The Finite Element Method: A Four-Article Series - Part 4
The following four-article series was published in a newsletter
of the American Society of Mechanical Engineers (ASME).
It serves as an introduction to the recent analysis discipline
known as the finite element method. The author is
an engineering consultant and expert witness specializing in
finite element analysis.
FINITE ELEMENT ANALYSIS: Post-processing
by Steve Roensch, President, Roensch & Associates
Last in a four-part series
After a finite element model has been prepared and checked,
boundary conditions have been applied, and the model has been
solved, it is time to investigate the results of the analysis.
This activity is known as the post-processing phase of the
finite element method.
Post-processing begins with a thorough check for problems that
may have occurred during solution. Most solvers provide a log
file, which should be searched for warnings or errors, and which
will also provide a quantitative measure of how well-behaved the
numerical procedures were during solution. Next, reaction loads
at restrained nodes should be summed and examined as a "sanity
check". Reaction loads that do not closely balance the applied
load resultant for a linear static analysis should cast doubt on
the validity of other results. Error norms such as strain energy
density and stress deviation among adjacent elements might be
looked at next, but for h-code analyses these quantities are
best used to target subsequent adaptive remeshing.
Once the solution is verified to be free of numerical problems,
the quantities of interest may be examined. Many display options
are available, the choice of which depends on the mathematical
form of the quantity as well as its physical meaning. For
example, the displacement of a solid linear brick element's node
is a 3-component spatial vector, and the model's overall
displacement is often displayed by superposing the deformed
shape over the undeformed shape. Dynamic viewing and animation
capabilities aid greatly in obtaining an understanding of the
deformation pattern. Stresses, being tensor quantities,
currently lack a good single visualization technique, and thus
derived stress quantities are extracted and displayed. Principal
stress vectors may be displayed as color-coded arrows,
indicating both direction and magnitude. The magnitude of
principal stresses or of a scalar failure stress such as the Von
Mises stress may be displayed on the model as colored bands.
When this type of display is treated as a 3D object subjected to
light sources, the resulting image is known as a shaded image
stress plot. Displacement magnitude may also be displayed by
colored bands, but this can lead to misinterpretation as a
stress plot.
An area of post-processing that is rapidly gaining popularity
is that of adaptive remeshing. Error norms such as strain energy
density are used to remesh the model, placing a denser mesh in
regions needing improvement and a coarser mesh in areas of
overkill. Adaptivity requires an associative link between the
model and the underlying CAD geometry, and works best if
boundary conditions may be applied directly to the geometry, as
well. Adaptive remeshing is a recent demonstration of the
iterative nature of h-code analysis.
Optimization is another area enjoying recent advancement. Based
on the values of various results, the model is modified
automatically in an attempt to satisfy certain performance
criteria and is solved again. The process iterates until some
convergence criterion is met. In its scalar form, optimization
modifies beam cross-sectional properties, thin shell thicknesses
and/or material properties in an attempt to meet maximum stress
constraints, maximum deflection constraints, and/or vibrational
frequency constraints. Shape optimization is more complex, with
the actual 3D model boundaries being modified. This is best
accomplished by using the driving dimensions as optimization
parameters, but mesh quality at each iteration can be a concern.
Another direction clearly visible in the finite element field
is the integration of FEA packages with so-called "mechanism"
packages, which analyze motion and forces of large-displacement
multi-body systems. A long-term goal would be real-time
computation and display of displacements and stresses in a
multi-body system undergoing large displacement motion, with
frictional effects and fluid flow taken into account when
necessary. It is difficult to estimate the increase in computing
power necessary to accomplish this feat, but 2 or 3 orders of
magnitude is probably close. Algorithms to integrate these
fields of analysis may be expected to follow the computing power
increases.
In summary, the finite element method is a relatively recent
discipline that has quickly become a mature method, especially
for structural and thermal analysis. The costs of applying this
technology to everyday design tasks have been dropping, while
the capabilities delivered by the method expand constantly. With
education in the technique and in the commercial software
packages becoming more and more available, the question has
moved from "Why apply FEA?" to "Why not?". The method is fully
capable of delivering higher quality products in a shorter
design cycle with a reduced chance of field failure, provided it
is applied by a capable analyst. It is also a valid indication
of thorough design practices, should an unexpected litigation
crop up. The time is now for industry to make greater use of
this and other analysis techniques.