Post on 03-Mar-2016
description
VARIATIONAL FORMULATION Strong Form Weak Form
Linear Equation
Inner Product
Expansion into Set of Basic Functions
Linear Equation in Terms of Basic Functions
Method of Weighted Residuals
Method of Weighted Residuals
Galerkin Method
Summary of Galerkin Method
Commonly Encountered Equations
Variational Formulations
One-Dimensional Model Problem
{0})( )0(, {1})( )1(
0,1on 0,
g
hx
xx
hugu
fuStrong Form:
Weak Form:
dxw f (w,f)
dx,uw, A(w,u) wherewhfwuwA
xx
1
0
1
0
)0(),(),(V wallfor such that Su Find
Galerkin:
),()0(),(),( V wallfor such that V vFind hhhh
hhhhhh gwAhwfwvwA
Matrix:
ne
),....(n-ee
g-k
h dxfN f
dxNN kwhere
} {fA{F}], {F} [kAe [K]{F} wher[K]{d}
e
a
a
ea
e
a
eb,xa,x
e
ab
en
enodal
en
e
121
0
2
1
11
Classical Linear Elastostatics
on
on
on 0
),(ij
gi
,
lkijklklijkl
hiijij
ii
ijij
uccwherehn
gu
fu
Strong Form:
Weak Form:
sd
h
n
i ii ii
(k,j)ijkl(i,j)
d hw (w,f) d fw f) (w,
ducw A(w,u) where (w,h)(w,f)A(w,u)
1
V wallfor such that Su Find
Galerkin:
),(),(),(),( V wallfor such that V vFind hhhh
hhhhhh gwAhwfwvwA
Matrix:
jbnqianp
gk dhN dfN f
dDNB k , ek e kwhere } {fA{F}F}], { [kAe [K] {F} wher[K]{d}
eded
n
q
e
qe
pqe
iae
iae
p
eba
e
abje
abT
ie
pq
en
enodal
en
e
en
elel
)1( ,)1(
1
11
enn
a
e
aa dxBxDx1
)()()( :point aat stress
Classical Linear Heat Conduction
, q on
on
on 0
i
g
,
jij
hii
ii
ukwherehnqgu
fq
Strong Form:
Weak Form:
dhw (w,f) dw f f) (w, dukw A(w,u) where
(w,h)(w,f)A(w,u)
jiji
,,
V wallfor such that Su Find
Galerkin:
),(),(),(),( V wallfor such that V vFind hhhh
hhhhhh gwAhwfwvwA
Matrix:
jbnqianp
gk dhN dfN f
dDNB kwhere } {fA{F}F}], { [kAe [K] {F} wher[K]{d}
eded
n
b
e
be
abe
ae
a
e
a
eba
e
ab
en
enodal
en
e
en
elel
)1( ,)1(
1
11
enn
a
e
aa dxBxDxq1
)()()( :point aat r flux vectoheat