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.

Model Files

Before you begin, copy the file(s) used in this problem to your working directory.
Figure 1. Cube subjected to uniform displacement


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
Figure 2. Patch test for solids


Table 1. Location of Inner Nodes
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 ( 2 x + y + z ) / 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIXaGaaGima8aadaahaaWcbeqaa8qacqGHsislcaaIZaaaaOWd amaabmaabaWdbiaaikdacaWG4bGaey4kaSIaamyEaiabgUcaRiaadQ haa8aacaGLOaGaayzkaaWdbiaac+cacaaIYaaaaa@421B@
v
10 − 3 ( x + 2 y + z ) / 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIXaGaaGima8aadaahaaWcbeqaa8qacqGHsislcaaIZaaaaOWd amaabmaabaWdbiaadIhacqGHRaWkcaaIYaGaamyEaiabgUcaRiaadQ haa8aacaGLOaGaayzkaaWdbiaac+cacaaIYaaaaa@421B@
w
10 − 3 ( x + y + 2 z ) / 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIXaGaaGima8aadaahaaWcbeqaa8qacqGHsislcaaIZaaaaOWd amaabmaabaWdbiaadIhacqGHRaWkcaWG5bGaey4kaSIaaGOmaiaadQ haa8aacaGLOaGaayzkaaWdbiaac+cacaaIYaaaaa@421B@

Results

ε x =   ε y =   ε z =   γ x y =   γ y z =   γ z x =   10 − 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaH1oqzpaWaaSbaaSqaa8qacaWG4baapaqabaGcpeGaeyypa0Ja aeiiaiabew7aL9aadaWgaaWcbaWdbiaadMhaa8aabeaak8qacqGH9a qpcaqGGaGaeqyTdu2damaaBaaaleaapeGaamOEaaWdaeqaaOWdbiab g2da9iaabccacqaHZoWzpaWaaSbaaSqaa8qacaWG4bGaamyEaaWdae qaaOWdbiabg2da9iaabccacqaHZoWzpaWaaSbaaSqaa8qacaWG5bGa amOEaaWdaeqaaOWdbiabg2da9iaabccacqaHZoWzpaWaaSbaaSqaa8 qacaWG6bGaamiEaaWdaeqaaOWdbiabg2da9iaabccacaaIXaGaaGim a8aadaahaaWcbeqaa8qacqGHsislcaaIZaaaaaaa@58E7@

σ x =  σ y =  σ z = 2000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHdpWCpaWaaSbaaSqaa8qacaWG4baapaqabaGcpeGaeyypa0Ja aeiiaiabeo8aZ9aadaWgaaWcbaWdbiaadMhaa8aabeaak8qacqGH9a qpcaqGGaGaeq4Wdm3damaaBaaaleaapeGaamOEaaWdaeqaaOWdbiab g2da9iaabccacaaIYaGaaGimaiaaicdacaaIWaGaai4oaaaa@4839@

τ xy =  τ yz =  τ zx = 400 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHepaDpaWaaSbaaSqaa8qacaWG4bGaamyEaaWdaeqaaOWdbiab g2da9iaabccacqaHepaDpaWaaSbaaSqaa8qacaWG5bGaamOEaaWdae qaaOWdbiabg2da9iaabccacqaHepaDpaWaaSbaaSqaa8qacaWG6bGa amiEaaWdaeqaaOWdbiabg2da9iaabccacaaI0aGaaGimaiaaicdaaa a@49C2@
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