J. N. Reddy

K. Olesen, B. Gervang, J. N. Reddy et al.

In this paper a mixed spectral element formulation is presented for planar, linear elasticity. The degrees of freedom for the stress are integrated traction components, i.e. surface force components. As a result the tractions between elements are continuous. The formulation is based on minimization of the complementary energy subject to the constraints that the stress field should satisfy equilibrium of forces and moments. The Lagrange multiplier which enforces equilibrium of forces is the displacement field and the Lagrange multiplier which enforces equilibrium of moments is the rotation. The formulation satisfies equilibrium of forces pointwise if the body forces are piecewise polynomial. Equilibrium of moments is weakly satisfied. Results of the method are given on orthogonal and curvilinear domains and an example with a point singularity is given.

G. S. Payette, K. B. Nakshatrala, J. N. Reddy

The main aim of this paper is to document the performance of $p$-refinement with respect to maximum principles and the non-negative constraint. The model problem is (steady-state) anisotropic diffusion with decay (which is a second-order elliptic partial differential equation). We considered the standard single-field formulation (which is based on the Galerkin formalism) and two least-squares-based mixed formulations. We have employed non-uniform Lagrange polynomials for altering the polynomial order in each element, and we have used $p = 1, ..., 10$. It will be shown that the violation of the non-negative constraint will not vanish with $p$-refinement for anisotropic diffusion. We shall illustrate the performance of $p$-refinement using several representative problems. The intended outcome of the paper is twofold. Firstly, this study will caution the users of high-order approximations about its performance with respect to maximum principles and the non-negative constraint. Secondly, this study will help researchers to develop new methodologies for enforcing maximum principles and the non-negative constraint under high-order approximations.