The Finite Element Method: A Four-Article Series
04 Jun 2011 | Category: uncategorized | Author: admin
FINITE ELEMENT ANALYSIS: Introduction
First in a four-part series
Finite element analysis (FEA) is a fairly recent discipline
crossing the boundaries of mathematics, physics, engineering
and computer science. The method has wide application and
enjoys extensive utilization in the structural, thermal and
fluid analysis areas. The finite element method is
comprised of three major phases:
(1) pre-processing, in
which the analyst develops a finite element mesh to divide
the subject geometry into subdomains for mathematical
analysis, and applies material properties and boundary
conditions,
(2) solution, during which the program derives
the governing matrix equations from the model and solves for
the primary quantities, and
(3) post-processing, in which
the analyst checks the validity of the solution, examines
the values of primary quantities (such as displacements and
stresses), and derives and examines additional quantities
(such as specialized stresses and error indicators).
The advantages of FEA are numerous and important. A new
design concept may be modeled to determine its real world
behavior under various load environments, and may therefore
be refined prior to the creation of drawings, when few
dollars have been committed and changes are inexpensive.
Once a detailed CAD model has been developed, FEA can
analyze the design in detail, saving time and money by
reducing the number of prototypes required. An existing
product which is experiencing a field problem, or is simply
being improved, can be analyzed to speed an engineering
change and reduce its cost. In addition, FEA can be
performed on increasingly affordable computer workstations
and personal computers, and professional assistance is
available.
It is also important to recognize the limitations of FEA.
Commercial software packages and the required hardware,
which have seen substantial price reductions, still require
a significant investment. The method can reduce product
testing, but cannot totally replace it. Probably most
important, an inexperienced user can deliver incorrect
answers, upon which expensive decisions will be based.
FEA is a demanding tool, in that the analyst must be
proficient not only in elasticity or fluids, but also in
mathematics, computer science, and especially the finite
element method itself.
Which FEA package to use is a subject that cannot possibly
be covered in this short discussion, and the choice involves
personal preferences as well as package functionality.
Where to run the package depends on the type of analyses
being performed. A typical finite element solution
requires a fast, modern disk subsystem for acceptable
performance. Memory requirements are of course dependent on
the code, but in the interest of performance, the more the
better, with 512 Mbytes to 8 Gbytes per user a representative
range. Processing power is the final link in the
performance chain, with clock speed, cache, pipelining and
multi-processing all contributing to the bottom line.
These analyses can run for hours on the fastest
systems, so computing power is of the essence.
One aspect often overlooked when entering the finite element
area is education. Without adequate training on the finite
element method and the specific FEA package, a new user will
not be productive in a reasonable amount of time, and may in
fact fail miserably. Expect to dedicate one to two weeks up
front, and another one to two weeks over the first year, to
either classroom or self-help education. It is also
important that the user have a basic understanding of the
computer's operating system.
Next month's article will go into detail on the
pre-processing phase of the finite element method.
© 1996-2005 Roensch & Associates. All rights reserved.
FINITE ELEMENT ANALYSIS: Pre-processing
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.
© 1996-2005 Roensch & Associates. All rights reserved.
FINITE ELEMENT ANALYSIS: Solution
Third in a four-part series
While the pre-processing and post-processing phases of the
finite element method are interactive and time-consuming for
the analyst, the solution is often a batch process, and is
demanding of computer resource. The governing equations are
assembled into matrix form and are solved numerically. The
assembly process depends not only on the type of analysis
(e.g. static or dynamic), but also on the model's element
types and properties, material properties and boundary
conditions.
In the case of a linear static structural analysis, the
assembled equation is of the form Kd = r, where K is the
system stiffness matrix, d is the nodal degree of freedom
(dof) displacement vector, and r is the applied nodal load
vector. To appreciate this equation, one must begin with
the underlying elasticity theory. The strain-displacement
relation may be introduced into the stress-strain relation
to express stress in terms of displacement. Under the
assumption of compatibility, the differential equations of
equilibrium in concert with the boundary conditions then
determine a unique displacement field solution, which in
turn determines the strain and stress fields. The chances
of directly solving these equations are slim to none for
anything but the most trivial geometries, hence the need for
approximate numerical techniques presents itself.
A finite element mesh is actually a displacement-nodal
displacement relation, which, through the element
interpolation scheme, determines the displacement anywhere
in an element given the values of its nodal dof.
Introducing this relation into the strain-displacement
relation, we may express strain in terms of the nodal
displacement, element interpolation scheme and differential
operator matrix. Recalling that the expression for the
potential energy of an elastic body includes an integral for
strain energy stored (dependent upon the strain field) and
integrals for work done by external forces (dependent upon
the displacement field), we can therefore express system
potential energy in terms of nodal displacement.
Applying the principle of minimum potential energy, we may
set the partial derivative of potential energy with respect
to the nodal dof vector to zero, resulting in: a summation
of element stiffness integrals, multiplied by the nodal
displacement vector, equals a summation of load integrals.
Each stiffness integral results in an element stiffness
matrix, which sum to produce the system stiffness matrix,
and the summation of load integrals yields the applied load
vector, resulting in Kd = r. In practice, integration rules
are applied to elements, loads appear in the r vector, and
nodal dof boundary conditions may appear in the d vector or
may be partitioned out of the equation.
Solution methods for finite element matrix equations are
plentiful. In the case of the linear static Kd = r,
inverting K is computationally expensive and numerically
unstable. A better technique is Cholesky factorization, a
form of Gauss elimination, and a minor variation on the
"LDU" factorization theme. The K matrix may be efficiently
factored into LDU, where L is lower triangular,
D is diagonal, and U is
upper triangular, resulting in LDUd = r.
Since L and D are easily inverted,
and U is upper
triangular, d may be determined by back-substitution.
Another popular approach is the wavefront method, which
assembles and reduces the equations at the same time. Some
of the best modern solution methods employ sparse matrix
techniques. Because node-to-node stiffnesses are non-zero
only for nearby node pairs, the stiffness matrix has a large
number of zero entries. This can be exploited to reduce
solution time and storage by a factor of 10 or more.
Improved solution methods are continually being developed.
The key point is that the analyst must understand the solution
technique being applied.
Dynamic analysis for too many analysts means normal modes.
Knowledge of the natural frequencies and mode shapes of a
design may be enough in the case of a single-frequency
vibration of an existing product or prototype, with FEA
being used to investigate the effects of mass, stiffness and
damping modifications. When investigating a future product,
or an existing design with multiple modes excited, forced
response modeling should be used to apply the expected
transient or frequency environment to estimate the
displacement and even dynamic stress at each time step.
This discussion has assumed h-code elements, for which the
order of the interpolation polynomials is fixed. Another
technique, p-code, increases the order iteratively until
convergence, with error estimates available after one
analysis. Finally, the boundary element method places
elements only along the geometrical boundary. These
techniques have limitations, but expect to see more of them
in the near future.
Next month's article will discuss the post-processing phase
of the finite element method.
© 1996-2005 Roensch & Associates. All rights reserved.
FINITE ELEMENT ANALYSIS: Post-processing
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.
© 1996-2005 Roensch & Associates. All rights reserved.
by Steve Roensch, President, Roensch & Associates
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