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，我们重点关注小应变范围内确定性和随机弹性和非弹性问题的两尺度建模，例如各向同性硬化的弹塑性。在微观尺度上，我们考虑基于快速傅立叶变换 (FFT) 的求解器，从而实现非常高效的基于三维图像的研究。基于我们之前在 Fourier-Galerkin 设置中基于变分 FFT 技术的结果，我们考虑了与 Moulinec-Suquet 算法等效的方案和基于精确积分的改进方案。对于宏观尺度问题，我们考虑采用类似 FE2 耦合程序的标准有限元方法以及单元网格 (MIEL) 方法，该方法允许在不假设尺度分离的情况下处理问题。该项目将进一步增加基于 FFT 的求解器（例如预处理、低阶技术加速）针对微观问题的计算效率，这些问题将使用随机材料场进行建模。在随机设置中，将开发变分设置内的离散化和求解程序，提供宏观材料属性的概率描述。基于 FFT 的求解器将考虑替代边界条件，以使其不仅能够在 FE2 框架中耦合，而且还能在 MIEL 方法中耦合；随机性从微观到宏观的转移将特别令人感兴趣。对于耦合随机问题，将开发以线搜索和置信域算法为重点的二尺度拟牛顿方法。因此，将开发基于 FFT 的二尺度非线性问题的离散化和求解程序微观尺度的求解器和宏观尺度的 FEM 求解器。我们期望与捷克共和国研究小组的合作将显着提高具有真实微观结构表示的两尺度模拟的效率，使它们可以在传统计算机平台上访问。