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One-phase Solidification Problem:
FEM approach via Parabolic Variational Inequalities
Ali Etaati
May 14th 2008
Content:
• One-phase Stefan problem-solidification example
• Derivation of a complementarity system
• Parabolic variational inequality
• Finite element method
One-Phase Stefan problem
1
2
)(t )(t )(0 t
)(0),(
)(0),(
)(0),( 0
txtxu
txtxu
txtxu
},)(:{)(
},)(:{)(
},)(:{)(0
txlxt
txlxt
txlxt
)0(0,0)(
),0()0(,0)(0xlxl
xxl
Solidification:
1
2
)(t )(t )(0 t
)()(
0
21
21
tt
tt
),0( TT
(T: final time)
Heat conduction equation (in the solid)
Txut
u,
Freezing temperature
0))(,( xlxu
Llxlxu solidliquid .))](,([
Energy balance (Stefan condition) at )(t
(L is the latent heat)
Solidification (cont’d):
Solidification (cont’d):
Initial and Boundary conditions (temperature distribution):
2
1
00
,0
,0),(
)0(,0
)0(),()0,(
xu
xtxgu
x
xxuxu
Initial enthalpy:
)0(,
)0(),()(
00
0xL
xxuxH
Freezing index:
)0(,),(
)0(),(,0
)0(),(,),(
),(
0
0
0
)(
xdxu
xxlt
xxltdxu
txUt
t
xl
Then:
)(,0,0
,0
),(
0
0
txUU
xU
xxHUt
U
T
T
2
1
0
,0
,),(~
xU
xdxgUt
xxU ,0)0,(
Linear complementarity system:
,0))((
,0
,0)(
0
0
UxHUt
U
U
xHUt
U
a.e. in T
Parabolic Variational Inequalities:
Let
}..:))(,,0({
}..),(:)({)(10
2
10
ToneaGvHTLvK
oneatxGvHvtk
where
)()),(,,0( 222TL
t
GHTLG
and),0(..0 ToneaG
Suppose: )(2 TLf
Define: vdxwvwawvdxvw .),(,),(
Parabolic Variational Inequalities (cont’d)
Problem 1:
Find with
such that
and for almost all and is such that:
for all with a.e. in (0,T).
))(;,0( 10
2 HTLw ))(;,0(/ 22 LTLtw
)0()0,( 0 kwxw
)()(),,0( tktwTt
),(),(),( wvfwvwawvt
w
))(;,0( 10
2 HTLv )()( tktv
Parabolic Variational Inequalities (cont’d)
Problem 2:
Find with such that
and
for all .
Kw ))(;,0(/ 22 LTLtw
)0()()0,( 0 kxwxw
0)},(),(),({0
dtwvfwvwawvt
wT
Kv
Parabolic Variational Inequalities (cont’d)
Equivalence:
),(,~),(,
tKv
twv
then
Consider a solution of Problem 2, for any and 0 ),,0( T
0)}~,()~,()~,({
dtwvfwvwawvt
w
We obtain the solution for Problem 1.
• Clearly a solution of Problem 1 solves Problem 2.
Parabolic Variational Inequalities (cont’d)
.0))((
,
,0
Gwfwt
w
Gw
fwt
w
a.e. in T
Theorem
Solution to Problem 2 satisfies the linear complementarity system:
Parabolic Variational Inequalities (cont’d)
Proof
For any non-negative and so, from Problem 2:
KwvC T ),(0
0)},(),(),({])[(0
dtwvfwvwawvt
wdxdtfw
t
w T
T
Which implies that
,0
fwt
wa.e. in T
Parabolic Variational Inequalities (cont’d)
T
dxdtfwt
wdtwvfwvwawv
t
wT
.])[()},(),(),({00
Hence,0
fwt
wa.e. in T
Proof (cont’d)
Now let Then for any
for sufficiently small so that
)}.,(),(:),{( txGtxwtx TT
KwvC T ),(0
Conversely, by noting that if satisfies the complementarity system, then, for
KwKv
,..,0))(( Tineawvfwt
w
It is then clear that w solves Problem 2.
Finite Element approximation (FEM)
: Space of continuous functions which are linear on each element and which vanish on the boundaries.
hV
General Discretisation by FEM for Problem 1:
hnnnn
hnnn wvfwvwawvkww ),(),(),/)(( 1111
For all )})1(,(:)({1 knxGvVxbvvkv iih
Iiii
nh
(for all interior points)
: a piecewise linear basis function.)(xbi
]1,0[,)1(1 nnn www
find such that11 nh
n kw
Finite Element approximation (cont’d)
In nkff )}({If f is continuous:
kn
nk
hn dtvtf
kvf
)1(
)),((1
),( Otherwise: (for all )
hVv
),(),(
,)(,)(
00
200
0
vwavwa
Hwifww I
(for all )hVv
In any case:
)(,)()( 100
02, HinwwLinftf Tkh
Time marching of the discrete system
(*)0)()(,0,0 111111 nnTnnnn GwzGwz
where
),(ˆ,),(
,ˆ/)( 11
jiijhjiij
nnnnn
bbaAbbM
fwAkwwMz
},,{ nnn Gfw are nodal vectors.
: is a symmetric positive definite matrix which
causes the problem (*) to have unique solution.
AkM ˆ
FEM; Stability
Let
kvvvvAvv
vvvvMuvuvvnnn
k
T
/)(,~ˆ~||||
,,||,~~,,~
122
2
and let’s assume:
)0(0)()( ** kandttfortktk
.0)0,(),(),( ** xGandttfortxGtxG
Or equivalently that
The complementarity problems are equivalent to
0)~~()~( 11 nTn wvZ for all 1~~ nGv (**)
We may take in (**), then nwv ~~
FEM; Stability (cont’d)
Stability theorem:
Providing the stability condition
2)]([)21( 2 hSk
holds when , there is a constant C, independent of space- and time-steps such that:
2/1
}||||{|||||||| 2
0
20212
0
jn
j
njk
n
j
fkwCwwk
FEM; Stability (cont’d)
Lemma:
is stable under the following conditions
}~~~){(
~~ 111 nnnnn GfkwkAIGw
The explicit method
0;,0;01 j
ijijii AjiAkA
0)(),(,0)(),( tGtxGxftxf
0)0()0,(0)(,0)( ''' GxwandtGtG
for which is bounded by}{ nw
0~~~~~ 11 nnnn wwGG
Reference
• Weak and variational methods for moving boundary problems, C M Elliott & J R Ockendon.
