Numerical algorithms for the simulation of finite plasticity with microstructures

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The occurrence of microstructures in solid mechanics and, in particular, in finite plasticity can be attributed to a loss of the convexity of the underlying energy potentials. While the material deforms macroscopically, structures in the form of shear bands, cracks or lami- nates arise on microscopic scales. Common to these examples is that their macroscopic simulations have to be based on the quasiconvexification of the energy functional.The projects within the research group either concern the modelling or the simulation. In contrast, the object of this project is the justification of computer simulations with an analysis of discretisation and design of converging adaptive mesh-refining algorithms. The mathematical justification concerns numerical simulations on the microscopic scale (a), on the macroscopic scale (b), for time-evolving microstructure (c).In the first funding period, an efficient algorithm on the numerical relaxation in single- crystal finite plasticity has been established in (a). The degenerate nature of the (quasi-) convexified variational model in (b) required novel stabilisation techniques on adapted finite element grids. The influence of the perturbation of the computed macroscopic energy density W is examined in the combination of (a) and (b). The main result in the analysis of perturbed minimisation problems guarantees convergence of an adaptive mesh-refining algorithm for asymptotically exact computation of energies.The project continues to investigate the convergence of numerical simulations of rate- independent evolution problems for the full time-space discretisation (c). The second funding period shall investigate the improvements of nonstandard finite element methods in (a), (b), and (c).