Goal-oriented adaptivity for nonlinear homogenization based on hierarchical models

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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.