Beiträge zur numerischen Homogenisierung mikroheterogener Materialien mittels Zwei-Skalen Finite Elemente Methoden

Abstract

Numerical homogenization includes methods for solving multiscale problems. The goal here is to solve the underlying problem on a macroscale, taking into account the influence of a heterogeneous microscale. Numerical homogenization methods thus provide a better understanding of the macroscopic problem while at the same time reducing the numerical effort compared to a complete analysis of the microscale. The Finite Element Heterogeneous Multiscale Method is one of these numerical homogenization methods and, through a complete a priori error analysis, provides a new and at the same time significant contribution to numerical homogenization by two-scale finite element methods. In this work and the publications on which it is based, the implementation of the Finite Element Heterogeneous Multiscale Method for vector-valued problems in the field of elasticity was described and numerous other aspects were examined. These include i) Confirmation of the a priori convergence statements for linear elasticity using numerous examples with different regularity with different coupling conditions of the two scales and linear and quadratic shape functions. ii) Comparison of the Finite Element Heterogeneous Multiscale Method with the FE² method. Although both methods are based on different approaches, the quantitative agreement of the results was shown using numerous examples, so that the a priori convergence statements can also be transferred to the FE² method. iii) While error analyzes in the literature usually deal with errors on the macroscale, this work also examined errors on the microscale. For this purpose, a distinction was made between the discretization error and a resolution error that takes the resolution of the microscale into account. iv) An investigation of the error estimation on the heterogeneous microscale provided an adapted error estimator algorithm with which the quality of the error estimation on the microscale can be significantly improved. v) Different adaptive and uniform mesh coarsening algorithms that can be used as a preprocessor to reduce the microscale degrees of freedom were examined based on their effects on the discretization and resolution errors. vi) In order to reduce the high numerical effort of non-linear two-scale problems, which results from the repeated solution of nested macro and micro equation systems, a new solution algorithm was introduced that can significantly speed up the calculations.