Challenges for the Least-Squares Finite Element Method in Solid Mechanics

Abstract

In this contribution we discuss the mixed least-squares finite element method (LSFEM) for applications in solid mechanics. The LSFEM is characterized by the minimization of the sum of the squared L2(B) norms of the residuals of a first order system of differential equations. For an analysis of the LSFEM, we discuss two applications in solid mechanics, with a focus on the challenges of the method. Here, we investigate the moderate performance for low order elements and the sensitive point of choosing the weighting factors and their balancing. Furthermore, we discuss the crucial point of the recalculation of support reactions. In this context we introduce two formulations, one classical approach and one with the balance of angular momentum as an additional constraint instead of simple introducing a condition for the symmetry of the stress tensor in order to fulfill the balance of angular momentum. It should be noted, that the symmetry of the stress tensor is not a priori fulfilled due to the application of Raviart-Thomas functions. A further aspect is the consistent approximation of stresses at material interfaces when utilizing Lagrange ansatz functions within the LSFEM. Therefore, we present a hybrid mixed formulation on the basis of a LS functional.