Paul M. Kurowski
President
Design Generator Inc.
London, Ontario, Canada
Edited by Paul Dvorak
Doing so
brings simplifying assumptions
that cause modeling errors to the
math model.
A correctly formulated mathematical
model captures the characteristics
of an object needed to make
design decisions. For example, analysis
of a compliant link under static load requires
nonlinear FEA to handle expected
large displacements. A cooling simulation
requires a thermal analysis (steady
state or transient), while a drop test would
call for a nonlinear dynamic analysis. So
the software must fit the problem.
The mathematical model must be discretized
into finite elements. We commonly call this discretization
process
meshing. While a
meshed model is easy
to illustrate, it may be
confusing because it
implies a mesh is just
imposed on model
geometry. Nodes connected
by lines show
finite elements, but
in fact, nothing of the
CAD model remains.
Continuous geometry
is replaced by nodes
and interactions between
nodes is defined
by elements
connecting the
nodes.
Meshing converts
a mathematical
model (geometry,
loads, and restraints)
into an FEA model.
That act ion compounds
errors, and
these are called discretization
errors. For
instance, an inappropriate
element may be
incapable of capturing
a real stress distribution.
Results such as
displacements and
stress distribution
may provide
useless
or insightful
information
depending
on how well elements
are used.
After creating an
FEA model, its solution
is a matter of
solving a large number
of linear equations.
The solution
introduces numerical
errors, although they
are usually low. Finally,
results are analyzed
to make design
decisions.
Each step takes us
further from reality,
which is where we started. Errors are introduced
at each step, some of them unavoidable,
while others amount
to FEA malpractice. Hence, it
makes sense to verify models
and validate results.
Although verification and
validation sound similar, there
is a difference FEA users should
know. Verification determines
the mathematical model, as submitted
for solution with FEA,
has been solved correctly.
Validation, on the other
hand, determines whether or
not FEA results correctly represents
reality from the perspective
of the intended use of the
model. It checks that results
correctly describe real-life behavior
of the object. The chart
Verification and validation further
highlights the difference
between the two.
But things can also go wrong
when validating and verifying
models. For instance, verification
fails when meshing is incorrect.
Models with meshing
errors, such as using one layer
of first-order solid elements in a
bending beam, or using a mesh
too coarse to capture a pattern,
would not pass verification tests. Models with incorrectly
defined loads would pass a verification test because
verification only tests for correctness of the finiteelement
solution, not that the mathematical model is
correct.
Establishing that a solution correctly predicts a real
object’s behavior is the task of validation, which should
follow verification. Validation fails when a model cannot
provide needed data. The two-link model from the
June FE Update cannot find or report on its stiffness.
And the square rod in the rubber block, also from the
previous column, cannot be used to find the deformed
shape of the rubber block.
FEA errors may be due to verification, validation, or
both. FEA software is not at fault in any of the cases discussed.
Place blame for the errors on not understanding
the modeling process and fundamental FEA assumptions
which let us attribute properties of real objects to
FEA models. Only proper training in FEA fundamentals
prevents errors.
Mesh size must be
small enough to
model stress gradients
and avoid excessive
element distortion.
The above illustration
shows incorrect results
produced by too
coarse a mesh. The
illustration to the right
shows correct results
produced by a refined
mesh. The models
here are prepared
with CosmosWorks
and available from
designgenerator.com/MD2008.
The steps in an FEA project: Create a
mathematical model, create the FEA model,
solve it, and analyze results.
Verification and validation of
FEA models should be part of
FEA projects.