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SUMMARYWeak form finite element models for the nonlinear quasi-static bending and extension of initially straight viscoelastic Euler-Bernoulli and Timoshenko beams are developed using the principle of virtual work. The mechanical properties of the beams are considered to be linear viscoelastic. However, large transverse displacements, moderate rotations and small strains are allowed by retaining the von Kármán strain components of the Green-Lagrange strain tensor in the formulation. The fully discretized finite element equations are developed using the trapezoidal rule in conjunction with a two-point recurrence relation. The resulting finite element equations, therefore, necessitate data storage from the previous time step only, and not the entire deformation history. Membrane locking is eliminated from the Euler-Bernoulli formulation through the use of selective reduced Gauss-Legendre quadrature. Membrane and shear locking are both circumvented in the Timoshenko beam finite element by employing a family of high-order Lagrange polynomials. A Newton-Raphson iterative scheme is used to solve the nonlinear finite element equations.

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Abstract. 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 (steadystate) 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.

We consider the application of high-order spectral/hp finite element technology to the numerical solution of boundary-value problems arising in the fields of fluid and solid mechanics. For many problems in these areas, high-order finite element procedures offer many theoretical and practical computational advantages over the low-order finite element technologies that have come to dominate much of the academic research and commercial software of the last several decades. Most notably, we may avoid various forms of locking which, without suitable stabilization, often plague low-order least-squares finite element models of incompressible viscous fluids as well as weak-form Galerkin finite element models of elastic and inelastic structures. 7.2 Transient flow of an incompressible viscous fluid inside a square cavity induced by a moving cylinder at t = 0.25, 0.50 and 0.70 (from left to right respectively): (a) velocity component v x (b) velocity component v y and (c) nondimensional pressure field p.