Dual-mixed hp finite element methods using first-order stress functions and rotations

Abstract

A two-field dual-mixed variational formulation of three-dimensional elasticity in terms of the non-symmetric stress tensor and the skew-symmetric rotation tensor is considered in this paper. The translational equilibrium equations are satisfied a priori by introducing the tensor of first-order stress functions. It is pointed out that the use of six properly chosen first-order stress function components leads to a (three-dimensional) weak formulation which is analogous to the displacement-pressure formulation of elasticity and the velocity-pressure formulation of Stokes flow. Selection of stable mixed hp finite element spaces is based on this analogy. Basic issues of constructing curvilinear dual-mixed p finite elements with higher-order stress approximation and continuous surface tractions are discussed in the two-dimensional case where the number of independent variables reduces to three, namely two components of a first-order stress function vector and a scalar rotation. Numerical performance of three quadrilateral dual-mixed hp finite elements is presented and compared to displacement-based hp finite elements when the Poisson's ratio converges to the incompressible limit of 0.5. It is shown that the dual-mixed elements developed in this paper are free from locking in the energy norm as well as in the stress computations, both for h- and p-extensions.