The inverse deformation mapping in the finite element method

Abstract

This paper aims at the exploitation of material forces to find an optimum mesh in the finite element method (FEM). The classical variational formulation provides the linear momentum equation in a Lagrangian description. A variational setting for the derivation of the canonical momentum equation in the Eulerian description is presented. The latter is based on an extremum principle for the total potential energy functional defined in terms of the inverse deformation function. This constitutes a theoretical framework which allows the formulation of the finite element method for the canonical momentum equation as well as the computation of the material forces arising from the discretization. Thus, apart from the finite element solution for the standard boundary value problem of elastostatics, a second one for the canonical momentum equation can be formulated and solved numerically. The former provides an optimum deformation by minimizing the standard total potential energy, namely solving the physical forces equilibrium equation. The latter provides an optimum discretization by minimizing the total potential energy in terms of the inverse deformation function, that is, solving the material force equilibrium equation. The latter provides an optimum discretization by minimizing the total potential energy in terms of the inverse deformation function, that is, solving the material forces equilibrium equation. The theoretical considerations are supported by providing a computational example.