fea session 01

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M.S Ramaiah School of Advanced Studies - Bangalore

Module Title:Finite Element Analysis


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Module Summary

Give the student an understanding of the basictheoretical techniques used in the solution of

engineering Problems

Use of Commercial Analysis codes and

Associated Pre and post processors

Areas Covered: Linear and Non-linear Statics,

thermal and Dynamic Analysis


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Module Learning Objectives

Apply the fundamentals of finite element formulation to problems inlinear stress analysis, heat transfer and dynamics

Develop stiffness matrices for simple one, two and three dimensional

elements, assemble system stiffness matrices, apply boundary

conditions and develop system equations

Demonstrate the role of Gauss numerical integration in elementformulation

Compare the banded and frontal Gauss elimination techniques in the

solution of system equations

Describe the advantages and features of higher order and

isoparametric elements

Replace distributed loading by equivalent nodal loading


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Module Learning Objectives(Contd.)

Use a pre-processor in a commercial finite element

software code to fully define a model in terms of mesh

design, element type, material properties and constraints

Use a postprocessor in a commercial finite element code tointerpret the results from analysis of a problem in stress

analysis, heat transfer or dynamics

Employ sub modelling and adaptive meshing techniques

within a commercial finite element code


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Module Syllabus

An overview of finite element method and

applications

Basics essential for understanding FEM

Matrix Algebra and Gaussian Elimination

Fundamentals of Elasticity

Behaviour of Materials

Variational and Weighted Residual Methods

Classification of Solid Mechanics Problems

Mesh Generation and Glossary of Terms


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Module Syllabus(contd.)

Finite Element Analysis- 1D Elasticity Problems

Finite Element Analysis 2D Elasticity Problems

Finite Element Analysis Axi-symmetric Problems

Finite Element Analysis Beams and Frames Finite Element Analysis 3D Elasticity Problems

Finite Element Analysis Dynamics Problems

Finite Element Analysis Heat transfer Problems

Introduction to Non Linear Analysis

Solution of Industrial Problems-Case studies


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Teaching and Learning Methods

Lecture Sessions

Practical Sessions using

ANSYS/NASTRAN/NISA/HM

Industrial problem solving


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Evaluation

PG : Assignment(100% assignment)

ACTP: Quiz, Written Exam, Assignment

(Quiz: 10%, Written Exam: 30%, Assignment: 60%)

MTP: Assignment(100%)


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Module Resources

Module Notes Reference Books:

K.J. Bathe, Finite Element Procedures, PHI, New Delhi, 1997

H.V.Lakshminarayana, The F.E.M. for Engg. Students, 1996

Chandraputla, T.R. and Ashok D. Belegundu, Introduction to Finite Elements in

Engineering,Second Edition, PHI, New Delhi, 2001 J.N. Reddy, An Introduction to the Finite Element Method, Second Edition, McGraw-Hill

International Editions, 1993

Vince Adams and Abraham Askenazi,Building Better Products with Finite Element

Analysis,Onward Press, 1998

L.J. Segerland, Applied Finite Element Analysis

M.J. Fagan, Finite Element Analysis, Theory and Practice, Longman Scientific and Technical,

1992 John o. Dow, A unified approach to FEM and Error Analysis Procedures, Academic Press,

1999

S. Rajashekaran, Finite Element Analysis in Engineering Design, Wheeler Publishing,1994

Zienkiewicz and Chung, FEM,1967


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http://www.dermotmonaghan.com/fea/htm/analysis_types/modal.htm

http://www.engineeringzones.com - A website created to educate people in the latestengineering technologies, manufacturing techniques and software tools. Excellent FEM links,including links to all commercial providers of FEM software.

http://www.comco.com/feaworld/feaworld.html - Extensive FEM links, categorized by analysistype (mechanical, fluids, electromagnetic, etc.)

http://femur.wpi.edu - Extensive collection of elementary and advanced material relating to the

FEM.

http://www.engr.usask.ca/%7Emacphed/finite/fe_resources/fe_resources.html - Lists manypublic domain and shareware programs.

http://sog1.me.qub.ac.uk/dermot/ferg/ferg.html#Finite - Home page of the the Finite ElementResearch Group at The Queen's University of Belfast. Excellent set of FEM links.

http://www.tenlinks.com/cae/ - Hundreds of links to useful and interesting CAE cited, including

FEM, CAE, free software, and career information.

http://www.geocities.com/SiliconValley/5978/fea.html - Extensive FEM links.

http://www.nafems.org/ - National Agency for Finite Element Methods and Standards(NAFEMS).

Selected FEM Resources on the Internet
http://www.dermotmonaghan.com/fea/htm/analysis_types/modal.htmhttp://www.dermotmonaghan.com/fea/htm/analysis_types/modal.htm


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Software and Manuals

ANSYS

NASTRAN/PATRAN

NISA

Hypermesh LS-DYNA

PRO/Mechanica

IDEAS/Simulation


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Session Topic

An overview of finite element method and applications

Session objectives is to learn about

Engineering Design Process

Definition of FEM

Sources of Error in the FEM

Advantages and Disadvantages of FEM

Classification of Solid-Mechanics Problems

Six Steps in the Finite Element Method

What's the difference between FEM & FEA

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Many problems in engineering and applied science aregoverned by differential or integral equations.

The solutions to these equations would provide an exact,closed-form solution to the particular problem beingstudied.

However, complexities in the geometry, properties and in

the boundary conditions that are seen in most real-worldproblems usually means that an exact solution cannot beobtained or obtained in a reasonable amount of time.

Finite Element Method Defined


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Current product design cycle times imply that engineersmust obtain design solutions in a short amount of time.

They are content to obtain approximate solutions that canbe readily obtained in a reasonable time frame, and withreasonable effort. The FEM is one such approximatesolution technique.

The FEM is a numerical procedure for obtainingapproximate solutions to many of the problems encounteredin engineering analysis.

Finite Element Method Defined (Contd.)


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In the FEM, a complex region defining a continuum isdiscretized into simple geometric shapes called elements.

The properties and the governing relationships are assumed

over these elements and expressed mathematically in terms ofunknown values at specific points in the elements called nodes.

An assembly process is used to link the individual elements tothe given system. When the effects of loads and boundary

conditions are considered, a set of linear or nonlinear algebraicequations is usually obtained.

Solution of these equations gives the approximate behavior ofthe continuum or system.


AC CA 404


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The continuum has an infinite number of degrees-of-freedom(DOF), while the discretized model has a finite number of

DOF. This is the origin of the name,finite elementmethod.

The number of equations is usually rather large for most real-world applications of the FEM, and requires the computational

power of the digital computer. The FEM has little practical

value if the digital computer were not available.

Advances in and ready availability of computers and software

has brought the FEM within reach of engineers working in

small industries, and even students.


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Two features of the finite element method are worth noting.

Thepiecewise approximation of the physical field (continuum) onfinite elements provides good precision even with simple

approximating functions. Simply increasing the number of elements

can achieve increasing precision.

Thelocality of the approximation leads to sparse equation systemsfor a discretized problem. This helps to ease the solution of problems

having very large numbers of nodal unknowns. It is not uncommon

today to solve systems containing a million primary unknowns.

Finite Element Method Defined (cont.)

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Degree of Freedom

Minimum number of independent coordinates

required to determine completely the positions ofall parts of a system at any instant of time

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Discrete and Continuous Systems

Systems with a finite number of degrees of freedom are

called discrete or lumped parameter systems Systems with an infinite number of degrees of freedom

are called continuous or distributed systems

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Comment

Most of the time, continuous systems are approximated as

discrete systems, and solutions are obtained in a simpler

manner

Practical systems are analysed as discrete systems Treatment of a system continuous gives exact results

Lumped Systems lead to ODE

Continuous Systems lead to PDE

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It is difficult to document the exact origin of the FEM, because thebasic concepts have evolved over a period of 150 or more years.

The termfinite elementwas first coined by Clough in 1960. In the

early 1960s, engineers used the method for approximate solution of

problems in stress analysis, fluid flow, heat transfer, and other

areas.

The first book on the FEM by Zienkiewicz and Chung was

published in 1967.

In the late 1960s and early 1970s, the FEM was applied to a wide

variety of engineering problems.

Origins of the Finite Element Method

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The 1970s marked advances in mathematical treatments, including

the development of new elements, and convergence studies.

Most commercial FEM software packages originated in the 1970s(ABAQUS, ADINA, ANSYS, MARK, PAFEC) and 1980s

(FENRIS, LARSTRAN 80, SESAM 80.)

The FEM is one of the most important developments in

computational methods to occur in the 20th century. In just a fewdecades, the method has evolved from one with applications in

structural engineering to a widely utilized and richly varied

computational approach for many scientific and technological areas.

Origins of the Finite Element Method (cont.)

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The FEM offers many important advantages to the design engineer:

Easily applied to complex, irregular-shaped objects composed

of several different materials and having complex boundary

conditions.

Applicable to steady-state, time dependent and eigenvalue

problems.

Applicable to linear and nonlinear problems.

One method can solve a wide variety of problems, includingproblems in solid mechanics, fluid mechanics, chemical reactions,

electromagnetics, biomechanics, heat transfer and acoustics, to

name a few.

How can the FEM Help the Design Engineer?

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General-purpose FEM software packages are available at

reasonable cost, and can be readily executed on

microcomputers, including workstations and PCs.

The FEM can be coupled to CAD programs to facilitate solid

modeling and mesh generation.

Many FEM software packages feature GUI interfaces, auto-

meshers, and sophisticated postprocessors and graphics to speedthe analysis and make pre and post-processing more user-

friendly.

How can the FEM Help the Design Engineer? (cont.)

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Simulation using the FEM also offers important business

advantages to the design organization:

Reduced testing and redesign costs thereby shortening

the product development time. Identify issues in designs before tooling is committed.

Refine components before dependencies to other

components prohibit changes.

Optimize performance before prototyping. Discover design problems before litigation.

Allow more time for designers to use engineering

judgment, and less time turning the crank.

How can the FEM Help the Design Organization?

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Several approaches can be used to transform the physical

formulation of a problem to its finite element discrete analogue.

If the physical formulation of the problem is described as adifferential equation, then the most popular solution method is

theMethod of Weighted Residuals.

If the physical problem can be formulated as the minimizationof a functional, then the Variational Formulation is usually

used.

Theoretical Basis: Formulating Element Equations

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The three main sources of error in a typical FEMsolution are discretization errors, formulation errors and

numerical errors.

Discretization error results from transforming the

physical system (continuum) into a finite element

model, and can be related to modeling the boundaryshape, the boundary conditions, etc.

Discretization error due to poor geometry

representation.

Discretization error effectively eliminated.

Sources of Error in the FEM

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Formulation error results from the use of elements that don'tprecisely describe the behavior of the physical problem.

Elements which are used to model physical problems for

which they are not suited are sometimes referred to as ill-

conditioned or mathematically unsuitable elements. For example a particular finite element might be formulated

on the assumption that displacements vary in a linear manner

over the domain. Such an element will produce no

formulation error when it is used to model a linearly varying

physical problem (linear varying displacement field in this

example), but would create a significant formulation error if it

used to represent a quadratic or cubic varying displacement

field.

Sources of Error in the FEM (cont.)

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Numerical error occurs as a result of numerical calculation

procedures, and includes truncation errors and round off errors.

Numerical error is therefore a problem mainly concerning the

FEM vendors and developers.

The user can also contribute to the numerical accuracy, for

example, by specifying a physical quantity, say Youngs modulus,

E, to an inadequate number of decimal places.


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Can readily handle complex geometry:

The heart and power of the FEM.

Can handle complex analysis types:

Vibration

Transients

Nonlinear Heat transfer

Fluids

Can handle complex loading:

Node-based loading (point loads).

Element-based loading (pressure, thermal, inertial forces). Time or frequency dependent loading.

Can handle complex restraints:

Indeterminate structures can be analyzed.

Advantages of the Finite Element Method

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Can handle bodies comprised of non-homogeneous materials:

Every element in the model could be assigned a different set ofmaterial properties.

Can handle bodies comprised of non-isotropic materials:

Orthotropic

Anisotropic

Special material effects are handled:

Temperature dependent properties.

Plasticity

Creep

Swelling Special geometric effects can be modeled:

Large displacements.

Large rotations.

Contact (gap) condition.

Advantages of the Finite Element Method (cont.)

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A specific numerical result is obtained for a specific problem. Ageneral closed-form solution, which would permit one to examine

system response to changes in various parameters, is not produced.

The FEM is applied to an approximation of the mathematical model of

a system (the source of so-called inherited errors.)

Experience and judgment are needed in order to construct a good finite

element model.

A powerful computer and reliable FEM software are essential.

Input and output data may be large and tedious to prepare and interpret.

Disadvantages of the Finite Element Method

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Numerical problems: Computers only carry a finite number of significant digits.

Round off and error accumulation.

Can help the situation by not attaching stiff (small) elements toflexible (large) elements.

Susceptible to user-introduced modeling errors: Poor choice of element types.

Distorted elements.

Geometry not adequately modeled.

Certain effects not automatically included:

Buckling

Large deflections and rotations.

Material nonlinearities .

Other nonlinearities.

Disadvantages of the Finite Element Method (cont.)

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Create elementsof the beam

Nodal displacement and forces

A FEM model in solid mechanics canbe thought of as a system of

assembled springs. When a load is

applied, all elements deform until all

forces balance.

F = KQ

K is dependent upon Youngs

modulus and Poissons ratio, as

well as the geometry.

Equations from discrete elements are

assembled together to form the global

stiffness matrix. Deflections are obtained by solving

the assembled set of linear equations.

Stresses and strains are calculated

from the deflections.

FEM Applied to Solid Mechanics Problems

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C C 0


Analysis of solids

Static Dynamics

Behavior of Solids

Linear Nonlinear

Material

Fracture

GeometricLarge Displacement

Instability

Plasticity

ViscoplasticityGeometric

Classification of solids

Skeletal Systems

1D Elements

Plates and Shells

2D Elements

Solid Blocks

3D Elements

TrussesCablesPipes

Plane StressPlane StrainAxisymmetricPlate BendingShells with flat elementsShells with curved elements

Brick ElementsTetrahedral ElementsGeneral Elements

Elementary Advanced

Stress Stiffening

Classification of Solid-Mechanics Problems

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[K] {q} = {Fapp} + {Fth} + {Fpr} + {Fma} + {Fpl} + {Fcr} + {Fsw} + {Fld}

[K] = total stiffness matrix

{q} = nodal displacement{Fapp} = applied nodal force load vector

{Fth} = applied element thermal load vector

{Fpr} = applied element pressure load vector

{Fma

} = applied element body force vector

{Fpl} = element plastic strain load vector

{Fcr} = element creep strain loadvector

{Fsw} = element swelling strain load vector

{Fld} = element large deflection load vector

Basic equation for a static analysis is as follows:

Governing Equation for Solid Mechanics Problems


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StartProblem

Definition

Reads or generates

nodes and elements(ex: ANSYS)

Reads or generatesmaterial property data.

Reads or generatesboundary conditions(loads andconstraints.)

Generates

element shape

functions

Calculates masterelement equations

Calculates

transformation

matrices

Maps element

equations into

global system Assembles

element equations

Introduces

boundary

conditions

Performs solution

procedures

Prints or plotscontours of stresscomponents.

Prints or plotscontours ofdisplacements.

Evaluates andprints errorbounds.

Analysis and

design decisionsStop

Process Flow in a Typical FEM Analysis

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The FEM has been applied to a richly diverse array of scientific andtechnological problems.

The next few slides present some examples of the FEM applied to a

variety of real-world design and analysis problems.

Variety of FEM Solutions is Wide and Growing Wider

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This example shows an intravenous pump modeled using

hexahedral elements.

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Car tires require sophisticated analysis because of their complex geometry,

large deformations, nonlinear material behavior, and varying contactconditions. Brick elements are used to represent the tread and steel bead,

while shell elements are used in the wall area. Membrane elements are used

to represent the tire cords.

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This forging example is a simulation of a bulk forming process

with multiple stages. This axisymmetric analysis begins with a

cylinder of metal meshed very simply.

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A 3-D finite element model of an instrumented canine cervical spine.

The model consisted of four vertebrae (C3-C6), a titanium alloy

plate, and two screws attached to the back of two vertebrae (C4-C5).

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Finite element analysis works on the premise that a complex structure like the

helicopter shown here can be simulated on a computer screen so that the

helicopter's physical properties can be studied to determine how well the design

will perform under real-world conditions. The computer models permit thedesign team to examine a wide range of options and to detect design flaws long

before the prototype stage.

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This guitar features two strips of graphite running the

length of the neck. This FEM model was used to

study how much the neck moved when string forces

were applied and moisture content changed.

Using the FEM calculations, designers could try

different reinforcement scenarios to increase neck

stability.

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The boats hull consists of a thick core material sandwiched between two

thinner layers of plys oriented in different directions. The initial analysiswork focused on maximizing the hull's overall stiffness by examining

different core-material densities and varying the ply thickness and

orientations.

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Dynamic analysis of a tuning fork, to find it's first eight modes of vibration.

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2

3

4

5

6

7

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Some Animations

Metal Forming

Crash Analysis

Crash analysis

Crash Analysis

Crash Analysis

Warhead

How about a woman?

Heat Transfer

Forging

Drop test

The Last one

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Other numerical solution methods:Finite differences

Approximates the derivatives in the differential equation using

difference equations. Useful for solving heat transfer and fluid mechanics problems. Works well for two-dimensional regions with boundaries parallel

to the coordinate axes. Cumbersome when regions have curved boundaries.

Weighted residual methods (not confined to a small subdomain): Collocation Subdomain

Least squares* Galerkins method*

Variational Methods* (not confined to a small subdomain)

* Denotes a method that has been used to formulate finite element

solutions.

Technologies That Compete With the FEM

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Prototype Testing

Reliable. Well-understood. Trusted by regulatory agencies (FAA, DOT, etc.) Results are essential for calibration of simulation software. Results are essential to verify modeled results from simulation. Non destructive testing (NDT) is lowering costs of testing in

general. Expensive, compared to simulation. Time consuming. Development programs that rely too much on testing are

increasingly less competitive in todays market. Faster product development schedules are pressuring the quality

of development test efforts. Data integrity is more difficult to maintain, compared to

simulation.

Technologies that Compete With the FEM (cont.)

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The FEM in particular, and simulation in general, are becomingintegrated with the entire product development process (rather than just

another task in the product development process):FEM cannot become the bottleneck.

A broader range of people are using the FEM:Not just hard-core analysts.

Increased data sharing between analysis data sources (CAD, testing,

FEM software, ERM software.)

FEM software is becoming easier to use:Improved GUIs, automeshers.Increased use of sophisticated shellscripts and wizards.

Future Trends in the FEM and Simulation

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Enhanced multiphysics capabilities are coming:Coupling between numerous physical phenomena.

Ex: Fluid-structural interaction is the most common example. Ex: Semiconductor circuits, EMI and thermal buildup vary with

current densities.Improved life predictors, improved service estimations. Increasing use of non-deterministic analysis and design methods:

Statistical modeling of material properties, tolerances, and

anticipated loads.Sensitivity analyses.

Faster and more powerful computer hardware. Massively parallel

processing.

Decreasing reliance on testing.

FEM and simulation software available via Internet subscription.

Future Trends in the FEM and Simulation (cont.)

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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.

What's the difference between FEM & FEA ??

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What's the difference between FEM & FEA ??(Cont.)

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 method (Galerkin) & the variational method

(Rayleigh-Ritz).

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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.

What's the difference between FEM & FEA ??(Cont.)

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Conclusions

Engineering Design Process have been dealt in brief

Sources of Error in the FEM have been dealt in brief

Advantages and Disadvantages of FEM have beendealt in brief

Covered the basic Six Steps in the Finite Element

Method

Explanation of difference between FEM & FEA