OS-V: 0760 MacNeal-Harder Solid Patch Test

Model Files

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.

Figure 2. Patch test for solids

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.

Results

${\epsilon }_x={\epsilon }_y={\epsilon }_z={\gamma }_{xy}={\gamma }_{yz}={\gamma }_{zx}={10}^{-3}$

${\sigma }_x={\sigma }_y={\sigma }_z=2000$

${\tau }_{xy}={\tau }_{yz}={\tau }_{zx}=400$

Figure 3. Elemental strains in all 6 direction plot

Figure 4. Elemental stresses in all 6 direction plot

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