The Finite Element Method: A Four-Article Series

FINITE ELEMENT ANALYSIS: IntroductionFirst in a				assumption of compatibility, the differential equations
four-part seriesFinite element analysis (FEA) is a fairly				of
recent discipline				equilibrium in concert with the boundary conditions
crossing the boundaries of mathematics, physics,				then
engineering				determine a unique displacement field solution, which
and computer science. The method has wide				in
application and				turn determines the strain and stress fields. The
enjoys extensive utilization in the structural, thermal				chances
and				of directly solving these equations are slim to none
fluid analysis areas. The finite element method is				for
comprised of three major phases:				anything but the most trivial geometries, hence the
(1) pre-processing, in				need for
which the analyst develops a finite element mesh to				approximate numerical techniques presents itself.A
divide				finite element mesh is actually a displacement-nodal
the subject geometry into subdomains for				displacement relation, which, through the element
mathematical				interpolation scheme, determines the displacement
analysis, and applies material properties and boundary				anywhere
conditions,				in an element given the values of its nodal dof.
(2) solution, during which the program derives				Introducing this relation into the strain-displacement
the governing matrix equations from the model and				relation, we may express strain in terms of the
solves for				nodal
the primary quantities, and				displacement, element interpolation scheme and
(3) post-processing, in which				differential
the analyst checks the validity of the solution,				operator matrix. Recalling that the expression for
examines				the
the values of primary quantities (such as				potential energy of an elastic body includes an
displacements and				integral for
stresses), and derives and examines additional				strain energy stored (dependent upon the strain
quantities				field) and
(such as specialized stresses and error				integrals for work done by external forces
indicators).The advantages of FEA are numerous and				(dependent upon
important. A new				the displacement field), we can therefore express
design concept may be modeled to determine its				system
real world				potential energy in terms of nodal
behavior under various load environments, and may				displacement.Applying the principle of minimum
therefore				potential energy, we may
be refined prior to the creation of drawings, when				set the partial derivative of potential energy with
few				respect
dollars have been committed and changes are				to the nodal dof vector to zero, resulting in: a
inexpensive.				summation
Once a detailed CAD model has been developed,				of element stiffness integrals, multiplied by the nodal
FEA can				displacement vector, equals a summation of load
analyze the design in detail, saving time and money				integrals.
by				Each stiffness integral results in an element stiffness
reducing the number of prototypes required. An				matrix, which sum to produce the system stiffness
existing				matrix,
product which is experiencing a field problem, or is				and the summation of load integrals yields the
simply				applied load
being improved, can be analyzed to speed an				vector, resulting in Kd = r. In practice, integration
engineering				rules
change and reduce its cost. In addition, FEA can be				are applied to elements, loads appear in the r vector,
performed on increasingly affordable computer				and
workstations				nodal dof boundary conditions may appear in the d
and personal computers, and professional assistance				vector or
is				may be partitioned out of the equation.Solution
available.It is also important to recognize the				methods for finite element matrix equations are
limitations of FEA.				plentiful. In the case of the linear static Kd = r,
Commercial software packages and the required				inverting K is computationally expensive and
hardware,				numerically
which have seen substantial price reductions, still				unstable. A better technique is Cholesky
require				factorization, a
a significant investment. The method can reduce				form of Gauss elimination, and a minor variation on
product				the
testing, but cannot totally replace it. Probably most				"LDU" factorization theme. The K matrix may be
important, an inexperienced user can deliver				efficiently
incorrect				factored into LDU, where L is lower triangular,
answers, upon which expensive decisions will be				D is diagonal, and U is
based.				upper triangular, resulting in LDUd = r.
FEA is a demanding tool, in that the analyst must be				Since L and D are easily inverted,
proficient not only in elasticity or fluids, but also in				and U is upper
mathematics, computer science, and especially the				triangular, d may be determined by back-substitution.
finite				Another popular approach is the wavefront method,
element method itself.Which FEA package to use is				which
a subject that cannot possibly				assembles and reduces the equations at the same
be covered in this short discussion, and the choice				time. Some
involves				of the best modern solution methods employ sparse
personal preferences as well as package				matrix
functionality.				techniques. Because node-to-node stiffnesses are
Where to run the package depends on the type of				non-zero
analyses				only for nearby node pairs, the stiffness matrix has
being performed. A typical finite element solution				a large
requires a fast, modern disk subsystem for				number of zero entries. This can be exploited to
acceptable				reduce
performance. Memory requirements are of course				solution time and storage by a factor of 10 or more.
dependent on				Improved solution methods are continually being
the code, but in the interest of performance, the				developed.
more the				The key point is that the analyst must understand
better, with 512 Mbytes to 8 Gbytes per user a				the solution
representative				technique being applied.Dynamic analysis for too
range. Processing power is the final link in the				many analysts means normal modes.
performance chain, with clock speed, cache,				Knowledge of the natural frequencies and mode
pipelining and				shapes of a
multi-processing all contributing to the bottom line.				design may be enough in the case of a
These analyses can run for hours on the fastest				single-frequency
systems, so computing power is of the essence.One				vibration of an existing product or prototype, with
aspect often overlooked when entering the finite				FEA
element				being used to investigate the effects of mass,
area is education. Without adequate training on the				stiffness and
finite				damping modifications. When investigating a future
element method and the specific FEA package, a				product,
new user will				or an existing design with multiple modes excited,
not be productive in a reasonable amount of time,				forced
and may in				response modeling should be used to apply the
fact fail miserably. Expect to dedicate one to two				expected
weeks up				transient or frequency environment to estimate the
front, and another one to two weeks over the first				displacement and even dynamic stress at each time
year, to				step.This discussion has assumed h-code elements,
either classroom or self-help education. It is also				for which the
important that the user have a basic understanding				order of the interpolation polynomials is fixed.
of the				Another
computer's operating system.Next month's article will				technique, p-code, increases the order iteratively until
go into detail on the				convergence, with error estimates available after
pre-processing phase of the finite element				one
method.Â© 1996-2005 Roensch & Associates. All				analysis. Finally, the boundary element method places
rights reserved.				elements only along the geometrical boundary.
FINITE ELEMENT ANALYSIS: Pre-processingSecond				These
in a four-part seriesAs discussed last month, finite				techniques have limitations, but expect to see more
element analysis is				of them
comprised of pre-processing, solution and				in the near future.Next month's article will discuss the
post-processing				post-processing phase
phases. The goals of pre-processing are to develop				of the finite element method.Â© 1996-2005
an				Roensch & Associates. All rights reserved.
appropriate finite element mesh, assign suitable				FINITE ELEMENT ANALYSIS: Post-processingLast in
material				a four-part seriesAfter a finite element model has
properties, and apply boundary conditions in the				been prepared and checked,
form of				boundary conditions have been applied, and the
restraints and loads.The finite element mesh				model has
subdivides the geometry into				been solved, it is time to investigate the results of
elements, upon which are found nodes.				the
The nodes, which are				analysis. This activity is known as the
really just point locations in space, are generally				post-processing
located				phase of the finite element method.Post-processing
at the element corners and perhaps near each				begins with a thorough check for problems
midside. For a				that may have occurred during solution. Most solvers
two-dimensional (2D) analysis, or a three-dimensional				provide a log file, which should be searched for
(3D)				warnings or
thin shell analysis, the elements are essentially 2D,				errors, and which will also provide a quantitative
but				measure
may be "warped" slightly to conform to a 3D				of how well-behaved the numerical procedures were
surface. An				during
example is the thin shell linear quadrilateral; thin shell				solution. Next, reaction loads at restrained nodes
implies essentially classical shell theory, linear defines				should
the interpolation of mathematical quantities across				be summed and examined as a "sanity check".
the				Reaction loads
element, and quadrilateral describes the geometry.				that do not closely balance the applied load resultant
For a 3D				for a
solid analysis, the elements have physical thickness in				linear static analysis should cast doubt on the validity
all				of
three dimensions. Common examples include solid				other results. Error norms such as strain energy
linear				density
brick and solid parabolic tetrahedral elements. In				and stress deviation among adjacent elements might
addition, there are many special elements, such as				be looked
axisymmetric elements for situations in which the				at next, but for h-code analyses these quantities are
geometry,				best
material and boundary conditions are all symmetric				used to target subsequent adaptive remeshing.Once
about an				the solution is verified to be free of numerical
axis.The model's degrees of freedom (dof) are				problems, the quantities of interest may be
assigned at the				examined. Many
nodes. Solid elements generally have three				display options are available, the choice of which
translational				depends
dof per node. Rotations are accomplished through				on the mathematical form of the quantity as well as
translations of groups of nodes relative to other				its
nodes.				physical meaning. For example, the displacement of a
Thin shell elements, on the other hand, have six dof				solid
per				linear brick element's node is a 3-component spatial
node: three translations and three rotations. The				vector,
addition				and the model's overall displacement is often
of rotational dof allows for evaluation of quantities				displayed by
through the shell, such as bending stresses due to				superposing the deformed shape over the
rotation				undeformed shape.
of one node relative to another. Thus, for structures				Dynamic viewing and animation capabilities aid greatly
in				in
which classical thin shell theory is a valid				obtaining an understanding of the deformation
approximation,				pattern.
carrying extra dof at each node bypasses the				Stresses, being tensor quantities, currently lack a
necessity of				good
modeling the physical thickness. The assignment of				single visualization technique, and thus derived stress
nodal				quantities are extracted and displayed. Principal
dof also depends on the class of analysis. For a				stress
thermal				vectors may be displayed as color-coded arrows,
analysis, for example, only one temperature dof				indicating
exists at				both direction and magnitude. The magnitude of
each node.Developing the mesh is usually the most				principal
time-consuming task				stresses or of a scalar failure stress such as the
in FEA. In the past, node locations were keyed in				Von Mises
manually				stress may be displayed on the model as colored
to approximate the geometry. The more modern				bands. When
approach is to				this type of display is treated as a 3D object
develop the mesh directly on the CAD geometry,				subjected to
which will be				light sources, the resulting image is known as a
(1) wireframe, with points and curves representing				shaded
edges,				image stress plot. Displacement magnitude may also
(2) surfaced, with surfaces defining boundaries, or				be
(3)				displayed by colored bands, but this can lead to
solid, defining where the material is. Solid geometry is				misinterpretation as a stress plot.An area of
preferred, but often a surfacing package can create				post-processing that is rapidly gaining
a				popularity is that of adaptive remeshing. Error norms
complex blend that a solids package will not handle.				such
As far				as strain energy density are used to remesh the
as geometric detail, an underlying rule of FEA is to				model,
"model				placing a denser mesh in regions needing
what is there", and yet simplifying assumptions				improvement and a
simply must				coarser mesh in areas of overkill. Adaptivity requires
be applied to avoid huge models. Analyst experience				an
is of				associative link between the model and the
the essence.The geometry is meshed with a				underlying CAD
mapping algorithm or an				geometry, and works best if boundary conditions
automatic free-meshing algorithm. The first maps a				may be
rectangular grid onto a geometric region, which must				applied directly to the geometry, as well. Adaptive
therefore have the correct number of sides. Mapped				remeshing is a recent demonstration of the iterative
meshes				nature
can use the accurate and cheap solid linear brick 3D				of h-code analysis.Optimization is another area
element, but can be very time-consuming, if not				enjoying recent advancement.
impossible,				Based on the values of various results, the model is
to apply to complex geometries. Free-meshing				modified automatically in an attempt to satisfy
automatically				certain
subdivides meshing regions into elements, with the				performance criteria and is solved again. The process
advantages of fast meshing, easy mesh-size				iterates until some convergence criterion is met. In
transitioning				its
(for a denser mesh in regions of large gradient), and				scalar form, optimization modifies beam
adaptive capabilities. Disadvantages include				cross-sectional
generation of				properties, thin shell thicknesses and/or material
huge models, generation of distorted elements, and,				properties in an attempt to meet maximum stress
in 3D,				constraints,
the use of the rather expensive solid parabolic				maximum deflection constraints, and/or vibrational
tetrahedral				frequency
element. It is always important to check elemental				constraints. Shape optimization is more complex,
distortion prior to solution. A badly distorted element				with the
will cause a matrix singularity, killing the solution. A				actual 3D model boundaries being modified. This is
less distorted element may solve, but can deliver				best
very poor				accomplished by using the driving dimensions as
answers. Acceptable levels of distortion are				optimization
dependent upon				parameters, but mesh quality at each iteration can
the solver being used.Material properties required				be a
vary with the type of solution.				concern.Another direction clearly visible in the finite
A linear statics analysis, for example, will require an				element
elastic modulus, Poisson's ratio and perhaps a density				field is the integration of FEA packages with
for				so-called
each material. Thermal properties are required for a				"mechanism" packages, which analyze motion and
thermal				forces of
analysis. Examples of restraints are declaring a nodal				large-displacement multi-body systems. A long-term
translation or temperature. Loads include forces,				goal
pressures				would be real-time computation and display of
and heat flux. It is preferable to apply boundary				displacements
conditions to the CAD geometry, with the FEA				and stresses in a multi-body system undergoing large
package				displacement motion, with frictional effects and fluid
transferring them to the underlying model, to allow				flow
for				taken into account when necessary. It is difficult to
simpler application of adaptive and optimization				estimate the increase in computing power necessary
algorithms.				to
It is worth noting that the largest error in the entire				accomplish this feat, but 2 or 3 orders of magnitude
process is often in the boundary conditions. Running				is
multiple cases as a sensitivity analysis may be				probably close. Algorithms to integrate these fields
required.Next month's article will discuss the solution				of
phase of the				analysis may be expected to follow the computing
finite element method.Â© 1996-2005 Roensch &				power
Associates. All rights reserved.				increases.In summary, the finite element method is a
FINITE ELEMENT ANALYSIS: SolutionThird in a				relatively recent
four-part seriesWhile the pre-processing and				discipline that has quickly become a mature method,
post-processing phases of the				especially for structural and thermal analysis. The
finite element method are interactive and				costs
time-consuming for				of applying this technology to everyday design tasks
the analyst, the solution is often a batch process,				have
and is				been dropping, while the capabilities delivered by the
demanding of computer resource. The governing				method expand constantly. With education in the
equations are				technique
assembled into matrix form and are solved				and in the commercial software packages becoming
numerically. The				more and
assembly process depends not only on the type of				more available, the question has moved from "Why
analysis				apply FEA?"
(e.g. static or dynamic), but also on the model's				to "Why not?". The method is fully capable of
element				delivering
types and properties, material properties and				higher quality products in a shorter design cycle with
boundary				a
conditions.In the case of a linear static structural				reduced chance of field failure, provided it is applied
analysis, the				by a
assembled equation is of the form Kd = r, where K				capable analyst. It is also a valid indication of
is the				thorough
system stiffness matrix, d is the nodal degree of				design practices, should an unexpected litigation crop
freedom				up.
(dof) displacement vector, and r is the applied nodal				The time is now for industry to make greater use
load				of this and
vector. To appreciate this equation, one must begin				other analysis techniques.Â© 1996-2005 Roensch
with				& Associates. All rights reserved.by Steve Roensch,
the underlying elasticity theory. The				President, Roensch & AssociatesSteve Roensch is a
strain-displacement				mechanical engineering consultant with more than 20
relation may be introduced into the stress-strain				years of professional experience. He has analyzed
relation				hundreds of product designs and has served as an
to express stress in terms of displacement. Under				expert witness across many industries, including giving
the				depositions and court testimony.

Brc inc

The Finite Element Method: A Four-Article Series