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ﺩﻭﺪﺤﻣ ﻥﺎﻤﻟﺍ ﺮﺑ ﯼﺍ ﻪﻣﺪﻘﻣ
An Introduction to Finite Element Analysis (FEA)
References:
1. FINITE ELEMENT ANALYSIS, THEORY AND APPLICATION WITH ANSYS, SAEED MOAENI, 3rd EDITTION, PRENTICE HALL
2. APPLIED FINITE ELEMENT ANALYSIS, 2nd Edition, LARRY J. SEGERLIND, JOHN WILEYAND SONS
3. THE FINITE ELEMENT METHOD IN ENGINEERING, 4TH EDITION, SINGIRESU S.RAO, ELSEVIER
4. FINITE ELEMENT METHOD, VOL. 1-3, O.C.ZIENKIEWICZ & R.L.TAYLOR, BUTTERWORTH HEINEMANN
5. FINITE ELEMENT MODELING FOR STRESS ANALYSIS, ROBERT D. COOK, JOHN WILEYAND SONS
Islamic Azad University, Najafabad Branch / Amir Atrian
3
Evaluation:
1. Homework (Manual & FE Solution of Some Simple Problems): 20-30%
Deadline for each problem: Deadline for each problem: 3 3 weeksweeks
EachEach problemproblem reportreport mustmust includeinclude thethe studentstudent name,name, fullfull solutionsolution procedureprocedure forfor both
both ofof manualmanual && FEFE inin wordword && PDFPDF filesfiles onlyonly..
TextText fontfont TimesTimes NewNew RomanRoman 1111 ptpt && BB--NazaninNazanin 1313 ptpt withwith singlesingle spacing,spacing, justifiedjustified
&
& editededited..
AbaqusAbaqus solution may be used to verify the manual solution.solution may be used to verify the manual solution.
Late & copied Late & copied homeworkshomeworks will be given zero credit.will be given zero credit.
2. Final (Computational): 70-80%
Computational exam covering the conceptsComputational exam covering the concepts discussed in lectures discussed in lectures
1. Introduction (2 weeks) - What’s FEM?
- Numerical Methods - Formulation
- Elements
- Coordinate Systems - More about FEM - About ABAQUS
2. Structural Problems (5 weeks) - Truss (Theory & ABAQUS)
- Beam (Theory & ABAQUS)
3. Solid Problems (3 weeks) - Theory of Elasticity
- Plane Stress/ Strain (Theory & ABAQUS) - Axisymmetric (ABAQUS)
4. Thermal Analysis (2 weeks) - Steady-state Analysis (ABAQUS)
- Transient (Time Dependent) Analysis (ABAQUS) 5. Dynamic Problems (? weeks)
Outline:
Islamic Azad University, Najafabad Branch / Amir Atrian
5
Chapter 1
Introduction
Islamic Azad University, Najafabad Branch / Amir Atrian
7 1.1.What is Finite Element?
- The finite element method (FEM) is a numerical procedure for obtaining solutions to many of the problems (steady, transient, linear & nonlinear) encountered in engineering analysis involving:
• Stress analysis
• Heat transfer
• Electromagnetism
• Fluid flow
Introduction:
3 methods to solve physical problems:
1- Exact solution
2-Numerical solution 3-Experimental solution
-In the finite element method, the solution region is considered as built up of many small, interconnected sub-regions called finite elements.
-The aim is to find the solution of a complicated problem by replacing it by a simpler one.
So, only an approximate solution rather than the exact solution can be found.
1.2.Historical background:
-Ancient mathematicians found the circumference of a circle by approximating it by the perimeter of a polygon.
- In recent times, the use of piecewise continuous functions defined over triangular regions, was first suggested by Courant in 1943 in the literature of applied mathematics.
- The basic ideas of the finite element method as known today were presented in the papers of Turner, Clough, Martin, and Topp and Argyris and Kelsey
-The name finite element was coined by Clough.
Introduction:
R
Area of one triangle:
Area of the circle:
Islamic Azad University, Najafabad Branch / Amir Atrian
9 1.3.Where does F.E. work?
Everywhere like….
Mechanical Engineering
Aerospace Engineering
Civil Engineering
Automotive Engineering
Stress analysis
Structure analysis
Introduction:
Where does F.E. work?
Everywhere like….
Heat transfer
Fluid dynamics
Biomechanics
Electromagnetism
…
Introduction:
Automotive application Exhaust manifold assembly
Islamic Azad University, Najafabad Branch / Amir Atrian
11 Where does F.E. work?
• And …
• Everywhere that an equation should be solved.
• F.E. is able to handle complex systems that defy “closed-form”analytical solutions.
Introduction:
1.4.Output of an F.E.A.
The following are some samples of what you can get from a specific analysis
Introduction:
• Fluid analysis Pressure
Gas temperature
Heat transfer parameters Velocities
• There are other types of analyses which have
been programmed by FEM Electromagnetic analysis Electric circuit analysis Poroelastic analysis Piezoelectric analysis
• Static analysis Deflection Stress
Strain Force Energy
• Dynamic analysis Frequencies
Deflection (mode shapes) Stress
Strain Force Energy
• Heat transfer analysis Temperature
Islamic Azad University, Najafabad Branch / Amir Atrian
13 1.5.Types of analysis:
Static analyses
Dynamic analyses (natural frequencies)
Transient dynamics
Heat transfer
Mechanisms
Fracture mechanics
Metal forming
Crashworthiness
Aerodynamics
Creep and plasticity analyses
Composite materials
Aeroelasticity
Introduction:
ABAQUS
ANSYS
LS-DYNA
DEFORM
SAP
Introduction:
1.6.How can the FEM Help the Design Organization?
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.
Islamic Azad University, Najafabad Branch / Amir Atrian
15 1.7.Numerical methods:
For many practical engineering problems, the exact solution can not be obtained.
This inability is may be attributed to either the complex nature of governing differential equations or due of the boundary or initial conditions.
In contrast to analytical solutions, which show the exact behavior of a system at any point within the system, numerical solutions approximate exact solutions only at discrete points called nodes.
Introduction:
Two common classes of numerical methods:
Introduction:
• Finite difference methods (FDM):
The differential equation is written for each node, and the derivatives are replaced by difference equations (for simple & isotropic problems)
This method is useful for solving heat transfer and fluid mechanics problems and works well for two-dimensional regions with boundaries parallel to the coordinate axes
• Finite element methods (FEM):
This method uses integral formulations rather than difference equations to create a system of algebraic equations.
An approximate continuous function is assumed to represent the solution for each element.
The complete solution is then generated by connecting or assembling the individual
Islamic Azad University, Najafabad Branch / Amir Atrian
17 1.8.Formulations:
• Strong form
• Weak form
Minimum total potential energy
Variational
Weighted residuals:Collacotion, Subdomain, Least squares, Galerkin
• Direct
Introduction:
- If the physical formulation of the problem is described as a differential equation, then the most popular solution method is the Method Method of of Weighted Weighted Residuals Residuals.
- If the physical problem can be formulated as the minimization of a functional, then
the Variational Variational Formulation Formulation is usually used.
Weighted residuals:
•• CollacotionCollacotion
•• SubdomainSubdomain
•• Least squaresLeast squares
•• GalerkinGalerkin
Introduction:
The weighted residual methods are based on assuming an approximate solution for the governing differential equation.
The assumed solution must satisfy the initial and boundary conditions of the given problem.
Because the assumed solution is not exact, substitution of the solution into the differential equation will lead to some residuals orerrors.
Islamic Azad University, Najafabad Branch / Amir Atrian
19
1.9.BASIC STEPS IN THE FINITE ELEMENT METHOD The
The basicbasic stepssteps involvedinvolved inin anyany finitefinite elementelement analysisanalysis consistconsist ofof thethe followingfollowing::
Introduction:
• Preprocessing Phase
1. Create and discretize the solution domain into finite elements; that is, subdivide the problem into nodes and elements.
2. Assume a shape function to represent the physical behavior of an element; that is, an approximate continuous function is assumed to represent the solution of an element.
3. Develop equations for an element.
4. Assemble the elements to present the entire problem. Construct the global stiffness matrix.
5. Apply boundary conditions. initial conditions, and loading.
•Solution Phase
6. Solve a set of linear or nonlinear algebraic equations simultaneously to obtain nodal results.
such as displacement values at different nodes or temperature values at different nodes in a heat transfer problem.
•Postprocessing Phase
7. Obtain other important information. At this point, you may be interested in values of principal stresses. heat fluxes, etc.
