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7/21/2019 Lecture note - XFEM and Meshfree_2.pdf
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f i
f ij
giI
I
f i = f f ij = f (·)
f igi = r = f ·g f ijklgkl = rij f : g = r
⊗ f igj = rij f ⊗ g = r
× f × g = ǫijk f i gk ǫijk
gi = (g1, g2, g3, g12, g13, g23) gij
Ω Γ Ω0 Γ0
x = φ(X, t),
x X
u(X, t) = x − X = φ(X, t) − x,
v(X, t) = ∂ u(X, t)
∂t = u
a(X, t) = ∂ 2u(X, t)
∂t2 = u
u v a
0
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a(X, t) = ∂ v(X, t)
∂t +
∂ vi(x, t)
∂xj
∂xi(X, t)
∂t
a(X, t) = ∂ v(X, t)
∂t +
∂ vi(x, t)
∂xjv
F = ∂ x∂ X
ǫ = ∂ u
X = I − F
D = 0.5
L + LT
L = vi,j = F · F−1
E = 0.5 F
T
F − I
σ E
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[[(·)]]
∂D(·)∂t , (·)
∂ (·)∂ X , ∇, (·),i
S
h
u
t
c
P
L
AL
std
enr
blnd
lin
(e)
0
max
min
ext
int
Q
a, b
diag
kin
E
G
K I , K II
x, x
X, X
u, u
d
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v, v
a, a
t, t
n
b
p, p
m, m
M, M
w
W
V
A
h
R
f
F
r
P, P
K
N, N
B
C
I
J
e
r, s
S
H
S
λ,λ
Λ, Λ
Π
β
∆
β
κ
K
ǫijk
ǫ, ǫ
σ,σ
σθθ
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ψ
Ψ
φ
Φ
δ,δ
ξ, η
Ω
Γ
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global
local
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•
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•
•
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X
ΦJ (X) p(X) uJ = p(XJ )
ΦJ (X) uJ = ΦJ (X) p(XJ ) = p(X)
completeness
reproducing conditions
J
ΦJ (X) = 1
J
ΦJ (X) X J = X J
ΦJ (X) Y J = Y
J
ΦJ (X) X Ji = X i
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J
ΦJ,X(X) = 0J
ΦJ,Y (X) = 0
J
ΦJ,X(X) X J = 1J
ΦJ,Y (X) X J = 0 J
ΦJ,X(X) Y J = 0J
ΦJ,Y (X) Y J = 1
J
ΦJ,i(X) = 0 J
ΦJ,i(X) X Jj = δ ij
ΦJ (x)
uJ = 1
J
ΦJ (x) = 1
partition of unities
D
Dt
I ∈S
mI vI
=
I ∈SmI vI = 0
mI v
mI vI = −J ∈S
∇ΦI (XJ ) · σ(XJ ) wJ
ΦI (XJ )
wJ
I ∈S
mI vI = −I ∈S
J ∈S
∇ΦI (XJ )·σ(XJ ) wJ = −J ∈S
I ∈S
∇ΦI (XJ )·σ(XJ ) wJ = 0
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I ∈S
∇ΦI (XJ ) = 0
D
Dt I mI
vI ×
XI = I m
I vI ×
XI
+ vI ×
vI
=0 = 0
×
D
Dt
I
mI vI × XI
=I
ǫijk
J
ΦI,m(XJ ) σmj(XJ )wJ
X Ik
ǫijk X Ik k − th
I
ǫijkJ I ΦI,m(XJ )X Ik δmk
σmj(XJ )wJ = ǫijkδ mkJ σmj(XJ )wJ
=J
ǫijmσmj(XJ ) =0
wJ = 0
k
k > 0
max i
|u(X i) − ui| ≤ Chk
C
h
Cn
n
h
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I
K
Support size of particle I
R_KR_I
lim h0→0
W (XI − XJ , h0) = δ (XI − XJ )
Ω0
W (XI − XJ , h0)dΩ0 = 1
W (XI − XJ , h0) = 0 ∀ XI − XJ ≥ R
δ h0
R
h0
h0
x x
h0
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h0
x
x
W (XI − XJ , h0) = W (XJ − XI , h0)
∇0W (XI − XJ , h0) = −∇0W (XJ − XI , h0)
W (X) = W 1D(X),
W (X) = W 1D(|X 1|) W 1D(|X 2|) W 1D(|X 3|)
X = (X 1, X 2, X 3) X =
X 21 + X 22 + X 23
=
C hD1 − 1.5z2 + 0.75z3 0 ≤ z < 1
C 4 hD
(2 − z)3
1 ≤ z ≤ 20 z > 2
D
z = r/h0
C
=
2/3 D = 1
10/(7 π) D = 21/π D = 3
h0
z
z = ||XI − XJ ||
∂W
∂X iJ =
∂W
∂ z
∂ z
∂X iJ
∂W ∂ z
=
3C hD+1
−z + 0.75z2
0 ≤ z < 1−3C
4 hD+1 (2 − z)2
1 ≤ z ≤ 20 z > 2
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−3 −2 −1 0 1 2 30
0.2
0.4
0.6
0.8
1
1.2
1.4
h/x = 1
−1 −0.5 0 0.5 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
u(x)u
rho(x)
h/x = 1
−3 −2 −1 0 1 2 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
h/x = 2
−1 −0.5 0 0.5 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
u(x)u
rho(x)
h/x = 2
−3 −2 −1 0 1 2 30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
h/x = 4
−1 −0.5 0 0.5 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
u(x)u
rho(x)
h/x41
u(x) = 1 − x2
x = 0.5
ωi = x
h/x = 1, 2, 4
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=
1 − 6z2 + 8z3 − 3z4 0 ≤ z < 1
0 1 ≤ z
=
x − xI ≡ r linear
z2 log z thin plate spline
e−z2/c2 Gaussian
z2 + R2q
multipolar
c R q
W J (x) = W (x − xJ (t), h(x, t))
h
h
ht+∆t = ht + h ∆t
h = 1/3
∇ · v
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v
h
F
h = h0 F
h
h0
h
W J (X) = W (X − XJ , h0)
xJ (t)
v(x, t) =I ∈S
W (x − xI (t)) vI (t),
a =I ∈S
W (x − xI (t)) vI + ∇W (x − xI (t)) xI · vI .
uh(X, t) =J ∈S
uJ (t) ΦJ (X)
uJ ΦJ (X)
S
ΦJ (X) = 0
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uh(xI ) = uI
ΦI (XJ ) = δ IJ δ IJ
H1
uh(X, t) =
Ω0
u(Y, t) W (X − Y, h0(Y)) dY
Ω0
Ω0
W (X − Y, h0(Y)) 1 d Y = 1
Ω0
W (X − Y, h0(Y)) Y dY = X
Ω0
W (X − Y, h0(Y)) X dY = X
Ω0
W (X − Y, h0(Y)) (X − Y ) d Y = 0
uh(X, t)
∇0uh(X, t) =
Ω0
∇0u(Y, t) W (X − Y, h0(Y)) dY
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∇0uh(X, t) =
Ω0
∇0 [u(Y, t) W (X − Y, h0(Y))] dY
−
Ω0
∇0u(Y, t) W (X − Y, h0(Y)) dY
∇0uh(X, t) = Γ0
u(Y, t) W (X − Y, h0(Y)) n0 dΓ0
−
Ω0
∇0u(Y, t) W (X − Y, h0(Y)) dY
∇0uh(X, t) = −
Ω0
∇0u(Y, t) W (X − Y, h0(Y)) dY
ΦJ (X) = W (X − XJ , h0) V 0J
V 0J
J
∇0uh(X) = −J ∈S
uJ ∇0ΦJ (X) with ∇0ΦJ = ∇0W (X − XJ , h0) V 0J
V 0J
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J ∈S
∇0W (X − XJ , h0) V 0J
uI ≡ 0
∇0uh(X) =
J ∈S(uJ − uI ) ∇0W (XI − XJ , h0) V 0
J
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∇0uh(X, t) =I ∈S
GI (X) uI (t)
uh,i(X, t) =I ∈S
GiI (X) uI (t)
GI
W S I (X) = W I (X)I ∈S
W I (X)
GI
GI (X) = a(X) · ∇0W S I (X) = aij(X)W S jI (X)
a(X)
I ∈S
GI (X) ⊗ XI = δ ij
A
a
A aT = I
I
= W S I,X X I W S I,Y X I
W S I,X Y I W S I,Y Y I
=
aXX aXY aYX aY Y
∇0uh(X, t) =I ∈S
a(X) · ∇0W S I (X) uI (t)
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ΦI = (a11(X) + a12(X) + a13(X)) W S I (X)
GXI = (a21(X) + a22(X) + a23(X)) W S I (X)
GY I = (a31(X) + a32(X) + a33(X)) W S I (X)
X
a A
=I
W S I (X) 1 X I − X Y I − Y
X I − X (X I − X )2 (X I − X )(Y I − Y )Y I − Y (X I − X )(Y I − Y ) (Y I − Y )2
3 × 3
ΦI = a11(X)W S I,X (X) + a12(X)W S I,Y (X) + a13(X)W S I (X)
GXI = a21(X)W S I,X (X) + a22(X)W S I,Y (X) + a23(X)W S I (X)
GY I = a31(X)W S
I,X (X) + a32(X)W S
I,Y (X) + a33(X)W S
I (X)
a
Φ
X a
=I
W S I,X (X) W S I,Y (X) W S I (X)W S I,X (X) X I W S I,Y (X) X I W S I (X) X I W S I,X (X) Y I W S I,Y (X) Y I W S I (X) Y I
O(h)
u(X)
X
u(XI ) = u(X) + u,X(X) (X I − X )
+ u,Y (X) (Y I − Y ) + 0.5u,XX (X) (X I − X )2
+ u,XY (X) (X I − X ) (Y I − Y )
+ 0.5u,Y Y (X) (Y I − Y )2 + O(h3)
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uh,X(X) − u,X
uh,X(X) − u,X =I
GXI (X) uI − u,X
=I
GXI (X) u(XI ) − u,X
uh,X(X) − u,X = u(X)I
GXI (X) + u,X(X)I
GXI (X)(X I − X ) − 1+ u,Y (X)
I
GXI (X) (Y I − Y )
+ 0.5 u ,XX (X)I
GXI (X)(X I − X )2
+ u,XY (X)I
GXI (X)(X I − X ) (Y I − Y )
+ 0.5 u ,Y Y (X)I
GXI (X)(Y I − Y )2
I GXI = 0 I GXI (X I
−X ) = 1
I
GXI (Y I − Y ) = 0
uh,X(X) − u,X = 0.5 u ,XX (X)I
GXI (X)(X I − X )2
+ u,XY (X)I
GXI (X)(X I − X ) (Y I − Y )
+ 0.5 u ,Y Y (X)I
GXI (X)(Y I − Y )2
|uh,X(X) − u,X | ≤ 0.5 |u,XX (X)| |I
GXI (X)(X I − X )2|
+ |u,XY (X)| |I
GXI (X)(X I − X ) (Y I − Y )|
+ 0.5 |u,Y Y (X)| |I
GXI (X)(Y I − Y )2|
d
X = (X Y )
|X I − X | ≤ d, |Y I − Y | ≤ d
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|uh,X(X) − u,X | ≤ (0.5 |u,XX (X)| + |u,XY (X)| + 0.5|u,Y Y (X)|) d2
|I
GXI (X)|
GXI
|GXI | ≤ C 1h0
h0 d = dh0
|uh,X(X) − u,X | ≤ C
(0.5 |u,XX (X)| + |u,XY (X)| + 0.5|u,Y Y (X)|) h0
h
Y
u B
∇0uh(X, t) =
−J ∈S
(uJ (t) − uI (t)) ∇0W (XJ − X, h0) V 0J
· B(X)
B(X) =
−J ∈S
(XJ − X) ⊗ ∇0W (XJ − X, h0) V 0J
−1
W (X − XJ , h) V 0J
B
B(X) = −J ∈SXJ ⊗ ∇0W
S
(XJ − X, h0)−1
B
u
∇0uh(X, t) =
−J ∈S
uJ (t) ∇0W S (XJ − X, h0) V 0J
· B(X)
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C (X, Y)
uh(X) =
ΩY
C (X, Y)W (X − Y)u(Y)dΩY
K (X, Y) = C (X, Y)W (X
−Y)
C (X, Y)
n
u(X) = pT(X)a
p(X)u(X) = p(X)pT(X)a
ΩY
p(Y)W (X − Y)u(Y)dΩY =
ΩY
p(Y)pT(Y)W (X − Y)dΩYa
a
uh(X) = pT(X)a
uh(X) = pT(X)
ΩY
p(Y)pT(Y)W (X−Y)dΩY
−1 ΩY
p(Y)w(X−Y)u(Y)dΩY
C (X, Y) = pT(X)
ΩY
p(Y)pT(Y)W (X − Y)dΩY
−1
p(Y)
= pT(X)[M(X)]−1p(Y)
uh(X) =
ΩY
C (X, Y)W (X − Y)u(Y)dΩY
=I ∈S
C (X, XI )w(X − YI )uI V 0I
= pT(X)[M(X)]−1I ∈S
p(XI )W (X − XI )uI V 0I
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M(X)
M(X) =
ΩY
p(Y)pT(Y)W (X − Y)dΩY
=I ∈S
p(XI )pT(XI )W (X − XI )V 0I
uh(x)
(xI , uI )
uI = u(xI )
uh(x)
m
u
h
(x) = a0 + a1x + a2x
2
+ ... + amx
m
uh(x) = pT(x)a
0
xi
Y
X
ui
xi
uh(xi)
uh(x)
a
uI
uh(xI )
J =nI =1
[uh(xI ) − uI ]2 =
nI =1
[pT(xI )a − uI ]2
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a
nI =1
p(xI )pT(xI )a =nI =1
p(xI )uI
a
uh(x)
xI
uI
pT(x) = [1 x] aT = [a0 a1]
3I =1
1 xI xI x2
I
a =
3I =1
1xI
uI
3 66 14
a =
6.516
a0 = −5/6 a1 = 1.5
uh(x) = −5
6 +
3
2x
a
X X
p
p(X) =
1 X Y ∀ X ∈ ℜ2
uh(X, t) =M I =1
pI (X) aI (X, t) = pT (X i) a(X i)
M a
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J(a(X i)) =N J =1
W (X − XJ , h0)
M I =1
pI (XJ )T aI (X, t) − u(XJ )
2
=
P(X) a(X) − u(X)T
W(X)
P(X) a(X) − u(X)
N W (X) = 0
uT
(ˆX) = u(
ˆX1) u(
ˆX2) ... u(
ˆXN )
P(X) =
p1(X1) p2(X1) ... pM (X1)
p1(X2) p2(X2) ... pM (X2)
p1(XN ) p2(XN ) ... pM (XN )
=
W (X − X1) 0 ... 0
0 W (X − X2) ... 0
0
0 0 ... W (X−
XN
)
a
∂ J(a(X i))
∂ a(X i) = −2PT (X) W(X) u(X)
+ 2PT (X) W(X) P(X) a(X) = 0
PT (X) W(X) u(X) = PT (X) W(X) P(X) a(X)
a
a(x) = PT (X) W(X) PT (X) =A∈RM ×M
PT (X) W(X) =B∈RM ×N
u(X)
uh(X, t) = pT (X) A−1(X) B(X) u(X)
uh(X, t) =M J =1
M K =1
N I =1
pJ (X) A−1JK (X) BKI (X) uI (X)
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ΦI (X)
ΦI (X, t) =M J =1
M K =1
pJ (X) A−1JK (X) BKI (X)
V 0I
A(X) =
P 11 ... P 1N
P M 1 ... P MN
W 1 ... 0
0 ... W N
P 11 ... P M 1
P 1N ... P MN
M = 1 p(X ) = 1
A(X) =
1 ... 1 W 1 ... 0
0 ... W N
1
1
A
p(x) = 1
ΦI (X) = W I (X)I ∈S
W I (X)
M = 3 p(X) = [1 X Y ]T
A
A(X) =
1 ... 1x1 ... xN y1 ... yN
W 1 ... 0
0 ... W N
1 x1 y1
1 xN yN
A 3 × 3
A
A
W(X) A
P A
N M
p(X) = [1 X Y ]
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a) b)
A A
A
A
κ = λmaxλmin
κ
κ → ∞
A
A
∂ Φ(X)
∂X i=
∂ pT (X)
∂X iA−1 B + pT (X)
∂ A−1(X)
∂X iB
+ pT (X) A−1(X)∂ B(X)
∂X i
∂ B(X)
∂X i= P(X)
∂ W(X)
∂X i
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A−1(X)
I = A−1(X) A(X)
0 = ∂ A−1(X)
∂X iA(X) + A−1(X)
∂ A(X)
∂X i
∂ A−
1(X)
∂X i = −A−1(X)∂ A(X)
∂X i A−1(X)
= A−1(X) P(X)∂ W(X)
∂X iPT (X) A−1(X)
∂ 2Φ(X)
∂X i∂X j=
∂ 2pT (X)
∂X i∂X jA−1(X) B(X)
+ 2∂ pT (X)
∂X i
∂ A−1(X)
∂X jB(X) + A−1(X)
∂ B(X)
∂X i
+ pT (X)
∂ 2A−1(X)
∂X i∂X jB(X) + A−1(X)
∂ 2B(X)
∂X i∂X j+
∂ A−1(X)
∂X i
∂ B(X)
∂X j + pT (X)
∂ A−1(X)
∂X j
∂ B(X)
∂X i
ΦJ
ΦJ (X) = γ (X) · p(XJ ) W (X − XJ , h0)
A(X) · γ (X) = p(XJ )
γ
A
∇0A(X) · γ (X) + A(X) · ∇0γ (X) = ∇0p(XJ )
∇0γ (X)
XI
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h0
ΦI (X) = W (XI , X) PT
XI − X
h0
γ (X),
W (Y, X) = W ((Y − X)/h0)
P(0) = I ∈S
ΦI (X) PXI − Xh0
γ (X)
A(X) γ (X) = P(0)
A(X) =J ∈S
W (XJ , X) PT
XJ − X
h0
P
XJ − X
h0
h0I h0I XI
W (XI , X) = W
XI − X
h0I
h0 P
h0 h0J
P
< f,g >X=J ∈S
W (XJ , X) f XJ − X
h0
gXJ − X
h0
X Z X
u
u(Z) ≃ uh(Z, X) = PT
Z − X
h0
c(X)
c
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F (X, Y ) = X 2 + Y 2
(R = 0.8) (R = 0.3) R
uh(X) =J ∈S
ΦJ (X)
uJ +
LK =1
pK (X) aJK
aJK
uh(X) =J ∈S
ΦJ (X) uJ +J ∈S
ΦJ (X)
LK =1
pK (X) aJK
global
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F (X, Y ) =X 2 + Y 2
(R = 0.8)
(R = 0.3) R
F (X, Y ) = X 2 + Y 2
25 × 25
R
R = 0.6
R = 1.6
R = 0.6
A
0.05%
X
Y
F x F ,X = 2X
0.005%
0.2%
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F (X, Y ) = sin
X 2 + Y 2
F 0 ≤ X ≤ π2
0 ≤ Y ≤ π2
F
x
π/300 R
x
V = d2
d
h
d < h <√
2d
x
u,X(X) = −N J =1
V J W J,X(X)uJ
u,X(X(5)) = −V J
W
(25),X u2 + W
(45),X u4 + W
(55),X u5 + W
(65),X u6 + W
(85),X u8
x
W (25),X = W (55)
,X =
W (85),X = 0
W IJ
u,X(X(5)) = V J
W (54),X u4 + W
(56),X u6
f (X ) = aX 2 + bX + c Y
y
F ,X = 2X cos`X 2 + Y 2
´
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F (X, Y ) = sin X 2
+ Y 2
F (X, Y ) = sin
X 2 + Y 2
(R = 0.6)
F (X, Y ) = sin
X 2 + Y 2
(R = 1.6)
F (X, Y ) = sin
X 2 + Y 2
(R = 1.6)
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f (X ) = aX 2 + bX + c
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f
x
w
w
f ,X(X(5)) = V J
W (54),X f 4 + W
(56),X f 6
= V J
−w
a (x(4))2 + bx(4) + c
+ w
a (x(6))2 + bx(6) + c
= V J w a(x
(5)
+ d)2
− (x(5)
− d)2+ b(x
(5)
+ d) − (x(5)
− d)= V J w
4 a d x(5) + 2 b d
= 2 V J w d
2 a x(5) + b
V J = d2
f
f ,X(X(I )) = 2 w d3
2 a x(I ) + b
a
b
f
d
h
d
h
x(I
)
errabs(d,h,x(I )) = 2 a x(I ) + b − 2 w(d, h) d3
2 a x(I ) + b
=
2 a x(I ) + b
1 − 2 w(d, h)d3
errrel(d,h,x(I )) =
2 a x(I ) + b
1 − 2 w(d, h)d3
2 a x(I ) + b
= 1 − 2 w(d, h)d3
1 − 2 w(d, h)d3 ≡ 0 ⇔ w(d, h) d3 = 0.5
(d, h)
d/h =√
2
35%
0.2%
w
d
h
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F (X, Y ) =X 2 + Y 2
(R = 1.6)
5%
10%
10%
25 × 25 = 625
V J = d2
21%
V new,J = (1.1d)2 = 1.21 V old,J
70%
∇0u(X(407))
V I ≡ −
J ∈S
∇0W (407)J (X) uJ
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10%
∇0W (407)J (X) uJ
K ∈S∇0W
(407)K (X) uK
∂W (407)J (X)∂X uJ = 0
10%
5% 10%
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F (X, Y ) =X 2 + Y 2
(R = 1.6)
F (X, Y ) =X 2 + Y 2
(R = 0.6)
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∇0 · P − b = ∅ ∀X ∈ Ω0
P P
b X
∇0
Ω0
u(X, t) = u(X, t) on Γu0
n0 · P(X, t) = t0(X, t) on Γt0
u
t0
Γu0
Γt0 = Γ0 , (Γu0
Γt0) = ∅
J = 0 J 0
u = 1
0∇0 · P + b on Ω0
e = 1
0F : PT
J
J 0
u
0 P
b
e
F = ∇u+I I
u(X, t) = u(X, t)
Γu0
n0 · P(X, t) = t0(X, t) Γt0
u
t0 n0
Γu0 ∪ Γt0 = Γ0 (Γu0 ∩ Γt0) = ∅
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C0
δ uh(X) =J ∈S
ΦJ (X) δ uJ
uh(X) =J ∈S
ΨJ (X) uJ
V = u(·, t)|u(·, t) ∈ H1, u(·, t) = u(t) on Γu0 ,V0 =
δ u|δ u ∈ H1, δ u = 0 on Γu0 ,
Ω0
∇0 · P · δ u dΩ0 +
Ω0
0 (b − u) · δ u dΩ0 = 0
Ω0 ∇0
·P
·δ u dΩ0 = Ω0 ∇
0
·(P
·δ u) dΩ0
− Ω0
(
∇0
⊗δ u)
T : P dΩ0
Ω0
∇0 · (P · δ u) dΩ0 =
Γt0
n0 · P · δ u dΓ0
t = n0 · P Ω0
∇0 · (P · δ u) dΩ0 =
Γt0
t · δ u dΓ0
Ω0
(∇0 ⊗ δ u)T : P dΩ0 − Ω0
0 b · δ u dΩ0 +
Γt0
t0 · δ u dΓ0
+
Ω0
0 δ u · u dΩ0 = 0
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J
mIJ uJ = f extI − f intI ,
f extI f intI
f extI =
Ω0
0 ΦI b dΩ0 +
Γt0
ΦI t dΓ0
f intI = Ω0
∇0ΦI · P dΩ0
mIJ =I ∈S
Ω0
0 ΨI (X) ΦJ (X) dΩ0.
ΦJ (X) = δ (X − XI )
ΨJ (X) = ΦJ (X)
ΨJ (X) = ΦJ (X)
Ω0
f (X) dΩ0 =J ∈S
f (XJ ) V 0J
V 0J J
f intI =J ∈S
V 0J ∇0ΦI (XJ ) · PJ
mI
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mI
V 0J
mI
mIJ
mIJ
mI =J ∈S
mIJ =J ∈S
Ω0
0 ΦI (X) ΨJ (X) dΩ0
= Ω0
0 ΦI (X)J ∈S
ΨJ (X) dΩ0
mI =
Ω0
0 ΦI (X) dΩ0
mI =
J ∈S J ΦI (X) V 0
J
M
N totI =1
mI =
N totI =1
J ∈S
J ΦI (X) V 0J =
J ∈S
J
N totI =1
ΦI (X)
V 0J =
J ∈S
J V 0J = M
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a) b)
Ω0
∇ΦI (X) dΩ0 =
Γ0
n0ΦI (X) dΓ0
ǫ XM
ǫ(XM ) =
Ω0
ǫ Ψ(X − XM ) dΩ0
ǫ ǫ = 0.5(ui,j + uj,i)
Ω0
Ψ(X − XM )
Ψ(X − XM ) ≥ 0 Ω0
Ψ(X − XM ) dΩ0 = 1
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stress point
stress point
stress point
particle particle particle particle
particleparticleparticleparticle
Ψ(X − XM ) = 1
AM ∀XM ∈ Ω0, otherwise Ψ(X − XM ) = 0
AM
ǫ(XM ) = 12AM
Ω0
(ui,j + uj,i) dΩ0
= 1
2AM
Γ0
(ui nj + uj ni) dΓ0
Ω0
f (X) dΩ0 =J ∈NP
f P J V 0P J +J ∈NS
f S J V 0S J
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master partic les
slave parti cles
N P
N S
XP I
uS I = J ∈S
ΦJ (XS I ) uP J , vS I =
J ∈SΦJ (XS
I ) vP J
S P
ΦJ (XS I ) J XS I
f intI =J ∈NP
V 0P J ∇0ΦI (XP J ) · PP J +J ∈NS
V 0S J ∇0ΦI (XS J ) · PS J
V 0P J V 0S J
V 0 = J ∈NP
V 0P J + J ∈NS
V 0S J
2nQ−1
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nQ
nQ
nQ =√
m + 2
m
Ω0
f (X) dΩ0 =
+1 −1
+1 −1
f (ξ, η) det Jξ(ξ, η) dξdη =mJ =1
wJ f (ξJ ) det Jξ(ξJ )
ξ = (ξ, η) m
wJ = w(ξ J ) w(ηJ )
ξ
η
det Jξ
Jξ = ∂ X
∂ ξ
f int =
mJ =1
wJ detJξ(ξJ ) ∇0Φ(X(ξJ ) − XP ) P(ξJ )
P
u ∈ V
δW = δW int − δW ext = 0 ∀δ u ∈ H1
δW int =
Ω0
(∇ ⊗ δ u)T
: P dΩ0
nQ = 2
nQ = 3
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a) b)
δW ext =
Ω0
0 δ u · b dΩ0 +
Γt0
δ u · t0 dΓ0
V = u(·, t)|u(·, t) ∈H1, u(·, t) = u(t) on Γu0 ,V0 =
δ u|δ u ∈ H1, δ u = 0 on Γu0 ,
K u = f ext
K
KIJ =
Ω0
BI CtBJ dΩ0
B
BI = ΦI,X 0
0 ΦI,Y ΦI,Y ΦI,X
f extI =
Γt0
ΦI (X) t0 dΓ0 +
Ω0
ΦI (X) b dΩ0
H1
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u ∈ V
δW = δW int − δW ext − δW u = 0 ∀δ u ∈ H1
δW int = Ω0\Γc0
(∇ ⊗ δ u)T : P dΩ0
δW ext =
Ω0\Γc0
0 δ u · b dΩ0 +
Γt0
δ u · t0 dΓ0
V =
u(·, t)|u(·, t) ∈ H1, u(·, t) = u(t) on Γu0 ,
V0 =
δ u|δ u ∈ H1, δ u = 0 on Γu0 ,
δW u
δW u
δW u = Γ0u
δ λ · (u − u) dΓ0 + Γ0
δ uλ dΓ0
λ
λ =J ∈S
ΦLJ (X) λJ
ΦLJ (X)
K GG 0 uλ = f
ext
q
K = KIJ
GIK = −
Γu
ΦI (X) ΦLK (X) S dΓ
qK = −
Γu
ΦLK (X) S u dΓ
S 2 × 2 S ij j = i
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•
• K
•
u λ
u
inf − sup
Π
Π = 2a21 − 2a1a2 + a2
2 + 18a1 + 6a2
a1 = a2
λ
Π = Π + λ(a1 − a2)
= 2a21 − 2a1a2 + a2
2 + 18a1 + 6a2 + λ(a1 − a2)
ai λ
∂ Π
∂a1= 0
∂ Π
∂a2= 0
∂ Π
∂λ = 0
a1 a2 λ
a1 = a2 = −12 λ = 6
δW u
δW u = 0.5 p
Γ0u
u − u2dΓ0
p
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uh(x) =N I =1
ΦI (xJ )uI
uh(x) xJ
uI
N
N
u = Du
D
N × N
u
u = D−1u
uh(x) =
nI =1
ΦI (x)D−1IJ uI
N ×N
Γu
N Ω
Γu N Γu
uh(x) =
N ΩI =1
ΦI (xJ ) uI Ω +
N ΓuI =1
ΦI (xJ ) uI Γu
Γu
u(xJ ) = g(xJ ), J = 1,...,N Γu
DΩuΩ (N Γu×N Ω)(N Ω×1)
+ DΓu uΓu (N Γu×N Γu )(N Γu×1)
= g
(N Γu×1)
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uΓu
uΓu =
DΓu−1
(g − DΩuΩ)
uh(x) =
N ΩI =1
ΦI (xJ ) uI Ω +
N ΓuI =1
ΦI (xJ )
[DΓu
IJ ]−1(gI − DΩ
IJ uΩJ )
uh(x) =
N Ωi=I
ΦI (x) − ΦI (x)[DΓu
IJ ]−1DΩ
IJ
uI +
N ΓuI =1
ΦI (x)[DΓu
IJ ]−1gJ
FE node
particle
particle boundaryparticle domain
blending region
element domain
element boundary
ΩP
ΩFE
ΓP
ΓFE
ΩB
ΩB ΩP ΩFE
ΓFE ΓP
uh = uFE (X) + R(X)
uP (X) − uFE (X)
, X ∈ ΩB
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uFE uP u
R(X)
R(X) = 1, X ∈ ΓP
R(X) = 0, X ∈ ΓFE
R(X) = 3 r 2(X) − 2 r3(X)
r(X) =J ∈S ΓP
N J (X)
S ΓP ΓP
uh(X) =I
N I (X)uI , XI ∈ ΩB
N I (X) = (1 − R(X)) N I (ξ (X)) + R(X) N I (X) X ∈ ΩB
˜N I (
X) = R(
X) N I (
X)
X /∈ Ω
B
ΓP
ΓFE
N I (X) = N I (X) X ∈ ΩB on ΓFE
N I (X) = 0 X /∈ ΩB on ΓFE
N I (X) = N (X) X /∈ ΩB on ΓP
R(X) = 1 ΓP R(X) = 0 ΓFE
W = W int − W ext + λT g
W int W ext
λ
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ΓFE 0
ΓP 0
Γ∗
0
ΩFE 0 Ω
P 0
g = uFE − uP
gh =
N J =1
N FE J (X, t) uFE J −J ∈S
N P J (X, t) uP J
δ λ
δ λP h (X, t) =N J =1
N FE J (X, t) δ ΛJ (t)
XL
XL = ΦI (ξ)XI
ξ
δ uh(X, t) =
N J =1
N FE J (X, t) δ uFE J (t) +J ∈S
N P J (X, t) δ uP J (t)
uh(X, t) =
N J =1
N FE J (X, t) uFE J (t) +J ∈S
N P J (X, t) uP J (t)
N FE (X, t) = 0 ∀ X ∈ ΩP 0
N P (X, t) = 0 ∀ X ∈ ΩFE 0
S
u λ
∂W
∂ u =
∂W int
∂ u − ∂ W ext
∂ u + λ
∂ g
∂ u = f int − f ext + λ
∂ g
∂ u = 0
∂W
∂ λ = g = 0
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W int
W ext u
f int =
ΩP 0 ∪ΩFE
0
(∇0 ⊗ δ u)T : P dΩ0
f ext =
ΩP 0 ∪ΩFE
0
δ u · b dΩ0 +
ΓP,t0 ∪ΓFE,t0
δ u · t0 dΓ0
λ ∂ g∂ u
0 = f int − f ext + λ∂ g
∂ u +
∂ f int
∂ u ∆u − ∂ f ext
∂ u ∆u +
∂ g
∂ u ∆λ + λ
∂ 2g
∂ u∂ u ∆u
0 = u + ∂ g
∂ u ∆u
KFE + λ ∂ 2g
∂ u∂ u 0
KFE −FE T
0 KP + λ ∂ 2g
∂ u∂ u
KFE −P T
KFE −FE KFE −P T 0
· ∆uFE J
∆uP J ∆Λ
=
f ext,FE − f int,FE − λT KFE −FE
f ext,P − f int,P − λT KFE −P
−g
KFE −FE KFE −P
g u
uFE uP KFE
KP u
b
t u
KFE −FE =
Γ∗0
NFE
T · NFE dΓ0
KFE −P = −
Γ∗0
NFE
T · NP dΓ0
KP =
ΩP 0
BP
T C BP dΩ0
KFE =
ΩFE0
BFE
T C BFE dΩ0
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f ext,FE =
ΩFE0
NFE
T b dΩ0 +
ΓFE,t0
NFE
T t0 dΓ0
f ext,P =
ΩP 0
NP
T b dΩ0 +
ΓP,t0
NP
T t0 dΓ0
f int,FE =
ΩFE0
BFE
T · P dΩ0
f
int,P
= ΩP 0B
P T · P dΩ0
K
∆u = u
∆λ = λ
∂ 2g∂ u∂ u
KFE 0
KFE −FE T 0 KP
KFE −P T
KFE −FE KFE −P T 0
· uFE J
uP J Λ
= f ext,FE
f ext,P
−g
Ω0
Γ0 Γ0 Γt0
Γu0
ΩFE 0 ΩP 0
Ωint0 Ωint
Ωint
Γα
0
α
α
α = l(X)l0
l(X) X
Γα0
α
Ωint0
Ωint0
W int =
ΩFE0
β FE FT · PdΩFE 0 +
ΩP 0
β P FT · PdΩP 0
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Γ 0
αΩ0
FE
Ω0
P
Ω0
int
α=1α=0
finite element node
particle
β
β FE (X) =
0 in ΩP 0
1 − α in Ωint0
1 in ΩFE 0 − Ωint0
β P (X) = 0 in Ω
FE
0α in Ωint0
1 in ΩP 0 − Ωint0
W ext =
ΩFE0
β FE ρ0b · udΩFE 0 +
ΩP 0
β P ρ0b · udΩP 0
+
ΓFE0
β FE t · udΓFE 0 +
ΓP 0
β P t · udΓP 0
Ωint0
N I (X)
wI (X)
uFE (X, t) =I
N I (X)uFE I (t)
uP (X, t) =I
wI (X)uP I (t)
Ωint0
gI = giI =
uFE iI − uP iI
=
J
N JI uFE iJ −
K
wKI uP iK
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ΛI (X)
λi(X, t) =I
ΛI (X)λiI (t)
ΛI (X)
N I (X) wI (X)
λi
λiI
W AL = W int − W ext + λT g + 1
2 pgT g
p
p = 0
W AL
uI
λI
∂W AL∂uFE iI = (F intiI − F extiI ) +
L
K
ΛKLλK N IL+ p
L
K
N KLuFE iK −K
wKLuP iK
N IL
= 0
∂W AL∂uP iI
= (f intiI − f extiI ) −L
K
ΛKLλK
wIL
− pL
K
N KLuFE iK −K
wKLuP iK
wIL
= 0
∂W AL
∂λiI = L ΛIL K N KLuFE iK
−K wKLuP iK = 0
N KI = N K (XI ) ΛKI = ΛK (XI )
Fint Fext
ΩFE 0
FintiI =
ΩFE0
β FE N I,j (X)Pji(X)dΩFE 0
FextiI =
ΩFE0
β FE N I (X)ρ0bidΩFE 0 +
Γt0
β FE N I (X)tidΓt0
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f int f ext
ΩP 0
f intiI =
ΩP 0
β P wI,j (X)Pji(X)dΩP 0
f extiI =
ΩP 0
β P wI (X)ρ0bidΩP 0 +
Γt0
β P wI (X)tidΓt0
d u
∆FintI =J
KFE IJ ∆uFE J or ∆Fint = KFE ∆dFE
∆f intI =J
KP IJ ∆uP J or ∆f int = KP ∆dP
KFE KP
KFE = KFE
11 KFE 12
KFE
21 KFE
22
KFE nn
KFE IJ = ∂ FintI
∂ uFE J
KP =
KP
11 KP 12
KP 21 KP
22
KP mm
KP IJ =
∂ f intI ∂ uP J
d
FE
=
dFE 1
dFE 2
dFE n
d
FE
I = uFE xI
uFE yI d
P
=
dP 1
dP 2
dP m
d
P
I = uP xI uP yI
A11 A12 LFE T
A21 A22 LP T
LFE LP 0
∆dFE
∆dP
∆λ
=
−rFE
−rP
−g
di ukP dj ulQ
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rFE = Fint − Fext + λT GFE + pgT GFE
rP = f int − f ext + λT GP + pgT GP
g = giI =
K
ΛIK giK
A11 = KFE + pGFE T GFE
A12 = pGFE T GP
A21 = pGP T GFE
A22 = KP + pGP T GP
λiI =K
ΛK (XI )λiK
KFE =
∂ Fint
∂ dFE
=
∂F intiI ∂uFE lQ
=
ΩFE0
β FE N I,jC jilkN Q,kdΩFE 0
KP =
∂ f int
∂ dP
=
∂f intiI ∂uP lQ
=
ΩP 0
β P wI,jC jilkwQ,kdΩP 0
LFE =
L
ΛIL∂ gL
∂dFE i
=
L
ΛIL∂gjL∂dFE i
=
L
ΛIL∂gL
∂uFE kP
=
L
ΛILN PI δ jk
LP =
L
ΛIL∂ gL∂dP i
=
L
ΛIL∂gjL∂dP i
=
L
ΛIL∂gL
∂uP kP
=
−L
ΛILwPI δ jk
GFE =
∂ gI ∂dFE i
=
∂gjI ∂uFE kP
= [N PI δ jk ]
GP =
∂ gI ∂dP i
=
∂ gjI ∂uP kP
= [−wPI δ jk ]
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c
u(x, t) = ˙u + H S [[ ˙u]](x, t)
u
∇ x r
s H S
δ S S
ǫ(x, t) = ∇S u = ∇S ˙u + H S ∇S [[ ˙u]] + δ S
[[ ˙u]] ⊗ n
S
weak
Ω
S
Ω+
Ω−
u(x, t) = ˙u + H Ωh(r, t)[[ ˙u]](s, t)
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H Ωh
H Ωh =
0 x ∈ Ω− \ Ωh
1 x ∈ Ω+ \ Ωh
s−s−s+−s− x ∈ Ωh
ǫ(x, t) = ∇S u = ∇S ˙u + H Ωh∇S [[ ˙u]] + ∇H Ωh [[ ˙u]]
∇
s
H Ωh
∇H Ωh = n
h(r)
h(r) n
h = s+−s−
a 1 X ∈ Ωh 0
ǫ(x, t) = ∇S u = ∇S ˙u + H Ωh∇S [[ ˙u]] + a
h
[[ ˙u]] ⊗ n
S
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undesired
hI 0
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CRACK
CRACK
CRACK
Visibility criterion
Diffraction criterion
Transparency criterion
Crack line
Crack line
Crack line
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crack
interdiscontinuities
I
crack
Domain of influence
I
crack crack
s0(x)
s2(x)
x
xI
xc
s1
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h0
undesired
hI 0
hI 0(X) =
s1 + s2(X)
s0(X)
λs0(X)
s0(X) = X − XI s1 = Xc − XI
s2(X) = X − Xc
λ
hI 0
∂W
∂X i=
∂W
∂h0I
∂h0I
∂X i
∂h0I
∂X i= λ
s1 + s2(X)
s0(X)
λ−1∂s2
∂X i+ (1 − λ)
s1 + s2(X)
s0(X)
λ∂s0
∂X i
∂s2
∂ X =
X − Xc
s2(X)
∂s0
∂ X =
X − XI
s0(X)
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I
crack
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X XI
h0I
h0I = s0(X) + hmI
sc(X)
sc
λ, λ ≥ 2
s0(X)
hmI
SI
sc(X)
sc = κh
κ
h
∂h0I
∂ X =
∂s0
X + λhmI
sλ−1c
sλc
∂sc∂ X
∂s0
∂ X =
X − XI
s0(X)
∂sc∂X 1
= −cos(θ) = X b − X c
sc(X)
∂sc∂X 2
= −sin(θ) = Y b − Y c
s2(X)
θ
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na
nB
ω
nA · nB ≤ β
nA · nB ≤ β β = 0o
β = 0o
ω = 90o
enrichment
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r θ
crack
u1 = 1
G
r
2G
K I Q
1I (θ) + K II Q
1II (θ)
u2 =
1
G
r
2G
K I Q
2I (θ) + K II Q
2II (θ)
G r θ
Q1I (θ) = κ − cos θ2 + sinθ sin θ2
Q2I (θ) = κ + sin
θ
2 + sinθ cos
θ
2
Q1II (θ) = κ + sin
θ
2 + sinθ cos
θ
2
Q2II (θ) = κ − cos
θ
2 − sinθ sin
θ
2
K I K II
κ = (3 − ν )/(1 + ν )
κ = (3 − 4ν )
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pT (X) =√
r sin(θ/2),√
r cos(θ/2),√
r sin(θ/2)sin(θ),√
r cos(θ/2)sin(θ)
p = [B1, B2, B3, B4]
02
46
810
0
5
10−3
−2
−1
0
1
2
3
B1 function
B1
02
46
810
0
5
100
0.5
1
1.5
2
2.5
B2 function
B2
02
46
810
0
5
10−3
−2
−1
0
1
2
3
B3 function
B3
02
46
810
0
5
10−1
0
1
2
3
B4 function
B4
p
p
pT (X) =
1, X , Y ,
√ r sin
θ
2,√
r cosθ
2,√
r sinθ
2sin(θ),
√ r cos
θ
2sin(θ)
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r
θ
ΦJ (X) = p(X)T · A(X)−1 · pJ (X) W (X − XJ , h)
A(X) =
J ∈S pJ (X) pT J (X) W (X − XJ , h)
A
A
A
uh(X) = R uenr(X) + (1 − R) ulin(X)
uenr
(X)
u
R
R
R = 1 − ξ R = 1 − 10ξ 3 + 15ξ 4 − 6ξ 5 ξ = (r − r1)(r2 − r1)
uh(X, t) =J ∈S
uJ (t) ΦJ (X)
ΦJ (X) = R ΦenrJ (X) + (1 − R)ΦlinJ (X)
ΦenrJ (X) ΦlinJ (X)
R = 1 − ξ
uh(X, t) =J ∈S
p(XJ )T a(X, t) +
ncK =1
kK I QK
I (XI ) + kK II QK II (XI )
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crack
Enriched
Transition
Linear
r1
r2
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nc uh u
p
n − th
kI
kII
kI
kII
J
a
L2
J = J ∈S
1
2 p(XJ )T a(X, t) +
ncK =1
kK I QK I + kK II QK II − uJ (t)2
W (X−XJ , h0)
J
A(X)a(X) =J ∈S
PJ (X)
uJ −
ncK =1
kK I QK I + kK II Q
K II
A(X) =J ∈S
p(XJ ) pT (XJ ) W (X − XJ , h0)
PJ (X) = [W (X − X1, h0)p(X1),...,W (X − Xn, h0)p(Xn)]
n a
a(X) =J ∈S
A−1(X)PJ (X)
uJ −
ncK =1
kK I QK I + kK II Q
K II
uh(X) =
J ∈SpT (X)A−1(X)PJ (X)
uJ −
nc
K =1 kK I QK
I + kK II QK II
+ncK =1
kK I QK I + kK II Q
K II
ΦJ (X) = pT (X)A−1(X)PJ (X)
uh(X) =J ∈S
ΦJ (X) uJ +
ncK =1
kK I QK I + kK II Q
K II
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uJ = uJ −ncK =1
kK I QK I + kK II Q
K II
kI
kII
uh(X) =I ∈S
ΦI (X)
uI +
J ∈Sc
bIJ pJ (X)
bIJ
Sc
local
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local
global
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Ω
ΩA
ΩB
Γ
ΩB
Ω = ΩA ∪ ΩB
ΩA∩
ΩB = ∅
Γ :
ΩA
ΩB
φ > 0
ΩA
φ < 0
φ = 0
Γ
n
φ(x)
φ(x) > 0 ∀ x ∈ ΩA
φ(x) < 0 ∀ x ∈ ΩB
φ(x) = 0 ∀ x ∈ Γ
Γ
φ(x)
φ(x, t)
n
Γ x ∈ Γ
n = ∇φ ∇φ
∇φ = 1 n = ∇φ n ΩB ΩA ΩB
φ ΩA φ
Γ x ∈ Γ
K = ni,i
∇φ = 1
K = ni,i = φ,ii
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Ω
f (x) Ω
ΩA ΩB Ω
f (x) =
ΩA
f (x) +
ΩB
f (x)
H (ξ )
H (ξ ) =
1 ∀ξ > 00 ∀ξ < 0
ΩA
ΩB
ΩA = x ∈ Ω/H (φ(x)) = 1
ΩB = x ∈ Ω/H (−φ(x)) = 1
Ω f (x
) = Ω f (x
)H (φ(x
)) + Ω f (x
)H (−φ(x
))
ΩA
ΩA ΩA
f ,i(x) =
Ω
f ,i(x)H (φ(x))
ΩA f ,i(x) = ∂ ΩA f (x)ni
ni ΩA
ΩA
f ,i(x) =
Ω
(f (x)H (φ(x))),i − f (x) (H (φ(x))),i
H (φ(x) H (φ(x))
,i
= φ,i(x)H ,i(φ(x)) = φ,i(x)δ (φ(x))
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Case 3:Case 2:Case 1:
ΩAΩAΩA ΩB
ΩBΩBΓΓΓ
∂ intΩA = Γ
∂ extΩA = ∂ Ω∂ extΩA = ∂ Ω
∂ ΩA = ∂ extΩA
∪∂ intΩA ∂ ΩA = Γ ∂ ΩA = ∂ extΩA
∂ Ω
δ (η)
φ φ,i(x) nB→A
H (φ(x))
,i
= nB→Ai on Γ
= 0 otherwise.
Ω
f ,i(x)H (φ(x)) =
Ω
f (x)H (φ(x))
,i
− Ω
f (x)
H (φ(x)),i
=
∂ Ω
f (x)H (φ(x))ni −
Γ
f (x)nB→Ai
=
∂ Ω
f (x)H (φ(x))ni +
Γ
f (x)nA→Bi
Ω
f ,i(x)H (φ(x)) =
∂ Ω =∂ extΩA
f (x) H (φ(x))
=1
ni +
Γ =∂ intΩA
f (x)nA→Bi
=
∂ ΩA
f (x)ni
Ω
f ,i(x)H (φ(x)) =
∂ Ω
f (x) H (φ(x)) =0
ni +
Γ
=∂ ΩA
f (x)nA→Bi
=
∂ ΩA
f (x)ni
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Ω
f ,i(x)H (φ(x)) =
∂ Ω
f (x) H (φ(x)) =1 onlyif x∈ΩA
ni +
Γ
f (x)nA→Bi
=
∂ ΩA
f (x)ni
H (φ) =
0 for φ < −ǫ12 + φ
2ǫ + 12π sin πφǫ for − ǫ < φ < ǫ
1 for ǫ < φ
H (φ) =
0 for φ < −ǫ12
+ 18
9φǫ − 5(φ
ǫ)3
for − ǫ < φ < ǫ
1 for ǫ < φ
ǫ
δ (φ) = 0 for φ < −ǫ
12ǫ
+ 12ǫ
sin πφǫ for − ǫ < φ < ǫ0 for ǫ < φ
d x
Γ
d = x − xΓ
xΓ x
Γ
φ(x)
φ(x) = d ΩA
φ(x) = −d
ΩB
φ(x) = min x∈Γ
x − x sign
n · (x − x)
∇φ = 1
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n
x
φ = 0
d
xΓ
Γ
φ < 0 φ > 0
N I (x)
I
S
φ(x) =I ∈S
N I (x)φI
φI
I
φ(x),i =I ∈S
N I,i(x)φI
φ
φ,i φ,i = 0
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φ
Dφ(x, t)
Dt = 0
v
∂φ(x, t)∂t + ∇φ(x, t) · v(x, t) = 0
φ + φ,ivi = 0
φn+1 − φn
∆t = −φn,iv
ni
φn+1 = φn − ∆t φn,ivni
∆t
φ vi
φ φ
|∇φ| = 1
Ω0
Γ0
• φ(X) = 0
φ(X) > 0
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f(X)=0 f X =0f(X)<0
voxels (background mesh) f(X)>0
activparticl
ΩCD
φ(X)
• φ(X) ≥ 0
XI
• I ∋ N act φ(XI ) ≥ 0 φ(XI ) = 0 XI
XI I
nsp
XI I
nip
φ(X) = 0
φ(X)
φ(X)
N I (X)
φ(X) =I ∈S
N I (X) XI
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B X
φ(X) > 0
ΩCD
φ(XI ) ≤ α h p
h p
α
uh(X) = J ∈S
N J (X) uJ + K ∈E
J ∈Sc
N K J (X) ψK (X) aK J
S
Sc
N J
N J
ψ(X)
aJ
E
K
N J (X ) = N J (X )
ψ S
S (ξ ) =
1 ∀ξ > 0−1 ∀ξ < 0
ψ(X)
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4321 crack
Shifting
crack
φ=0φ>0φ<0
N 2(X) N 3(X)
N 2(X)H (f (X))
N 3(X)H (f (X))
N 2(X) (H (f (X)) − H (f (X2)))
N 3(X) (H (f (X)) − H (f (X3)))
uh(X ) =J ∈S
N J (X ) uJ +J ∈Sc
N J (X ) S (φ(X )) aJ
N 1 = 0.5(1 − r) N 2 = 0.5(1 + r)
r
N 2(X ) N 3(X )
X c
φ(X c) = 0 X c
φ(X ) < 0 X < X c φ(X ) > 0 X > X c
X 2 < X c S (φ(X 2)) = −1
S (φ(X 3)) = 1
X 3 > X c N J (X ) S (X )
u(X ) K ∈ Sc
u(X K ) = uK + S (φ(X K )) aJ
uK
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uh(X ) =J ∈S
N J (X ) uJ +J ∈Sc
N J (X ) (S (φ(X )) − S (φ(X J ))) aJ
u(X K ) = uK
[[uh
(X )]] = u(X +
) − u(X −)=
J ∈S
N J (X +) uJ +J ∈Sc
N J (X +)
S (φ(X +))
aJ
−J ∈S
N J (X −) uJ +J ∈Sc
N J (X −)
S (φ(X −))
aJ
=J ∈Sc
N J (X )
S (φ(X +)) − S (φ(X −))
aJ
= 2J ∈Sc
N J (X ) aJ
N J (X −) = N J (X
+
)
[[uh(X )]] =J ∈Sc
N J (X )
H (φ(X +)) − H (φ(X −))
aJ
=J ∈Sc
N J (X ) aJ
J ∈Sc
N J (X ) aJ 2 J ∈Sc
N J (X ) aJ
ψ
φ
ψJ (x, t) = |φ(x, t)| − |φ(xJ , t)|
vh(x) =J ∈S
N J (x) vJ (t) +J ∈Sc
N J (x) ψJ (φ(x), t) aJ (t)
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4321
φ=0φ>0φ<0
interface
1 2 3 4
N 2(X) N 3(X)
Ψ2(X) Ψ3(X)
N 2(X) (H (f (X)) − H (f (X2))) N 3(X) (H (f (X)) − H (f (X3)))
∇Ψ2(X)
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Sc
v
u
ψ
ψ
N 2(x, t) ψ2(x, t) N 3(x, t) ψ3(x, t)
∇vh(x) =J ∈S
∇N J (x)vJ (t)
+J ∈Sc
(∇N J (x) ψJ (φ(x), t) + N J (x) ∇ψJ (φ(x), t)) aJ (t)
∇ψJ (x, t) = sign(φ) ∇φ = sign(φ)nint
nint
∇ψJ (x, t)
[[∇vh(X )]] = 2J ∈Sc
N J (X ) aJ nint
[[∇vh(X )nint]] = 2J ∈Sc
N J (X ) aJ
−1 1
uh(X ) =2I =1
N I (X ) [uI + aI (H (X − X c) − H (X I − X c))]
= u1 N 1 + u2 N 2 + a1 N 1 H (X − X c)
+ a2 N 2 [H (X − X c) − 1]
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H
N I = N I H (X −X c)+N I (1 − H (X − X c))
I = 1, 2
uh(X ) = (u1 + a1) N 1 H (X − X c) + u1 N 1 (1 − H (X − X c))
+ (u2 − a2) N 2 (1 − H (X − X c)) + u2 N 2 H (X − X c)
element1
u1
1 = u1
u12 = u2 − a2
element2
u2
1 = u1 + a1
u22 = u2
uh(X ) = u11 N 1 (1 − H (X − X c)) + u1
2N 2 (1 − H (X − X c))
+ u21 N 1 H (X − X c) + u2
2 N 2 H (X − X c)
X < X c
(1 − H (X − X c))
X > X c
H (X −X c)
[[uh(X )]]X=Xc = lim ǫ→0
[u(X + ǫ) − u(X − ǫ)]X=Xc
= N 1(X c)
u21 − u1
1
+ N 2(X c)
u2
2 − u12
= a1 N 1(X c) + a2 N 2(X c)
u12
u21
Ω
uhi (x) =4I =1
N I (x)uIi +3J =1
N J (x)ψ(x)aJi
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XC
φ<0 φ>0
1 4XC
φ<0
23
23
1 2
crack
crack
φ>0
1
4
2
3
1 4
N 2(X )
N 1(X )
N 1(X )
N 4(X )
u+ u+
u− u−
I
N I uI I
N I uI
[[u]] [[u]]
N 1(X ) (H (X − X c) − H (X 1 − X c))
N 2(X ) (H (X − X c) − H (X 2 − X c))
ψ(x)
uIi = 0
aJi = 1
(N 1, N 2, N 3)
3J =1
N J (x) = 1.
I ∈N
N I (x) = 1
Ψ(x)
I ∈N
N I (x)Ψ(x) = Ψ(x)
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000000111111
000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111
00
00
00
11
11
11
00
00
00
11
11
11
00
00
00
11
11
11
0000
00
1111
11
0000
00
1111
11
0000
00
1111
11
0000
00
1111
11
0000
00
1111
11
0000
00
1111
11
0000
00
1111
11
0000
00
1111
11
0000
00
1111
11
00
00
00
11
11
11000000111111000000111111 000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111
Ω
Senr
Ω p.e.
N I (x) f i(x) ψ(x) f i(x)×ψ(x)
st
st
st
st
st
st
st
st
Ψ N I Ψ
Ωstd
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Ωenr
Ωblnd
000000000111111111000000000000111111111111000000000000111111111111 000000000111111111000000000000111111111111000000000000111111111111000000000000111111111111000000000111111111000000000000111111111111
Ωenr
Ωblnd
Ωstd
Ωenr
Ωblnd Ωstd
uI = 0 aJ = 1
uh(x) =
J ∈N enr
N J (x)Ψ(x) = Ψ(x) ∀x ∈ ΩenrN J (x)Ψ(x) = Ψ(x) ∀x ∈ Ωblnd
N J (x)Ψ(x) = 0 ∀x ∈ Ωstd
Ωenr
Ωstd
N J Ψ
Ψ(x) = xH (x)
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H
x = 0
uh(x) =2I =1
N I (x) + N 1(x)(xH (x) − x1H (x1))a1
uh(ξ ) = u1(1 − ξ ) + u2ξ + a1ξh(1 − ξ )
ξ = x − x1
h
h
uh
e
e ≡ u − uint
x
e,x|x ≡ d
dxe(x) = 0
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x
e(x) = e(x) + e,x|x(x − x) + 1
2e,xx|x(x − x)2 + O(h3)
e(x) = e(x) + 1
2e,xx|x(x − x)2
x = x1 e(x1) = 0 uh
uh(xI ) = u(xI )
e(x) = −1
2e,xx|x(x − x)2
e(x) = u,xx + 2a1
h
1
2(x − x1)2 ≤ 1
8h2
e(x) ≤ 1
8 h2
max(u,xx +
2a1
h )
2a1/h
h2 h
n ξ n n > 1
e(x)
≤ 1
8
h2max(u,xx + 2a1
hn
)
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r
s
r
y
x
(1, 1)
(1,−1)
(−1, 1)
(−1,−1)
s = 1
r = 1r = −1
s = −1
s
1 : (x1, y1)
2 : (x2, y2)
3 : (x3, y3)
4 : (x4, y4)
N I , I = 1...4
N 1(r, s) = 1
4(1 − r)(1 − s)
N 2(r, s) = 1
4(1 + r)(1 − s)
N 3(r, s) = 1
4(1 + r)(1 + s)
N 4(r, s) = 1
4(1 − r)(1 + s)
r s
ue(M ) =
» uxuy
– =
» N 1 N 2 N 3 N 4 0 0 0 0
0 0 0 0 N 1 N 2 N 3 N 4
–
266666666664
ux1ux2ux3ux4uy1uy2uy3uy4
377777777775
= Nestd(M ) qe
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ue(M ) =
» uxuy
– =
» N 1 N 2 N 3 N 4 0 0 0 0
0 0 0 0 N 1 N 2 N 3 N 4. . .
. . . N 1ψ1 N 2ψ2 N 3ψ3 N 4ψ4 0 0 0 0
0 0 0 0 N 1ψ1 N 2ψ2 N 3ψ3 N 4ψ4
–
2666666666666666666666666664
ux1ux2ux3ux4uy1uy2uy3uy4ax1ax2ax3ax4ay1ay2ay3ay4
3777777777777777777777777775
ue(M ) = [ Nestd(M ) Ne
enr(M ) ] qe
ue(M ) = Ne(M ) qe
Ne(M ) = [Nestd(M ) Ne
enr(M )]
ψ(x)
ψI
ψI (x) = ψ(x) − ψ(xI )
ǫ =
ǫxxǫyy
2ǫxy
= Due(M )
D =
∂
∂x 0
0 ∂
∂y∂
∂y
∂
∂x
ue(M )
ǫ = DNe(M ) qe = Be(M ) qe
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Be(M )
Be(M ) = [Bestd(M ) Beenr(M )]
Bestd(M )
Bestd =
N 1,x N 2,x N 3,x N 4,x 0 0 0 00 0 0 0 N 1,y N 2,y N 3,y N 4,y
N 1,y N 2,y N 3,y N 4,y N 1,x N 2,x N 3,x N 4,x
Be
enr(M )
Beenr =
24 (N 1ψ1),x (N 2ψ2),x (N 3ψ3),x (N 4ψ4),x 0 0 0 0
0 0 0 0 (N 1ψ1),y (N 2ψ2),y (N 3ψ3),y (N 4ψ4),y(N 1ψ1),y (N 2ψ2),y (N 3ψ3),y (N 4ψ4),y (N 1ψ1),x (N 2ψ2),x (N 3ψ3),x (N 4ψ4),x
35
uhi,j =I ∈S
N J,i(x) ujJ +I ∈S
(N J (x)H (φ(x))),i ajJ
=I ∈S
N J,i(x) ujJ +I ∈S
(N J,i(x)H (φ(x)) + N J (x)H ,i(φ(x))) ajJ
H ,i(φ(x)) = δ
H ,i = 1 H ,i = 0
Beenr =
24 N 1,xψ1 N 2,xψ2 N 3,xψ3 N 4,xψ4 0 0 0 0
0 0 0 0 N 1,yψ1 N 2,yψ2 N 3,yψ3 N 4,yψ4
N 1,xψ1 N 2,xψ2 N 3,xψ3 N 4,xψ4 N 1,yψ1 N 2,yψ2 N 3,yψ3 N 4,yψ4
35
ψ(x) = |φ(x)|
ψ(x)
ψ(x),i
= sign(φ(x)) φ,i(x)
φ(x)
φ(x) = [ N 1 N 2 N 3 N 4 ]
φ1
φ2
φ3
φ4
x
φ(x),x = [ N 1,x N 2,x N 3,x N 4,x ]
φ1
φ2
φ3
φ4
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y
φ(x),y = [ N 1,y N 2,y N 3,y N 4,y ]
φ1
φ2
φ3
φ4
∂N I ∂x
= ∂N I
∂r
∂r
∂x +
∂N I ∂s
∂s
∂x∂N I ∂y
= ∂N I
∂r
∂r
∂y +
∂ N I ∂s
∂s
∂y
N I
N ,x N ,y = N ,r N ,s
∂r
∂x
∂r
∂y
∂s∂x
∂s∂y
= J−1
J N I (r, s)
r s
N 1,r = −1
4(1 − s) N 1,s = −1
4(1 − r)
N 2,r = 1
4(1 − s) N 2,s = −1
4(1 + r)
N 3,r = 1
4
(1 + s) N 3,s = 1
4
(1 + r)
N 4,r = −1
4(1 + s) N 4,s =
1
4(1 − r)
J =
∂x
∂r
∂x
∂s
∂y
∂r
∂y
∂s
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x =4I =1
N I xI , ∂x
∂r =
4I =1
∂N I ∂r
xI , ∂x
∂s =
4I =1
∂N I ∂s
xI
∂x
∂r =
N 1,r N 2,r N 3,r N 4,r
x1
x2
x3
x4
∂x
∂s =
N 1,s N 2,s N 3,s N 4,s
x1
x2
x3
x4
y =4I =1
N I yI , ∂y
∂r =
4I =1
∂N I ∂r
yI , ∂y
∂s =
4I =1
∂N I ∂s
yI
∂y
∂r =
N 1,r N 2,r N 3,r N 4,r y1
y2
y3
y4
∂y
∂s =
N 1,s N 2,s N 3,s N 4,s
y1
y2
y3
y4
Ke = Ωe
BeT
(M ) Ce Be(M ) dΩ = 1
−1 1
−1
BeT
(r, s) Ce Be(r, s) det J dr ds
Ce
8 × 8
Kel =
Ωe
BeT
std(M )CeBestd(M )
Ωe
BeT
std(M )CeBeenr(M )
Ωe
BeT
enr(M )CeBestd(M )
Ωe
BeT
enr(M )CeBeenr(M )
16 × 16
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crack
background cell
1
2
3
4
5
6
7
8
9
1011
crack
5
9
6
7
8
1
2
3
4
background cellCrack path produced
by level set Crack path recognized by the code
φ
F
F =
Ω−
F (X)dΩ +
Ω+
F (X)dΩ
=
Ω−
F (X(ξ)) detJ−(ξ)dΩ +
Ω+
F (X(ξ)) detJ+(ξ) dΩ
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Voronoi cells
Delaunay triangulation
Crack
Gauss point
Node
A2A1 A3
A4 A5 A6
A8A7 A9
A−i
A+i
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enriched nodes
not enriched nodes
crack crack tip crack
crack tip
∇ (F jψi)· ∇
(F lψk) dx
r−0.5
∇F i
G :
xy
←
x yy
ξ w
ξ = G(ξ ) , w = w det(
∇G)
∇0 · P − b = ∅ ∀X ∈ Ω0 \ Γc0
u(X, t) = u(X, t) on Γu0
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n0 · P(X, t) = t0(X, t) on Γt0
n0 · P(X, t) = 0 on Γc0
u
t0
Γc0
Γu0
Γt0
Γc0 = Γ0 , (Γu0
Γt0)
(Γt0
Γc0)
(Γu0
Γc0) =∅
u ∈ V
δW = δW int − δW ext = 0 ∀δ u
δW int =
Ω0
(∇ ⊗ δ u)T : P dΩ0
δW ext =
Ω0
δ u · b dΩ0 +
Γt0
δ u · t0 dΓ0
V =
u(·, t)|u(·, t) ∈ H1, u(·, t) = u(t) on Γu0 , u discontinuous on Γc0
V0 =
δ u|δ u ∈ H1, δ u = 0 on Γu0 , δ u discontinuous on Γc0
Space of Bounded Deformations
•
•
•
•
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23
12
12
13
23
udisc = ξ∗3 Ψ3(ξ∗) a3
ξ∗ = [ξ ∗1 ξ ∗2 ξ ∗3 ] 23P
ξ ∗3 = 1 − ξ ∗1 − ξ ∗2 Ψ3(ξ∗) = sign(φ(ξ∗)) − sign(φ3) ξ∗ ξ
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2
3 1P13
2
P
N 3(ξ) = 1 − ξ 1 − ξ 2
N 1(ξ) = ξ 1
N 2(ξ) = ξ 2
ξ 1ξ 1
ξ 2ξ 2
ξ ∗1 = ξ 1ξ 1P
, ξ ∗2 = ξ 2
ξ 1P
P
31
udisc = ξ∗2 Ψ2(ξ∗) a2
ξ ∗1 = ξ 1 − ξ 1P ξ 2P
ξ 2, ξ ∗2 = ξ 2ξ 2P
Ψ2(ξ∗) = sign (φ(ξ∗)) − sign(φ2) a3 = aP = 0
udisc = I ξ∗I ΨI (ξ
∗) aI
aI
udisc
Ωenr Ωenr
Ωenr
Ωenr
B
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crack tip enrichment
Heaviside enrichment
B = [B1 B2 B3 B4]
=
√ r sin
θ
2,√
r cosθ
2,√
r sinθ
2sin(θ),
√ r cos
θ
2sin(θ)
B
r = 0
uh(X) =I ∈S
N I (X) uI +
I ∈Sc(X)
N I (X) H (f I (X)) aI
+
I ∈St(X)
N I (X)K
BK (X) bKI
St
B
a b c d
a
p
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0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
aa
crack
b
cd
A+
A−
r+ r−
r+ = A+
A+ + A− , r− = A−
A+ + A−
a b c d a b
KuuIJ Kua
IJ KubIJK
KauIJ Kaa
IJ KabIJK
KbuIJK Kba
IJK KbbIJK
uJ aJ
bJK
=
f extI f extI f extIK
K d = f ext
K d = u a bT
f ext =
f u f a f bT
f b =
f b1 f b2 f b3 f b4
f uI =
Ω
N I b dΩ +
Γt
N I t dΓ
f aI =
Ω
N I (H (φ(X)) − H (φ(XI ))) b dΩ+
Γt
N I (H (φ(X)) − H (φ(XI ))) t dΓ
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f blI =
Ω
N I
BlI (X) − BlI (XI )
b dΩ+
Γt
N I
BlI (X) − BlI (XI )
t dΓ
K =
Ω
BT C B dΩ
B
BuI = N I,X 0
0 N I,Y
N I,Y N I,X
BaI =
N I,X (H (φ(X)) − H (φ(XI ))) 00 N I,Y (H (φ(X)) − H (φ(XI )))
N I,Y (H (φ(X)) − H (φ(XI ))) N I,X (H (φ(X)) − H (φ(XI )))
BblI |l=1,2,3,4 =
N I
BlK (X) − BlK (XI ),X
0
0
N I
BlK (X) − BlK (XI ),Y
N I
BlK (X) − BlK (XI )
,Y
N I
BlK (X) − BlK (XI )
,X
N I BlK (X)
,i
= N I,i BlK (X) + N I BlK (X),i
α
Bl,i = B l,r r,i + Bl,θ θ,i
θ
r
, i
Bl
,r
Bl
,θ
B1,r =
sin(θ/2)
2√
2B1,θ =
√ 2cos(θ/2)
2
B2,r =
cos(θ/2)
2√
2B2,θ = −
√ 2sin(θ/2)
2
B3,r =
sin(θ/2) sin(θ)
2√
2B3,θ =
√ r
cos(θ/2) sin(θ)
2 + sin(θ/2) cos(θ)
B4,r =
cos(θ/2) sin(θ)
2√
2B4,θ =
√ r
sin(θ/2) sin(θ)
2 + cos(θ/2) cos(θ)
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X
Y
α
X
Y
r
θ
r, X = cos(θ) θ, X = −sin/r
r,Y = sin(θ) θ,Y = cos/r
B1, X =
sin(θ/2)
2√
2B1,Y =
cos(θ/2)
2√
2
B2, X =
cos(θ/2)
2√
2B2,Y =
sin(θ/2)
2√
2
B3, X =
−sin(3θ/2) sin(θ)
2√ 2B3,Y =
sin(θ/2) + sin(3θ/2) cos(θ)
2√ 2B4, X = −cos(3θ/2) sin(θ)
2√
2B4,Y =
cos(θ/2) + cos(3θ/2) cos(θ)
2√
2
B,X = B, X cos(α) + B,Y sin(α)
B,Y = B, X sin(α) + B,Y cos(α)
α
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Branching discontinuityIntersecting discontinuity
φ1(x) = 0φ1(x) = 0
φ2(x) = 0φ2(x) = 0
S1c
φ1(X) = 0
S2c φ2(X) = 0
S3c = S1
c
S2c
S1t
S2t
uh(X) =I ∈S(X)
N I (X) uI +
I ∈S1c(X)
N I (X) H (φ1(X)) a(1)I
+
I ∈S2c(X)
N I (X) H (φ2(X)) a(2)I
+ I ∈S3c(X)
N I (X) H (φ1(X)) H (φ2(X)) a
(3)
I
+
I ∈S1t (X)
N I (X)K
B(1)K (X) b
(1)KI
+
I ∈S2t (X)
N I (X)K
B(2)K (X) b
(2)KI
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(φ1 < 0, φ2 < 0)
(φ1 >0, φ2 > 0)
(φ1 > 0, φ2 < 0)
(φ1 > 0, φ2 < 0)
(φ1 > 0, φ2 >0) (φ1 < 0, φ2 < 0) 1 X
φ1(X) =
φ0
1(X),
φ02(X1) φ0
2(X) > 0φ0
2(X),
φ02(X1) φ0
2(X) < 0
0
uh(X) =I ∈S(X)
N I (X) uI +
ncn=1
I ∈Sc(X)
N I (X) H (φ(n)I (X)) a
(n)I
+
mtm=1
I ∈St(X)
N I (X)K
B(m)K (X) b
(m)KI
nc mt
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∇0 · P − b = ∅ ∀X ∈ Ω0 \ Γc0
u(X, t) = u(X, t) on Γu0
n0 · P(X, t) = t0(X, t) on Γt0
n0 · P(X, t) = 0 on Γc0 if not in contact
t+0t = t−0t = 0, t+
0N = −t−0N on Γc0 if in contact
[[uN ]] ≤ 0 on Γc0
[[n · P]] = 0 on Γc0
t0N = n · P · n
t0t
[[uN ]] = u+ · n+ = u− · n− ≤ 0
n+ = n−
Ω0
(∇ ⊗ δ u)T
: P dΩ0 −
Ω0
δ u · b dΩ0 −
Γt0
δ u · t0 dΓ0 + δ
Γc0
λ [[uN ]] dΓ0 ≥ 0
C−1
C0
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•
•
•
•
K II
K II
θc
σθθ
vc
σθθ = K I √
2πrf I h(θ, vc) +
K II √ 2πr
f II h (θ, vc)
f I h f II h
vc
σcθθ
σcθθ
σcθθ = K cI √
2πr
K cI
K I sinθc + K II (3cosθc − 1) = 0
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θc = 2arctan
K I −
K 2I + 8K 2II
4K II
mdiag = m
nnodes
1
mes(Ωel)
Ωel
ψ2 dΩel
Ωel m mes(Ω)el
nnodes Ω ψ
M
lumped
II = J M
consistent
IJ , or
MlumpedII = m
MconsistentII
J
MconsistentIJ
∆t ≤ ∆tc = 2/ωmax
uh(X) = N 1 u1 + N 1 φ1 a1 + N 2 u2 + N 2 φ2a2
lumped =
m1 0 0 00 m2 0 00 0 m3 00 0 0 m4
ωmax det(K− ω M) K
M
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mi
E hkin = 0.5uT Mlumped u
E kin = 0.5
Ωel v2 dΩ
¯u
a
E hkin = 0.5
m1 u21 + m2 u2
2
= 0.5 ˙u
2(m1 + m2)
E kin = 0.5 m ˙u2
= E hkin m1 = m2 = 0.5 m m
˙u = aφ1(x)
u
E h
kin = 0.5 m3 a2
1 + m4 a2
2 = 0.5 a2
(m3 + m4)
E kin = 0.5
a2
Ωel
ψ21 dΩel
m3 m4
m3 = m4 = m
2 mes(Ω)el
Ωel
ψ21 dΩel
l
N 1(x) = 1 − x
l
N 2(x) = x
l
FE = A l
1/3 1/61/6 1/3
,
FE = E A
l
1 −1−1 1
E
A
∆tc,FE = 2
ωmax= l
3E
lumpedFE =
A l
1/2 0
0 1/2
∆tlumpedc,FE = l
E
=√
3∆tc,FE
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s
s
uh(x) = N 1(x) u1 + N 1(x) S (x − s) a1
+ N 2(x) u2 + N 2(x) S (x − s)a2
XFEM = A l 1/3 1/6
1/6 1/32s2 − 2s + 1/3 − 2/3s3 1/6 − s2 + 2/3s3
1/6 − s2 + 2/3s3 1/3 − 21
. . .
. . .
2s2 − 2s + 1/3 − 2/3s3 1/6 − s2 + 2/3s3
1/6 − s2 + 2/3s3 1/3 − 2/3s3
1/3 1/61 − 2s 2s − 1
1/6 1/3
XFEM = E A
l
1 −1 1 − 2s 2s − 1−1 1 2s − 1 1 − 2s
1−
2s 2s−
1 1 −
12s − 1 1 − 2s −1 1
lumpedXFEM = 0.5
A l
1 0 0 00 1 0 00 0 1 00 0 0 1
s
x = 0 x = l
0
l
x = 0
x = l
∆tlumpedc,XFEM = 1√ 2
∆tlumpedc,FE
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crack crack
crack
effective crack length
a) b)
c) d)
1 1 2
34
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Sc
√ 2sin(θ/2)
Sc
a b
a · n0 = b · n0 = 0
n0
a b
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crack
T (X)
a · ∇0T = ∇0T · a = 0 in Ω0
b· ∇0
T = ∇0
T ·
b = 0 in Ω0
∂φ
∂t + v · ∇φ = 0
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r = φ2 + ψ2 ψ φ
θ = arctan(φ/ψ)
θ
θ = ±π
φ = 0
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P
ΦI
Φ∗J (X) = pT (X) · A∗(X)−1 · D∗(XJ )
A∗(X) =J
p(XJ ) pT (XJ ) W (r∗J ; h∗)
D∗(XJ ) =J
p(XJ ) W (r∗J ; h∗)
h∗
3
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Γc,ext
Γc,ext
strong embedded elements
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Γc,ext
P
a) b) c)
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a) b)
uh(X) =I ∈S
N I (X) uJ +M(e)s (X) [[u
(e)I (X )]]
u
u
(e)
M(e)s
M(e)s (X) =
0 ∀(e) /∈ S
H (e)s − ρ(e) ∀(e) ∈ S
ρ(e) =N +e
I =1
N +I (X)
H s
S
N +e
(e)
Ω+0
ǫh(X) =I ∈S
(∇0N I (X) ⊗ uI )S −∇ρ(e) ⊗ [[u
(e)I (X )]]
S +
η(e)s
k
[[u
(e)I (X )]] ⊗ n
S
S η(e)s /k
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η(e)s
η(e)s =
1 ∀X ∈ S ke0 ∀X /∈ S ke
k
K
(e)uu K
(e)uu
K(e)
uu K
(e)
uu u(e)
[[u(e)
I ]] =
FextI 0
K(e)uu =
Ω0
BT C B dΩ0
K(e)uu =
Ω0
BT C B dΩ0
K(e)uu =
Ω0
BT ∗ C B dΩ0
K(e)uu =
Ω0
BT ∗ C B dΩ0
C
B
∇ρ(e) =
∂ρ(e)
∂x 0
0 ∂ρ(e)
∂y∂ρ(e)
∂y∂ρ(e)
∂x
n(e) =
nx 00 ny
ny nx
B
B∗ = B
B∗ = B
[[u(e)I ]]
[[u(e)I ]] = −
K
(e)uu
−1
K(e)uu u(e)
[[u(e)I ]]
K u = f
K = Kuu − Kuu K−1uu Kuu
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S
Ω− Ω+
S
interelement − separation methods
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