Benchmark cases for gradient-based optimization in linear elastic solid mechanics using lattice Boltzmann methods
Abstract
Gradient-based optimization (GBO) of problems constrained by partial differential equations is central to parameter identification and structural design in solid mechanics. Lattice Boltzmann method (LBM) schemes for linear elastic solids have only recently been derived. Combined with automatically generated discrete adjoint collision kernels, they make GBO possible with a purely LBM-based primal and adjoint solver. We propose, to our knowledge, the first optimization benchmarks for solid LBM: three inverse problems, each with a manufactured solution of the forward problem and a known reference control. These are the identification of two scalar amplitudes of a body force field on a periodic domain, the distributed control of the same field with up to $12800$ degrees of freedom, and the identification of Young's modulus on an elliptic plate with Dirichlet and Neumann boundaries. The primal scheme converges with second order on the periodic domain and with first order on the elliptic plate. The recovered controls follow these rates, with experimental orders of convergence of $1.92$ and $1.06$, respectively. Gradients obtained via forward-mode automatic differentiation (f-AD) and the discrete adjoint LBM (ALBM) require the same number of optimization steps and yield control errors agreeing to at least eleven significant digits. The ALBM gradient is further validated node-wise against a finite difference quotient, with deviations decreasing under mesh refinement. The benchmarks are implemented in the open-source LBM framework OpenLB and provide a verification basis for optimization in LBM-based solid mechanics and its extension to fluid-structure interaction.
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BibTeX
@misc{grafen2026benchmark,
title = {{Benchmark cases for gradient-based optimization in linear elastic solid mechanics using lattice Boltzmann methods}},
author = {Johannes L. Grafen and Florian Kaiser and Stephan Simonis and Shota Ito and Mathias J. Krause},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31347},
eprint = {2609.31347},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31347},
}
RIS
TY - UNPB
TI - Benchmark cases for gradient-based optimization in linear elastic solid mechanics using lattice Boltzmann methods
AU - Johannes L. Grafen
AU - Florian Kaiser
AU - Stephan Simonis
AU - Shota Ito
AU - Mathias J. Krause
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31347
UR - https://arxiv.org/abs/2609.31347
AB - Gradient-based optimization (GBO) of problems constrained by partial differential equations is central to parameter identification and structural design in solid mechanics. Lattice Boltzmann method (LBM) schemes for linear elastic solids have only recently been derived. Combined with automatically generated discrete adjoint collision kernels, they make GBO possible with a purely LBM-based primal and adjoint solver. We propose, to our knowledge, the first optimization benchmarks for solid LBM: three inverse problems, each with a manufactured solution of the forward problem and a known reference control. These are the identification of two scalar amplitudes of a body force field on a periodic domain, the distributed control of the same field with up to $12800$ degrees of freedom, and the identification of Young's modulus on an elliptic plate with Dirichlet and Neumann boundaries. The primal scheme converges with second order on the periodic domain and with first order on the elliptic plate. The recovered controls follow these rates, with experimental orders of convergence of $1.92$ and $1.06$, respectively. Gradients obtained via forward-mode automatic differentiation (f-AD) and the discrete adjoint LBM (ALBM) require the same number of optimization steps and yield control errors agreeing to at least eleven significant digits. The ALBM gradient is further validated node-wise against a finite difference quotient, with deviations decreasing under mesh refinement. The benchmarks are implemented in the open-source LBM framework OpenLB and provide a verification basis for optimization in LBM-based solid mechanics and its extension to fluid-structure interaction.
ER -