Variational formulation of the finite calculus equations in solid mechanics and diffusion-reaction problems

Abstract

We present a variational formulation of the finite calculus (FIC) equations for problems in mechanics governed by differential equations with symmetric operators. Applications considered include solid mechanics, diffusion-transport and diffusion-reaction problems. The key of the variational formulation is the identification of the FIC governing equations with the classical differential equations of mechanics written in terms of modified non-local variables. A total potential energy (TPE) functional is found in terms of the modified variables. The FIC equations in the domain and the boundary are recovered as the Euler-Lagrange equations and the natural boundary condition of the TPE functional, respectively. Symmetric finite element equations are obtained after discretization of the TPE functional, therefore preserving the symmetry of the governing infinitesimal equations. The variational FIC expression is reinterpreted as a Petrov Galerkin weighted residual form of the original FIC equations with non-local weighting functions. The analogy of the variational FIC-FEM formulation with a discontinuous Galerkin method is recognized. Extensions to multidimensional linear elastostatics and diffusion-reaction problems are presented.