应用于参数化有限元模型局部超还原的子空间自适应权重立方法

IF 2.7 3区 工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY International Journal for Numerical Methods in Engineering Pub Date : 2024-09-03 DOI:10.1002/nme.7590
J. R. Bravo, J. A. Hernández, S. Ares de Parga, R. Rossi
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引用次数: 0

摘要

本文关注的是能够处理多个子空间函数的正交/立方规则,其方式是所有子空间的积分点是共同的,而(非负)权重则是为每个特定子空间量身定制的。这些子空间自适应权重立方规则可用于加速计算力学应用,这些应用要求高效评估空间积分,其积分函数可在多个预计算子空间之间动态切换。其中一种应用是局部超还原阶建模(HROM),其中解流形近似表示为基矩阵集合,每个基矩阵对应参数空间中的不同区域。所提出的优化框架在积分点位置方面是离散的,也就是说,积分点是从给定有限元网格的高斯点中选择的,而函数的目标子空间则由正交基矩阵表示,正交基矩阵是利用奇异值分解(SVD)从这些高斯点的函数值中构造出来的。这种离散框架使我们也能处理积分近似为每个有限元贡献的加权和的问题,如 C. Farhat 及其合作者的能量守恒采样和加权方法。本文研究了两种不同的求解策略。第一种是基于作者在其他地方开发的经验立方法(ECM)增强版的贪婪策略(我们称之为子空间自适应权重立方法,简称 SAW-ECM),而第二种方法则基于立方问题的凸化,因此可以通过线性规划算法来解决。我们在一个涉及多项式函数积分的玩具问题中表明,SAW-ECM 在计算成本和最优性方面都明显优于另一种方法。另一方面,我们说明了 SAW-ECM 在高度非线性平衡问题(大应变机制)中构建局部 HROMs 的性能。我们证明,只要子空间转换误差可以忽略不计,使用自适应权重进行超还原的相关误差就可以通过用于确定基矩阵的 SVD 的截断公差来控制。我们还证明,随着子空间数量的增加,积分点的数量也会明显减少,而且在使用尽可能多的子空间作为快照的极限情况下,SAW-ECM 提供的规则的积分点数量仅取决于解流形的内在维度和避免子空间转换误差所需的重叠程度。拟议的 SAW-ECM 的 Python 源代码可在公共存储库 https://github.com/Rbravo555/localECM 中公开访问。
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A subspace-adaptive weights cubature method with application to the local hyperreduction of parameterized finite element models

This article is concerned with quadrature/cubature rules able to deal with multiple subspaces of functions, in such a way that the integration points are common for all the subspaces, yet the (nonnegative) weights are tailored to each specific subspace. These subspace-adaptive weights cubature rules can be used to accelerate computational mechanics applications requiring efficiently evaluating spatial integrals whose integrand function dynamically switches between multiple pre-computed subspaces. One of such applications is local hyperreduced-order modeling (HROM), in which the solution manifold is approximately represented as a collection of basis matrices, each basis matrix corresponding to a different region in parameter space. The proposed optimization framework is discrete in terms of the location of the integration points, in the sense that such points are selected among the Gauss points of a given finite element mesh, and the target subspaces of functions are represented by orthogonal basis matrices constructed from the values of the functions at such Gauss points, using the singular value decomposition (SVD). This discrete framework allows us to treat also problems in which the integrals are approximated as a weighted sum of the contribution of each finite element, as in the energy-conserving sampling and weighting method of C. Farhat and co-workers. Two distinct solution strategies are examined. The first one is a greedy strategy based on an enhanced version of the empirical cubature method (ECM) developed by the authors elsewhere (we call it the subspace-adaptive weights ECM, SAW-ECM for short), while the second method is based on a convexification of the cubature problem so that it can be addressed by linear programming algorithms. We show in a toy problem involving integration of polynomial functions that the SAW-ECM clearly outperforms the other method both in terms of computational cost and optimality. On the other hand, we illustrate the performance of the SAW-ECM in the construction of a local HROMs in a highly nonlinear equilibrium problem (large strains regime). We demonstrate that, provided that the subspace-transition errors are negligible, the error associated to hyperreduction using adaptive weights can be controlled by the truncation tolerances of the SVDs used for determining the basis matrices. We also show that the number of integration points decreases notably as the number of subspaces increases, and that, in the limiting case of using as many subspaces as snapshots, the SAW-ECM delivers rules with a number of integration points only dependent on the intrinsic dimensionality of the solution manifold and the degree of overlapping required to avoid subspace-transition errors. The Python source codes of the proposed SAW-ECM are openly accessible in the public repository https://github.com/Rbravo555/localECM.

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来源期刊
CiteScore
5.70
自引率
6.90%
发文量
276
审稿时长
5.3 months
期刊介绍: The International Journal for Numerical Methods in Engineering publishes original papers describing significant, novel developments in numerical methods that are applicable to engineering problems. The Journal is known for welcoming contributions in a wide range of areas in computational engineering, including computational issues in model reduction, uncertainty quantification, verification and validation, inverse analysis and stochastic methods, optimisation, element technology, solution techniques and parallel computing, damage and fracture, mechanics at micro and nano-scales, low-speed fluid dynamics, fluid-structure interaction, electromagnetics, coupled diffusion phenomena, and error estimation and mesh generation. It is emphasized that this is by no means an exhaustive list, and particularly papers on multi-scale, multi-physics or multi-disciplinary problems, and on new, emerging topics are welcome.
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