introduction to fem - university of tulsapersonal.utulsa.edu/~meh097/lecture2.pdfrrm (continued) 1....
TRANSCRIPT
![Page 1: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/1.jpg)
Introduction to FEM
Session 2 (01/17/2013)
![Page 2: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/2.jpg)
Variational Methods of ApproximationNumerical Methods which utilize approximate solution
to the differential equations in an attempt to minimize the residual error. Techniques that attempt to minimize the weighted residual error over the domain are called WEIGHTED RESIDUAL METHODS.
The various methods differ from each other in the choice of the weight function, approximate function, and integral formulation used.
Assume an approximate solution for the dependent variable
N
jjjN xCuxu
10)()(
![Page 3: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/3.jpg)
Classic Variational Methods
• RAYLEIGH-RITZ (based on weak form)• PETROV-GALERKIN (based on integral form)• GALERKIN (based on integral form)• LEAST SQUARES (based on integral form)• COLLOCATION
![Page 4: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/4.jpg)
Rayleigh-Ritz Method (RRM)• Uses weak form of the problem
– Equivalent to the original differential equation and includes the NBCs
– Places less restrictive continuity requirements on the dependent variables
• For a symbolic development of the method, the weak form of a BVP may be written as
• In the RRM we choose the weight function to be the same family as the approximate functions
0)(, wluwB
Niw i ,...,1
![Page 5: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/5.jpg)
RRM (cont.) On substitution of the weight and approximate function
If the functional B(0,0) is bilinear we may separate the arguments of the approximate solution and factor out the Ritz coefficient
On substitution and rearranging
B is typically symmetric
01
, ( ) ( )N
i j j ij
B C x l
1,...,i N
0 01 1
, ( ) , ,N N
i j j i j j ij j
B C x B C B
01
, ,N
i j j i ij
B C l B
FCBFCB ij
N
jij
1
![Page 6: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/6.jpg)
RRM (continued)1. φ0 must satisfy the specified EBCs2. φi must satisfy the homogenous form of
specified EBCs3. φi must be sufficiently differentiable as
required by the weak form4. φi must be a linearly independent set
Rows/columns of B must be linearly indepnednt for a solution to exist (necessary for [B]-1 to exist)
5. φi must be a complete setContains all terms of the lowest order admissible up to
highest order desired
![Page 7: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/7.jpg)
RRM (continued)
Satisfies EBCs
Does not satisfy homogenous form of EBCs
not complete set. can’t generate linear terms
not linearly independent
satisfies all conditions
xL
100
, , : 0, , : (0) 0, ( ) 0jii j
ddB a dx L EBC u u L
dx dx
12
2
x
x
21
32
x x L
x x L
1
2 10
x x L
x x L
1
22
x x L
x x L
![Page 8: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/8.jpg)
Methods of Weighted ResidualsMethods based on the weighted-integral form of the
differential equationsMethods maybe generally described by considering
the operator equation A(u)=f in ΩA() : Linear or Nonlinear operator, often a differential
operator acting on the dependent variable Linear operator satisfies A(αu+βv)=αA(u)+βA(v)
Linear in dependent variable uNonlinear
f: Function of independent variables
( ) d duA u a cu
dx dx
( ) d duA u u
dx dx
![Page 9: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/9.jpg)
MWR (continued)
• Once again, we will seek the approximate solution
• we define the residual of the approximation as
• The parameters Cj are determined by requiring the residual to vanish in a weighted-integral sense
• Ψi: weight function
N
jjjN xCuxu
10)()(
01
( ) ( )N
N j jj
R A u f A C x f
0Rdi
![Page 10: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/10.jpg)
MWR (continued)• The requirements on Φ0 and Φj for the weighted-residual
methods are different from those for Rayleigh-Ritz• Differentiability requirements are dictated by the
weighted-integral statement. Thus, Φj must have nonzero derivative up to the order appearing in the operator A().
• Unlike the Rayleigh-Ritz method (based on weak form), the approximate solution uN must satisfy both EBCs and NBCs
• Φ0 satisfies all specified boundary conditions• Φj satisfies homogenous form of all specified boundary
conditions
![Page 11: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/11.jpg)
Petrov-Galerkin MethodThe approximate function Ψi are not equal to the
approximation functions Φ.
When A() is linear in its arguments
[A] is not symmetric
01
0 0N
i i j jj
Rd A C f d
1
N
ij j ij
A C F A C F
01
N
i j j ij
A d C f A d
![Page 12: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/12.jpg)
Galerkin MethodThe approximate function Ψi are equal to the
approximation functions Φ.With a linear operator A()
[A] is not symmetricAlthough both the Rayleigh-Ritz and the Galerkin
methods both assume the same form for the weight function, the two methods are different due to the former using the weak form and later, the weighted integral.
The two methods yield the same solution when the same approximation function are used and all BCs are EBCs.
1
N
ij j ij
A C F
0,ij i j i iA A d F f A d
![Page 13: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/13.jpg)
Least Square Method Parameter Cj of the approximate solution are
determined by minimizing the integral of the square of the residual
The minimal is found asDenoting this is a weighted-residual method If A() is a linear operator
[A] is symmetric
2R d
2 0 0
i i
RR d R d
C C
i
i
R
C
i ii
RA
C
0,ij i j i iA A A d F A f A d
1
N
ij j ij
A C F
![Page 14: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/14.jpg)
Collocation MethodUnknown parameters Cj are found by requiring the
residual to be identically zero at N selected points Xi, i=1,…N in the domain Ω
Choosing N distinct points will yield N equations for the N unknowns R(Xi,Cj)=0
This method is a weighted-residual method with the weight function chosen as the Dirac delta function. Ψi=δ(X-Xi)
, , 0i i j i jRd X X R X C d R X C
fdXfX )(
![Page 15: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/15.jpg)
Variational Methods of ApproximationConsider the following 2nd order BVP
With algebraic polynomials find the two-parameter approximation and compare with exact solution using:
Rayleigh-RitzGalerkin Petrov-GalerkinLeast-SquaresCollocation
2
0 1
0 0 (1) 0 10
d dua x q x
dx dx
u q x u a
41120exactu x x
![Page 16: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/16.jpg)
![Page 17: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/17.jpg)
RAYLEIGH-RITZ METHOD
Uses the weak form of the DEUses approximation solutionWe previously found
Recall that the weight functions must vanish at boundary points where EBCs are specified. That is, w satisfies the homogenous form of the specified EBCs. u(0)=u(1)=0; w(0)=w(1)=0
01
( )N
N j jj
u u C x
00
0 0L
x L x
dw du du dua wq dx w L a w a
dx dx dx dx
![Page 18: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/18.jpg)
RRM (continued)
The weak form becomesUtilizing the example given in previous lecture,
the admissible set for a 2-parameetr Ritz approximation was
The weak form may be cast in functional form
0
0L dw du
a wq dxdx dx
21 20, ,x x L x x L
1
0
1
0
, ( ) 0
, ,
B w u l w
dw duB w u a dx Bilinear Symmetric Functional
dx dx
l w wqdx Linear Functional
![Page 19: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/19.jpg)
RRM (continued)The weight function is chosen to be the same family as
the approximation functions w=Φi.On substitution and rearranging
1
001
1 1 000 0
, , ,
, , 0
Nji
ij j i ij i i ij
ii i i
ddB C F B a dx F l B
dx dx
d dl qdx B a dx
dx dx
21 2
2 21 2
0, ,
2 1, 3 2 , 10,
x x L x x L
x x x a q x
![Page 20: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/20.jpg)
![Page 21: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/21.jpg)
![Page 22: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/22.jpg)
![Page 23: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/23.jpg)
![Page 24: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/24.jpg)
![Page 25: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/25.jpg)
![Page 26: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/26.jpg)
![Page 27: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/27.jpg)
![Page 28: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/28.jpg)
![Page 29: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/29.jpg)
![Page 30: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/30.jpg)
![Page 31: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/31.jpg)
![Page 32: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/32.jpg)
![Page 33: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/33.jpg)
![Page 34: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/34.jpg)
![Page 35: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/35.jpg)
![Page 36: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/36.jpg)
![Page 37: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/37.jpg)
![Page 38: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/38.jpg)
![Page 39: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/39.jpg)
![Page 40: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/40.jpg)
![Page 41: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/41.jpg)
![Page 42: Introduction to FEM - University of Tulsapersonal.utulsa.edu/~meh097/Lecture2.pdfRRM (continued) 1. φ 0 must satisfy the specified EBCs 2. φ i must satisfy the homogenous form of](https://reader034.vdocument.in/reader034/viewer/2022051205/5adc9db37f8b9a9a768bc471/html5/thumbnails/42.jpg)