mane 4240 & civl 4240 introduction to finite elements
DESCRIPTION
MANE 4240 & CIVL 4240 Introduction to Finite Elements. Prof. Suvranu De. Numerical Integration in 1D. Reading assignment: Lecture notes, Logan 10.4. Summary: Newton-Cotes Integration Schemes Gaussian quadrature. y. F. x. x. x=L. x=0. 2. 1. d 2x. d 1x. Axially loaded elastic bar. - PowerPoint PPT PresentationTRANSCRIPT
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MANE 4240 & CIVL 4240Introduction to Finite Elements
Numerical Integration in 1D
Prof. Suvranu De
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Reading assignment:
Lecture notes, Logan 10.4
Summary:
• Newton-Cotes Integration Schemes•Gaussian quadrature
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Axially loaded elastic bar
A(x) = cross section at xb(x) = body force distribution (force per unit length)E(x) = Young’s modulusx
y
x=0 x=Lx
F
2
1
TB EB dxx
xk A
Element stiffness matrix2
1i jB EB dx
x
ij xk A
1 2
d1xd2x
For each elementx1
x2
where i
( )B idN x
dx
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dxbN2
1x
x
Tbf
Element nodal load vector
dxbN2
1i
x
xibf
2 2 2
1 1 1
T
2 2
2 1 2 1
1 1 1 11 1B EB dx AEdx AEdx
1 1 1 1x x x x
x x x
x x xA
Only for a linear finite element
Question: How do we compute these integrals using a computer?
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Any integral from x1 to x2 can be transformed to the following integral on (-1, 1)
1
1)( dfI
Use the following change of variables
21 2
1
2
1xxx
Goal: Obtain a good approximate value of this integral1. Newton-Cotes Schemes (trapezoidal rule, Simpson’s rule, etc)2. Gauss Integration Schemes
NOTE: Integration schemes in 1D are referred to as “quadrature rules”
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Trapezoidal rule: Approximate the function f by a straight line gthat passes through the end points and integrate the straight line
f
-1 1
f
fg
)1(2
1)1(
2
1)( ffg
)1()1()()(1
1
1
1
ffdgdfI
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•Requires the function f(x) to be evaluated at 2 points (-1, 1) • Constants and linear functions are exactly integrated• Not good for quadratic and higher order polynomials
How can I make this better?
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Simpson’s rule: Approximate the function f by a parabola gthat passes through the end points and through f(0) and integrate the parabola
( 1) (1 )( ) ( 1) (1 )(1 ) (0) (1)
2 2g f f f
)1(3
1)0(
3
4)1(
3
1)()(
1
1
1
1
fffdgdfI
f
-1 1
f
fg
f(
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•Requires the function f(x) to be evaluated at 3 points (-1,0, 1) • Constants, linear functions and parabolas are exactly integrated• Not good for cubic and higher order polynomials
How to generalize this formula?
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Notice that both the integration formulas had the general form
Trapezoidal rule:M=2
11
11
22
11
W
W
Simpson’s rule:M=3
Accurate for polynomial of degree at most 1 (=M-1)
13/1
03/413/1
22
22
11
W
WW Accurate for polynomial of
degree at most 2 (=M-1)
M
iii fWdfI
1
1
1)()(
Weight Integration point
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Generalization of these two integration rules: Newton-Cotes
• Divide the interval (-1,1) into M-1 equal intervals using M points• Pass a polynomial of degree M-1 through these M points (the value of this polynomial will be equal to the value of the function at these M-1 points)• Integrate this polynomial to obtain an approximate value of the integral
f
-1 1
f
f
g
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With ‘M’ points we may integrate a polynomial of degree ‘M-1’exactly.
Is this the best we can do ?
With ‘M’ integration points and ‘M’ weights, I should be able to integrate a polynomial of degree 2M-1 exactly!!Gauss integration rule
See table 10-1 (p 405) of Logan
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Gauss quadrature
M
iii fWdfI
1
1
1)()(
Weight Integration point
How can we choose the integration points and weights to exactly integrate a polynomial of degree 2M-1?
Remember that now we do not know, a priori, the location of the integration points.
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Example: M=1 (Midpoint qudrature)
How can we choose W1 and 1 so that we may integrate a (2M-1=1) linear polynomial exactly?
10)( aaf
)()( 11
1
1 fWdfI
0
1
12)( adf
But we want
101011
1
1)()( WaWafWdf
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Hence, we obtain the identity
1111002 WaWaa
For this to hold for arbitrary a0 and a1 we need to satisfy 2 conditions
0:2
2:1
11
1
WCondition
WCondition
i.e., 0;2 11 W
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f
-1 1
f
g
For M=1 )0(2)(1
1fdfI
Midpoint quadrature rule: • Only one evaluation of fis required at the midpoint of the interval.• Scheme is accurate for constants and linear polynomials (compare with Trapezoidal rule)
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Example: M=2
How can we choose W1 ,W2 1 and 2 so that we may integrate a polynomial of degree (2M-1=4-1=3) exactly?
33
2210)( aaaaf
)()()( 2211
1
1 fWfWdfI
20
1
1 3
22)( aadf
But we want
322
3113
222
211222111210
2211
1
1)()()(
WWaWWaWWaWWa
fWfWdf
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Hence, we obtain the 4 conditions to determine the 4 unknowns (W1 ,W2 1 and 2 )
0:4
3
2:3
0:2
2:1
322
311
222
211
2211
21
WWCondition
WWCondition
WWCondition
WWCondition
Check that the following is the solution
3
1;
3
1
1
21
21
WW
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f
-1 1
For M=2 )3
1()
3
1()(
1
1ffdfI
• Only two evaluations of fis required.• Scheme is accurate for polynomials of degree at most 3 (compare with Simpson’s rule)
)3
1(f )
3
1(f
**
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Exercise: Derive the 6 conditions required to find the integration points and weights for a 3-point Gauss quadrature rule
Newton-Cotes Gauss quadrature
1. ‘M’ integration points are necessary to exactly integrate a polynomial of degree ‘M-1’ 2. More expensive
1. ‘M’ integration points are necessary to exactly integrate a polynomial of degree ‘2M-1’ 2. Less expensive3. Exponential convergence, error proportional to M
M
2
2
1
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Example
1 3 2
1f( )d f( )I where
Exact integration
2
3I Integrate and check!
Newton-Cotes
To exactly integrate this I need a 4-point Newton-Cotes formula. Why?
Gauss To exactly integrate this I need a 2-point Gauss formula. Why?
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1
1f( )d
1 1f(- ) f( )
3 32
3
I
Gauss quadrature:
Exact answer!
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Comparison of Gauss quadrature and Newton-Cotes for the integral
1
1)2cos( dxxI
Newton-Cotes
Gauss quadrature
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In FEM we ALWAYS use Gauss quadrature
Linear Element1 2
1 1
1 1 1T
1 1 1
1 1 1 11 1B EB d AEd AEd
1 1 1 14 4k A
Stiffness matrix
dbN1
1 Tbf dbN
1
1 iibf
Nodal load vector
Usually a 2-point Gauss integration is used. Note that if A, E and b are complex functions of x, they will not be accurately integrated
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Quadratic Element
2
1 1T T
1 1B EB d E B B dk A A
Stiffness matrix
Nodal shape functions 1
1 10
3
1
22
3
( ) 12
( ) 1
( ) 12
N
N
N
You should be able to derive these!
Assuming E and A are constants
31 2 1 1B 2 1 2 2 1
2 2
dNdN dNdN
d d d d
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1 1T T
1 1
2 2
12
1 22
B EB d E B B d
1/ 2 2 1/ 2 1/ 4
2 1/ 2 4 2 1/ 2
1/ 4 2 1/ 2 1/ 2
k A A
AE d
Need to exactly integrate quadratic terms.Hence we need a 2-point Gauss quadrature scheme..Why?