A mixed least‐squares finite element formulation with explicit consideration of the balance of moment of momentum, a numerical study

Abstract

Important conditions in structural analysis are the fulfillment of the balance of linear momentum (vanishing resultant forces) and the balance of angular momentum (vanishing resultant moment), which is not a priori satisfied for arbitrary element formulations. In this contribution, we analyze a mixed least-squares (LS) finite element formulation for linear elasticity with explicit consideration of the balance of angular momentum. The considered stress-displacement ( − u) formulation is based on the squared L 2 ()-norm minimization of the residuals of a first-order system of differential equations. The formulation is constructed by means of two residuals, that is, the balance of linear momentum and the constitutive equation. Motivated by the crucial point of weighting factors within LS formulations, a scale independent formulation is constructed. The displacement approximation is performed by standard Lagrange polynomials and the stress approximation with Raviart-Thomas functions. The latter ansatz functions do not a priori fulfill the symmetry of the Cauchy stress tensor. Therefore, a redundant residual, the balance of angular momentum ((x − x 0 ) × (div + f ) + axl[ − T ]), is introduced and the results are discussed from the engineering point of view, especially for coarse mesh discretizations.However, this formulation shows an improvement compared to standard LS − u formulations, which is considered here in a numerical study. K E Y W O R D Sbalance of moment of momentum, linear elasticity, mixed least-squares finite element method 1 The consideration of finite element formulations using standard displacement methods are well established in structural analysis of engineering problems within the frameworks of small and finite deformations. Unfortunately, the applicability of standard displacement formulations is limited by certain constraints, which can lead to locking behavior and unreliable results in the displacement and stress fields, see for example, Babuška and Suri. [1] An alternative to the primal finite element method are mixed variational principles where for example, displacements and stresses are approximated directly. One major advantage of mixed methods is the robustness in the presence of certain limiting situations, for example, incompressible or nearly This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes.