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TABLE 6.4
Boundary Conditions and Theoretical Solutions for the
Patch Tests in Fig. 6.23
a.
Quadrilateral Thin, Flat Element
(In-plane deformation)
Boundary Conditions:
10
−
3
u
=
(
x
+
y
/
2
)
v
=
10
−
3
(
y
+
x
/
2
)
Theoretical Solutions:
∗
x
=
y
=
γ
xy
=
10
−
3
σ
x
=
σ
y
=
1333;
τ
xy
=
400
b.
Solid Element
Boundary Conditions:
10
−
3
u
=
(
2
x
+
y
+
z
)/
2
v
=
10
−
3
(
x
+
2
y
+
z
)/
2
10
−
3
w
=
(
x
+
y
+
2
z
)/
2
Theoretical Solutions:
∗
x
=
y
=
z
=
γ
xy
=
γ
xz
=
10
−
3
σ
x
=
σ
y
=
σ
z
=
2000;
τ
xy
=
τ
xz
=
τ
yz
=
400
∗
The strains are constant throughout the elements.
For the usual patch test, the calculated displacements, strains, and stresses in the inte-
rior of the patch should be consistent with the constant strain state. If so, the element is
considered to be viable; if not, the element formulation is suspect, and it can be anticipated
that the results obtained using it may not converge correctly. The success of the patch test
may depend on the geometry of the element, i.e., on the topology of the element layout,
and on the boundary conditions. Thus, a given element should be tested using more than
one geometry, mesh layout, and strain state. A patch test is considered to be useful in the
study of nonconforming elements. If the element cannot pass the patch test, it can still be
acceptable.
Accuracy Test [MacNeal and Harder, 1985]
In addition to the patch test, which is designed to check the convergence of the element,
other benchmark tests to verify the accuracy of the element may be important. The design
of a comprehensive set of tests should take into account the parameters which affect the
accuracy, e.g., element geometry, loading, problem geometry, and material properties.
Normally, in an accuracy test, each element has a standard shape: for a two-dimensional
element, a square, and for a three-dimensional element, a cube. Of course, the standard
shapes cannot always be used in a structural analysis. In some meshes, they have to be
distorted. In a test problem, distorted, i.e., nonstandard shaped, elements should also be
tested. The distorted shaped elements should be checked with several kinds of loading.
For the loading, the benchmark test problem should account for all load cases that can
cause all possible deformations of the structure. For example, for a beam, the load should
include a shear force and bending moment in all coordinate directions, along with an axial
force and a twisting moment.
In the case of geometry, structures of different shapes, e.g., straight, curved, or twisted,
should be tested. An element may give excellent results for one structural geometry, but
may behave poorly for another geometry.
Poisson's ratio may have a strong effect on element accuracy as its value approaches 0.5.
For some materials, such an effect should be considered.
The basic guideline for the test problem design is to use the element to be tested in
different kinds of structures under different loading conditions. Table 6.5 lists several kinds
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