Upscaling and reliable two-scale Fourier/finite element-based simulations

升级且可靠的基于两尺度傅里叶/有限元的模拟

基本信息

批准号：
324231889
负责人：
Professor Dr. Hermann Georg Matthies
金额：
--
依托单位：
Institut für Wissenschaftliches Rechnen
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2017
资助国家：
德国
起止时间：
2016-12-31 至 2021-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/324231889?language=en
关键词：
Upscaling reliable two scale Fourier

项目摘要

In this project, we focus on the two-scale modelling of deterministic and also stochastic elastic and inelastic problems in the small strain regime, such as elasto-plasticity with isotropic hardening. On the micro-scale we consider Fast Fourier Transform (FFT)-based solvers, enabling very efficient three-dimensional image-based studies. Building on our previous results on variational FFT-based techniques in the Fourier-Galerkin setting, we consider a scheme equivalent to the Moulinec-Suquet algorithm and an improved scheme based on exact integration. For a macro-scale problem, we consider the standard finite element method with FE2-like coupling procedures and also the mesh-in-element (MIEL) method, which allows to treat problems without the assumption of scale separation.The project will further increase the computational efficiency of FFT-based solvers (e.g. preconditioning, acceleration by low-rank techniques) for micro-scale problems, which will be modelled with random material fields. In the stochastic setting, discretisation and solution procedures within a variational setting will be developed, providing a probabilistic description of macro material properties. Alternative boundary conditions will be considered for FFT-based solvers to enable its coupling not only in the FE2-framework, but also in the MIEL method; the transfer of randomness from micro- to macro-scale will be of particular interest. For those coupled stochastic problems, two-scale quasi-Newton methods with emphasis on line-search and trust-region algorithms will be developed.As a result, the discretisation and solution procedures will be developed for two-scale nonlinear problems with FFT-based solvers on the micro-scale and FEM solvers on the macro-scale. We expect that the collaboration with the research group in the Czech Republic will lead to a significant increase in the efficiency of two-scale simulations with realistic microstructural representations, making them accessible on conventional computer platforms.

在这个项目中，我们着重于小应变状态中确定性和随机弹性和非弹性问题的两尺度建模，例如以及各向同性硬化的弹性塑性。在微观尺度上，我们考虑了基于快速的傅立叶变换（FFT）的求解器，从而实现了非常有效的三维基于图像的研究。在我们先前基于傅立叶 - 盖尔金环境中基于变异FFT的技术的结果的基础上，我们考虑了一种相当于Moulinec-Suquet算法的方案，并且基于精确集成的改进方案。对于宏观尺度问题，我们考虑使用类似Fe2的耦合程序以及元件（MIEL）方法的标准有限元方法，该方法允许在不假定规模分离的情况下处理问题。项目将进一步提高基于FFT的solvers的计算效率（例如，在较低的材料领域）对材料进行材料的随机范围（例如，对型号的材料）进行了模型，该材料会导致材料的模型。在随机设置中，将开发在变异环境中的离散和解决方案程序，从而提供对宏观材料属性的概率描述。将考虑基于FFT的求解器的替代边界条件，不仅可以在Fe2-Framework中，还可以在MIEL方法中启用其耦合。从微尺度到宏观尺度的随机性将特别感兴趣。对于那些耦合的随机问题，将开发出两尺度的准Newton方法，重点是搜索线和信任区域算法。结果，将开发出针对宏观规模上的基于FFT的两个规模的非线性问题的离散化和解决方案程序。我们预计，与捷克共和国的研究小组的合作将导致具有现实的微观结构表示的两尺度模拟的效率显着提高，从而使其在传统的计算机平台上访问。