Lecture note

7/21/2019 Lecture note - XFEM and Meshfree_2.pdf

http://slidepdf.com/reader/full/lecture-note-xfem-and-meshfree2pdf 1/147

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),

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

a(X, t) = ∂ v(X, t)

∂t +

∂ vi(x, t)

∂xi(X, 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

F − I

[[(·)]]

∂D(·)∂t , (·)

∂ (·)∂ X , ∇, (·),i

K I , K II

Λ, Λ

ǫ, ǫ

σθθ

ξ, η

global

ΦJ (X) p(X) uJ = p(XJ )

ΦJ (X) uJ = ΦJ (X) p(XJ ) = p(X)

completeness

reproducing conditions

ΦJ (X) = 1

ΦJ (X) X J = X J

ΦJ (X) Y J = Y

ΦJ (X) X Ji = X i

ΦJ,X(X) = 0J

ΦJ,Y (X) = 0

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

ΦJ,i(X) = 0 J

ΦJ,i(X) X Jj = δ ij

ΦJ (x)

uJ = 1

ΦJ (x) = 1

partition of unities

I ∈S

I ∈SmI vI = 0

mI vI = −J ∈S

∇ΦI (XJ ) · σ(XJ ) wJ

ΦI (XJ )

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

Dt I mI

XI = I m

I vI ×

+ vI ×

=0 = 0

mI vI × XI

ΦI,m(XJ ) σmj(XJ )wJ

ǫijk X Ik k − th

ǫijkJ I ΦI,m(XJ )X Ik δmk

σmj(XJ )wJ = ǫijkδ mkJ σmj(XJ )wJ

ǫijmσmj(XJ ) =0

wJ = 0

|u(X i) − ui| ≤ Chk

Support size of particle I

R_KR_I

lim h0→0

W (XI − XJ , h0) = δ (XI − XJ )

W (XI − XJ , h0)dΩ0 = 1

W (XI − XJ , h0) = 0 ∀ XI − XJ ≥ R

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

z = r/h0

2/3 D = 1

10/(7 π) D = 21/π D = 3

z = ||XI − XJ ||

∂X iJ =

∂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

h/x = 1

−1 −0.5 0 0.5 1−0.2

rho(x)

h/x = 1

−3 −2 −1 0 1 2 30

h/x = 2

−1 −0.5 0 0.5 1−0.2

rho(x)

h/x = 2

−3 −2 −1 0 1 2 30

h/x = 4

−1 −0.5 0 0.5 1−0.2

rho(x)

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

W J (x) = W (x − xJ (t), h(x, t))

ht+∆t = ht + h ∆t

h = 1/3

∇ · v

h = h0 F

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)

ΦJ (X) = 0

uh(xI ) = uI

ΦI (XJ ) = δ IJ δ IJ

uh(X, t) =

u(Y, t) W (X − Y, h0(Y)) dY

W (X − Y, h0(Y)) 1 d Y = 1

W (X − Y, h0(Y)) Y dY = X

W (X − Y, h0(Y)) X dY = X

W (X − Y, h0(Y)) (X − Y ) d Y = 0

uh(X, t)

∇0uh(X, t) =

∇0u(Y, t) W (X − Y, h0(Y)) dY

∇0uh(X, t) =

∇0 [u(Y, t) W (X − Y, h0(Y))] dY

∇0u(Y, t) W (X − Y, h0(Y)) dY

∇0uh(X, t) = Γ0

u(Y, t) W (X − Y, h0(Y)) n0 dΓ0

∇0u(Y, t) W (X − Y, h0(Y)) dY

∇0uh(X, t) = −

∇0u(Y, t) W (X − Y, h0(Y)) dY

ΦJ (X) = W (X − XJ , h0) V 0J

∇0uh(X) = −J ∈S

uJ ∇0ΦJ (X) with ∇0ΦJ = ∇0W (X − XJ , h0) V 0J

J ∈S

∇0W (X − XJ , h0) V 0J

uI ≡ 0

∇0uh(X) =

J ∈S(uJ − uI ) ∇0W (XI − XJ , h0) V 0

∇0uh(X, t) =I ∈S

GI (X) uI (t)

uh,i(X, t) =I ∈S

GiI (X) uI (t)

W S I (X) = W I (X)I ∈S

W I (X)

GI (X) = a(X) · ∇0W S I (X) = aij(X)W S jI (X)

I ∈S

GI (X) ⊗ XI = δ ij

A aT = 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)

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

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

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

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)

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

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|

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

GXI (X)|

|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

∇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

W (X − XJ , h) V 0J

B(X) = −J ∈SXJ ⊗ ∇0W

(XJ − X, h0)−1

∇0uh(X, t) =

−J ∈S

uJ (t) ∇0W S (XJ − X, h0) V 0J

· B(X)

C (X, Y)

uh(X) =

C (X, Y)W (X − Y)u(Y)dΩY

K (X, Y) = C (X, Y)W (X

C (X, Y)

u(X) = pT(X)a

p(X)u(X) = p(X)pT(X)a

p(Y)W (X − Y)u(Y)dΩY =

p(Y)pT(Y)W (X − Y)dΩYa

uh(X) = pT(X)a

uh(X) = pT(X)

p(Y)pT(Y)W (X−Y)dΩY

−1 ΩY

p(Y)w(X−Y)u(Y)dΩY

C (X, Y) = pT(X)

p(Y)pT(Y)W (X − Y)dΩY

= pT(X)[M(X)]−1p(Y)

uh(X) =

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) =

p(Y)pT(Y)W (X − Y)dΩY

=I ∈S

p(XI )pT(XI )W (X − XI )V 0I

(xI , uI )

uI = u(xI )

(x) = a0 + a1x + a2x

+ ... + amx

uh(x) = pT(x)a

uh(xi)

uh(xI )

J =nI =1

[uh(xI ) − uI ]2 =

[pT(xI )a − uI ]2

p(xI )pT(xI )a =nI =1

p(xI )uI

pT(x) = [1 x] aT = [a0 a1]

1 xI xI x2

3 66 14

a0 = −5/6 a1 = 1.5

uh(x) = −5

p(X) =

1 X Y ∀ X ∈ ℜ2

uh(X, t) =M I =1

pI (X) aI (X, t) = pT (X i) a(X i)

J(a(X i)) =N J =1

W (X − XJ , h0)

M I =1

pI (XJ )T aI (X, t) − u(XJ )

P(X) a(X) − u(X)T

P(X) a(X) − u(X)

N W (X) = 0

(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 ... W (X−

∂ 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(x) = PT (X) W(X) PT (X) =A∈RM ×M

PT (X) W(X) =B∈RM ×N

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)

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

p(x) = 1

ΦI (X) = W I (X)I ∈S

W I (X)

M = 3 p(X) = [1 X Y ]T

A(X) =

1 ... 1x1 ... xN y1 ... yN

W 1 ... 0

0 ... W N

1 x1 y1

1 xN yN

A 3 × 3

W(X) A

p(X) = [1 X Y ]

κ = λmaxλmin

κ → ∞

∂ 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−

∂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 (X) = γ (X) · p(XJ ) W (X − XJ , h0)

A(X) · γ (X) = p(XJ )

∇0A(X) · γ (X) + A(X) · ∇0γ (X) = ∇0p(XJ )

∇0γ (X)

ΦI (X) = W (XI , X) PT

XI − X

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

h0I h0I XI

W (XI , X) = W

XI − X

h0 h0J

< f,g >X=J ∈S

W (XJ , X) f XJ − X

gXJ − X

u(Z) ≃ uh(Z, X) = PT

Z − X

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

(R = 0.8) (R = 0.3) R

uh(X) =J ∈S

ΦJ (X)

pK (X) aJK

uh(X) =J ∈S

ΦJ (X) uJ +J ∈S

ΦJ (X)

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 = 0.6

R = 1.6

R = 0.6

F x F ,X = 2X

0.005%

F (X, Y ) = sin

X 2 + Y 2

F 0 ≤ X ≤ π2

0 ≤ Y ≤ π2

π/300 R

V = d2

d < h <√

u,X(X) = −N J =1

V J W J,X(X)uJ

u,X(X(5)) = −V J

(25),X u2 + W

(45),X u4 + W

(55),X u5 + W

(65),X u6 + W

(85),X u8

W (25),X = W (55)

W (85),X = 0

u,X(X(5)) = V J

W (54),X u4 + W

(56),X u6

f (X ) = aX 2 + bX + c Y

F ,X = 2X cos`X 2 + Y 2

F (X, Y ) = sin X 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(X(5)) = V J

W (54),X f 4 + W

(56),X f 6

a (x(4))2 + bx(4) + c

a (x(6))2 + bx(6) + c

= V J w a(x

− (x(5)

− d)2+ b(x

+ 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 ,X(X(I )) = 2 w d3

2 a x(I ) + b

errabs(d,h,x(I )) = 2 a x(I ) + b − 2 w(d, h) d3

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 =√

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

(R = 1.6)

25 × 25 = 625

V J = d2

V new,J = (1.1d)2 = 1.21 V old,J

∇0u(X(407))

V I ≡ −

J ∈S

∇0W (407)J (X) uJ

K ∈S∇0W

(407)K (X) uK

∂W (407)J (X)∂X uJ = 0

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

u(X, t) = u(X, t) on Γu0

n0 · P(X, t) = t0(X, t) on Γt0

Γt0 = Γ0 , (Γu0

Γt0) = ∅

J = 0 J 0

0∇0 · P + b on Ω0

0F : PT

F = ∇u+I I

u(X, t) = u(X, t)

n0 · P(X, t) = t0(X, t) Γt0

Γu0 ∪ Γt0 = Γ0 (Γu0 ∩ Γt0) = ∅

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 ,

∇0 · P · δ u dΩ0 +

0 (b − u) · δ u dΩ0 = 0

Ω0 ∇0

·δ u dΩ0 = Ω0 ∇

·δ u) dΩ0

− Ω0

⊗δ u)

T : P dΩ0

∇0 · (P · δ u) dΩ0 =

n0 · P · δ u dΓ0

t = n0 · P Ω0

∇0 · (P · δ u) dΩ0 =

t · δ u dΓ0

(∇0 ⊗ δ u)T : P dΩ0 − Ω0

0 b · δ u dΩ0 +

t0 · δ u dΓ0

0 δ u · u dΩ0 = 0

mIJ uJ = f extI − f intI ,

f extI f intI

f extI =

0 ΦI b dΩ0 +

ΦI t dΓ0

f intI = Ω0

∇0ΦI · P dΩ0

mIJ =I ∈S

0 ΨI (X) ΦJ (X) dΩ0.

ΦJ (X) = δ (X − XI )

ΨJ (X) = ΦJ (X)

f (X) dΩ0 =J ∈S

f (XJ ) V 0J

V 0J J

f intI =J ∈S

V 0J ∇0ΦI (XJ ) · PJ

mI =J ∈S

mIJ =J ∈S

0 ΦI (X) ΨJ (X) dΩ0

0 ΦI (X)J ∈S

ΨJ (X) dΩ0

0 ΦI (X) dΩ0

J ∈S J ΦI (X) V 0

N totI =1

J ∈S

J ΦI (X) V 0J =

J ∈S

N totI =1

ΦI (X)

V 0J =

J ∈S

J V 0J = M

∇ΦI (X) dΩ0 =

n0ΦI (X) dΓ0

ǫ(XM ) =

ǫ Ψ(X − XM ) dΩ0

ǫ ǫ = 0.5(ui,j + uj,i)

Ψ(X − XM )

Ψ(X − XM ) ≥ 0 Ω0

Ψ(X − XM ) dΩ0 = 1

stress point

particle particle particle particle

particleparticleparticleparticle

Ψ(X − XM ) = 1

AM ∀XM ∈ Ω0, otherwise Ψ(X − XM ) = 0

ǫ(XM ) = 12AM

(ui,j + uj,i) dΩ0

(ui nj + uj ni) dΓ0

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

uS I = J ∈S

ΦJ (XS I ) uP J , vS I =

J ∈SΦJ (XS

I ) vP J

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 =√

f (X) dΩ0 =

+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 )

det Jξ

Jξ = ∂ X

∂ ξ

f int =

wJ detJξ(ξJ ) ∇0Φ(X(ξJ ) − XP ) P(ξJ )

u ∈ V

δW = δW int − δW ext = 0 ∀δ u ∈ H1

δW int =

(∇ ⊗ δ u)T

: P dΩ0

nQ = 2

nQ = 3

δW ext =

0 δ u · b dΩ0 +

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

BI CtBJ dΩ0

BI = ΦI,X 0

0 ΦI,Y ΦI,Y ΦI,X

f extI =

ΦI (X) t0 dΓ0 +

ΦI (X) b dΩ0

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 +

δ u · t0 dΓ0

u(·, t)|u(·, t) ∈ H1, u(·, t) = u(t) on Γu0 ,

δ u|δ u ∈ H1, δ u = 0 on Γu0 ,

δW u = Γ0u

δ λ · (u − u) dΓ0 + Γ0

δ uλ dΓ0

λ =J ∈S

ΦLJ (X) λJ

ΦLJ (X)

K GG 0 uλ = f

K = KIJ

GIK = −

ΦI (X) ΦLK (X) S dΓ

qK = −

ΦLK (X) S u dΓ

S 2 × 2 S ij j = i

inf − sup

Π = 2a21 − 2a1a2 + a2

2 + 18a1 + 6a2

a1 = a2

Π = Π + λ(a1 − a2)

= 2a21 − 2a1a2 + a2

2 + 18a1 + 6a2 + λ(a1 − a2)

∂ Π

∂a1= 0

∂ Π

∂a2= 0

∂ Π

∂λ = 0

a1 a2 λ

a1 = a2 = −12 λ = 6

δW u = 0.5 p

u − u2dΓ0

uh(x) =N I =1

ΦI (xJ )uI

uh(x) xJ

u = Du

N × N

u = D−1u

uh(x) =

ΦI (x)D−1IJ uI

Γu N Γu

uh(x) =

N ΩI =1

ΦI (xJ ) uI Ω +

N ΓuI =1

ΦI (xJ ) uI 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)

(N Γu×1)

uΓu =

DΓu−1

(g − DΩuΩ)

uh(x) =

N ΩI =1

ΦI (xJ ) uI Ω +

N ΓuI =1

ΦI (xJ )

IJ ]−1(gI − DΩ

IJ uΩJ )

uh(x) =

N Ωi=I

ΦI (x) − ΦI (x)[DΓu

IJ ]−1DΩ

N ΓuI =1

ΦI (x)[DΓu

IJ ]−1gJ

FE node

particle

particle boundaryparticle domain

blending region

element domain

element boundary

ΩB ΩP ΩFE

ΓFE ΓP

uh = uFE (X) + R(X)

uP (X) − uFE (X)

, X ∈ ΩB

uFE uP u

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 /∈ Ω

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

ΓFE 0

ΩFE 0 Ω

g = uFE − uP

N J =1

N FE J (X, t) uFE J −J ∈S

N P J (X, t) uP J

δ λP h (X, t) =N J =1

N FE J (X, t) δ ΛJ (t)

XL = ΦI (ξ)XI

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

∂ u =

∂W int

∂ u − ∂ W ext

∂ u + λ

∂ u = f int − f ext + λ

∂ u = 0

∂ λ = g = 0

W ext u

f int =

ΩP 0 ∪ΩFE

(∇0 ⊗ δ u)T : P dΩ0

f ext =

ΩP 0 ∪ΩFE

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 +

∂ 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

KFE −FE KFE −P

uFE uP KFE

KFE −FE =

Γ∗0

T · NFE dΓ0

KFE −P = −

Γ∗0

T · NP dΓ0

T C BP dΩ0

T C BFE dΩ0

f ext,FE =

T b dΩ0 +

ΓFE,t0

T t0 dΓ0

f ext,P =

T b dΩ0 +

ΓP,t0

T t0 dΓ0

f int,FE =

T · P dΩ0

= ΩP 0B

P T · P dΩ0

∆u = u

∆λ = λ

∂ 2g∂ u∂ u

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

Γ0 Γ0 Γt0

ΩFE 0 ΩP 0

Ωint0 Ωint

α = l(X)l0

l(X) X

Ωint0

W int =

β FE FT · PdΩFE 0 +

β P FT · PdΩP 0

α=1α=0

finite element node

particle

β FE (X) =

0 in ΩP 0

1 − α in Ωint0

1 in ΩFE 0 − Ωint0

β P (X) = 0 in Ω

0α in Ωint0

1 in ΩP 0 − Ωint0

W ext =

β FE ρ0b · udΩFE 0 +

β P ρ0b · udΩP 0

β FE t · udΓFE 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

N JI uFE iJ −

wKI uP iK

ΛI (X)

λi(X, t) =I

ΛI (X)λiI (t)

ΛI (X)

N I (X) wI (X)

W AL = W int − W ext + λT g + 1

2 pgT g

∂W AL∂uFE iI = (F intiI − F extiI ) +

ΛKLλK N IL+ p

N KLuFE iK −K

wKLuP iK

∂W AL∂uP iI

= (f intiI − f extiI ) −L

ΛKLλK

− pL

N KLuFE iK −K

wKLuP iK

∂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 =

β FE N I,j (X)Pji(X)dΩFE 0

FextiI =

β FE N I (X)ρ0bidΩFE 0 +

β FE N I (X)tidΓt0

f int f ext

f intiI =

β P wI,j (X)Pji(X)dΩP 0

f extiI =

β P wI (X)ρ0bidΩP 0 +

β P wI (X)tidΓt0

∆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

21 KFE

KFE nn

KFE IJ = ∂ FintI

∂ uFE J

11 KP 12

KP 21 KP

KP IJ =

∂ f intI ∂ uP J

I = uFE xI

uFE yI d

I = uP xI uP yI

A11 A12 LFE T

A21 A22 LP T

LFE LP 0

∆dFE

−rFE

di ukP dj ulQ

rFE = Fint − Fext + λT GFE + pgT GFE

rP = f int − f ext + λT GP + pgT GP

g = giI =

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

∂ Fint

∂ dFE

∂F intiI ∂uFE lQ

β FE N I,jC jilkN Q,kdΩFE 0

∂ f int

∂ dP

∂f intiI ∂uP lQ

β P wI,jC jilkwQ,kdΩP 0

ΛIL∂ gL

∂dFE i

ΛIL∂gjL∂dFE i

ΛIL∂gL

∂uFE kP

ΛILN PI δ jk

ΛIL∂ gL∂dP i

ΛIL∂gjL∂dP i

ΛIL∂gL

∂uP kP

ΛILwPI δ jk

∂ gI ∂dFE i

∂gjI ∂uFE kP

= [N PI δ jk ]

∂ gI ∂dP i

∂ gjI ∂uP kP

= [−wPI δ jk ]

u(x, t) = ˙u + H S [[ ˙u]](x, t)

∇ x r

δ S S

ǫ(x, t) = ∇S u = ∇S ˙u + H S ∇S [[ ˙u]] + δ S

[[ ˙u]] ⊗ n

u(x, t) = ˙u + H Ωh(r, t)[[ ˙u]](s, t)

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]]

∇H Ωh = n

h(r) n

h = s+−s−

a 1 X ∈ Ωh 0

ǫ(x, t) = ∇S u = ∇S ˙u + H Ωh∇S [[ ˙u]] + a

[[ ˙u]] ⊗ n

undesired

Visibility criterion

Diffraction criterion

Transparency criterion

Crack line

interdiscontinuities

Domain of influence

crack crack

undesired

hI 0(X) =

s1 + s2(X)

λs0(X)

s0(X) = X − XI s1 = Xc − XI

s2(X) = X − Xc

∂X i=

∂h0I

∂X i

∂h0I

∂X i= λ

s1 + s2(X)

λ−1∂s2

∂X i+ (1 − λ)

s1 + s2(X)

λ∂s0

∂X i

∂ X =

X − Xc

∂ X =

X − XI

h0I = s0(X) + hmI

λ, λ ≥ 2

sc = κh

∂h0I

∂ X =

X + λhmI

sλ−1c

∂sc∂ X

∂ X =

X − XI

∂sc∂X 1

= −cos(θ) = X b − X c

∂sc∂X 2

= −sin(θ) = Y b − Y c

nA · nB ≤ β

nA · nB ≤ β β = 0o

β = 0o

ω = 90o

enrichment

u1 = 1

1I (θ) + K II Q

1II (θ)

2I (θ) + K II Q

2II (θ)

G r θ

Q1I (θ) = κ − cos θ2 + sinθ sin θ2

Q2I (θ) = κ + sin

2 + sinθ cos

Q1II (θ) = κ + sin

2 + sinθ cos

Q2II (θ) = κ − cos

2 − sinθ sin

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]

10−3

B1 function

B2 function

10−3

B3 function

10−1

B4 function

pT (X) =

1, X , Y ,

√ r sin

r cosθ

r sinθ

2sin(θ),

√ r cos

2sin(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)

uh(X) = R uenr(X) + (1 − R) ulin(X)

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 )

Enriched

Transition

Linear

nc uh u

n − th

J = J ∈S

2 p(XJ )T a(X, t) +

ncK =1

kK I QK I + kK II QK II − uJ (t)2

W (X−XJ , h0)

A(X)a(X) =J ∈S

PJ (X)

uJ −

ncK =1

kK I QK I + kK II Q

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)]

a(X) =J ∈S

A−1(X)PJ (X)

uJ −

ncK =1

kK I QK I + kK II Q

uh(X) =

J ∈SpT (X)A−1(X)PJ (X)

uJ −

K =1 kK I QK

I + kK II QK II

+ncK =1

kK I QK I + kK II Q

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

uJ = uJ −ncK =1

kK I QK I + kK II Q

uh(X) =I ∈S

ΦI (X)

J ∈Sc

bIJ pJ (X)

global

Ω = ΩA ∪ ΩB

ΩA∩

ΩB = ∅

φ > 0

φ < 0

φ = 0

φ(x) > 0 ∀ x ∈ ΩA

φ(x) < 0 ∀ x ∈ ΩB

φ(x) = 0 ∀ x ∈ Γ

φ(x, t)

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

f (x) Ω

ΩA ΩB Ω

f (x) =

f (x) +

H (ξ )

H (ξ ) =

1 ∀ξ > 00 ∀ξ < 0

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

f ,i(x) =

f ,i(x)H (φ(x))

ΩA f ,i(x) = ∂ ΩA f (x)ni

ni ΩA

f ,i(x) =

(f (x)H (φ(x))),i − f (x) (H (φ(x))),i

H (φ(x) H (φ(x))

= 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))

= nB→Ai on Γ

= 0 otherwise.

f ,i(x)H (φ(x)) =

f (x)H (φ(x))

− Ω

H (φ(x)),i

∂ Ω

f (x)H (φ(x))ni −

f (x)nB→Ai

∂ Ω

f (x)H (φ(x))ni +

f (x)nA→Bi

f ,i(x)H (φ(x)) =

∂ Ω =∂ extΩA

f (x) H (φ(x))

Γ =∂ intΩA

f (x)nA→Bi

∂ ΩA

f (x)ni

f ,i(x)H (φ(x)) =

∂ Ω

f (x) H (φ(x)) =0

=∂ ΩA

f (x)nA→Bi

∂ ΩA

f (x)ni

f ,i(x)H (φ(x)) =

∂ Ω

f (x) H (φ(x)) =1 onlyif x∈ΩA

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

9φǫ − 5(φ

for − ǫ < φ < ǫ

1 for ǫ < φ

δ (φ) = 0 for φ < −ǫ

+ 12ǫ

sin πφǫ for − ǫ < φ < ǫ0 for ǫ < φ

d = x − xΓ

φ(x) = d ΩA

φ(x) = −d

φ(x) = min x∈Γ

x − x sign

n · (x − x)

∇φ = 1

φ = 0

φ < 0 φ > 0

N I (x)

φ(x) =I ∈S

N I (x)φI

φ(x),i =I ∈S

N I,i(x)φI

φ,i φ,i = 0

Dφ(x, t)

Dt = 0

∂φ(x, t)∂t + ∇φ(x, t) · v(x, t) = 0

φ + φ,ivi = 0

φn+1 − φn

∆t = −φn,iv

φn+1 = φn − ∆t φn,ivni

|∇φ| = 1

• φ(X) = 0

φ(X) > 0

f(X)=0 f X =0f(X)<0

voxels (background mesh) f(X)>0

activparticl

• φ(X) ≥ 0

• I ∋ N act φ(XI ) ≥ 0 φ(XI ) = 0 XI

φ(X) = 0

N I (X)

φ(X) =I ∈S

N I (X) XI

φ(X) > 0

φ(XI ) ≤ α h p

uh(X) = J ∈S

N J (X) uJ + K ∈E

J ∈Sc

N K J (X) ψK (X) aK J

N J (X ) = N J (X )

S (ξ ) =

1 ∀ξ > 0−1 ∀ξ < 0

4321 crack

Shifting

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)

N 2(X ) N 3(X )

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

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

(X )]] = u(X +

) − u(X −)=

J ∈S

N J (X +) uJ +J ∈Sc

N J (X +)

S (φ(X +))

−J ∈S

N J (X −) uJ +J ∈Sc

N J (X −)

S (φ(X −))

=J ∈Sc

N J (X )

S (φ(X +)) − S (φ(X −))

= 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 −))

=J ∈Sc

N J (X ) aJ

J ∈Sc

N J (X ) aJ 2 J ∈Sc

N J (X ) aJ

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)

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)

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

∇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]

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

1 = u1

u12 = u2 − a2

element2

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

+ N 2(X c)

2 − u12

= a1 N 1(X c) + a2 N 2(X c)

uhi (x) =4I =1

N I (x)uIi +3J =1

N J (x)ψ(x)aJi

φ<0 φ>0

N 2(X )

N 1(X )

N 4(X )

u− u−

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))

uIi = 0

aJi = 1

(N 1, N 2, N 3)

N J (x) = 1.

I ∈N

N I (x) = 1

I ∈N

N I (x)Ψ(x) = Ψ(x)

000000111111

000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111

11000000111111000000111111 000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111000000111111

Ω p.e.

N I (x) f i(x) ψ(x) f i(x)×ψ(x)

Ψ N I Ψ

Ωblnd

000000000111111111000000000000111111111111000000000000111111111111 000000000111111111000000000000111111111111000000000000111111111111000000000000111111111111000000000111111111000000000000111111111111

Ωblnd

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

N J Ψ

Ψ(x) = xH (x)

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

e ≡ u − uint

e,x|x ≡ d

dxe(x) = 0

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

2(x − x1)2 ≤ 1

e(x) ≤ 1

max(u,xx +

n ξ n n > 1

h2max(u,xx + 2a1

(1, 1)

(1,−1)

(−1, 1)

(−1,−1)

r = 1r = −1

s = −1

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)

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 )]

ψI (x) = ψ(x) − ψ(xI )

ǫxxǫyy

= Due(M )

∂x 0

∂y∂

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

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

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

ψ(x) = |φ(x)|

ψ(x),i

= sign(φ(x)) φ,i(x)

φ(x) = [ N 1 N 2 N 3 N 4 ]

φ(x),x = [ N 1,x N 2,x N 3,x N 4,x ]

φ(x),y = [ N 1,y N 2,y N 3,y N 4,y ]

∂N I ∂x

= ∂N I

∂x +

∂N I ∂s

∂x∂N I ∂y

= ∂N I

∂y +

∂ N I ∂s

N ,x N ,y = N ,r N ,s

∂s∂x

∂s∂y

= J−1

J N I (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

(1 + s) N 3,s = 1

(1 + r)

N 4,r = −1

4(1 + s) N 4,s =

4(1 − r)

x =4I =1

N I xI , ∂x

∂r =

∂N I ∂r

xI , ∂x

∂s =

∂N I ∂s

∂r =

N 1,r N 2,r N 3,r N 4,r

∂s =

N 1,s N 2,s N 3,s N 4,s

y =4I =1

N I yI , ∂y

∂r =

∂N I ∂r

yI , ∂y

∂s =

∂N I ∂s

∂r =

N 1,r N 2,r N 3,r N 4,r y1

∂s =

N 1,s N 2,s N 3,s N 4,s

Ke = Ωe

(M ) Ce Be(M ) dΩ = 1

−1 1

(r, s) Ce Be(r, s) det J dr ds

8 × 8

std(M )CeBestd(M )

std(M )CeBeenr(M )

enr(M )CeBestd(M )

enr(M )CeBeenr(M )

16 × 16

background cell

background cellCrack path produced

by level set Crack path recognized by the code

F (X)dΩ +

F (X)dΩ

F (X(ξ)) detJ−(ξ)dΩ +

F (X(ξ)) detJ+(ξ) dΩ

Voronoi cells

Delaunay triangulation

Gauss point

A2A1 A3

A4 A5 A6

A8A7 A9

enriched nodes

not enriched nodes

crack crack tip crack

crack tip

∇ (F jψi)· ∇

(F lψk) dx

r−0.5

∇F i

ξ = G(ξ ) , w = w det(

∇0 · P − b = ∅ ∀X ∈ Ω0 \ Γc0

n0 · P(X, t) = t0(X, t) on Γt0

n0 · P(X, t) = 0 on Γc0

Γc0 = Γ0 , (Γu0

Γc0) =∅

u ∈ V

δW = δW int − δW ext = 0 ∀δ u

δW int =

(∇ ⊗ δ u)T : P dΩ0

δW ext =

δ u · b dΩ0 +

δ u · t0 dΓ0

u(·, t)|u(·, t) ∈ H1, u(·, t) = u(t) on Γu0 , u discontinuous on Γc0

δ u|δ u ∈ H1, δ u = 0 on Γu0 , δ u discontinuous on Γc0

Space of Bounded Deformations

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

3 1P13

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

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

Ωenr Ωenr

crack tip enrichment

Heaviside enrichment

B = [B1 B2 B3 B4]

√ r sin

r cosθ

r sinθ

2sin(θ),

√ r cos

2sin(θ)

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

a b c d

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 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 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 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

1 1 1 1 1 1

0 0 0 0 0 0

1 1 1 1 1 1

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

f extI f extI f extIK

K d = f ext

K d = u a bT

f ext =

f u f a f bT

f b1 f b2 f b3 f b4

f uI =

N I b dΩ +

N I t dΓ

f aI =

N I (H (φ(X)) − H (φ(XI ))) b dΩ+

N I (H (φ(X)) − H (φ(XI ))) t dΓ

f blI =

BlI (X) − BlI (XI )

b dΩ+

BlI (X) − BlI (XI )

BT C B dΩ

BuI = N I,X 0

0 N I,Y

N I,Y N I,X

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 =

BlK (X) − BlK (XI ),X

BlK (X) − BlK (XI ),Y

BlK (X) − BlK (XI )

N I BlK (X)

= N I,i BlK (X) + N I BlK (X),i

Bl,i = B l,r r,i + Bl,θ θ,i

B1,r =

sin(θ/2)

2B1,θ =

√ 2cos(θ/2)

B2,r =

cos(θ/2)

2B2,θ = −

√ 2sin(θ/2)

B3,r =

sin(θ/2) sin(θ)

2B3,θ =

cos(θ/2) sin(θ)

2 + sin(θ/2) cos(θ)

B4,r =

cos(θ/2) sin(θ)

2B4,θ =

sin(θ/2) sin(θ)

2 + cos(θ/2) cos(θ)

r, X = cos(θ) θ, X = −sin/r

r,Y = sin(θ) θ,Y = cos/r

B1, X =

sin(θ/2)

2B1,Y =

cos(θ/2)

B2, X =

cos(θ/2)

2B2,Y =

sin(θ/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)

2B4,Y =

cos(θ/2) + cos(3θ/2) cos(θ)

B,X = B, X cos(α) + B,Y sin(α)

B,Y = B, X sin(α) + B,Y cos(α)

Branching discontinuityIntersecting discontinuity

φ1(x) = 0φ1(x) = 0

φ2(x) = 0φ2(x) = 0

φ1(X) = 0

S2c φ2(X) = 0

S3c = S1

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

I ∈S1t (X)

N I (X)K

B(1)K (X) b

I ∈S2t (X)

N I (X)K

B(2)K (X) b

(φ1 < 0, φ2 < 0)

(φ1 >0, φ2 > 0)

(φ1 > 0, φ2 < 0)

(φ1 > 0, φ2 >0) (φ1 < 0, φ2 < 0) 1 X

φ1(X) =

φ02(X1) φ0

2(X) > 0φ0

φ02(X1) φ0

2(X) < 0

uh(X) =I ∈S(X)

N I (X) uI +

I ∈Sc(X)

N I (X) H (φ(n)I (X)) a

I ∈St(X)

N I (X)K

B(m)K (X) b

∇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

[[uN ]] = u+ · n+ = u− · n− ≤ 0

n+ = n−

(∇ ⊗ δ u)T

: P dΩ0 −

δ u · b dΩ0 −

δ u · t0 dΓ0 + δ

λ [[uN ]] dΓ0 ≥ 0

σθθ

σθθ = K I √

2πrf I h(θ, vc) +

K II √ 2πr

f II h (θ, vc)

f I h f II h

σcθθ

σcθθ = K cI √

K I sinθc + K II (3cosθc − 1) = 0

θc = 2arctan

K I −

K 2I + 8K 2II

mdiag = m

nnodes

mes(Ωel)

ψ2 dΩel

Ωel m mes(Ω)el

nnodes Ω ψ

lumped

II = J M

consistent

IJ , or

MlumpedII = m

MconsistentII

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

E hkin = 0.5uT Mlumped u

E kin = 0.5

Ωel v2 dΩ

E hkin = 0.5

m1 u21 + m2 u2

= 0.5 ˙u

2(m1 + m2)

E kin = 0.5 m ˙u2

= E hkin m1 = m2 = 0.5 m m

˙u = aφ1(x)

kin = 0.5 m3 a2

1 + m4 a2

2 = 0.5 a2

(m3 + m4)

E kin = 0.5

ψ21 dΩel

m3 = m4 = m

2 mes(Ω)el

ψ21 dΩel

N 1(x) = 1 − x

N 2(x) = x

FE = A l

1/3 1/61/6 1/3

FE = E A

1 −1−1 1

∆tc,FE = 2

ωmax= l

lumpedFE =

∆tlumpedc,FE = l

3∆tc,FE

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

1 −1 1 − 2s 2s − 1−1 1 2s − 1 1 − 2s

2s 2s−

1 1 −

12s − 1 1 − 2s −1 1

lumpedXFEM = 0.5

1 0 0 00 1 0 00 0 1 00 0 0 1

x = 0 x = l

∆tlumpedc,XFEM = 1√ 2

∆tlumpedc,FE

crack crack

effective crack length

√ 2sin(θ/2)

a · n0 = b · n0 = 0

a · ∇0T = ∇0T · a = 0 in Ω0

b· ∇0

T = ∇0

b = 0 in Ω0

∂t + v · ∇φ = 0

r = φ2 + ψ2 ψ φ

θ = arctan(φ/ψ)

θ = ±π

φ = 0

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∗)

Γc,ext

strong embedded elements

Γc,ext

a) b) c)

uh(X) =I ∈S

N I (X) uJ +M(e)s (X) [[u

(e)I (X )]]

M(e)s (X) =

0 ∀(e) /∈ S

H (e)s − ρ(e) ∀(e) ∈ S

ρ(e) =N +e

N +I (X)

ǫh(X) =I ∈S

(∇0N I (X) ⊗ uI )S −∇ρ(e) ⊗ [[u

(e)I (X )]]

η(e)s

(e)I (X )]] ⊗ n

S η(e)s /k

η(e)s

η(e)s =

1 ∀X ∈ S ke0 ∀X /∈ S ke

(e)uu K

uu u(e)

[[u(e)

I ]] =

FextI 0

K(e)uu =

BT C B dΩ0

K(e)uu =

BT C B dΩ0

K(e)uu =

BT ∗ C B dΩ0

K(e)uu =

BT ∗ C B dΩ0

∇ρ(e) =

∂ρ(e)

∂x 0

0 ∂ρ(e)

∂y∂ρ(e)

n(e) =

nx 00 ny

B∗ = B

[[u(e)I ]]

[[u(e)I ]] = −

K(e)uu u(e)

[[u(e)I ]]

K u = f

K = Kuu − Kuu K−1uu Kuu

Ω− Ω+

interelement − separation methods

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