The Finite Element Method - TAMU...
Transcript of 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
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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.
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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.
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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.
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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
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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
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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
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APPROXIMATE SOLUTIONS
point boundary aat)(
)(in)()()(
Puubdxdua
Lxfuxcdxduxa
dxd
=−+
=Ω=−+
−
0
,00
Model Problem
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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¥ ¥= - =
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ENGINEERING EXAMPLES OF THE MODEL PROBLEM IN 1-D
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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
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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
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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
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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
=
≈ = +
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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
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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
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Methods of Approximation(Continued)
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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
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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:
Methods of Approximation(Continued)
( , ) ( ), 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
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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
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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 = =
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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
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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
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F(x)
xba
Actual function
Example 2: Determine the integral of a function
=b
a
dxxFI )(
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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
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EXAMPLE 2 (continued) – Refined mesh
F(x)
x
Actual function
1xa = 7b x=
Approximation
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Approximation of a curved surface with a plane
(Triangular element)
(represents the solution)(temperature profile)
Domain
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••
•
•
•
•
(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•• •
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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.
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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.
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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)
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