THE FINITE ELEMENT METHOD

 

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           THE FINITE ELEMENT METHOD: A FOUR-ARTICLE SERIES

        The following four-article series was published recently

        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 special-

        izing in Finite Element Analysis, and may be reached at:

                     Roensch Engineering Consulting

                     634 Lake Shore Road

                     Grafton, WI 53024-9723

                     414-375-2228

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FINITE ELEMENT ANALYSIS: Introduction

by Steve Roensch, Roensch Engineering Consulting

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First in a four-part series

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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 professional assistance is available.

It is also important to recognize the limitations of FEA.  Commercial

software packages and the required hardware, while coming down in price,

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, on the other hand, is

becoming increasingly clear.  A typical finite element solution creates a

temporary matrix file as large as 1 Gbyte, with 50 to 100 Mbytes common,

thus requiring 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 8 to 32 Mbytes per user

a representative range.  Processing power is the final link in the

performance chain, with clock speed, cache, pipelining and vector

processing all contributing to the bottom line.  All in all, today's user

needs a minimum of 1 or 2 Mflops (millions of double-precision

floating-point operations per second) sustained performance, with 10 or 20

Mflops being all the better.  These analyses can run for hours or even days

on the fastest systems, so computing power is of the essence.  Given these

requirements, performing FEA on a PC may be suitable for teaching the

method, but is likely to be found insufficient for dedicated analysis.

Until very recently, only an expensive host could fulfill the needs of a

full-time analyst.  (Unfortunately, unleashing several solutions without

careful priority control could all but kill the interactive productivity of

time-shared users.)  Today, however, powerful engineering workstations

provide an affordable platform for FEA, and are rapidly becoming the system

of choice.  Expect to pay $50K to $200K for station and software, depending

on hardware performance and software capability.

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.

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Steve Roensch is an engineering consultant specializing in finite element

analysis.  (C) 1991 Roensch Engineering Consulting, 414-375-2228

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FINITE ELEMENT ANALYSIS: Pre-processing

by Steve Roensch, Roensch Engineering Consulting

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Second in a four-part series

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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.  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.

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Steve Roensch is an engineering consultant specializing in finite element

analysis.  (C) 1991 Roensch Engineering Consulting, 414-375-2228

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FINITE ELEMENT ANALYSIS: Solution

by Steve Roensch, Roensch Engineering Consulting

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Third in a four-part series

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While the pre-processing and post-processing phases of the finite element

method are interactive and time-consuming for the analyst, the solution is

usually 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 the "LDU"

factorization theme.  The K matrix may be efficiently factorized 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.  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.

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Steve Roensch is an engineering consultant specializing in finite element

analysis.  (C) 1991 Roensch Engineering Consulting, 414-375-2228

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FINITE ELEMENT ANALYSIS: Post-processing

by Steve Roensch, Roensch Engineering Consulting

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Last in a four-part series

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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.  Principle stress vectors may be displayed as color-coded

arrows, indicating both direction and magnitude.  The magnitude of

principle 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.

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.

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Steve Roensch is an engineering consultant specializing in finite element

analysis.  (C) 1991 Roensch Engineering Consulting, 414-375-2228

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