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Transcript of FEA Session 01
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ACTP MCAD-404T
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.
Finite Element Method Defined (Contd.)
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.
Finite Element Method Defined (Contd.)
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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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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.
Sources of Error in the FEM (cont.)
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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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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.
1
2
3
4
5
6
7
8
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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