Goal-oriented adaptivity for nonlinear homogenization based on hierarchical models

基于分层模型的非线性均质化目标导向自适应性

• 批准号: 352532268
• 负责人: Dr.-Ing. Ismail Caylak, since 3/2024
• 金额: --
• 依托单位: Lehrstuhl für Technische Mechanik (LTM)
• 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 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/352532268?language=en
• 关键词: Goal; oriented; adaptivity; nonlinear; homogenization

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.

使用新型材料开发和生产创新产品需要深入了解用于安全设计部件和机器的模拟方法。复合材料等非均质材料在工业实践中的使用越来越多，使得使用均匀化技术的有限元（FE）模拟成为一种被广泛接受甚至是不可避免的工具。机器部件被设计为在宏观层面上服务，其可以通过标准有限元方法（FEM）来模拟，其（空间）离散化误差可以通过自适应网格细化来容易地控制。当材料在一定的长度尺度上（比如说微米）具有固有的不均匀性时，它就变得复杂得多。 人们必须在该尺度上处理这些异质性，然后执行尺度转换以获得宏观尺度上的整体行为，这通常被称为均匀化。为了减少与均匀化相关的计算成本，已经付出了相当大的努力，通常被认为是降阶均匀化。然而，所产生的模型误差通常不受控制。该项目涉及非线性均匀化问题的数值效率，仅在需要的局部宏域上选择性地使用耗时，精确的均匀化方法。为此，模型自适应，作为一个有前途的方法，将开发塑性问题。与自适应FEM非常相似，它从一个负担得起的均匀化方法开始，然后通过逐圈误差控制，执行到精确均匀化方法的局部切换以提高精度（称为模型细化）。与自适应有限元的层次有限元结构一样，为了模型自适应性，需要建立层次模型。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间依赖性都给误差估计带来了一些困难。为了实现完全误差控制，模型自适应性将与自适应时空有限元耦合。