finite element method intro

8/6/2019 Finite Element Method Intro

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Finite Element Method

An Introduction

K.H.J. Mangala


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What is FEM?

The finite element method is a numerical method for solving problems of engineering and mathematical

physics.Useful for problems with complicated geometries, loadings, and material properties where a n a l y t ic a l s o l u t i o n scan not be obtained.


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Discretizations

Model body by dividing it into anequivalent system of smaller bodies orunits (finite elements) interconnected at

points common to two or moreelements (nodes or nodal points)and/or boundary lines and/or surfaces.


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History

Grew out of aerospace industry.

Post-WW II jets, missiles, space flight

Need for light weight structures

Required accurate stress analysis

Paralleled growth of computers


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IntroductionFinite element analysis is a powerful tool that allows engineersto quickly analyze and refine a design .
http://en.wikipedia.org/wiki/Image:FAE_visualization.jpg


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IntroductionFEA can be applied to problems involving:

structural design


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Introduction Vibrations


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IntroductionHeat Transfer


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Introduction Sheet metal parts


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IntroductionProduct Design


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IntroductionProduct Design


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IntroductionIn a structural simulation, FEM helps tremendously in producingstiffness and strength visualizations and also in minimizingweight, materials, and costsFEM allows detailed visualization of where structures bend ortwist, and indicates the distribution of stresses and

displacements.
http://en.wikipedia.org/wiki/Image:FAE_visualization.jpg


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IntroductionFEA is extensively used in Aerospace industry, automobileindustry, computer hardware development,
http://www.nenastran.com/newnoran/animation


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Introduction

Obtain a set of algebraic equations tosolve for unknown nodal quantity(displacement).

Secondary quantities (stresses andstrains) are expressed in terms of nodalvalues of primary quantity


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Advantages

Irregular BoundariesGeneral LoadsDifferent Materials

Boundary Conditions Variable Element SizeEasy Modification

DynamicsNonlinear Problems (Geometric or Material)


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Steps in the Process

Discretize and Select Element TypeSelect a Displacement FunctionDefine Strain/Displacement and Stress/Strain

RelationshipsDerive Element Stiffness Matrix & Eqs.

Assemble Equations and Introduce B.C.s

Solve for the Unknown Degrees of FreedomSolve for Element Stresses and StrainsInterpret the Results


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Computer Programs

Small Special Purpose ProgramsTrussFramePlane StressHeat Transfer

Large General Purpose Program(Commercial)


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Programs

ALGOR ANSYSCOSMOS/MSTARDYNEIMAGES-3DMSC/NASTRAN

SAP90 ADINA NISA


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General Purpose Programs Advantages

Easy input - preprocessor.Solves many types of problems

Modular design - fluids, dynamics,heat, etc.Can run on PCs now.

Relatively low cost.


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General Purpose ProgramsDisadvantages

High development costs.Less efficient than smaller programs,

Often proprietary. User access to codelimited.


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FEA Concept contd.


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FEA Concept contd.


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FEA Concept contd.


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Displacement-Based contd.


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Two-Spring Model contd.


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Finite Element Solution Procedure contd.


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Nodes & Elements


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A Typical FE Mesh


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Boundary Conditions


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Finite Element Analysis (FEA)

FEA is a tool of design analysis Design Analysis : process of investigating certain properties of parts, assemblies or structures

Historical Note

The finite element m ethod of structuralanalysis w as created by academic andindustrial researchers during the 1950s and1960s.

The underlying theory is over 100 y ears old,and w as the basis for pen-and-papercalculations in the evaluation of suspensionbridges and steam boilers.


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Finite Element Analysis (FEA) - 1Design analysis can be conducted on:

Real objectsModels that represents certain aspects of a real object

If models are used instead of real objects, theanalysis can be conducted earlier in the designprocess before the final product or prototypes arebuiltModels can be physical models or mathematicalmodels

Simple mathematical models can be solvedanalytically, but more complex models require theuse of numerical methods

Design Analysis


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Design Analysis

Real Objects Models

Physical ModelsMathematical Models

Numerical Analytical

Finite Element Method

Finite Difference Method

Boundary Element Method

Design Analysis


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Finite Element Analysis (FEA) - 2

Finite Element Analysis (FEA) is one of those numerical methodsused to solve complex mathematical modelsFEA users require familiarity with the mechanics of materials,engineering design, and other topics as required in anyMechanical Engineering curriculum

FEA is based on CAD modelsThe ultimate objective of using FEA as a design tool is tochange the design process from iterative cycles of design prototype test into a streamlined process where prototypesare used only for final design verification


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Design Process: Traditional vs FEA-driven

Design

Prototyping

Testing

Production

CAD FEA

Traditional P roductDesign Process

FEA-Driven ProductDesign Process

Prototyping

Testing

Production

With the use of FEA, design iterations are moved from the physical space of prototyping and testing into virtual space of computer-based-simulations


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Why is FEA needed?

To reduce the amount of prototype testingComputer simulation allows multiple what-if scenarios tobe tested quickly and effectively.

To simulate designs that are not suitable forprototype testingEg: Surgical implants, such as an artificial knee

The bottom line:Cost savingsTime savings reduce time to market!Create more reliable, better-quality designs


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FEA Modelling Process: Structural Analysis

Pre-processingGeometry creation CAD modelDefining material propertiesDefining boundary conditions (Loads and Restraints)Meshing

CADModel

FEA Model

Analysisof

results

Pre-processing Solving Post-processing

Example with COSMOSXpress


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Pre-processing: Geometry Creation CADModel

Can use any CAD software tool2D or 3D modelImport CAD model to FEA environment using data exchangestandards such as IGES (Initial Graphics ExchangeSpecification) or STEP

Reliability of geometry exchange?Integration of CAD and FEA software

Assigning materialSelection from existing materials from the library

Defining and assigning new material properties


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The Finite Element ModelOften, CAD geometry will not satisfy the requirements of FEA

geometryCAD geometry contain all information necessary for part orassembly manufacturingCAD geometry can serve as a starting point in the process of FEmodel preparation

FEA geometry m ust be : Meshable Allow for creation of a meshthat is solvable within areasonable time

Converting CAD geometry intoFEA-specific geom etry

Defeaturing Idealization


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The Finite Element Model: Meshing 1

Meshing (discretization): A process of splitting the solutiondomain into simply shaped subdomains called Finite Elements. A continuous body has an infinite number of points or, in otherwords, an infinite number of degrees of freedom . Discretizationreplaces it with a system that has a finite number of degrees of

freedom possible to solve with numerical methods


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Types of finite elements


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The Finite Element Model: Meshing 33D tetrahedral solid elements

Nodes

Linear solid element Parabolic solid element

DOFTranslation

Rotation

In general, for the same mesh density (number of elements), parabolicelements yield better results than linear elements because: 1) they representcurved boundaries more accurately, and 2) they produce better mathematical

approximations. However, parabolic elements require greater computationalresources than linear elements

2D shell elements


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Meshing TechniquesManual meshing Automatic meshing

Only practical meshing choice for complex modelsMeshing is done in the background and not visible to the userThe user can control the element size

finite element method intro

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