Finite element modeling

The Finite Element Method: A Four-Article Series

FINITE ELEMENT ANALYSIS: IntroductionFirst in				relation may be introduced into the
a four-part seriesFinite element analysis				stress-strain relation
(FEA) is a fairly recent discipline
				to express stress in terms of displacement.
crossing the boundaries of mathematics,				Under the
physics, engineering
				assumption of compatibility, the
and computer science. The method has wide				differential equations of
application and
				equilibrium in concert with the boundary
enjoys extensive utilization in the				conditions then
structural, thermal and
				determine a unique displacement field
fluid analysis areas. The finite element				solution, which in
method is
				turn determines the strain and stress
comprised of three major phases:				fields. The chances

(1) pre-processing, in				of directly solving these equations are slim
				to none for
which the analyst develops a finite element
mesh to divide				anything but the most trivial geometries,
				hence the need for
the subject geometry into subdomains for
mathematical				approximate numerical techniques presents
				itself.A finite element mesh is actually a
analysis, and applies material properties				displacement-nodal
and boundary
				displacement relation, which, through the
conditions,				element

(2) solution, during which the program				interpolation scheme, determines the
derives				displacement anywhere

the governing matrix equations from the				in an element given the values of its nodal
model and solves for				dof.

the primary quantities, and				Introducing this relation into the
				strain-displacement
(3) post-processing, in which
				relation, we may express strain in terms of
the analyst checks the validity of the				the nodal
solution, examines
				displacement, element interpolation scheme
the values of primary quantities (such as				and differential
displacements and
				operator matrix. Recalling that the
stresses), and derives and examines				expression for the
additional quantities
				potential energy of an elastic body includes
(such as specialized stresses and error				an integral for
indicators).The advantages of FEA are
numerous and important. A new				strain energy stored (dependent upon the
				strain field) and
design concept may be modeled to determine
its real world				integrals for work done by external forces
				(dependent upon
behavior under various load environments,
and may therefore				the displacement field), we can therefore
				express system
be refined prior to the creation of
drawings, when few				potential energy in terms of nodal
				displacement.Applying the principle of
dollars have been committed and changes are				minimum potential energy, we may
inexpensive.
				set the partial derivative of potential
Once a detailed CAD model has been				energy with respect
developed, FEA can
				to the nodal dof vector to zero, resulting
analyze the design in detail, saving time				in: a summation
and money by
				of element stiffness integrals, multiplied
reducing the number of prototypes required.				by the nodal
An existing
				displacement vector, equals a summation of
product which is experiencing a field				load integrals.
problem, or is simply
				Each stiffness integral results in an
being improved, can be analyzed to speed an				element stiffness
engineering
				matrix, which sum to produce the system
change and reduce its cost. In addition,				stiffness matrix,
FEA can be
				and the summation of load integrals yields
performed on increasingly affordable				the applied load
computer workstations
				vector, resulting in Kd = r. In practice,
and personal computers, and professional				integration rules
assistance is
				are applied to elements, loads appear in the
available.It is also important to recognize				r vector, and
the limitations of FEA.
				nodal dof boundary conditions may appear in
Commercial software packages and the				the d vector or
required hardware,
				may be partitioned out of the
which have seen substantial price				equation.Solution methods for finite element
reductions, still require				matrix equations are

a significant investment. The method can				plentiful. In the case of the linear static
reduce product				Kd = r,

testing, but cannot totally replace it.				inverting K is computationally expensive and
Probably most				numerically

important, an inexperienced user can deliver				unstable. A better technique is Cholesky
incorrect				factorization, a

answers, upon which expensive decisions will				form of Gauss elimination, and a minor
be based.				variation on the

FEA is a demanding tool, in that the analyst				"LDU" factorization theme. The K matrix may
must be				be efficiently

proficient not only in elasticity or fluids,				factored into LDU, where L is lower
but also in				triangular,

mathematics, computer science, and				D is diagonal, and U is
especially the finite
				upper triangular, resulting in LDUd = r.
element method itself.Which FEA package to
use is a subject that cannot possibly				Since L and D are easily inverted,

be covered in this short discussion, and the				and U is upper
choice involves
				triangular, d may be determined by
personal preferences as well as package				back-substitution.
functionality.
				Another popular approach is the wavefront
Where to run the package depends on the type				method, which
of analyses
				assembles and reduces the equations at the
being performed. A typical finite element				same time. Some
solution
				of the best modern solution methods employ
requires a fast, modern disk subsystem for				sparse matrix
acceptable
				techniques. Because node-to-node
performance. Memory requirements are of				stiffnesses are non-zero
course dependent on
				only for nearby node pairs, the stiffness
the code, but in the interest of				matrix has a large
performance, the more the
				number of zero entries. This can be
better, with 512 Mbytes to 8 Gbytes per user				exploited to reduce
a representative
				solution time and storage by a factor of 10
range. Processing power is the final link				or more.
in the
				Improved solution methods are continually
performance chain, with clock speed, cache,				being developed.
pipelining and
				The key point is that the analyst must
multi-processing all contributing to the				understand the solution
bottom line.
				technique being applied.Dynamic analysis for
These analyses can run for hours on the				too many analysts means normal modes.
fastest
				Knowledge of the natural frequencies and
systems, so computing power is of the				mode shapes of a
essence.One aspect often overlooked when
entering the finite element				design may be enough in the case of a
				single-frequency
area is education. Without adequate
training on the finite				vibration of an existing product or
				prototype, with FEA
element method and the specific FEA package,
a new user will				being used to investigate the effects of
				mass, stiffness and
not be productive in a reasonable amount of
time, and may in				damping modifications. When investigating a
				future product,
fact fail miserably. Expect to dedicate one
to two weeks up				or an existing design with multiple modes
				excited, forced
front, and another one to two weeks over the
first year, to				response modeling should be used to apply
				the expected
either classroom or self-help education. It
is also				transient or frequency environment to
				estimate the
important that the user have a basic
understanding of the				displacement and even dynamic stress at each
				time step.This discussion has assumed h-code
computer's operating system.Next month's				elements, for which the
article will go into detail on the
				order of the interpolation polynomials is
pre-processing phase of the finite element				fixed. Another
method.Â© 1996-2005 Roensch & Associates.
All rights reserved.				technique, p-code, increases the order
				iteratively until
FINITE ELEMENT ANALYSIS:
Pre-processingSecond in a four-part seriesAs				convergence, with error estimates available
discussed last month, finite element analysis				after one
is
				analysis. Finally, the boundary element
comprised of pre-processing, solution and				method places
post-processing
				elements only along the geometrical
phases. The goals of pre-processing are to				boundary. These
develop an
				techniques have limitations, but expect to
appropriate finite element mesh, assign				see more of them
suitable material
				in the near future.Next month's article will
properties, and apply boundary conditions in				discuss the post-processing phase
the form of
				of the finite element method.Â© 1996-2005
restraints and loads.The finite element mesh				Roensch & Associates. All rights reserved.
subdivides the geometry into
				FINITE ELEMENT ANALYSIS: Post-processingLast
elements, upon which are found nodes.				in a four-part seriesAfter a finite element
				model has been prepared and checked,
The nodes, which are
				boundary conditions have been applied, and
really just point locations in space, are				the model has
generally located
				been solved, it is time to investigate the
at the element corners and perhaps near each				results of the
midside. For a
				analysis. This activity is known as the
two-dimensional (2D) analysis, or a				post-processing
three-dimensional (3D)
				phase of the finite element
thin shell analysis, the elements are				method.Post-processing begins with a thorough
essentially 2D, but				check for problems

may be "warped" slightly to conform to a 3D				that may have occurred during solution.
surface. An				Most solvers

example is the thin shell linear				provide a log file, which should be searched
quadrilateral; thin shell				for warnings or

implies essentially classical shell theory,				errors, and which will also provide a
linear defines				quantitative measure

the interpolation of mathematical quantities				of how well-behaved the numerical procedures
across the				were during

element, and quadrilateral describes the				solution. Next, reaction loads at
geometry. For a 3D				restrained nodes should

solid analysis, the elements have physical				be summed and examined as a "sanity check".
thickness in all				Reaction loads

three dimensions. Common examples include				that do not closely balance the applied load
solid linear				resultant for a

brick and solid parabolic tetrahedral				linear static analysis should cast doubt on
elements. In				the validity of

addition, there are many special elements,				other results. Error norms such as strain
such as				energy density

axisymmetric elements for situations in				and stress deviation among adjacent elements
which the geometry,				might be looked

material and boundary conditions are all				at next, but for h-code analyses these
symmetric about an				quantities are best

axis.The model's degrees of freedom (dof)				used to target subsequent adaptive
are assigned at the				remeshing.Once the solution is verified to be
				free of numerical
nodes. Solid elements generally have three
translational				problems, the quantities of interest may be
				examined. Many
dof per node. Rotations are accomplished
through				display options are available, the choice of
				which depends
translations of groups of nodes relative to
other nodes.				on the mathematical form of the quantity as
				well as its
Thin shell elements, on the other hand, have
six dof per				physical meaning. For example, the
				displacement of a solid
node: three translations and three
rotations. The addition				linear brick element's node is a 3-component
				spatial vector,
of rotational dof allows for evaluation of
quantities				and the model's overall displacement is
				often displayed by
through the shell, such as bending stresses
due to rotation				superposing the deformed shape over the
				undeformed shape.
of one node relative to another. Thus, for
structures in				Dynamic viewing and animation capabilities
				aid greatly in
which classical thin shell theory is a valid
approximation,				obtaining an understanding of the
				deformation pattern.
carrying extra dof at each node bypasses the
necessity of				Stresses, being tensor quantities, currently
				lack a good
modeling the physical thickness. The
assignment of nodal				single visualization technique, and thus
				derived stress
dof also depends on the class of analysis.
For a thermal				quantities are extracted and displayed.
				Principal stress
analysis, for example, only one temperature
dof exists at				vectors may be displayed as color-coded
				arrows, indicating
each node.Developing the mesh is usually the
most time-consuming task				both direction and magnitude. The magnitude
				of principal
in FEA. In the past, node locations were
keyed in manually				stresses or of a scalar failure stress such
				as the Von Mises
to approximate the geometry. The more
modern approach is to				stress may be displayed on the model as
				colored bands. When
develop the mesh directly on the CAD
geometry, which will be				this type of display is treated as a 3D
				object subjected to
(1) wireframe, with points and curves
representing edges,				light sources, the resulting image is known
				as a shaded
(2) surfaced, with surfaces defining
boundaries, or (3)				image stress plot. Displacement magnitude
				may also be
solid, defining where the material is.
Solid geometry is				displayed by colored bands, but this can
				lead to
preferred, but often a surfacing package can
create a				misinterpretation as a stress plot.An area
				of post-processing that is rapidly gaining
complex blend that a solids package will not
handle. As far				popularity is that of adaptive remeshing.
				Error norms such
as geometric detail, an underlying rule of
FEA is to "model				as strain energy density are used to remesh
				the model,
what is there", and yet simplifying
assumptions simply must				placing a denser mesh in regions needing
				improvement and a
be applied to avoid huge models. Analyst
experience is of				coarser mesh in areas of overkill.
				Adaptivity requires an
the essence.The geometry is meshed with a
mapping algorithm or an				associative link between the model and the
				underlying CAD
automatic free-meshing algorithm. The first
maps a				geometry, and works best if boundary
				conditions 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 meshes				remeshing is a recent demonstration of the
				iterative nature
can use the accurate and cheap solid linear
brick 3D				of h-code analysis.Optimization is another
				area enjoying recent advancement.
element, but can be very time-consuming, if
not impossible,				Based on the values of various results, the
				model is
to apply to complex geometries.
Free-meshing automatically				modified automatically in an attempt to
				satisfy certain
subdivides meshing regions into elements,
with the				performance criteria and is solved again.
				The process
advantages of fast meshing, easy mesh-size
transitioning				iterates until some convergence criterion is
				met. In its
(for a denser mesh in regions of large
gradient), and				scalar form, optimization modifies beam
				cross-sectional
adaptive capabilities. Disadvantages
include generation of				properties, thin shell thicknesses and/or
				material
huge models, generation of distorted
elements, and, in 3D,				properties in an attempt to meet maximum
				stress constraints,
the use of the rather expensive solid
parabolic tetrahedral				maximum deflection constraints, and/or
				vibrational frequency
element. It is always important to check
elemental				constraints. Shape optimization is more
				complex, with the
distortion prior to solution. A badly
distorted element				actual 3D model boundaries being modified.
				This is best
will cause a matrix singularity, killing the
solution. A				accomplished by using the driving dimensions
				as optimization
less distorted element may solve, but can
deliver very poor				parameters, but mesh quality at each
				iteration can be a
answers. Acceptable levels of distortion
are dependent upon				concern.Another direction clearly visible in
				the finite element
the solver being used.Material properties
required vary with the type of solution.				field is the integration of FEA packages
				with so-called
A linear statics analysis, for example, will
require an				"mechanism" packages, which analyze motion
				and forces of
elastic modulus, Poisson's ratio and perhaps
a density for				large-displacement multi-body systems. A
				long-term goal
each material. Thermal properties are
required for a thermal				would be real-time computation and display
				of displacements
analysis. Examples of restraints are
declaring a nodal				and stresses in a multi-body system
				undergoing large
translation or temperature. Loads include
forces, pressures				displacement motion, with frictional effects
				and fluid flow
and heat flux. It is preferable to apply
boundary				taken into account when necessary. It is
				difficult to
conditions to the CAD geometry, with the FEA
package				estimate the increase in computing power
				necessary to
transferring them to the underlying model,
to allow for				accomplish this feat, but 2 or 3 orders of
				magnitude is
simpler application of adaptive and
optimization algorithms.				probably close. Algorithms to integrate
				these fields of
It is worth noting that the largest error in
the entire				analysis may be expected to follow the
				computing power
process is often in the boundary conditions.
Running				increases.In summary, the finite element
				method is a relatively recent
multiple cases as a sensitivity analysis may
be required.Next month's article will discuss				discipline that has quickly become a mature
the solution phase of the				method,

finite element method.Â© 1996-2005 Roensch				especially for structural and thermal
& Associates. All rights reserved.				analysis. The costs

FINITE ELEMENT ANALYSIS: SolutionThird in a				of applying this technology to everyday
four-part seriesWhile the pre-processing and				design tasks have
post-processing phases of the
				been dropping, while the capabilities
finite element method are interactive and				delivered by the
time-consuming for
				method expand constantly. With education in
the analyst, the solution is often a batch				the technique
process, and is
				and in the commercial software packages
demanding of computer resource. The				becoming more and
governing equations are
				more available, the question has moved from
assembled into matrix form and are solved				"Why apply FEA?"
numerically. The
				to "Why not?". The method is fully capable
assembly process depends not only on the				of delivering
type of analysis
				higher quality products in a shorter design
(e.g. static or dynamic), but also on the				cycle with a
model's element
				reduced chance of field failure, provided it
types and properties, material properties				is applied by a
and boundary
				capable analyst. It is also a valid
conditions.In the case of a linear static				indication of thorough
structural analysis, the
				design practices, should an unexpected
assembled equation is of the form Kd = r,				litigation crop up.
where K is the
				The time is now for industry to make greater
system stiffness matrix, d is the nodal				use of this and
degree of freedom
				other analysis techniques.Â© 1996-2005
(dof) displacement vector, and r is the				Roensch & Associates. All rights reserved.by
applied nodal load				Steve Roensch, President, Roensch &
				AssociatesSteve Roensch is a mechanical
vector. To appreciate this equation, one				engineering consultant with more than 20
must begin with				years of professional experience. He has
				analyzed hundreds of product designs and has
the underlying elasticity theory. The				served as an expert witness across many
strain-displacement				industries, including giving depositions and
				court testimony.

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Introduction to biomechanics

The Finite Element Method: A Four-Article Series