Goal-oriented adaptivity for nonlinear homogenization based on hierarchical models

使用新材料的创新产品的开发和生产需要对组件和机器的安全设计的模拟方法有深入的了解。在工业实践中，复合材料等非均质材料的使用越来越多，使得使用均质技术的有限元（FE）模拟成为一种被广泛接受的甚至是不可避免的工具。机械部件的设计服务于宏观层面，可以用标准有限元方法（FEM）进行模拟，其（空间）离散化误差可以通过自适应网格细化控制。当材料在一定的长度尺度上（比如微尺度）具有固有的非均质性时，问题就变得复杂得多。人们必须在那个尺度上处理这些异质性，然后进行尺度转换，以获得宏观尺度上的整体行为，这通常被称为均质化。人们已经付出了相当大的努力来减少与均质化相关的计算成本，通常被认为是降阶均质化。然而，产生的模型误差通常是不受控制的。本项目处理非线性均匀化问题的数值效率，选择使用耗时，精确的均匀化方法，只在需要的局部宏观域上。为此，模型自适应作为一种很有前途的方法，将在塑性问题中得到发展。与自适应有限元法非常相似，它从负担得起的均质方法开始，然后通过循环误差控制，局部切换到精确的均质方法，以提高精度（称为模型精化）。与自适应有限元的分层有限元结构一样，为了实现模型的自适应，需要建立分层模型。涵盖了各种各样的非线性均匀化方法，我们将考虑平均场，模型降阶和计算方法。塑性问题的非线性和时效性给误差估计带来了一定的困难。为了实现全误差控制，将模型自适应与自适应时空有限元相结合。

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.