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

在这个项目中，我们专注于小应变区确定性和随机性弹性和非弹性问题的双尺度模拟，例如具有各向同性硬化的弹塑性问题。在微观上，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图像的三维研究。基于我们以前在傅立叶-伽辽金环境下基于变分FFT技术的结果，我们考虑了一种等价于Moulinec-Suquite算法的方案和一种基于精确积分的改进方案。对于宏观尺度的问题，我们考虑了具有类Fe2耦合过程的标准有限元方法和网格中单元(MIEL)方法，该方法可以在不假设尺度分离的情况下处理问题。该项目将进一步提高基于FFT的求解器(例如，预处理、低阶加速技术)用于微观尺度问题的计算效率，这些问题将被建模为随机物质场。在随机环境中，将开发变分环境中的离散化和求解程序，提供宏观材料属性的概率描述。对于基于FFT的求解器，将考虑不同的边界条件，以使其不仅在Fe2框架内，而且在MIEL方法中实现耦合；将随机性从微观尺度转移到宏观尺度将是特别有趣的。对于这些耦合的随机问题，将发展双尺度拟牛顿方法，重点是线搜索和信赖域算法，并在微观尺度上开发基于FFT的解算器和宏观尺度上的有限元解算器的两尺度非线性问题的离散化和求解程序。我们预计，与捷克共和国研究小组的合作将大大提高具有真实微观结构表示的双尺度模拟的效率，使其能够在传统的计算机平台上使用。

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.