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Optimization-based model order reduction of fluid-structure interaction problems

Tommaso Taddei, Xuejun Xu, Lei Zhang

TL;DR

This work develops an optimization-based framework for fluid-structure interaction that couples full-order and reduced-order models through an implicit equality-constrained problem, using the interface flux as control and SQP for solution. It adopts projection-based MOR with Galerkin for the solid and LSPG for the fluid, complemented by an enrichment strategy to stabilize the coupled reduced system and allow independent reduction of fluid and solid components. Numerical results on three model problems (including a Turek-type benchmark) demonstrate accurate state reconstruction, robustness of the ROM and hybrid ROM-FOM solvers, and the benefits of SQP over DtN-like approaches for moderate control dimensions. The framework offers a path toward efficient many-query FSI simulations and hybrid solver strategies, while identifying challenges in long-time stability and suggesting avenues such as energy-conserving time integrators and space-time MOR to further enhance performance.

Abstract

We introduce optimization-based full-order and reduced-order formulations of fluid structure interaction problems. We study the flow of an incompressible Newtonian fluid which interacts with an elastic body: we consider an arbitrary Lagrangian Eulerian formulation of the fluid problem and a fully Lagrangian formulation of the solid problem; we rely on a finite element discretization of both fluid and solid equations. The distinctive feature of our approach is an implicit coupling of fluid and structural problems that relies on the solution to a constrained optimization problem with equality constraints. We discuss the application of projection-based model reduction to both fluid and solid subproblems: we rely on Galerkin projection for the solid equations and on least-square Petrov-Galerkin projection for the fluid equations. Numerical results for three model problems illustrate the many features of the formulation.

Optimization-based model order reduction of fluid-structure interaction problems

TL;DR

This work develops an optimization-based framework for fluid-structure interaction that couples full-order and reduced-order models through an implicit equality-constrained problem, using the interface flux as control and SQP for solution. It adopts projection-based MOR with Galerkin for the solid and LSPG for the fluid, complemented by an enrichment strategy to stabilize the coupled reduced system and allow independent reduction of fluid and solid components. Numerical results on three model problems (including a Turek-type benchmark) demonstrate accurate state reconstruction, robustness of the ROM and hybrid ROM-FOM solvers, and the benefits of SQP over DtN-like approaches for moderate control dimensions. The framework offers a path toward efficient many-query FSI simulations and hybrid solver strategies, while identifying challenges in long-time stability and suggesting avenues such as energy-conserving time integrators and space-time MOR to further enhance performance.

Abstract

We introduce optimization-based full-order and reduced-order formulations of fluid structure interaction problems. We study the flow of an incompressible Newtonian fluid which interacts with an elastic body: we consider an arbitrary Lagrangian Eulerian formulation of the fluid problem and a fully Lagrangian formulation of the solid problem; we rely on a finite element discretization of both fluid and solid equations. The distinctive feature of our approach is an implicit coupling of fluid and structural problems that relies on the solution to a constrained optimization problem with equality constraints. We discuss the application of projection-based model reduction to both fluid and solid subproblems: we rely on Galerkin projection for the solid equations and on least-square Petrov-Galerkin projection for the fluid equations. Numerical results for three model problems illustrate the many features of the formulation.

Paper Structure

This paper contains 26 sections, 1 theorem, 77 equations, 17 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Approximation of fluid structure interaction problems
  3. Optimization-based model reduction of fluid structure interaction problems
  4. Governing equations
  5. Strong form of the equations
  6. Weak formulation
  7. Discrete formulation
  8. Finite element spaces
  9. Semi-discrete formulation
  10. Fully-discrete formulation
  11. Optimal control formulation
  12. Discussion
  13. Reduced-order formulation
  14. Data compression
  15. Definition of the local reduced-order models
  16. ...and 11 more sections

Key Result

Lemma 1

Let the coupled fluid-structure system eq:weak_formulation_semidiscrete be isolated; i.e., ${u}_{\rm f}=0$ on $\partial \Omega_{\rm f}(t)\setminus\Gamma(t)$, $\widetilde{\sigma}_{\rm s} \widetilde{n}_{\rm s}=0$ on $\partial \widetilde{\Omega}_{\rm s}\setminus\widetilde{\Gamma}$. Assume that $\wideti

Figures (17)

  • Figure 1: referential and current domains of FSI problems. Here, $\Phi_{\rm f}(t)$ is the ALE bijection that maps the reference fluid domain $\widetilde{\Omega}_{\rm f}$ into the current domain $\Omega_{\rm f}(t)$.
  • Figure 2: elastic beam; computational meshes.
  • Figure 3: elastic beam; behavior of the streamwise velocity for three time instants.
  • Figure 4: elastic beam; performance of the global ROM. (a) energy content of the discarded POD modes. (b)-(c) relative error and number of modes for different choices of the tolerance ${\rm tol}_{\rm pod}$ with ${\rm tol}_{\rm en}=0.1$.
  • Figure 5: elastic beam; performance of the global ROM. (a) horizontal displacement of the top-right corner of the structure. (b)-(c) drag and lift force.
  • ...and 12 more figures

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof