An engineers' defence of the patch test

Abstract

This convergence criterion is proved for any 'legalizable' finite element formulation, i.e. any model for which a conforming displacement version is possible, using the same nodal variables. Engineers may therefore take control of (and responsibility for) convergence, for most of the elements in service today. Stummel's counter-example is explained as a technical misunderstanding, due to inadequate documentation of the patch test. It is suggested, without proof, that the test (properly applied) is universal, provided that it is always combined with an adequate test of stability. INTRODUCTION: 'NECESSITY'The patch test for convergence is and will remain an invaluable tool for practical engineers, a simple and decisive check not of the competitive performance of a program, but of its reliability and correctness. It is as primitive and uncomplicated a device as any other test of consistency, e.g. that in finite differences. We surmise that the ongoing debate regarding the necessity and sufficiency of the test may well lead ultimately to a settlement on semanfic grounds: what is 'necessity' and what precisely do we mean by a 'finite element'? 'Sufficiency' under what circumstances? and first and foremost, to avoid misunderstanding, how exactly should we do a 'patch test'?To d o a 'patch test' is to adopt the following general strategy:g (a) choose a mesh of arbitrary geometry, as in Figure 1; (b) 'target' on to some arbitrary field(s) of constant stress; (c) run job(s) on the computer (using the standard program, as for real jobs) with data such that the mesh would reasonably be expected to respond with each targeted state of uniform stress, in turn; (d) examine the output and check carefully that everything conforms to expectations. Thus we engineers rely upon it to establish the accuracy, not merely of the algebraic manipulations in some published paper, but of the computer program itself. Again, in assembling the data for (b), we try to provide as many opportunities as possible for the elements to misbehave, and, most important, for such misdemeanours to be evident in the output. To take an example: if Figure 1 represented meshes of plate bending elements, with w, w, and w, as nodal variables, then every internal node (i.e. any node completely surrounded by elements) should be in equilibrium, i.e. free and unloaded. We therefore fix the values of w, w, and w y at every external node, with no applied loads. (Or, slightly better, we might fix w, and w, at every t Professor. t: Consultant, Control Data Canada.$The bulk of this paper is too sophisticated for teaching. The best strategy in the classroom is that adopted by Hinton: Take a plate bending element that fails. (Many fail.) Set two assignments, one that succeeds and one that does not: both should model a square plate, simply supported at three corners, and with a lateral point load at the fourth. (The exact solution is M,, =constant.) When the students compare the results, interest in that particular element evaporates: and faith in the patch test is born, wit...

Curvature-based energy for simulation and variational modeling

“…Thus we introduce an extra angular variable for each edge. The mean curvature estimate in this case is quite similar to (16):…”

“…For conforming elements, convergence is assured by the general FEM theory. and most practical nonconforming elements are shown to satisfy the patch test, [17] which is analogous to our conditions C1 and C2 (whether various forms of the patch test are sufficient or necessary for convergence is discussed in a number of books and papers: [41,31,16,32,34]). …”

Numerical Method for Darcy Flow Derived Using Discrete Exterior Calculus

“…The first results shown in Figure 4 are for patch tests [31]. It is desirable that for simple meshes a numerical method for Darcy flow should reproduce constant velocity and linear pressure exactly up to machine precision.…”

Letter to the Editor

“…[3,4,5,7]. The patch test determines whether an element is capable of reproducing a constant stress state for arbitrary element geometries.…”