FINITE ELEMENT ANALYSIS: Pre-processing
by Steve Roensch, President, Roensch & Associates
...Second in a four-part series
As discussed last month, finite element analysis is
comprised of pre-processing, solution and post-processing
phases. The goals of pre-processing are to develop an
appropriate finite element mesh, assign suitable material
properties, and apply boundary conditions in the form of
restraints and loads.
The finite element mesh subdivides the geometry into
elements, upon which are found nodes.
The nodes, which are
really just point locations in space, are generally located
at the element corners and perhaps near each midside. For a
two-dimensional (2D) analysis, or a three-dimensional (3D)
thin shell analysis, the elements are essentially 2D, but
may be "warped" slightly to conform to a 3D surface. An
example is the thin shell linear quadrilateral; thin shell
implies essentially classical shell theory, linear defines
the interpolation of mathematical quantities across the
element, and quadrilateral describes the geometry. For a 3D
solid analysis, the elements have physical thickness in all
three dimensions. Common examples include solid linear
brick and solid parabolic tetrahedral elements. In
addition, there are many special elements, such as
axisymmetric elements for situations in which the geometry,
material and boundary conditions are all symmetric about an
axis.
The model's degrees of freedom (dof) are assigned at the
nodes. Solid elements generally have three translational
dof per node. Rotations are accomplished through
translations of groups of nodes relative to other nodes.
Thin shell elements, on the other hand, have six dof per
node: three translations and three rotations. The addition
of rotational dof allows for evaluation of quantities
through the shell, such as bending stresses due to rotation
of one node relative to another. Thus, for structures in
which classical thin shell theory is a valid approximation,
carrying extra dof at each node bypasses the necessity of
modeling the physical thickness. The assignment of nodal
dof also depends on the class of analysis. For a thermal
analysis, for example, only one temperature dof exists at
each node.
Developing the mesh is usually the most time-consuming task
in FEA. In the past, node locations were keyed in manually
to approximate the geometry. The more modern approach is to
develop the mesh directly on the CAD geometry, which will be
(1) wireframe, with points and curves representing edges,
(2) surfaced, with surfaces defining boundaries, or (3)
solid, defining where the material is. Solid geometry is
preferred, but often a surfacing package can create a
complex blend that a solids package will not handle. As far
as geometric detail, an underlying rule of FEA is to "model
what is there", and yet simplifying assumptions simply must
be applied to avoid huge models. Analyst experience is of
the essence.
The geometry is meshed with a mapping algorithm or an
automatic free-meshing algorithm. The first maps a
rectangular grid onto a geometric region, which must
therefore have the correct number of sides. Mapped meshes
can use the accurate and cheap solid linear brick 3D
element, but can be very time-consuming, if not impossible,
to apply to complex geometries. Free-meshing automatically
subdivides meshing regions into elements, with the
advantages of fast meshing, easy mesh-size transitioning
(for a denser mesh in regions of large gradient), and
adaptive capabilities. Disadvantages include generation of
huge models, generation of distorted elements, and, in 3D,
the use of the rather expensive solid parabolic tetrahedral
element. It is always important to check elemental
distortion prior to solution. A badly distorted element
will cause a matrix singularity, killing the solution. A
less distorted element may solve, but can deliver very poor
answers. Acceptable levels of distortion are dependent upon
the solver being used.
Material properties required vary with the type of solution.
A linear statics analysis, for example, will require an
elastic modulus, Poisson's ratio and perhaps a density for
each material. Thermal properties are required for a thermal
analysis. Examples of restraints are declaring a nodal
translation or temperature. Loads include forces, pressures
and heat flux. It is preferable to apply boundary
conditions to the CAD geometry, with the FEA package
transferring them to the underlying model, to allow for
simpler application of adaptive and optimization algorithms.
It is worth noting that the largest error in the entire
process is often in the boundary conditions. Running
multiple cases as a sensitivity analysis may be required.
Next month's article will discuss the solution phase of the
finite element method.
© 2008 Roensch & Associates. All rights reserved.
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This 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.
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