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Numerical Methods for Shape Optimal Design of Fluid-Structure Interaction Problems

Johannes Haubner, Michael Ulbrich

TL;DR

This paper develops a numerical framework for shape optimization of unsteady fluid-structure interaction (FSI) problems using the method of mappings. Shapes are parameterized by a bi-Lipschitz transformation and optimized on a fixed reference domain, enabling exact discrete gradients and compatibility with general-purpose solvers like IPOPT. An ALE-based FSI model is formulated, with a weak formulation on the nominal domain and a carefully designed objective (volume drag) and penalties to maintain mesh quality and regularity. Numerical results on the FSI2 benchmark demonstrate substantial drag reductions when optimizing the obstacle boundary and the interface, validating the approach and its potential for robust, mesh-controlled shape optimization in complex FSI settings.

Abstract

We consider the method of mappings for performing shape optimization for unsteady fluid-structure interaction (FSI) problems. In this work, we focus on the numerical implementation. We model the optimization problem such that it takes several theoretical results into account, such as regularity requirements on the transformations and a differential geometrical point of view on the manifold of shapes. Moreover, we discretize the problem such that we can compute exact discrete gradients. This allows for the use of general purpose optimization solvers. We focus on an FSI benchmark problem to validate our numerical implementation. The method is used to optimize parts of the outer boundary and the interface. The numerical simulations build on FEniCS, dolfin-adjoint and IPOPT. Moreover, as an additional theoretical result, we show that for a linear special case the adjoint attains the same structure as the forward problem but reverses the temporal flow of information.

Numerical Methods for Shape Optimal Design of Fluid-Structure Interaction Problems

TL;DR

This paper develops a numerical framework for shape optimization of unsteady fluid-structure interaction (FSI) problems using the method of mappings. Shapes are parameterized by a bi-Lipschitz transformation and optimized on a fixed reference domain, enabling exact discrete gradients and compatibility with general-purpose solvers like IPOPT. An ALE-based FSI model is formulated, with a weak formulation on the nominal domain and a carefully designed objective (volume drag) and penalties to maintain mesh quality and regularity. Numerical results on the FSI2 benchmark demonstrate substantial drag reductions when optimizing the obstacle boundary and the interface, validating the approach and its potential for robust, mesh-controlled shape optimization in complex FSI settings.

Abstract

We consider the method of mappings for performing shape optimization for unsteady fluid-structure interaction (FSI) problems. In this work, we focus on the numerical implementation. We model the optimization problem such that it takes several theoretical results into account, such as regularity requirements on the transformations and a differential geometrical point of view on the manifold of shapes. Moreover, we discretize the problem such that we can compute exact discrete gradients. This allows for the use of general purpose optimization solvers. We focus on an FSI benchmark problem to validate our numerical implementation. The method is used to optimize parts of the outer boundary and the interface. The numerical simulations build on FEniCS, dolfin-adjoint and IPOPT. Moreover, as an additional theoretical result, we show that for a linear special case the adjoint attains the same structure as the forward problem but reverses the temporal flow of information.
Paper Structure (29 sections, 76 equations, 4 figures, 2 tables)

This paper contains 29 sections, 76 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Method of mappings
  3. Theoretical challenges
  4. Algorithmic realization
  5. Requirements on software framework
  6. Related work
  7. Fluid-Structure Interaction Problem
  8. Outline
  9. FSI Model for Numerical Simulations
  10. ALE Transformation
  11. Strong ALE Formulation
  12. Weak ALE Formulation
  13. Model of Shape Optimization Problem for FSI
  14. Transformation of FSI Equations to Nominal Domain
  15. Choice of Objective Function
  16. ...and 14 more sections

Figures (4)

  • Figure 1: Schematic illustration of the method of mappings combined with an ALE transformation
  • Figure 2: Shape reference domain $\tilde{\Omega}$
  • Figure 3: Optimization of shape of obstacle (Section \ref{['sec::61']}): Comparison of vertical displacement of the tip of the flap for the initial (top left) and optimized (bottom left) design
  • Figure 4: Optimization of interface (Section \ref{['sec::62']}): Comparison of vertical displacement of the tip of the flap for the initial (top left) and optimized (bottom left) design. The initial displacement of the tip of the flap for the optimized design is not plotted as zero, because the tip of the optimized flap is located at (0.597718, 0.20417) instead of (0.6,0.2).

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5