Numerical settings for mechanically driven mass diffusion

Mechanically driven mass diffusion is a fully coupled problem: changes in mass concentration induce strains in the solid (swelling effect), as well as strain gradients induce mass fluxes (piezodiffusive effect). In the standard finite element setting, the presence of strain gradients in the piezo-diffusive term demands C1 continuous displacement interpolation. In the literature, the most used strategies to avoid C1 continuous shape functions are: a mixed approach, where the volumetric strain is interpolated as an independent variable (early explored in [1]), and a staggered solution strategy with a smoothing L2 projection in the entire domain [2]. In this paper, two new formulations which require standard continuous interpolation for both displacement and concentration are presented, basing respectively on a discontinuous Galerkin approach (that has been successfully used for strain gradient models in [3]) and on a staggered solution strategy in conjunction with a smoothing superconvergent patch-based recovery procedure, proposed in [4]. These two approaches exhibit good convergence properties, comparable or superior to those of the mixed formulation, and allow to drastically reduce the computational cost. Some benchmarks are proposed to validate the formulations. References [1] Girrens SP, Smith FW, Finite element analysis of coupled constituent diffusion in thermoelastic solids, Comp. Meth. Appl. Mech. Eng, 62, 1987, 209-223. [2] Garikipati K, Bassman L, Deal M, A lattice-based micromechanical continuum formulation for stress-driven mass transport in polycrystalline solids, J. Mech. Phys. Sol., 49, 2001, 1209-1237. [3] Molari L, Wells GN, Garikipati K, Ubertini F, A discontinuous Galerkin method for strain gradient-dependent damage: Study of interpolations and convergence, Comp. Meth. Appl. Mech. Eng., 195, 2006, 1480-1498. [4] Ubertini F, Patch recovery based on complementary energy, Int. J. Num. Meth. Eng., 59, 2004, 1501-1538.