发表机构
Karlsruhe Institute of Technology; ETH Zürich（卡尔斯鲁厄理工学院; 苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对线弹性固体LBM提出首个梯度优化基准，含三个反问题，验证了ALBM与f-AD梯度一致性及二阶/一阶收敛，并在OpenLB中实现。
AI 中文摘要
受偏微分方程约束问题的梯度优化（GBO）是固体力学中参数识别与结构设计的核心。针对线弹性固体的格子玻尔兹曼方法（LBM）格式直至近期才被推导出来。结合自动生成的离散伴随碰撞核，这些格式使得纯LBM基的原问题与伴随求解器能够实现GBO。我们提出（据我们所知）首个针对固体LBM的优化基准：三个反问题，每个均具有正问题的人为构造解和已知参考控制。这些基准包括：周期域上体力场两个标量幅值的识别、同一场（自由度高达12800）的分布控制，以及具有Dirichlet和Neumann边界的椭圆板上杨氏模量的识别。原问题格式在周期域上具有二阶收敛性，在椭圆板上具有一阶收敛性。恢复的控制遵循这些收敛阶，实验收敛阶分别为1.92和1.06。通过前向模式自动微分（f-AD）和离散伴随LBM（ALBM）获得的梯度所需优化步数相同，且控制误差至少一致到十一位有效数字。ALBM梯度进一步针对有限差分商逐节点验证，偏差随网格细化而减小。这些基准已在开源LBM框架OpenLB中实现，为基于LBM的固体力学优化及其向流固耦合的扩展提供了验证基础。
英文摘要
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.
