fem lecture
TRANSCRIPT
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
11.5
22.5
33.5
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
T3 T4 T5
Piecewise Quadratic Approximationx
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
dxduau
qcuaudxd
,
:ionSpecificat CondtionsBoundary
0)(
: EquationalDifferenti
)(,,,
:ionSpecificat CondtionsBoundary
)()(
: EquationalDifferenti
2
2
2
2
2
2
2
2
dxwdb
dxd
dxwdb
dxdww
xfdx
wdbdxd
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
nyvC
xuCn
xv
yuCT
nxv
yuCn
yvC
xuCT
Fyv
xu
yEv
Fyv
xu
xEu
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
FF
uu
kkkk
kukuFkukuF
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
EEedx
duxdxd
dxdue
uxu
kkk
kkk
L TT
j
iTL TT fdxAPP
dxEA00
][}{}{}{][][}{ NδdδddBBδd
L TL T fdxAdxEA
00][}{}{][][ NPdBB
Vectorent DisplacemNodal}{
Vector Loading][}{
MatrixStiffness][][][
0
0
j
i
L T
j
i
L T
uu
fdxAPP
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
dxduAEP i
i dxdu
AEP jj
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
dN
dN
ntDisplacemeElasticeApproximat
uu
Lx
Lx
uu
u
uxux
uLxu
Lxx
Luuuu
Laauau
xaau
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}{
11
2][}{
111111
1
1
][][][][][
2
1
2
1
02
1
02
1
00
uu
LAfPP
dx
LxLx
AfPP
fdxAPP
LAEL
LLL
LAEdxAEdxEAK
oL
o
L T
LTL T
d
NF
BBBB
11
21111
}{}]{[2
1
2
1 LAfPP
uu
LAE oFdK
Quadratic Approximation Scheme
}]{[
)()()(
42
3
2
1
321
332211
23213
2
3212
11
2321
dN
ntDisplacemeElasticeApproximat
uuu
u
uxuxuxuLaLaau
LaLaau
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
7818168
187
3FFF
uuu
LAE
EquationElement
Lagrange Interpolation FunctionsUsing Natural or Normalized Coordinates
11
(1) (2) )1(21
)1(21
2
1
)1(21
)1)(1(
)1(21
3
2
1
)1)(31)(
31(
169
)31)(1)(1(
1627
)31)(1)(1(
1627
)31)(
31)(1(
169
4
3
2
1
(1) (2) (3)
(1) (2) (3) (4)
jiji
ji ,0,1
)(
Simple ExampleP
A1,E1,L1 A2,E2,L2
(1) (3)(2)
1 2
0000011011
1 Element EquationGlobal
)1(2
)1(1
3
2
1
1
11 PP
UUU
LEA
)2(2
)2(1
3
2
1
2
22
0
110110
0002 Element EquationGlobal
PP
UUU
LEA
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
PPP
PPP
P
UUU
LEA
LEA
LEA
LEA
LEA
LEA
LEA
LEA
0 Loadinged DistributZero Take
f
Simple Example ContinuedP
A1,E1,L1 A2,E2,L2
(1) (3)(2)
1 2
0
0ConditionsBoundary
)2(1
)1(2
)2(2
1
PP
PP
U
P
P
UU
LEA
LEA
LEA
LEA
LEA
LEA
LEA
LEA
00
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
PUU
LEA
LEA
LEA
LEA
LEA
0
3
2
2
22
2
22
2
22
2
22
1
11
LEA ,,Properties mFor Unifor
PU
UL
AE 01112
3
2
PPAEPLU
AEPLU )1(
132 ,2, Solving
One-Dimensional Beam ElementDeflection of an Elastic Beam
22423
11211
22
2
42
2
2
3
12
2
21
2
2
1
,,,
,
,
dxdwuwu
dxdwuwu
dxwdEIQ
dxwdEI
dxdQ
dxwdEIQ
dxwdEI
dxdQ
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 TLdVfwQuQuQuQdxEI
0443322110][}{][][ NdBB T
4
3
2
1
}{
uuuu
d ][][][ 4321
dxd
dxd
dxd
dxd
dxd
NB
22
22
30
2333636
3233636
2][][][
LLLLLL
LLLLLL
LEIdxEI
LBBK T
L
LfL
QQQQ
uuuu
LLLLLL
LLLLLL
LEI
6
6
122333636
3233636
2
4
3
2
1
4
3
2
1
22
22
3
L
LfLdxfdxfLL T
6
6
12][
0
4
3
2
1
0N
FEA Beam Problemf
a b
Uniform EI
00
00
6
6
12
00000000000000/2/3/1/300/3/6/3/600/1/3/2/300/3/6/3/6
2)1(
4
)1(3
)1(2
)1(1
6
5
4
3
2
1
22
2323
22
2323
QQQQ
a
a
fa
UUUUUU
aaaaaaaaaaaaaaaa
EI
1Element
)2(4
)2(3
)2(2
)2(1
6
5
4
3
2
1
22
2323
22
2323
00
/2/3/1/300/3/6/3/600/1/3/2/300/3/6/3/600000000000000
2
QQQQ
UUUUUU
bbbbbbbbbbbbbbbb
EI
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
00
6
6
12
/2/3/6/1/3/2/2/3/6/3/3/6/600/1/3/200/3/6/3/6
2
QQQQ
a
a
fa
UUUUUU
aaaaabaaababa
aaaaaaa
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
0000
00
6
12/2
/3/6/1/3/2/2/3/6/3/3/6/6
2
4
3
2
1
23
2
332233
afa
UUUU
aaaaabaaababa
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
460260
61206120
0000
260460
61206120
0000
QQPQQP
wu
wu
LEI
LEI
LEI
LEI
LEI
LEI
LEI
LEI
LAE
LAE
LEI
LEI
LEI
LEI
LEI
LEI
LEI
LEI
LAE
LAE
Element Equation Can Then Be Rotated to Accommodate Arbitrary Orientation
Some Standard FEA ReferencesBathe, 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.