DFG Priority Programme 1748: Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretization Methods, Mechanical and Mathematical Analysis
Reliability of efficient approximation schemes for material discontinuities described by functions of bounded variation
Spaces of functions of bounded variations provide an attractive framework to describe material discontinuities such as damage and fracture. In the recent past, suitable notions of solution and general existence theories for corresponding evolutionary model problems have been established and numerical methods for discretizing and iteratively solving variational problems involving the total variation norm have been developed and analyzed.
In a first funding phase, abstract existence results for coupled rate-dependent/rate-independent systems, delamination processes in visco-elastodynamics, and phasefield descriptions of damage evolutions have been investigated analytically.
Algorithmic contributions have been made to the iterative solution of model problems on functions of bounded variation, adaptive approximation of discontinuous functions based on fully computable a posteriori error estimates, and the convergent finite element simulation of a BV-regularized damage model.
The aim is now to combine, refine and extend these results to complex models describing damage and fracture evolution. The envisaged models shall capture material discontinuities either in BV directly or as a scaling limit.
It is planned to derive a priori and a posteriori error estimates, construct adaptive approximation schemes intertwined with results based on existence theory and Gamma-convergence, and to implement and apply them to specific benchmark problems in mechanics. Particular attention will be paid to the reliability of efficient methods, e.g., convergence of suitable time-stepping and adaptive approximation schemes, intertwined with results based on existence theory and convergence.