq:continuous_formulation} as done in \cite[Chap. 9]{Quarteroni2009Cardiovascular}.
\begin{proposition}
\label{prop:energy_continuous} \cite[Chap. 9]{Quarteroni2009Cardiovascular}
Provided $(\mathbf{u}(t), p(t))$ a solution of the formulation \eqref{eq:continuous_formulation}, the following energy balance holds:
\begin{equation}
\label{eq:continuous_energy_estimate}
%\begin{aligned}
\partial_{t} \int_{\Omega_0} \frac{\rho}{2} J^t \vert \mathbf{u} \vert^2 = - \int_{\Omega_0} J^t 2\mu \vert \epsilon^t (\mathbf{u}) \vert^2 %- (\alpha - 1) \int_{\Omega_0} \frac{\rho}{2} \partial_{t} J^t - \int_{\Omega_0} \frac{\rho}{2} Div \left( J^t F^{-1}_t \left( (\beta - 1) \mathbf{u} - (\alpha -1) \mathbf{w} \right)\right)
%\end{aligned}
\end{equation}
\end{proposition}
\begin{remark}
Proposition \ref{prop:energy_continuous} makes use of the \textit{Geometric Conservation Law} (GCL) $\partial_t J^t = Div\left( J^t F_{t}^{-1} \mathbf{w}\right)$.
\end{remark}
\section{Monolithic schemes (first order in time)}
\label{sec:monolithic_schemes}
%Different \CB{time} discretization approaches for Problem \eqref{eq:continuous_formulation} can be taken into account, nevertheless if we fix a first order discretization for the time-derivative term, the schemes used in the literature can be characterized within a single formulation.
Most of the numerical schemes for Problem \eqref{eq:continuous_formulation} reported in the literature are first order and can be written as follows.
Given a conforming finite element space $\mathbf{V} \times Q$ of $\mathbf{H}^1_0(\Omega_0) \times L^2_0(\Omega_0)$ for velocity and pressure fields, the discrete problem of interest reads:
Find $(\mathbf{u}^{n+1}, p^{n+1}) \in \mathbf{V} \times Q$ s.t.
\begin{equation}
\label{eq:discretized_monolithic_formulation}
\mathcal{A}(\mathbf{u}^{n+1},\mathbf{v}) - \mathcal{B}(\mathbf{v},p^{n+1}) + \mathcal{B}(\mathbf{u}^{n+1},q) = F(\mathbf{v}) \quad \forall (\mathbf{v}, q) \in \mathbf{V} \times Q
\end{equation}
being
\begin{equation}
\label{eq:lhs_bilinear_form_A}
\begin{aligned}
\mathcal{A} (\mathbf{u}, \mathbf{v}) := & \int_{\Omega_0} \rho \frac{J^{\star\star}}{\tau} \mathbf{u} \cdot \mathbf{v} + \int_{\Omega_0} \rho J^{\star} Grad(\mathbf{u}) F^{-1}_{\star} (\mathbf{u}^{\ast} - \mathbf{w}^{\ast\ast}) \cdot \mathbf{v} + \int_{\Omega_0} J^{\star} 2\mu \epsilon^{\star}(\mathbf{u}):\epsilon^{\star}(\mathbf{v}) \\
& + \alpha \int_{\Omega_0} \frac{\rho}{2} \left( \frac{J^{n+1} - J^{n}}{\tau} - Div\left( J^{\star} F^{-1}_{\star} \mathbf{w}^{\ast\ast}_{h} \right) \right) \mathbf{u} \cdot \mathbf{v} + \beta \int_{\Omega_0} \frac{\rho}{2} Div\left( J^{\star} F^{-1}_{\star} \mathbf{u}^{\ast} \right) \mathbf{u} \cdot \mathbf{v}
\end{aligned}
\end{equation}
with $\alpha, \beta \in \{0, 1\}$ given parameters, and
%The matrix $B \in \mathbb{R}^{m^{Q}\times m^{\mathbf{V}}}$ and vector $F_n \in \mathbb{R}^n$ correspond also to the discretization of the bilinear and linear forms:
\begin{equation}
\label{eq:remaining_forms}
\begin{aligned}
\mathcal{B}(\mathbf{u}, q) & := \int_{\Omega_0} Div\left( J^{\star} F^{-1}_{\star} \mathbf{u} \right) q \quad \forall q \in Q, \quad \mathcal{F}(\mathbf{v}) := \int_{\Omega_0} \rho \frac{J^n}{\tau} \mathbf{u}^n \cdot \mathbf{v} \quad \forall \mathbf{v} \in \mathbf{V}
\end{aligned}
\end{equation}
\begin{remark}
The term multiplying $\alpha$ is the discrete residual of GCL, while the one multiplying $\beta$ is a strongly consistent term vanishing for incompressible velocity fields.
\end{remark}
%\begin{equation}
%\label{eq:discretized_monolithic_formulation}
%\begin{aligned}
% \int_{\Omega_0} \rho J^{\star \star} \frac{\mathbf{u}^{n+1} - \mathbf{u}^n}{\tau} \cdot \mathbf{v} + \int_{\Omega_0} \rho J^{\star} Grad(\mathbf{u}^{n+1}) F_{\star}^{-1} (\mathbf{u}^{\ast} - \mathbf{w}^{\ast \ast}) \cdot \mathbf{v} +\int_{\Omega_0} J^{\star} 2\mu \epsilon^{\star} (\mathbf{u}^{n+1}):\epsilon^{\star} (\mathbf{v}) & \\
% - \int_{\Omega_0} Div(J^{\star} F_{\star}^{-1} \mathbf{v}) p^{n+1} + \int_{\Omega_0} Div(J^{\star} F_{\star}^{-1} \mathbf{u}^{n+1}) q & \\
% \color{red}{ + \alpha \int_{\Omega_0} \frac{\rho}{2} \left( \frac{J^{n+1} - J^n}{\tau} - Div\left( J^{\star} F_{\star}^{-1}\mathbf{w}^{\ast\ast} \right) \right) \mathbf{u}^{n+1} \cdot \mathbf{v} + \beta \int_{\Omega_0} \frac{\rho}{2} Div\left( J^{\star} F_{\star}^{-1} \mathbf{u}^{\ast} \right) \mathbf{u}^{n+1} \cdot \mathbf{v} } & = 0
%\end{aligned}
%\end{equation}
%Within \CB{Formulation \eqref{eq:discretized_monolithic_formulation}}, an explicit domain treatment is done by \cite{Basting2017} whom linearize the convective term with $(\star, \star\star, \ast, \ast\ast) = (n, n, k, n)$, where $k$ the Newton's iteration index while \cite{Murea2016} takes $(\star, \star\star, \ast, \ast \ast) = (n, n, n, n)$, both using $\alpha = \beta = 0$; similarly \cite{Lozovskiy2018} use an implicit coupling with $(\star, \star\star, \ast, \ast \ast) = (n, n+1, n, n+1)$ whereas \cite{smaldone2014} uses an explicit one with $(\star, \star\star,\ast, \ast\ast) = (n, n+1, n, n)$ where $\alpha = \beta = 1$.
%If the domain treatment is implicit \cite{Langer2016} takes the approach with $(\star, \star\star, \ast, \ast\ast) = (n+1, n+1, n+1, n+1), \, \alpha = \beta = 0$ by applying a Newton solver for the fully nonlinear problem, same formulation is also taken by \cite{LeTallec2001}, \cite{Wall2009} and \cite{Liu2018} in the context of space-dependent densities with different choices of $\alpha, \beta$; if the coupling is explicit \cite{Landajuela2016} takes $(\star, \star\star, \ast, \ast\ast) = (n+1, n+1, n, n+1)$ with $\alpha = \beta = 0$ providing a linearization of the convective term, while \cite{Wang2020} utilizes an alternative weak formulation where the domain velocity $\mathbf{w}$ is obtain as solution to a specific weak form with $(\star, \star\star, \ast) = (n+1, n, n+1)$, $\alpha = \beta = 1$.
Formulation \eqref{eq:discretized_monolithic_formulation} contains a wide family of reported methods:
\begin{itemize}
\item Using $\alpha = \beta = 0$: $(\star, \star\star, \ast, \ast\ast) = (n, n, n+1, n)$ is used in \cite{Basting2017}, $(\star, \star\star, \ast, \ast \ast) = (n, n, n, n)$ in \cite{Murea2016} and $(\star, \star\star, \ast, \ast\ast) = (n+1, n+1, n+1, n+1)$ in \cite{Langer2016}, and $(\star, \star\star, \ast, \ast\ast) = (n, n+1, n, n+1)$ in \cite{Landajuela2016}.
\item Using $\alpha = \beta = 1$: $(\star, \star\star, \ast, \ast \ast) = (n, n+1, n, n+1)$ in \cite{Lozovskiy2018}, $(\star, \star\star,\ast, \ast\ast) = (n, n+1, n, n)$ in \cite{smaldone2014} and $(\star, \star\star, \ast, \ast\ast) = (n+1, n, n+1, n+1)$ in \cite{LeTallec2001, Wang2020}.
\end{itemize}
\begin{remark}
Other methods such as second order approximations can be found in \cite{Tallec2003, Liu2018} and Crank-Nicolson approaches in \cite{failer2020impact, Hessenthaler2017}.
% and more general surveys in \cite{Wall2009}.
% Notice that, whenever we take $\star \star = n+1$ the energy equality associated to the problem cannot be easily stabilized with consistent terms.
\end{remark}
%\subsection{Well posedness at each time step}
%In the following, well-posedness of Problem \eqref{eq:discretized_monolithic_formulation} is assessed using the standard techniques for saddle-point problems. As standard in literature, the problem is studied based on its matrix formulation that reads for each time step:
%\begin{equation}
%\label{eq:matrix_formulation}
%\begin{bmatrix}
%A & -B^T \\
%B & 0
%\end{bmatrix}
%\begin{bmatrix}
%U^{n+1} \\
%P^{n+1}
%\end{bmatrix}
%=\begin{bmatrix}
%F_n \\
%0
%\end{bmatrix},\quad U^{0} = U_{init}
%\end{equation}
%
%where $(U^{n+1}, P^{n+1}) \in \mathbb{R}^{m^{\mathbf{V}}} \times \mathbb{R}^{m^{Q}}$ denote the coefficients associated to the finite element basis with $(m^{\mathbf{V}}, m^{Q})$ the degrees of freedom pair for velocity/pressure elements and $U_{init}$ the coefficients associated the discretization of $\mathbf{u}_{init}$.
%\CB{The matrix $A_n \in \mathbb{R}^{m^{\mathbf{V}}\times m^{\mathbf{V}}}$ corrresponds to the discretization of the bilinear form:} % \rightarrow \mathbb{R}$ is given by
%\begin{equation}
%\label{eq:lhs_bilinear_form_A}
%\begin{aligned}
%\mathcal{A} (\mathbf{u}, \mathbf{v}) = & \int_{\Omega_0} \rho \frac{J^{\star\star}}{\tau} \mathbf{u}^{n+1} \cdot \mathbf{v} + \int_{\Omega_0} \rho J^{\star} Grad(\mathbf{u}^{n+1}) F^{-1}_{\star} (\mathbf{u}^{n} - \mathbf{w}^{\ast\ast}) \cdot \mathbf{v} + \int_{\Omega_0} J^{\star} 2\mu \epsilon^{\star}(\mathbf{u}^{n+1}):\epsilon^{\star}(\mathbf{v}) \\
%& + \alpha \int_{\Omega_0} \frac{\rho}{2} \left( \frac{J^{n+1} - J^{n}}{\tau} - Div\left( J^{\star} F^{-1}_{\star} \mathbf{w}^{\ast\ast}_{h} \right) \right) \mathbf{u}^{n+1} \cdot \mathbf{v} + \beta \int_{\Omega_0} \frac{\rho}{2} Div\left( J^{\star} F^{-1}_{\star} \mathbf{u}^{n} \right) \mathbf{u}^{n+1} \cdot \mathbf{v}
%\end{aligned}
%\end{equation}
%The matrix $B \in \mathbb{R}^{m^{Q}\times m^{\mathbf{V}}}$ and vector $F_n \in \mathbb{R}^n$ correspond also to the discretization of the bilinear and linear forms:
%\begin{equation}
%\label{eq:remaining_forms}
%\begin{aligned}
%B(\mathbf{u}, q) & := \int_{\Omega_0} Div\left( J^{\star} F^{-1}_{\star} \mathbf{u} \right) q \quad \forall q \in Q, \quad F_n(\mathbf{v}) := \int_{\Omega_0} \rho \frac{J^n}{\tau} \mathbf{u}^n \cdot \mathbf{v} \quad \forall \mathbf{v} \in \mathbf{V}
%\end{aligned}
%\end{equation}
\begin{proposition}
\label{prop:monolithic_schemes}
By assuming well-posed, orientation-preserving deformation mappings, i.e. $(J^n)_{n \in \mathbb{N}}$ bounded in $L^{\infty}(\Omega_0)$, $J^n > 0$ for each $n \geq 0$, Problem \eqref{eq:discretized_monolithic_formulation} has unique solution for inf-sup stable finite element spaces if $\left( 2J^{\star\star} + J^{n+1} - J^{n} \right) > 0$ and $\alpha = \beta = 1$.
\end{proposition}
\begin{proof}
%The matrix system \eqref{eq:matrix_formulation} is solvable whenever $A_n^{-1}$ exists and $BA^{-1}_nB^{T}$ is invertible.
Since all operators are bounded, and inf-sup stable elements are used for velocity and pressure, it is enough to ensure that the bilinear form $\mathcal{A}$ is coercive.
%Thus its enough to ensure $\text{rank}(B) = m$ and $A_n$ positive definite.
%As standard in literature, let us evaluate \eqref{eq:lhs_bilinear_form_A} in $\mathbf{v} = \mathbf{u}$ and $q = p$. Integrating by parts the convective term and joining expressions, the following equality holds:
Indeed:
\begin{equation}
\label{eq:proof_monolithic_case}
\begin{aligned}
\mathcal{A}(\mathbf{u}, \mathbf{u}) = & \int_{\Omega_0} \frac{J^{\star}}{2\tau} \left( \frac{2 J^{\star\star}}{J^{\star}} + \alpha \frac{J^{n+1} - J^{n}}{J^{\star}} \right) \vert \mathbf{u} \vert^2 + % \int_{\Omega_0}
J^{\star} 2\mu \vert \epsilon^{\star}(\mathbf{u}) \vert^2 %\\
%&
+ %\int_{\Omega_0}
\frac{\rho}{2} Div\left( J^{\star} F^{-1}_{\star} \left( (\beta -1) \mathbf{u}^{\ast} - (\alpha-1) \mathbf{w}^{\ast\ast} \right) \right) \vert \mathbf{u} \vert^2
\end{aligned}
\end{equation}
being the last quantity strictly positive under the stated assumptions.
\qed
\end{proof}
\begin{corollary}
Assuming $\alpha = \beta = 1$, Problem \eqref{eq:discretized_monolithic_formulation} is well posed when:
\begin{itemize}
\item $3J^{n+1} - J^{n} > 0$ if $\star \star = n+1$, i.e. a restriction on the time step size.
\item $J^{n+1} + J^{n} > 0$ if $\star \star = n$, i.e. no restriction on the time step size.
\end{itemize}
No restrictions apply to $\star,\ast, \ast\ast$.
\end{corollary}
%\subsection{Energy balance}
\begin{proposition}
\label{prop:energy_estimate_monolithic}
Under assumptions of Proposition \ref{prop:monolithic_schemes} and $\alpha = \beta = 1, \star\star = n$, the scheme \eqref{eq:discretized_monolithic_formulation} is unconditionally stable.
\end{proposition}
\begin{proof}
By setting $\mathbf{v} = \mathbf{u}^{n+1}$ in the bi-linear form \eqref{eq:lhs_bilinear_form_A}, $q = p^{n+1}$ in forms \eqref{eq:remaining_forms} and manipulating terms as standard in literature, the energy equality follows:
\begin{equation}
\begin{aligned}
\int_{\Omega_0} \rho \frac{J^{n+1}}{2\tau} \vert \mathbf{u}^{n+1} \vert^2 - \int_{\Omega_0} \rho \frac{J^{n}}{2\tau} \vert \mathbf{u}^n \vert^2 = & \int_{\Omega_0} \frac{\rho}{2\tau} (J^{n+1} - J^{\star\star}) \vert \mathbf{u}^{n+1} \vert^2 + \int_{\Omega_0} \frac{\rho}{2\tau} (J^{\star\star} - J^n) \vert \mathbf{u}^n \vert^2 - \int_{\Omega_0} 2\mu J^{\star} \vert \epsilon^{\star} (\mathbf{u}^{n+1}) \vert^2 \\
& - \int_{\Omega_0} \frac{\rho}{2\tau} J^{\star\star} \vert \mathbf{u}^{n+1} - \mathbf{u}^{n} \vert^2 + \int_{\Omega_0} \frac{\rho}{2} Div(J^{\star} F^{-1}_{\star} (\mathbf{u}^{\ast} - \mathbf{w}^{\ast\ast})) \vert \mathbf{u}^{n+1} \vert^2 \\
& - \int_{\Omega_0} \frac{\rho}{2} \alpha \frac{J^{n+1} - J^{n}}{\tau} \vert \mathbf{u}^{n+1} \vert^2 + \int_{\Omega_0} \frac{\rho}{2} Div\left( J^{\star}F^{-1}_{\star} (\beta \mathbf{u}^{\ast} - \alpha \mathbf{w}^{\ast\ast}) \right) \vert \mathbf{u}^{n+1} \vert^2
\end{aligned}
\end{equation}
Thus, for $\alpha=\beta=1$ and $\star\star = n$ the result follows.
\qed
\end{proof}
%\begin{remark}
%In the general case where Dirichlet and Neumann boundary conditions appear, $\Gamma_N \cup \Gamma_D = \partial \Omega_0$ with $\Gamma_N \neq \emptyset$, the same result also holds true with the addition of backflow stabilization term.
%\end{remark}
\section{Chorin-Temam schemes}
\label{sec:chorin_temam_schemes}
In the following, we describe a family of Chorin-Temam (CT) schemes for the iNSE-ALE problem, as we did for the monolithic case. %Such description keeps the freedom of choice for certain coefficients, which must be restricted to ensure unconditional stability.
Given $\mathbf{\widetilde V} $ a conforming space of $\mathbf{H}^1_0 (\Omega_0)$ and $ \widetilde Q$ a conforming space of $ L^2_0 (\Omega_0) \cap H^1(\Omega_0)$, $\tilde{\mathbf{u}}^{0} \in \mathbf{\widetilde V}$, for $n\geq0$:% the proposed two-step CT schemes reads
\begin{enumerate}[1.]
\item \textbf{Pressure-Projection Step $(\text{PPS})_{n}$}
%Seeks a pressure term allowing the projection of $\tilde{\mathbf{u}}^{n+1}$ in the space of divergence-free solutions, in the form:
Find $p^{n} \in \widetilde{Q}$ s.t.
\begin{equation}
\label{eq:chorin_temam_pps}
\int_{\Omega_0} \frac{\tau}{\rho} J^{\circ} Grad(p^{n}) F^{-1}_{\circ} : Grad(q) F^{-1}_{\circ} = - \int_{\Omega_0} Div\left( J^{\circ} F^{-1}_{\circ} \tilde{\mathbf{u}}^{n}\right) q \quad \forall q \in \widetilde{Q}
\end{equation}
\item \textbf{Fluid-Viscous Step $(\text{FVS})_{n+1}$} %Seeks a tentative velocity field $\tilde{\mathbf{u}}^{n+1}$ with pressure given explicitly:
Find $\tilde{\mathbf{u}}^{n+1} \in \mathbf{\widetilde{V}}$ s.t.
\begin{equation}
\label{eq:chorin_temam_fvs}
\begin{aligned}
\int_{\Omega_0} \rho J^{\star\star} \frac{\tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^n}{\tau} \cdot \mathbf{v} + \int_{\Omega_0} \rho J^{\star} Grad(\tilde{\mathbf{u}}^{n+1}) F_{\star}^{-1} (\tilde{\mathbf{u}}^{n} - \mathbf{w}^{\ast\ast}) \cdot \mathbf{v} +\int_{\Omega_0} J^{\star} 2 \mu \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) : \epsilon^{\star} (\mathbf{v}) & \\
-\int_{\Omega_0} Div(J^{\circ \circ} F_{\circ \circ}^{-1} \mathbf{v}) p^n
+ \int_{\Omega_0} \frac{\rho}{2} \frac{J^{n+1} - J^{n}}{\tau} \tilde{\mathbf{u}}^{n+1} \cdot \mathbf{v} + \int_{\Omega_0} \frac{\rho}{2} Div\left( J^{\star} F_{\star}^{-1}(\tilde{\mathbf{u}}^{n} - \mathbf{w}^{\ast\ast})\right) \tilde{\mathbf{u}}^{n+1} \cdot \mathbf{v} & = 0 \quad \forall \mathbf{v} \in \mathbf{\widetilde{V}}
\end{aligned}
\end{equation}
% In particular, the corrected velocity is obtained through the update:
% \begin{equation}
% \begin{aligned}
% \mathbf{u}^{n+1} & = \tilde{\mathbf{u}}^{n+1} - \frac{\tau}{\rho} Grad(p^{n+1}) F^{-1}_{\circ + 1} \text{ in } \Omega_0
% \end{aligned}
% \end{equation}
\end{enumerate}
%\begin{remark}
%The CT scheme iNSE above, uses a linearization of the convective term $\ast = n$, easily extended to the case $\ast = n+1$.. In particular, the stabilization terms are active $\alpha = \beta = 1$, with freedom in the choice of $\circ\circ$. The objective in the following will be to show which values achieve unconditional stability in our formulation.
%\end{remark}
%\subsection{Time Stability}
The following energy estimate can be obtained under suitable conditions:
\begin{proposition}
\label{prop:energy_estimate_chorin_temam}
Under assumptions $\circ = \circ\circ = \star \star = n$, the solution to scheme \eqref{eq:chorin_temam_pps}-\eqref{eq:chorin_temam_fvs} is unconditionally stable with
%is unconditionally energy stable if $\circ = \circ \circ = \star\star = n$, without condition over $\star$. Moreover, the energy estimate is given by
\begin{equation}
\begin{aligned}
\int_{\Omega_0} \rho \frac{J^{n+1}}{2\tau} \vert \tilde{\mathbf{u}}^{n+1} \vert^2 - \int_{\Omega_0} \rho \frac{J^{n}}{2\tau} \vert \tilde{\mathbf{u}}^{n} \vert^2 \leq & - \int_{\Omega_0} J^{\star} 2\mu \vert \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) \vert^2 - \int_{\Omega_0} J^{n} \frac{\tau}{2\rho} \vert Grad(p^n) F^{-1}_{n} \vert^2 .
\end{aligned}
\end{equation}
\end{proposition}
\begin{proof}
As standard in literature, let us take $\mathbf{v} = \tilde{\mathbf{u}}^{n+1}$ in $(\text{FVS})_{n+1}$, and $q = p^n$ in $(\text{PPS})_{n}$. Adding both equalities and rewriting expressions, it follows:
\begin{equation}
\begin{aligned}
\int_{\Omega_0} \rho \frac{J^{n+1}}{2\tau} \vert \tilde{\mathbf{u}}^{n+1} \vert^2 - \int_{\Omega_0} \rho \frac{J^{n}}{2\tau} \vert \tilde{\mathbf{u}}^{n} \vert^2 = & \int_{\Omega_0} \frac{\rho}{2\tau}(J^{n+1} - J^{\star\star}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2 + \int_{\Omega_0} \frac{\rho}{2\tau}(J^{\star\star} - J^{n}) \vert \tilde{\mathbf{u}}^{n} \vert^2 \\
& - \int_{\Omega_0} \frac{\rho}{2\tau} J^{\star\star} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2 - \int_{\Omega_0} J^{\star} 2\mu \vert \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) \vert^2 \\
& + \int_{\Omega_0} Div\left( J^{\circ\circ} F^{-1}_{\circ\circ} (\tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n}) \right) p^n + \int_{\Omega_0} Div\left( (J^{\circ\circ}F^{-1}_{\circ\circ} - J^{\circ}F^{-1}_{\circ}) \tilde{\mathbf{u}}^n \right) p^n \\
& - \int_{\Omega_0} \frac{\tau}{\rho} J^{\circ} \vert F^{-T}_{\circ} Grad(p^n) \vert^2 - \int_{\Omega_0} \frac{\rho}{2\tau} (J^{n+1} - J^{n}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2
\end{aligned}
\end{equation}
Bounding the first divergence term using integration by parts and Cauchy-Schwartz inequality, it follows
\begin{equation}
\begin{aligned}
\int_{\Omega_0} Div\left( J^{\circ\circ} F^{-1}_{\circ\circ} (\tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n}) \right) p^n & \leq \int_{\Omega_0} \frac{\rho}{2\tau} J^{\circ\circ} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2 + \int_{\Omega_0} \frac{\tau}{2\rho} J^{\circ\circ} \vert F^{-T}_{\circ\circ} Grad(p^n) \vert^2
\end{aligned}
\end{equation}
Thus, the following energy estimate can be obtained:
\begin{equation}
\label{eq:energy_estimate_proof}
\begin{aligned}
\int_{\Omega_0} \rho \frac{J^{n+1}}{2\tau} \vert \tilde{\mathbf{u}}^{n+1} \vert^2 - \int_{\Omega_0} \rho \frac{J^{n}}{2\tau} \vert \tilde{\mathbf{u}}^{n} \vert^2 \leq & \int_{\Omega_0} \frac{\rho}{2\tau} (J^{n+1} - J^{\star\star}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2 + \int_{\Omega_0} \frac{\rho}{2\tau} (J^{\star\star} - J^{n}) \vert \tilde{\mathbf{u}}^{n} \vert^2 \\
& - \int_{\Omega_0} \frac{\rho}{2\tau} J^{\star\star} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2 - \int_{\Omega_0} J^{\star} 2\mu \vert \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) \vert^2 \\
& + \int_{\Omega_0} \frac{\rho}{2\tau} J^{\circ\circ} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2 + \int_{\Omega_0} \frac{\tau}{2\rho} J^{\circ\circ} \vert F^{-T}_{\circ\circ} Grad(p^n) \vert^2 \\
& + \int_{\Omega_0} Div\left( (J^{\circ\circ} F^{-1}_{\circ\circ} - J^{\circ}F^{-1}_{\circ}) \tilde{\mathbf{u}}^{n}\right) p^n - \int_{\Omega_0} \frac{\tau}{\rho} J^{\circ} \vert F^{-T}_{\circ} Grad(p^n) \vert^2 \\
& - \int_{\Omega_0} \frac{\rho}{2\tau} (J^{n+1} - J^{n}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2
\end{aligned}
\end{equation}
From estimate \eqref{eq:energy_estimate_proof} it follows that whenever $\circ = \circ \circ = \star \star = n$ unconditional energy stability is attained, where $\star$ remains free of choice.
\qed
\end{proof}
%\begin{remark}
%In the general case where $\Gamma_N \cup \Gamma_D = \partial \Omega_0$ with $\Gamma_N \neq \emptyset$ a similar bound is attained with the addition of backflow stabilization term, where the same procedure as above holds for all terms except the one on $\Gamma_N$.
%\end{remark}
\section{Numerical examples}
\label{sec:numerical_examples}
We consider a rectangular domain with opposite vertices $\{ (0, -1), (6, 1) \}$ where the iNSE-ALE formulation \eqref{eq:continuous_formulation} will be simulated over the interval $[0, 2] \, [s]$ with non-zero initial condition of the form $ \mathbf{u}(0) := \gamma (1 - \mathbf{X}_1^2) \mathbf{X}_0 (6 - \mathbf{X}_0), \, \gamma = 0.001$.
The domain is deformed using $\mathcal{X}(\mathbf{X}, t) := \big( (1 + 0.9 sin(8 \pi t)) \mathbf{X}_0,\, \mathbf{X}_1 \big)$. %, i.e. an oscillation with initial expansion and frequency of $4 \pi$.
Discretization setup for Formulation \eqref{eq:discretized_monolithic_formulation} and \eqref{eq:chorin_temam_pps}-\eqref{eq:chorin_temam_fvs} is done choosing a time step $\tau = 0.01$ and space triangulation with elements diameter $h \approx 0.01 $, implemented through FEniCS \cite{FEniCS2015} using Python for interface and postprocessing.
To exemplify the theoretical results from previous sections, four schemes are taken into account. Monolitic (M) Formulation \eqref{eq:discretized_monolithic_formulation} is taken with linearized convective term and implicit treatment, i.e., $(\star, \ast, \ast\ast) = (n+1, n, n+1)$ where for $\star\star$ we consider two choices, denoted $\text{M}\, \star\star = n$ and $\text{M}\, \star\star = n+1$.
For both cases the space discretization is carried out with $ \mathbf{V}/Q = [\mathbb{P}_2]^d/\mathbb{P}_1$ Lagrange finite elements. Similarly, Chorin-Temam (CT) scheme \eqref{eq:chorin_temam_fvs}-\eqref{eq:chorin_temam_pps} is taken with linearized convective term and implicit treatment, i.e. $(\star, \ast\ast, \circ, \circ\circ) = (n+1, n+1, n, n)$ with $\star\star$ as before, denoting each scheme by $\text{CT}\, \star\star = n$ and $\text{CT}\, \star\star = n+1$ with space discretization done through $\mathbf{\widetilde{V}}/\widetilde{Q} = [\mathbb{P}_1]^d/\mathbb{P}_1$ elements. In all cases $\alpha=\beta=1$.
The results are assessed using time-dependent normalized parameters $ \hat{\delta}_{\text{M}}:= \delta_{\text{M}}/E_{st}^{\star}, \hat{\delta}_{\text{CT}}:= \delta_{\text{CT}}/E_{st}^{\star}$ defined as: %, where $E_{st}^{\star}$ denote the \CB{\mod{strain energy}{viscous dissipation}} and $\delta_{M}, \delta_{CT}$ residual errors defined as:
\begin{equation}
\label{eq:energy_error}
\begin{aligned}
\delta_{M}^{n+1} &:= D^{n+1} + E_{st}^{\star} + \int_{\Omega_0} \frac{\rho J^{\star\star}}{2\tau} \vert \mathbf{u}^{n+1} - \mathbf{u}^{n} \vert^2, \quad \delta_{CT}^{n+1} := D^{n+1} + E^{\star}_{st} + \int_{\Omega_0} \frac{\tau J^{\circ}}{2 \rho} \vert F^{-T}_{\circ} Grad(p^n) \vert^2 \\
D^{n+1} &:= \int_{\Omega_0} \frac{\rho}{2\tau} \left( J^{n+1} \vert \mathbf{u}^{n+1} \vert^2 - J^n \vert \mathbf{u}^n \vert^2 \right), \quad E^{\star}_{st} = \int_{\Omega_0} 2 \mu J^{\star} \vert \epsilon^{\star} (\mathbf{u}^{n+1}) \vert^2
\end{aligned}
\end{equation}
Figure \ref{fig:delta_hat_oscs} shows $\hat{\delta}_{\text{M}}, \hat{\delta}_{\text{CT}}$ values for each tested scheme. Propositions \ref{prop:energy_estimate_monolithic} and \ref{prop:energy_estimate_chorin_temam} are confirmed since $\hat{\delta}_{\text{M}}=0$ and $\hat{\delta}_{\text{CT}} \leq 0$ if $\star\star = n$. For $\star \star = n+1$, peaks appearing throughout the simulation are defined by the sign change of domain velocity, i.e. in the change from expansion to contraction.
Importantly, the spurious numerical energy rate related to discretization of the GCL condition appear to be positive in expansion, therefore being a potential source of numerical instabilities.
%Moreover the expected energy decay from Propositions \ref{prop:energy_estimate_monolithic}, \ref{prop:energy_estimate_chorin_temam} for the choice $\star \star = n$ in $\text{M, CT}$ schemes is obtained.
%\CB{Recall that $\hat{\delta}_{\text{M}}, \hat{\delta}_{\text{CT}}$ in \eqref{eq:energy_error} are taken in such form since on scheme $\text{M}\, \star\star = n$ it follows $\hat{\delta}_{MT} = 0$ and similarly on $\text{CT}\, \star\star = n$ it follows $\hat{\delta}_{\text{CT}} \sim 0$, defining the base case.}
\begin{figure}[!hbtp]
\centering
\includegraphics[width=\textwidth]{figs/Comparison_Delta_Hat_Value_GCL_True_solver_LU.png}
\caption{Summary of the numerical experiment in terms of energy balance. Left: Monolithic residual error values $\hat{\delta}_{\text{M}}$;
Right: Chorin-Temam residual error values $\hat{\delta}_{\text{CT}}$. %Both cases are simulated with deformation mapping $\mathcal{A}$ defined previously and shown on the interval $[0, 1] \, [s]$.}
}
\label{fig:delta_hat_oscs}
\end{figure}
\section{Conclusion}
Reported first order time discretization schemes for the iNSE-ALE have been reviewed, theoretically analyzed and numerically assessed. For the monolithic case, two schemes lead to well-posed energy-stable problems whenever $\alpha=\beta=1$ with $\star \star = n$ as studied by \cite{LeTallec2001, Lozovskiy2018, smaldone2014, Wang2020}.
To the best of the authors knowledge, the unconditionally stable Chorin-Temam scheme derived in this work has not been reported yet. %, and moreover the numerical experiments studied here validate the energy stable propositions.
%\appendix
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Latest modifications done by: R.A. Arostica
We are using Elsevier's journal article template.
The main tex file containing the article under writing in paper.tex
I've chosen the single column, but we can easily change it for a double column if necessary.

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%%
%% Copyright 2019-2020 Elsevier Ltd
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%% Template article for cas-dc documentclass for
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%\documentclass[a4paper,fleqn,longmktitle]{cas-dc}
\documentclass[a4paper,fleqn]{cas-dc}
%\usepackage[authoryear,longnamesfirst]{natbib}
%\usepackage[authoryear]{natbib}
\usepackage[numbers]{natbib}
%%%Author definitions
\def\tsc#1{\csdef{#1}{\textsc{\lowercase{#1}}\xspace}}
\tsc{WGM}
\tsc{QE}
\tsc{EP}
\tsc{PMS}
\tsc{BEC}
\tsc{DE}
%%%
\begin{document}
\let\WriteBookmarks\relax
\def\floatpagepagefraction{1}
\def\textpagefraction{.001}
\shorttitle{Leveraging social media news}
\shortauthors{CV Radhakrishnan et~al.}
\title [mode = title]{This is a specimen $a_b$ title}
\tnotemark[1,2]
\tnotetext[1]{This document is the results of the research
project funded by the National Science Foundation.}
\tnotetext[2]{The second title footnote which is a longer text matter
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another line in the footnotes area of the first page.}
\author[1,3]{CV Radhakrishnan}[type=editor,
auid=000,bioid=1,
prefix=Sir,
role=Researcher,
orcid=0000-0001-7511-2910]
\cormark[1]
\fnmark[1]
\ead{cvr_1@tug.org.in}
\ead[url]{www.cvr.cc, cvr@sayahna.org}
\credit{Conceptualization of this study, Methodology, Software}
\address[1]{Elsevier B.V., Radarweg 29, 1043 NX Amsterdam, The Netherlands}
\author[2,4]{Han Theh Thanh}[style=chinese]
\author[2,3]{CV Rajagopal}[%
role=Co-ordinator,
suffix=Jr,
]
\fnmark[2]
\ead{cvr3@sayahna.org}
\ead[URL]{www.sayahna.org}
\credit{Data curation, Writing - Original draft preparation}
\address[2]{Sayahna Foundation, Jagathy, Trivandrum 695014, India}
\author%
[1,3]
{Rishi T.}
\cormark[2]
\fnmark[1,3]
\ead{rishi@stmdocs.in}
\ead[URL]{www.stmdocs.in}
\address[3]{STM Document Engineering Pvt Ltd., Mepukada,
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\fntext[fn1]{This is the first author footnote. but is common to third
author as well.}
\fntext[fn2]{Another author footnote, this is a very long footnote and
it should be a really long footnote. But this footnote is not yet
sufficiently long enough to make two lines of footnote text.}
\nonumnote{This note has no numbers. In this work we demonstrate $a_b$
the formation Y\_1 of a new type of polariton on the interface
between a cuprous oxide slab and a polystyrene micro-sphere placed
on the slab.
}
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details of various options of the {natbib} package, please take a
look at the {natbib} documentation, which is part of any standard
\LaTeX{} installation.
\appendix
\section{My Appendix}
Appendix sections are coded under \verb+\appendix+.
\verb+\printcredits+ command is used after appendix sections to list
author credit taxonomy contribution roles tagged using \verb+\credit+
in frontmatter.
\printcredits
%% Loading bibliography style file
%\bibliographystyle{model1-num-names}
\bibliographystyle{cas-model2-names}
% Loading bibliography database
\bibliography{cas-refs}
%\vskip3pt
\bio{}
Author biography without author photo.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
\endbio
\bio{figs/pic1}
Author biography with author photo.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
\endbio
\bio{figs/pic1}
Author biography with author photo.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
Author biography. Author biography. Author biography.
\endbio
\end{document}

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%%
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%%
%% This file is part of the 'CAS Bundle'.
%% ......................................
%%
%% It may be distributed under the conditions of the LaTeX Project Public
%% License, either version 1.2 of this license or (at your option) any
%% later version. The latest version of this license is in
%% http://www.latex-project.org/lppl.txt
%% and version 1.2 or later is part of all distributions of LaTeX
%% version 1999/12/01 or later.
%%
%% The list of all files belonging to the 'CAS Bundle' is
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%%
%% $Id: cas-dc.cls 49 2020-03-14 09:05:10Z rishi $
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\let\comma\@empty
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@article{Formaggia2004,
doi = {10.1016/j.cma.2003.09.028},
url = {https://doi.org/10.1016/j.cma.2003.09.028},
year = {2004},
month = oct,
publisher = {Elsevier {BV}},
volume = {193},
number = {39-41},
pages = {4097--4116},
author = {Luca Formaggia and Fabio Nobile},
title = {Stability analysis of second-order time accurate schemes for {ALE}{\textendash}{FEM}},
journal = {Computer Methods in Applied Mechanics and Engineering}
}
@book{Ern2004,
doi = {10.1007/978-1-4757-4355-5},
url = {https://doi.org/10.1007/978-1-4757-4355-5},
year = {2004},
publisher = {Springer New York},
author = {Alexandre Ern and Jean-Luc Guermond},
title = {Theory and Practice of Finite Elements}
}
@article{FEniCS2015,
author = {Aln\ae{}s, Martin and Blechta, Jan and Hake, Johan and Johansson, August and Kehlet, Benjamin and Logg, Anders and Richardson, Chris and Ring, Johannes and Rognes, Marie E and Wells, Garth N},
language = {eng},
title = {The FEniCS Project Version 1.5},
journal = {Archive of Numerical Software},
volume = {Vol 3},
pages = {Starting Point and Frequency. Year: 2013},
publisher = {University Library Heidelberg},
year = {2015},
copyright = {Authors who publish with this journal agree to the following terms: Authors retain copyright and grant the journal right of first publication with the descriptive part of the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal. The code part of the work is licensed under a suitable OSI approved Open Source license. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.}
}
@book{Quarteroni2009Cardiovascular,
year = {2009},
publisher = {Springer Milan},
editor = {Luca Formaggia and Alfio Quarteroni and Alessandro Veneziani},
title = {Cardiovascular Mathematics}
}
@book{Richter2017,
year = {2017},
publisher = {Springer International Publishing},
author = {Thomas Richter},
title = {Fluid-structure Interactions}
}
@article{bertoglio2018benchmark,
title={Benchmark problems for numerical treatment of backflow at open boundaries},
author={Bertoglio, Crist{\'o}bal and Caiazzo, Alfonso and Bazilevs, Yuri and Braack, Malte and Esmaily, Mahdi and Gravemeier, Volker and L. Marsden, Alison and Pironneau, Olivier and E. Vignon-Clementel, Irene and A. Wall, Wolfgang},
journal={International journal for numerical methods in biomedical engineering},
volume={34},
number={2},
pages={e2918},
year={2018},
publisher={Wiley Online Library}
}
@article{Bert2018,
author = {Bertoglio, C. and Nu{\~n}ez, R. and Galarce, F. and Nordsletten, D. and Osses, A.},
title = {Relative pressure estimation from velocity measurements in blood flows: State-of-the-art and new approaches},
journal = {International Journal for Numerical Methods in Biomedical Engineering},
volume = {2018;34:e2925. https://doi.org/10.1002/cnm.2925},
year = {2018}
}
@article{Lozovskiy2018,
year = {2018},
month = may,
publisher = {Elsevier {BV}},
volume = {333},
pages = {55--73},
author = {Alexander Lozovskiy and Maxim A. Olshanskii and Yuri V. Vassilevski},
title = {A quasi-Lagrangian finite element method for the Navier{\textendash}Stokes equations in a time-dependent domain},
journal = {Computer Methods in Applied Mechanics and Engineering}
}
@article{Liu2018,
year = {2018},
month = aug,
publisher = {Elsevier {BV}},
volume = {337},
pages = {549--597},
author = {Ju Liu and Alison L. Marsden},
title = {A unified continuum and variational multiscale formulation for fluids, solids, and fluid{\textendash}structure interaction},
journal = {Computer Methods in Applied Mechanics and Engineering}
}
@article{Nordsletten2008,
year = {2008},
publisher = {Wiley},
volume = {56},
number = {8},
pages = {1457--1463},
author = {D. A. Nordsletten and P. J. Hunter and N. P. Smith},
title = {Conservative and non-conservative arbitrary Lagrangian{\textendash}Eulerian forms for ventricular flows},
journal = {International Journal for Numerical Methods in Fluids}
}
@article{Landajuela2016,
year = {2016},
month = jul,
publisher = {Wiley},
volume = {33},
number = {4},
pages = {e2813},
author = {Mikel Landajuela and Marina Vidrascu and Dominique Chapelle and Miguel A. Fern{\'{a}}ndez},
title = {Coupling schemes for the {FSI} forward prediction challenge: Comparative study and validation},
journal = {International Journal for Numerical Methods in Biomedical Engineering}
}
@article{Burtschell2017,
year = {2017},
month = apr,
publisher = {Elsevier {BV}},
volume = {182},
pages = {313--324},
author = {Bruno Burtschell and Dominique Chapelle and Philippe Moireau},
title = {Effective and energy-preserving time discretization for a general nonlinear poromechanical formulation},
journal = {Computers {\&} Structures}
}
@article{Basting2017,
year = {2017},
month = feb,
publisher = {Elsevier {BV}},
volume = {331},
pages = {312--336},
author = {Steffen Basting and Annalisa Quaini and Sun{\v{c}}ica {\v{C}}ani{\'{c}} and Roland Glowinski},
title = {Extended {ALE} Method for fluid{\textendash}structure interaction problems with large structural displacements},
journal = {Journal of Computational Physics}
}
@article{Deparis2016,
year = {2016},
month = dec,
publisher = {Elsevier {BV}},
volume = {327},
pages = {700--718},
author = {Simone Deparis and Davide Forti and Gwenol Grandperrin and Alfio Quarteroni},
title = {{FaCSI}: A block parallel preconditioner for fluid{\textendash}structure interaction in hemodynamics},
journal = {Journal of Computational Physics}
}
@incollection{Colciago2017,
year = {2017},
month = nov,
publisher = {De Gruyter},
author = {Claudia M. Colciago and Simone Deparis and Davide Forti},
editor = {Stefan Frei and B\"{a}rbel Holm and Thomas Richter and Thomas Wick and Huidong Yang},
title = {7. Fluid-structure interaction for vascular flows: From supercomputers to laptops},
booktitle = {Fluid-Structure Interaction}
}
@article{LeTallec2001,
year = {2001},
month = mar,
publisher = {Elsevier {BV}},
volume = {190},
number = {24-25},
pages = {3039--3067},
author = {P. Le Tallec and J. Mouro},
title = {Fluid structure interaction with large structural displacements},
journal = {Computer Methods in Applied Mechanics and Engineering}
}
@phdthesis{smaldone2014,
TITLE = {{Numerical analysis and simulations of coupled problems for the cariovascular system}},
AUTHOR = {Smaldone, Saverio},
SCHOOL = {{L'UNIVERSIT{\'E} PIERRE ET MARIE CURIE - Paris VI }},
YEAR = {2014},
MONTH = Oct,
TYPE = {Theses},
HAL_ID = {tel-01287506},
HAL_VERSION = {v1},
}
@article{Balzani2015,
year = {2015},
month = dec,
publisher = {Wiley},
volume = {32},
number = {10},
pages = {e02756},
author = {Daniel Balzani and Simone Deparis and Simon Fausten and Davide Forti and Alexander Heinlein and Axel Klawonn and Alfio Quarteroni and Oliver Rheinbach and Joerg Schr\"{o}der},
title = {Numerical modeling of fluid-structure interaction in arteries with anisotropic polyconvex hyperelastic and anisotropic viscoelastic material models at finite strains},
journal = {International Journal for Numerical Methods in Biomedical Engineering}
}
@misc{langer2014numerical,
title={Numerical Simulation of Fluid-Structure Interaction Problems with Hyperelastic Models: A Monolithic Approach},
author={Ulrich Langer and Huidong Yang},
year={2014},
eprint={1408.3737},
archivePrefix={arXiv},
primaryClass={math.NA}
}
@article{Langer2018,
doi = {10.1016/j.matcom.2016.07.008},
url = {https://doi.org/10.1016/j.matcom.2016.07.008},
year = {2018},
month = mar,
publisher = {Elsevier {BV}},
volume = {145},
pages = {186--208},
author = {Ulrich Langer and Huidong Yang},
title = {Numerical simulation of fluid{\textendash}structure interaction problems with hyperelastic models: A monolithic approach},
journal = {Mathematics and Computers in Simulation}
}
@misc{failer2020impact,
title={On the Impact of Fluid Structure Interaction in Blood Flow Simulations: Stenotic Coronary Artery Benchmark},
author={Lukas Failer and Piotr Minakowski and Thomas Richter},
year={2020},
eprint={2003.05214},
archivePrefix={arXiv},
primaryClass={physics.comp-ph}
}
@article{Langer2016,
year = {2016},
month = feb,
publisher = {Wiley},
volume = {108},
number = {4},
pages = {303--325},
author = {Ulrich Langer and Huidong Yang},
title = {Robust and efficient monolithic fluid-structure-interaction solvers},
journal = {International Journal for Numerical Methods in Engineering}
}
@article{Boffi2004,
year = {2004},
month = oct,
publisher = {Elsevier {BV}},
volume = {193},
number = {42-44},
pages = {4717--4739},
author = {Daniele Boffi and Lucia Gastaldi},
title = {Stability and geometric conservation laws for {ALE} formulations},
journal = {Computer Methods in Applied Mechanics and Engineering}
}
@article{Murea2016,
year = {2016},
month = jun,
publisher = {Wiley},
volume = {109},
number = {8},
pages = {1067--1084},
author = {Cornel Marius Murea and Soyibou Sy},
title = {Updated Lagrangian/Arbitrary Lagrangian-Eulerian framework for interaction between a compressible neo-Hookean structure and an incompressible fluid},
journal = {International Journal for Numerical Methods in Engineering}
}
@article{Hessenthaler2017,
year = {2017},
month = feb,
publisher = {Wiley},
volume = {33},
number = {8},
pages = {e2845},
author = {Andreas Hessenthaler and Oliver R\"{o}hrle and David Nordsletten},
title = {Validation of a non-conforming monolithic fluid-structure interaction method using phase-contrast {MRI}},
journal = {International Journal for Numerical Methods in Biomedical Engineering}
}
@Inbook{Wall2009,
author="Wall, W. A.
and Gerstenberger, A.
and Mayer, U. M.",
editor="Eberhardsteiner, Josef
and Hellmich, Christian
and Mang, Herbert A.
and P{\'e}riaux, Jacques",
title="Advances in Fixed-Grid Fluid Structure Interaction",
bookTitle="ECCOMAS Multidisciplinary Jubilee Symposium: New Computational Challenges in Materials, Structures, and Fluids",
year="2009",
publisher="Springer Netherlands",
address="Dordrecht",
pages="235--249",
isbn="978-1-4020-9231-2",
doi="10.1007/978-1-4020-9231-2_16",
url="https://doi.org/10.1007/978-1-4020-9231-2_16"
}
@inproceedings{Tallec2003,
author = {Tallec, Le and Hauret, Patrice},
year = {2003},
month = {01},
pages = {},
title = {Energy conservation in fluid-structure interactions}
}
@misc{Wang2020,
title={An energy stable one-field monolithic arbitrary Lagrangian-Eulerian formulation for fluid-structure interaction},
author={Yongxing Wang and Peter K. Jimack and Mark A. Walkley and Olivier Pironneau},
year={2020},
eprint={2003.03819},
archivePrefix={arXiv},
primaryClass={cs.CE}
}

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%% ......................................
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%% later version. The latest version of this license is in
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\begin{document}
\def\testa{This is a specimen document. }
\def\testc{\testa\testa\testa\testa}
\def\testb{\testc\testc\testc\testc\testc}
\long\def\test{\testb\par\testb\par\testb\par}
\pinclude{\copy\contbox\printSq{\LastPage}}
\title{Documentation for Elsevier's Complex Article Service (CAS)
\LaTeX\ template}
\author{Elsevier Ltd}
\contact{elsarticle@stmdocs.in}
\version{1.0}
\date{\today}
\maketitle
\section{Introduction}
Two classfiles namely \file{cas-sc.cls} and \file{cas-dc.cls} were
written for typesetting articles submitted in journals of Elsevier's
Complex Article Service (CAS) workflow.
\subsection{Usage}
\begin{enumerate}
\item \file{cas-sc.cls} for single column journals.
\begin{vquote}
\documentclass[<options>]{cas-sc}
\end{vquote}
\item \file{cas-dc.cls} for single column journals.
\begin{vquote}
\documentclass[<options>]{cas-dc}
\end{vquote}
\end{enumerate}
and have an option longmktitle to handle long front matter.
\section{Front matter}
\begin{vquote}
\title [mode = title]{This is a specimen $a_b$ title}
\tnotemark[1,2]
\tnotetext[1]{This document is the results of the research
project funded by the National Science Foundation.}
\tnotetext[2]{The second title footnote which is a longer text
matter to fill through the whole text width and overflow into
another line in the footnotes area of the first page.}
\author[1,3]{CV Radhakrishnan}[type=editor,
auid=000,bioid=1,
prefix=Sir,
role=Researcher,
orcid=0000-0001-7511-2910]
\cormark[1]
\fnmark[1]
\ead{cvr_1@tug.org.in}
\ead[url]{www.cvr.cc, cvr@sayahna.org}
\end{vquote}
\begin{vquote}
\credit{Conceptualization of this study, Methodology,
Software}
\address[1]{Elsevier B.V., Radarweg 29, 1043 NX Amsterdam,
The Netherlands}
\author[2,4]{Han Theh Thanh}[style=chinese]
\author[2,3]{CV Rajagopal}[%
role=Co-ordinator,
suffix=Jr,
]
\fnmark[2]
\ead{cvr3@sayahna.org}
\ead[URL]{www.sayahna.org}
\credit{Data curation, Writing - Original draft preparation}
\address[2]{Sayahna Foundation, Jagathy, Trivandrum 695014,
India}
\author[1,3]{Rishi T.}
\cormark[2]
\fnmark[1,3]
\ead{rishi@stmdocs.in}
\ead[URL]{www.stmdocs.in}
\address[3]{STM Document Engineering Pvt Ltd., Mepukada,
Malayinkil, Trivandrum 695571, India}
\cortext[cor1]{Corresponding author}
\cortext[cor2]{Principal corresponding author}
\fntext[fn1]{This is the first author footnote. but is common
to third author as well.}
\fntext[fn2]{Another author footnote, this is a very long
footnote and it should be a really long footnote. But this
footnote is not yet sufficiently long enough to make two lines
of footnote text.}
\end{vquote}
\begin{vquote}
\nonumnote{This note has no numbers. In this work we
demonstrate $a_b$ the formation Y\_1 of a new type of
polariton on the interface between a cuprous oxide slab
and a polystyrene micro-sphere placed on the slab.
}
\begin{abstract}[S U M M A R Y]
This template helps you to create a properly formatted
\LaTeX\ manuscript.
\noindent\texttt{\textbackslash begin{abstract}} \dots
\texttt{\textbackslash end{abstract}} and
\verb+\begin{keyword}+ \verb+...+ \verb+\end{keyword}+
which contain the abstract and keywords respectively.
Each keyword shall be separated by a \verb+\sep+ command.
\end{abstract}
\begin{keywords}
quadrupole exciton \sep polariton \sep \WGM \sep \BEC
\end{keywords}
\maketitle
\end{vquote}
\begin{figure}
\includegraphics[width=\textwidth]{sc-sample.pdf}
\caption{Single column output (classfile: cas-sc.cls).}
\end{figure}
\begin{figure}
\includegraphics[width=\textwidth]{dc-sample.pdf}
\caption{Double column output (classfile: cas-dc.cls).}
\end{figure}
\subsection{Title}
\verb+\title+ command have the below options:
\begin{enumerate}
\item \verb+title:+ Document title
\item \verb+alt:+ Alternate title
\item \verb+sub:+ Sub title
\item \verb+trans:+ Translated title
\item \verb+transsub:+ Translated sub title
\end{enumerate}
\begin{vquote}
\title[mode=title]{This is a title}
\title[mode=alt]{This is a alternate title}
\title[mode=sub]{This is a sub title}
\title[mode=trans]{This is a translated title}
\title[mode=transsub]{This is a translated sub title}
\end{vquote}
\subsection{Author}
\verb+\author+ command have the below options:
\begin{enumerate}
\item \verb+auid:+ Author id
\item \verb+bioid:+ Biography id
\item \verb+alt:+ Alternate author
\item \verb+style:+ Style of author name chinese
\item \verb+prefix:+ Prefix Sir
\item \verb+suffix:+ Suffix
\item \verb+degree:+ Degree
\item \verb+role:+ Role
\item \verb+orcid:+ ORCID
\item \verb+collab:+ Collaboration
\item \verb+anon:+ Anonymous author
\item \verb+deceased:+ Deceased author
\item \verb+twitter:+ Twitter account
\item \verb+facebook:+ Facebook account
\item \verb+linkedin:+ LinkedIn account
\item \verb+plus:+ Google plus account
\item \verb+gplus:+ Google plus account
\end{enumerate}
\begin{vquote}
\author[1,3]{Author Name}[type=editor,
auid=000,bioid=1,
prefix=Sir,
role=Researcher,
orcid=0000-0001-7511-2910,
facebook=<facebook id>,
twitter=<twitter id>,
linkedin=<linkedin id>,
gplus=<gplus id>]
\end{vquote}
\subsection{Various Marks in the Front Matter}
The front matter becomes complicated due to various kinds
of notes and marks to the title and author names. Marks in
the title will be denoted by a star ($\star$) mark;
footnotes are denoted by super scripted Arabic numerals,
corresponding author by of an Conformal asterisk (*) mark.
\subsubsection{Title marks}
Title mark can be entered by the command, \verb+\tnotemark[<num>]+
and the corresponding text can be entered with the command
\verb+\tnotetext[<num>]+ \verb+{<text>}+. An example will be:
\begin{vquote}
\title[mode=title]{Leveraging social media news to predict
stock index movement using RNN-boost}
\tnotemark[1,2]
\tnotetext[1]{This document is the results of the research
project funded by the National Science Foundation.}
\tnotetext[2]{The second title footnote which is a longer
text matter to fill through the whole text width and
overflow into another line in the footnotes area of
the first page.}
\end{vquote}
\verb+\tnotetext+ and \verb+\tnotemark+ can be anywhere in
the front matter, but shall be before \verb+\maketitle+ command.
\subsubsection{Author marks}
Author names can have many kinds of marks and notes:
\begin{vquote}
footnote mark : \fnmark[<num>]
footnote text : \fntext[<num>]{<text>}
affiliation mark : \author[<num>]
email : \ead{<emailid>}
url : \ead[url]{<url>}
corresponding author mark : \cormark[<num>]
corresponding author text : \cortext[<num>]{<text>}
\end{vquote}
\subsubsection{Other marks}
At times, authors want footnotes which leave no marks in
the author names. The note text shall be listed as part of
the front matter notes. Class files provides
\verb+\nonumnote+ for this purpose. The usage
\begin{vquote}
\nonumnote{<text>}
\end{vquote}
\noindent and should be entered anywhere before the \verb+\maketitle+
command for this to take effect.
\subsection{Abstract and Keywords}
Abstract shall be entered in an environment that starts
with \verb+\begin{abstract}+ and ends with
\verb+\end{abstract}+. Longer abstracts spanning more than
one page is also possible in Class file even in double
column mode. We need to invoke longmktitle option in the
class loading line for this to happen smoothly.
The key words are enclosed in a \verb+{keyword}+
environment.
\begin{vquote}
\begin{abstract}
This is a abstract. \lipsum[3]
\end{abstract}
\begin{keywords}
First keyword \sep Second keyword \sep Third
keyword \sep Fourth keyword
\end{keywords}
\end{vquote}
\section{Main Matter}
\subsection{Tables}
\subsubsection{Normal tables}
\begin{vquote}
\begin{table}
\caption{This is a test caption.}
\begin{tabular*}{\tblwidth}{@{} LLLL@{} }
\toprule
Col 1 & Col 2\\
\midrule
12345 & 12345\\
12345 & 12345\\
12345 & 12345\\
\bottomrule
\end{tabular*}
\end{table}
\end{vquote}
\subsubsection{Span tables}
\begin{vquote}
\begin{table*}[width=.9\textwidth,cols=4,pos=h]
\caption{This is a test caption.}
\begin{tabular*}{\tblwidth}{@{} LLLLLL@{} }
\toprule
Col 1 & Col 2 & Col 3 & Col4 & Col5 & Col6 & Col7\\
\midrule
12345 & 12345 & 123 & 12345 & 123 & 12345 & 123 \\
12345 & 12345 & 123 & 12345 & 123 & 12345 & 123 \\
12345 & 12345 & 123 & 12345 & 123 & 12345 & 123 \\
\bottomrule
\end{tabular*}
\end{table*}
\end{vquote}
\subsection{Figures}
\subsubsection{Normal figures}
\begin{vquote}
\begin{figure}
\centering
\includegraphics[scale=.75]{Fig1.pdf}
\caption{The evanescent light - $1S$ quadrupole coupling
($g_{1,l}$) scaled to the bulk exciton-photon coupling
($g_{1,2}$). The size parameter $kr_{0}$ is denoted as $x$ and
the \PMS is placed directly on the cuprous oxide sample ($\delta
r=0$, See also Fig. \protect\ref{FIG:2}).}
\label{FIG:1}
\end{figure}
\end{vquote}
\subsubsection{Span figures}
\begin{vquote}
\begin{figure*}
\centering
\includegraphics[width=\textwidth,height=2in]{Fig2.pdf}
\caption{Schematic of formation of the evanescent polariton on
linear chain of \PMS. The actual dispersion is determined by
the ratio of two coupling parameters such as exciton-\WGM
coupling and \WGM-\WGM coupling between the microspheres.}
\label{FIG:2}
\end{figure*}\end{vquote}
\subsection{Theorem and theorem like environments}
CAS class file provides a few hooks to format theorems and
theorem like environments with ease. All commands the
options that are used with \verb+\newtheorem+ command will work
exactly in the same manner. Class file provides three
commands to format theorem or theorem like environments:
\begin{enumerate}
\item \verb+\newtheorem+ command formats a theorem in
\LaTeX's default style with italicized font for theorem
statement, bold weight for theorem heading and theorem
number typeset at the right of theorem heading. It also
optionally accepts an argument which will be printed as an
extra heading in parentheses. Here is an example coding and
output:
\begin{vquote}
\newtheorem{theorem}{Theorem}
\begin{theorem}\label{thm}
The \WGM evanescent field penetration depth into the
cuprous oxide adjacent crystal is much larger than the
\QE radius:
\begin{equation*}
\lambda_{1S}/2 \pi \left({\epsilon_{Cu2O}-1}
\right)^{1/2} = 414 \mbox{ \AA} \gg a_B = 4.6
\mbox{ \AA}
\end{equation*}
\end{theorem}
\end{vquote}
\item \verb+\newdefinition+ command does exactly the same
thing as with except that the body font is up-shape instead
of italic. See the example below:
\begin{vquote}
\newdefinition{definition}{Definition}
\begin{definition}
The bulk and evanescent polaritons in cuprous oxide
are formed through the quadrupole part of the light-matter
interaction:
\begin{equation*}
H_{int} = \frac{i e }{m \omega_{1S}} {\bf E}_{i,s}
\cdot {\bf p}
\end{equation*}
\end{definition}
\end{vquote}
\item \verb+\newproof+ command helps to define proof and
custom proof environments without counters as provided in
the example code. Given below is an example of proof of
theorem kind.
\begin{vquote}
\newproof{pot}{Proof of Theorem \ref{thm}}
\begin{pot}
The photon part of the polariton trapped inside the \PMS
moves as it would move in a micro-cavity of the effective
modal volume $V \ll 4 \pi r_{0}^{3} /3$. Consequently, it
can escape through the evanescent field. This evanescent
field essentially has a quantum origin and is due to
tunneling through the potential caused by dielectric
mismatch on the \PMS surface. Therefore, we define the
\emph{evanescent} polariton (\EP) as an evanescent light -
\QE coherent superposition.
\end{pot}
\end{vquote}
\end{enumerate}
\subsection{Enumerated and Itemized Lists}
CAS class files provides an extended list processing macros
which makes the usage a bit more user friendly than the
default LaTeX list macros. With an optional argument to the
\verb+\begin{enumerate}+ command, you can change the list
counter type and its attributes. You can see the coding and
typeset copy.
\begin{vquote}
\begin{enumerate}[1.]
\item The enumerate environment starts with an optional
argument `1.' so that the item counter will be suffixed
by a period as in the optional argument.
\item If you provide a closing parenthesis to the number in the
optional argument, the output will have closing
parenthesis for all the item counters.
\item You can use `(a)' for alphabetical counter and `(i)' for
roman counter.
\begin{enumerate}[a)]
\item Another level of list with alphabetical counter.
\item One more item before we start another.
\begin{enumerate}[(i)]
\item This item has roman numeral counter.
\end{vquote}
\begin{vquote}
\item Another one before we close the third level.
\end{enumerate}
\item Third item in second level.
\end{enumerate}
\item All list items conclude with this step.
\end{enumerate}
\section{Biography}
\verb+\bio+ command have the below options:
\begin{enumerate}
\item \verb+width:+ Width of the author photo (default is 1in).
\item \verb+pos:+ Position of author photo.
\end{enumerate}
\begin{vquote}
\bio[width=10mm,pos=l]{tuglogo.jpg}
\textbf{Another Biography:}
Recent experimental \cite{HARA:2005} and theoretical
\cite{DEYCH:2006} studies have shown that the \WGM can travel
along the chain as "heavy photons". Therefore the \WGM
acquires the spatial dispersion, and the evanescent
quadrupole polariton has the form (See Fig.\ref{FIG:3}):
\endbio
\end{vquote}
\section[CRediT...]{CRediT authorship contribution statement}
Give the authorship contribution after each author as
\begin{vquote}
\credit{Conceptualization of this study, Methodology,
Software}
\end{vquote}
To print the details use \verb+\printcredits+
\begin{vquote}
\author[1,3]{V. {{\=A}}nand Rawat}[auid=000,
bioid=1,
prefix=Sir,
role=Researcher,
orcid=0000-0001-7511-2910]
\end{vquote}
\begin{vquote}
\cormark[1]
\fnmark[1]
\ead{cvr_1@tug.org.in}
\ead[url]{www.cvr.cc, www.tug.org.in}
\credit{Conceptualization of this study, Methodology,
Software}
\address[1]{Indian \TeX{} Users Group, Trivandrum 695014,
India}
\author[2,4]{Han Theh Thanh}[style=chinese]
\author[2,3]{T. Rishi Nair}[role=Co-ordinator,
suffix=Jr]
\fnmark[2]
\ead{rishi@sayahna.org}
\ead[URL]{www.sayahna.org}
\credit{Data curation, Writing - Original draft preparation}
. . .
. . .
. . .
\printcredits
\end{vquote}
\section{Bibliography}
For CAS categories, two reference models are recommended.
They are \file{model1-num-names.bst} and \file{model2-names.bst}.
Former will format the reference list and their citations according to
numbered scheme whereas the latter will format according name-date or
author-year style. Authors are requested to choose any one of these
according to the journal style. You may download these from
The above bsts are available in the following location for you to
download:
\url{https://support.stmdocs.in/wiki/index.php?title=Model-wise_bibliographic_style_files}
\hfill $\Box$
\end{document}

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# $Id: makefile 37 2020-02-25 09:06:02Z rishi $
file=elsdoc-cas
all: pdf out
make pdf
make pdf
out:
if [ -f $(file).out ] ; then cp $(file).out tmp.out; fi ;
sed 's/BOOKMARK/dtxmark/g;' tmp.out > x.out; mv x.out tmp.out ;
pdf:
pdflatex $(file).tex
index:
makeindex -s gind.ist -o $(file).ind $(file).idx
changes:
makeindex -s gglo.ist -o $(file).gls $(file).glo
xview:
xpdf -z 200 $(file).pdf &>/dev/null
view:
open -a 'Adobe Reader.app' $(file).pdf
ins:
latex $(file).ins
diff:
diff $(file).sty ../$(file).sty |less
copy:
cp $(file).sty ../

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%%
%% pdfwidgets.sty
%%
%% $Id: pdfwidgets.sty,v 1.2 2007-10-22 09:45:17 cvr Exp $
%%
%% (c) C. V. Radhakrishnan <cvr@river-valley.org>
%%
%% This package may be distributed under the terms of the LaTeX Project
%% Public License, as described in lppl.txt in the base LaTeX distribution.
%% Either version 1.0 or, at your option, any later version.
%%
%\RequirePackage[oldstyle]{minion}
%\RequirePackage[scaled=.8]{prima}
%\RequirePackage[scaled=.9]{lfr}
\usepackage[dvipsnames,svgnames]{xcolor}
\RequirePackage{graphicx}
\RequirePackage{tikz}
\usetikzlibrary{backgrounds}
%\def\thesection{\ifnum\c@section<10
% \protect\phantom{0}\fi\arabic{section}}
\newdimen\lmrgn
\def\rulecolor{orange}
\def\rulewidth{1pt}
\pgfdeclareshape{filledbox}{%
\inheritsavedanchors[from=rectangle] % this is nearly a rectangle
\inheritanchorborder[from=rectangle]
\inheritanchor[from=rectangle]{center}
\inheritanchor[from=rectangle]{north}
\inheritanchor[from=rectangle]{south}
\inheritanchor[from=rectangle]{west}
\inheritanchor[from=rectangle]{east}
% ... and possibly more
\backgroundpath{% this is new
% store lower right in xa/ya and upper right in xb/yb
\southwest \pgf@xa=\pgf@x \pgf@ya=\pgf@y
\northeast \pgf@xb=\pgf@x \pgf@yb=\pgf@y
% compute corner of ``flipped page''
\pgf@xc=\pgf@xb \advance\pgf@xc by-5pt % this should be a parameter
\pgf@yc=\pgf@yb \advance\pgf@yc by-5pt
% construct main path
\pgfsetlinewidth{\rulewidth}
\pgfsetstrokecolor{\rulecolor}
\pgfpathmoveto{\pgfpoint{\pgf@xa}{\pgf@ya}}
\pgfsetcornersarced{\pgfpoint{9pt}{9pt}}
\pgfpathlineto{\pgfpoint{\pgf@xa}{\pgf@yb}}
% \pgfsetcornersarced{\pgforigin}
\pgfsetcornersarced{\pgfpoint{9pt}{9pt}}
\pgfpathlineto{\pgfpoint{\pgf@xb}{\pgf@yb}}
\pgfsetcornersarced{\pgfpoint{9pt}{9pt}}
\pgfpathlineto{\pgfpoint{\pgf@xb}{\pgf@ya}}
\pgfsetcornersarced{\pgforigin}
\pgfpathclose ;
% \draw(\pgf@xa,\pgf@ya) -- (\pgf@xa,\pgf@yb) ;
}%
}
\pgfdeclareshape{roundedbox}{%
\inheritsavedanchors[from=rectangle] % this is nearly a rectangle
\inheritanchorborder[from=rectangle]
\inheritanchor[from=rectangle]{center}
\inheritanchor[from=rectangle]{north}
\inheritanchor[from=rectangle]{south}
\inheritanchor[from=rectangle]{west}
\inheritanchor[from=rectangle]{east}
% ... and possibly more
\backgroundpath{% this is new
% store lower right in xa/ya and upper right in xb/yb
\southwest \pgf@xa=\pgf@x \pgf@ya=\pgf@y
\northeast \pgf@xb=\pgf@x \pgf@yb=\pgf@y
% compute corner of ``flipped page''
\pgf@xc=\pgf@xb \advance\pgf@xc by-5pt % this should be a parameter
\pgf@yc=\pgf@yb \advance\pgf@yc by-5pt
% construct main path
\pgfsetlinewidth{\rulewidth}
\pgfsetstrokecolor{\rulecolor}
\pgfpathmoveto{\pgfpoint{\pgf@xa}{\pgf@ya}}
\pgfsetcornersarced{\pgfpoint{4pt}{4pt}}
\pgfpathlineto{\pgfpoint{\pgf@xa}{\pgf@yb}}
% \pgfsetcornersarced{\pgforigin}
\pgfsetcornersarced{\pgfpoint{4pt}{4pt}}
\pgfpathlineto{\pgfpoint{\pgf@xb}{\pgf@yb}}
\pgfsetcornersarced{\pgfpoint{4pt}{4pt}}
\pgfpathlineto{\pgfpoint{\pgf@xb}{\pgf@ya}}
% \pgfsetcornersarced{\pgforigin}
\pgfsetcornersarced{\pgfpoint{4pt}{4pt}}
\pgfpathclose ;
% \draw(\pgf@xa,\pgf@ya) -- (\pgf@xa,\pgf@yb) ;
}%
}
\pgfdeclareshape{buttonbox}{%
\inheritsavedanchors[from=rectangle] % this is nearly a rectangle
\inheritanchorborder[from=rectangle]
\inheritanchor[from=rectangle]{center}
\inheritanchor[from=rectangle]{north}
\inheritanchor[from=rectangle]{south}
\inheritanchor[from=rectangle]{west}
\inheritanchor[from=rectangle]{east}
% ... and possibly more
\backgroundpath{% this is new
% store lower right in xa/ya and upper right in xb/yb
\southwest \pgf@xa=\pgf@x \pgf@ya=\pgf@y
\northeast \pgf@xb=\pgf@x \pgf@yb=\pgf@y
% compute corner of ``flipped page''
\pgf@xc=\pgf@xb \advance\pgf@xc by-5pt % this should be a parameter
\pgf@yc=\pgf@yb \advance\pgf@yc by-5pt
% construct main path
\pgfsetlinewidth{1pt}
\pgfsetstrokecolor{blue!10}
\pgfpathmoveto{\pgfpoint{\pgf@xa}{\pgf@ya}}
\pgfsetcornersarced{\pgfpoint{4pt}{4pt}}
\pgfpathlineto{\pgfpoint{\pgf@xa}{\pgf@yb}}
% \pgfsetcornersarced{\pgforigin}
\pgfsetcornersarced{\pgfpoint{4pt}{4pt}}
\pgfpathlineto{\pgfpoint{\pgf@xb}{\pgf@yb}}
\pgfsetcornersarced{\pgforigin}
% \pgfsetcornersarced{\pgfpoint{9pt}{9pt}}
\pgfpathlineto{\pgfpoint{\pgf@xb}{\pgf@ya}}
\pgfsetcornersarced{\pgforigin}
\pgfpathclose ;
% \draw(\pgf@xa,\pgf@ya) -- (\pgf@xa,\pgf@yb) ;
}%
}
\pgfdeclareshape{quotedbox}{%
\inheritsavedanchors[from=rectangle] % this is nearly a rectangle
\inheritanchorborder[from=rectangle]
\inheritanchor[from=rectangle]{center}
\inheritanchor[from=rectangle]{north}
\inheritanchor[from=rectangle]{south}
\inheritanchor[from=rectangle]{west}
\inheritanchor[from=rectangle]{east}
% ... and possibly more
\backgroundpath{% this is new
% store lower right in xa/ya and upper right in xb/yb
\southwest \pgf@xa=\pgf@x \pgf@ya=\pgf@y
\northeast \pgf@xb=\pgf@x \pgf@yb=\pgf@y
% compute corner of ``flipped page''
\pgf@xc=\pgf@xb \advance\pgf@xc by-5pt % this should be a parameter
\pgf@yc=\pgf@yb \advance\pgf@yc by-5pt
% construct main path
\pgfsetlinewidth{\rulewidth}
\pgfsetstrokecolor{\rulecolor}
\pgfpathmoveto{\pgfpoint{\pgf@xa}{\pgf@ya}}
\pgfsetcornersarced{\pgfpoint{9pt}{9pt}}
\pgfpathlineto{\pgfpoint{\pgf@xa}{\pgf@yb}}
\pgfsetcornersarced{\pgforigin}
% \pgfsetcornersarced{\pgfpoint{4pt}{4pt}}
\pgfpathlineto{\pgfpoint{\pgf@xb}{\pgf@yb}}
\pgfsetcornersarced{\pgforigin}
% \pgfsetcornersarced{\pgfpoint{9pt}{9pt}}
\pgfpathlineto{\pgfpoint{\pgf@xb}{\pgf@ya}}
\pgfsetcornersarced{\pgforigin}
\pgfpathclose ;
% \draw(\pgf@xa,\pgf@ya) -- (\pgf@xa,\pgf@yb) ;
}%
}
\newcounter{clip}
\newdimen\mywidth
\mywidth=\linewidth
\def\src#1{\gdef\@src{#1}}\let\@src\@empty
\def\includeclip{\@ifnextchar[{\@includeclip}{\@includeclip[]}}
\def\@includeclip[#1]#2#3#4{\par
% \vskip.75\baselineskip plus 3pt minus 1pt
\computeLinewidth{\mywidth}%
\begingroup\color{white}%
\noindent%
\begin{tikzpicture}
%\node[fill=black!10,draw,shape=filledbox,
\node[fill=black!10,%
draw,
shade,%
top color=blue!10,
bottom color=cyan!5,
shape=filledbox,
inner sep=\Sep,
text width=\Linewidth] (x)
{\parbox{\Linewidth}
{\ifx\@src\@empty\else\refstepcounter{clip}\label{clip\theclip}%
{\par\vskip6pt\color{orange}\sffamily\small
~Clip \theclip:\space\@src.}%
\par\vskip3pt\fi\normalcolor
\includegraphics[width=\Linewidth,page={#2},%
viewport={#3},clip=true,#1]{#4}}
\hspace*{-10pt}};
\end{tikzpicture}
\endgroup
% \par\vskip.5\baselineskip
% plus 3pt minus 1pt
}
%%
%% include clippings from a pdf document:
%% #1 => Optional argument for \includegraphics
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%% #3 => co-ordinates
%% #4 => file name
\newenvironment{quoted}{%\bigskip
\computeLinewidth{.95\linewidth}%
\global\setbox0=\hbox\bgroup
\begin{minipage}{.95\linewidth}\color{brown}%
\footnotesize\ttfamily\obeyspaces\obeylines}
{\end{minipage}\egroup
\vskip12pt plus 3pt minus 3pt\noindent\begin{tikzpicture}
\node[fill=blue!10,draw,shade,top color=orange!10,
bottom color=white,shape=filledbox,
inner sep=8pt,text width=\Linewidth] (x) {\box0} ;
\end{tikzpicture}%
\vskip12pt plus 3pt minus 3pt}
\newdimen\Linewidth
\newdimen\Sep
\def\computeLinewidth#1{\global\setlength\Linewidth{#1}%
\global\addtolength{\Linewidth}{-2\Sep}}
\newdimen\npskip
\npskip=0mm
\long\def\NavigationPanel{%
\global\setbox0=\hbox\bgroup
\begin{minipage}[t][.8125\panelheight][t]{.9\panelwidth}\color{brown}%
%\centering
\ifx\@pinclude\empty\relax\par\vfill\else
\@pinclude\fi
%River Valley Technologies
\end{minipage}\egroup
\Sep=.5cm
\@tempdima=\panelwidth
\advance\@tempdima-1cm
\computeLinewidth{\@tempdima}%
\def\rulewidth{.2pt}%
\noindent\begin{tikzpicture}
\node[fill=blue!10,draw,shade,bottom color=brown!30,
top color=white,shape=filledbox,
inner sep=\the\Sep,text width=\Linewidth] (x)
{\hspace*{\npskip}\box0} ;
\end{tikzpicture}%
\vspace*{.0125\panelheight}
}
\long\def\pinclude#1{\gdef\@pinclude{#1}}
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\def\Strut{\vrule depth 2pt height 10pt width 0pt}
\def\pdfButton#1#2{\begin{tikzpicture}
\node[fill=blue!10,draw,shade,top color=blue!50,
bottom color=white,shape=buttonbox,
inner sep=2pt,text width=#1](x)
{\parbox{#1}{\centering\Strut#2}}; \end{tikzpicture}}
\def\vpanel{\def\@linkcolor{blue}%
\def\@urlcolor{blue}%
\def\@menucolor{blue}%
\begin{minipage}[t][\vpanelheight][c]{\paperwidth}%
\normalsfcodes%
\hspace*{.25cm}
\begin{minipage}[c][\vpanelheight][c]{17cm}
\parbox[c][27mm][b]{15mm}%
% {\includegraphics[width=15mm]{logo4.pdf}}\hfill%\hspace{1cm}
{\def\rulecolor{Goldenrod}%
\def\rulewidth{1pt}%
\begin{tikzpicture}%
%\node[fill=black!10,draw,shape=filledbox,
\node[fill=white!10,%
draw,
% shade,%
% top color=blue!10,
% bottom color=white,
shape=roundedbox,
inner sep=2mm,
text width=13mm] (x)
{\includegraphics[width=13mm]{els-logo.pdf}};
\end{tikzpicture}}\hfill
%
\parbox[c][24mm][b]{145mm}%
{{\fontsize{30}{30}\selectfont\textsf{\color{white}elsarticle.cls}}
\quad{\fontsize{14}{14}\selectfont\sffamily\color{blue!50}
A better way to format your submission}}
\end{minipage}
\hfill
\begin{minipage}[c][\vpanelheight][b]{7.9cm}
\sffamily\footnotesize
\pdfButton{2cm}{\href{mailto:elsarticle@river-valley.com}{BUGS}}
\pdfButton{2cm}{\href{http://support.river-valley.com}{SUPPORT}}
\pdfButton{2cm}%
{\href{http://www.elsevier.com/locate/latex}%
{RESOURCES}}
% \pdfButton{2cm}{\Acrobatmenu{GoToPage}{GoTo}}
\end{minipage}\\
\rule{\paperwidth}{0.1pt}
\end{minipage}%
}
\@ifundefined{backgroundcolor}%
{\def\backgroundcolor#1{\gdef\@backgroundcolor{#1}}}{}
\colorlet{panelbackground}{orange!10}
\backgroundcolor{orange!10}
\def\@urlcolor{brown}
\def\@linkcolor{brown}
\def\@menucolor{brown}
\RequirePackage{moreverb}
\newenvironment{vquote}%
{\medskip
\verbatimwrite{tmp.tex}}
{\endverbatimwrite
\aftergroup\printBox}
\def\printBox{\bgroup\def\rulecolor{orange}%
\def\rulewidth{.2pt}%
\noindent\begin{tikzpicture}
\node[fill=blue!10,draw,shade,top color=white!10,
bottom color=cyan!5,shape=quotedbox,
inner sep=8pt,text width=.95\linewidth]
{\color{orange}\vspace*{-1pc}%
\verbatiminput{tmp.tex}%
\vspace*{-\baselineskip}%
} ;
\end{tikzpicture}%
\egroup
\medskip
}
\def\red{\color{Sepia}}
\def\verbatim@font{\red\normalfont\ttfamily}
\def\verbatimcontinuewrite{%
\@bsphack
% \verbatim@out=#1
\let\do\@makeother\dospecials
\obeyspaces\catcode`\^^M\active \catcode`\^^I=12
\def\verbatim@processline{%
\immediate\write\verbatim@out
{\the\verbatim@line}}%
\verbatim@start}
\def\@@@lbr{\expandafter\@gobble\string\{}
\def\@@@rbr{\expandafter\@gobble\string\}}
\def\@@@pcr{\expandafter\@gobble\string\%}
%\immediate\write18{touch mytool.tex
% ^^J rm mytool.tex ^^J touch mytool.tex}
\newenvironment{toolwrite}[1]%
{\@tempdima=#1
\verbatimwrite{xx}}
{\endverbatimwrite
\immediate\write18{echo
"\string\Clear\@@@lbr\the\@tempdima\@@@rbr\@@@lbr\@@@pcr">>mytool.tex^^J
cat xx.tex >> mytool.tex ^^J
echo "\@@@rbr" >> mytool.tex}}
\tikzstyle{place}=[scale=.39,rectangle,draw=blue!90,fill=blue!30,thin,%
minimum height=1mm,minimum width=13mm]
\tikzstyle{trans}=[scale=.39,rectangle,draw=Olive,fill=Olive!20,thin,%
minimum height=1mm,minimum width=13mm]
\tikzstyle{past}=[scale=.39,rectangle,draw=Olive,fill=Olive!60,thin,%
minimum height=1mm,minimum width=13mm]
\def\printSq#1{\parbox{107mm}{\@tempcnta=1
\let\printfill\@empty
\loop\ifnum\@tempcnta<#1
{\printfill\ifnum\c@page=\@tempcnta
\tikz\node at(0,0) [place]{};\else
\ifnum\c@page<\@tempcnta
\hyperlink{page.\the\@tempcnta}{\tikz\node at(0,0)
[trans]{};}%
\else
\hyperlink{page.\the\@tempcnta}{\tikz\node at(0,0)
[past]{};}%
\fi\fi}%
\advance\@tempcnta 1 \let\printfill\,\repeat}}
\endinput

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%
%
% File: rvdtx.sty
%
% Auxiliary package to format *.dtx documents.
%
% Copyright (c) 2008-2020 CV Radhakrishnan <cvr@stmdocs.in>,
%
% This file may be distributed and/or modified under the conditions
% of the LaTeX Project Public License, either version 1.2 of this
% license or (at your option) any later version. The latest version
% of this license is in:
%
% http://www.latex-project.org/lppl.txt
%
% and version 1.2 or later is part of all distributions of LaTeX
% version 1999/12/01 or later.
%
%
\newcounter{colorscheme}
\newif\if@xcolor \@xcolorfalse
\newif\if@mylogo \@mylogofalse
\DeclareOption{mylogo}{\global\@mylogotrue}
\DeclareOption{green}{\setcounter{colorscheme}{1}}
\DeclareOption{orange}{\setcounter{colorscheme}{0}}
\DeclareOption{xcolor}{\global\@xcolortrue}
\DeclareOption{qone}{\AtEndOfPackage{\global\let\dtxmark\dtxmarkone}}
\DeclareOption{qtwo}{\AtEndOfPackage{\global\let\dtxmark\dtxmarktwo}}
\ProcessOptions
\def\loadXcolor{\if@xcolor\RequirePackage[dvipsnames,svgnames]{xcolor}\fi}
\loadXcolor
\ifcase\thecolorscheme
%
% Orange color spec (default)
%
\colorlet{itemcolor}{brown}
\colorlet{verbcolor}{Sepia}
\colorlet{botrulecolor}{orange!25}
\colorlet{botbgcolor}{orange!15}
\colorlet{botcolor}{orange!80}
\colorlet{pgrulecolor}{orange}
\colorlet{pgbgcolor}{white}
\colorlet{quicklinkrulecolor}{orange!40}
\colorlet{quicklinkcolor}{brown}
\colorlet{topverticalrule}{brown}
\colorlet{titlecolor}{brown}
\colorlet{hlinkcolor}{brown}
\colorlet{hlinktricolor}{orange!70}
\colorlet{linkcolor}{brown}
\colorlet{urlcolor}{brown}
% \colorlet{arrayrulecolor}{olive!30}
\colorlet{seccolor}{brown}
\colorlet{toprulecolor}{orange!30}
\colorlet{topbgcolor}{orange!10}
\colorlet{topcolor}{brown!80}
%
%
\or% Green color specs
%
%
\colorlet{itemcolor}{OliveGreen}
\colorlet{verbcolor}{OliveGreen}
\colorlet{botrulecolor}{GreenYellow!25}
\colorlet{botbgcolor}{GreenYellow!30}
\colorlet{botcolor}{Green!80}
\colorlet{pgrulecolor}{GreenYellow}
\colorlet{pgbgcolor}{white}
\colorlet{quicklinkrulecolor}{Green!40}
\colorlet{quicklinkcolor}{Green}
\colorlet{topverticalrule}{Green}
\colorlet{titlecolor}{DarkOliveGreen}
\colorlet{hlinkcolor}{DarkOliveGreen}
\colorlet{hlinktricolor}{Green!70}
\colorlet{linkcolor}{OliveGreen}
\colorlet{urlcolor}{OliveGreen}
% \colorlet{arrayrulecolor}{olive!30}
\colorlet{seccolor}{OliveGreen}
\colorlet{toprulecolor}{GreenYellow!50}
\colorlet{topbgcolor}{GreenYellow!20}
\colorlet{topcolor}{GreenYellow!80}
\fi
\def\floatpagefraction{.99}
\usepackage{geometry}
\geometry{top=2in,
bottom=1in,
left=2in,
right=1in,
a4paper}
%\DeclareRobustCommand{\LaTeX}{L\kern-.25em%
% {\sbox\z@ T%
% \vbox to\ht\z@{%
% {\check@mathfonts
% \fontsize\sf@size\z@
% \math@fontsfalse\selectfont
% A}%
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% }%-.10em%
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\DeclareRobustCommand{\LaTeX}{L\kern-.25em%
{\sbox\z@ T%
\vbox to\ht\z@{%
\hbox{%
\check@mathfonts
\fontsize\sf@size\z@
\math@fontsfalse\selectfont
A}%
\vss}%
}%
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\TeX}
\RequirePackage{pdfwidgets}
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\def\xml{\textsc{xml}\xspace}
\def\latex{\LaTeX\xspace}
\def\pdf{\textsc{pdf}\xspace}
\def\pdfa{\textsc{pdf/a-1}b\xspace}
\def\pdfx{\textsc{pdf/x-1}a\xspace}
\def\xmp{\textsc{xmp}\xspace}
\def\pdftex{\textsc{pdf\TeX}\xspace}
\def\defmacro#1{\texttt{\@bsl#1}}
\def\thanh{H\`an Th\^e Th\`anh\xspace}
\def\gnulinux{\textsc{gnu/linux}\xspace}
\let\@DRAFTout@Hook\@empty
\newcommand{\DRAFTout}{\g@addto@macro\@DRAFTout@Hook}
\newcommand{\@DRAFTout@Out}{%
\afterassignment\@DRAFTout@Test
\global\setbox\@cclv=
}
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\ifvoid\@cclv\relax
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\@DRAFTout@Output
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\@DRAFTout@Hook%
\@DRAFTout@Org@Out\box\@cclv%
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\newcommand*{\@DRAFTout@Init}{%
\let\@DRAFTout@Org@Out\shipout
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\addtolength\OHeight{\headheight}
\addtolength\OHeight{\headsep}
\addtolength\OHeight{\footskip}
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\AtBeginDocument{\@DRAFTout@Init}
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\newcommand{\AddToDraftOverlay}{\g@addto@macro\@DraftOverlay@Hook}
\newcommand{\ClearDraftOverlay}{\let\@DraftOverlay@Hook\@empty}
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\else
\bgroup
\@tempdima=1in
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\ifoverlay
\global\setbox\@cclv\vbox{%
\box\@cclv
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\@DraftOverlay@Hook%
\endpicture}}%
\else
\global\setbox\@cclv\vbox{%
\vbox{\let\protect\relax%
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\box\@cclv}%
\fi
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}
\definecolor{gray30}{gray}{.7}
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\DRAFTout{\@DraftOverlay}
\long\def\puttext(#1)#2{\AddToDraftOverlay{%
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\RequirePackage{shortvrb}
\MakeShortVerb{\|}
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\IfFileExists{lfr.sty}{\RequirePackage[scaled=.85]{lfr}}{}
%\IfFileExists{prima.sty}{\RequirePackage[scaled=.8]{prima}}{}
\def\theCodelineNo{\reset@font\tiny\arabic{CodelineNo}}
\def\@seccntformat#1{\llap{\csname the#1\endcsname.\hspace*{6pt}}}
\def\section{\@startsection {section}{1}{\z@}%
{-3.5ex \@plus -1ex \@minus -.2ex}%
{2.3ex \@plus.2ex}%
{\normalfont\large\bfseries\color{seccolor}}}
\def\subsection{\@startsection{subsection}{2}{\z@}%
{-2.25ex\@plus -1ex \@minus -.2ex}%
{1.5ex \@plus .2ex}%
{\normalfont\normalsize\bfseries\color{seccolor}}}
\def\subsubsection{\@startsection{subsubsection}{3}{\z@}%
{-1.25ex\@plus -1ex \@minus -.2ex}%
{1.5ex \@plus .2ex}%
{\normalfont\normalsize\bfseries\color{seccolor}}}
%\RequirePackage[draft]{pdfdraftcopy}
% \draftstring{}
\puttext(0,36){\botstring}%
\puttext(0,840){\copy\topbox}
\if@mylogo
\puttext(531,829){\cvrlogo}
\fi
\RequirePackage{colortbl}
%\arrayrulecolor{arrayrulecolor}
\let\shline\hline
\def\hline{\noalign{\vskip3pt}\shline\noalign{\vskip4pt}}
\RequirePackage[pdftex,colorlinks]{hyperref}
\def\Hlink#1#2{\hyperlink{#2}{\color{hlinktricolor}%
$\blacktriangleright$~\color{hlinkcolor}#1}}
\def\@linkcolor{linkcolor}
\def\@urlcolor{urlcolor}
\pagestyle{empty}
\def\version#1{\gdef\@version{#1}}
\def\@version{1.0}
\def\contact#1{\gdef\@contact{#1}}
\def\author#1{\gdef\@author{#1}}
\def\@author{STM Document Engineering Pvt Ltd.}
\def\@contact{\texttt{support@stmdocs.in}}
\def\keywords#1{\gdef\@keywords{#1}}
\def\@keywords{\LaTeX, \xml}
\long\def\Hrule{\\[-4pt]\hspace*{-3em}%
{\color{quicklinkrulecolor}\rule{\linewidth}{.1pt}}\\}
\long\def\dtxmarkone[#1][#2]#3#4#5{\def\next{#1}%
\ifcase\next\or\Hlink{#4}{#3}\Hrule \fi}
\newcounter{dtx}
\long\def\dtxmarktwo[#1][#2]#3#4#5{\def\next{#1}%
\stepcounter{dtx}\parbox{.45\linewidth}%
{\ifcase\next\or\Hlink{#4}{#3}\fi}%
\ifodd\thedtx\relax\else\Hrule\fi}
\let\dtxmark\dtxmarkone
\newbox\topbox
\long\def\maketitle{\global\setbox\topbox=\vbox{\hsize=\paperwidth
\parindent=0pt
\fcolorbox{toprulecolor}{topbgcolor}%
{\parbox[t][2in][c]{\paperwidth}%
{\hspace*{15mm}%
\parbox[c]{.35\paperwidth}{\fontsize{18pt}{20pt}%
\raggedright\normalfont\sffamily \selectfont
\color{titlecolor} \@title\\[6pt]
{\normalsize\rmfamily\scshape\@author}}%
% {\footnotesize\textsc{keywords:} \@keywords}}%
\hfill
\parbox[c][2in][c]{1mm}{\color{topverticalrule}%
\rule{.1pt}{2in}}%
\hfill
\parbox[c][2in][c]{.35\paperwidth}%
{\normalfont\footnotesize\sffamily\color{quicklinkcolor}%
\advance\baselineskip-3pt%
\vspace*{6pt} QUICK LINKS\Hrule
\IfFileExists{tmp.out}{\input tmp.out}{}%
}\hspace*{5mm}%
}%
}%
}%
}
\gdef\botstring{\fcolorbox{botrulecolor}{botbgcolor}%
{\parbox[t][.5in][t]{\paperwidth}%
{\normalfont\sffamily\footnotesize%
\color{botcolor}%
\hspace*{5mm}\parbox[c][.5in][c]{.45\paperwidth}%
{\raggedright \textcopyright\ 2019, Elsevier Ltd.
Bugs, feature requests, suggestions and comments %\\
shall be mailed to \href{mailto:elsarticle@stmdocs.in}
{$<$elsarticle@stmdocs.in$>$}.
}\hfill%
\parbox[c][.5in][c]{1cm}
{\centering\sffamily\mdseries
\fcolorbox{pgrulecolor}{pgbgcolor}{\thepage}%
}\hfill
\parbox[c][.5in][c]{.45\paperwidth}
{\raggedleft\begin{tabular}{rl}%
Version:&\@version\\
Date:&\@date\\
Contact:&\@contact
\end{tabular}\hspace*{5mm}%
}%
}%
}%
}
\def\MacroFont{\fontencoding\encodingdefault
\fontfamily\ttdefault
\fontseries\mddefault
\fontshape\updefault
\color{verbcolor}\small}%
\def\verbatim@font{\normalfont\color{verbcolor}\ttfamily}
\def\verb{\relax\ifmmode\hbox\else\leavevmode\null\fi
\bgroup
\verb@eol@error \let\do\@makeother \dospecials
\verbatim@font\@noligs
\@ifstar\@sverb\@verb}
\def\@lbr{\expandafter\@gobble\string\{}
\def\@rbr{\expandafter\@gobble\string\}}
\def\@bsl{\expandafter\@gobble\string\\}
\def\@Bsl#1{\texttt{\@bsl#1}\xspace}
\def\trics#1{\protect\@Bsl{#1}}
\def\onecs#1{\protect\@Bsl{#1}}
%\let\trics\onecs
\@ifundefined{c@Glossary}{}{\c@GlossaryColumns=1
\c@IndexColumns=2}
\def\index@prologue{\section{Index}%
\markboth{Index}{Index}%
% Numbers written in italic refer to the page
% where the corresponding entry is described;
% numbers underlined refer to the
% \ifcodeline@index
% code line of the
% \fi
% definition; numbers in roman refer to the
% \ifcodeline@index
% code lines
% \else
% pages
% \fi
% where the entry is used.
}
\@ifundefined{theglossary}{}{%
\renewenvironment{theglossary}{%
\glossary@prologue%][\GlossaryMin]%
\GlossaryParms \let\item\@idxitem \ignorespaces}%
{}}
\newenvironment{decl}[1][]%
{\par\small\addvspace{1.5ex plus 1ex}%
\vskip -\parskip
\ifx\relax#1\relax
\def\@decl@date{}%
\else
\def\@decl@date{\NEWfeature{#1}}%
\fi
\noindent%\hspace{-\leftmargini}%
\begin{tabular}{l}\hline\ignorespaces}%
{\\\hline\end{tabular}\nobreak\@decl@date\par\nobreak
\vspace{0.75ex}\vskip -\parskip\ignorespacesafterend\noindent}
\newif\ifhave@multicol
\newif\ifcodeline@index
\IfFileExists{multicol.sty}{\have@multicoltrue
\RequirePackage{multicol}%
}{}
\newdimen\IndexMin \IndexMin = 80pt
\newcount\c@IndexColumns \c@IndexColumns = 2
\ifhave@multicol
\renewenvironment{theindex}
{\begin{multicols}\c@IndexColumns[\index@prologue][\IndexMin]%
\IndexParms \let\item\@idxitem \ignorespaces}%
{\end{multicols}}
\else
\typeout{Can't find multicol.sty -- will use normal index layout if
necessary.}
\def\theindex{\@restonecoltrue\if@twocolumn\@restonecolfalse\fi
\columnseprule \z@ \columnsep 35\p@
\twocolumn[\index@prologue]%
\IndexParms \let\item\@idxitem \ignorespaces}
\def\endtheindex{\if@restonecol\onecolumn\else\clearpage\fi}
\fi
\long\def\IndexPrologue#1{\@bsphack\def\index@prologue{#1}\@esphack}
\@ifundefined{index@prologue}
{\def\index@prologue{\section{Index}%
\markboth{Index}{Index}%
% Numbers written in italic refer to the page
% where the corresponding entry is described;
% numbers underlined refer to the
% \ifcodeline@index
% code line of the
% \fi
% definition; numbers in roman refer to the
% \ifcodeline@index
% code lines
% \else
% pages
% \fi
% where the entry is used.
}}{}
\@ifundefined{IndexParms}
{\def\IndexParms{%
\parindent \z@
\columnsep 15pt
\parskip 0pt plus 1pt
\rightskip 15pt
\mathsurround \z@
\parfillskip=-15pt
\footnotesize
\def\@idxitem{\par\hangindent 30pt}%
\def\subitem{\@idxitem\hspace*{15pt}}%
\def\subsubitem{\@idxitem\hspace*{25pt}}%
\def\indexspace{\par\vspace{10pt plus 2pt minus 3pt}}%
}}{}
\def\efill{\hfill\nopagebreak}%
\def\dotfill{\leaders\hbox to.6em{\hss .\hss}\hskip\z@ plus 1fill}%
\def\dotfil{\leaders\hbox to.6em{\hss .\hss}\hfil}%
\def\pfill{\unskip~\dotfill\penalty500\strut\nobreak
\dotfil~\ignorespaces}%
\let\scan@allowedfalse\relax
\def\tlformat#1{\begingroup\Large
\parbox[c][1.25em][c]{1.25em}{\centering\fontfamily{phv}
\fontseries{m}%
\selectfont\color{white}\huge#1}%
\endgroup}
\def\tlFormat#1{\begingroup\Large
\parbox[c][1.25em][c]{1.25em}{\centering\fontfamily{phv}
\fontseries{m}%
\selectfont\color{black}\huge#1}%
\endgroup}
\def\cvrlogo{\begingroup\fboxsep=2pt
\colorbox{olive}{\tlformat{c}}%
\colorbox{blue}{\tlformat{v}}%
\colorbox{red}{\tlformat{r}}
\endgroup}
\endinput
%%
%% End of file 'rvdtx.sty'
%%

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@ -0,0 +1,94 @@
% Copyright 2019-2020 Elsevier Ltd
%
% This file is part of the 'CAS Bundle'.
% --------------------------------------
%
% It may be distributed and/or modified under the
% conditions of the LaTeX Project Public License, either version 1.2
% of this license or (at your option) any later version.
% The latest version of this license is in
% http://www.latex-project.org/lppl.txt
% and version 1.2 or later is part of all distributions of LaTeX
% version 1999/12/01 or later.
%
% The list of all files belonging to the LaTeX 'CAS Bundle' is
% given in the file `manifest.txt'.
%
% CONTENTS OF THE CAS BUNDLE
% ==========================
Directory elsevier-cas-template/
cas-sc.cls
Classfile to be used for single column format
cas-dc.cls
Classfile to be used for double column format
model2-names.bst
BibTeX style file
cas-sc-template.tex
TeX template
cas-sc-template.pdf
PDF output of the above template
cas-dc-template.tex
TeX template
cas-dc-template.pdf
PDF output of the above template
manifest.txt
this file
README
small readme documentation
Directory doc/
The following files are graphic files needed for creating pdf output
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\begin{document}
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\shorttitle{Incompressible flows in moving domains}
\shortauthors{Ar\'ostica and Bertoglio}
%\begin{frontmatter}
\title[mode = title]{On monolithic and Chorin-Temam schemes for incompressible flows in moving domains}
\tnotetext[1]{This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 852544 - CardioZoom).}
\author[1]{Reidmen Ar\'{o}stica}
\ead{r.a.arostica.barrera@rug.nl}
%\ead[url]{https://sites.google.com/view/cvmath/people/ar\'{o}stica?authuser=0}
%\credit{Conceptualization, Analysis, Implementation}
\author[1]{Crist\'{o}bal Bertoglio}
\ead{c.a.bertoglio@rug.nl}
%\ead[url]{https://sites.google.com/view/cvmath/people/bertoglio?authuser=0}
%\credit{Conceptualization, Methodology, Guidance}
\address[1]{Bernoulli Institute, University of Groningen, Groningen, 9747AG, The Netherlands}
%\cortext[cor1]{Corresponding author}
%\cortext[cor2]{Principal corresponding author}
\begin{abstract}
%When working with numerical schemes for the incompressible Navier-Stokes equations (iNSE), several schemes can be proposed and diverse techniques be used, from monolithic approaches to Chorin-Temam methods. We study a standard but general monolithic scheme that characterizes several approaches used in literature, where unconditional energy stability will holds under constrains with the addition of consistent terms. With such finding, we derive a Chorin-Temam scheme and show the unconditional stability of it.
Several time discretization schemes for the incompressible Navier-Stokes equations (iNSE) in moving domains have been proposed. Here we introduce them in a unified fashion, allowing a common well possedness and time stability analysis. It can be therefore shown that only a particular choice of the numerical scheme ensures such properties. The analysis is performed for monolithic and Chorin-Temam schemes. Results are supported by numerical experiments.
\end{abstract}
%\begin{graphicalabstract}
%%\includegraphics{figs/grabs.pdf}
%\end{graphicalabstract}
%\begin{highlights}
%\item Research highlights item 1
%\item Research highlights item 2
%\item Research highlights item 3
%\end{highlights}
\begin{keywords}
incompressible flows \sep moving domains \sep monolithic coupling \sep Chorin-Temam \sep stability analysis %\\
\end{keywords}
% EDITING COMMANDS
\newcommand{\del}[1]{\sout{#1}}
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\maketitle
\section{Introduction}
%When working with flows from a simulation point of view, several schemes are proposed depending on its applications, suitable for specific requirement e.g. high spatio-temporal resolution, fidelity, stability or fast implementation/simulation times.
%The literature is vast, but to our knowledge there is not a overview trying to summarize their approaches in a single scheme.
Several works have been reported dealing with the numerical solution of the iNSE in moving domains within an Arbitrary Lagrangian Eulerian formulation (ALE), primarily in the context of fluid-solid coupling.
In particular different choices of time discretization have been reported on \cite{Basting2017, Murea2016, Landajuela2016,Lozovskiy2018,smaldone2014,Langer2018,LeTallec2001,Liu2018,failer2020impact,Hessenthaler2017}.
% In \cite{Basting2017, Murea2016, Landajuela2016} the FSI within Arbitrary Lagrangian-Eulerian (ALE) formalism in a linearized approach is studied, and \cite{Lozovskiy2018,smaldone2014} also include mass conservation terms. \cite{Langer2018,LeTallec2001,Liu2018} propose fully non-linear FSI schemes, and extensions using Crank-Nicholson are used by \cite{failer2020impact,Hessenthaler2017}. \CB{No entiendo la clasificacion de los papers, me parece que le falta un poco de coherencia. No deberian tambien haber mas papers, o estos son todos? }
%Thougthful
To the best of the authors knowledge, only a few monolithic schemes have been thoroughly analyzed, e.g. in \cite{Lozovskiy2018, Burtschell2017, LeTallec2001, smaldone2014}, while no analysis has been reported for Chorin-Temam (CT) methods.
The goal of this work is therefore to assess well-posedness and unconditional energy balance of the iNSE-ALE for all reported monolithic and CT discretization schemes within a single formulation.
% Maybe we need to add the works of \cite{Boffi2004} for general surveys of ALE schemes.
%State-of-the art: monolithic (what is proven), CT (not proven, just "used") \\
%, which as it will be seen later on, holds under some restrictions with help of consistent stabilization terms.
%The findings for the monolithic case are then used to introduce a Chorin-Temam scheme, for which unconditional energy stability holds.
The reminder of this paper is structured as follows:
Section \ref{sec:continuous_problem} provides the continuous problem that will be studied. Section \ref{sec:monolithic_schemes} introduces a general monolithic scheme that characterizes several approaches used in literature, well-posedness and energy stability are studied and discussed. Section \ref{sec:chorin_temam_schemes} introduces the Chorin-Temam schemes where time stability is analyzed. Finally, Section \ref{sec:numerical_examples} provides numerical examples testing our results.
\section{The continuous problem}
\label{sec:continuous_problem}
In the following, let us consider a domain $\Omega^0 \subset \mathbb{R}^d$ with $d = 2,3$ and a deformation mapping $\mathcal{X}: \mathbb{R}^d\times \mathbb{R}_{+}\mapsto\mathbb{R}^d$ that defines the time evolving domain $\Omega^t := \mathcal{X}(\Omega^0, t)$.
We assume $\mathcal{X}$ a {\reva{$\mathcal{C}^1$}} mapping in both coordinates, 1-to-1 with {\reva{ $\mathcal{C}^1$}} inverse. We denote $\mathbf{X} \in \mathbb{R}^d$ the cartesian coordinate system in $\Omega^0$ and $\mathbf{x}^t := \mathcal{X}(\mathbf{X}, t)$ the one in $\Omega^t$, by $F^t := \frac{\partial \mathbf{x}^t}{\partial \mathbf{X}}$ the deformation gradient, $H^t := (F^t)^{-1}$ its inverse and $J^t := det(F^t)$ its {\reva{Jacobian}}.
Similarly, $Grad(\mathbf{\mathbf{f}}) := \frac{\partial \mathbf{f}}{\partial \mathbf{X}}$, $Div(\mathbf{f}) := \frac{\partial}{\partial \mathbf{X}} \cdot \mathbf{f}$ denote the gradient and divergence operators respectively and $\epsilon^t(\mathbf{f}) := \frac{1}{2}( Grad(\mathbf{f}) H^t + (H^t)^{T} Grad(\mathbf{f})^{T})$ the symmetric gradient, for $\mathbf{f}$ a well-defined vector function.
% {\reva{By $\mathbf{H}^1(\Omega_0), \mathbf{H}^1(\Omega)_0, \mathbf{H}^1_{\mathbf{g}}$ we denote the standard Sobolev spaces of vector fields $\mathbf{u}$ defined in $\Omega_0$ with values in $\mathbb{R}^d$ for the first one, such that $\mathbf{u} = \mathbf{0}$ on $\partial \Omega_0$ for the second and $\mathbf{u} = \mathbf{g}$ on $\partial \Omega_0$ for the third one, being $\mathbf{g} \in \mathbf{H}^1(\Omega_0)$ and
% with $L^2_0(\Omega_0)$ the standard square integrable space of functions $p$ s.t. $\int_{\Omega_0} p \, \text{d}\mathbf{X}= 0$.
% }}
{\reva{
By $\mathbf{H}^1_0 (\Omega^0)$ we denote the standard Sobolev space of vector fields $\mathbf{u}$ defined in $\Omega^0$ with values in $\mathbb{R}^d$ such that $\mathbf{u} = \mathbf{0}$ on $\partial \Omega^0$, by $L^2_0(\Omega^0)$ the standard square integrable space of functions $r$ defined in $\Omega^0$ with values in $\mathbb{R}$ s.t. $\int_{\Omega^0} r \, \text{d}\mathbf{X} = 0$ and $T > 0$ a final time.
}}
We consider the weak form of the iNSE in ALE form \cite[Ch. 5]{Richter2017}: Find $(\mathbf{u}(t), p(t)) \in \mathbf{H}^{1}_{0}(\Omega^0) \times L^2_0(\Omega^0)$ for $t \in (0, T)$ with $\mathbf{u}(0) = \mathbf{u}_{init}$ s.t.
\begin{equation}
\label{eq:continuous_formulation}
\begin{aligned}
\int_{\Omega^0} \rho J^t \frac{\partial \mathbf{u} }{\partial t} \cdot \mathbf{v} + \rho J^t Grad(\mathbf{u}) H^t (\mathbf{u} - \mathbf{w}) \cdot \mathbf{v} + J^t 2\mu \, \epsilon^t(\mathbf{u}):\epsilon^t(\mathbf{v}) - Div(J^t H^t \mathbf{v}) p + Div(J^t H^t \mathbf{u})q \, \text{d}\mathbf{X} = 0%& \\
\end{aligned}
\end{equation}
for all $(\mathbf{v}, q) \in \mathbf{H}^1_0(\Omega^0) \times L^2_0(\Omega^0)$, $\mathbf{u}_{init} \in \mathbf{H}^1_0(\Omega^0)$ given initial {\revb{ and $\mathbf{w} := \frac{\partial \mathcal{X}}{\partial t}$ time-varying}} domain velocities. \reva{For the sake of simplicity, we omit the time-dependency on the fields $\mathbf{u}, p$}.
Notice that the \reva{velocity} flow at time $t$ is given by $\mathbf{u} \circ \mathcal{X}^{-1}(\cdot, t)$.
%
%\begin{remark}
%Formulation \eqref{eq:continuous_formulation} includes strongly consistent \textit{Geometric Conservation Law} (GCL) and \textit{Temam} stabilization terms multiplied by scalars $\alpha$ and $\beta$ respectively. This allows for a unified analysis for all reported methods.
%\end{remark}
%An energy estimate can be deduced from \eqref{eq:continuous_formulation} as done in \cite[Chap. 9]{Quarteroni2009Cardiovascular}.
\begin{proposition}
\label{prop:energy_continuous} \cite[Chap. 9]{Quarteroni2009Cardiovascular}
Provided \reva{$(\mathbf{u}(t), p(t)) \in \mathbf{H}^1_0(\Omega^0) \times L^2_0(\Omega^0)$} a solution of Problem \eqref{eq:continuous_formulation}, % and $\mathbf{u}_{\mathbf{g}}(t) \in \mathbf{H}^1_{\mathbf{g}}(\Omega^0)$ the extension of $g \in L^2(\partial \Omega^0)$
the following energy balance holds:
\begin{equation}
\label{eq:continuous_energy_estimate}
\begin{aligned}
\frac{\partial}{\partial t} \int_{\Omega^0} \frac{\rho}{2} J^t \vert \mathbf{u} \vert^2 \, \text{d}\mathbf{X} =& - \int_{\Omega^0} J^t 2\mu \vert \epsilon^t (\mathbf{u}) \vert^2 \, \text{d}\mathbf{X}. %- \int_{\partial\Omega_0} \rho J_t F^{-1}_{t} (\mathbf{u} - \mathbf{w}) \cdot \mathbf{N} \, \text{d}\mathbf{S} + \mathcal{D}(\mathbf{u}; \mathbf{u}_{\mathbf{g}})
\end{aligned}
\end{equation}
% where $\mathcal{D}$ denotes the mechanical work performed by the non-homogeneous boundary condition $g$, and defined as:
% \begin{equation}
% \label{eq:remainder_energy_estimate}
% \mathcal{D}(\mathbf{u}; \mathbf{u}_{\mathbf{g}}) := \int_{\Omega_0} \rho J_t \partial_{t} \mathbf{u} \cdot \mathbf{u}_{\mathbf{g}} + \rho J_t Grad(\mathbf{u}) F^{-1}_{t} (\mathbf{u} - \mathbf{w}) \cdot \mathbf{u}_{\mathbf{g}} + J_t 2\mu \epsilon^{t} (\mathbf{u}) : \epsilon^{t} (\mathbf{u}_{\mathbf{g}}) + Div\big(J_t F^{-1}_{t} \mathbf{u}_{\mathbf{g}} \big)p \, \text{d}\mathbf{X}
% \end{equation}
\end{proposition}
\begin{remark}
Proposition \ref{prop:energy_continuous} makes use of the \textit{Geometric Conservation Law} (GCL) $\frac{\partial J^t}{\partial t} = Div\left( J^t F_{t}^{-1} \mathbf{w}\right)$.
\end{remark}
\begin{remark}
\reva{In the general case with non-homogeneous Dirichlet boundary conditions,
the energy balance also includes flow intensification due to the moving boundary. In such case, the intensification term appearing on the energy balance \eqref{eq:continuous_energy_estimate} in given by:
\begin{equation}
\label{eq:extension_energy_continuous}
\int_{\partial\Omega^0} \rho \frac{\vert \mathbf{u} \vert^2}{2} J^t H^t (\mathbf{u} - \mathbf{w}) \cdot \mathbf{N} \, \text{d}\mathbf{S}
\end{equation}
where $\mathbf{N} \in \mathbb{R}^d$ denotes the outward normal.
}
\end{remark}
\begin{remark}
\reva{Although Dirichlet boundary conditions are used throughout this work, it can be extended straightforwardly to the Neumann case by including the so called \textit{backflow stabilizations}, see e.g. \cite{bertoglio2018benchmark}.}
\end{remark}
\section{Monolithic schemes (first order in time)}
\label{sec:monolithic_schemes}
%Different \CB{time} discretization approaches for Problem \eqref{eq:continuous_formulation} can be taken into account, nevertheless if we fix a first order discretization for the time-derivative term, the schemes used in the literature can be characterized within a single formulation.
Most of the numerical schemes for Problem \eqref{eq:continuous_formulation} reported in the literature are first order and can be written as follows. \reva{Let $(t^n)_{n \in \mathbb{N}}$ be a uniform discretization of the time interval $(0, T)$ with step size $\tau > 0$
and let $H^n := H^{t^n}, J^n := J^{t^n}, w^n := w(t^n)$ be discrete sequences.}
Given a conforming finite element space $\mathbf{V} \times Q$ of $\mathbf{H}^1_0(\Omega^0) \times L^2_0(\Omega^0)$ for velocity and pressure fields, the discrete problem of interest reads:
Find $(\mathbf{u}^{n+1}, p^{n+1}) \in \mathbf{V} \times Q$ s.t.
\begin{equation}
\label{eq:discretized_monolithic_formulation}
\mathcal{A}(\mathbf{u}^{n+1},\mathbf{v}) - \mathcal{B}(\mathbf{v},p^{n+1}) + \mathcal{B}(\mathbf{u}^{n+1},q) = \mathcal{F}(\mathbf{v}) \quad \forall (\mathbf{v}, q) \in \mathbf{V} \times Q
\end{equation}
being
\begin{equation}
\label{eq:lhs_bilinear_form_A}
\begin{aligned}
\mathcal{A} (\mathbf{u}, \mathbf{v}) := & \int_{\Omega^0} \rho \frac{J^{\star\star}}{\tau} \mathbf{u} \cdot \mathbf{v} \, \text{d}\mathbf{X} + \int_{\Omega^0} \rho J^{\star} Grad(\mathbf{u}) H^{\star} (\mathbf{u}^{\ast} - \mathbf{w}^{\ast\ast}) \cdot \mathbf{v} \, \text{d}\mathbf{X} + \int_{\Omega^0} J^{\star} 2\mu \epsilon^{\star}(\mathbf{u}):\epsilon^{\star}(\mathbf{v}) \, \text{d}\mathbf{X} \\
& + \alpha \int_{\Omega^0} \frac{\rho}{2} \left( \frac{J^{n+1} - J^{n}}{\tau} - Div\left( J^{\star} H^{\star} \mathbf{w}^{\ast\ast} \right) \right) \mathbf{u} \cdot \mathbf{v} \, \text{d}\mathbf{X} + \beta \int_{\Omega^0} \frac{\rho}{2} Div\left( J^{\star} H^{\star} \mathbf{u}^{\ast} \right) \mathbf{u} \cdot \mathbf{v} \, \text{d}\mathbf{X}
\end{aligned}
\end{equation}
with $\alpha, \beta \in \{0, 1\}$ given parameters, and
%The matrix $B \in \mathbb{R}^{m^{Q}\times m^{\mathbf{V}}}$ and vector $F_n \in \mathbb{R}^n$ correspond also to the discretization of the bilinear and linear forms:
\begin{equation}
\label{eq:remaining_forms}
\begin{aligned}
\mathcal{B}(\mathbf{u}, q) & := \int_{\Omega^0} Div\left( J^{\star} H^{\star} \mathbf{u} \right) q \, \text{d}\mathbf{X} \quad \forall q \in Q, \quad \mathcal{F}(\mathbf{v}) := \int_{\Omega^0} \rho \frac{J^{\star\star}}{\tau} \mathbf{u}^n \cdot \mathbf{v} \, \text{d}\mathbf{X} \quad \forall \mathbf{v} \in \mathbf{V}
\end{aligned}
\end{equation}
\begin{remark}
The term multiplying $\alpha$ is the discrete residual of GCL, while the one multiplying $\beta$ is a strongly consistent term vanishing for incompressible velocity fields.
\end{remark}
%\begin{equation}
%\label{eq:discretized_monolithic_formulation}
%\begin{aligned}
% \int_{\Omega_0} \rho J^{\star \star} \frac{\mathbf{u}^{n+1} - \mathbf{u}^n}{\tau} \cdot \mathbf{v} + \int_{\Omega_0} \rho J^{\star} Grad(\mathbf{u}^{n+1}) F_{\star}^{-1} (\mathbf{u}^{\ast} - \mathbf{w}^{\ast \ast}) \cdot \mathbf{v} +\int_{\Omega_0} J^{\star} 2\mu \epsilon^{\star} (\mathbf{u}^{n+1}):\epsilon^{\star} (\mathbf{v}) & \\
% - \int_{\Omega_0} Div(J^{\star} F_{\star}^{-1} \mathbf{v}) p^{n+1} + \int_{\Omega_0} Div(J^{\star} F_{\star}^{-1} \mathbf{u}^{n+1}) q & \\
% \color{red}{ + \alpha \int_{\Omega_0} \frac{\rho}{2} \left( \frac{J^{n+1} - J^n}{\tau} - Div\left( J^{\star} F_{\star}^{-1}\mathbf{w}^{\ast\ast} \right) \right) \mathbf{u}^{n+1} \cdot \mathbf{v} + \beta \int_{\Omega_0} \frac{\rho}{2} Div\left( J^{\star} F_{\star}^{-1} \mathbf{u}^{\ast} \right) \mathbf{u}^{n+1} \cdot \mathbf{v} } & = 0
%\end{aligned}
%\end{equation}
%Within \CB{Formulation \eqref{eq:discretized_monolithic_formulation}}, an explicit domain treatment is done by \cite{Basting2017} whom linearize the convective term with $(\star, \star\star, \ast, \ast\ast) = (n, n, k, n)$, where $k$ the Newton's iteration index while \cite{Murea2016} takes $(\star, \star\star, \ast, \ast \ast) = (n, n, n, n)$, both using $\alpha = \beta = 0$;
%similarly \cite{Lozovskiy2018} use an implicit coupling with $(\star, \star\star, \ast, \ast \ast) = (n, n+1, n, n+1)$ whereas \cite{smaldone2014} uses an explicit one with $(\star, \star\star,\ast, \ast\ast) = (n, n+1, n, n)$ where $\alpha = \beta = 1$.
%If the domain treatment is implicit \cite{Langer2016} takes the approach with $(\star, \star\star, \ast, \ast\ast) = (n+1, n+1, n+1, n+1), \, \alpha = \beta = 0$ by applying a Newton solver for the fully nonlinear problem, same formulation is also taken by \cite{LeTallec2001}, \cite{Wall2009} and \cite{Liu2018} in the context of space-dependent densities with different choices of
%$\alpha, \beta$; if the coupling is explicit \cite{Landajuela2016} takes $(\star, \star\star, \ast, \ast\ast) = (n+1, n+1, n, n+1)$ with $\alpha = \beta = 0$ providing a linearization of the convective term, while \cite{Wang2020} utilizes an alternative weak formulation where the domain velocity $\mathbf{w}$ is obtain as solution to a specific weak form with $(\star, \star\star, \ast) = (n+1, n, n+1)$, $\alpha = \beta = 1$.
Formulation \eqref{eq:discretized_monolithic_formulation} contains a wide family of reported methods:
\begin{itemize}
\item Using $\alpha = \beta = 0$: $(\star, \star\star, \ast, \ast\ast) = (n, n, n+1, n)$ is used in \cite{Basting2017}, $(\star, \star\star, \ast, \ast \ast) = (n, n, n, n)$ in \cite{Murea2016} and $(\star, \star\star, \ast, \ast\ast) = (n+1, n+1, n+1, n+1)$ in \cite{Langer2016}, and $(\star, \star\star, \ast, \ast\ast) = (n+1, n+1, n, n+1)$ in \cite{Landajuela2016}.
\item Using $\alpha = \beta = 1$: \reva{$(\star, \star\star, \ast, \ast \ast) = (n+1, n, n, n+1)$} in \cite{Lozovskiy2018}, $(\star, \star\star,\ast, \ast\ast) = (n+1, n, n, n)$ in \cite{smaldone2014} and $(\star, \star\star, \ast, \ast\ast) = (n+1, n, n+1, n+1)$ in \cite{LeTallec2001, Wang2020}.
\end{itemize}
%\subsection{Well posedness at each time step}
%In the following, well-posedness of Problem \eqref{eq:discretized_monolithic_formulation} is assessed using the standard techniques for saddle-point problems. As standard in literature, the problem is studied based on its matrix formulation that reads for each time step:
%\begin{equation}
%\label{eq:matrix_formulation}
%\begin{bmatrix}
%A & -B^T \\
%B & 0
%\end{bmatrix}
%\begin{bmatrix}
%U^{n+1} \\
%P^{n+1}
%\end{bmatrix}
%=\begin{bmatrix}
%F_n \\
%0
%\end{bmatrix},\quad U^{0} = U_{init}
%\end{equation}
%
%where $(U^{n+1}, P^{n+1}) \in \mathbb{R}^{m^{\mathbf{V}}} \times \mathbb{R}^{m^{Q}}$ denote the coefficients associated to the finite element basis with $(m^{\mathbf{V}}, m^{Q})$ the degrees of freedom pair for velocity/pressure elements and $U_{init}$ the coefficients associated the discretization of $\mathbf{u}_{init}$.
%\CB{The matrix $A_n \in \mathbb{R}^{m^{\mathbf{V}}\times m^{\mathbf{V}}}$ corrresponds to the discretization of the bilinear form:} % \rightarrow \mathbb{R}$ is given by
%\begin{equation}
%\label{eq:lhs_bilinear_form_A}
%\begin{aligned}
%\mathcal{A} (\mathbf{u}, \mathbf{v}) = & \int_{\Omega_0} \rho \frac{J^{\star\star}}{\tau} \mathbf{u}^{n+1} \cdot \mathbf{v} + \int_{\Omega_0} \rho J^{\star} Grad(\mathbf{u}^{n+1}) F^{-1}_{\star} (\mathbf{u}^{n} - \mathbf{w}^{\ast\ast}) \cdot \mathbf{v} + \int_{\Omega_0} J^{\star} 2\mu \epsilon^{\star}(\mathbf{u}^{n+1}):\epsilon^{\star}(\mathbf{v}) \\
%& + \alpha \int_{\Omega_0} \frac{\rho}{2} \left( \frac{J^{n+1} - J^{n}}{\tau} - Div\left( J^{\star} F^{-1}_{\star} \mathbf{w}^{\ast\ast}_{h} \right) \right) \mathbf{u}^{n+1} \cdot \mathbf{v} + \beta \int_{\Omega_0} \frac{\rho}{2} Div\left( J^{\star} F^{-1}_{\star} \mathbf{u}^{n} \right) \mathbf{u}^{n+1} \cdot \mathbf{v}
%\end{aligned}
%\end{equation}
%The matrix $B \in \mathbb{R}^{m^{Q}\times m^{\mathbf{V}}}$ and vector $F_n \in \mathbb{R}^n$ correspond also to the discretization of the bilinear and linear forms:
%\begin{equation}
%\label{eq:remaining_forms}
%\begin{aligned}
%B(\mathbf{u}, q) & := \int_{\Omega_0} Div\left( J^{\star} F^{-1}_{\star} \mathbf{u} \right) q \quad \forall q \in Q, \quad F_n(\mathbf{v}) := \int_{\Omega_0} \rho \frac{J^n}{\tau} \mathbf{u}^n \cdot \mathbf{v} \quad \forall \mathbf{v} \in \mathbf{V}
%\end{aligned}
%\end{equation}
\begin{proposition}
\label{prop:monolithic_schemes}
By assuming well-posed, orientation-preserving deformation mappings, i.e. $(J^n)_{n \in \mathbb{N}}$ bounded in $L^{\infty}(\Omega^0)$, $J^n > 0$ for each $n \geq 0$, Problem \eqref{eq:discretized_monolithic_formulation} has unique solution for inf-sup stable finite element spaces if $\left( 2J^{\star\star} + J^{n+1} - J^{n} \right) > 0$ and $\alpha = \beta = 1$.
\end{proposition}
\begin{proof}
%The matrix system \eqref{eq:matrix_formulation} is solvable whenever $A_n^{-1}$ exists and $BA^{-1}_nB^{T}$ is invertible.
Since all operators are bounded and inf-sup stable elements are used for velocity and pressure, it is enough to ensure that the bilinear form $\mathcal{A}$ is coercive.
%Thus its enough to ensure $\text{rank}(B) = m$ and $A_n$ positive definite.
%As standard in literature, let us evaluate \eqref{eq:lhs_bilinear_form_A} in $\mathbf{v} = \mathbf{u}$ and $q = p$. Integrating by parts the convective term and joining expressions, the following equality holds:
Indeed:
\begin{equation}
\label{eq:proof_monolithic_case}
\begin{aligned}
\mathcal{A}(\mathbf{u}, \mathbf{u}) = & \int_{\Omega^0} \frac{J^{\star}}{2\tau} \left( \frac{2 J^{\star\star}}{J_{\star}} + \alpha \frac{J^{n+1} - J^{n}}{J^{\star}} \right) \vert \mathbf{u} \vert^2 + % \int_{\Omega^0}
J^{\star} 2\mu \vert \epsilon^{\star}(\mathbf{u}) \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} \frac{\rho}{2} Div\bigg( J^{\star} H^{\star} \big( (\beta -1) \mathbf{u}^{\ast} - (\alpha-1) \mathbf{w}^{\ast\ast} \big) \bigg) \vert \mathbf{u} \vert^2 \, \text{d}\mathbf{X}
\end{aligned}
\end{equation}
being the last quantity strictly positive under the stated assumptions.
\qed
\end{proof}
\begin{remark}
\reva{
The extension of Proposition \ref{prop:monolithic_schemes} to the case with non-homogeneous Dirichlet boundary conditions follows from the trace theorem by assuming $\Omega^0$ a Lipschitz bounded open set \cite{Ern2004}.
}%under appropriate restrictions on the computation domain, e.g. by assuming $\Omega_0$ a lipschitz bounded open set \cite{Ern2004}.}
\end{remark}
\begin{corollary}
Assuming $\alpha = \beta = 1$, Problem \eqref{eq:discretized_monolithic_formulation} is well posed when:
\begin{itemize}
\item $3J^{n+1} - J^{n} > 0$ if $\star \star = n+1$, i.e. a restriction on the time step size.
\item $J^{n+1} + J^{n} > 0$ if $\star \star = n$, i.e. no restriction on the time step size{\reva{, since we assume orientation preserving deformation mappings.}}
\end{itemize}
No restrictions apply to $\star,\ast, \ast\ast$.
\end{corollary}
%\subsection{Energy balance}
\begin{proposition}
\label{prop:energy_estimate_monolithic}
Under assumptions of Proposition \ref{prop:monolithic_schemes} and $\alpha = \beta = 1, \star\star = n$, the scheme \eqref{eq:discretized_monolithic_formulation} is unconditionally \reva{ energy stable with energy estimate:}
\begin{equation}
\label{eq:energy_estimate_monolithic}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau} \vert \mathbf{u}^{n+1} \vert^2 \, \text{d} \mathbf{X} - \int_{\Omega^0} \rho \frac{J^n}{2\tau} \vert \mathbf{u}^n \vert^2 \, \text{d}\mathbf{X}
= -\int_{\Omega^0} 2\mu J^{\star} \vert \epsilon^{\star}(\mathbf{u}^{n+1}) \vert^2 \, \text{d} \mathbf{X} - \int_{\Omega^0} \frac{\rho}{2\tau} J^{n} \vert \mathbf{u}^{n+1} - \mathbf{u}^{n} \vert^2 \, \text{d} \mathbf{X}.
\end{equation}
\end{proposition}
\begin{proof}
By setting $\mathbf{v} = \mathbf{u}^{n+1}$ in the bi-linear form \eqref{eq:lhs_bilinear_form_A}, $q = p^{n+1}$ in forms \eqref{eq:remaining_forms} and manipulating terms as standard in literature, the energy equality follows:
\begin{equation}
\begin{aligned}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau} \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \rho \frac{J^{n}}{2\tau} \vert \mathbf{u}^n \vert^2 \, \text{d}\mathbf{X} = & \int_{\Omega^0} \frac{\rho}{2\tau} (J^{n+1} - J^{\star\star}) \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X} +
\int_{\Omega^0} \frac{\rho}{2\tau} (J^{\star\star} - J^n) \vert \mathbf{u}^n \vert^2 \, \text{d}\mathbf{X} & \\
& - \int_{\Omega^0} 2\mu J^{\star} \vert \epsilon^{\star} (\mathbf{u}^{n+1}) \vert^2 \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2\tau} J^{\star\star} \vert \mathbf{u}^{n+1} - \mathbf{u}^{n} \vert^2 \, \text{d}\mathbf{X} + \int_{\Omega^0} \frac{\rho}{2} Div(J^{\star} H^{\star} (\mathbf{u}^{\ast} - \mathbf{w}^{\ast\ast})) \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2} \alpha \frac{J^{n+1} - J^{n}}{\tau} \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} \frac{\rho}{2} Div\left( J^{\star} H^{\star} (\beta \mathbf{u}^{\ast} - \alpha \mathbf{w}^{\ast\ast}) \right) \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X}
\end{aligned}
\end{equation}
Thus, for $\alpha=\beta=1$ and $\star\star = n$ the result follows.
\qed
\end{proof}
\begin{remark}
\revb{This works focuses on first-order schemes in time. The reason is that second order schemes, although stable in fixed domain, has been shown to be only conditionally stable in ALE form, as it was shown in \cite{Formaggia2004} for the advection-diffusion problem for Crank-Nicolson (CN) and BDF(2). Therefore, we do not analyze here the schemes used in \cite{ failer2020impact, Hessenthaler2017,Tallec2003} -- based on CN and used in the context of fluid-solid interaction -- since their analysis repeats from \cite{Formaggia2004}. Also in the same context, some authors have used the generalized $\alpha$-methods since it is a popular scheme for elastodynamics \cite{Liu2018}. However, there is no reported stability analysis even for the the fixed domain setting, and its stability properties are usually assumed to be transferred from the linear setting.}% for such cases even in the fixed domain setting.}
%Just to mention a few, second order schemes based on the Generalized $\alpha$-method as in \cite{Liu2018} and Crank-Nicolson (CN) approaches e.g. in \cite{Tallec2003, failer2020impact, Hessenthaler2017} have been proposed.
%\reva{However, for the Generalized $\alpha$-method, further research on its unconditional energy stability for iNSE is required and as obtained in \cite{Formaggia2004} for the advection-diffusion problem, both CN and BDF(2) are only conditionally stable, depending on the domain velocity. % and thus out of the scope of this work.
%Obtaining a higher order in time, unconditionally energy stable scheme for the advection-diffusion equations in ALE form (what includes the iNSE) still remains an open problem.}
% and more general surveys in \cite{Wall2009}.
% Notice that, whenever we take $\star \star = n+1$ the energy equality associated to the problem cannot be easily stabilized with consistent terms.
%Just to mention a few, second order schemes based on the Generalized $\alpha$-method as in \cite{Liu2018} and Crank-Nicolson (CN) approaches e.g. in \cite{Tallec2003, failer2020impact, Hessenthaler2017} have been proposed.
%\reva{However, for the Generalized $\alpha$-method, further research on its unconditional energy stability for iNSE is required and as obtained in \cite{Formaggia2004} for the advection-diffusion problem, both CN and BDF(2) are only conditionally stable, depending on the domain velocity. % and thus out of the scope of this work.
%Obtaining a higher order in time, unconditionally energy stable scheme for the advection-diffusion equations in ALE form (what includes the iNSE) still remains an open problem.}
%% and more general surveys in \cite{Wall2009}.
%% Notice that, whenever we take $\star \star = n+1$ the energy equality associated to the problem cannot be easily stabilized with consistent terms.
\end{remark}
%\begin{remark}
%In the general case where Dirichlet and Neumann boundary conditions appear, $\Gamma_N \cup \Gamma_D = \partial \Omega_0$ with $\Gamma_N \neq \emptyset$, the same result also holds true with the addition of backflow stabilization term.
%\end{remark}
\section{Chorin-Temam schemes}
\label{sec:chorin_temam_schemes}
In the following, we describe a family of Chorin-Temam (CT) schemes for the iNSE-ALE problem, as we did for the monolithic case. %Such description keeps the freedom of choice for certain coefficients, which must be restricted to ensure unconditional stability.
Given $\mathbf{\widetilde V} $ a conforming space of $\mathbf{H}^1_0 (\Omega^0)$ and $ \widetilde Q$ a conforming space of $ L^2_0 (\Omega^0) \cap H^1(\Omega^0)$, $\tilde{\mathbf{u}}^{0} \in \mathbf{\widetilde V}$, for $n\geq0$:% the proposed two-step CT schemes reads
\begin{enumerate}[1.]
\item \textbf{Pressure-Projection Step $(\text{PPS})_{n}$}
%Seeks a pressure term allowing the projection of $\tilde{\mathbf{u}}^{n+1}$ in the space of divergence-free solutions, in the form:
Find $p^{n} \in \widetilde{Q}$ s.t.
\begin{equation}
\label{eq:chorin_temam_pps}
\int_{\Omega^0} \frac{\tau}{\rho} J^{\circ} Grad(p^{n}) H^{\circ} : Grad(q) H^{\circ} \, \text{d}\mathbf{X} = - \int_{\Omega^0} Div\left( J^{\circ} H^{\circ} \tilde{\mathbf{u}}^{n}\right) q \, \text{d}\mathbf{X} \quad \forall q \in \widetilde{Q}
\end{equation}
\item \textbf{Fluid-Viscous Step $(\text{FVS})_{n+1}$} %Seeks a tentative velocity field $\tilde{\mathbf{u}}^{n+1}$ with pressure given explicitly:
Find $\tilde{\mathbf{u}}^{n+1} \in \mathbf{\widetilde{V}}$ s.t.
\begin{equation}
\label{eq:chorin_temam_fvs}
\begin{aligned}
\int_{\Omega^0} \rho J^{\star\star} \frac{\tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^n}{\tau} \cdot \mathbf{v} \, \text{d}\mathbf{X}
+ \int_{\Omega^0} \rho J^{\star} Grad(\tilde{\mathbf{u}}^{n+1}) H^{\star} (\tilde{\mathbf{u}}^{n} - \mathbf{w}^{\ast\ast}) \cdot \mathbf{v} \, \text{d}\mathbf{X}
+ \int_{\Omega^0} J^{\star} 2 \mu \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) : \epsilon^{\star} (\mathbf{v}) \, \text{d}\mathbf{X} & \\
- \int_{\Omega^0} Div(J^{\circ \circ} H^{\circ \circ} \mathbf{v}) p^n \, \text{d}\mathbf{X}
+ \int_{\Omega^0} \frac{\rho}{2} \frac{J^{n+1} - J^{n}}{\tau} \tilde{\mathbf{u}}^{n+1} \cdot \mathbf{v} \, \text{d}\mathbf{X}
+ \int_{\Omega^0} \frac{\rho}{2} Div\left( J^{\star} H^{\star}(\tilde{\mathbf{u}}^{n} - \mathbf{w}^{\ast\ast})\right) \tilde{\mathbf{u}}^{n+1} \cdot \mathbf{v} \, \text{d}\mathbf{X} & = 0 \quad \forall \mathbf{v} \in \mathbf{\widetilde{V}}
\end{aligned}
\end{equation}
% In particular, the corrected velocity is obtained through the update:
% \begin{equation}
% \begin{aligned}
% \mathbf{u}^{n+1} & = \tilde{\mathbf{u}}^{n+1} - \frac{\tau}{\rho} Grad(p^{n+1}) F^{-1}_{\circ + 1} \text{ in } \Omega_0
% \end{aligned}
% \end{equation}
\end{enumerate}
%\begin{remark}
%The CT scheme iNSE above, uses a linearization of the convective term $\ast = n$, easily extended to the case $\ast = n+1$.. In particular, the stabilization terms are active $\alpha = \beta = 1$, with freedom in the choice of $\circ\circ$. The objective in the following will be to show which values achieve unconditional stability in our formulation.
%\end{remark}
%\subsection{Time Stability}
The following energy estimate can be obtained under suitable conditions:
\begin{proposition}
\label{prop:energy_estimate_chorin_temam}
Under assumptions $\circ = \circ\circ = \star \star = n$, the solution to scheme \eqref{eq:chorin_temam_pps}-\eqref{eq:chorin_temam_fvs} is unconditionally stable, i.e.
%is unconditionally energy stable if $\circ = \circ \circ = \star\star = n$, without condition over $\star$. Moreover, the energy estimate is given by
\reva{
\begin{equation}
\begin{aligned}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau} \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \rho \frac{J^{n}}{2\tau} \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d} \mathbf{X} \leq & - \int_{\Omega^0} J^{\star} 2\mu \vert \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} J^{n} \frac{\tau}{2\rho} \vert Grad(p^n) H^{n} \vert^2 \, \text{d}\mathbf{X} .
\end{aligned}
\end{equation}}
\end{proposition}
\begin{proof}
As standard in literature, let us take $\mathbf{v} = \tilde{\mathbf{u}}^{n+1}$ in $(\text{FVS})_{n+1}$, and $q = p^n$ in $(\text{PPS})_{n}$. Adding both equalities and rewriting expressions, it follows:
\begin{equation}
\begin{aligned}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau} \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \rho \frac{J^{n}}{2\tau} \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} = & \int_{\Omega^0} \frac{\rho}{2\tau}(J^{n+1} - J^{\star\star}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X}
+ \int_{\Omega^0} \frac{\rho}{2\tau}(J^{\star\star} - J^{n}) \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2\tau} J^{\star\star} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} J^{\star} 2\mu \vert \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} Div\left( J^{\circ\circ} H^{\circ\circ} (\tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n}) \right) p^n \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} Div\left( (J^{\circ\circ} H^{\circ\circ} - J^{\circ} H^{\circ}) \tilde{\mathbf{u}}^n \right) p^n \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\tau}{\rho} J^{\circ} \vert (H^{\circ})^T Grad(p^n) \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \frac{\rho}{2\tau} (J^{n+1} - J^{n}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X}
\end{aligned}
\end{equation}
Bounding the first divergence term using integration by parts and \reva{ Cauchy-Schwarz } inequality, it follows
\begin{equation}
\begin{aligned}
\int_{\Omega^0} Div\left( J^{\circ\circ} H^{\circ\circ} (\tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n}) \right) p^n \, \text{d}\mathbf{X} & \leq \int_{\Omega^0} \frac{\rho}{2\tau} J^{\circ\circ} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} + \int_{\Omega^0} \frac{\tau}{2\rho} J^{\circ\circ} \vert (H^{\circ\circ})^T Grad(p^n) \vert^2 \, \text{d}\mathbf{x}
\end{aligned}
\end{equation}
Thus, the following energy estimate can be obtained:
\begin{equation}
\label{eq:energy_estimate_proof}
\begin{aligned}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau} \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \rho \frac{J^{n}}{2\tau} \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} \leq & \int_{\Omega^0} \frac{\rho}{2\tau} (J^{n+1} - J^{\star\star}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X}
+ \int_{\Omega^0} \frac{\rho}{2\tau} (J^{\star\star} - J^{n}) \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2\tau} J^{\star\star} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} J^{\star} 2\mu \vert \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} \frac{\rho}{2\tau} J^{\circ\circ} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} + \int_{\Omega^0} \frac{\tau}{2\rho} J^{\circ\circ} \vert (H^{\circ\circ})^{T} Grad(p^n) \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} Div\left( (J^{\circ\circ} H^{\circ\circ} - J^{\circ} H^{\circ}) \tilde{\mathbf{u}}^{n}\right) p^n \, \text{d}\mathbf{X} - \int_{\Omega^0} \frac{\tau}{\rho} J^{\circ} \vert (H^{\circ})^{T} Grad(p^n) \vert^2 \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2\tau} (J^{n+1} - J^{n}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X}
\end{aligned}
\end{equation}
From estimate \eqref{eq:energy_estimate_proof} it follows that whenever $\circ = \circ \circ = \star \star = n$ unconditional energy stability is attained, where $\star$ remains free of choice.
\qed
\end{proof}
%\begin{remark}
%In the general case where $\Gamma_N \cup \Gamma_D = \partial \Omega_0$ with $\Gamma_N \neq \emptyset$ a similar bound is attained with the addition of backflow stabilization term, where the same procedure as above holds for all terms except the one on $\Gamma_N$.
%\end{remark}
\section{Numerical examples}
\label{sec:numerical_examples}
We consider a rectangular domain with opposite vertices $\{ (0, -1), (6, 1) \}$ where the iNSE-ALE formulation \eqref{eq:continuous_formulation} will be simulated over the interval $(0,\, 2) \, [s]$ with non-zero initial condition of the form \revb{ $ \mathbf{u}(0) := \big(\gamma (1 - \mathbf{X}_1^2) \mathbf{X}_0 (6 - \mathbf{X}_0), 0\big) ,\, \gamma = 0.001$}.
The domain is deformed using $\mathcal{X}(\mathbf{X}, t) := \big( (1 + 0.9 sin(8 \pi t)) \mathbf{X}_0,\, \mathbf{X}_1 \big)$. %, i.e. an oscillation with initial expansion and frequency of $4 \pi$.
Discretization setup for Formulation \eqref{eq:discretized_monolithic_formulation} and \eqref{eq:chorin_temam_pps}-\eqref{eq:chorin_temam_fvs} is done choosing a time step $\tau = 0.01$ and space triangulation with elements diameter $h \approx 0.01 $, implemented through FEniCS \cite{FEniCS2015} using Python for interface and postprocessing.
To exemplify the theoretical results from previous sections, four schemes are taken into account. Monolithic (M) Formulation \eqref{eq:discretized_monolithic_formulation} is taken with linearized convective term and implicit treatment, i.e., $(\star, \ast, \ast\ast) = (n+1, n, n+1)$ where for $\star\star$ we consider two choices, denoted $\text{M}\, \star\star = n$ and $\text{M}\, \star\star = n+1$.
For both cases the space discretization is carried out with $ \mathbf{V}/Q = [\mathbb{P}_2]^d/\mathbb{P}_1$ Lagrange finite elements. Similarly, Chorin-Temam (CT) scheme \eqref{eq:chorin_temam_fvs}-\eqref{eq:chorin_temam_pps} is taken with linearized convective term and implicit treatment, i.e. $(\star, \ast\ast, \circ, \circ\circ) = (n+1, n+1, n, n)$ with $\star\star$ as before, denoting each scheme by $\text{CT}\, \star\star = n$ and $\text{CT}\, \star\star = n+1$ with space discretization done through $\mathbf{\widetilde{V}}/\widetilde{Q} = [\mathbb{P}_1]^d/\mathbb{P}_1$ elements.
\reva{In all cases homogeneous (equal to $\mathbf{0}$) boundary conditions are imposed for the velocity, zero-mean on the pressure and $\alpha=\beta=1$}.
The results are assessed using time-dependent normalized parameters $ \hat{\delta}_{\text{M}}:= \delta_{\text{M}}/E_{st}^{\star}, \hat{\delta}_{\text{CT}}:= \delta_{\text{CT}}/E_{st}^{\star}$ defined as: %, where $E_{st}^{\star}$ denote the \CB{\mod{strain energy}{viscous dissipation}} and $\delta_{M}, \delta_{CT}$ residual errors defined as:
\begin{equation}
\label{eq:energy_error}
\begin{aligned}
\delta_{M}^{n+1} &:= D^{n+1} + E_{st}^{\star} + \int_{\Omega_0} \frac{\rho J^{\star\star}}{2\tau} \vert \mathbf{u}^{n+1} - \mathbf{u}^{n} \vert^2 \, \text{d}\mathbf{X} , \quad \delta_{CT}^{n+1} := D^{n+1} + E^{\star}_{st} + \int_{\Omega_0} \frac{\tau J^{\circ}}{2 \rho} \vert (H^{\circ})^T Grad(p^n) \vert^2 \, \text{d}\mathbf{X} \\
D^{n+1} &:= \int_{\Omega_0} \frac{\rho}{2\tau} \left( J^{n+1} \vert \mathbf{u}^{n+1} \vert^2 - J^n \vert \mathbf{u}^n \vert^2 \right) \, \text{d}\mathbf{X} , \quad E^{\star}_{st} = \int_{\Omega_0} 2 \mu J^{\star} \vert \epsilon^{\star} (\mathbf{u}^{n+1}) \vert^2 \, \text{d}\mathbf{X}.
\end{aligned}
\end{equation}
Figure \ref{fig:delta_hat_oscs} shows $\hat{\delta}_{\text{M}}, \hat{\delta}_{\text{CT}}$ values for each tested scheme. Propositions \ref{prop:energy_estimate_monolithic} and \ref{prop:energy_estimate_chorin_temam} are confirmed since $\hat{\delta}_{\text{M}}=0$ and $\hat{\delta}_{\text{CT}} \leq 0$ if $\star\star = n$. For $\star \star = n+1$, peaks appearing throughout the simulation are defined by the sign change of domain velocity, i.e. in the change from expansion to contraction.
Importantly, the spurious numerical energy rate related to discretization of the GCL condition appear to be positive in expansion, therefore being a potential source of numerical instabilities.
%Moreover the expected energy decay from Propositions \ref{prop:energy_estimate_monolithic}, \ref{prop:energy_estimate_chorin_temam} for the choice $\star \star = n$ in $\text{M, CT}$ schemes is obtained.
%\CB{Recall that $\hat{\delta}_{\text{M}}, \hat{\delta}_{\text{CT}}$ in \eqref{eq:energy_error} are taken in such form since on scheme $\text{M}\, \star\star = n$ it follows $\hat{\delta}_{MT} = 0$ and similarly on $\text{CT}\, \star\star = n$ it follows $\hat{\delta}_{\text{CT}} \sim 0$, defining the base case.}
\begin{figure}[!hbtp]
\centering
\includegraphics[width=\textwidth]{figs/Comparison_Delta_Hat_Value_GCL_True_solver_LU.png}
\caption{Summary of the numerical experiment in terms of energy balance. Left: Monolithic residual error values $\hat{\delta}_{\text{M}}$;
Right: Chorin-Temam residual error values $\hat{\delta}_{\text{CT}}$. %Both cases are simulated with deformation mapping $\mathcal{A}$ defined previously and shown on the interval $[0, 1] \, [s]$.}
}
\label{fig:delta_hat_oscs}
\end{figure}
\section{Conclusion}
\reva{Several} reported time discretization schemes for the iNSE-ALE have been reviewed \reva{ and analyzed in terms of their well posedness at each time step and time stability. The stability analysis is confirmed by numerical experiments}. For the monolithic case, two schemes lead to well-posed energy-stable problems whenever $\alpha=\beta=1$ with $\star \star = n$ as studied in \cite{LeTallec2001, Lozovskiy2018, smaldone2014, Wang2020}.
To the best of the authors knowledge, the unconditionally stable Chorin-Temam scheme derived in this work has not been reported yet. %, and moreover the numerical experiments studied here validate the energy stable propositions.
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% \newcommand{\tsum}{{\textstyle\sum\limits}} % for small sums in large equations
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% \newcommand{\grad}{\nabla} % for the gradient
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% \newcommand{\abs}[1]{| #1 |} % \abs{x} for the absolute value of x
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\begin{document}
\parindent = 0pt
%
\begin{center}
{\bf Response to the reviewer's comments } \\ (\manuscript)
\end{center}
Dear Professor Tucker,\\
We thank you and the reviewers for the comments and suggestions to improve our manuscript. In the next pages, we replied to each of the comments and included modifications in a new version accordingly.
We hope that after our response the manuscript can be accepted for publication in \journal. \\
%, in particular for the questions. \\% We will address the comment in the following lines.\\
%In response to the first reviewer, indeed updates on the pressure space definition and notation have been done to correctly defined our problem
%in the continuum setting. Also comments on the space-time regularity and restrictions for the
%computational domain were added, e.g. in Remark 6. \\
%
%Our analysis is done throughtout the article using homogeneous Dirichlet boundary conditions and thus our derivations do not contain the mechanical work as expected.
%We assume such case for matter of convenience since its extension follows by using standard techniques found in literature. Nevertheless, comments
%about flow intensification were added in Remark 2. \\
%
%Finally, we point out the inaccurate choice of words for our conclusion, which in this new version was updated. \\
%
%In response to the second reviewer, regarding the concern about limitations of first order time discretizations, we studied analogously to the work [1] second order
%approaches such as Crank-Nicolson and BDF(2) schemes for the monolithic iNSE-ALE problem. To our knowledge, only conditionally energy stable discrete formulations can be
%obtained with the usual techniques and extensions to pressure correction schemes still lack in unconditional energy stable results. \\
%
%Further, the question of unconditionally energy stable higher order iNSE-ALE formulation still remains open and further research must be done. \\
%We hope that after our response the manuscript can be accepted for publication in \journal. \\
Sincerely yours \\
Reidmen Ar\'{o}stica and Crist\'obal Bertoglio
%\end{document}
%%%% NEW REVIEWER
\setcounter{countResp}{0}
\newpage
%\setcounter{page}{1}
\begin{center}
{\bf Detailed reply to Referee \#1} \\ (\manuscript)
\end{center}
We thank the reviewer for her/his suggestions and comments and we are happy to read the positive comments about our work. Below, we reply in detail to all the raised issues.
For the sake of convenience, all changes in the revised manuscript version are marked in \textcolor{orange}{orange}.
\ReviewComments{p.1 If a function $p$ belongs to the functional space $L^2$, it is not possibble to fix its value at a point. Pressure should be fixed by zero-ing its mean value.}
{
Thanks for pointing this to us, we corrected it in the new version.
}
\ReviewComments{p.2 On Remark 1: if you address spatial discretizations here, you should mention the problem of convective stabilization.
For heart chambers, for instance, this is the crucial issue. If you address non-zero boundary conditions and refer to the trace theorem, you should mention
the restrictions of the computational mesh imposed by the trace theorem in finite element spaces.}
{
We thank the reviewer for the comments regarding the restrictions on the computational mesh. Remark 6 has been updated including them.
}
\ReviewComments{p.2 I do not understand Proposition 1 for a domain with moving boundary: it states that the change of the kinetic energy is spent to viscous dissipation.
However, the motion of the boundary performs a mechanical work, and the energy balance should account flow intensification due to the moving boundary, see, e.g. [12].}
{
We fully agree with the reviewer's comment. In our case, the mechanical work of the boundary does not appear provided the homogeneous Dirichlet boundary conditions originally imposed to the problem.
In this version, we added a comment regarding such fact in Remark 2.
}
\ReviewComments{p.2 after Remark 3: check carefully classification of all known methods. I suspect that it contains errors. At least, the paper [12] has another set of indices n+1, n, n, n+1
and this is important since it is proved to be unconditionally stable whereas Corollary 3 states that it has restrictions on the time step.}
{
We thank the reviewer for this comment. We noticed a typo in the discrete formulation (5) which could have induced misundestanding throughout the article.
We reviewed carefully the discrete formulation in [12], where $k$ there corresponds to $n+1$ in our manuscript.
Therefore, we updated the indices to (n+1, n, n, n+1) to match the proposed scheme in [12].
}
\ReviewComments{p.4 line -10: moving domain formulation is weird: $X_0 = 0$ is mapped to 1, $X_0$ = 6 is mapped to 1+5.4sin(8$\pi$t), i.e. spans the interval [-4.4, 6.4], which means degeneration
and tangling of the physical domain.}
{
The reviewer is right that $X_0 = 0$ is mapped to 1, nevertheless $X_0 = 6$ is mapped to $6 (1 + 0.9 sin(8 \pi t))$, spanning the interval $[0.6, 11.4]$ thus no mesh degeneration.
}
\ReviewComments{p.4 Fig 1: these curves imply that you lost boundary flow intensification term in the energy balance.}
{
We agree that this would be indeed the general case. However, since homogeneous Dirichlet boundary conditions are used in the numerical experiment, the energy balance does not include intensification terms.
}
\ReviewComments{p.4, first line of Conclusion: the authors are too ambitious to claim that they theoretically analyzed time discretization schemes.
Actually, they suggest hints to a future stability analysis.}
{
We agree that our choice of words was inaccurate. This has been updated.
}
\ReviewComments{General: in mathematical journals, integrals are completed by dx, dt, ds, etc.}
{
Thanks for this remark. It has been updated.
}
\ReviewComments{p1, line 4 of section 2: $J^t$; p2. (4): $w^{**}$ no subscript $_h$ here!; p.4. Boundary conditions are not set in section 5.; p.5 Fig. 1: use $M \star\star$ rather than M**.
p.1. Jacobian, is the standard, p.4. Schwarz, Monolithic.}
{
We acknowledge the comments regarding figure styling and typos. They have been updated in this version.
}
\pagebreak
%%%%%%%%%%%%%%%% NEXT REVIEWER
\newpage
\setcounter{countResp}{0}
%\setcounter{page}{1}
\begin{center}
{\bf Detailed reply to Referee \#2} \\ (\manuscript)
\end{center}
We thank the reviewer for her/his positive impression of the manuscript and for his/her suggestions and comments. Below, we reply in detail to all the raised issues.
For the sake of convenience, all changes in the revised manuscript version are marked in \textcolor{red}{red}.
%\vspace{-.5cm}
\ReviewComments{My main concern however is the limitation to first order method. In applications I do not think that first order schemes are of much interest.
At least in FSI, mostly higher BFD schemes [1] (also includin a stability analysis), the fractional step theta scheme [2] or variants of the Crank-Nicolson method [3]
are applied since they all give second order. Is it possible to include at least one of these schemes in the analysis?}
{
We thank the reviewer for the comment. Analogous to [1] second-order monolithic schemes for the iNSE-ALE problem can be proposed, e.g. using Crank-Nicolson and BDF(2) approaches. However, as obtained in [1] for the advection-diffusion problem, both CN and BDF(2) are only conditionally stable, depending on the domain velocity. % and thus out of the scope of this work.
We believe that obtaining a higher order in time, unconditionally energy stable scheme for the advection-diffusion equations in ALE form (what includes the iNSE) still remains an open problem.
We have now extended Remark 6 in order to explain so and refer to [1] for more details.
}
\ReviewComments{Also, there is much research on pressure correction schemes of higher order, (I guess not for ALE), but it would be worth to extend these
ideas to ALE formulations.}
{
We agree with the reviewer that there is a lack of higher order fractional-step schemes for the iNSE problem in ALE formalism. In this article, we provide
a Chorin-Temam approach using our finding for the first order monolithic schemes since unconditional energy stability can be archieved.
As mentioned in the previous comment, already with second order schemes, e.g. Crank-Nicolson and BDF(2), unconditional energy stability is an open problem for the monolithic case,
thus its extension to pressure correction schemes can only be expected with conditional energy stability if current techniques are used, a topic beyond the scope of this paper.
}
\ReviewComments{Notation: the index 't' is sometimes a subscript, sometimes a superscript, e.g. in Jt in line 4 of Section 2, but $J^t$ in (1). Maybe this could be unified?}
{
Thanks for pointing that inconsistency out, we acknowledge the lack of unification in the notation. We have updated the notation to consider all time indexes as superscripts.
}
\ReviewComments{Problem description: Do you not require any temporal regularity of the domain velocity w(t)?}
{
We fully agree with the reviewer, a temporal regularity assumption for the domain velocity was lacking. We have updated the regularity of the deformation mapping $\mathcal{X}$ to be at least $\mathcal{C}^1$ is space and time,
thus for $\frac{\partial \mathcal{X}}{\partial t} := \mathbf{w}$ to be $\mathcal{C}^1$ in space and $\mathcal{C}^0$ in time.
}
\ReviewComments{I think the assumption of an initial pressure p(0)=0 might cause problems. Is this compatible with the configuration that you use in your test case, i.e.
does it fit to the initial velocity and the right hand side, the Dirichlet data? How do you check that?}
{
Indeed it was a typo. It was removed and updated to the correct
pressure space of $L^2(\Omega^0)$ functions with zero-mean.
}
\ReviewComments{The notation based on stars is difficult to undestand. I see that it is very general and has advantages. But, if you refer to specific shcemes, I could not undestand
what exactly is meant without wrtiting it down. Maybe you can give details on the most relevant schemes that you are considering.}
{
We thank the reviewer for the comment and agree with it. Nevertheless it provides advantages when describing a wide range of schemes.
In particular, the items after Remark 4 where we adress specific choices of indices $(\star, \star\star, \ast, \ast\ast)$ are based on relevant schemes applied in different scenarios.
}
\ReviewComments{Proposition 4. What exactly is the role of (7). You derive it by testing with proper functions, thats fine. And I see that many terms cancel for the specific choice of alpha, beta
and star-star. But where does it go to?}
{
Thanks for pointing that out. We notice the lack of a detailed description which has been corrected in this version of the article. (7) was intended to describe the unconditional energy stability obtained
for the monolithic case whenever suitable conditions are applied (in this case: $\alpha = \beta = 1, \star\star = n$). Now, Proposition 4 includes the specific bound obtained from the analysis.
}
\ReviewComments{Numerical examples: Is u(0) a scalar? I guess, 2nd component is zero? Does this initial value fit to a zero pressure at time t=0?}
{
We thank the reviewer for the question. It was a typo in the definition of $u(0)$ that has been updated to the correct expression.
As guessed by the reviewer, the 2nd component indeed is 0.\\
As clarified above, there is no initial condition for the pressure since it was a typo.
%For the monolithic scheme, only the initial velocity is required for the time iteration whereas for the Chorin-Temam scheme, the initial pressure is computed solving $(FVS)_{0}$.
%Thus, no direct computation of $p_0$ is required.
}
\ReviewComments{Reference [1] has some html code.}
{
Thanks for pointing that out. It has been corrected.
}
\ReviewComments{Reference [9] has been published in 2018 in Mathematics and Computer in Simulation.}
{
Thanks, it has been updated.
}
\vspace{1cm}
{
[1] L. Formaggia, F. Nobile. Stability analysis of second-order time
accurate schemes for ALE-FEM, Comp. Meth. Appl. Math. Engrg. 193, 2004
[2] M. Razzaq, H. Damanik, J. Hron, A. Ouazzi, S. Turek, FEM multigrid
techniques for fluid-structure interaction with application to
hemodynamics, Applied Numerical Mathematics, 2012
[3] T. Richter and T. Wick. In T. Carraro and M. Geiger and S. Körkel
and R. Rannacher (Eds.), On time discretizations of fluid-structure
interactions, Multiple Shooting and Time Domain Decomposition
Methods. Springer, 2015
}
\end{document}

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title={An energy stable one-field monolithic arbitrary Lagrangian-Eulerian formulation for fluid-structure interaction},
author={Yongxing Wang and Peter K. Jimack and Mark A. Walkley and Olivier Pironneau},
year={2020},
eprint={2003.03819},
archivePrefix={arXiv},
primaryClass={cs.CE}
}

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\title{On monolithic and Chorin-Temam schemes for incompressible flows in moving domains}
\author{
Reidmen Ar\'{o}stica, Crist\'{o}bal Bertoglio \\
Bernoulli Institute, University of Groningen,
9747AG Groningen, The Netherlands
}
\date{\today}
\begin{document}
\maketitle
\begin{abstract}
Several time discretization schemes for the incompressible Navier-Stokes equations (iNSE) in moving domains have been proposed.
Here we introduce them in a unified fashion, allowing a common well possedness and time stability analysis. It can be therefore shown that only a particular choice of the numerical scheme ensures such properties. The analysis is performed for monolithic and Chorin-Temam schemes. Results are supported by numerical experiments.
\end{abstract}
\section{Introduction}
%When working with flows from a simulation point of view, several schemes are proposed depending on its applications, suitable for specific requirement e.g. high spatio-temporal resolution, fidelity, stability or fast implementation/simulation times.
%The literature is vast, but to our knowledge there is not a overview trying to summarize their approaches in a single scheme.
Several works have been reported dealing with the numerical solution of the iNSE in moving domains within an Arbitrary Lagrangian Eulerian formulation (ALE), primarily in the context of fluid-solid coupling.
In particular different choices of time discretization have been reported on \cite{Basting2017, Murea2016, Landajuela2016,Lozovskiy2018,smaldone2014,langer2014numerical,LeTallec2001,Liu2018,failer2020impact,Hessenthaler2017}.
% In \cite{Basting2017, Murea2016, Landajuela2016} the FSI within Arbitrary Lagrangian-Eulerian (ALE) formalism in a linearized approach is studied, and \cite{Lozovskiy2018,smaldone2014} also include mass conservation terms. \cite{langer2014numerical,LeTallec2001,Liu2018} propose fully non-linear FSI schemes, and extensions using Crank-Nicholson are used by \cite{failer2020impact,Hessenthaler2017}. \CB{No entiendo la clasificacion de los papers, me parece que le falta un poco de coherencia. No deberian tambien haber mas papers, o estos son todos? }
%Thougthful
To the best of the authors knowledge, only a few monolithic schemes have been thoroughly analyzed, e.g. in \cite{Lozovskiy2018, Burtschell2017, LeTallec2001, smaldone2014}, while no analysis has been reported for Chorin-Temam (CT) methods.
The goal of this work is therefore to assess well-posedness and unconditional energy balance of the iNSE-ALE for all reported monolithic and CT discretization schemes within a single formulation.
% Maybe we need to add the works of \cite{Boffi2004} for general surveys of ALE schemes.
%State-of-the art: monolithic (what is proven), CT (not proven, just "used") \\
%, which as it will be seen later on, holds under some restrictions with help of consistent stabilization terms.
%The findings for the monolithic case are then used to introduce a Chorin-Temam scheme, for which unconditional energy stability holds.
The reminder of this paper is structured as follows:
Section \ref{sec:continuous_problem} provides the continuous problem that will be studied. Section \ref{sec:monolithic_schemes} introduces a general monolithic scheme that characterizes several approaches used in literature, well-posedness and energy stability are studied and discussed. Section \ref{sec:chorin_temam_schemes} introduces the Chorin-Temam schemes where time stability is analyzed. Finally, Section \ref{sec:numerical_examples} provides numerical examples testing our results.
\section{The continuous problem}
\label{sec:continuous_problem}
In the following, let us consider a domain $\Omega^0 \subset \mathbb{R}^d$ with $d = 2,3$ and a deformation mapping $\mathcal{X}: \mathbb{R}^d\times \mathbb{R}_{+}\mapsto\mathbb{R}^d$ that defines the time evolving domain $\Omega^t := \mathcal{X}(\Omega^0, t)$.
We assume $\mathcal{X}$ a $\mathcal{C}^1$ mapping in both coordinates, 1-to-1 with $\mathcal{C}^1$ inverse. We denote $\mathbf{X} \in \mathbb{R}^d$ the cartesian coordinate system in $\Omega^0$ and $\mathbf{x}^t := \mathcal{X}(\mathbf{X}, t)$ the one in $\Omega^t$, by $F^t := \frac{\partial \mathbf{x}^t}{\partial \mathbf{X}}$ the deformation gradient, $H^t := (F^t)^{-1}$ its inverse and $J^t := det(F^t)$ its Jacobian.
Similarly, $Grad(\mathbf{\mathbf{f}}) := \frac{\partial \mathbf{f}}{\partial \mathbf{X}}$, $Div(\mathbf{f}) := \frac{\partial}{\partial \mathbf{X}} \cdot \mathbf{f}$ denote the gradient and divergence operators respectively and $\epsilon^t(\mathbf{f}) := \frac{1}{2}( Grad(\mathbf{f}) H^t + (H^t)^{T} Grad(\mathbf{f})^{T})$ the symmetric gradient, for $\mathbf{f}$ a well-defined vector function.
By $\mathbf{H}^1_0 (\Omega^0)$ we denote the standard Sobolev space of vector fields $\mathbf{u}$ defined in $\Omega^0$ with values in $\mathbb{R}^d$ such that $\mathbf{u} = \mathbf{0}$ on $\partial \Omega^0$, by $L^2_0(\Omega^0)$ the standard square integrable space of functions $r$ defined in $\Omega^0$ with values in $\mathbb{R}$ s.t. $\int_{\Omega^0} r \, \text{d}\mathbf{X} = 0$ and $T > 0$ a final time.
We consider the weak form of the iNSE in ALE form \cite[Ch. 5]{Richter2017}: Find $(\mathbf{u}(t), p(t)) \in \mathbf{H}^{1}_{0}(\Omega^0) \times L^2_0(\Omega^0)$ for $t \in (0, T)$ with $\mathbf{u}(0) = \mathbf{u}_{init}$ s.t.
\begin{equation}
\label{eq:continuous_formulation}
\begin{aligned}
\int_{\Omega^0} \rho J^t \frac{\partial \mathbf{u} }{\partial t} \cdot \mathbf{v} + \rho J^t Grad(\mathbf{u}) H^t (\mathbf{u} - \mathbf{w}) \cdot \mathbf{v} + J^t 2\mu \, \epsilon^t(\mathbf{u}):\epsilon^t(\mathbf{v}) \dif{X} & \\
- \int_{\Omega^0} Div(J^t H^t \mathbf{v}) p \dif{X} + \int_{\Omega^0} Div(J^t H^t \mathbf{u})q \dif{X} &= 0%& \\
\end{aligned}
\end{equation}
for all $(\mathbf{v}, q) \in \mathbf{H}^1_0(\Omega^0) \times L^2_0(\Omega^0)$, $\mathbf{u}_{init} \in \mathbf{H}^1_0(\Omega^0)$ given initial and $\mathbf{w} := \frac{\partial \mathcal{X}}{\partial t}$ time-varying domain velocities. For the sake of simplicity, we omit the time-dependency on the fields $\mathbf{u}, p$.
Notice that the velocity flow at time $t$ is given by $\mathbf{u} \circ \mathcal{X}^{-1}(\cdot, t)$.
\begin{proposition}
\label{prop:energy_continuous} \cite[Chap. 9]{Quarteroni2009Cardiovascular}
Provided $(\mathbf{u}(t), p(t)) \in \mathbf{H}^1_0(\Omega^0) \times L^2_0(\Omega^0)$ a solution of Problem \eqref{eq:continuous_formulation}, % and $\mathbf{u}_{\mathbf{g}}(t) \in \mathbf{H}^1_{\mathbf{g}}(\Omega^0)$ the extension of $g \in L^2(\partial \Omega^0)$
the following energy balance holds:
\begin{equation}
\label{eq:continuous_energy_estimate}
\begin{aligned}
\frac{\partial}{\partial t} \int_{\Omega^0} \frac{\rho}{2} J^t \vert \mathbf{u} \vert^2 \, \text{d}\mathbf{X} =& - \int_{\Omega^0} J^t 2\mu \vert \epsilon^t (\mathbf{u}) \vert^2 \, \text{d}\mathbf{X}. %- \int_{\partial\Omega_0} \rho J_t F^{-1}_{t} (\mathbf{u} - \mathbf{w}) \cdot \mathbf{N} \, \text{d}\mathbf{S} + \mathcal{D}(\mathbf{u}; \mathbf{u}_{\mathbf{g}})
\end{aligned}
\end{equation}
\end{proposition}
\begin{remark}
Proposition \ref{prop:energy_continuous} makes use of the \textit{Geometric Conservation Law} (GCL) $\frac{\partial J^t}{\partial t} = Div\left( J^t F_{t}^{-1} \mathbf{w}\right)$.
\end{remark}
\begin{remark}
In the general case with non-homogeneous Dirichlet boundary conditions,
the energy balance also includes flow intensification due to the moving boundary. In such case, the intensification term appearing on the energy balance \eqref{eq:continuous_energy_estimate} in given by:
\begin{equation}
\label{eq:extension_energy_continuous}
\int_{\partial\Omega^0} \rho \frac{\vert \mathbf{u} \vert^2}{2} J^t H^t (\mathbf{u} - \mathbf{w}) \cdot \mathbf{N} \, \text{d}\mathbf{S}
\end{equation}
where $\mathbf{N} \in \mathbb{R}^d$ denotes the outward normal.
\end{remark}
\begin{remark}
Although Dirichlet boundary conditions are used throughout this work, it can be extended straightforwardly to the Neumann case by including the so called \textit{backflow stabilizations}, see e.g. \cite{bertoglio2018benchmark}.
\end{remark}
\section{Monolithic schemes (first order in time)}
\label{sec:monolithic_schemes}
Most of the numerical schemes for Problem \eqref{eq:continuous_formulation} reported in the literature are first order and can be written as follows.
Let $(t^n)_{n \in \mathbb{N}}$ be a uniform discretization of the time interval $(0, T)$ with step size $\tau > 0$
and let $H^n := H^{t^n}, J^n := J^{t^n}, w^n := w(t^n)$ be discrete sequences.
Given a conforming finite element space $\mathbf{V} \times Q$ of $\mathbf{H}^1_0(\Omega^0) \times L^2_0(\Omega^0)$ for velocity and pressure fields, the discrete problem of interest reads:
Find $(\mathbf{u}^{n+1}, p^{n+1}) \in \mathbf{V} \times Q$ s.t.
\begin{equation}
\label{eq:discretized_monolithic_formulation}
\mathcal{A}(\mathbf{u}^{n+1},\mathbf{v}) - \mathcal{B}(\mathbf{v},p^{n+1}) + \mathcal{B}(\mathbf{u}^{n+1},q) = \mathcal{F}(\mathbf{v}) \quad \forall (\mathbf{v}, q) \in \mathbf{V} \times Q
\end{equation}
being
\begin{equation}
\label{eq:lhs_bilinear_form_A}
\begin{aligned}
\mathcal{A} (\mathbf{u}, \mathbf{v}) := & \int_{\Omega^0} \rho \frac{J^{\star\star}}{\tau} \mathbf{u} \cdot \mathbf{v} \, \text{d}\mathbf{X} + \int_{\Omega^0} \rho J^{\star} Grad(\mathbf{u}) H^{\star} (\mathbf{u}^{\ast} - \mathbf{w}^{\ast\ast}) \cdot \mathbf{v} \, \text{d}\mathbf{X} + \int_{\Omega^0} J^{\star} 2\mu \epsilon^{\star}(\mathbf{u}):\epsilon^{\star}(\mathbf{v}) \, \text{d}\mathbf{X} \\
& + \alpha \int_{\Omega^0} \frac{\rho}{2} \left( \frac{J^{n+1} - J^{n}}{\tau} - Div\left( J^{\star} H^{\star} \mathbf{w}^{\ast\ast} \right) \right) \mathbf{u} \cdot \mathbf{v} \, \text{d}\mathbf{X} + \beta \int_{\Omega^0} \frac{\rho}{2} Div\left( J^{\star} H^{\star} \mathbf{u}^{\ast} \right) \mathbf{u} \cdot \mathbf{v} \, \text{d}\mathbf{X}
\end{aligned}
\end{equation}
with $\alpha, \beta \in \{0, 1\}$ given parameters, and
%The matrix $B \in \mathbb{R}^{m^{Q}\times m^{\mathbf{V}}}$ and vector $F_n \in \mathbb{R}^n$ correspond also to the discretization of the bilinear and linear forms:
\begin{equation}
\label{eq:remaining_forms}
\begin{aligned}
\mathcal{B}(\mathbf{u}, q) & := \int_{\Omega^0} Div\left( J^{\star} H^{\star} \mathbf{u} \right) q \, \text{d}\mathbf{X} \quad \forall q \in Q, \quad \mathcal{F}(\mathbf{v}) := \int_{\Omega^0} \rho \frac{J^{\star\star}}{\tau} \mathbf{u}^n \cdot \mathbf{v} \, \text{d}\mathbf{X} \quad \forall \mathbf{v} \in \mathbf{V}
\end{aligned}
\end{equation}
\begin{remark}
The term multiplying $\alpha$ is the discrete residual of GCL, while the one multiplying $\beta$ is a strongly consistent term vanishing for incompressible velocity fields.
\end{remark}
Formulation \eqref{eq:discretized_monolithic_formulation} contains a wide family of reported methods:
\begin{itemize}
\item Using $\alpha = \beta = 0$: $(\star, \star\star, \ast, \ast\ast) = (n, n, n+1, n)$ is used in \cite{Basting2017}, $(\star, \star\star, \ast, \ast \ast) = (n, n, n, n)$ in \cite{Murea2016} and $(\star, \star\star, \ast, \ast\ast) = (n+1, n+1, n+1, n+1)$ in \cite{Langer2016}, and $(\star, \star\star, \ast, \ast\ast) = (n+1, n+1, n, n+1)$ in \cite{Landajuela2016}.
\item Using $\alpha = \beta = 1$: $(\star, \star\star, \ast, \ast \ast) = (n+1, n, n, n+1)$ in \cite{Lozovskiy2018}, $(\star, \star\star,\ast, \ast\ast) = (n+1, n, n, n)$ in \cite{smaldone2014} and $(\star, \star\star, \ast, \ast\ast) = (n+1, n, n+1, n+1)$ in \cite{LeTallec2001, Wang2020}.
\end{itemize}
\begin{proposition}
\label{prop:monolithic_schemes}
By assuming well-posed, orientation-preserving deformation mappings, i.e. $(J^n)_{n \in \mathbb{N}}$ bounded in $L^{\infty}(\Omega^0)$, $J^n > 0$ for each $n \geq 0$, Problem \eqref{eq:discretized_monolithic_formulation} has unique solution for inf-sup stable finite element spaces if $\left( 2J^{\star\star} + J^{n+1} - J^{n} \right) > 0$ and $\alpha = \beta = 1$.
\end{proposition}
\begin{proof}
%The matrix system \eqref{eq:matrix_formulation} is solvable whenever $A_n^{-1}$ exists and $BA^{-1}_nB^{T}$ is invertible.
Since all operators are bounded and inf-sup stable elements are used for velocity and pressure, it is enough to ensure that the bilinear form $\mathcal{A}$ is coercive.
%Thus its enough to ensure $\text{rank}(B) = m$ and $A_n$ positive definite.
%As standard in literature, let us evaluate \eqref{eq:lhs_bilinear_form_A} in $\mathbf{v} = \mathbf{u}$ and $q = p$. Integrating by parts the convective term and joining expressions, the following equality holds:
Indeed:
\begin{equation}
\label{eq:proof_monolithic_case}
\begin{aligned}
\mathcal{A}(\mathbf{u}, \mathbf{u}) = & \int_{\Omega^0} \frac{J^{\star}}{2\tau} \left( \frac{2 J^{\star\star}}{J_{\star}} + \alpha \frac{J^{n+1} - J^{n}}{J^{\star}} \right) \vert \mathbf{u} \vert^2 + % \int_{\Omega^0}
J^{\star} 2\mu \vert \epsilon^{\star}(\mathbf{u}) \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} \frac{\rho}{2} Div\bigg( J^{\star} H^{\star} \big( (\beta -1) \mathbf{u}^{\ast} - (\alpha-1) \mathbf{w}^{\ast\ast} \big) \bigg) \vert \mathbf{u} \vert^2 \, \text{d}\mathbf{X}
\end{aligned}
\end{equation}
being the last quantity strictly positive under the stated assumptions.
\end{proof}
\begin{remark}
The extension of Proposition \ref{prop:monolithic_schemes} to the case with non-homogeneous Dirichlet boundary conditions follows from the trace theorem by assuming $\Omega^0$ a Lipschitz bounded open set \cite{Ern2004}.
\end{remark}
\begin{corollary}
Assuming $\alpha = \beta = 1$, Problem \eqref{eq:discretized_monolithic_formulation} is well posed when:
\begin{itemize}
\item $3J^{n+1} - J^{n} > 0$ if $\star \star = n+1$, i.e. a restriction on the time step size.
\item $J^{n+1} + J^{n} > 0$ if $\star \star = n$, i.e. no restriction on the time step size, since we assume orientation preserving deformation mappings.
\end{itemize}
No restrictions apply to $\star,\ast, \ast\ast$.
\end{corollary}
\begin{proposition}
\label{prop:energy_estimate_monolithic}
Under assumptions of Proposition \ref{prop:monolithic_schemes} and $\alpha = \beta = 1, \star\star = n$, the scheme \eqref{eq:discretized_monolithic_formulation} is unconditionally energy stable with energy estimate:
\begin{equation}
\label{eq:energy_estimate_monolithic}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau} \vert \mathbf{u}^{n+1} \vert^2 \, \text{d} \mathbf{X} - \int_{\Omega^0} \rho \frac{J^n}{2\tau} \vert \mathbf{u}^n \vert^2 \, \text{d}\mathbf{X}
= -\int_{\Omega^0} 2\mu J^{\star} \vert \epsilon^{\star}(\mathbf{u}^{n+1}) \vert^2 \, \text{d} \mathbf{X} - \int_{\Omega^0} \frac{\rho}{2\tau} J^{n} \vert \mathbf{u}^{n+1} - \mathbf{u}^{n} \vert^2 \, \text{d} \mathbf{X}.
\end{equation}
\end{proposition}
\begin{proof}
By setting $\mathbf{v} = \mathbf{u}^{n+1}$ in the bi-linear form \eqref{eq:lhs_bilinear_form_A}, $q = p^{n+1}$ in forms \eqref{eq:remaining_forms} and manipulating terms as standard in literature, the energy equality follows:
\begin{equation}
\begin{aligned}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau} \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \rho \frac{J^{n}}{2\tau} \vert \mathbf{u}^n \vert^2 \, \text{d}\mathbf{X} = & \int_{\Omega^0} \frac{\rho}{2\tau} (J^{n+1} - J^{\star\star}) \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X} +
\int_{\Omega^0} \frac{\rho}{2\tau} (J^{\star\star} - J^n) \vert \mathbf{u}^n \vert^2 \, \text{d}\mathbf{X} & \\
& - \int_{\Omega^0} 2\mu J^{\star} \vert \epsilon^{\star} (\mathbf{u}^{n+1}) \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \frac{\rho}{2\tau} J^{\star\star} \vert \mathbf{u}^{n+1} - \mathbf{u}^{n} \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} \frac{\rho}{2} Div(J^{\star} H^{\star} (\mathbf{u}^{\ast} - \mathbf{w}^{\ast\ast})) \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2} \alpha \frac{J^{n+1} - J^{n}}{\tau} \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} \frac{\rho}{2} Div\left( J^{\star} H^{\star} (\beta \mathbf{u}^{\ast} - \alpha \mathbf{w}^{\ast\ast}) \right) \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X}
\end{aligned}
\end{equation}
Thus, for $\alpha=\beta=1$ and $\star\star = n$ the result follows.
\end{proof}
\begin{remark}
This works focuses on first-order schemes in time.
The reason is that second order schemes, although stable in fixed domain, has been shown to be only conditionally stable in ALE form, as it was shown in \cite{Formaggia2004} for the advection-diffusion problem for Crank-Nicolson (CN) and BDF(2).
Therefore, we do not analyze here the schemes used in \cite{ failer2020impact, Hessenthaler2017,Tallec2003} -- based on CN and used in the context of fluid-solid interaction -- since their analysis repeats from \cite{Formaggia2004}.
Also in the same context, some authors have used the generalized $\alpha$-methods since it is a popular scheme for elastodynamics \cite{Liu2018}. However, there is no reported stability analysis even for the the fixed domain setting,
and its stability properties are usually assumed to be transferred from the linear setting.
\end{remark}
\section{Chorin-Temam schemes}
\label{sec:chorin_temam_schemes}
In the following, we describe a family of Chorin-Temam (CT) schemes for the iNSE-ALE problem, as we did for the monolithic case. %Such description keeps the freedom of choice for certain coefficients, which must be restricted to ensure unconditional stability.
Given $\mathbf{\widetilde V} $ a conforming space of $\mathbf{H}^1_0 (\Omega^0)$ and $ \widetilde Q$ a conforming space of $ L^2_0 (\Omega^0) \cap H^1(\Omega^0)$, $\tilde{\mathbf{u}}^{0} \in \mathbf{\widetilde V}$, for $n\geq0$:% the proposed two-step CT schemes reads
\begin{enumerate}
\item \textbf{Pressure-Projection Step $(\text{PPS})_{n}$}
%Seeks a pressure term allowing the projection of $\tilde{\mathbf{u}}^{n+1}$ in the space of divergence-free solutions, in the form:
Find $p^{n} \in \widetilde{Q}$ s.t.
\begin{equation}
\label{eq:chorin_temam_pps}
\int_{\Omega^0} \frac{\tau}{\rho} J^{\circ} Grad(p^{n}) H^{\circ} : Grad(q) H^{\circ} \, \text{d}\mathbf{X} = - \int_{\Omega^0} Div\left( J^{\circ} H^{\circ} \tilde{\mathbf{u}}^{n}\right) q \, \text{d}\mathbf{X} \quad \forall q \in \widetilde{Q}
\end{equation}
\item \textbf{Fluid-Viscous Step $(\text{FVS})_{n+1}$} %Seeks a tentative velocity field $\tilde{\mathbf{u}}^{n+1}$ with pressure given explicitly:
Find $\tilde{\mathbf{u}}^{n+1} \in \mathbf{\widetilde{V}}$ s.t.
\begin{equation}
\label{eq:chorin_temam_fvs}
\begin{aligned}
\int_{\Omega^0} \rho J^{\star\star} \frac{\tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^n}{\tau} \cdot \mathbf{v} \, \text{d}\mathbf{X}
+ \int_{\Omega^0} \rho J^{\star} Grad(\tilde{\mathbf{u}}^{n+1}) H^{\star} (\tilde{\mathbf{u}}^{n} - \mathbf{w}^{\ast\ast}) \cdot \mathbf{v} \, \text{d}\mathbf{X} & \\
+ \int_{\Omega^0} J^{\star} 2 \mu \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) : \epsilon^{\star} (\mathbf{v}) \, \text{d}\mathbf{X}
- \int_{\Omega^0} Div(J^{\circ \circ} H^{\circ \circ} \mathbf{v}) p^n \, \text{d}\mathbf{X} & \\
+ \int_{\Omega^0} \frac{\rho}{2} \frac{J^{n+1} - J^{n}}{\tau} \tilde{\mathbf{u}}^{n+1} \cdot \mathbf{v} \, \text{d}\mathbf{X}
+ \int_{\Omega^0} \frac{\rho}{2} Div\left( J^{\star} H^{\star}(\tilde{\mathbf{u}}^{n} - \mathbf{w}^{\ast\ast})\right) \tilde{\mathbf{u}}^{n+1} \cdot \mathbf{v} \, \text{d}\mathbf{X} & = 0 \quad \forall \mathbf{v} \in \mathbf{\widetilde{V}}
\end{aligned}
\end{equation}
% In particular, the corrected velocity is obtained through the update:
% \begin{equation}
% \begin{aligned}
% \mathbf{u}^{n+1} & = \tilde{\mathbf{u}}^{n+1} - \frac{\tau}{\rho} Grad(p^{n+1}) F^{-1}_{\circ + 1} \text{ in } \Omega_0
% \end{aligned}
% \end{equation}
\end{enumerate}
The following energy estimate can be obtained under suitable conditions:
\begin{proposition}
\label{prop:energy_estimate_chorin_temam}
Under assumptions $\circ = \circ\circ = \star \star = n$, the solution to scheme \eqref{eq:chorin_temam_pps}-\eqref{eq:chorin_temam_fvs} is unconditionally stable, i.e.
%is unconditionally energy stable if $\circ = \circ \circ = \star\star = n$, without condition over $\star$. Moreover, the energy estimate is given by
\begin{equation}
\begin{aligned}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau} \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \rho \frac{J^{n}}{2\tau} \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d} \mathbf{X} \leq & - \int_{\Omega^0} J^{\star} 2\mu \vert \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} J^{n} \frac{\tau}{2\rho} \vert Grad(p^n) H^{n} \vert^2 \, \text{d}\mathbf{X} .
\end{aligned}
\end{equation}
\end{proposition}
\begin{proof}
As standard in literature, let us take $\mathbf{v} = \tilde{\mathbf{u}}^{n+1}$ in $(\text{FVS})_{n+1}$, and $q = p^n$ in $(\text{PPS})_{n}$. Adding both equalities and rewriting expressions, it follows:
\begin{equation}
\begin{aligned}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau} \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \rho \frac{J^{n}}{2\tau} \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} = & \int_{\Omega^0} \frac{\rho}{2\tau}(J^{n+1} - J^{\star\star}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X}
+ \int_{\Omega^0} \frac{\rho}{2\tau}(J^{\star\star} - J^{n}) \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2\tau} J^{\star\star} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} J^{\star} 2\mu \vert \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} Div\left( J^{\circ\circ} H^{\circ\circ} (\tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n}) \right) p^n \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} Div\left( (J^{\circ\circ} H^{\circ\circ} - J^{\circ} H^{\circ}) \tilde{\mathbf{u}}^n \right) p^n \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\tau}{\rho} J^{\circ} \vert (H^{\circ})^T Grad(p^n) \vert^2 \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2\tau} (J^{n+1} - J^{n}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X}
\end{aligned}
\end{equation}
Bounding the first divergence term using integration by parts and Cauchy-Schwarz inequality, it follows
\begin{equation}
\begin{aligned}
\int_{\Omega^0} Div\left( J^{\circ\circ} H^{\circ\circ} (\tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n}) \right) p^n \, \text{d}\mathbf{X} & \leq \int_{\Omega^0} \frac{\rho}{2\tau} J^{\circ\circ} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} + \int_{\Omega^0} \frac{\tau}{2\rho} J^{\circ\circ} \vert (H^{\circ\circ})^T Grad(p^n) \vert^2 \, \text{d}\mathbf{x}
\end{aligned}
\end{equation}
Thus, the following energy estimate can be obtained:
\begin{equation}
\label{eq:energy_estimate_proof}
\begin{aligned}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau} \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \rho \frac{J^{n}}{2\tau} \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} \leq & \int_{\Omega^0} \frac{\rho}{2\tau} (J^{n+1} - J^{\star\star}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X}
+ \int_{\Omega^0} \frac{\rho}{2\tau} (J^{\star\star} - J^{n}) \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2\tau} J^{\star\star} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} J^{\star} 2\mu \vert \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} \frac{\rho}{2\tau} J^{\circ\circ} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} \frac{\tau}{2\rho} J^{\circ\circ} \vert (H^{\circ\circ})^{T} Grad(p^n) \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} Div\left( (J^{\circ\circ} H^{\circ\circ} - J^{\circ} H^{\circ}) \tilde{\mathbf{u}}^{n}\right) p^n \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\tau}{\rho} J^{\circ} \vert (H^{\circ})^{T} Grad(p^n) \vert^2 \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2\tau} (J^{n+1} - J^{n}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X}
\end{aligned}
\end{equation}
From estimate \eqref{eq:energy_estimate_proof} it follows that whenever $\circ = \circ \circ = \star \star = n$ unconditional energy stability is attained, where $\star$ remains free of choice.
\end{proof}
\section{Numerical examples}
\label{sec:numerical_examples}
We consider a rectangular domain with opposite vertices $\{ (0, -1), (6, 1) \}$ where the iNSE-ALE formulation \eqref{eq:continuous_formulation} will be simulated over the interval $(0,\, 2) \, [s]$ with non-zero initial condition of the form $ \mathbf{u}(0) := \big(\gamma (1 - \mathbf{X}_1^2) \mathbf{X}_0 (6 - \mathbf{X}_0), 0\big) ,\, \gamma = 0.001$.
The domain is deformed using $\mathcal{X}(\mathbf{X}, t) := \big( (1 + 0.9 sin(8 \pi t)) \mathbf{X}_0,\, \mathbf{X}_1 \big)$. %, i.e. an oscillation with initial expansion and frequency of $4 \pi$.
Discretization setup for Formulation \eqref{eq:discretized_monolithic_formulation} and \eqref{eq:chorin_temam_pps}-\eqref{eq:chorin_temam_fvs} is done choosing a time step $\tau = 0.01$ and space triangulation with elements diameter $h \approx 0.01 $, implemented through FEniCS \cite{FEniCS2015} using Python for interface and postprocessing.
To exemplify the theoretical results from previous sections, four schemes are taken into account. Monolithic (M) Formulation \eqref{eq:discretized_monolithic_formulation} is taken with linearized convective term and implicit treatment, i.e., $(\star, \ast, \ast\ast) = (n+1, n, n+1)$ where for $\star\star$ we consider two choices, denoted $\text{M}\, \star\star = n$ and $\text{M}\, \star\star = n+1$.
For both cases the space discretization is carried out with $ \mathbf{V}/Q = [\mathbb{P}_2]^d/\mathbb{P}_1$ Lagrange finite elements. Similarly, Chorin-Temam (CT) scheme \eqref{eq:chorin_temam_fvs}-\eqref{eq:chorin_temam_pps} is taken with linearized convective term and implicit treatment, i.e. $(\star, \ast\ast, \circ, \circ\circ) = (n+1, n+1, n, n)$ with $\star\star$ as before, denoting each scheme by $\text{CT}\, \star\star = n$ and $\text{CT}\, \star\star = n+1$ with space discretization done through $\mathbf{\widetilde{V}}/\widetilde{Q} = [\mathbb{P}_1]^d/\mathbb{P}_1$ elements.
In all cases homogeneous (equal to $\mathbf{0}$) boundary conditions are imposed for the velocity, zero-mean on the pressure and $\alpha=\beta=1$.
The results are assessed using time-dependent normalized parameters $ \hat{\delta}_{\text{M}}:= \delta_{\text{M}}/E_{st}^{\star}, \hat{\delta}_{\text{CT}}:= \delta_{\text{CT}}/E_{st}^{\star}$ defined as: %, where $E_{st}^{\star}$ denote the \CB{\mod{strain energy}{viscous dissipation}} and $\delta_{M}, \delta_{CT}$ residual errors defined as:
\begin{equation}
\label{eq:energy_error}
\begin{aligned}
\delta_{M}^{n+1} &:= D^{n+1} + E_{st}^{\star} + \int_{\Omega_0} \frac{\rho J^{\star\star}}{2\tau} \vert \mathbf{u}^{n+1} - \mathbf{u}^{n} \vert^2 \, \text{d}\mathbf{X} , \quad \delta_{CT}^{n+1} := D^{n+1} + E^{\star}_{st} + \int_{\Omega_0} \frac{\tau J^{\circ}}{2 \rho} \vert (H^{\circ})^T Grad(p^n) \vert^2 \, \text{d}\mathbf{X} \\
D^{n+1} &:= \int_{\Omega_0} \frac{\rho}{2\tau} \left( J^{n+1} \vert \mathbf{u}^{n+1} \vert^2 - J^n \vert \mathbf{u}^n \vert^2 \right) \, \text{d}\mathbf{X} , \quad E^{\star}_{st} = \int_{\Omega_0} 2 \mu J^{\star} \vert \epsilon^{\star} (\mathbf{u}^{n+1}) \vert^2 \, \text{d}\mathbf{X}.
\end{aligned}
\end{equation}
Figure \ref{fig:delta_hat_oscs} shows $\hat{\delta}_{\text{M}}, \hat{\delta}_{\text{CT}}$ values for each tested scheme. Propositions \ref{prop:energy_estimate_monolithic} and \ref{prop:energy_estimate_chorin_temam} are confirmed since $\hat{\delta}_{\text{M}}=0$ and $\hat{\delta}_{\text{CT}} \leq 0$ if $\star\star = n$. For $\star \star = n+1$, peaks appearing throughout the simulation are defined by the sign change of domain velocity, i.e. in the change from expansion to contraction.
Importantly, the spurious numerical energy rate related to discretization of the GCL condition appear to be positive in expansion, therefore being a potential source of numerical instabilities.
%Moreover the expected energy decay from Propositions \ref{prop:energy_estimate_monolithic}, \ref{prop:energy_estimate_chorin_temam} for the choice $\star \star = n$ in $\text{M, CT}$ schemes is obtained.
%\CB{Recall that $\hat{\delta}_{\text{M}}, \hat{\delta}_{\text{CT}}$ in \eqref{eq:energy_error} are taken in such form since on scheme $\text{M}\, \star\star = n$ it follows $\hat{\delta}_{MT} = 0$ and similarly on $\text{CT}\, \star\star = n$ it follows $\hat{\delta}_{\text{CT}} \sim 0$, defining the base case.}
\begin{figure}[!hbtp]
\centering
\includegraphics[width=\textwidth]{figs/Comparison_Delta_Hat_Value_GCL_True_solver_LU.png}
\caption{Summary of the numerical experiment in terms of energy balance. Left: Monolithic residual error values $\hat{\delta}_{\text{M}}$;
Right: Chorin-Temam residual error values $\hat{\delta}_{\text{CT}}$. %Both cases are simulated with deformation mapping $\mathcal{A}$ defined previously and shown on the interval $[0, 1] \, [s]$.}
}
\label{fig:delta_hat_oscs}
\end{figure}
\section{Conclusion}
Several reported time discretization schemes for the iNSE-ALE have been reviewed and analyzed in terms of their well posedness at each time step and time stability. The stability analysis is confirmed by numerical experiments. For the monolithic case, two schemes lead to well-posed energy-stable problems whenever $\alpha=\beta=1$ with $\star \star = n$ as studied in \cite{LeTallec2001, Lozovskiy2018, smaldone2014, Wang2020}.
To the best of the authors knowledge, the unconditionally stable Chorin-Temam scheme derived in this work has not been reported yet. %, and moreover the numerical experiments studied here validate the energy stable propositions.
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