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

在这个项目中，我们专注于小应变状态下确定性和随机弹性和非弹性问题的双尺度建模，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里叶变换（FFT）的求解器，实现非常有效的基于三维图像的研究。基于我们之前在傅里叶-伽廖金环境下基于变分fft的技术的结果，我们考虑了一种相当于Moulinec-Suquet算法的方案和一种基于精确积分的改进方案。对于宏观尺度问题，我们考虑了具有FE2-like耦合过程的标准有限元方法以及允许在不假设尺度分离的情况下处理问题的单元网格（MIEL）方法。该项目将进一步提高基于fft的求解器（例如预处理，低秩加速技术）对微尺度问题的计算效率，这些问题将用随机材料场建模。在随机设置中，将开发变分设置中的离散化和解决程序，提供宏观材料特性的概率描述。将考虑基于fft的求解器的备选边界条件，以使其不仅在fe2框架中耦合，而且在MIEL方法中耦合；将随机性从微观尺度转移到宏观尺度将特别有趣。对于那些耦合的随机问题，将发展以线搜索和信任域算法为重点的双尺度拟牛顿方法。因此，将发展两尺度非线性问题的离散化和求解程序，在微观尺度上使用基于fft的求解器，在宏观尺度上使用FEM求解器。我们期望与捷克共和国研究小组的合作将导致具有真实微观结构表示的双尺度模拟效率的显着提高，使它们能够在传统的计算机平台上访问。

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.