4.5 The patch test

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Irons proposed the patch test originally to check the convergence of nonconforming

elements [125]. Although passing a patch test—theoretically at

least—is neither necessary nor sufficient for the convergence of an FE solution

[235], it is a very good test to check and compare elements.

The patch test is based on the observation that the stress distribution

becomes more and more uniform the smaller the elements become. Therefore

convergence can only be expected if an FE program can solve load cases with

uniform stress states exactly.

In a wider sense, a patch test is simply a test to reproduce a certain stress

distribution on a given mesh.

Wilson’s improved bilinear element Q4 + 2 often yields better results than

the original bilinear element Q4, even though it is a nonconforming element.

To study the behavior of these two elements side by side, a cantilever plate

was subjected to three standard load cases, producing

4.5 The patch test 345

Fig. 4.17. These meshes should have no difficulty reproducing simple stress states:

a) regular mesh, b) irregular mesh

• a constant moment

• linear bending moments = constant shear forces

• quadratic bending moments = linear shear forces

Each load case was solved on a relatively coarse mesh consisting of eight

rectangular elements, and alternatively on a distorted mesh; see Fig. 4.17 b.

When the plate is stretched uniformly, this will produce a constant normal

force, and this load case must be solved exactly. This is the original patch test.

Both elements passed this test. But in the other three load cases, distinct

deviations from the beam solution appeared; see Table 4.4. As expected the

Table 4.4. Normal stress σ (kN/m2), R = regular mesh, I = irregular mesh,

= bilinear element, Q4 + 2 = Wilson

Moment Constant Linear Quadratic

Mesh x = 0.0 x = l/2 x = 0.0 x = l/2 x = 0.0 x = l/2

Exact 1500 1500 1200 600 1200 300

R. Q4 + 2 1500 1500 1051 600 940 337

I. Q4 + 2 1322 1422 940 701 773 452

R. Q4 1072 1072 745 428 659 240

I. Q4 687 578 454 187 393 172

stresses at edge nodes were not as accurate as stresses at internal nodes, but

nevertheless it is remarkable how much difficulty the bilinear element had

in modeling the bending states on such a coarse mesh. In particular, the

errors in the shear stresses were large; see Table 4.5. On the regular mesh

Q4

346 4 Plane problems

Table 4.5. Shear stress τ (kN/m2)

Moment Constant Linear Quadratic

Mesh x = 0.0 x = l/2 x = 0.0 x = l/2 x = 0.0 x = l/2

Exact 0 0 50 50 100 50

R. Q4 + 2 0 0 50 50 87.5 50

I. Q4 + 2 58 28 65 80 130 73

R. Q4 438 0 364 8 376 8

I. Q4 502 220 380 294 366 11

Fig. 4.18. Displacements produced on the irregular mesh under constant horizontal

volume forces: a) nonconforming element Q4+2 (Wilson), b) conforming element

Q4 (bilinear)

the nonconforming Wilson element yielded the exact solution (the value 87.5

instead of 100 in the last column is due to the fact that some of the load is

reduced directly into the support nodes). In the bilinear element the incorrect

shear forces have nearly the same magnitude as the normal stresses. In the

nonconforming solution they are a factor of 4 to 10 smaller.

The poor properties of the bilinear element also become apparent if a

constant horizontal volume force is applied; see Fig. 4.18. Though the stresses

of the two solutions, Q4 and Q4 + 2, are similar, the lateral displacements of

the bilinear elements must cause concern. These displacements are due to a

Poisson ratio ν > 0. They are not that large, but they cause asymmetries—

even in the stresses—and if the structure is statically indeterminate, we should

be very careful.

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