Energy principles and finite element methods for pure traction linear elasticity

Abstract

A conforming finite element discretization of the pure traction elasticity boundary value problem results in a singular linear system of equations. The singularity of the linear system is removed through various methods. In this report, we consider an alternative approach in which discrete finite element formulations are derived directly from the principle of minimum potential energy, rather than from the weak form of the differential equations. This point of view turns out to be particularly well suited to handle the rigid body modes, which are the source of the singularity in the finite element linear system. Our approach allows us to formulate a regularized potential energy principle and prove that the weak problem corresponding to its first-order optimality condition is coercive in H1(Ω). This guarantees nonsingular algebraic problems, enables simplified solution algorithms and leads to more efficient and robust preconditioners for the iterative solution linear equations. © 2011 Institute of Mathematics.