Goal-oriented adaptivity for nonlinear homogenization based on hierarchical models

开发和生产使用新材料的创新产品需要深入了解用于部件和机器安全设计的模拟方法。随着复合材料等非均质材料在工业实践中的日益广泛应用，使用均匀化技术进行有限元模拟已成为公认的、甚至是不可避免的工具。机械零件设计用于宏观层面，可用标准有限元方法进行模拟，其(空间)离散化误差可通过自适应网格加密轻松控制。当材料在一定的长度尺度(比如微观)上具有固有的异质性时，情况就变得复杂得多。人们必须处理该尺度上的这些异质性，然后进行尺度转换，以获得宏观尺度上的整体行为，这通常被称为同质化。人们已经付出了相当大的努力来降低与齐化相关的计算成本，该齐化通常被认为是降阶齐化。然而，由此产生的模型误差通常是不受控制的。这个项目只在需要的地方使用耗时的、精确的均匀化方法来处理非线性齐化问题的数值效率。为此，模型自适应作为一种很有前途的方法，将被开发用于解决塑性问题。与自适应有限元方法非常相似的是，它从一种负担得起的均匀化方法开始，然后通过循环误差控制，局部切换到精确的均匀化方法以提高精度(称为模型精化)。与自适应有限元的层次化有限元结构一样，需要建立层次化模型以保证模型的自适应性。涵盖了大量的非线性齐化方法，我们将讨论平均场、模型降阶和计算方法。塑性问题的非线性和时变性都给误差估计带来了一些困难。为了实现完全误差控制，模型自适应将与自适应时空有限元相耦合。

The development and the production of innovative products using novel materials require in-depth knowledge of simulation methods for a safe design of components and machines. The increasing use of heterogeneous materials like composites in the industrial praxis has made the finite element (FE) simulation using homogenization techniques a well-accepted and often even inevitable tool. A machine component is designed to serve on a macro level, which may be simulated by means of standard finite element methods (FEM), whose (spatial) discretization errors can be easily controlled by an adaptive mesh refinement. It becomes much more complicated, when the material possesses inherent heterogeneities on a certain length scale (say micro). One has to deal with these heterogeneities on that scale and then perform scale transition to obtain the overall behavior on the macro scale, which is often referred to as homogenization. Considerable effort has been paid to reduce the computational cost associated with homogenization, often regarded as reduced order homogenization. However, the resulting model error is usually left uncontrolled. This project deals with the numerical efficiency of nonlinear homogenization problems with a selective use of time-consuming, accurate homogenization methods only on local macro domains where needed. To this end, model adaptivity, as a promising methodology, will be developed for plasticity problems. Quite similarly to the adaptive FEM, it starts with an affordable homogenization method, and then through a loopwise error control, a local switch to accurate homogenization methods is performed to enhance the accuracy (referred to as model refinement). Like hierarchical FE structures for adaptive FEM, hierarchical models need to be established for model adaptivity. Covering a large variety of nonlinear homogenization methods, we will consider mean-field, model order reduction and computational methods. Both nonlinearities and time-dependency of the plasticity problem rise some difficulties for error estimate to be addressed in this project. Towards a full error control, model adaptivity will be coupled to adaptive space-time finite elements.