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Φ − Φ = γv · (G − G ) ,
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7 @`˳= 7‰Ä7 Ã8= QŸÂ>7 Ì>?A2KÒ1Ço2¥Ã8ͳ7‰H
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0
Φ(r, t) = γB0 T + γv · G ,
0
1
1
2
1
2
2
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κ Γ
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and κ = ∇ ·
.
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CÈ 1Ê
φ + u · ∇φ = 0 ⇔ φ + u |∇φ| = 0 ,
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φ(x, t)
= 0 on Γ

> 0 outside Γ .
t
t
n
n
c
x
√
y
z
T
q
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|v| =
v·v =
vx2 + vy2 + vz2 .
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v
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vx vx vy vx vz
vx Mv = vvT = vy  vx vy vz = vx vy vy2 vy vz  .
vz
vx vz vy vz vz2
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2
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M
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Z
Z
CÈ 1Ê
−
R(x)dS
,
E(Γ ) =
χ(v)dx
| Ω {z
| Γ {z
c
}
}
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χ(v) = χ(|v|) = H(|v| − δ) = H (|v|) ,
Õͳ?A58? δ ∈ R Ò³PO?A58? H 7‰HOML?ÚE2³?XM$0>H¨7 2ªà áXÏkâ
flow outside the surface
c
discontinuities at the surface
2
δ
+

1
H(x) = 0
1
2 [1
+
x
+
1
π
x>
x < −
sin( πx
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Èná>ákÊ
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CÈ 1Ê
Ó?XÄ;>@`?XH
Z
Z
Ènákî>Ê
E(φ) = χ(v)H(φ)dx − R(x)|∇φ|δ(φ)dx ,
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7 2CÃ8ͳ?–HJ?XÄ;>2EMC7 21Ã8?AÂ>503=K7‰H
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Ã8ͳ?OHJG³5JÑÉ0>Ä?¨0358?X0œ?A= ?A@`?A21Ã6032EM Ωδ(·) 7‰H(Ã8ͳ?¨;>2³?ML|∇φ|δ(φ)dx
7 @`?A2EHJ7 ;>2E03=³ML?A= Ã0œÑƒG³2EÄrÃ87 ;>2KdÒ D³58;>@#Ã8ͳ?
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ML?XH8Ä?A21ÃXBL7‰H¨Ã8ͳ?A2
ÈnáXÏ1Ê
φ = −(χ + κR + ∇R · n)|∇φ| ,
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c
t
t
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Discontinuity measure
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
324 5 ’„³’ Z(T1[jX_AW ›1[
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0.8
10
0.7
20
0.6
0.5
30
0.4
40
0.3
0.2
50
0.1
60
10
20
30
40
50
60
0
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