Ill-conditioned and multiscale partial differential equations (PDEs) arise in many fields. It is a very challenging problem to compute a resolved, fine-scale solution or to find a robust low-dimensional approximation. In this paper, the first large-scale application of multiscale-spectral generalized finite element methods (MS-GFEM) to composite aero-structures is presented. A problem-specific coarse space generated from local eigenproblems yields a spectral element-type method with excellent approximation properties at low basis size, even for challenging multiscale problems. The implementation of the framework in the DUNE software package, as well as a detailed description of all components of the method are presented and exemplified on a composite laminated beam under compressive loading. The excellent parallel scalability of the method, as well as its superior performance compared to the related, previously introduced GenEO method are demonstrated on two realistic application cases. Further, by allowing low-cost approximate solves for closely related models or geometries this efficient, novel technology provides the basis for future applications in optimisation or uncertainty quantification on challenging problems in composite aero-structures.
翻译:在许多领域都出现了有限制和多尺度的局部偏差方程(PDE),计算一个已解决的、微小的解决方案或找到一个稳健的低维近似值是一个极具挑战性的问题。本文介绍了对复合空气结构首次大规模应用多光谱通用有限元素方法(MS-GFEM)的情况。由本地的单质问题生成的一个因问题而异的空间产生一种光谱元素型方法,其光准特性极小,即使是对具有挑战性的多尺度问题也是如此。DUNE软件包中的框架的实施,以及该方法所有组成部分的详细说明,都以综合压载层成形束为示例。该方法的极平行性及其优异性,在两个现实的应用案例中得到了证明。此外,由于允许为密切相关的模型或近似特性提供低成本的近似解决办法,这一高效的新技术为今后在复合空气结构中具有挑战性的问题进行优化或不确定性的量化提供了基础。