Fundamentals of Fea

10/10/2014 FUNDAMENTALS OF FEA - Mechanical Engg. Blog | CAD-CAM-CAE Blog | Civil Engg. Blog-Pramit Kumar Senapati

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Mechanical Engg. Blog | CAD-CAM-CAE Blog | Civil Engg. Blog-Pramit Kumar Senapati » F.E.A. » FUNDAMENTALS OF FEA

FUNDAMENTALS OF FEAANALYSIS GUIDE BY Pramit Kumar Senapati:–

(CAE Project Consultant) at www.bemechanicalproject.com

Chief SME (Subject Matter Expert) at www.smartlearningindia.com

What Is FEA??

Finite element analysis was first developed for use in the aerospace and nuclear industries where the safety

of structures is critical. Today, the growth in usage of the method is directly attributable to the rapid

advances in computer technology in recent years. As a result, commercial finite element packages exist that

are capable of solving the most sophisticated problems, not just in structural analysis, but for a wide range of

phenomena such as steady state and dynamic temperature distributions, fluid flow and manufacturing

processes such as injection molding and metal forming.

FEA consists of a computer model of a material or design that is loaded and analyzed for specific results. It is

used in new product design, and existing product refinement. A company is able to verify that a proposed

design will be able to perform to the client’s specifications prior to manufacturing or construction. Modifying

an existing product or structure is utilised to qualify the product or structure for a new service condition. In

case of structural failure, FEA may be used to help determine the design modifications to meet the new

condition.

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Mathematically, the structure to be analyzed is subdivided into a mesh of finite sized elements of simple

shape. Within each element, the variation of displacement is assumed to be determined by simple

polynomial shape functions and nodal displacements. Equations for the strains and stresses are developed in

terms of the unknown nodal displacements. From this, the equations of equilibrium are assembled in a

matrix form which can be easily be programmed and solved on a computer. After applying the appropriate

boundary conditions, the nodal displacements are found by solving the matrix stiffness equation. Once the

nodal displacements are known, element stresses and strains can be calculated

Within each of these modeling schemes, the programmer can insert numerous algorithms (functions) which

may make the system behave linearly or non-linearly. Linear systems are far less complex and generally

ignore many subtleties of model loading & behaviour. Non-linear systems can account for more realistic

behaviour such as plastic deformation, changing loads etc. and is capable of testing a component all the way

to failure.

Despite the proliferation and power of commercial software packages available, it is essential to have an

understanding of the technique & physical processes involved in the analysis. Only then can an appropriate

& accurate analysis model be selected, correctly defined and subsequently interpreted.

History of FEM & FEA

Finite Element Analysis (FEA) was first developed in 1943 by R. Courant, who utilized the Ritz method of

numerical analysis and minimization of variational calculus to obtain approximate solutions to vibration

systems. Shortly thereafter, a paper published in 1956 by Turner, Clough, Martin, & Topp established a

broader definition of numerical analysis. This paper centered on the “stiffness and deflection of complex

structures”.

By the early 70’s, FEA was limited to expensive mainframe computers generally owned by the aeronautics,

automotive, defense, and nuclear industries, and the scope of analyses were considerably limited. Finite

Element technology was further enhanced during the 70’s by such people as Zeinkiewicz & Cheung, when

they applied the technology to general problems described by Laplace & Poisson’s equations.

Mathematicians were developing better solution algorithms, the Galerkin, Ritz & Rayleigh-Ritz methods

emerged as the optimum solutions for certain categories of general type problems. Later, considerable

research was carried out into the modelling & solution of non-linear problems, Hinton & Crisfield being





major contributors.

While considerable strides were made in the development of the finite element method, other areas did not

remain static. Very powerful mesh generation algorithms have been developed. Commercial generators have

the capability of meshing all but the most difficult geometry. Superior CAE concepts have also emerged, it is

not unusual to have a single CAD model for producing engineering drawings, carrying out kinematic &

assembly analysis, as well as being used for finite element modelling.

Due to the rapid decline in the cost of computers and the phenomenal increase in computing power, present

day desktop computers are capable of producing accurate results for all kinds of parameters (standard PC’s

are over 10 times more powerful than the best supercomputers of the early 90’s).

The finite element method now has it’s roots in many disciplines, the end result is a technology that is so

advanced that it is almost indinguishable from magic. The vast catalog of capability that comprises FEA, will

no doubt grow considerably larger in the future. CAE is here to stay, but in order to harness it’s true power,

the user must be familiar with many concepts, including the mechanics of the problem being modelled. All

analyses require time, experience & most importantly, careful planning.

Application Areas

In essence, the finite element is a mathematical method for solving ordinary & partial differential equations.

Because it is a numerical method, it has the ability to solve complex problems that can be represented in

differential equation form. As these types of equations occur naturally in virtually all fields of the physical

sciences, the applications of the finite element method are limitless as regards the solution of practical

design problems.





Due to the high cost of computing power of years gone by, FEA has a history of being used to solve complex

& cost critical problems. Classical methods alone usually cannot provide adequate information to determine

the safe working limits of a major civil engineering construction. If a tall building, a large suspension bridge

or a neuclear reactor failed catastrophically, the economic & social costs would be unacceptably high.

In recent years, FEA has been used almost universally to solve structural engineering problems. One

discipline that has relied heavily on the technology is the aerospace industry. Due to the extreme demands

for faster, stronger, lighter & more efficient aircrafts, manufacturers have to rely on the technique to stay

competitive. But more importantly, due to safety, high manufacturing costs of components & the high media

coverage that the industry is exposed to, aircraft companies need to ensure that none of their components

fail, that is to cease providing the service that the design intended.

FEA has been used routinely in high volume production & manufacturing industries for many years, as to get

a product design wrong would be detrimental. For example, if a large manufacturer had to recall one model

alone due to a piston design fault, they would end up having to replace up to 10 million pistons. Similarly, if

an oil platform had to shut down due to one of the major components failing (platform frame, turrets, etc..),

the cost of lost revenue is far greater than the cost of fixing or replacing the components, not to mention the

huge enviornmental & safety costs that such an incident could incurr.

The finite element method is a very important tool for those involved in engineering design, it is now used

routinely to solve problems in the following areas:

Structural strength design

Structural interation with fluid flows

Analysis of Shock (underwater & in materials)

Acoustics

Thermal analysis

Vibrations

Crash simulations

Fluid flows

Electrical analyses

Mass diffusion

Buckling problems

Dynamic analyses




Electromagnetic evaluations

Metal forming

Coupled analyses

Nowadays, even the most simple of products rely on the finite element method for design evaluation. This is

because contemporary design problems usually cannot be solved as accurately & cheaply using any other

method that is currently available. Physical testing was the norm in years gone by, but now it is simply too

expensive.

A Typical Analysis

In the real world, no analysis is typical, as there are usually facets that cause it to differ from others. There is

however a main procedure that most FE investigations take. This procedure is detailed below:

Planning the Analysis

This is arguably the most important part of any analysis, as it helps ensure the success of the simulation.

Oddly enough, it is usually the one analysts leave out. The purpose of an FE analysis is to model the

behaviour of a structure under a system of loads. In order to do so, all influencing factors must be considered

& determined wether their effects are considerable or negligable on the final result. The degree of accuracy

to which any system can be modelled is very much dependant on the level of planning that has been carried

out. Answers to many questions need to be found. ‘Planning an analysis’ is dealt with in detail in the

‘improving results’ section of this site.

Pre-Processor

The preprocessor stage in general FE packages involves the following:

Specifying the title, that is the name of the problem. This is optional but very useful, especially if a

number of design iterations are to be completed on the same base model.



Setting the type of analysis to be used, e.g. structural, fluid, thermal or electromagnetic, etc.

(sometimes this can only be done by selecting a particular element type).

Creating the model. The model is drawn in 1D, 2D or 3D space in the appropriate units (M, mm, in,

etc..). The model may be created in the pre-processor, or it can be imported from another CAD

drafting package via a neutral file format (IGES, STEP, ACIS, Parasolid, DXF, etc.). If a model is drawn in

mm for example and the material properties are defined in SI units, then the results will be out of

scale by factors of 10 . The same units should be applied in all directions, otherwise results will be

difficult to interpret, or in extreme cases the results will not show up mistakes made during the

loading and restraining of the model.

Defining the element type, this may be 1D, 2D or 3D, and specific to the analysis type being carried

out (you need thermal elements to do thermal analyses).

Applying a Mesh. Mesh generation is the process of dividing the analysis continuum into a number of

discrete parts or finite elements. The finer the mesh, the better the result, but the longer the analysis

time. Therefore, a compromise between accuracy & solution speed is usually made. The mesh may be

created manually, such as the one on the right, or generated automatically like the one below. In the

manually created mesh, you will notice that the elements are smaller at the joint. This is known as

mesh refinement, and it enables the stresses to be captured at the geometric discontinuity (the

junction).

Manual meshing is a long & tedious process for models with any degree of geometric complication, but with

useful tools emerging in pre-processors, the task is becoming easier. Automatic mesh generators are very

useful & popular. The mesh is created automatically by a mesh engine, the only requirement is to define the

mesh density along the model’s edges. Automatic meshing has limitations as regards mesh quality &

solution accuracy. Automatic brick element(hex) meshers are limited in function, but are steadily improving.

Any mesh is usually applied to the model by simply selecting the mesh command on the preprocessor list of

the gui.

Assigning properties. Material properties (Young’s modulus, Poissons ratio, the density, & if

applicable, coefficients of expansion, friction, thermal conductivity, damping effect, specific heat etc.)

will have to be defined. In addition element properties may need to be set. If 2D elements are being

6



used, the thickness property is required. 1D beam elements require area, I , I , I , J, & a direction

cosine property which defines the direction of the beam axis in 3D space. Shell elements, which are

2½D in nature (2D elements in 3D space), require orientation & neutral surface offset parameters to

be defined. Special elements (mass, contact, spring, gap, coupling, damper etc.) require properties

(specific to the element type) to be defined for their use.

Apply Loads. Some type of load is usually applied to the analysis model. The loading may be in the

form of a point load, a pressure or a displacement in a stress (displacement) analysis, a temperature or

a heat flux in a thermal analysis & a fluid pressure or velocity in a fluid analysis. The loads may be

applied to a point, an edge, a surface or a even a complete body. The loads should be in the same

units as the model geometry & material properties specified. In the cases of modal (vibration) &

buckling analyses, a load does not have to be specified for the analysis to run.

Applying Boundary Conditions. If you apply a load to the model, then in order to stop it accelerating

infinitely through the computer’s virtual ether (mathematically known as a zero pivot), at least one

constraint or boundary condition must be applied. Structural boundary conditions are usually in the

form of zero displacements, thermal BCs are usually specified temperatures, fluid BCs are usually

specified pressures. A boundary condition may be specified to act in all directions (x,y,z), or in certain

directions only. They can be placed on nodes, keypoints, areas or on lines. BC’s on lines can be in the

form of symmetric or anti-symmetric type boundary conditions, one allowing in plane rotations and

out of plane translations, the other allowing in plane translations and out of plane rotations for a

given line. The application of correct boundary conditions are a critical to the accurate solution of the

design problem. At least one BC has to be applied to every model, even modal & buckling analyses

with no loads applied. See the ‘Advanced BCs’ section for explanations on more advanced boundary

condition types.Solution

Thankfully, this part is fully automatic. The FE solver can be logically divided into three main parts, the

pre-solver, the mathematical-engine & the post-solver. The pre-solver reads in the model created by

the pre-processor and formulates the mathematical representation of the model. All parameters

defined in the pre-processing stage are used to do this, so if you left something out, chances are the

pre-solver will complain & cancel the call to the mathematical-engine. If the model is correct the

solver proceeds to form the element-stiffness matrix for the problem & calls the mathematical-engine

which calculates the result (displacement, temperatures, pressures, etc.). The results are returned to

the solver & the post-solver is used to calculate strains, stresses, heat fluxes, velocities, etc.) for each

xx yy xy



node within the component or continuum. All these results are sent to a results file which may be read

by the post-processor.Post-Processor

Here the results of the analysis are read & interpreted. They can be presented in the form of a table, a

contour plot, deformed shape of the component or the mode shapes and natural frequencies if

frequency analysis is involved. Other results are available for fluids, thermal and electrical analysis

types. Most post-processors provide an animation service, which produces an animation & brings

your model to life.Contour plots are usually the most effective way of viewing results for structural

type problems. Slices can be made through 3D models to facilite the viewing of internal stress

patterns.All post-processors now include the calculation of stress & strains in any of the x, y or z

directions, or indeed in a direction at an angle to the coordinate axes. The principal stresses and

strains may also be plotted, or if required the yield stresses and strains according to the main theories

of failure (von mises, St. Venant, Tresca etc.). Other information such as the strain energy, plastic strain

and creep strain may be obtained for certain types of analyses.A Final Word

The finite element method extremely powerful. However, with comforting contour plots, one can be

easily fooled into thinking that a superior result has been achieved. The quality of the result is totally

dependent on the quality of the analysis model & how accurately it represents the physical problem

being investigated. Remember, careful planning is the key to a successful analysis. Sometimes an

analysis is not required, as some problems have analytical or imperical solutions, others may be

determined using spreadsheets.



Hiring FEA Consulting Services

Paying a company to carry out FEA services for you is simple, most companies out there are keen to have

your money. However, the costly service provided may not furnish you with any extra benefit as regards

results or design information. In the worst event it may even disimprove your design and leave you with a

poor view of FEA technology.

This section is intended to help engineers, who are unfamiliar with FEA, hire an FE consulting firm to appraise

or enhance their products. The following information should be kept in mind when considering approaching

FE consultants:

The Costs of FE Consulting Services

Currently (August 2001), FEA services costs vary from £25 & £60 Sterling per hour. Lower rates may be

available for large quantities of work, so it is worthwhile asking.

Is FEA Really Required ??

Despite FEA being an indispensible tool when improving the design of a product, it is still not the only way of

obtaining high quality solutions. Many design problems may be solved using alternative methods such as

analytical, imperical & other numerical techniques. Using spreadsheets to formulate your solutions can be

very effective in certain circumstances. Be clear on the alternatives to FEA, and the advantages &

disadvantages of each solution.

Defining the Objectives

Before any consultants are contacted, it worthwhile thinking about what the objectives of the work would be:

How will this analysis help your product to evolve ?

Do you require highly detailed data or general results ?

Is the simulation likely to be non-linear or time dependant ?

What format would you like the results in (report, data CD, raw result text files, presentation, etc.) ?

Are the simulation results you require realistically achievable within the time frame & budget available

?

Finding a Consulting Firm

This can be either a difficult or a very easy task. For smaller companies, my advice would be to contact

NAFEMS. They are an organisation who govern finite element methods & standards, and have lists of

registered member suppliers & also have analysts that are approved via their accredited analyst schemes.

However, if you work for a large organisation, the company may have an approved list of suppliers that you

are obliged to use, problem solved.

Wherever you source the firm, ensure that they are technically capable & have sufficient experience in the

discipline being investigated. Some may have quality certifications, it is worthwhile to check.

FEA & Errors


http://www.nafems.org/



Despite FEA being based around approximate solutions, the solver will usually return quite accurate results to

the questions posed. However, if the model is inferior, the wrong questions will be asked & the software will

return a result that is not representative of the physical phenomena under investigation. Correct results can

only be assured when the analyst has a sound knowledge of both FEA (& its limitations) and the physical

processes that are being simulated. This type of knowledge can only be gained by having both exposure to

the method at a high level, and an initimate appreciation of the processes being simulated.

Quotes & Orders

When requesting a quote, ensure that you have furnished sufficient detail to allow the consultants to provide

an accurate quotation. Indications of time scales, solution requirements & deliverables are always useful. It

may be useful to specify a penalty clause for late deliverables and how you intend to pay (half during the

work period, half after deliverables & project agreements have been satisfied). Always include a statement

saying that you intend to monitor the work as it is being completed, as sometimes oversights may only

become apparant after initial stages of the work has been completed. This will enable you to modify the job

specification before it is too late.

On receiving a quotation, ensure that there is sufficient detail there that keeps the analysis within certain

rigid bounds. Ensure that the analysts have sufficient understanding of the commissioning objectives.

Anything that is not in the quotation & should be may cause problems later, so ensure that there are clearly

defined timeframes & deliverables. The quote should also indicate any portions of the investigation that may

be too ambitious or simply not feasible. This information will be indispensible to you when deciding the best

design route to take. If there is no mention of such details, it may be worth contacting them to make sure.

Most importantly, ensure that there are details relating to model quality, such as their implementation of

planning checklists etc..

Before confirming an order, be sure that you are happy with all aspects of the proposal furnished by the

supplier. If there were additional agreements after the quotation had been sent, ensure that you receive

confirmation of this in writing or via email before placing the order. Ensure anything to do with time & cost

are clearly defined, it will save much time, money & embarrassment to both parties at a later stage.

Post Analysis

On viewing the results furnished to you, decide if the project objectives have been achieved for you. If the

analysis ran later than anticipated or the results are lack lustre, hold out on your final payment. At this stage

contact the company & try to determine why there is a deviation between what was agreed & what was

delivered. If you still receive no satisfaction, the least you can do is implement your penalty clause & the

most you can do is seek legal advice and renege on your final payment.

Charlatan Services

Believe it or not, there are companies out there with beautiful websites that claim to have extensive

experience of carrying out analyses in a particular discipline. They may even go on to say that they have

carried out analyses for several high profile companies relating to the discipline. However, the truth may be a

little different, as they may be trying to expand their portfolio of work, and are being over enthusiastic of

their capability & the level knowledge required when carrying out work in the new discipline. If there are no



companies named in their claims, do not be afraid to ask the analysts to provide detailed information on

their experience, the nature of the work carried out, the problems encountered & the companies they

successfully carried out work for. If they are genuine, there will be no problem with such a request & they

would probably be delighted to share their past experiences & achievements with you.

Buying Fea Software

The selection of a Finite Element package for carrying out evaluation and certification of engineering designs

is not as difficult as it may initially seem. The basic considerations are how accurate your analyses need to be,

how much you are willing to spend and what type of users/analysts you intend to use the package.

Most advanced solid modelling packages have some sort of finite element capability, examples are

Unigraphics, I-deas, Solid Works, Pro-Engineer and Catia. The finite element capabilities however are limited

to evaluation use only, i.e. after coming up with a suitable design, the FE capability can be used to determine

whether parameters in the design (materials, dimensions, etc..) need to be increased or reduced. Some don’t

even allow you to view the mesh generated, which is of crucial importance in a precise analysis. They also

usually do not have any capabilities for manual mesh generation, which is required again for quality results

and also for large analyses, where the mesh density should be as low as possible for the model, but in a

manner that does not reduce the accuracy of the results obtained.

Some of the considerations are listed below:

1. Type of user

If the user is an experienced Finite Element Analyst, then a more robust and flexible package is required,

which means more expense, but the quality of analysis results is superior. On the other hand if the user is a

design engineer who requires approximate values for stress, displacement or strain (within 30% of the

actual), then a solid modeller with FE capabilities is more suitable.

2. Type of Analysis

Safety critical components or devices require analysis using full FE capabilities. This sometimes involves

predicting thermal-stress relationships (transient and steady state), Elastic stress distributions, buckling

modes, analysis to plastic collapse, natural frequencies, evaluation of fatigue life. This is so that the design

can be assessed accurately and efficiently. These types of analyses are usually carried out by experienced

analysts and are relatively expensive.

3. Hardware

Most PC versions of FE packages for professional use (as opposed to academic) require Windows NT4 or

Windows 2000, with the latest service packs and also with an Intel premium type chip, Pentium 2, 3 or 4 (not

a Celeron). On Unix based platforms, the Sun operating system is the most common, but most vendors

provide cover on almost all platform types. The platform used is usually a function of the cost the user is

willing to pay for the hardware it will run on.

4. Pre & Post Processor

These are used for generating the finite element model and reviewing the results. Most packages come with

some sort of Pre and Post processor facilities. One of the most important aspects of a pre-processor is the

ability to deal with different file types (dwg, prt, sat, xmt, stl, iges, dxf, par). Also, interfacing to spreadsheet



packages and word processors can be important. The most common pre and post processors are

Hypermesh, Patran and Ansys. Ansys pre and post are built in to the complete FE package, and Patran is

usually used with Nastran. Most advanced analyses require editing the model by hand in a texteditor. In

which case, expert knowledge is usually required. Another important feature is the mesh generator. There are

standard mesh generators, which mesh manually sub-divided models. Automatic mesh generators attempt

to mesh the region at hand using the best approximation. The main types of automatic mesh generators are:

Delaunay; This creates triangular and tetrahedral based meshes. The most famous and robust is

created by Paul Louis George of INRIA, France. This algorithm is suitable for both generating

meshes, where density varies from point to point, and adaptive refinement of initial meshes.

Octree; With this technique the meshing proceeds by subdividing the shape containing the model

into small octants, within which tetrahedral meshes are generated. Very complex parts can be

meshed with this technique.

Advancing front; The advancing front technique works by building a 3D mesh from a boundary

(surface) mesh while progressing inwards towards the centre of the model. The technique has been

applied to generating a tetrahedral mesh of a volume from a surface mesh of triangles, and also to

brick mesh generation, where it is known as plastering, Plastering is an attempt to extend the

paving concept to 3 dimensions. Starting with a closed outside boundary of quadrilateral (square)

elements, each element is projected into the volume to form a hexahedron (brick element). As with

paving, the elements are projected in layer by layer to create an advancing front mesh, which has

the desirable characteristic of placing high quality elements near the volume boundaries. As the

element layers build up, complex interior voids may be produced, which in some cases are

impossible to fill with all hex elements.

5. What is an Expert ??

This question often pops up in conversation with engineers who are considering purchasing an FE package.

An expert on FE for engineering analyses would be somebody who has the ability to carry out any given

analysis and stand by their results with confidence (be able to argue with a stress engineer on the nature of a

problem, the path to solution and the results obtained). This requires knowing your engineering theories,

knowing the facilities available in FE packages (or more importantly, their limitations) and knowing the effects

of carrying out idealisations on the model (detail suppression, dimensional reduction, use of specialist

elements, mixed dimensional coupling and planes of symmetry [planar, cyclic, axial & repetitive], loading and

boundary conditions).

6. Advanced modelling options

Most commercial FE packages offer advanced modelling options such as constraint equations, elastic-plastic

capabilities, Frequency (modal) analysis, Buckling, sub-modelling, sub-structuring, contact analysis, user sub-



routines, coupled analyses (static-thermal) etc.. Basic packages that are available with solid modellers usually

do not have any of these options. Some packages have the ability to carry out adaptive analyses. There are

two main types of adaptive procedures, mesh refinement and optimisation. In mesh refinement the mesh is

modified in a manner that lends itself to more accurate results. There are three types of mesh refinement. H-

Type; where the number of elements are increased. This is commonly achieved by sub-dividing existing

elements. R-Type; where the nodes are re-located to more suitable locations so that the size of the elements

are reduced at the more critical areas of a model. P-Type; where the order of the elements is increased. This

usually involves increasing the elements using one-higher isoparametric type formulations. In design

optimisation, the package will change the geometry in a manner that is determined by previous analysis

results. This involves many iterations, but the result is a design which has evolved to be the optimum as

regards strength, stiffness and weight.

7. Cost ??

The cost of FEA is small compared with material tests, but it is expensive none the less. In the UK, an

experienced analyst would expect an annual remuneration of £30k plus. The cost of hardware varies from

about £1,500 to whatever you have to spend. The cost of analysis packages are also expensive (anywhere up

to 30K for one seat). They are charged annually, but most companyies will allow it to be paid in monthly

instalments, usually for a 10% extra charge. As with all software packages with a relatively small customer

base, documentation, support and training are frightfully expensive.

8. Performance

As regards FEA as an analysis tool (as opposed to an evaluation tool), performance is of vital importance. The

package should:

Be easy to use

Have the technology to calculate parameters (stress, temperatures, etc..) fast and efficiently

Be accurate with all calculations

Robust (no crashing out without good reason & stating the reason)

9. Most common FEA Packages

If you are designing neuclear reactors, you will need a high performance analysis package. There are many

on the market, the big name ones are Abaqus, Nastran & Ansys. Abaqus has a vast amount of element types,

has more advanced options & has a name for providing fast, high quality non-linear solutions.

What’s the difference between FEM & FEA ??

This is a very contentious issue, one that academics love to debate over a cool long-neck of a friday evening.

I am going to stick my head on the block here & try to explain the difference, happy chopping my academic

friends.

The terms ‘finite element method’ & ‘finite element analysis’ seem to be used interchanably in most

documentation, so the question arises is there a difference between FEM & FEA ??

The answer is yes, there is a difference, albeit a subtle one that is not really important enough to loose sleep



over.

The finite element method is a mathematical method for solving ordinary & elliptic partial differential

equations via a piecewise polynomial interpolation scheme. Put simply, FEM evaluates a differential equation

curve by using a number of polynomial curves to follow the shape of the underlying & more complex

differential equation curve. Each polynomial in the solution can be represented by a number of points and so

FEM evaluates the solution at the points only. A linear polynomial requires 2 points, while a quadratic

requires 3. The points are known as node points or nodes. There are essentially three mathematical ways that

FEM can evaluate the values at the nodes, there is the non-variational method (Ritz), the residual mehod

(Galerkin) & the variational method (Rayleigh-Ritz).

FEA is an implementation of FEM to solve a certain type of problem. For example if we were intending to

solve a 2D stress problem. For the FEM mathematical solution, we would probably use the minimum

potential energy principle, which is a variational solution. As part of this, we need to generate a suitable

element for our analysis. We may choose a plane stress, plane strain or an axisymmetric type formulation,

with linear or higher order polynomials. Using a piecewise polynomial solution to solve the underlying

differential equation is FEM, while applying the specifics of element formulation is FEA, e.g. a plane strain

triangular quadratic element.

Using FEA: A Word of Warning

Introduction

FEA is an extremely potent engineering design utility, but one which should be used with great care. Despite

years of research by some of the earth’s most intelligent mathematicians & scientists, it can only answer the

questions asked of it. So as the saying goes, ask a stupid question…

The Frothy Solution

Current CAD vendors are now selling suites which have cut down versions of FEA engines integrated with

computer aided design software. The notion is to allow ordinary rank and file designers to analyse as they

design and change & update models to reach workable solutions much earlier in the design process. This

kind of approach is commonly referred to as the pushbutton solution.

Pensive analysts are petrified of pushbutton analysis. This is because of the colossal errors that can be made

at the push of a button. The errors are usually uncontrollable and oftentimes undetectable. Some vendors are

even selling FEA plug-ins where it is not possible to view the mesh (this is ludicrous).

The oblivious among us may say that analysts are afraid of pushbutton solutions due to the job loss factor, or

perhaps they are terrified of being cast out of the ivory towers in which they reside. Such arguments are

nonsensical, there will always be real problems & design issues to solve (Would you enter the Superbike

Class Isle of Man TT on a moped with an objective to win, even if it had the wheels of the latest & greatest

superbike ??).

The temptation to analyse components is almost irresistible for the inexperienced, especially in an

enviornment of one-click technology coupled with handsome & comforting contour plots. The bottom line is

that FEA is not a trivial process, no level of automation and pre & post processing can make analyses easy, or

more importantly, correct.



The Analysis Titan

If you have recently been awarded an engineering degree, Congratulations, but remember it does not qualify

you to carry out FE analyses. If it did, then a sailing course should be adequate to become Captain aboard

the Blue Marlin (world’s largest transporter vessel).

This is not to say that regular engineers cannot become top rate analysts without a Ph.D.. Some analysts have

a Masters degree, but most have no more than a batchelors degree. The key to good analyses is a

knowledge of the limitations of the method & an understanding of the physical phenomena under

investigation.

Superior results are usually difficult to achieve without years of high level exposure to fields that comprise

FEA technology (differential equations, numerical analysis, vector calculus, etc..). Expertise in such disciplines

is required to both fully understand the requirements of any particular design circumstance, and to be able

to quantify the accuracy of the analysis (or more importantly, inaccuracy) with reasonable success.

To Conclude

Finite element computer programs have become common tools in the hands of design engineers.

Unfortunately, many engineers who lack the proper training or understanding of the underlying concepts

have been using these tools. Given the oppertunity, FEA will confess to anything, the essence of any session

should be to interrogate the solver with well formed & appropriate questons

Documents

Fundamentals of Fea