Stable hp mixed finite elements based on the Hellinger–Reissner principle

AbstractIn the Hellinger–Reissner formulation for linear elasticity, both the displacement u and the stress σ are taken as unknowns, giving rise to a saddle point problem. We present new pairings of quadrilateral ‘trunk’ finite element spaces for this method and prove stability (and optimality) in terms of both h and p. The effect of mesh shape regularity on the stability constant is explicitly tracked. Our results provide a theoretical basis for recent numerical experiments (in the context of a mixed p formulation for viscoelasticity) that showed these spaces worked well computationally

On the asymptotic behavior of the discrete spectrum in buckling problems for thin plates

Version 24.08.2004We consider the buckling problem for a family of thin plates with thickness parameter \epsilon. This involves finding the least positive multiple \lambda_min(\epsilon) of the load that makes the plate buckle, a value that can be expressed in terms of an eigenvalue problem involving a non-compact operator. We show that under certain assumptions on the load, we have \lambda_\min(\epsilon) = O(\epsilon^2). This guarantees that provided the plate is thin enough, this minimum value can be numerically approximated without the spectral pollution that is possible due to the presence of the non-compact operator. We provide numerical computations illustrating some of our theoretical results

Analytic and Computational Assessment of Locking in the hp Finite Element Method

Locking is the phenomenon by which the numerical approximation of parameter-dependent problems deteriorates for values of the parameter close to a limiting value. In this paper, we give a definition of locking and develop precise computable and analytic ways to quantify it. Using the example of nearly incompressible elasticity, we show by means of computational and theoretical results, the difference between the h version and p=hp version in combatting locking. Our results establish the superiority of high order elements (both h; p and hp) when the standard variational form is used. We also discuss other issues such as curved elements, mixed methods, and locking phenomena for problems over anisotropic materials and over thin domains. Key Words: locking, h version, p version, hp version, finite element method To appear in Computer Methods in Applied Mechanics and Engineering, 1995. (1) Research partially supported by the Air Force Office of Scientific Research, Bolling AFB, DC, under G..

The p and hp Versions of the Finite Element Method for Problems with Boundary Layers

DedicatedtoProfessorIvoBabuˇska on the occasion of his seventieth birthday Abstract. We study the uniform approximation of boundary layer functions exp(−x/d) forx∈(0, 1), d ∈ (0, 1], by the p and hp versions of the finite element method. For the p version (with fixed mesh), we prove super-exponential convergence in the range p +1/2>e/(2d). We also establish, for this version, an overall convergence rate of O(p −1 √ ln p) in the energy norm error which is uniform in d, and show that this rate is sharp (up to the √ ln p term) when robust estimates uniform in d ∈ (0, 1] are considered. For the p version with variable mesh (i.e., the hp version), we show that exponential convergence, uniform in d ∈ (0, 1], is achieved by taking the first element at the boundary layertobeofsizeO(pd). Numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates, even when few degrees of freedom are used and when d is as small as, e.g., 10 −8. They also illustrate the superiority of the hp approach over other methods, including a low-order h version with optimal “exponential ” mesh refinement. The estimates established in this paper are also applicable in the context of corresponding spectral element methods. 1

On the Selection of a Locking-free hp Element for Elasticity Problems

. We consider the design of robust elements that can be used in hp codes for elasticity problems. Our goal is to identify an element that is free of locking for the case of a nearly incompressible material, both in terms of the h version and the p version. By this we mean that the displacements as well as the stresses should converge optimally. For the standard finite element method, locking is observed in the stresses for the p version and in both the stresses and displacements for the h version. Hence, we recommend a mixed formulation. Several mixed methods are tested, and a method that is optimal both in terms of h and p is identified. KEY WORDS: p version, hp version, mixed methods, locking, robust. 1. Introduction There are several p and hp version commercial programs currently available for industrial use. Codes with hp capabilities allow the user to selectively employ a combination of h-refinement and p-refinement to achieve accuracy. For example, a highly refined mesh with l..