Approximation of fracture energies with p-growth via piecewise affine finite elements

Sergio Conti, Matteo Focardi, Flaviana Iurlano
2018 E S A I M: Control, Optimisation and Calculus of Variations  
The modeling of fracture problems within geometrically linear elasticity is often based on the space of generalized functions of bounded deformation GSBD p (Ω), p ∈ (1, ∞), their treatment is however hindered by the very low regularity of those functions and by the lack of appropriate density results. We construct here an approximation of GSBD p functions, for p ∈ (1, ∞), with functions which are Lipschitz continuous away from a jump set which is a finite union of closed subsets of C 1
more » ... ets of C 1 hypersurfaces. The strains of the approximating functions converge strongly in L p to the strain of the target, and the area of their jump sets converge to the area of the target. The key idea is to use piecewise affine functions on a suitable grid, which is obtained via the Freudenthal partition of a cubic grid.
doi:10.1051/cocv/2018021 fatcat:ukvjrcrhmrdklablanh2uz4tuy