fem lecture
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MCE 565Wave Motion & Vibration in Continuous Media
Spring 2005Professor M. H. Sadd
Introduction to Finite Element Methods
Need for Computational Methods
• Solutions Using Either Strength of Materials or Theory of Elasticity Are Normally Accomplished for Regions and Loadings With Relatively Simple Geometry
• Many Applicaitons Involve Cases with Complex Shape, Boundary Conditions and Material Behavior
• Therefore a Gap Exists Between What Is Needed in Applications and What Can Be Solved by Analytical Closed-form Methods
• This Has Lead to the Development of Several Numerical/Computational Schemes Including: Finite Difference, Finite Element and Boundary Element Methods
Introduction to Finite Element AnalysisThe finite element method is a computational scheme to solve field problems in engineering and science. The technique has very wide application, and has been used on problems involving stress analysis, fluid mechanics, heat transfer, diffusion, vibrations, electrical and magnetic fields, etc. The fundamental concept involves dividing the body under study into a finite number of pieces (subdomains) called elements (see Figure). Particular assumptions are then made on the variation of the unknown dependent variable(s) across each element using so-called interpolation or approximation functions. This approximated variation is quantified in terms of solution values at special element locations called nodes. Through this discretization process, the method sets up an algebraic system of equations for unknown nodal values which approximate the continuous solution. Because element size, shape and approximating scheme can be varied to suit the problem, the method can accurately simulate solutions to problems of complex geometry and loading and thus this technique has become a very useful and practical tool.
Advantages of Finite Element Analysis
- Models Bodies of Complex Shape
- Can Handle General Loading/Boundary Conditions
- Models Bodies Composed of Composite and Multiphase Materials
- Model is Easily Refined for Improved Accuracy by Varying
Element Size and Type (Approximation Scheme)
- Time Dependent and Dynamic Effects Can Be Included
- Can Handle a Variety Nonlinear Effects Including Material
Behavior, Large Deformations, Boundary Conditions, Etc.
Basic Concept of the Finite Element Method
Any continuous solution field such as stress, displacement, temperature, pressure, etc. can be approximated by a discrete model composed of a set of piecewise continuous functions defined over a finite number of subdomains.
Exact Analytical Solution
x
T
Approximate Piecewise Linear Solution
x
T
One-Dimensional Temperature Distribution
Two-Dimensional Discretization
-1-0.5
00.5
11.5
22.5
3
1
1.5
2
2.5
3
3.5
4-3
-2
-1
0
1
2
xy
u(x,y)
Approximate Piecewise Linear Representation
Discretization Concepts
x
T
Exact Temperature Distribution, T(x)
Finite Element Discretization
Linear Interpolation Model (Four Elements)
Quadratic Interpolation Model (Two Elements)
T1
T2 T2 T3 T3 T4 T4T5
T1
T2
T3T4 T5
Piecewise Linear Approximation
T
x
T1
T2T3 T3
T4 T5
T
T1
T2
T3T4 T5
Piecewise Quadratic Approximation
x
Temperature Continuous but with Discontinuous Temperature Gradients
Temperature and Temperature GradientsContinuous
Common Types of Elements
One-Dimensional ElementsLine
Rods, Beams, Trusses, Frames
Two-Dimensional ElementsTriangular, Quadrilateral
Plates, Shells, 2-D Continua
Three-Dimensional ElementsTetrahedral, Rectangular Prism (Brick)
3-D Continua
Discretization Examples
One-Dimensional Frame Elements
Two-Dimensional Triangular Elements
Three-Dimensional Brick Elements
Basic Steps in the Finite Element MethodTime Independent Problems
- Domain Discretization- Select Element Type (Shape and Approximation)- Derive Element Equations (Variational and Energy Methods)- Assemble Element Equations to Form Global System
[K]{U} = {F} [K] = Stiffness or Property Matrix {U} = Nodal Displacement Vector {F} = Nodal Force Vector - Incorporate Boundary and Initial Conditions - Solve Assembled System of Equations for Unknown Nodal Displacements and Secondary Unknowns of Stress and Strain Values
Common Sources of Error in FEA
• Domain Approximation• Element Interpolation/Approximation• Numerical Integration Errors
(Including Spatial and Time Integration)• Computer Errors (Round-Off, Etc., )
Measures of Accuracy in FEA
Accuracy
Error = |(Exact Solution)-(FEM Solution)|
Convergence
Limit of Error as:
Number of Elements (h-convergence) or
Approximation Order (p-convergence)
Increases
Ideally, Error 0 as Number of Elements or Approximation Order
Two-Dimensional Discretization Refinement
(Discretization with 228 Elements)
(Discretization with 912 Elements)
(Triangular Element)
(Node)
One Dimensional ExamplesStatic Case
1 2
u1 u2
Bar Element
Uniaxial Deformation of BarsUsing Strength of Materials Theory
Beam Element
Deflection of Elastic BeamsUsing Euler-Bernouli Theory
1 2
w1w2
2
1
dx
duau
qcuaudx
d
,
:ionSpecificat CondtionsBoundary
0)(
: EquationalDifferenti
)(,,,
:ionSpecificat CondtionsBoundary
)()(
: EquationalDifferenti
2
2
2
2
2
2
2
2
dx
wdb
dx
d
dx
wdb
dx
dww
xfdx
wdb
dx
d
Two Dimensional Examples
u1
u2
1
2
3 u3
v1
v2
v3
1
2
3
Triangular Element
Scalar-Valued, Two-Dimensional Field Problems
Triangular Element
Vector/Tensor-Valued, Two-Dimensional Field Problems
yx ny
nxdn
d
yxfyx
,
:ionSpecificat CondtionsBoundary
),(
: Equationial DifferentExample
2
2
2
2
yxy
yxx
y
x
ny
vC
x
uCn
x
v
y
uCT
nx
v
y
uCn
y
vC
x
uCT
Fy
v
x
u
y
Ev
Fy
v
x
u
x
Eu
221266
661211
2
2
Conditons Boundary
0)1(2
0)1(2
ents Displacemof Terms in Equations FieldElasticity
Development of Finite Element Equation• The Finite Element Equation Must Incorporate the Appropriate Physics of the Problem
• For Problems in Structural Solid Mechanics, the Appropriate Physics Comes from Either Strength of Materials or Theory of Elasticity
• FEM Equations are Commonly Developed Using Direct, Variational-Virtual Work or Weighted Residual Methods
Variational-Virtual Work Method
Based on the concept of virtual displacements, leads to relations between internal and external virtual work and to minimization of system potential energy for equilibrium
Weighted Residual Method
Starting with the governing differential equation, special mathematical operations develop the “weak form” that can be incorporated into a FEM equation. This method is particularly suited for problems that have no variational statement. For stress analysis problems, a Ritz-Galerkin WRM will yield a result identical to that found by variational methods.
Direct Method
Based on physical reasoning and limited to simple cases, this method is worth studying because it enhances physical understanding of the process
Simple Element Equation ExampleDirect Stiffness Derivation
1 2k
u1 u2
F1 F2
}{}]{[
rm Matrix Foinor
2 Nodeat mEquilibriu
1 Nodeat mEquilibriu
2
1
2
1
212
211
FuK
F
F
u
u
kk
kk
kukuF
kukuF
Stiffness Matrix Nodal Force Vector
Common Approximation SchemesOne-Dimensional Examples
Linear Quadratic Cubic
Polynomial Approximation
Most often polynomials are used to construct approximation functions for each element. Depending on the order of approximation, different numbers of element parameters are needed to construct the appropriate function.
Special Approximation
For some cases (e.g. infinite elements, crack or other singular elements) the approximation function is chosen to have special properties as determined from theoretical considerations
One-Dimensional Bar Element
udVfuPuPedV jjii
}]{[ : LawStrain-Stress
}]{[}{][
)( :Strain
}{][)( :ionApproximat
dB
dBdN
dN
EEe
dx
dux
dx
d
dx
due
uxu
kkk
kkk
L TT
j
iTL TT fdxAP
PdxEA
00][}{}{}{][][}{ NδdδddBBδd
L TL T fdxAdxEA
00][}{}{][][ NPdBB
Vectorent DisplacemNodal}{
Vector Loading][}{
MatrixStiffness][][][
0
0
j
i
L T
j
i
L T
u
u
fdxAP
P
dxEAK
d
NF
BB
}{}]{[ FdK
One-Dimensional Bar Element
A = Cross-sectional AreaE = Elastic Modulus
f(x) = Distributed Loading
dVuFdSuTdVe iV iS in
iijV ijt
Virtual Strain Energy = Virtual Work Done by Surface and Body Forces
For One-Dimensional Case
udVfuPuPedV jjii
(i) (j)
Axial Deformation of an Elastic Bar
Typical Bar Element
dx
duAEP i
i dx
duAEP j
j iu ju
L
x
(Two Degrees of Freedom)
Linear Approximation Scheme
Vectorent DisplacemNodal}{
Matrix FunctionionApproximat][
}]{[1
)()(
1
2
1
2
121
2211
2112
1
212
1121
d
N
dN
ntDisplacemeElasticeApproximat
u
u
L
x
L
xu
uu
uxux
uL
xu
L
xx
L
uuuu
Laau
auxaau
x (local coordinate system)(1) (2)
iu ju
L
x(1) (2)
u(x)
x(1) (2)
1(x) 2(x)
1
k(x) – Lagrange Interpolation Functions
Element EquationLinear Approximation Scheme, Constant Properties
Vectorent DisplacemNodal}{
1
1
2][}{
11
11111
1
][][][][][
2
1
2
1
02
1
02
1
00
u
u
LAf
P
Pdx
L
xL
x
AfP
PfdxA
P
P
L
AEL
LLL
LAEdxAEdxEAK
oL
o
L T
LTL T
d
NF
BBBB
1
1
211
11}{}]{[
2
1
2
1 LAf
P
P
u
u
L
AE oFdK
Quadratic Approximation Scheme
}]{[
)()()(
42
3
2
1
321
332211
23213
2
3212
11
2321
dN
ntDisplacemeElasticeApproximat
u
uu
u
uxuxuxu
LaLaau
La
Laau
au
xaxaau
x(1) (3)
1u 3u
(2)
2u
L
u(x)
x(1) (3)(2)
x(1) (3)(2)
1
1(x) 3(x)2(x)
3
2
1
3
2
1
781
8168
187
3F
F
F
u
u
u
L
AE
EquationElement
Lagrange Interpolation FunctionsUsing Natural or Normalized Coordinates
11
(1) (2) )1(2
1
)1(2
1
2
1
)1(2
1
)1)(1(
)1(2
1
3
2
1
)1)(3
1)(
3
1(
16
9
)3
1)(1)(1(
16
27
)3
1)(1)(1(
16
27
)3
1)(
3
1)(1(
16
9
4
3
2
1
(1) (2) (3)
(1) (2) (3) (4)
ji
jiji ,0
,1)(
Simple ExampleP
A1,E1,L1 A2,E2,L2
(1) (3)(2)
1 2
0000
011
011
1 Element EquationGlobal
)1(2
)1(1
3
2
1
1
11 P
P
U
U
U
L
EA
)2(2
)2(1
3
2
1
2
22
0
110
110
000
2 Element EquationGlobal
P
P
U
U
U
L
EA
3
2
1
)2(2
)2(1
)1(2
)1(1
3
2
1
2
22
2
22
2
22
2
22
1
11
1
11
1
11
1
11
0
0
EquationSystem Global Assembled
P
P
P
P
PP
P
U
U
U
L
EA
L
EAL
EA
L
EA
L
EA
L
EAL
EA
L
EA
0
Loadinged DistributZero Take
f
Simple Example ContinuedP
A1,E1,L1 A2,E2,L2
(1) (3)(2)
1 2
0
0
ConditionsBoundary
)2(1
)1(2
)2(2
1
PP
PP
U
P
P
U
U
L
EA
L
EAL
EA
L
EA
L
EA
L
EAL
EA
L
EA
0
0
0
0
EquationSystem Global Reduced
)1(1
3
2
2
22
2
22
2
22
2
22
1
11
1
11
1
11
1
11
PU
U
L
EA
L
EAL
EA
L
EA
L
EA0
3
2
2
22
2
22
2
22
2
22
1
11
LEA ,,Properties
mFor Unifor
PU
U
L
AE 0
11
12
3
2
PPAE
PLU
AE
PLU )1(
132 ,2
, Solving
One-Dimensional Beam ElementDeflection of an Elastic Beam
22423
11211
2
2
2
4
2
2
2
3
1
2
2
2
1
2
2
1
,,,
,
,
dx
dwuwu
dx
dwuwu
dx
wdEIQ
dx
wdEI
dx
dQ
dx
wdEIQ
dx
wdEI
dx
dQ
I = Section Moment of InertiaE = Elastic Modulus
f(x) = Distributed Loading
(1) (2)
Typical Beam Element1w
L
2w1 2
1M 2M
1V 2V
x
Virtual Strain Energy = Virtual Work Done by Surface and Body Forces
wdVfwQuQuQuQedV 44332211
L TL
dVfwQuQuQuQdxEI0443322110
][}{][][ NdBB T
(Four Degrees of Freedom)
Beam Approximation FunctionsTo approximate deflection and slope at each node requires approximation of the form
34
2321)( xcxcxccxw
Evaluating deflection and slope at each node allows the determination of ci thus leading to
FunctionsionApproximat Cubic Hermite theare where
,)()()()()( 44332211
i
uxuxuxuxxw
Beam Element Equation
L TL
dVfwQuQuQuQdxEI0443322110
][}{][][ NdBB T
4
3
2
1
}{
u
u
u
u
d ][][
][ 4321
dx
d
dx
d
dx
d
dx
d
dx
d
NB
22
22
30
233
3636
323
3636
2][][][
LLLL
LL
LLLL
LL
L
EIdxEI
LBBK T
L
LfL
Q
Q
Q
Q
u
u
u
u
LLLL
LL
LLLL
LL
L
EI
6
6
12
233
3636
323
3636
2
4
3
2
1
4
3
2
1
22
22
3
L
LfLdxfdxf
LL T
6
6
12][
0
4
3
2
1
0N
FEA Beam Problemf
a b
Uniform EI
0
0
0
0
6
6
12
000000
000000
00/2/3/1/3
00/3/6/3/6
00/1/3/2/3
00/3/6/3/6
2)1(
4
)1(3
)1(2
)1(1
6
5
4
3
2
1
22
2323
22
2323
Q
Q
Q
Q
a
a
fa
U
U
U
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U
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EI
1Element
)2(4
)2(3
)2(2
)2(1
6
5
4
3
2
1
22
2323
22
2323
0
0
/2/3/1/300
/3/6/3/600
/1/3/2/300
/3/6/3/600
000000
000000
2
Q
Q
Q
Q
U
U
U
U
U
U
bbbb
bbbb
bbbb
bbbbEI
2Element
(1) (3)(2)
1 2
FEA Beam Problem
)2(4
)2(3
)2(2
)1(4
)2(1
)1(3
)1(2
)1(1
6
5
4
3
2
1
23
2
232233
2
2323
0
0
6
6
12
/2
/3/6
/1/3/2/2
/3/6/3/3/6/6
00/1/3/2
00/3/6/3/6
2
Q
Q
Q
Q
a
a
fa
U
U
U
U
U
U
a
aa
aaba
aababa
aaa
aaaa
EI
System AssembledGlobal
0,0,0 )2(4
)2(3
)1(12
)1(11 QQUwU
Conditions Boundary
0,0 )2(2
)1(4
)2(1
)1(3 QQQQ
Conditions Matching
0
0
0
0
0
0
6
12
/2
/3/6
/1/3/2/2
/3/6/3/3/6/6
2
4
3
2
1
23
2
332233
afa
U
U
U
U
a
aa
aaba
aababa
EI
System Reduced
Solve System for Primary Unknowns U1 ,U2 ,U3 ,U4
Nodal Forces Q1 and Q2 Can Then Be Determined
(1) (3)(2)
1 2
Special Features of Beam FEA
Analytical Solution GivesCubic Deflection Curve
Analytical Solution GivesQuartic Deflection Curve
FEA Using Hermit Cubic Interpolation Will Yield Results That Match Exactly With Cubic Analytical Solutions
Truss ElementGeneralization of Bar Element With Arbitrary Orientation
x
y
k=AE/L
cos,sin cs
Frame ElementGeneralization of Bar and Beam Element with Arbitrary Orientation
(1) (2)
1w
L
2w1 2
1M 2M
1V 2V
2P1P1u 2u
4
3
2
2
1
1
2
2
2
1
1
1
22
2323
22
2323
460
260
6120
6120
0000
260
460
6120
6120
0000
Q
Q
P
Q
Q
P
w
u
w
u
L
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EIL
AE
L
AEL
EI
L
EI
L
EI
L
EIL
EI
L
EI
L
EI
L
EIL
AE
L
AE
Element Equation Can Then Be Rotated to Accommodate Arbitrary Orientation
Some Standard FEA References
Bathe, K.J., Finite Element Procedures in Engineering Analysis, Prentice-Hall, 1982, 1995.Beer, G. and Watson, J.O., Introduction to Finite and Boundary Element Methods for Engineers, John Wiley, 1993Bickford, W.B., A First Course in the Finite Element Method, Irwin, 1990.Burnett, D.S., Finite Element Analysis, Addison-Wesley, 1987.Chandrupatla, T.R. and Belegundu, A.D., Introduction to Finite Elements in Engineering, Prentice-Hall, 2002.Cook, R.D., Malkus, D.S. and Plesha, M.E., Concepts and Applications of Finite Element Analysis, 3rd Ed., John Wiley, 1989.Desai, C.S., Elementary Finite Element Method, Prentice-Hall, 1979.Fung, Y.C. and Tong, P., Classical and Computational Solid Mechanics, World Scientific, 2001.Grandin, H., Fundamentals of the Finite Element Method, Macmillan, 1986.Huebner, K.H., Thorton, E.A. and Byrom, T.G., The Finite Element Method for Engineers, 3rd Ed., John Wiley, 1994.Knight, C.E., The Finite Element Method in Mechanical Design, PWS-KENT, 1993.Logan, D.L., A First Course in the Finite Element Method, 2nd Ed., PWS Engineering, 1992.Moaveni, S., Finite Element Analysis – Theory and Application with ANSYS, 2nd Ed., Pearson Education, 2003.Pepper, D.W. and Heinrich, J.C., The Finite Element Method: Basic Concepts and Applications, Hemisphere, 1992.Pao, Y.C., A First Course in Finite Element Analysis, Allyn and Bacon, 1986.Rao, S.S., Finite Element Method in Engineering, 3rd Ed., Butterworth-Heinemann, 1998.Reddy, J.N., An Introduction to the Finite Element Method, McGraw-Hill, 1993.Ross, C.T.F., Finite Element Methods in Engineering Science, Prentice-Hall, 1993.Stasa, F.L., Applied Finite Element Analysis for Engineers, Holt, Rinehart and Winston, 1985.Zienkiewicz, O.C. and Taylor, R.L., The Finite Element Method, Fourth Edition, McGraw-Hill, 1977, 1989.