Approximations of energy minimization in cell-induced phase transitions of fibrous biomaterials: $Γ$-convergence analysis

We consider a model of energy minimization arising in the study of the mechanical behavior caused by cell contraction within a fibrous biological medium. The macroscopic model is based on the theory of non rank-one convex nonlinear elasticity for phase transitions. We study appropriate numerical approximations based on the discontinuous Galerkin treatment of higher gradients and used succesfully in numerical simulations of experiments. We show that the discrete minimizers converge in the limit to minimizers of the continuous problem. This is achieved by employing the theory of $\Gamma$-convergence of the approximate energy functionals to the continuous model when the discretization parameter tends to zero. The analysis is involved due to the structure of numerical approximations which are defined in spaces with lower regularity than the space where the minimizers of the continuous variational problem are sought. This fact leads to the development of a new approach to $\Gamma$-convergence, appropriate for discontinuous finite element discretizations, which can be applied to quite general energy minimization problems. Furthermore, the adoption of exponential terms penalising the interpenetration of matter requires a new framework based on Orlicz spaces for discontinuous Galerkin methods which is developed in this paper as well.