| This article was
published in Machine Design, "Converging
on An Accurate Solution,"
April 13, 2006 In finite element analysis (FEA), a finer mesh often
results in a more accurate solution. While engineers cannot typically obtain
the exact solution for a model, an approximation can be obtained with very
high accuracy using finite-element methods. However, as a mesh is made finer
and accuracy increased, computational intensity also increases, often
leading to longer solution times. This conundrum begs the question: Just how
fine of a mesh is fine enough to accurately represent the real-world event?
The question cannot be answered categorically. To find the solution that
best balances computational capacity and accuracy, you should perform a mesh
convergence study.
A mesh convergence study is an empirical process that
compares the results of one meshed model with those of another. As such, one
of the easiest and best ways to start is with the fewest, yet reasonable
number of elements. That is, begin by meshing your model as coarsely as
seems reasonable and analyze it. Then, recreate the mesh with a denser
element distribution, re-analyze it and compare the results to those of the
more coarsely-meshed model. Are the results similar? If not, then the coarse
mesh is not very accurate. Increase the mesh density and re-analyze the
model.
Keep increasing the mesh density and re-analyzing the
model until the results converge satisfactorily. That is, when you reach a
point at which finer meshing no longer yields appreciably different results,
the mesh may be considered fine enough. This type of mesh convergence study
can help you obtain an accurate solution with a mesh that is sufficiently
dense and yet not overly demanding of computer resources.
Figure 1
A precision contour display gives a visual
indication of the effects of the finite element mesh on accuracy.
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Figure 2
A stainless steel plate (4" x 4" x 0.1") with fixed boundary
conditions on all sides is subjected to a uniform pressure load of
100 psi normal to the element faces. A mesh convergence study is
performed using an n x n mesh where n = 2, 4, 8, 16 and 32 plate
elements.
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Bob Williams
Product Manager
ALGOR, Inc.
Pittsburgh, PA
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To determine when results have
converged satisfactorily and accurately, you can use the following methods:
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Display precision contours, which
show a graphical representation of the stepped changes in results
from one element to the next. This contour can be used to determine
the effect of the mesh on accuracy and as guidance for the locations
needing mesh refinement.
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Display unsmoothed result contours to
see the stepped changes in the results between adjacent elements.
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Display residual forces in the model
and check the reactions at supports to make sure they balance or
otherwise meet expectations based on engineering judgment.
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Inquire on the result values at the
same location (e.g., the center).
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