A fractal patch test

Abstract

SUMMARYElement consistency is generally checked using the patch test on an element patch of finite size. This condition may in certain cases be too restrictive, and disqualifies elements that appear to be convergent. A method termed 'fractal patch test' is presented, in which the patch size is maintained constant while the distorted mesh is refined. Examples are given for four-node quadrilateral elements used in plane stress and strain analysis, and for plate bending elements.

Closed-Form Isoparametric Shape Functions of Four-Node Convex Finite Elements

“…This strong geometrical criterion yields excellent accuracy, as has been widely observed, vide ref. [1]. The independent variables are the coordinates of the first and second nodes, and the horizontal and vertical offsets a and b, respectively, shown in Figure 7.…”

“…a o + a 1 x + a 2 y = 0; for two-dimensional cases (1) a o + a 1 x + a 2 y + a 3 z = 0; for three-dimensional cases (2) The challenge to postulate shape functions for plane quadrilaterals and threedimensional hexahedrals were adequately met by the ingenious formulation of Taig, ref. [9] -well-known in the literature as the isoparametric formulation.…”

A set of pathological tests to validate new finite elements

“…New tests for assessing above requirements. Eigendata test, sub-divisibility test for membrane finite elements Belytschko & Lasry (1988) A fractal patch test White & Abel (1989) Testing of shell finite element accuracy and robustness Babuska & Scapollo (1989) Bench mark computation and performance evaluation of rhombic plate bending problem White & Abel (1989) A suite of tests to determine the accuracy and robustness of shell finite elements for linear elastic and geometrically nonlinear problems. Smith (1990) Benchmark tests for geometrically non-linear two dimensional beams.…”

Imaging of pancreatic islet cells

“…The patch test has been used for over four decades as a condition for convergence of finite elements and as a verification tool for algorithms 10–20. In addition, alternative forms of the patch test have been discussed for various applications 21–30.…”