the finite element method - tamu...

JN Reddy

The Finite Element Method

Read: Chapters 1 and 2

GENERAL INTRODUCTION

• Engineering and analysis• Simulation of a physical process• Examples mathematical model

development• Approximate solutions and methods

of approximation • The basic features of the finite element

method• Examples• Finite element discretization• Terminology• Steps involved in the finite element

model development

CONTENTS

JN Reddy Introduction: 2

INTRODUCTORY REMARKS

What we do as engineers?

Knowing the fundamentals associated with each engineering problem you set out to tackle, not only makes you a better engineer but also empowers you as an engineer.

• develop mathematical models,• conduct physical experiments, • carry out numerical simulations to help

designer, and• design and build systems to achieve a (1)

functionality in (2) most economical way.


INTRODUCTORY REMARKS

Engineering is the discipline, art, and professionof acquiring and applying technical, scientific,and mathematical knowledge to design andimplement materials, structures, machines,devices, systems, and processes that safelyrealize a desired objective.

Engineering is a problem-solving discipline, and solution requires an understanding of the phenomena that occurs in the system or process.


AnalysisAnalysis is an aid to design and manufacturing, and not an end in itself.Analysis involves the following steps:

identifying the problem and nature of the response to be determined,

selecting the mathematical model (i.e., governing equations),

solving the problem with a solution method (e.g., FEM), and

evaluating the results in light of the design parameters.


MODELING OF APhysical Process

Physical SystemAssumptions

concerningthe system

Laws of physics(conservation

principles)

Mathematical Model

(BVP, IVP)

Numerical Simulations

Computationaldevice

Numerical method(FEM,FDM,BEM,etc.)

BVP – Boundary value problems (equilibrium problems)IVP – Initial value problems (time-dependent problems)

FEM – Finite Element Method

FDM – Finite Difference Method

BEM – Boundary Element Method

JN Reddy

EXAMPLES OF MATHEMATICAL MODEL DEVELOPMENT

Rectangular fins(a)

Body from which heat has to be extracted

(b)

Lateral surface and right end areexposed to ambient temperature, T∞

ah

L

( )P T Tβ ¥-Convection, , P = perimeter

x

yz

( )xqA ( )x xqAΔ+

l

a

( )

02

0

( ) ( ) ( )

( ) ,

x x xx x x h

h

A AqA qA P x T T r x

d dTAq P T T r A q kdx dx

β ρ

β ρ

ΔΔ Δ Δ+

+ ¥

¥

æ ö+ ÷ç- - - + =÷ç ÷÷çè ø

- - - + = =-

Objective: Determine heat flow in a heat exchanger fin

Basic Concepts: 6

3D to 2D

2D to 1D

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EXAMPLE OF ENGINEERING MODEL DEVELOPMENT (continued)

( ) hd dTAk P T T r Adx dx

β ρ¥

æ ö÷ç- + - =÷ç ÷çè ø

, , , hu T T a Ak c P r A fd dua cu fdx dx

β ρ¥= - = = =

æ ö÷ç- + =÷ç ÷çè ø

( )0 0

0

( ) ( ) ,

, ,

x x xdA A f A x A f Adx

du d duE AE fdx dx dx

σ σ σ

σ ε ε

Δ Δ+ - + = + =

æ ö÷ç= = - + =÷ç ÷çè ø

, force per unit length

Lx Δx

f

Δx

( )x xAσ Δ+( )xAσf

Determine: Axial deformation of a bar

Basic Concepts: 7

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APPROXIMATE SOLUTIONS

point boundary aat)(

)(in)()()(

Puubdxdua

Lxfuxcdxduxa

dxd

=−+

=Ω=−+

−

0

,00

Model Problem

Basic Concepts: 8

E, A

b=k L

u(L)

u0

P

x, uc

Elastic deformation of a bar

( )f xa EA=

Heat transfer in a fin

uninsulated bara kAc Pβ== ( )f x

xL

0,u T T u T¥ ¥= - =

JN Reddy

ENGINEERING EXAMPLES OF THE MODEL PROBLEM IN 1-D

Basic Concepts: 9

Flow of viscous fluid through a channel

0 0in ( , )xdvd f bdy dy

μ − − = Ω =

( ) ( ), horizontal velocitypressure gradient, / (constant)

( ) fluid viscosityshear stresselocity of the top surface

xu y v yf dp dx

yQU v

μ

==

===

0U ³y

b

x

2b

Couette flow

Poiseuille flow

JN Reddy

Exact and Approximate Solutions

Approximation of the actual solution over the entire domainActual solution

Approximate solution

An exact solution satisfies (a) the differential equation at every point of the domain and (b) boundary conditions on the boundary. An approximate solution satisfies the differential equation as well as the specified boundary conditions in some “acceptable sense” (to be made clearer shortly). We seek the approximate solution as a linear combination of unknown parameters ci and known functions

01

( ) ( ) ( ) ( )N

N i ii

u x u x c x x=

» = +å f f0

andi

f f

Basic Concepts: 10

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Determining Approximate Solutions(continued)

Then ci are determined such that the residual, R(x), is zero in some sense.

0NN

dud a x c x u f xdx dx

æ ö÷ç- + - =÷ç ÷÷çè ø( ) ( ) ( )

0NN

dud a( x ) c( x )u f ( x ) R( x )dx dx

æ ö÷ç- + - º ¹÷ç ÷÷çè ø

will result in a non-zero function on the left side of the equality:

1. Suppose that is selected to satisfy the boundary conditionsexactly. Then substitution of uN(x) into the differentialequation

if

Basic Concepts: 11

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METHODS OF APPROXIMATION

1. One sense in which the residual, R(x), can be made zero is to require it to be zero at selected number of points. The number of points should be equal to the number of unknowns in the approximate solution

This way of determining ci is known as the Collocation method. We obtain N algebraic equations in N unknown C’s

NixR i ,,2,1,0 ==)(

01

0

( ) ( ) ( )

( ) and ( ) are functions to be selected to satisfy thespecified boundary conditions and are parameters to be determined such that the residual is zero insome sense.

f f

f f

N

N j jj

j

j

u x u c x x

x xcmade

=

≈ = +

Basic Concepts: 12

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Methods of Approximation(Continued)

2. Another approach in which the residual, R(x), can be made zero is in a least-squares sense; i.e., minimize the integral of the square of the residual with respect to C’s.

02

),,,(

0

0

221

=∂∂=

∂∂

=

dxcRR

cJ

dxRcccJ

L

ii

L

N

or

Minimize

This method is known as the least-Squares method. We obtain N algebraic equations in N unknown C’s

00

=∂∂

dxcRR

L

iBasic Concepts: 13

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3. Yet, another approach in which the residual, R(x), can be made zero is in a weighted-residual sense

00 , 1,2, ,

where are linearly independent set of functions

L

i

i

Rdx i Nò= y =

y

This method is known as the Weighted-Residual method. We obtain N algebraic equations in N unknown C’s. In general, weight functions are not the same as the approximation functions . Various special cases are

Petrov-Galerkin Method:Galerkin Method:

i i

i i

y ¹ f

y = f

iψ

iϕ

Basic Concepts: 14


JN Reddy

WEIGHTED-INTEGRAL FORMULATIONSin the Numerical Solution of Differential Eqs.

The approximation methods discussed earlier can be viewed as special cases of the weighted-residual methods of approximation. In particular, we have

• Collocation method

• Least-squares method

♦ Petrov-Galerkin method

♦ Galerkin Method

= -( ) ( )i ix x xy d

¶=

¶( )i

i

Rxc

y

¹( ) ( )i ix xy f

( ) ( )i ix x=y f

Basic Concepts: 15


4. Another approach in which the governing equation is castin a weak-form and the weight function is taken the sameas the approximation function is known as the Ritz method:


( , ) ( ), 1,2, ,i N iB u i Nf f= =

The Ritz method is the most commonly used method forall commercial software. In this method, satisfies onlythe specified essential (geometric) boundary conditionswhile satisfies the homogeneous form of the specifiedessential boundary conditions. The specified natural (force)boundary condition are included in the weak form.

0f

if


WORKING EXAMPLE

0

0

0, 0

(0) ,x L

d dua f x Ldx dx

duu u a Pdx =

æ ö÷ç- - = < <÷ç ÷çè ø

= =

For a weighted-residual method:

0f

ifsatisfy the actual specified BCs

satisfy the homogeneous form of the actualspecified BCs

For the Ritz method:

0f

ifsatisfy the actual specified essential BCs

satisfy the homogeneous form of the actualspecified essential BCs


WORKING EXAMPLE (CONTINUED)

00 0 0 0

1

0

0 0 0 2

( ) ,

( ) , ( )

x L

ii

x L

d Pu a P u xdx a

da x L xdx

ff f

ff f

=

=

= = = +

= = = -

For a weighted-residual method:

For the Ritz (or weak-form Galerkin) method:

0 00 0 0( ) , ( )iuf f= =

0 0, ( ) iiu x xf f = =

JN Reddy

BASIC FEATURES OF THE FINITE ELEMENT METHOD (FEM)

Divide whole into parts (finite element mesh) Set up the `problem’ over a typical part

(derive a set of relationships between primary and secondary variables)

Assemble the parts to obtain the solution to the whole

Basic Concepts: 19

JN Reddy

1 1 1

1 1 1

, ,

N N N

i i i i i ii i i

N N N

i i ii i i

m x m y m zX Y Z

m m m= = =

= = =

= = =

1. See the simplicity in the complicated (see the geometric shapes that are simple to identify the center of mass).

2. Determine the center of mass of each part.3. Put the parts together to obtain the required solution.

Example 1: Determine the center of massof a 3D machine part

Basic Concepts: 20

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F(x)

xba

Actual function

Example 2: Determine the integral of a function

=b

a

dxxFI )(

Basic Concepts: 21

JN ReddyBasic Concepts:

22

F(x)

x

Actual functionApproximation

EXAMPLE 2 (continued)

≤≤+≤≤+≤≤+

≈bxx,xbaxxx,xbaxxa,xba

xF

333

3222

211

)(

1xa = 4xb =2x 3x

++

+==11 i

i

i

i

x

x

ii

x

x

ii dxxbadxxFI )()(

321 IIII ++≈

1I 2I 3I

JN Reddy Basic Concepts: 23

EXAMPLE 2 (continued) – Refined mesh

F(x)

x

Actual function

1xa = 7b x=

Approximation

JN Reddy

Approximation of a curved surface with a plane

(Triangular element)

(represents the solution)(temperature profile)

Domain

Basic Concepts: 24

JN Reddy Basic Concepts: 25

••

•

•

•

•

(b) Finite element mesh

hΩDomain,

Boundary, Γ°°

°°

Boundary fluxΩe

•

°

ΩDomain,

Finite Element Discretization

Elements Nodes

(a) Given domain

(c) Typical element with boundary fluxes

(d) Discretized domain

••

••

•

••

•

eΩ•• •

JN Reddy

Some Examples of Real-World Finite Element Discretizations

Introduction: 26

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FEM Terminology

Element A geometric sub-domain of the region being simulated, with the property that it allows a unique (1) representation of its geometry and (2) derivation of the approximation (interpolation) functions.

Node A geometric location in the element which plays a role in the derivation of the interpolation functions and it is the point at which solution is sought.

Mesh A collection of elements (or nodes) that replaces the actual domain.

Weak Form An integral statement equivalent to the governing equations and natural boundary conditions. More to come.

Basic Concepts: 27

JN Reddy

FEM Terminology (continued)

• Finite Element Model A set of algebraic equations relating the nodal values of the primary variables (e.g., displacements) to the nodal values of the secondary variables (e.g., forces) in an element.

Finite element model is NOT the same as the finite element method. There is only one finite element method but there can be more than one finite element model of a problem (depending on the approximate method used to derive the algebraic equations).

Numerical Simulation Evaluation of the mathematical model (i.e., solution of the governing equations) using a numerical method and computer.

Basic Concepts: 28

JN Reddy

Major Steps of Finite Element Model Development

Begin with the governing equations of the problem

Develop its weak form over a typical finite element

Approximate the solution over each finite element

Obtain algebraic relations among the quantities of interest over each finite element (i.e., finite element model)

Basic Concepts: 29

JN Reddy

Major Steps of Finite Element Model

Development

Finite Element Model Development

Governing (Differential) Equations

Engineering ProblemFormulation

Virtual work statements

Weak Form Development

Basic Concepts: 30

Solid mechanics

the finite element method - tamu...

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