The finite element patch test revisited a computer test for convergence, validation and error estimates

The patch test and the general convergence criteria of the finite element method

One of the central considerations in the theory of Markov chains is their convergence to an equilibrium. Coefficients of ergodicity provide an efficient method for such an analysis. Besides giving sufficient and sometimes necessary conditions for convergence, they additionally measure its rate. In this paper we explore coefficients of ergodicity for the case of imprecise Markov chains. The latter provide a convenient way of modelling dynamical systems where parameters are not determined precisely. In such cases a tool for measuring the rate of convergence is even more important than in the case of precisely determined Markov chains, since most of the existing methods of estimating the limit distributions are iterative. We define a new coefficient of ergodicity that provides necessary and sufficient conditions for convergence of the most commonly used class of imprecise Markov chains. This so-called weak coefficient of ergodicity is defined through an endowment of the structure of a metric space to the class of imprecise probabilities. Therefore we first make a detailed analysis of the metric properties of imprecise probabilities.

This paper discusses various forms of the Irons patch test and shows again that its satisfaction is a necessary convergence condition being equivalent to consistency, Further, it is shown that the test can be applied for verification of stability requirements, thus providing also a sufficient condition of convergence. The patch test is useful in testing the order of asymptotic convergence and the satisfaction of the test gives a guide for development of certain incompatible element forms.

9 - The patch test, reduced integration, and non-conforming elements

On the convergence of the MAOR method

The patch test is shown to be contained in the variational formulations of the finite element methods at the assembling level, all of which require the vanishing of the virtual work of interface connection loads. By a systematic introduction of stress generating functions, attention is drawn to the fact that any given finite element model can be assembled in two different ways: either by identification of a set of boundary displacements (leading to the direct stiffness method), or by identification of a set of local stress function values (leading to the dual direct flexibility method). Looking at any conjugate couple (generalized displacement‐generalized surface traction) at an interface, one is strongly transmitted, the other weakly. Discretization of the zero virtual work condition at an interface of plate bending models, by means of Legendre polynomial expansions, allows a systematic construction of so‐called ‘non‐conforming’ elements that pass the patch test. They are in fact identified with weakly conforming, but strongly diffusive, hybrids, and the lowest degree element (quadratic) is in fact the Morley constant‐moment element. Examples are given for higher degree displacement fields.The case of plate stretching elements can be handled by duality, the difficulties being here associated with the requirements for diffusivity. Non‐diffusive elements that pass the zero interface virtual work test can be constructed systematically and are identifiable with weakly diffusive, but strongly conforming, hybrids of the type first proposed by T. H. H. Pian.

Automatic generation of triangular finite element meshes

In order to solve numerically a second-order elliptic boundary-value problem in a bounded domain, ..cap omega.., by use of the finite element method, it is often necessary in practice to violate certain assumptions of the standard variational formulation. Two of these variational crimes are considered here. It is shown that optimal L infinity estimates still hold. The first crime occurs when a nonconforming finite element method is employed, so that smoothness requirements are violated at interelement boundaries. The second crime occurs when numerical integration is employed, so that the bilinear form is perturbed. Abstract error estimates are established. These estimates are first applied to treat a nonconforming finite element method and then to treat the case of numerical integration. In both cases, the patch test is crucial to the proof of L infinity estimates, just as it was in the case of mean square estimates.

Analysis and computation of a least-squares method for consistent mesh tying

A mixed finite element formulation is developed for the general 2D plane strain, linear isotropic gradient elasticity problem. Form II of the dipolar strain gradient theory for micro-structured solids is considered. The main variables are the double stress tensor μ and the displacement field vector u . Standard C 0 −continuous, high polynomial order hierarchical basis functions are employed for the finite element solution spaces (p-extension). The formulation is numerically validated against the standard axial tension patch test and the Mode I crack problem. The theoretical convergence rates of the uniform h - and p -extensions are confirmed using a benchmark problem where only double stresses appear. Results for the crack problem demonstrate that proper mesh refinement at areas of steep gradients ensures reproduction of the exact solution behaviour at different length scales. More specifically, the asymptotic exponents of the crack face opening displacement and the crack head true stress solutions of the Mode I crack problem are recovered. Finally, the upper bound of the true tensile normal stress near the crack tip is estimated. This upper bound is of major importance since the nature of the exact solution may change radically as we proceed from the macro- to micro-scale.

We present a method to assess the stability of pairs of interpolation spaces for mixed formulations. The method is based on a straightforward calculation of the eigenvalues of the discrete matrices through Fourier decomposition in plane waves and is intended to give, via straightforward numerical computations, a sharper determination of stability than the well‐known ‘patch test’ of Zienkiewicz et al. Special attention is devoted to the study of stability and accuracy of equal‐order interpolations.

In this work, a displacement welding technique is proposed to analyze the whole model which consists of independently modeled finite element substructures. In the proposed method, the incompatible displacement fields in the interfaces of independently modeled substructures are directly welded together through a blended function that is newly defined in the mismatching interface region. To construct the blended function, the moving least squares function, which does not require well-defined nodal connectivity, is utilized along with the original finite element shape function. The meshless character of the moving least squares function makes it possible to efficiently handle the mismatching interfaces of independently modeled finite element substructures. To prove the validity of the proposed method, the patch tests and convergence tests are carried out for various mismatching models. To assess the performance of the proposed welding technique, several numerical examples are worked out including beam type problem, plate with circular hole, and L-shaped beam type problem.

This work presents a new technique for joining dissimilar meshes in non‐linear finite element analysis, with the novel result that the adjoining surfaces do not need to be spatially coincident for patch tests to be passed. The work builds in some sense on results previously presented by Dohrmann et al. (International Journal for Numerical Methods in Engineering 1991; 40:1205), but considerably generalizes upon the results presented there. Most importantly, it is shown that consistency of the tied finite element formulation may be achieved such that implementation of element formulations is not affected, through the use of a three‐field variational principle applied over the contacting region. A key conclusion of the work is that although the approach advocated bears some resemblance to mortar element schemes for contact analysis (see, for example, McDevitt and Laursen, International Journal for Numerical Methods in Engineering 2000; 48:1525), a straightforward application of a mortaring scheme is not sufficient to guarantee patch test passage when dissimilar meshing of curved surfaces leads to local open volumes (or overlaps) between tied surfaces in their reference configurations. The proposed algorithm is demonstrated through a number of numerical examples performed within a matrix‐free non‐linear equation solving framework. Published in 2003 by John Wiley & Sons, Ltd.

Two methods are presented for connecting dissimilar three-dimensional finite element meshes. The first method combines the concept of master and slave surfaces with the uniform strain approach for finite elements. By modifying the boundaries of elements on a slave surface, corrections are made to element formulations such that first-order patch tests are passed. The second method is based entirely on constraint equations, but only passes a weaker form of the patch test for non-planar surfaces. Both methods can be used to connect meshes with different element types. In addition, master and slave surfaces can be designated independently of relative mesh resolutions. Example problems in three-dimensional linear elasticity are presented. Published in 2000 by John Wiley & Sons, Ltd.

The present paper describes the formulation of a new moderately thick plate bending triangular finite element based on Mindlin–Reissner plate theory. It is called a Great Triangular Moderately Thick Plate Finite Element, or GTMTPFE. The formulation is based on the strain approach, on solution of Airy’s function and on the analytical integration in the construction of the stiffness matrix. The strengths associated with this approach consist of: • automatic verification of equilibrium conditions and kinematic compatibility conditions, • the enrichment of the degrees of the interpolation polynomials of displacements, strains and constraints (refinement p), • the consideration distortions sections related to Poisson effects, • the treatment of blocking phenomena related to transverse shear. In general, this approach results in a competitive, robust and efficient new moderately thick plate finite element. This is visible, on the one hand, through its stability against patch tests (constant twists, state of constants moments, transverse shear locking phenomenon, isotropy test). This is visible, through its good response to the patch tests to which it is subjected (constant torsions, state of constant moments, phenomenon of blocking in transverse shears, isotropy test). As has excellent convergence to the reference solution. Thus, it exhibits better performance behavior than other existing plate elements in the literature, particularly for moderately thick plates and for thin plates (L/h ratio greater than 4).