lecture note - xfem and meshfree_2.pdf
TRANSCRIPT
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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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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)
![Page 44: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/44.jpg)
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f (X ) = aX 2 + bX + c
![Page 45: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/45.jpg)
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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
![Page 46: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/46.jpg)
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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
![Page 48: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/48.jpg)
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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)
![Page 50: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/50.jpg)
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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 ]
![Page 71: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/71.jpg)
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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 )
![Page 85: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/85.jpg)
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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
![Page 93: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/93.jpg)
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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
![Page 95: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/95.jpg)
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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
![Page 97: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/97.jpg)
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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)
![Page 98: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/98.jpg)
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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)
![Page 101: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/101.jpg)
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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
![Page 103: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/103.jpg)
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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)
![Page 104: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/104.jpg)
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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
![Page 105: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/105.jpg)
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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)
![Page 106: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/106.jpg)
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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
![Page 107: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/107.jpg)
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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
)
![Page 108: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/108.jpg)
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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
![Page 109: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/109.jpg)
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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
![Page 111: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/111.jpg)
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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
![Page 112: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/112.jpg)
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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
![Page 113: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/113.jpg)
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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
![Page 114: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/114.jpg)
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Voronoi cells
Delaunay triangulation
Crack
Gauss point
Node
A2A1 A3
A4 A5 A6
A8A7 A9
A−i
A+i
![Page 116: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/116.jpg)
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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
![Page 117: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/117.jpg)
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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
•
•
•
•
![Page 118: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/118.jpg)
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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
![Page 119: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/119.jpg)
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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
![Page 120: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/120.jpg)
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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
![Page 121: Lecture note - XFEM and Meshfree_2.pdf](https://reader035.vdocuments.mx/reader035/viewer/2022081802/5695d13e1a28ab9b0295baac/html5/thumbnails/121.jpg)
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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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