Introduction to FEA and FEM

7/26/2019 Introduction to FEA and FEM

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Introduction

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 extensie

utili!ation in the structural, thermal and fluid analysis areas. The finite element method is comprised of

three major phases" (#) pre-processing, in which the analyst deelops a finite element mesh to diide the

subject geometry into subdomains for mathematical analysis, and applies material properties and

boundary conditions, ($)solution, during which the program deries the goerning matrix e%uations from

the model and soles for the primary %uantities, and (&) post-processing, in which the analyst chec's the

alidity of the solution, examines the alues of primary %uantities (such as displacements and stresses),

and deries and examines additional %uantities (such as speciali!ed stresses and error indicators).

The adantages of FEA are numerous and important. A new design concept may be modeled to determine

its real world behaior under arious load enironments, and may therefore be refined prior to the

creation of drawings, when few dollars hae been committed and changes are inexpensie. nce a

detailed A* model has been deeloped, FEA can analy!e the design in detail, saing time and money by

reducing the number of prototypes re%uired. An existing product which is experiencing a field problem, oris simply being improed, can be analy!ed to speed an engineering change and reduce its cost. +n

addition, FEA can be performed on increasingly affordable computer wor'stations and personal

computers, and professional assistance is aailable.

+t is also important to recogni!e the limitations of FEA. ommercial software pac'ages and the re%uired

hardware, which hae seen substantial price reductions, still re%uire a significant inestment. The method

can reduce product testing, but cannot totally replace it. robably most important, an inexperienced user

can delier incorrect answers, upon which expensie 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.

-hich FEA pac'age to use is a subject that cannot possibly be coered in this short discussion, and the

choice inoles personal preferences as well as pac'age functionality. -here to run the pac'age depends

on the type of analyses being performed. A typical finite element solution re%uires a fast, modern dis'

subsystem for acceptable performance. emory re%uirements are of course dependent on the code, but in

the interest of performance, the more the better, with a representatie range measured in gigabytes per

user. rocessing power is the final lin' in the performance chain, with cloc' 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.

ne aspect often oerloo'ed when entering the finite element area is education. -ithout ade%uate trainingon the finite element method and the specific FEA pac'age, a new user will not be productie in a

reasonable amount of time, and may in fact fail miserably. Expect to dedicate one to two wee's up front,

and another one to two wee's oer the first year, to either classroom or self/help education. +t is also

important that the user hae a basic understanding of the computer0s operating system.

1ext month0s article will go into detail on the pre/processing phase of the finite element method.


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Pre-Processing

As discussed last month, finite element analysis is comprised of pre/processing, solution and post/

processing phases. The goals of pre/processing are to deelop an appropriate finite element mesh, assign

suitable material properties, and apply boundary conditions in the form of restraints and loads.

The finite element mesh subdiides the geometry into elements, upon which are foundnodes. 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 ($*) analysis, or a three/dimensional (&*) thin shell analysis,

the elements are essentially $*, but may be 2warped2 slightly to conform to a &* surface. An example is

the thin shell linear %uadrilateral3 thin shell implies essentially classical shell theory, lineardefines the

interpolation of mathematical %uantities across the element, and quadrilateraldescribes the geometry. For

a &* solid analysis, the elements hae physical thic'ness in all three dimensions. ommon examples

include solid linear bric' and solid parabolic tetrahedral elements. +n 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 model0s degrees of freedom (dof) are assigned at the nodes. 4olid elements generally hae three

translational dof per node. 5otations are accomplished through translations of groups of nodes relatie to

other nodes. Thin shell elements, on the other hand, hae six dof per node" three translations and three

rotations. The addition of rotational dof allows for ealuation of %uantities through the shell, such as

bending stresses due to rotation of one node relatie to another. Thus, for structures in which classical thin

shell theory is a alid approximation, carrying extra dof at each node bypasses the necessity of modeling

the physical thic'ness. 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.

*eeloping the mesh is usually the most time/consuming tas' in FEA. +n the past, node locations were

'eyed in manually to approximate the geometry. The more modern approach is to deelop the mesh

directly on the A* geometry, which will be (#) wireframe, with points and cures representing edges,

($) surfaced, with surfaces defining boundaries, or (&)solid, defining where the material is. 4olid

geometry is preferred, but often a surfacing pac'age can create a complex blend that a solids pac'age will

not handle. As far as geometric detail, an underlying rule of FEA is to 2model what is there2, and yet

simplifying assumptions simply must be applied to aoid 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 hae the correct number of sides.

apped meshes can use the accurate and cheap solid linear bric' &* element, but can be ery time/consuming, if not impossible, to apply to complex geometries. Free/meshing automatically subdiides

meshing regions into elements, with the adantages of fast meshing, easy mesh/si!e transitioning (for a

denser mesh in regions of large gradient), and adaptie capabilities. *isadantages include generation of

huge models, generation of distorted elements, and, in &*, the use of the rather expensie solid parabolic

tetrahedral element. +t is always important to chec' elemental distortion prior to solution. A badly

distorted element will cause a matrix singularity, 'illing the solution. A less distorted element may sole,

but can delier ery poor answers. Acceptable leels of distortion are dependent upon the soler being


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used.

aterial properties re%uired ary with the type of solution. A linear statics analysis, for example, will

re%uire an elastic modulus, oisson0s ratio and perhaps a density for each material. Thermal properties are

re%uired for a thermal analysis. Examples of restraints are declaring a nodal translation or temperature.

6oads include forces, pressures and heat flux. +t is preferable to apply boundary conditions to the A*geometry, with the FEA pac'age transferring them to the underlying model, to allow for simpler

application of adaptie and optimi!ation algorithms. +t is worth noting that the largest error in the entire

process is often in the boundary conditions. 5unning multiple cases as a sensitiity analysis may be

re%uired.

Solution

-hile the pre/processing and post/processing phases of the finite element method are interactie and

time/consuming for the analyst, the solution is often a batch process, and is demanding of computer

resource. The goerning e%uations are assembled into matrix form and are soled numerically. The

assembly process depends not only on the type of analysis (e.g. static or dynamic), but also on the model0selement types and properties, material properties and boundary conditions.

+n the case of a linear static structural analysis, the assembled e%uation is of the form Kd = r, where Kis

the system stiffness matrix, d is the nodal degree of freedom (dof) displacement ector, and ris the

applied nodal load ector. To appreciate this e%uation, 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. 7nder the assumption of compatibility, the differential e%uations of e%uilibrium

in concert with the boundary conditions then determine a uni%ue displacement field solution, which in

turn determines the strain and stress fields. The chances of directly soling these e%uations are slim to

none for anything but the most triial geometries, hence the need for approximate numerical techni%ues

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 gien the alues of its nodal

dof. +ntroducing this relation into the strain/displacement relation, we may express strain in terms of the

nodal displacement, element interpolation scheme and differential operator matrix. 5ecalling 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 wor' 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 deriatie of potential energywith respect to the nodal dof ector to !ero, resulting in" a summation of element stiffness integrals,

multiplied by the nodal displacement ector, e%uals 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 ector, resulting in Kd = r. +n practice, integration

rules are applied to elements, loads appear in the rector, and nodal dof boundary conditions may appear

in the dector or may be partitioned out of the e%uation.


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4olution methods for finite element matrix e%uations are plentiful. +n the case of the linear static Kd = r,

inerting Kis computationally expensie and numerically unstable. A better techni%ue is holes'y

factori!ation, a form of 8auss elimination, and a minor ariation on the 26*72 factori!ation theme.

The Kmatrix may be efficiently factored into LDU, where Lis lower triangular, Dis diagonal, and Uis

upper triangular, resulting in LDUd = r. 4ince LandDare easily inerted, and Uis upper

triangular, dmay be determined by bac'/substitution. Another popular approach is the waefront method,which assembles and reduces the e%uations at the same time. 4ome of the best modern solution methods

employ sparse matrix techni%ues. 9ecause node/to/node stiffnesses are non/!ero only for nearby node

pairs, the stiffness matrix has a large number of !ero entries. This can be exploited to reduce solution time

and storage by a factor of #: or more. +mproed solution methods are continually being deeloped. The

'ey point is that the analyst must understand the solution techni%ue being applied.

*ynamic analysis for too many analysts means normal modes. ;nowledge of the natural fre%uencies and

mode shapes of a design may be enough in the case of a single/fre%uency ibration of an existing product

or prototype, with FEA being used to inestigate the effects of mass, stiffness and damping modifications.

-hen inestigating a future product, or an existing design with multiple modes excited, forced response

modeling should be used to apply the expected transient or fre%uency enironment to estimate the

displacement and een dynamic stress at each time step.

This discussion has assumed h/code elements, for which the order of the interpolation polynomials is

fixed. Another techni%ue, p/code, increases the order iteratiely until conergence, with error estimates

aailable after one analysis. Finally, the boundary element method places elements only along the

geometrical boundary. These techni%ues hae limitations, but expect to see more of them in the near

future.

Post processor

After a finite element model has been prepared and chec'ed, boundary conditions hae been applied, and

the model has been soled, it is time to inestigate the results of the analysis. This actiity is 'nown as the

post/processing phase of the finite element method.

ost/processing begins with a thorough chec' for problems that may hae occurred during solution. ost

solers proide a log file, which should be searched for warnings or errors, and which will also proide a

%uantitatie measure of how well/behaed the numerical procedures were during solution. 1ext, reaction

loads at restrained nodes should be summed and examined as a 2sanity chec'2. 5eaction loads that do not

closely balance the applied load resultant for a linear static analysis should cast doubt on the alidity of

other results. Error norms such as strain energy density and stress deiation among adjacent elements

might be loo'ed at next, but for h/code analyses these %uantities are best used to target subse%uentadaptie remeshing.

nce the solution is erified to be free of numerical problems, the %uantities of interest may be examined.

any display options are aailable, the choice of which depends on the mathematical form of the %uantity

as well as its physical meaning. For example, the displacement of a solid linear bric' element0s node is a

&/component spatial ector, and the model0s oerall displacement is often displayed by superposing the

deformed shape oer the undeformed shape. *ynamic iewing and animation capabilities aid greatly in


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obtaining an understanding of the deformation pattern. 4tresses, being tensor %uantities, currently lac' a

good single isuali!ation techni%ue, and thus deried stress %uantities are extracted and displayed.

rincipal stress ectors 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

Documents

Introduction to FEA and FEM