Variationally consistent computational homogenization of chemomechanical problems with stabilized weakly periodic boundary conditions
Journal article, 2021

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.


Dirichlet and Neumann RVE-conditions

variationally consistent

computational homogenization

weak periodicity

chemomechanical coupling


Stefan Kaessmair

University of Erlangen-Nuremberg (FAU)

Kenneth Runesson

Chalmers, Industrial and Materials Science, Material and Computational Mechanics

Paul Steinmann

University of Erlangen-Nuremberg (FAU)

R. Janicke

Technische Universität Braunschweig

Fredrik Larsson

Chalmers, Industrial and Materials Science, Material and Computational Mechanics

International Journal for Numerical Methods in Engineering

0029-5981 (ISSN) 1097-0207 (eISSN)

Vol. 122 22 6429-6454

Subject Categories

Applied Mechanics

Computational Mathematics

Mathematical Analysis



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4/5/2022 5