The Finite Element Method: A Four-Article Series

FINITE ELEMENT ANALYSIS: IntroductionFirst in aassumption of compatibility, the differential equations
four-part seriesFinite element analysis (FEA) is a fairlyof
recent disciplineequilibrium in concert with the boundary conditions
crossing the boundaries of mathematics, physics,then
engineeringdetermine a unique displacement field solution, which
and computer science. The method has widein
application andturn determines the strain and stress fields. The
enjoys extensive utilization in the structural, thermalchances
andof directly solving these equations are slim to none
fluid analysis areas. The finite element method isfor
comprised of three major phases:anything but the most trivial geometries, hence the
(1) pre-processing, inneed for
which the analyst develops a finite element mesh toapproximate numerical techniques presents itself.A
dividefinite element mesh is actually a displacement-nodal
the subject geometry into subdomains fordisplacement relation, which, through the element
mathematicalinterpolation scheme, determines the displacement
analysis, and applies material properties and boundaryanywhere
conditions,in an element given the values of its nodal dof.
(2) solution, during which the program derivesIntroducing this relation into the strain-displacement
the governing matrix equations from the model andrelation, we may express strain in terms of the
solves fornodal
the primary quantities, anddisplacement, element interpolation scheme and
(3) post-processing, in whichdifferential
the analyst checks the validity of the solution,operator matrix. Recalling that the expression for
examinesthe
the values of primary quantities (such aspotential energy of an elastic body includes an
displacements andintegral for
stresses), and derives and examines additionalstrain energy stored (dependent upon the strain
quantitiesfield) and
(such as specialized stresses and errorintegrals for work done by external forces
indicators).The advantages of FEA are numerous and(dependent upon
important. A newthe displacement field), we can therefore express
design concept may be modeled to determine itssystem
real worldpotential energy in terms of nodal
behavior under various load environments, and maydisplacement.Applying the principle of minimum
thereforepotential energy, we may
be refined prior to the creation of drawings, whenset the partial derivative of potential energy with
fewrespect
dollars have been committed and changes areto 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 candisplacement vector, equals a summation of load
analyze the design in detail, saving time and moneyintegrals.
byEach stiffness integral results in an element stiffness
reducing the number of prototypes required. Anmatrix, which sum to produce the system stiffness
existingmatrix,
product which is experiencing a field problem, or isand the summation of load integrals yields the
simplyapplied load
being improved, can be analyzed to speed anvector, resulting in Kd = r. In practice, integration
engineeringrules
change and reduce its cost. In addition, FEA can beare applied to elements, loads appear in the r vector,
performed on increasingly affordable computerand
workstationsnodal dof boundary conditions may appear in the d
and personal computers, and professional assistancevector or
ismay be partitioned out of the equation.Solution
available.It is also important to recognize themethods for finite element matrix equations are
limitations of FEA.plentiful. In the case of the linear static Kd = r,
Commercial software packages and the requiredinverting K is computationally expensive and
hardware,numerically
which have seen substantial price reductions, stillunstable. A better technique is Cholesky
requirefactorization, a
a significant investment. The method can reduceform of Gauss elimination, and a minor variation on
productthe
testing, but cannot totally replace it. Probably most"LDU" factorization theme. The K matrix may be
important, an inexperienced user can deliverefficiently
incorrectfactored into LDU, where L is lower triangular,
answers, upon which expensive decisions will beD is diagonal, and U is
based.upper triangular, resulting in LDUd = r.
FEA is a demanding tool, in that the analyst must beSince L and D are easily inverted,
proficient not only in elasticity or fluids, but also inand U is upper
mathematics, computer science, and especially thetriangular, d may be determined by back-substitution.
finiteAnother popular approach is the wavefront method,
element method itself.Which FEA package to use iswhich
a subject that cannot possiblyassembles and reduces the equations at the same
be covered in this short discussion, and the choicetime. Some
involvesof the best modern solution methods employ sparse
personal preferences as well as packagematrix
functionality.techniques. Because node-to-node stiffnesses are
Where to run the package depends on the type ofnon-zero
analysesonly for nearby node pairs, the stiffness matrix has
being performed. A typical finite element solutiona large
requires a fast, modern disk subsystem fornumber of zero entries. This can be exploited to
acceptablereduce
performance. Memory requirements are of coursesolution time and storage by a factor of 10 or more.
dependent onImproved solution methods are continually being
the code, but in the interest of performance, thedeveloped.
more theThe key point is that the analyst must understand
better, with 512 Mbytes to 8 Gbytes per user athe solution
representativetechnique being applied.Dynamic analysis for too
range. Processing power is the final link in themany analysts means normal modes.
performance chain, with clock speed, cache,Knowledge of the natural frequencies and mode
pipelining andshapes 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 fastestsingle-frequency
systems, so computing power is of the essence.Onevibration of an existing product or prototype, with
aspect often overlooked when entering the finiteFEA
elementbeing used to investigate the effects of mass,
area is education. Without adequate training on thestiffness and
finitedamping modifications. When investigating a future
element method and the specific FEA package, aproduct,
new user willor an existing design with multiple modes excited,
not be productive in a reasonable amount of time,forced
and may inresponse modeling should be used to apply the
fact fail miserably. Expect to dedicate one to twoexpected
weeks uptransient or frequency environment to estimate the
front, and another one to two weeks over the firstdisplacement and even dynamic stress at each time
year, tostep.This discussion has assumed h-code elements,
either classroom or self-help education. It is alsofor which the
important that the user have a basic understandingorder of the interpolation polynomials is fixed.
of theAnother
computer's operating system.Next month's article willtechnique, p-code, increases the order iteratively until
go into detail on theconvergence, with error estimates available after
pre-processing phase of the finite elementone
method.© 1996-2005 Roensch & Associates. Allanalysis. Finally, the boundary element method places
rights reserved.elements only along the geometrical boundary.
FINITE ELEMENT ANALYSIS: Pre-processingSecondThese
in a four-part seriesAs discussed last month, finitetechniques have limitations, but expect to see more
element analysis isof them
comprised of pre-processing, solution andin the near future.Next month's article will discuss the
post-processingpost-processing phase
phases. The goals of pre-processing are to developof the finite element method.© 1996-2005
anRoensch & Associates. All rights reserved.
appropriate finite element mesh, assign suitableFINITE ELEMENT ANALYSIS: Post-processingLast in
materiala four-part seriesAfter a finite element model has
properties, and apply boundary conditions in thebeen prepared and checked,
form ofboundary conditions have been applied, and the
restraints and loads.The finite element meshmodel has
subdivides the geometry intobeen solved, it is time to investigate the results of
elements, upon which are found nodes.the
The nodes, which areanalysis. This activity is known as the
really just point locations in space, are generallypost-processing
locatedphase of the finite element method.Post-processing
at the element corners and perhaps near eachbegins with a thorough check for problems
midside. For athat may have occurred during solution. Most solvers
two-dimensional (2D) analysis, or a three-dimensionalprovide 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
butmeasure
may be "warped" slightly to conform to a 3Dof how well-behaved the numerical procedures were
surface. Anduring
example is the thin shell linear quadrilateral; thin shellsolution. Next, reaction loads at restrained nodes
implies essentially classical shell theory, linear definesshould
the interpolation of mathematical quantities acrossbe summed and examined as a "sanity check".
theReaction loads
element, and quadrilateral describes the geometry.that do not closely balance the applied load resultant
For a 3Dfor a
solid analysis, the elements have physical thickness inlinear static analysis should cast doubt on the validity
allof
three dimensions. Common examples include solidother results. Error norms such as strain energy
lineardensity
brick and solid parabolic tetrahedral elements. Inand stress deviation among adjacent elements might
addition, there are many special elements, such asbe looked
axisymmetric elements for situations in which theat next, but for h-code analyses these quantities are
geometry,best
material and boundary conditions are all symmetricused to target subsequent adaptive remeshing.Once
about anthe solution is verified to be free of numerical
axis.The model's degrees of freedom (dof) areproblems, the quantities of interest may be
assigned at theexamined. Many
nodes. Solid elements generally have threedisplay options are available, the choice of which
translationaldepends
dof per node. Rotations are accomplished throughon the mathematical form of the quantity as well as
translations of groups of nodes relative to otherits
nodes.physical meaning. For example, the displacement of a
Thin shell elements, on the other hand, have six dofsolid
perlinear brick element's node is a 3-component spatial
node: three translations and three rotations. Thevector,
additionand the model's overall displacement is often
of rotational dof allows for evaluation of quantitiesdisplayed by
through the shell, such as bending stresses due tosuperposing the deformed shape over the
rotationundeformed shape.
of one node relative to another. Thus, for structuresDynamic viewing and animation capabilities aid greatly
inin
which classical thin shell theory is a validobtaining an understanding of the deformation
approximation,pattern.
carrying extra dof at each node bypasses theStresses, being tensor quantities, currently lack a
necessity ofgood
modeling the physical thickness. The assignment ofsingle visualization technique, and thus derived stress
nodalquantities are extracted and displayed. Principal
dof also depends on the class of analysis. For astress
thermalvectors may be displayed as color-coded arrows,
analysis, for example, only one temperature dofindicating
exists atboth direction and magnitude. The magnitude of
each node.Developing the mesh is usually the mostprincipal
time-consuming taskstresses or of a scalar failure stress such as the
in FEA. In the past, node locations were keyed inVon Mises
manuallystress may be displayed on the model as colored
to approximate the geometry. The more modernbands. When
approach is tothis type of display is treated as a 3D object
develop the mesh directly on the CAD geometry,subjected to
which will belight sources, the resulting image is known as a
(1) wireframe, with points and curves representingshaded
edges,image stress plot. Displacement magnitude may also
(2) surfaced, with surfaces defining boundaries, orbe
(3)displayed by colored bands, but this can lead to
solid, defining where the material is. Solid geometry ismisinterpretation as a stress plot.An area of
preferred, but often a surfacing package can createpost-processing that is rapidly gaining
apopularity is that of adaptive remeshing. Error norms
complex blend that a solids package will not handle.such
As faras strain energy density are used to remesh the
as geometric detail, an underlying rule of FEA is tomodel,
"modelplacing a denser mesh in regions needing
what is there", and yet simplifying assumptionsimprovement and a
simply mustcoarser mesh in areas of overkill. Adaptivity requires
be applied to avoid huge models. Analyst experiencean
is ofassociative link between the model and the
the essence.The geometry is meshed with aunderlying CAD
mapping algorithm or angeometry, and works best if boundary conditions
automatic free-meshing algorithm. The first maps amay be
rectangular grid onto a geometric region, which mustapplied directly to the geometry, as well. Adaptive
therefore have the correct number of sides. Mappedremeshing is a recent demonstration of the iterative
meshesnature
can use the accurate and cheap solid linear brick 3Dof h-code analysis.Optimization is another area
element, but can be very time-consuming, if notenjoying recent advancement.
impossible,Based on the values of various results, the model is
to apply to complex geometries. Free-meshingmodified automatically in an attempt to satisfy
automaticallycertain
subdivides meshing regions into elements, with theperformance criteria and is solved again. The process
advantages of fast meshing, easy mesh-sizeiterates until some convergence criterion is met. In
transitioningits
(for a denser mesh in regions of large gradient), andscalar form, optimization modifies beam
adaptive capabilities. Disadvantages includecross-sectional
generation ofproperties, 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 parabolicmaximum deflection constraints, and/or vibrational
tetrahedralfrequency
element. It is always important to check elementalconstraints. Shape optimization is more complex,
distortion prior to solution. A badly distorted elementwith the
will cause a matrix singularity, killing the solution. Aactual 3D model boundaries being modified. This is
less distorted element may solve, but can deliverbest
very pooraccomplished by using the driving dimensions as
answers. Acceptable levels of distortion areoptimization
dependent uponparameters, but mesh quality at each iteration can
the solver being used.Material properties requiredbe a
vary with the type of solution.concern.Another direction clearly visible in the finite
A linear statics analysis, for example, will require anelement
elastic modulus, Poisson's ratio and perhaps a densityfield is the integration of FEA packages with
forso-called
each material. Thermal properties are required for a"mechanism" packages, which analyze motion and
thermalforces of
analysis. Examples of restraints are declaring a nodallarge-displacement multi-body systems. A long-term
translation or temperature. Loads include forces,goal
pressureswould be real-time computation and display of
and heat flux. It is preferable to apply boundarydisplacements
conditions to the CAD geometry, with the FEAand stresses in a multi-body system undergoing large
packagedisplacement motion, with frictional effects and fluid
transferring them to the underlying model, to allowflow
fortaken into account when necessary. It is difficult to
simpler application of adaptive and optimizationestimate the increase in computing power necessary
algorithms.to
It is worth noting that the largest error in the entireaccomplish this feat, but 2 or 3 orders of magnitude
process is often in the boundary conditions. Runningis
multiple cases as a sensitivity analysis may beprobably close. Algorithms to integrate these fields
required.Next month's article will discuss the solutionof
phase of theanalysis may be expected to follow the computing
finite element method.© 1996-2005 Roensch &power
Associates. All rights reserved.increases.In summary, the finite element method is a
FINITE ELEMENT ANALYSIS: SolutionThird in arelatively recent
four-part seriesWhile the pre-processing anddiscipline that has quickly become a mature method,
post-processing phases of theespecially for structural and thermal analysis. The
finite element method are interactive andcosts
time-consuming forof applying this technology to everyday design tasks
the analyst, the solution is often a batch process,have
and isbeen dropping, while the capabilities delivered by the
demanding of computer resource. The governingmethod expand constantly. With education in the
equations aretechnique
assembled into matrix form and are solvedand in the commercial software packages becoming
numerically. Themore and
assembly process depends not only on the type ofmore available, the question has moved from "Why
analysisapply FEA?"
(e.g. static or dynamic), but also on the model'sto "Why not?". The method is fully capable of
elementdelivering
types and properties, material properties andhigher quality products in a shorter design cycle with
boundarya
conditions.In the case of a linear static structuralreduced chance of field failure, provided it is applied
analysis, theby a
assembled equation is of the form Kd = r, where Kcapable analyst. It is also a valid indication of
is thethorough
system stiffness matrix, d is the nodal degree ofdesign practices, should an unexpected litigation crop
freedomup.
(dof) displacement vector, and r is the applied nodalThe time is now for industry to make greater use
loadof this and
vector. To appreciate this equation, one must beginother analysis techniques.© 1996-2005 Roensch
with& Associates. All rights reserved.by Steve Roensch,
the underlying elasticity theory. ThePresident, Roensch & AssociatesSteve Roensch is a
strain-displacementmechanical engineering consultant with more than 20
relation may be introduced into the stress-strainyears of professional experience. He has analyzed
relationhundreds of product designs and has served as an
to express stress in terms of displacement. Underexpert witness across many industries, including giving
thedepositions and court testimony.