Lecture 27
General variation inequality:
H-Hilbert’s Space:
Properties:
1. Vector Space
2. Inner Product
a. < > :Η×Η →R
b. For Rn , inner product is defined as: <xy>=xTy
c. For Function Space Η1 : fg>=0Lfxgx+ fg
3. u=<u,u12
COMPLETE:
Cauchy criteria characterize convergence:
um→u in Η
iff um-u ↓0
Completeness Property:
um→ u
iff um-un < ∈ for n,m≥N∈
a : Η×Η→R
l : Η→R
au,ϕ=lϕ for all ϕ
Ju=12au,u-lu
Η=Η01
au,ϕ=T0Luxϕx
lϕ=0Lfϕ
K is contained in H:
1. Convex:
Any line joining two points should lie in the set K.
2. Closed:
K contains its boundary.
Definitions:
Energy: u∈K
Ju=min Iv v∈K
Variational Inequality :
au,ϕ-u≥lϕ-u
u+tϕ-u=1-tu+tϕ
Ju=Ju+tϕ-u
Assumptions:
· a:Η×Η →R Symmetric Bi-linear
· l:Η→R Linear
Ju+tϕ-u=⋯
⋮
=Ju+tau,ϕ-u+t2aϕ-u,ϕ-u-tlϕ-u
Ju+tϕ-u-Jut=au,ϕ-u-lϕ-u+taϕ-u,ϕ-u≥0
Let t=1
Jϕ-Ju=au,ϕ-u-lϕ-u+aϕ-u,ϕ-u
≥0 +
Jϕ-Ju≥0
EXISTENCE:
Use Energy and Cauchy criteria
Assume:
· H is Hilbert space
· K is convex
· K is closed
· a(u,v) is symmetric
· a(u,v) is bilinear
· a(u,v) is coercive
· a(u,v) is bounded
· l(u) is linear
· L(u) is bounded
a:Η×Η →R Symmetric Bi-linear
c1|u|2≤a(u,u)≤c2|u|2
Coercive Bounded
l:H R
Linear: |lu|≤c3|u|
Bounded
Outline
-M≤Jv ∀ v ε K
d≤Jum & Jum↓d
2*Ju+v2+au-v2,u-v2=Ju+J(v)
Where u um and v vm
Jum Jun
d d+ ε
c1um-un22≤aum-un2,um-un2≤d+ε+d+ε-2d=2ε
So, um u
· FEM for membrane with an obstacle
Obstacle region where membrane adheres to the obstacle.
Obstacle: Ψ0(x,y)
u(x,y)≥Ψ0(x,y)
If u is NOT on the obstacle, then :-Tuxx+uyy=f≤0
for
u(x,y)≥Ψ0(x,y)
i.e. above the obstacle.
au,∅=T(ux∅x+uy∅y)
l∅=f∅
H=u ∈ L2(Ω)|(ux2+uy2+u)<∞
K=u ∈H|u≥Ψ0,u≡given on ∂Ω
A=a∅i,∅j
d=l(ϕi)
K=u ∈ Rn|uj≥Ψoj
H→R
au,∅→uTA∅
l(∅)→dT∅
Take A→reduced system matrix
Store the non-zero terms in the reduced stiffness matrix to improve the SOR. Otherwise, all the extra terms get solved.
um+13 from G.S
um+2/3 from SOR
um+1 from projection
=maxΨ0,ujm+2/3
Only one extra line is required in the code for SOR to make it SOR projection.
Use fem2dsurelt.m to find surrounding elements and surrounding nodes.