基于流形自适应权重的经验求积法非线性有限元模型的维度超降维
Dimensional hyperreduction of nonlinear finite element models via empirical cubature with manifold-adaptive weights
浏览论文内容
中文总结 AI 辅助
该研究提出MAW-ECM方法，结合非线性流形表示，可大幅减少非线性有限元模型的采样积分点数量，同时基本保持精度。
中文摘要 AI 辅助
针对参数化有限元问题的非线性流形降阶模型，在广义（潜）坐标数量上可实现大幅压缩，且通过采样-加权超降维方法，采样单元/积分点的数量也可压缩。然而，当前采样-加权方法采用的权重在解流形上保持固定，我们认为这一限制导致超降维潜力未被充分挖掘：允许权重随潜坐标连续且非线性变化，可进一步减少采样空间实体的数量。为利用这一可能性，我们提出流形自适应权重经验求积法（MAW-ECM）。该方法从可行的固定权重ECM规则出发，通过凸二次权重再分配问题实施贪婪剪枝策略，在满足局部条件和正性的前提下移除采样实体。我们在两个非线性基准问题上评估该方法：表现出负增量刚度的超材料单胞均匀化问题，以及依赖历史的连续体损伤问题。在这两个问题中，非线性流形均通过初始线性压缩构建，随后根据输入信息将潜坐标识别为保留模态振幅的一般线性组合，相关时融入图信息以确定解流形的本征维度。研究表明，将非线性流形表示与MAW-ECM结合，相比对应标准线性降阶模型，采样积分点数量减少两个数量级以上；仅与固定权重流形模型相比，自适应权重在均匀化基准中消除了剩余约80%的点，在损伤基准中消除了剩余97%以上的点，同时基本保持精度。
英文摘要
Nonlinear-manifold reduced-order models for parametrized finite element problems can achieve substantial compression both in the number of generalized (latent) coordinates and, through sampling-and-weighting hyperreduction, in the number of sampled elements/integration points. Yet current sampling-and-weighting approaches employ weights that remain fixed over the solution manifold. We contend that this restriction leaves hyperreduction potential untapped: allowing the weights to vary continuously and nonlinearly with the latent coordinates can further decrease the number of sampled spatial entities. To exploit this possibility, we propose the Manifold-Adaptive-Weight Empirical Cubature Method (MAW-ECM). Starting from a feasible fixed-weight ECM rule, a greedy pruning strategy removes sampled entities through convex quadratic weight-redistribution problems enforcing local conditions and positivity. The method is assessed on two nonlinear benchmarks: homogenization of a metamaterial unit cell exhibiting negative incremental stiffness, and a history-dependent continuum-damage problem. In both cases, the nonlinear manifold is constructed from an initial linear compression followed by an input-informed identification of the latent coordinates as general linear combinations of the retained modal amplitudes, incorporating graph information when relevant to seek the intrinsic dimensionality of the solution manifold. We show that combining the nonlinear-manifold representation with MAW-ECM reduces the number of sampled integration points by more than two orders of magnitude relative to the corresponding standard linear reduced model. Relative to the fixed-weight manifold models alone, the adaptive weights eliminate approximately 80% of the remaining points in the homogenization benchmark and more than 97% in the damage benchmark, while essentially preserving their accuracy.
发表机构
- Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE)（国际工程数值方法中心）
- Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
- Stanford University（斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。
