weigted residual methods
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Weigted Residual Methods
Spring 2003 Weighted Residuals 2
Approximate solutions, including FE solutions, can be constructed from governing differential equations. One approach is the Galerkin method. This can be applied to non-structural problems.
Spring 2003 Weighted Residuals 3
Galerkin Method
Notation:
x independent variables, e.g. coordinates of a point
u u x dependent variables, e.g. displacements of a point
u u x approximate solution
f function of x (may be constant or zero)
D differential operat
or
Spring 2003 Weighted Residuals 4
Problem Statement
In domain V:
Du - f 0
with appropriate B.C.
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Problem Statement
Residual in domain V:
R Du - f
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Approximate Solution
th
i
u is a linear combination
of basis functions
u is a typically a polynomial
of n terms whose
i term is muliplied
by a generalized d.o.f. a
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Solution
i
Minimize residual w.r.t. weights
Best Approximation:
W Rdv 0 i 1,2, ,n
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One-Dimensional Example
q cx P
TL
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One-Dimensional Example
x x
2
2
x
duE E
dx
d uAE q 0
dxd
A q 0dx
Spring 2003 Weighted Residuals 10
One-Dimensional Example
2
2
T
d u cx0
dx AE
duAE P at x = L
dx
Spring 2003 Weighted Residuals 11
Exact Solution
3
2
T
T
2
2
d u cx0
dx AEdu
AE P at x
cL
= Ld
P cu x x x
AE 2AE 6A
x
E
Spring 2003 Weighted Residuals 12
Galerkin ProblemTL 2
i 20
ii
d u cxW dx 0
dx AE
duW
da
Spring 2003 Weighted Residuals 13
Integration by Parts
d uv udv vdu
udv uv vdu
Spring 2003 Weighted Residuals 14
Galerkin ProblemT
T T
L 2
i 20
L L2
i i20 0
i
2
2
i
d u cxW dx
dx AE
d u cxW dx W d
d u dudv dx v
dx d
dWu W
x 0dx
x
A
dux
E
d
Spring 2003 Weighted Residuals 15
Galerkin Problem
T
T T
L 2
i 20
L L
ii
00
d uW dx
dx
dWdu duW dx
dx dx dx
Spring 2003 Weighted Residuals 16
Galerkin Problem
T
T T
T T
L 2
i 20
L L
L L
ii i
00
2
i i20 0
dWdu du cxW W dx
d u cxW dx
dx AE
d u cxW dx W dx 0
dx AE
dx dx dx AE
Spring 2003 Weighted Residuals 17
Assumed Function2
1 2
11
1
2 22
2
u a x a x
du dWW x 1da dxdu dW
W x 2xda dx
Spring 2003 Weighted Residuals 18
Integrals
T
T
L
1 2 T
0
L2
1 2 T
0
cx P1 a 2a x x dx L 0
AE AE
cx P2x a 2a x x dx L 0
AE AE
Spring 2003 Weighted Residuals 19
Solution
2T
1
T2
7cLPa
AE 12AE
cLa
4AE
Spring 2003 Weighted Residuals 20
Solution
22T T
2T T
7cL cLPu x x x
AE 12AE 4AE
7cL cLdu PE x
dx A 12A 2A2
Spring 2003 Weighted Residuals 21
Solution2
23T
2T T7cL cLPu x x x
AE 12AE 4AE
cLP cu x x x
AE 2AE 6AE
Spring 2003 Weighted Residuals 22
Comparison of Results
Quantity Location Exact Galerkinu L/2 0.2292 0.2292u L 0.3333 0.3333u,x 0 0.5000 0.5833u,x L/2 0.3750 0.3333u,x L 0.0000 0.0833
Spring 2003 Weighted Residuals 23
Galerkin FEM Formulation:Uniform Bar, Axial Load
2
,xx2
,x
d uAE q(x) AEu q(x) 0
dxF AEu
L
q q x
x,u
A,E
Spring 2003 Weighted Residuals 24
Galerkin FEM Formulation:Uniform Bar, Axial Load
T
1 2
u N d
L x xN
L L
d u u
Spring 2003 Weighted Residuals 25
Galerkin FEM Formulation:Uniform Bar, Axial Load
1 1 2 2
i ii
u N u N u
uW N
d
L x xN
L L
Spring 2003 Weighted Residuals 26
Galerkin FEM Formulation:Uniform Bar, Axial Load
jels
LN
i ,xxj 1 0
N AEu q dx 0
Spring 2003 Weighted Residuals 27
Galerkin FEM Formulation
j
j
L
i ,xx
0
LL
i ,x i,x ,x00
For EA constant:
N AEu dx
N AEu N AEu dx
Spring 2003 Weighted Residuals 28
B.C.
,xF=AEu
For ends of the element
Spring 2003 Weighted Residuals 29
Galerkin FEM Formulation:Uniform Bar, Axial Load
jels
jels els
LN
i ,xxj 1 0
LN NL
i,x ,x i i ,x 0j 1 j 10
N AEu q dx 0
N AEu N q dx N AEu 0
Spring 2003 Weighted Residuals 30
,x
1,x
2
B N
u1 1u B d
uL L
Spring 2003 Weighted Residuals 31
jels
els
els
els els
LN
i,x ,x ij 1 0
NL
i ,x 0j 1
LNT
jj 1 0
LN N LT T
0j 1 j 10
N AEu N q dx
N AEu 0
B AE B dx d
N qdx N F 0
Spring 2003 Weighted Residuals 32
els
els
els
LNT
jj 1 0
LNT
e jj 1 0
N LT
0j 1
k B AE B dx
r N qdx
P N F 0
Spring 2003 Weighted Residuals 33
Beam Dynamics
4 2
L4 2
,xxxx L
d v d vEI (x) 0
dx dtEIv v 0
Spring 2003 Weighted Residuals 34
B.C.
,xx B
,xxx B
EIv -M =0
EIv -V =0
Spring 2003 Weighted Residuals 35
Shape Functions
3 2 31 3
3 2 2 32 3
3 23 3
3 2 24 3
1ˆ ˆN 2x 3x L L
L1
ˆ ˆ ˆN x L 2x L xLL1
ˆ ˆN 2x 3x LL1
ˆ ˆN x L x LL
Spring 2003 Weighted Residuals 36
Shape Functions for Beam Element
-0.500
0.000
0.500
1.000
0
N1 N3
N2
N4
L
Spring 2003 Weighted Residuals 37
1 1 2 1 3 2 4 2
1
1
2
2
i ii
v N v N N v N
v
dv
vW N
d
Spring 2003 Weighted Residuals 38
jL
T
,xxxx L
0
N EIv v dx 0
Spring 2003 Weighted Residuals 39
LT
,xxxx
0
L LT TT
,xx ,xx ,xxx ,x ,xx00
EI constant Integration
(by parts twice!)
N v dx
N v dx N v N v
Spring 2003 Weighted Residuals 40
jLT
,xxxx L
0
LT T
,xx ,xx L
0
LTT
,xxx ,x ,xx0
N EIv v dx 0
N EI v N v dx
N v N v 0
Spring 2003 Weighted Residuals 41
L
T T
,xx ,xx L
0
LTT
,xxx ,x ,xx0
N EI v N v dx
N v N v
Spring 2003 Weighted Residuals 42
,xx B
,xxx B
LT T
,xx ,xx L
0
LTT B B
,x0
EIv -M =0
EIv -V =0
N EI v N v dx
V MN N
EI EI
Spring 2003 Weighted Residuals 43
,xx
,xx
v N d
v B d
B N
Spring 2003 Weighted Residuals 44
LT
0
LT
L
0
LTT B B
,x0
k B EI B dx
m N N dx
V MR N N
EI EI
Spring 2003 Weighted Residuals 45
Heat Flow in a Bar
,x
dAkT
dx
Af Af d Af
Spring 2003 Weighted Residuals 46
T
eT N T
Spring 2003 Weighted Residuals 47
1 1 2 2
i ii
T N T N T
TW N
d
L x xN
L L
Spring 2003 Weighted Residuals 48
L
T
,x
0
0dN AkT dx
dx 0
Spring 2003 Weighted Residuals 49
L
T T
,x ,x
0
0N AkT dx N Af
0
Spring 2003 Weighted Residuals 50
,x ,x e
LT 1 1
,x ,x e2 20
T N T
A FN Ak N T dx
A F
Spring 2003 Weighted Residuals 51
LT
T ,x ,x e
0
T
k N Ak N T dx
1 1k kA
1 1
Spring 2003 Weighted Residuals 52
Two Dimensional Problems
x ,x y ,y
x ,x y ,y B
In volume V: k k Q 0x y
is knownOn boundary S: either
lk mk f 0
Spring 2003 Weighted Residuals 53
Poisson’s Equation
x ,x y ,y
x y
2
k k Q 0x y
If k k k constant:
k Q 0
Spring 2003 Weighted Residuals 54
Shape Functions
eN
Shape functions depend on 2D element:L
CST, LST, Quad, Quadratic, etc.
Spring 2003 Weighted Residuals 55
Galerkin Residuals
T
x ,x y ,y
0
0N k k Q dxdy
x y
0
Spring 2003 Weighted Residuals 56
Integration by Parts
T
x ,x
T
,x x ,x
T
x ,x
N k dxdyx
N k dxdy
N k ldS
Spring 2003 Weighted Residuals 57
Integration by Parts
T
y ,y
T
,y y ,y
T
y ,y
N k dxdyy
N k dxdy
N k mdS
Spring 2003 Weighted Residuals 58
,x ,x e
,y ,y e
N
N
Spring 2003 Weighted Residuals 59
Galerkin Residuals
TT
,x x ,x ,y y ,y e
T T
B
N k N N k N dxdy
N Qdxdy N f dS
Spring 2003 Weighted Residuals 60
Galerkin Residuals
TT
,x x ,x ,y y ,y
T T
B
k N k N N k N dxdy
r N Qdxdy N f dS
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