Lecture note - XFEM and Meshfree_2.pdf

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



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



[[(·)]]

∂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



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



ψ

Ψ

φ

Φ

δ,δ

ξ, η

Ω

Γ







•

global

local

•

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•

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•

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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



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



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



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



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



−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



=

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



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



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



∇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



J ∈S

∇0W (X − XJ , h0) V 0J

uI ≡ 0

∇0uh(X) =

J ∈S(uJ − uI ) ∇0W (XI − XJ , h0) V 0

J



∇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)



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)



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



|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)



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



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



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



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)



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 ]



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



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



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



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



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%



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

´



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)



f (X ) = aX 2 + bX + c



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





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



10%

∇0W (407)J (X) uJ

K ∈S∇0W

(407)K (X) uK

∂W (407)J (X)∂X uJ = 0

10%

5% 10%



F (X, Y ) =X 2 + Y 2

(R = 1.6)

F (X, Y ) =X 2 + Y 2

(R = 0.6)



∇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) = ∅



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



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



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



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



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



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



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



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



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



•

• 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



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)



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



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



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



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



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



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



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



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



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 ]



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)





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





undesired

hI 0



CRACK

CRACK

CRACK

Visibility criterion

Diffraction criterion

Transparency criterion

Crack line

Crack line

Crack line



crack

interdiscontinuities

I

crack

Domain of influence

I

crack crack

s0(x)

s2(x)

x

xI

xc

s1



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)



I

crack



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



na

nB

ω

nA · nB ≤ β

nA · nB ≤ β β = 0o

β = 0o

ω = 90o

enrichment



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 )



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)



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 )



crack

Enriched

Transition

Linear

r1

r2



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



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





local

global



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



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))



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



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



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



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



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



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)



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



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)



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)



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]



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



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)



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



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)



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



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

)



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



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



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



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



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



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





Voronoi cells

Delaunay triangulation

Crack

Gauss point

Node

A2A1 A3

A4 A5 A6

A8A7 A9

A−i

A+i



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




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

•

•

•

•



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



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



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



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



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)



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



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



(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



∇0 · P − b = ∅ ∀X ∈ Ω0 \ Γc0


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



•

•

•

•

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



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



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



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



crack crack

crack

effective crack length

a) b)

c) d)

1 1 2

34



Sc

√ 2sin(θ/2)

Sc

a b

a · n0 = b · n0 = 0

n0

a b



crack

T (X)

a · ∇0T = ∇0T · a = 0 in Ω0

b· ∇0

T = ∇0

T ·

b = 0 in Ω0

∂φ

∂t + v · ∇φ = 0



r = φ2 + ψ2 ψ φ

θ = arctan(φ/ψ)

θ

θ = ±π

φ = 0







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



Γc,ext

Γc,ext

strong embedded elements





Γc,ext

P

a) b) c)



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



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



S

Ω− Ω+

S

interelement − separation methods









Documents

Lecture note - XFEM and Meshfree_2.pdf