Numerical Relaxation for Isotropic Damage
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Finite element simulations of pseudo-time-incremental damage models in the finite strain setting often suffer from mesh dependence and stability issues caused by the non-convexity of the underlying energy densities. This talk will focus on the computational relaxation of such a model by (semi)convexification. Therefore, algorithms for the approximation of semi-convex hulls, such as the polyconvex or the rank-one convex envelope, will be discussed. The computational complexity of (semi-)convexification algorithms is substantially affected by the computational dimension. Operating within the d x d dimensional space of the deformation gradient makes it unfeasible to perform a large finite element simulation at this stage due to the computational resources required. We present feasible methods to determine semiconvex envelopes in the presence of isotropy. These methods are based on the characterisation of the energy density in terms of signed singular values. In the case of polyconvexity, the practical algorithm only requires computing the lower convex envelope on a d-dimensional manifold, which can be achieved using computational geometry algorithms or an optimization approach. Reducing the dimensionality from d x d (matrix space) to d (signed singular value space), computation is significantly accelerated and this speedup makes the complete simulation of boundary value problems feasible. The numerical results of the relaxed isotropic damage model are presented with regard to the mesh sensitivity of the computed approximations.
