OS-V: 0760 MacNeal-Harder Solid Patch Test

MacNeal-Harder TestThe patch test is a classical benchmark problem for the element. If it produces correct results for the test, the result for any problem solved with the element will converge toward the correct solution. The intended purpose of the proposed problem set is to ascertain the accuracy of finite element in various applications.

Benchmark Model

The outer dimension have a unit cube of 1 mm size. There is a mesh of the cube with node locations as mentioned in the table with first order CHEXA elements. The eight corners of the cube are constrained in all three translational direction and free in all three rotational directions. Displacement is enforced using SPCD on the eight nodes of cylinder in X, Y and Z translation directions of the cube.

The material properties are:
Material Properties
Value
Young's Modulus
1 x 106 Pa
Poisson's Ratio
0.25

x y z
1 0.249 0.342 0.192
2 0.826 0.288 0.288
3 0.850 0.649 0.263
4 0.273 0.750 0.230
5 0.320 0.186 0.643
6 0.677 0.305 0.683
7 0.788 0.693 0.644
8 0.165 0.745 0.702

The arbitrarily distorted element shapes are an essential part of the test. The principal virtue of a patch test is that if an element produces correct results for the test, the results for any problem solved with the element will converge toward the correct solution as the elements are subdivided. On the other hand, passing the patch test does not guarantee satisfactory results, since the rate of convergence may be too slow for practical use. The above patch test is an extension of Robinson’s patch test to three dimensions.

Displacement boundary conditions for the test are:
u
${10}^{-3}\left(2x+y+z\right)/2$
v
${10}^{-3}\left(x+2y+z\right)/2$
w
${10}^{-3}\left(x+y+2z\right)/2$

Results

The results CHEXA elements agree with the reference results.

Reference

MacNeal, R.H., and Harder, R.L., A Proposed Standard Set of Problems to Test Finite Element Accuracy, Finite Elements in Analysis and Design, 1 (1985) 3-20