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Learning Mesh Motion Techniques with Application to Fluid-Structure Interaction

Johannes Haubner, Ottar Hellan, Marius Zeinhofer, Miroslav Kuchta

TL;DR

This work tackles mesh degeneration in ALE-based FSI and PDE-constrained shape optimization by learning boundary-extension operators. It introduces two learning paradigms: (i) a hybrid PDE-NN where a neural network parametrizes a coefficient in a nonlinear PDE to extend boundary deformations, with well-posedness ensured by convexity-inspired designs and lagging nonlinearities; and (ii) an NN-corrected harmonic extension that adds a boundary-preserving, learned interior correction to the harmonic extension. Both approaches are embedded in a three-subsystem splitting of the monolithic FSI problem to isolate mesh motion, and are evaluated on the FSI benchmark II with training data generated from biharmonic and artificial deformers. Results indicate that learned operators can produce high-quality, bijective-like mesh mappings and can match or exceed traditional biharmonic extensions in stability, with the hybrid method showing strong generalization in gravity and membrane tests, albeit at higher computational cost. The study highlights the practical potential of learned mesh-motion operators for FSI and offers a foundation for further unsupervised or geometry-portable extensions.

Abstract

Mesh degeneration is a bottleneck for fluid-structure interaction (FSI) simulations and for shape optimization via the method of mappings. In both cases, an appropriate mesh motion technique is required. The choice is typically based on heuristics, e.g., the solution operators of partial differential equations (PDE), such as the Laplace or biharmonic equation. Especially the latter, which shows good numerical performance for large displacements, is expensive. Moreover, from a continuous perspective, choosing the mesh motion technique is to a certain extent arbitrary and has no influence on the physically relevant quantities. Therefore, we consider approaches inspired by machine learning. We present a hybrid PDE-NN approach, where the neural network (NN) serves as parameterization of a coefficient in a second order nonlinear PDE. We ensure existence of solutions for the nonlinear PDE by the choice of the neural network architecture. Moreover, we present an approach where a neural network corrects the harmonic extension such that the boundary displacement is not changed. In order to avoid technical difficulties in coupling finite element and machine learning software, we work with a splitting of the monolithic FSI system into three smaller subsystems. This allows to solve the mesh motion equation in a separate step. We assess the quality of the learned mesh motion technique by applying it to a FSI benchmark problem. In addition, we discuss generalizability and computational cost of the learned mesh motion operators.

Learning Mesh Motion Techniques with Application to Fluid-Structure Interaction

TL;DR

This work tackles mesh degeneration in ALE-based FSI and PDE-constrained shape optimization by learning boundary-extension operators. It introduces two learning paradigms: (i) a hybrid PDE-NN where a neural network parametrizes a coefficient in a nonlinear PDE to extend boundary deformations, with well-posedness ensured by convexity-inspired designs and lagging nonlinearities; and (ii) an NN-corrected harmonic extension that adds a boundary-preserving, learned interior correction to the harmonic extension. Both approaches are embedded in a three-subsystem splitting of the monolithic FSI problem to isolate mesh motion, and are evaluated on the FSI benchmark II with training data generated from biharmonic and artificial deformers. Results indicate that learned operators can produce high-quality, bijective-like mesh mappings and can match or exceed traditional biharmonic extensions in stability, with the hybrid method showing strong generalization in gravity and membrane tests, albeit at higher computational cost. The study highlights the practical potential of learned mesh-motion operators for FSI and offers a foundation for further unsupervised or geometry-portable extensions.

Abstract

Mesh degeneration is a bottleneck for fluid-structure interaction (FSI) simulations and for shape optimization via the method of mappings. In both cases, an appropriate mesh motion technique is required. The choice is typically based on heuristics, e.g., the solution operators of partial differential equations (PDE), such as the Laplace or biharmonic equation. Especially the latter, which shows good numerical performance for large displacements, is expensive. Moreover, from a continuous perspective, choosing the mesh motion technique is to a certain extent arbitrary and has no influence on the physically relevant quantities. Therefore, we consider approaches inspired by machine learning. We present a hybrid PDE-NN approach, where the neural network (NN) serves as parameterization of a coefficient in a second order nonlinear PDE. We ensure existence of solutions for the nonlinear PDE by the choice of the neural network architecture. Moreover, we present an approach where a neural network corrects the harmonic extension such that the boundary displacement is not changed. In order to avoid technical difficulties in coupling finite element and machine learning software, we work with a splitting of the monolithic FSI system into three smaller subsystems. This allows to solve the mesh motion equation in a separate step. We assess the quality of the learned mesh motion technique by applying it to a FSI benchmark problem. In addition, we discuss generalizability and computational cost of the learned mesh motion operators.
Paper Structure (24 sections, 8 theorems, 78 equations, 21 figures, 2 tables, 2 algorithms)

This paper contains 24 sections, 8 theorems, 78 equations, 21 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Approach 1: Hybrid NN-PDE Approach
  3. Choice of bar alpha
  4. Choice of the neural network
  5. Approximation Properties of Input Convex Neural Network Architectures
  6. Approach 2: NN-corrected Harmonic Extension Approach
  7. FSI example
  8. Splitting the FSI problem into smaller subproblems
  9. Discretization
  10. Numerical Results for FSI benchmark problem II
  11. Generation of Training Data
  12. Hybrid PDE-NN Approach
  13. NN-corrected harmonic extension approach
  14. Parameter study
  15. Generalizability
  16. ...and 9 more sections

Key Result

Lemma 1

Let $\Omega$ be a bounded Lipschitz domain, $p \geq 2$, $g \in W^{1,p}(\Omega)$. Moreover, let Then, there exists a unique minimizer of the optimization problem pLopt and the solution of the optimization problem is characterized by

Figures (21)

  • Figure 1: Mesh degeneration at the top corner of the elastic structure for harmonic extension (left), but not for the learned extension operator (middle; here shown for hybrid PDE-NN approach with artificial training data) and biharmonic extension (right).
  • Figure 2: Comparison of solutions to \ref{['eq:mask_poisson']} with $f=1$ (left) and the hand-tuned $f$\ref{['eq:mask_rhs_hand_tuned']} (right).
  • Figure 3: Numerical results for harmonic and biharmonic extensions.
  • Figure 4: The body deformations caused by the six base load configurations the artificial dataset is based on.
  • Figure 5: Visualization of learned neural net $\alpha(\theta, s)$ in the hybrid PDE-NN approach based on training data via FSI benchmark II (left) and artificial training data (right).
  • ...and 16 more figures

Theorems & Definitions (18)

  • Lemma 1: see lindqvist2017
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Remark 1
  • ...and 8 more