Transcript FEA Short Course - University of Rhode Island

Introduction to Finite Element Methods
MCE 565
Wave Motion & Vibration in Continuous Media
Spring 2005
Professor M. H. Sadd
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 Closedform 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 Analysis
The 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.
One-Dimensional Temperature Distribution
T
T
Approximate Piecewise
Linear Solution
Exact Analytical Solution
x
x
Two-Dimensional Discretization
2
1
0
u(x,y)
-1
-2
-3
4
3.5
3
Approximate Piecewise
Linear Representation
2.5
2
1.5
y
1
-1
-0.5
0
0.5
x
1
1.5
2
2.5
3
Discretization Concepts
T
Exact Temperature Distribution, T(x)
x
Finite Element Discretization
Linear Interpolation Model
(Four Elements)
T1
T2
T2
T3 T3
Quadratic Interpolation Model
(Two Elements)
T
1
T2
T4 T4
T3
T5
T
T3
T4
T5
T
T1
T1
T2
T2
T3
T4
T3
T5
T4
T5
x
Piecewise Linear Approximation
x
Piecewise Quadratic Approximation
Temperature Continuous but with
Discontinuous Temperature Gradients
Temperature and Temperature Gradients
Continuous
Common Types of Elements
One-Dimensional Elements
Line
Rods, Beams, Trusses, Frames
Two-Dimensional Elements
Triangular, Quadrilateral
Plates, Shells, 2-D Continua
Three-Dimensional Elements
Tetrahedral, 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 Method
Time 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
(Node)

(Discretization with 228 Elements)


(Triangular Element)
(Discretization with 912 Elements)
One Dimensional Examples
Static Case
Bar Element
Beam Element
Uniaxial Deformation of Bars
Using Strength of Materials Theory
Deflection of Elastic Beams
Using Euler-Bernouli Theory
u1
w1
u2
1
2
Differential Equat ion:
d
 ( au)  cu  q  0
dx
Boundary CondtionsSpecification :
du
u,a
dx
w2
q2
q1
1
2
Differential Equation:
d2
d 2w
 2 (b 2 )  f ( x )
dx
dx
BoundaryCondtionsSpecification :
dw
d 2w d
d 2w
w,
,b 2 ,
(b
)
dx
dx
dx dx2
Two Dimensional Examples
Triangular Element
Triangular Element
Scalar-Valued, Two-Dimensional
Field Problems
Vector/Tensor-Valued, TwoDimensional Field Problems
v3
f3
u3
3
3
2
v2
f2
2
v1
1
f1
1
u2
u1
ElasticityField Equationsin T ermsof Displacements
ExampleDifferential Equation:
 2f  2f

 f ( x, y )
x 2 y 2
BoundaryCondtionsSpecification :
f,
df f
f

nx  n y
dn x
y
 2 u 
E
  u v 
    Fx  0
2(1  ) x  x y 
 2 v 
E
  u v 
    Fy  0
2(1  ) y  x y 
BoundaryConditons

 u v 
u
v 
Tx   C11
 C12 n x  C66   n y
x
y 

 y x 
 u v 

u
v 
T y  C66   n x   C12
 C22 n y
x
y 
 y x 

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, VariationalVirtual Work or Weighted Residual 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
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.
Simple Element Equation Example
Direct Stiffness Derivation
u1
u2
F1
F2
1
2
k
Equilibrium at Node1  F1  ku1  ku2
Equilibrium at Node 2  F2  ku1  ku2
or in Matrix Form
k
 k

 k   u1   F1 
  

k  u2   F2 
Stiffness Matrix
[ K ]{u}  {F }
Nodal Force Vector
Common Approximation Schemes
One-Dimensional Examples
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.
Linear
Quadratic
Cubic
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
Approximation : u    k ( x )uk  [ N ]{d }
k
du
d
d[N ]
   k ( x )u k 
{d }  [ B]{d }
dx k dx
dx
Stress - Strain Law :   Ee  E[ B]{d }
Strain : e 
 edV  P u

i i
 Pj u j   fudV 

L
L
 Pi 
{δd}T  A[ B]T E[ B]dx{d}  {δd}T    {δd}T  A[ N ]T fdx 
0
0
 Pj 
L
L
T
T
A
[
B
]
E
[
B
]
dx
{
d
}

{
P
}

A
[
N
]
fdx


0
0
L
[ K ]   A[ B]T E[ B]dx  Stiffness Matrix
0
[ K ]{d}  {F }
 Pi  L
{F }      A[ N ]T fdx  LoadingVector
0
 Pj 
 ui 
{d }     Nodal Displacement Vector
u j 
One-Dimensional Bar Element
Axial Deformation of an Elastic Bar
x
A = Cross-sectional Area
E = Elastic Modulus
f(x) = Distributed Loading
Typical Bar Element
ui
du
Pi   AE i
dx
uj

(i)
L
(j)
Pj   AE
du j
dx
(Two Degrees of Freedom)
Virtual Strain Energy = Virtual Work Done by Surface and Body Forces

V
ij eij dV   Ti n ui dS   Fi ui dV
St
V
For One-Dimensional Case
 edV  P u

i i
 Pj u j   fudV

Linear Approximation Scheme
ui
(1)
uj
L
(2)
x (local coordinate system)
u(x)
Approximate Elastic Displacement
u  a1  a 2 x 
u1  a1
u2  a1  a 2 L
u  u1
x

x
 u  u1  2
x  1  u1   u2
L
L

 L
 1 ( x )u1   2 ( x )u2
u  1
 u1   x
 2    1 
u2   L
(1)
1(x)
x
(2)
2(x)
x   u1 
   [ N ]{d }
L  u2 
1
[ N ] Approximation FunctionMatrix
{d }  Nodal Displacement Vector
(1)
(2)
x
k(x) – Lagrange Interpolation Functions
Element Equation
Linear Approximation Scheme, Constant Properties
 1
L
L
 L   1 1 
AE  1  1
T
T
[ K ]   A[ B] E[ B]dx  AE[ B] [ B] dx  AE 

L

 1 1 
0
0
1   L L 
L




 L 
 x
 
L
 P1  L
 P1 
 P  Af L 1
T
{F }      A[ N ] fdx     Afo   L dx   1   o  
0
x
2 1
 P2  0
 P2 
 P2 


 L 
u 
{d }   1   Nodal Displacement Vector
u2 
AE  1 1   u1   P1  Afo L 1
[ K ]{d}  {F } 
   



L  1  1 u2   P2 
2 1
Quadratic Approximation Scheme
u2
u1
(1)
(2)
u3
(3)
x
L
u(x)
Approximate Elastic Displacement
u1  a1
u  a1  a 2 x  a3 x
2
L
L2
 u2  a1  a 2  a3
2
4
2
u3  a1  a 2 L  a3 L
(1)
u  1 ( x )u1   2 ( x )u2   3 ( x )u3
u  1  2
 u1 
 
 3 u2   [ N ]{d }
u3 
(2)
2(x)
1(x)
(3)
x
3(x)
1
Element Equation
 7  8 1   u1   F1 
AE 
   
 8 16  8 u2    F2 

3L 
 1  8 7  u3   F3 
(1)
(2)
(3)
x
Lagrange Interpolation Functions
Using Natural or Normalized Coordinates
1 , i  j
i ( j )  
0 , i  j

(1)
1    1
(2)

(2)
(1)
1
(1  )
2
1
 2  (1  )
2
1 
(3)
1
1   (1  )
2
 2  (1  )(1  )
3 
9
1
1
(1  )(  )(  )
16
3
3
27
1
2 
(1  )(1  )(  )
16
3
27
1
3 
(1  )(1  )(  )
16
3
9 1
1
 4   (  )(  )(1  )
16 3
3
1  

(1)
(2)
(3)
(4)
1
(1  )
2
Simple Example
P
A1,E1,L1
A2,E2,L2
1
2
(1)
T akeZero Distributed Loading
f 0
(3)
(2)
Global Equation Element1
Global EquationElement2
(1)
 1  1 0 U 1   P1 
A1 E1 
  

 1 1 0 U 2    P2(1) 

L1 
0 0 U 3   0 
 0
0  U1   0 
0 0
A2 E2 
  

0 1  1 U 2    P1( 2 ) 

L2 
0  1 1  U 3   P2( 2 ) 
Assembled Global System Equation
 A1 E1
 L
1

A
  1 E1
 L1

 0

A1 E1
L1
A1 E1 A2 E 2

L1
L2
AE
 2 2
L2



(1)
  P1 
 U 1   P1
A2 E 2     (1)
  

U 2    P2  P1( 2 )    P2 

L2    
( 2)
 P 
  3
A2 E 2  U 3   P2

L2 
0
Simple Example Continued
P
A1,E1,L1
A2,E2,L2
1
2
(1)
(3)
(2)
Reduced Global System Equation
BoundaryCondit ions
U1  0
P2( 2 )  P
P2(1)  P1( 2 )  0
 A1 E1 A2 E 2
 L  L
2
 1
A
E
  2 2

L2
A2 E 2 
L2  U 2   0 


A2 E 2  U 3   P 
L2 

 A1 E1
 L
1

  A1 E1
 L1

 0

A1 E1
L1
A1 E1 A2 E 2

L1
L2
AE
 2 2
L2

For Uniform
PropertiesA, E , L
Solving  U 2 


(1)
  0   P1 
AE   

 2 2  U 2    0 
L2    

A2 E 2  U 3   P 

L2 
0
AE  2  1 U 2   0 
  
L  1 1  U 3   P 
PL
2 PL
, U3 
, P1(1)   P
AE
AE
One-Dimensional Beam Element
Deflection of an Elastic Beam
f(x) = Distributed Loading
x
I = Section Moment of Inertia
E = Elastic Modulus
Typical Beam Element
w2
w1
q1
q2

(1)
L
(2)
M1
V1
M2
V2
(Four Degrees of Freedom)
 d 2w 
d  d 2w 
 EI 2  , Q2   EI 2 
Q1 
dx 
dx 1
dx 1

 d 2w 
d  d 2w 
Q3    EI 2  , Q4   EI 2 
dx 
dx  2
dx  2

dw
dw
u1  w1 , u2  q1  
, u3  w2 , u4  q2  
dx 1
dx 2
Virtual Strain Energy = Virtual Work Done by Surface and Body Forces
 edV  Q u

1 1
 Q2 u2  Q3u3  Q4 w4   fwdV 

L
L
0
0
EI  [ B]T [ B]dx{d}  Q1u1  Q2u2  Q3u3  Q4 w4   f [ N ]T dV
Beam Approximation Functions
To approximate deflection and slope at each
node requires approximation of the form
w( x)  c1  c2 x  c3 x 2  c4 x 3
Evaluating deflection and slope at each node
allows the determination of ci thus leading to
w( x)  f1 ( x)u1  f2 ( x)u2  f3 ( x)u3  f4 ( x)u4 ,
where fi are theHermiteCubic Approximation Functions
Beam Element Equation
L
L
0
0
EI  [ B]T [ B]dx{d}  Q1u1  Q2u2  Q3u3  Q4 w4   f [ N ]T dV
 u1 
u 
 
{d }   2 
u3 
u 4 
[ B] 
df df df df
d[N ]
[ 1 2 3 4 ]
dx
dx dx dx dx
 3L  6  3L
 6
 3L 2 L2 3L L2 
L
2
EI

[ K ]  EI  [ B]T [ B]dx  3 
0
3L
6
3L 
L  6


2
3L 2 L2 
 3L L

L
0
 f1 
 6 
 
 L 
L f 2 
fL
 
f [ N ]T dx  f   dx 
 
0
f
12
3
 
 6 
f4 
 L 
 3L  6  3L  u1  Q1 
 6 
 6
2

3L
L2  u2  Q2  fL  L 
2 EI  3L 2 L
    
 
3
3L
6
3L  u3  Q3  12  6 
L  6


2
 L 
3L 2 L2  u4  Q4 
  3L L
FEA Beam Problem
f
Uniform EI
a
b
1
2
(2)
(1)
(3)
Ele m e n t1
 6 / a3

2
 3 / a
 6 / a 3
2 EI 
2
 3 / a
 0

 0
 3 / a2
 6 / a3
 3 / a2
2/a
3 / a2
1/ a
2
3
3 / a2
1/ a
3 / a2
2/a
0
0
0
0
0
0
3/ a
6/a
0 0 U1 
 6  Q1(1) 
 
 a   (1) 
0 0 U 2 
  Q2 
0 0 U 3 
fa  6  Q3(1) 


 
 

12  a  Q4(1) 
0 0 U 4 
0  0 
0 0 U 5 
 
  

 0   0 
0 0 U 6 
Ele m e n t2
0
0

0
2 EI 
0
0

0
0
0
0
0
0
6 / b3
0
0
 3 / b2
0
0
 6 / b3
0  3 / b2
0  6 / b3
2/b
3 / b2
3 / b2
6 / b3
0  3 / b2
1/ b
3 / b2
0  U1   0 
0  U 2   0 
 3 / b 2  U 3  Q1( 2 ) 
   

1 / b  U 4  Q2( 2 ) 
3 / b 2  U 5  Q3( 2 ) 
  

2 / b  U 6  Q4( 2 ) 
FEA Beam Problem
1
2
(2)
(1)
(3)
GlobalAsse m ble d
Syste m
6 / a 3

 
 
2 EI 
 
 

 
 3 / a2
 6 / a3
2/a
3 / a2

6/a 6/b


2/a  2/b
3 / a2



6 / a3




3
 3 / a2
1/ a
3
3/ a 3/ b
2
 U1 
 6   Q1(1) 

 
 a  
(1)
0  U 2 

   Q2
(1)
( 2) 
2





U
6
3/ a  3
fa   Q3  Q1 
         (1)

12  a  Q4  Q2( 2) 
1 / a  U 4 
 0   Q3( 2) 
3 / a 2  U 5 
 
  

 0   Q4( 2) 
2 / a  U 6 
0
0
0
2
 6 / a3
BoundaryConditions
MatchingConditions
U1  w1(1)  0 , U 2  q1(1)  0 , Q3( 2)  Q4( 2)  0
Q3(1)  Q1( 2)  0 , Q4(1)  Q2( 2)  0
Re du ce dS yste m
6 / a 3  6 / b 3


2 EI 





3 / a 2  3 / b2
2/a  2/b
 6 / a3
3 / a2

6 / a3


 3 / a 3  U1 
6  0
 
   
1 / a  U 2 
fa a  0
      
12 0  0
3 / a 2  U 3 
 
0  0
2 / a  U 4 
Solve System for Primary Unknowns U1 ,U2 ,U3 ,U4
Nodal Forces Q1 and Q2 Can Then Be Determined
Special Features of Beam FEA
Analytical Solution Gives
Cubic Deflection Curve
Analytical Solution Gives
Quartic Deflection Curve
FEA Using Hermit Cubic Interpolation
Will Yield Results That Match Exactly
With Cubic Analytical Solutions
Truss Element
Generalization of Bar Element With Arbitrary Orientation
y
k=AE/L
x
s  sin q , c  cos q
Frame Element
Generalization of Bar and Beam Element with Arbitrary Orientation
w1
w2
q
q1
2
u1 
u2
P1
P2
(1)
(2)
L
M2
M1
V1
V2
 AE
 L

 0

 0

 AE

 L
 0

 0

0
0
12EI
L3
6 EI
L2
6 EI
L2
4 EI
L
0
0

12EI
L3
6 EI
L2
6 EI
L2
2 EI
L


AE
L
0
0
AE
L
0
0
0
12EI
L3
6 EI
 2
L

0
12EI
L3
6 EI
 2
L


6 EI   u   P 
1
 1
L2   w  Q 
2 EI   1   1 
L   q1   Q2 
  u2   P2 
0    
 w2  Q3 

6 EI
 2   q2  Q4 
L 
4 EI 
L 
0
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, 1993
Bickford, 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.