FEM Lecture Notes-4
of 16
-
Upload
macynthia26 -
Category
Documents
-
view
24 -
download
2
description
Finite Element
Transcript of FEM Lecture Notes-4
-
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
