316 lines
16 KiB
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316 lines
16 KiB
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\begin{document}
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\parindent = 0pt
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%
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\begin{center}
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{\bf Response to the reviewer's comments } \\ (\manuscript)
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\end{center}
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Dear Professor Tucker,\\
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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.
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We hope that after our response the manuscript can be accepted for publication in \journal. \\
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%, in particular for the questions. \\% We will address the comment in the following lines.\\
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%In response to the first reviewer, indeed updates on the pressure space definition and notation have been done to correctly defined our problem
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%in the continuum setting. Also comments on the space-time regularity and restrictions for the
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%computational domain were added, e.g. in Remark 6. \\
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%
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%Our analysis is done throughtout the article using homogeneous Dirichlet boundary conditions and thus our derivations do not contain the mechanical work as expected.
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%We assume such case for matter of convenience since its extension follows by using standard techniques found in literature. Nevertheless, comments
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%about flow intensification were added in Remark 2. \\
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%
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%Finally, we point out the inaccurate choice of words for our conclusion, which in this new version was updated. \\
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%
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%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
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%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
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%obtained with the usual techniques and extensions to pressure correction schemes still lack in unconditional energy stable results. \\
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%
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%Further, the question of unconditionally energy stable higher order iNSE-ALE formulation still remains open and further research must be done. \\
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%We hope that after our response the manuscript can be accepted for publication in \journal. \\
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Sincerely yours \\
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Reidmen Ar\'{o}stica and Crist\'obal Bertoglio
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%\end{document}
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%%%% NEW REVIEWER
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\setcounter{countResp}{0}
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\newpage
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\begin{center}
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{\bf Detailed reply to Referee \#1} \\ (\manuscript)
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\end{center}
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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.
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For the sake of convenience, all changes in the revised manuscript version are marked in \textcolor{orange}{orange}.
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\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.}
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{
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Thanks for pointing this to us, we corrected it in the new version.
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}
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\ReviewComments{p.2 On Remark 1: if you address spatial discretizations here, you should mention the problem of convective stabilization.
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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
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the restrictions of the computational mesh imposed by the trace theorem in finite element spaces.}
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{
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We thank the reviewer for the comments regarding the restrictions on the computational mesh. Remark 6 has been updated including them.
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}
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\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.
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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].}
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{
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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.
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In this version, we added a comment regarding such fact in Remark 2.
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}
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\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
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and this is important since it is proved to be unconditionally stable whereas Corollary 3 states that it has restrictions on the time step.}
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{
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We thank the reviewer for this comment. We noticed a typo in the discrete formulation (5) which could have induced misundestanding throughout the article.
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We reviewed carefully the discrete formulation in [12], where $k$ there corresponds to $n+1$ in our manuscript.
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Therefore, we updated the indices to (n+1, n, n, n+1) to match the proposed scheme in [12].
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}
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\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
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and tangling of the physical domain.}
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{
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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.
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}
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\ReviewComments{p.4 Fig 1: these curves imply that you lost boundary flow intensification term in the energy balance.}
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{
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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.
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}
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\ReviewComments{p.4, first line of Conclusion: the authors are too ambitious to claim that they theoretically analyzed time discretization schemes.
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Actually, they suggest hints to a future stability analysis.}
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{
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We agree that our choice of words was inaccurate. This has been updated.
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}
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\ReviewComments{General: in mathematical journals, integrals are completed by dx, dt, ds, etc.}
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{
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Thanks for this remark. It has been updated.
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}
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\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**.
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p.1. Jacobian, is the standard, p.4. Schwarz, Monolithic.}
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{
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We acknowledge the comments regarding figure styling and typos. They have been updated in this version.
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}
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\pagebreak
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%%%%%%%%%%%%%%%% NEXT REVIEWER
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\newpage
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\setcounter{countResp}{0}
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%\setcounter{page}{1}
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\begin{center}
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{\bf Detailed reply to Referee \#2} \\ (\manuscript)
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\end{center}
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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.
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For the sake of convenience, all changes in the revised manuscript version are marked in \textcolor{red}{red}.
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%\vspace{-.5cm}
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\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.
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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]
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are applied since they all give second order. Is it possible to include at least one of these schemes in the analysis?}
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{
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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.
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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.
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We have now extended Remark 6 in order to explain so and refer to [1] for more details.
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}
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\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
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ideas to ALE formulations.}
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{
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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
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a Chorin-Temam approach using our finding for the first order monolithic schemes since unconditional energy stability can be archieved.
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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,
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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.
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}
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\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?}
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{
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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.
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}
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\ReviewComments{Problem description: Do you not require any temporal regularity of the domain velocity w(t)?}
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{
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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,
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thus for $\frac{\partial \mathcal{X}}{\partial t} := \mathbf{w}$ to be $\mathcal{C}^1$ in space and $\mathcal{C}^0$ in time.
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}
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\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.
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does it fit to the initial velocity and the right hand side, the Dirichlet data? How do you check that?}
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{
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Indeed it was a typo. It was removed and updated to the correct
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pressure space of $L^2(\Omega^0)$ functions with zero-mean.
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}
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\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
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what exactly is meant without wrtiting it down. Maybe you can give details on the most relevant schemes that you are considering.}
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{
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We thank the reviewer for the comment and agree with it. Nevertheless it provides advantages when describing a wide range of schemes.
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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.
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}
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\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
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and star-star. But where does it go to?}
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{
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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
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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.
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}
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\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?}
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{
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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.
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As guessed by the reviewer, the 2nd component indeed is 0.\\
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As clarified above, there is no initial condition for the pressure since it was a typo.
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%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 $(FVS)_{0}$.
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%Thus, no direct computation of $p_0$ is required.
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}
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\ReviewComments{Reference [1] has some html code.}
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{
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Thanks for pointing that out. It has been corrected.
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}
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\ReviewComments{Reference [9] has been published in 2018 in Mathematics and Computer in Simulation.}
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{
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Thanks, it has been updated.
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}
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\vspace{1cm}
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{
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[1] L. Formaggia, F. Nobile. Stability analysis of second-order time
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accurate schemes for ALE-FEM, Comp. Meth. Appl. Math. Engrg. 193, 2004
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[2] M. Razzaq, H. Damanik, J. Hron, A. Ouazzi, S. Turek, FEM multigrid
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techniques for fluid-structure interaction with application to
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hemodynamics, Applied Numerical Mathematics, 2012
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[3] T. Richter and T. Wick. In T. Carraro and M. Geiger and S. Körkel
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}
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\end{document}
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