Variationally consistent computational homogenization of chemomechanical problems with stabilized weakly periodic boundary conditions
Stefan Kaessmair, Kenneth Runesson, Paul Steinmann, Ralf Jänicke, Fredrik Larsson · International Journal for Numerical Methods in Engineering · 2021
Abstract A variationally consistent model‐based computational homogenization approach for transient chemomechanically coupled problems is developed based on the classical assumption of first‐order prolongation of the displacement, chemical potential, and (ion) concentration fields within a representative volume element (RVE). The presence of the chemical potential and the concentration as primary global fields represents a mixed formulation, which has definite advantages. Nonstandard diffusion, governed by a Cahn–Hilliard type of gradient model, is considered under the restriction of miscibility. Weakly periodic boundary conditions on the pertinent fields provide the general variational setting for the uniquely solvable RVE‐problem(s). These boundary conditions are introduced with a novel approach in order to control the stability of the boundary discretization, thereby circumventing the need to satisfy the LBB‐condition: the penalty stabilized Lagrange multiplier formulation, which enforces stability at the cost of an additional Lagrange multiplier for each weakly periodic field (three fields for the current problem). In particular, a neat result is that the classical Neumann boundary condition is obtained when the penalty becomes very large. In the numerical examples, we investigate the following characteristics: the mesh convergence for different boundary approximations, the sensitivity for the choice of penalty parameter, and the influence of RVE‐size on the macroscopic response.