Discontinuous Galerkin approach for mechanically driven mass diffusion in elastic solids

Mechanically driven mass diffusion is characterized by a two-way interaction between mechanical and diffusive quantities: changes in mass concentration induce volumetric strain in the solid (swelling effect) and, vice versa, volumetric strain induces mass fluxes (piezo-diffusive effect). In the standard finite element setting, the presence of strain gradients in the piezo-diffusive coupling term demands C1 continuous displacement interpolation. Various techniques can be used to avoid C1 continuous shape functions. In the literature, the most used strategies for the solution of the problem are: a mixed approach, where the volumetric strain is interpolated as an independent variable (early explored in [1]), and a staggered solution strategy in conjunction with a smoothing L2 projection in the entire domain [2]. In this paper, a new formulation, which requires standard C0 interpolation for both displacement and concentration, is presented, basing on a discontinuous Galerkin approach. Recently, these approaches have been successfully used for strain gradient models [3]. The new formulation is discussed and compared with a mixed one and with a staggered solution strategy used employing a smoothing superconvergent patch-based recovery procedure, proposed in [4]. An error analysis is carried out to show the convergence rate. The discontinuous formulation exhibits convergence properties comparable to those of the mixed formulation, but allows to drastically reduce the computational cost with respect the other approaches. Some benchmarks are proposed to validate the formulation. References: 1. S. P. Girrens, F. W. Smith, "Finite element analysis of coupled constituent diffusion in thermoelastic solids", Computer Methods in Applied Mechanics in Engineering, v. 62, p. 209-223, 1987. 2. K. Garikipati, L. Bassman, M. Deal, "A lattice-based micromechanical continuum formulation for stress-driven mass transport in polycrystalline solids", Journal of the Mechanics and Physics of Solids, v. 49, p. 1209-1237, 2001. 3. L. Molari, G. N. Wells, K. Garikipati, F. Ubertini, "A Discontinuous Galerkin method for a strain gradient-dependent damage: Study of interpolations, convergence", Computer Methods in Applied Mechanics in Engineering, v.195, p. 1480-1498, 2006. 4. F. Ubertini, "Patch recovery based on complementary energy", International Journal for Numerical Methods in Engineering, v. 59, p. 1501-1538, 2004.