{"id":"2e43acd5-5ecd-4ea5-8b62-3756bd640c58","arxiv_id":"2412.18916","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"An implicit, optimization-based fluid-structure interaction formulation with separate Galerkin and least-squares Petrov-Galerkin reduced models is demonstrated to reproduce full-order solutions on three 2D benchmark-style problems.","lead":"This paper introduces an optimization-based coupling for fluid-structure interaction, in which the interface stress is treated as a control variable in a constrained optimization problem. It adds separate reduced-order models for the fluid and the solid, and shows on three 2D test cases that the reduced models can reproduce the full-order results.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The δ→0 limit of the regularized problem (25) to (24) is cited but unverified and δ is unreported, so the FOM/ROM reference solution may be for a nearby problem.","rationale":"The reader identifies the undocumented convergence of (25) to (24) as the weakest assumption; I agree. This assumption is load-bearing: the optimization formulation is the paper's central methodological innovation, and the supposed equivalence to the original FSI system is what justifies treating the FOM results as reference solutions and the ROM errors as errors against the true FSI state. The paper has no delta-sensitivity study, and no delta values appear in Sections 5.1–5.3. Moreover, the regularizer is an H¹(eΓ) seminorm, which has a nontrivial kernel (constant functions); convergence theorems for such penalties typically require a norm or additional control of the kernel. The cited works [33,34,35] analyze optimization-based decoupling for FSI, but the hypotheses—e.g., linear or weakly nonlinear settings or norm regularization—are not shown to hold for the ALE formulation with Newmark/BDF2 time stepping and Saint-Venant Kirchhoff solids. A secondary concern is that the convergence study in Table 1 neglects the ALE deformation (the manufactured solution imposes transmission conditions while keeping Ω_f fixed), so second-order space-time accuracy of the full ALE coupling is not demonstrated by that table; however, the Turek benchmark provides some mesh-convergence evidence. The delta concern is more foundational because it affects the correctness of the reference solution itself. A direct numerical test with a sequence of δ values, or a comparison to a monolithic solve of (24), would settle the matter. The paper is otherwise coherent and candid about limitations (e.g., ROM instabilities, lack of hyper-reduction), so the appropriate verdict remains CONDITIONAL: accept only after the delta-convergence issue and reproducibility gaps are addressed.","tokens_in":26974,"tokens_out":10667,"duration_ms":102586,"concrete_test":"Rerun the Section 5.1 manufactured problem at fixed h=1/60, Δt=0.0025 with δ = 10^-2, 10^-4, 10^-6, 10^-8, 10^-10 and, for comparison, solve (24) directly by imposing the kinematic condition Pf uf = Ps Ds,Δt ds as a strong constraint (e.g., via a monolithic Newton solve). If the errors in Table 1 stabilize as δ→0 to the values of the monolithic solve, the concern is resolved; if the solution depends on δ across this range, report the δ used in the paper and verify that the experimental results lie in the converged regime.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.4 asserts that solutions of (25) converge to those of (24) as δ→0, citing [33,34,35]. No convergence study is provided, no δ values are reported in any experiment, and the regularization uses the H¹(eΓ) seminorm, which is not a norm: it does not penalize constant interface tractions. The cited analyses may not apply to the present ALE formulation with a time-dependent fluid domain and a nonlinear hyperelastic solid, and the hypotheses (e.g., well-posedness of the control-to-state map and coercivity of the regularizer) are not checked. Since all FOM results in Tables 1–4 and Figure 10, and every ROM error in Figures 4–6 and 11–17, is computed against a solution of (25) for an unspecified δ, a non-negligible or non-vanishing δ would mean the 'full-order' reference is the solution of a regularized nearby problem, not of the FSI system (24). The central claim of the paper—that the ROM reproduces the full-order FSI solution—would then be measured against the wrong target.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an optimization-based partitioned formulation for fluid-structure interaction (FSI) in an arbitrary Lagrangian-Eulerian (ALE) framework. The interface flux is introduced as a control variable, and the implicitly coupled fluid-solid system is reformulated as the constrained optimization problem (25), solved by sequential quadratic programming (SQP). Reduced-order models are then built separately for the fluid state, the solid state, and the control using proper orthogonal decomposition (POD), with Galerkin projection for the solid and least-squares Petrov-Galerkin (LSPG) projection for the fluid, together with an enrichment strategy. Numerical experiments include a manufactured-solution convergence study, an elastic-beam test, the Turek-Hron benchmark, and ROM and hybrid ROM-FOM reconstructions. The paper's central claim is that this optimization-based coupling enables seamless coupling of independent full-order and reduced-order subproblems without compatibility conditions.","tokens_in":27245,"tokens_out":6633,"duration_ms":63921,"significance":"If the formulation is sound, it provides a practically relevant alternative to Dirichlet-to-Neumann and monolithic coupling for partitioned FSI solvers, especially in the model-reduction setting where the fluid and solid reduced spaces can be constructed independently. The SQP-vs-DtN comparison in Table 2 and the agreement with the Turek benchmark in Section 5.3 are useful and credible. The manufactured-solution convergence rates in Table 1 are also encouraging. The main weaknesses are that the convergence of the regularized problem (25) to the original coupled system (24) as delta tends to zero is asserted but not verified for the present ALE formulation, that the regularization parameter delta is not reported in any experiment, and that the fully discrete stability analysis is deferred. Because every FOM and ROM error is measured against a solution of (25), the missing delta documentation is a load-bearing reproducibility and correctness gap.","major_comments":[{"comment":"The statement that solutions of (25) converge to solutions of (24) as delta approaches 0 is cited to [33,34,35] but is not verified for the present ALE formulation with a time-dependent fluid domain and a nonlinear hyperelastic solid. Moreover, the regularizer in (25) is the H1(eGamma) seminorm, which is not coercive on the control space G because constant interface tractions are not penalized. No delta values are reported anywhere in Section 5, so the FOM reference used in Tables 1-4 and Figures 4-6 and 10-17 is a solution of (25) for an unspecified delta. This is load-bearing because every ROM error is measured against that reference rather than against a solution of (24). Please provide a delta-convergence study (e.g., delta = 10^-2, 10^-4, 10^-6) or prove the convergence under the paper's assumptions, and report the delta used in each numerical experiment.","section":"Section 3.4, Eq. (25)"},{"comment":"The convergence study in Table 1 refines h and Delta t simultaneously while delta is held fixed (and unreported). If delta is fixed positive, the error to the exact solution of (2) includes a regularization error that does not vanish with h and Delta t unless the exact control happens to lie in the kernel of the regularizer. The observed second-order rates are therefore not by themselves a complete convergence statement for the coupled system (24). The authors should either report the value of delta used, demonstrate that the rates are independent of delta, or include a study in which delta is sent to zero along with the discretization parameters.","section":"Section 5.1, Table 1"},{"comment":"The paper states that the enrichment strategy 'ensures algebraic stability of the coupled problem' (Section 6) and that enrichment and Petrov-Galerkin projection are 'key to ensure ... stability' (Section 5.2), but the fully discrete stability analysis is explicitly deferred in Section 3.3. The numerical evidence in Figures 14 and 17 shows that stability is not universal: the Galerkin ROM and the LSPG ROM without enrichment are unstable from early times, and the hybrid ROM on mesh1 develops oscillations after about 7 seconds. The stability claim should be restricted to the enriched LSPG configuration and, ideally, supported by a discrete energy estimate or a precise numerical stability criterion rather than presented as a general property of the enrichment strategy.","section":"Section 4.1 and Section 6"}],"minor_comments":[{"comment":"The POD control inner product in (31) is the full H1(eGamma) inner product, while the regularizer in (25) uses only the H1(eGamma) seminorm. The relation between these two choices should be clarified, since the POD inner product is a norm but the regularizer is not.","section":"Eq. (31) vs Section 3.4"},{"comment":"The dimensions of the control matrix eEs are stated as RNu x NGamma, but the definition (eEs)_{j,l} = integral over eGamma of phi^f_{i_f_l} * phi^s_j (with j as a solid index) suggests the dimension should be RNs x NGamma. Please correct the typo.","section":"Appendix B"},{"comment":"The text says 'The results of the ROM are in good agreement with the HF results for all tolerances considered' and then immediately notes that tolpod = 10^-4 leads to an unstable ROM. Since Figure 13 shows only tolpod = 10^-5, 10^-6, 10^-7, either include the 10^-4 case in the figure or rephrase the sentence to avoid an apparent contradiction.","section":"Section 5.2, Figure 13 context"},{"comment":"The pseudo-elastic parameters mu_m and lambda_m for the ALE mesh deformation are said to be discussed in Section 5, but no values are reported. These parameters affect the mesh motion and hence the FOM solution; please report them for each test case.","section":"Section 2.1 and Section 5"},{"comment":"The statement 'we divide the equations by rho_f' is not reflected in the governing equations or in the reported parameter values; please specify the nondimensionalization so that the numerical setup is reproducible.","section":"Section 5.3.1"},{"comment":"The manuscript does not state whether the implementation or data are available. Given the number of algorithmic parameters (tolsqp, tolen, tolpod, delta, mesh parameters), a code/data availability statement would substantially improve reproducibility.","section":"Global"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and presents a useful algorithmic contribution. The main issue is the unreported and unverified regularization parameter delta, which affects the interpretation of all numerical results. This is fixable with a delta-convergence study and proper reporting. The stability claims should also be moderated. I do not see grounds for rejection, but the current manuscript requires nontrivial additional numerical or analytical work before it can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the full-order method is credible and the paper is honest about its ROM being a solution-reproduction prototype. The thing to fix before serious use is the regularization parameter δ: the paper asserts convergence of (25) to (24) as δ→0 citing [33–35], but reports no δ values, no convergence study, and uses an H1(eΓ) seminorm that does not penalize constant interface tractions. All FOM and ROM errors are measured against a solution of (25), so an unreported, non-negligible δ would mean the reference itself is for a nearby problem.\n\nWhat is actually new: the specific combination of optimization-based coupling with separate Galerkin (solid) and LSPG (fluid) reduced spaces plus enrichment, solved by SQP, and the hybrid ROM-FOM variant. Previous work did optimization-based FSI at full order only, or optimization-based MOR for flows without solids. The authors cite Kuberry-Lee and their own earlier work correctly.\n\nCredit where due: the manufactured-solution convergence rates in Table 1 are clean, and the Turek benchmark comparison in Table 4 is reasonable for all three meshes. The paper states at the outset that the ROM part is solution reproduction, reports long-time instabilities, and does not claim speedups because hyper-reduction is not addressed. That candor is real. The semi-discrete energy identity in Lemma 1 is proved in the appendix and is fine.\n\nSoft spots, roughly in order: (1) the δ issue above is the main one; it is fixable by reporting δ, running a δ-convergence check, and either replacing the seminorm or justifying why constant controls cannot spoil the limit. (2) The fully discrete stability proof is explicitly deferred; acceptable for a numerical paper but a limit. (3) ROM evaluation is in-sample: POD spaces come from the same trajectory that is reconstructed, so low errors are partly baked in. The authors label this correctly as a solution-reproduction problem. (4) Minor: pseudo-elastic mesh parameters and δ are not reported, which hurts reproducibility. No code or data are provided.\n\nBottom line: this is a serious numerical methods paper, not a breakthrough and not a failure. It deserves a real referee. A revision that reports δ, adds a δ-convergence experiment, and includes one ROM test on parameters not used to build the spaces would substantially raise its value. If I worked in FSI or component-based MOR, I would cite the full-order SQP/optimization treatment.","headline":"A credible optimization-based partitioned FSI/MOR formulation with honest FOM tests, but unreported δ and in-sample ROM results keep it from being fully convincing.","tokens_in":27757,"tokens_out":3901,"would_cite":true,"duration_ms":37753,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M60","65M99","74F10","76D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Fluid-structure interaction can be recast as an optimization problem with the interface flux as control, allowing separate reduced models for fluid and solid.","keywords":["model order reduction","fluid-structure interaction","partitioned method","optimization-based method","arbitrary Lagrangian-Eulerian formulation","proper orthogonal decomposition","least-squares Petrov-Galerkin","sequential quadratic programming"],"falsifier":"Solve the exact-solution test of Section 5.1 with the regularization parameter $\\delta$ set to several values from large to tiny, and compute the error against the exact solution; if the error does not decrease toward the reported second-order convergence as $\\delta\\to 0$, the full-order solution used throughout is not the fluid-structure solution but a nearby regularized one.","tokens_in":26748,"feed_emoji":"🌊","tokens_out":10828,"duration_ms":89844,"temperature":0.7,"pith_summary":"This paper tries to establish that fluid-structure interaction can be solved as a constrained optimization problem rather than through alternating Dirichlet-to-Neumann iterations. At each time step the unknowns are the fluid state, the solid state, and an interface flux acting as a control variable; the objective weakly enforces the velocity match and a regularizer penalizes excessive control. If this formulation is right, one can couple full-order and reduced-order models of the fluid and solid without any compatibility condition between their approximation spaces, because matching is handled by the optimization. That would make model reduction of FSI a matter of combining established fluid and solid reduction techniques in a partitioned, implicit way.","feed_headline":"Fluid-structure coupling becomes an optimization problem","feed_subtitle":"Interface flux acts as control, so fluid and solid reduced models can be built separately and coupled implicitly.","key_machinery":"The load-bearing object is the optimization problem (25), whose objective punishes the mismatch between the fluid velocity on the interface and the time-discrete solid velocity, plus $\\frac{\\delta}{2}$ times the $H^1(\\tilde{\\Gamma})$ seminorm of the control $g$; the constraints are the discrete residuals of the fluid and solid. The control $g$ represents the normal flux at the interface. For the reduced-order model the same structure is kept, with POD spaces for fluid state, solid state, and control; enrichment modes are added to the state spaces and the fluid ROM uses least-squares Petrov-Galerkin projection so the coupled problem stays stable.","core_discovery":"The central claim is that the discrete fluid-structure system (24) is equivalent, in the limit of vanishing regularization, to the constrained optimization problem (25), with the interface flux $g$ as control; the objective is the squared mismatch between the fluid velocity on the interface and the Newmark time-discrete solid velocity, plus a $\\delta$-weighted $H^1(\\tilde{\\Gamma})$ control penalty. Since $g$ enters linearly in both residuals, static condensation turns each SQP subproblem into a least-squares problem for $g$, so fluid and solid subproblems can be solved independently and then coupled through the control. The paper argues this removes the need for compatibility conditions between fluid and solid reduced spaces, and demonstrates the setup on three two-dimensional problems: an exact-solution test showing second-order convergence, an elastic-beam configuration where SQP needs far fewer iterations than Dirichlet-Neumann, and the benchmark case where LSPG-plus-enrichment ROMs and a ROM-FOM hybrid reproduce the high-fidelity results.","pith_inferences":["The same optimization coupling could be applied to non-conforming fluid and solid meshes; the paper says compatibility is not required but only tests conforming meshes.","Because the cost of forming the sensitivity matrices grows with the control dimension, reducing the control space by POD or adaptive selection of interface degrees is a natural next step toward larger or three-dimensional problems.","The reported ROM errors are measured against the regularized reference solution, so their practical meaning depends on the unverified $\\delta\\to 0$ limit; a $\\delta$-convergence study would settle this.","The enrichment strategy that stabilizes the ROM could double as an indicator of where the reduced fluid space is deficient, since it is built from the interface control response."],"forward_implications":["The full-order partitioned solver is second-order accurate in space and time when BDF2 and Newmark are used, based on the exact-solution test.","SQP with the inexact Jacobian is faster than Dirichlet-Neumann iteration in the elastic-beam tests, with about a ten-fold speedup on the coarse mesh.","Fluid and solid reduced spaces can be built independently with POD, and the optimization enforces interface coupling without compatibility conditions.","Enrichment of the state spaces and LSPG for the fluid are both needed: Galerkin-only or no-enrichment ROMs become unstable or inaccurate in the benchmark test.","A hybrid configuration with a reduced-order solid and full-order fluid remains accurate over a longer integration window in the benchmark test, although the paper reports spurious force oscillations can appear on the finer mesh."],"supporting_citations":[{"why":"Introduces the optimization-based domain decomposition formulation for incompressible flows that this paper extends to FSI.","marker":"[26]"},{"why":"One of the cited analyses for convergence of the regularized optimization solution to the FSI solution as the control penalty goes to zero.","marker":"[33]"},{"why":"Analysis of an FSI problem recast in an optimal-control setting, cited for the same $\\delta\\to 0$ convergence.","marker":"[34]"},{"why":"Convergence of a Neumann-control decoupling over a single time step, also cited for the regularization limit.","marker":"[35]"},{"why":"The authors' previous component-based MOR work that supplies the SQP and enrichment strategy used here.","marker":"[52]"},{"why":"Least-squares Petrov-Galerkin projection, the fluid ROM formulation whose stability is demonstrated in the experiments.","marker":"[7, 54]"},{"why":"Provides the exact-solution FSI test used to measure the reported convergence rates.","marker":"[2]"},{"why":"The benchmark FSI test case used to evaluate the high-fidelity solver and the ROM/hybrid configurations.","marker":"[57]"}],"fun_headline_variants":["Optimization turns fluid-structure coupling into a least-squares fit","Fluid and solid reduced models coupled via optimization","SQP needs fewer iterations for fluid-structure coupling","Least-squares control separates fluid and solid ROMs","Optimization-based reduction avoids compatibility conditions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method's reference solution comes from a regularized optimization problem; the paper assumes this problem converges to the real fluid-structure system as the penalty term shrinks, citing earlier work but giving no check or penalty values for its own moving-mesh tests.","fun_headline_variants_meta":{"raw":{"variants":["Optimization turns fluid-structure coupling into a least-squares fit","Fluid and solid reduced models coupled via optimization","SQP needs fewer iterations for fluid-structure coupling","Least-squares control separates fluid and solid ROMs","Optimization-based reduction avoids compatibility conditions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000268,"raw_usage":{"total_tokens":1583,"prompt_tokens":874,"completion_tokens":709,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":490,"completion_tokens_details":{"reasoning_tokens":634}},"tokens_in":490,"tokens_out":709,"duration_ms":6408,"temperature":1.0,"reasoning_tokens":634,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T04:20:48.461089+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the exact-solution test of Section 5.1 with the regularization parameter $\\delta$ set to several values from large to tiny, and compute the error against the exact solution; if the error does not decrease toward the reported second-order convergence as $\\delta\\to 0$, the full-order solution used throughout is not the fluid-structure solution but a nearby regularized one.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the optimization-based domain decomposition formulation for incompressible flows that this paper extends to FSI."},{"cited_title":"Kuberry and H","cited_arxiv_id":null,"evidence_quote":"One of the cited analyses for convergence of the regularized optimization solution to the FSI solution as the control penalty goes to zero."},{"cited_title":"Kuberry and H","cited_arxiv_id":null,"evidence_quote":"Analysis of an FSI problem recast in an optimal-control setting, cited for the same $\\delta\\to 0$ convergence."},{"cited_title":"Kuberry and H","cited_arxiv_id":null,"evidence_quote":"Convergence of a Neumann-control decoupling over a single time step, also cited for the regularization limit."},{"cited_title":"Taddei, X","cited_arxiv_id":null,"evidence_quote":"The authors' previous component-based MOR work that supplies the SQP and enrichment strategy used here."},{"cited_title":"Astorino and C","cited_arxiv_id":null,"evidence_quote":"Provides the exact-solution FSI test used to measure the reported convergence rates."},{"cited_title":"Turek and J","cited_arxiv_id":null,"evidence_quote":"The benchmark FSI test case used to evaluate the high-fidelity solver and the ROM/hybrid configurations."}],"review_version":1}