The Finite Element Method: A Four-Article Series - Part 2
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: 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.