lecture xfem meshfree

8/3/2019 Lecture Xfem Meshfree

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fi fij giI I fi = f fij = f () figi = r = fg fijklgkl = rij f : g = r figj = rij f g = r f g = ijk fi gk ijk

gi = (g1, g2, g3, g12, g13, g23) gij

0 0

x = (X, t),

x X

u(X, t) = x X = (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 = xX

=u

X= I F

D = 0.5

L + LT

L = vi,j = F F1

E = 0.5 FTF I

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

KI, KII 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

, ,

K

ijk , ,


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, ,


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global

local


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X

J(X) p(X) uJ = p(XJ)

J(X) uJ = J(X) p(XJ) = p(X)

completeness

reproducing conditions

J

J(X) = 1 J

J(X) XJ = XJ

J(X) YJ = Y

J

J(X) XJi = Xi


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J

J,X(X) = 0J

J,Y(X) = 0 J

J,X(X) XJ = 1J

J,Y(X) XJ = 0 J

J,X(X) YJ = 0J

J,Y(X) YJ = 1

J

J,i(X) = 0 J

J,i(X) XJj = ij

J(x)

uJ = 1 J

J(x) = 1

partition of unities

D

Dt

IS

mIvI

=

ISmIvI = 0

mI v

mIvI = JS

I(XJ) (XJ) wJ

I(XJ) wJ IS

mIvI = IS

JS

I(XJ)(XJ) wJ = JS

IS

I(XJ)(XJ) wJ = 0


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IS

I(XJ) = 0

D

Dt I mIvI XI = I mIvI XI+ vI vI =0 = 0

D

Dt

I

mIvI XI

=I

ijk

J

I,m(XJ) mj(XJ)wJ

XIk

ijk XIk k th I

ijkJ I I,m(XJ)XIk mk

mj(XJ)wJ = ijkmkJ mj(XJ)wJ=J

ijmmj(XJ) =0

wJ = 0

k

k > 0

maxi

|u(Xi) ui| Chk

C

h

Cn

n

h


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I

K

Support size of particle I

R_KR_I

limh00

W(XI XJ, h0) = (XI XJ)

0W(XI XJ, h0)d0 = 1

W(XI XJ, h0) = 0 XI XJ R

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) = W1D(X),

W(X) = W1D(|X1|) W1D(|X2|) W1D(|X3|) X = (X1, X2, X3) X =

X21 + X

22 + X

23

=

ChD1 1.5z2 + 0.75z3 0 z < 1

C4 hD

(2 z)3 1 z 20 z > 2

D

z = r/h0 C

=

2/3 D = 1

10/(7 ) D = 21/ D = 3

h0 z

z = ||XI XJ|| W

XiJ=

W

z

z

XiJ

Wz

=

3ChD+1

z + 0.75z2 0 z < 13C

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 10.2

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 10.2

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 10.2

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

i = 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

ez2/c2 Gaussian

z2 + R2q

multipolar

c R q

WJ(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

WJ(X) = W(X XJ, h0)

xJ(t)

v(x, t) =IS

W(x xI(t)) vI(t),

a =IS

W(x xI(t)) vI+ W(x xI(t)) xI vI.

uh(X, t) =JS

uJ(t) J(X)

uJ J(X) S J(X) = 0


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uh(xI) = uI

I(XJ) = IJ IJ

H1

uh(X, t) =

0

u(Y, t) W(X Y, h0(Y)) dY

0

0

W(X Y, h0(Y)) 1 dY = 1

0

W(X Y, h0(Y)) Y dY = X

0

W(X Y, h0(Y)) X dY = X

0

W(X Y, h0(Y)) (X Y) dY = 0

uh(X, t)

0uh(X, t) =

0

0u(Y, t) W (X Y, h0(Y)) dY


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0uh(X, t) =

0

0 [u(Y, t) W (X Y, h0(Y))] dY

0

0u(Y, t) W(X Y, h0(Y)) dY

0uh(X, t) = 0

u(Y, t) W (X Y, h0(Y)) n0 d0

0

0u(Y, t) W (X Y, h0(Y)) dY

0uh(X, t) =

0

0u(Y, t) W (X Y, h0(Y)) dY

J(X) = W(X XJ, h0) V0J

V0J

J

0uh(X) = JS

uJ0J(X) with 0J = 0W(X XJ, h0) V0J

V0J


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JS

0W(X XJ, h0) V0J

uI 0

0uh(X) =

JS(uJ uI) 0W(XI XJ, h0) V0J


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0uh(X, t) =IS

GI(X) uI(t)

uh,i(X, t) =IS

GiI(X) uI(t)

GI

WSI (X) =WI(X)

ISWI(X)

GI

GI(X) = a(X) 0WSI (X) = aij(X)WSjI(X) a(X)

IS

GI(X) XI = ij

A

a

A aT = I

I

= WSI,X XI WSI,Y XI

WSI,X YI WSI,Y YI

=

aXX aXYaYX aY Y

0uh(X, t) =IS

a(X) 0WSI (X) uI(t)


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I = (a11(X) + a12(X) + a13(X)) WSI (X)

GXI = (a21(X) + a22(X) + a23(X)) WSI (X)

GY I = (a31(X) + a32(X) + a33(X)) WSI (X)

X

a A

=I

WSI (X) 1 XI X YI YXI X (XI X)2 (XI X)(YI Y)

YI Y (XI X)(YI Y) (YI Y)2

3 3

I = a11(X)WSI,X (X) + a12(X)W

SI,Y (X) + a13(X)W

SI (X)

GXI = a21(X)WSI,X (X) + a22(X)W

SI,Y (X) + a23(X)W

SI (X)

GY I = a31(X)WS

I,X (X) + a32(X)WS

I,Y (X) + a33(X)WS

I (X)

a

X a

=I

WSI,X (X) WSI,Y (X) WSI (X)WSI,X (X) XI WSI,Y (X) XI WSI (X) XIWSI,X (X) YI W

SI,Y (X) YI W

SI (X) YI

O(h)

u(X) X

u(XI) = u(X) + u,X(X) (XI X)+ u,Y(X) (YI Y) + 0.5u,XX (X) (XI X)2+ u,XY(X) (XI X) (YI Y)+ 0.5u,Y Y(X) (YI 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)(XI X) 1+ u,Y(X)

I

GXI(X) (YI Y)

+ 0.5 u,XX (X)I

GXI(X)(XI X)2

+ u,XY (X)I

GXI(X)(XI X) (YI Y)

+ 0.5 u,Y Y(X)I

GXI(X)(YI Y)2

I GXI = 0 I GXI(XIX) = 1

I

GXI(YI Y) = 0

uh,X(X) u,X = 0.5 u,XX (X)I

GXI(X)(XI X)2

+ u,XY(X)I

GXI(X)(XI X) (YI Y)

+ 0.5 u,Y Y(X)I

GXI(X)(YI Y)2

|uh,X(X) u,X | 0.5 |u,XX (X)| |I

GXI(X)(XI X)2|

+ |u,XY(X)| |I

GXI(X)(XI X) (YI Y)|

+ 0.5 |u,Y Y(X)| |I

GXI(X)(YI Y)2|

d

X = (X Y)

|XI X| d, |YI 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| C1h0

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) =

JS

(uJ(t) uI(t)) 0W(XJ X, h0) V0J

B(X)

B(X) =

JS

(XJ X) 0W(XJ X, h0) V0J1

W(X XJ, h) V0J B

B(X) = JSX

J 0WS

(XJ X, h0)1

B

u

0uh(X, t) =

JS

uJ(t) 0WS(XJ X, h0) V0J

B(X)


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C(X, Y)

uh(X) =

Y

C(X, Y)W(X Y)u(Y)dY

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

Y

p(Y)W(X Y)u(Y)dY =

Y

p(Y)pT(Y)W(X Y)dYa

a

uh(X) = pT(X)a

uh(X) = pT(X)

Y

p(Y)pT(Y)W(XY)dY1

Y

p(Y)w(XY)u(Y)dY

C(X, Y) = pT(X)

Y

p(Y)pT(Y)W(X Y)dY1

p(Y)

= pT(X)[M(X)]1p(Y)

uh(X) =

Y

C(X, Y)W(X Y)u(Y)dY

=IS

C(X, XI)w(X YI)uIV0I

= pT(X)[M(X)]1IS

p(XI)W(X XI)uIV0I


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M(X)

M(X) =

Y

p(Y)pT(Y)W(X Y)dY

=IS

p(XI)pT(XI)W(X XI)V0I

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 x2I

a =

3I=1

1

xI

uI

3 66 14

a =

6.516

a0 = 5/6 a1 = 1.5

uh(x) = 56

+3

2x

a

X X

p

p(X) =

1 X Y X 2

uh(X, t) =MI=1

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

M a


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J(a(Xi)) =NJ=1

W(X XJ, h0)MI=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

(X) = u( X1) u( X2) ... u( XN)

P(X) =

p1(X1) p2(X1) ... pM(X1)

p1(X2) p2(X2) ... pM(X2)

p1(XN) p2(XN) ... pM(XN)

=

W(X X1) 0 ... 00 W(X X2) ... 0

0

0 0 ... W (X

XN

)

a

J(a(Xi))

a(Xi)= 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) =ARMM

PT(X) W(X) =BRMN

u(X)

uh(X, t) = pT(X) A1(X) B(X) u(X)

uh(X, t) =MJ=1

MK=1

NI=1

pJ(X) A1JK(X) BKI(X) uI(X)


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I(X)

I(X, t) =MJ=1

MK=1

pJ(X) A1JK (X) BKI(X)

V0I

A(X) =

P11 ... P 1N

PM1 ... P MN

W1 ... 0

0 ... W N

P11 ... P M1

P1N ... P MN

M = 1 p(X) = 1

A(X) =

1 ... 1 W1 ... 0

0 ... W N

1

1

A

p(x) = 1

I(X) =WI(X)

ISWI(X)

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

A(X) =

1 ... 1x1 ... xNy1 ... yN

W1 ... 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

=maxmin

A

A

(X)

Xi=

pT(X)

XiA1 B + pT(X)

A1(X)Xi

B

+ pT(X) A1(X)B(X)

Xi

B(X)

Xi= P(X)

W(X)

Xi


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A1(X)

I = A1(X) A(X)

0 =A1(X)

XiA(X) + A1(X)

A(X)

Xi

A

1(X)

Xi = A1(X)A(X)

Xi A1(X)

= A1(X) P(X)W(X)

XiPT(X) A1(X)

2(X)

XiXj=

2pT(X)

XiXjA1(X) B(X)

+ 2pT(X)

Xi

A1(X)

XjB(X) + A1(X)

B(X)

Xi

+ pT(X)

2A1(X)XiXj

B(X) + A1(X)2B(X)

XiXj+

A1(X)Xi

B(X)

Xj + pT(X)

A1(X)Xj

B(X)

Xi

J

J(X) = (X) p(XJ) W(X XJ, h0)

A(X) (X) = p(XJ)

A

0A(X) (X) + A(X) 0(X) = 0p(XJ) 0(X)

XI


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h0

I(X) = W(XI, X) PT

XI X

h0

(X),

W(Y, X) = W ((Y X)/h0)

P(0) = IS

I(X) PXI Xh0

(X)

A(X) (X) = P(0)

A(X) =JS

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=JS

W(XJ, X) fXJ X

h0

gXJ X

h0

X Z X

u

u(Z) uh(Z, X) = PT

Z Xh0

c(X)

c


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F(X, Y) = X2 + Y2 (R = 0.8) (R = 0.3) R

uh(X) =JS

J(X)

uJ+

LK=1

pK(X) aJK

aJK

uh(X) =JS

J(X) uJ+JS

J(X)

LK=1

pK(X) aJK

global


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F(X, Y) =X2 + Y2 (R = 0.8) (R = 0.3) R

F(X, Y) = X2 + Y2

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

X2 + Y2

F 0 X 2 0 Y

2

F

x

/300 R

x

V = d2

d

h

d < h 0

A

< 0

= 0

n

(x)

(x) > 0 x A(x) < 0 x B

(x) = 0 x

(x)

(x, t)

n x

n =

= 1 n = n B A B A

x

K = ni,i

= 1 K = ni,i = ,ii


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f(x) A B

f(x) =

A

f(x) +

B

f(x)

H()

H() =

1 > 00 < 0

A B

A = {x /H((x)) = 1}

B = {x /H((x)) = 1}

f(x) = f(x)H((x)) + f(x)H((x)) A

A A

f,i(x) =

f,i(x)H((x))

A f,i(x) = A f(x)ni ni A

A

f,i(x) =

(f(x)H((x))),i f(x) (H((x))),i

H((x) H((x))

,i

= ,i(x)H,i((x)) = ,i(x)((x))


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Case 3:Case 2:Case 1:

AAA B

BB

intA =

extA = extA =

A = extA

intA A = A =

extA

() ,i(x) n

BA

H((x))

,i

= nBAi on

= 0 otherwise.

f,i(x)H((x)) =

f(x)H((x))

,i

f(x)

H((x)),i

=

f(x)H((x))ni

f(x)nBAi

=

f(x)H((x))ni +

f(x)nABi

f,i(x)H((x)) =

=extA

f(x) H((x))

=1ni +

=intA

f(x)nABi

=

A

f(x)ni

f,i(x)H((x)) =

f(x) H((x)) =0

ni +

=A

f(x)nABi

=

A

f(x)ni


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f,i(x)H((x)) =

f(x) H((x)) =1 onlyif xA

ni +

f(x)nABi

=

A

f(x)ni

H() =

0 for < 12 +

2 +

12 sin

for < <

1 for <

H() =

0 for < 12

+ 18

9 5( )3

for < <

1 for <

() = 0 for < 12 + 12 sin for < <

0 for <

d x

d = x x x x (x)

(x) = d A

(x) = d B

(x) = minx

x x sign

n (x x)

= 1


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n

x

= 0

d

x

< 0 > 0

NI(x) I

S

(x) =IS

NI(x)I

I I

(x),i =IS

NI,i(x)I

,i ,i = 0


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D(x, t)

Dt= 0

v

(x, t)t + (x, t) v(x, t) = 0

+ ,ivi = 0

n+1 nt

= n,ivni

n+1 = n t n,ivni t

vi || = 1

0 0

(X) = 0 (X) > 0


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

activparticl

CD

(X)

(X) 0

XI

I Nact (XI) 0 (XI) = 0 XI

XI I

nsp XI I nip

(X) = 0

(X) (X)

NI(X)

(X) =IS

NI(X) XI


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B X

(X) > 0 CD

(XI) hp hp

uh(X) = JS

NJ(X) uJ+ KE

JSc

NKJ (X) K(X) aKJ

S

Sc

NJ NJ (X)

aJ

E

K

NJ(X) = NJ(X)

S

S() =

1 > 0

1 < 0

(X)


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4321 crack

Shifting

crack

=0>0 0 X > Xc X2 < Xc S((X2)) = 1 S((X3)) = 1 X3 > Xc NJ(X) S(X)

u(X) K Sc u(XK) = uK + S((XK)) aJ

uK


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uh(X) =JS

NJ(X) uJ+JSc

NJ(X) (S((X)) S((XJ))) aJ

u(XK) = uK

[[uh

(X)]] = u(X+

) u(X)=

JS

NJ(X+) uJ+

JSc

NJ(X+)

S((X+))

aJ

JS

NJ(X) uJ+

JSc

NJ(X)

S((X))

aJ

=JSc

NJ(X)

S((X+)) S((X)) aJ= 2

JSc

NJ(X) aJ

NJ(X) = NJ(X+

)

[[uh(X)]] =JSc

NJ(X)

H((X+)) H((X)) aJ=

JSc

NJ(X) aJ

JSc

NJ(X) aJ 2JSc

NJ(X) aJ

J(x, t) = |(x, t)| |(xJ, t)|

vh(x) =JS

NJ(x) vJ(t) +JSc

NJ(x) J((x), t) aJ(t)


100/147

4321

=0>0


101/147

Sc

v

u

N2(x, t) 2(x, t) N3(x, t) 3(x, t)

vh(x) =JS

NJ(x)vJ(t)

+JSc

(NJ(x) J((x), t) + NJ(x) J((x), t)) aJ(t)

J(x, t) = sign() = sign()nint nint

J(x, t)

[[vh(X)]] = 2JSc

NJ(X) aJ nint

[[vh(X)nint]] = 2JSc

NJ(X) aJ

1 1

uh(X) =2I=1

NI(X) [uI+ aI (H(X Xc) H(XI Xc))]

= u1 N1 + u2 N2 + a1 N1 H(X Xc)+ a2 N2 [H(X Xc) 1]


102/147

H

NI = NIH(XXc)+NI (1 H(X Xc)) I = 1, 2

uh(X) = (u1 + a1) N1 H(X Xc) + u1 N1 (1 H(X Xc))+ (u2 a2) N2 (1 H(X Xc)) + u2 N2 H(X Xc)

element1

u11 = u1

u12 = u2 a2

element2

u21 = u1 + a1

u22 = u2

uh(X) = u11 N1 (1 H(X Xc)) + u12N2 (1 H(X Xc))+ u21 N1 H(X Xc) + u22 N2 H(X Xc)

X < Xc

(1 H(X Xc)) X > Xc H(XXc)

[[uh(X)]]X=Xc = lim0

[u(X+ ) u(X )]X=Xc

= N1(Xc)

u21 u11

+ N2(Xc)

u22 u12

= a1 N1(Xc) + a2 N2(Xc)

u12 u21

uhi (x) =4I=1

NI(x)uIi +3J=1

NJ(x)(x)aJi


103/147

XC

0

1 4XC

0

1

4

2

3

1 4

N2(X)

N1(X)

N1(X)

N4(X)

u+ u+

u u

I

NI uII

NI uI

[[u]] [[u]]

N1(X) (H(X Xc) H(X1 Xc))

N2(X) (H(X Xc) H(X2 Xc))

(x) uIi = 0 aJi = 1 (N1, N2, N3)

3J=1

NJ(x) = 1.

IN

NI(x) = 1

(x)

INNI(x)(x) = (x)


104/147

0 00 00 01 11 11 1

0 00 00 01 11 11 1 0 00 00 01 11 11 10 00 00 01 11 11 10 00 00 01 11 11 1 0 00 00 01 11 11 1 0 00 00 01 11 11 10 00 00 01 11 11 1 0 00 00 01 11 11 1 0 00 00 01 11 11 1 0 00 00 01 11 11 1 0 00 00 01 11 11 1 0 00 00 01 11 11 1 0 00 00 01 11 11 10 00 00 01 11 11 1 0 00 00 01 11 11 10 00 00 01 11 11 1 0 00 00 01 11 11 10 00 00 01 11 11 10 00 00 01 11 11 10 00 00 01 11 11 10 00 00 01 11 11 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

0 0

0 0

0 0

1 1

1 1

1 1

0 00 0

0 0

1 11 1

1 1

0 00 0

0 0

1 11 1

1 1

0 00 0

0 0

1 11 1

1 1

0 00 0

0 0

1 11 1

1 1

0 00 0

0 0

1 11 1

1 1

0 00 0

0 0

1 11 1

1 1

0 00 0

0 0

1 11 1

1 1

0 00 0

0 0

1 11 1

1 1

0 00 0

0 0

1 11 1

1 1

0 0

0 0

0 0

1 1

1 1

1 10 00 00 01 11 11 10 00 00 01 11 11 1 0 00 00 01 11 11 1 0 00 00 01 11 11 1 0 00 00 01 11 11 10 00 00 01 11 11 1 0 00 00 01 11 11 1 0 00 00 01 11 11 10 00 00 01 11 11 1 0 00 00 01 11 11 10 00 00 01 11 11 10 00 00 01 11 11 10 00 00 01 11 11 10 00 00 01 11 11 10 00 00 01 11 11 10 00 00 01 11 11 10 00 00 01 11 11 10 00 00 01 11 11 1

Senr

p.e.

NI(x) fi(x) (x) fi(x)(x)

st

st

st

st

st

st

st

st

NI

std


105/147

enr

blnd

0 0 00 0 00 0 01 1 11 1 11 1 10 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 10 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 1 0 0 00 0 00 0 01 1 11 1 11 1 10 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 10 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 10 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 10 0 00 0 00 0 01 1 11 1 11 1 10 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 1

enr

blnd

std

enr blnd std

uI = 0 aJ = 1

uh(x) =

JNenr

NJ(x)(x) = (x) x enrNJ(x)(x) = (x) x blnd

NJ(x)(x) = 0 x std

enr std

NJ

(x) = xH(x)


106/147

H

x = 0

uh(x) =2I=1

NI(x) + N1(x)(xH(x) x1H(x1))a1

uh() = u1(1 ) + u2 + a1h(1 )

=x x1

h

h

uh

e

e u uint

x

e,x|x ddx e(x) = 0


107/147

x

e(x) = e(x) + e,x|x(x x) + 12

e,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) = 12

e,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 +

2a1h )

2a1/h h2 h

n n n > 1

e(x)

1

8

h2max(u,xx +2a1

hn

)


108/147

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)

NI, I = 1...4

N1(r, s) =1

4(1 r)(1 s)

N2(r, s) =1

4(1 + r)(1 s)

N3(r, s) =1

4(1 + r)(1 + s)

N4(r, s) =1

4(1 r)(1 + s)

r s

ue(M) =

uxuy

=

N1 N2 N3 N4 0 0 0 00 0 0 0 N1 N2 N3 N4

266666666664

ux1ux2ux3ux4uy1uy2uy3uy4

377777777775

= Nestd(M) qe


109/147

ue(M) =

uxuy

=

N1 N2 N3 N4 0 0 0 00 0 0 0 N1 N2 N3 N4

. . .

. . .N11 N22 N33 N44 0 0 0 0

0 0 0 0 N11 N22 N33 N44

2666666666666666666666666664

ux1ux2ux3ux4uy1uy2uy3uy4ax1ax2ax3ax4ay1ay2ay3ay4

3777777777777777777777777775

ue(M) = [ Nestd(M) Neenr(M) ] q

e

ue(M) = Ne(M) qe

Ne(M) = [Nestd(M) Neenr(M)]

(x) I

I(x) = (x) (xI)

=

xxyy

2xy

= Due(M)

D =

x0

0

y

y

x

ue(M)

= DNe(M) qe = Be(M) qe


110/147

Be(M)

Be(M) = [Bestd(M) Beenr(M)]

Bestd(M)

Bestd =

N1,x N2,x N3,x N4,x 0 0 0 00 0 0 0 N1,y N2,y N3,y N4,yN1,y N2,y N3,y N4,y N1,x N2,x N3,x N4,x

Be

enr(M)

Beenr =

24 (N11),x (N22),x (N33),x (N44),x 0 0 0 00 0 0 0 (N11),y (N22),y (N33),y (N44),y

(N11),y (N22),y (N33),y (N44),y (N11),x (N22),x (N33),x (N44),x

35

uhi,j =IS

NJ,i(x) ujJ +IS

(NJ(x)H((x))),i ajJ

=IS

NJ,i(x) ujJ +IS

(NJ,i(x)H((x)) + NJ(x)H,i((x))) ajJ

H,i((x)) =

H,i = 1 H,i = 0

Beenr =

24 N1,x1 N2,x2 N3,x3 N4,x4 0 0 0 00 0 0 0 N1,y1 N2,y2 N3,y3 N4,y4

N1,x1 N2,x2 N3,x3 N4,x4 N1,y1 N2,y2 N3,y3 N4,y4

35

(x) = |(x)| (x)

(x),i

= sign((x)) ,i(x)

(x)

(x) = [ N1 N2 N3 N4 ]

1234

x

(x),x = [ N1,x N2,x N3,x N4,x ]

1234


111/147

y

(x),y = [ N1,y N2,y N3,y N4,y ]

1234

NIx

=NIr

r

x+

NIs

s

xNIy

=NIr

r

y+

NIs

s

y

NI

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

r

x

r

y

sx

sy

= J1

J NI(r, s) r s

N1,r = 14

(1 s) N1,s = 14

(1 r)

N2,r =1

4(1 s) N2,s = 1

4(1 + r)

N3,r =1

4

(1 + s) N3,s =1

4

(1 + r)

N4,r = 14

(1 + s) N4,s =1

4(1 r)

J =

x

r

x

s

y

r

y

s


112/147

x =4I=1

NIxI ,x

r=

4I=1

NIr

xI ,x

s=

4I=1

NIs

xI

x

r=

N1,r N2,r N3,r N4,r

x1x2x3

x4

x

s=

N1,s N2,s N3,s N4,s

x1x2x3x4

y =4I=1

NIyI ,y

r=

4I=1

NIr

yI ,y

s=

4I=1

NIs

yI

y

r=

N1,r N2,r N3,r N4,r y1y2y3

y4

y

s=

N1,s N2,s N3,s N4,s

y1y2y3y4

Ke = e BeT(M) Ce Be(M) d =

1

1 1

1

BeT

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

Ce

8 8

Kel =

eBe

T

std(M)CeBestd(M)

e

BeT

std(M)CeBeenr(M)

e

BeT

enr(M)CeBestd(M)

e

BeT

enr(M)CeBeenr(M)

16 16


113/147

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

F

F =

F(X)d +

+

F(X)d

=

F(X()) detJ()d +

+F(X()) detJ+() d


114/147

F =

F(X()()) detJ(()) detJ()d

+

+

F(X()()) detJ+(()) detJ+()d

F=n

GPI=1

F(I) detJ() detJ()wI+

n

+

GPI=1

F(I) detJ+() detJ+()wI

nGP n+GP

+

wI

A+

A

w+I = wA+IAI

wI

= wAI

AI

at the time

before


115/147


116/147

enriched nodes

not enriched nodes

crack crack tip crack

crack tip

(Fji) (Flk) dx

r0.5

Fi

G :

xy

x yy

w

= G() , w = w det(

G)

0 P b = X 0 \ c0

u(X, t) = u(X, t) on u0


117/147

n0 P(X, t) = t0(X, t) on t0

n0 P(X, t) = 0 on c0 u t0

c0 u0

t0

c0 = 0 , (u0

t0)

(t0

c0)

(u0

c0) =

u V

W = Wint Wext = 0 u

Wint =

0

( u)T : P d0

Wext =

0

u b d0 +

t0

u t0 d0

V =

u(, t)|u(, t) H1, u(, t) = u(t) on u0 , u discontinuous on c0

V0 =

u|u H1, u = 0 on u0 , u discontinuous on c0

Space of Bounded Deformations


118/147

23

12

12

13

23

udisc = 3 3(

) a3

= [1 2

3 ] 23P

3 = 1 1 2 3(

) = sign (()) sign(3)


119/147

2

3 1P13

2

P

N3() = 1 1 2N1() = 1

N2() = 2

11

22

1 =1

1P, 2 = 2

1P P

31

udisc = 2 2(

) a2

1 = 1 1P2P

2, 2 =

22P

2() = sign(()) sign(2) a3 = aP = 0

udisc = I I I(

) aI

aI

udisc

enr enr enr

enr

B


120/147

crack tip enrichment

Heaviside enrichment

B = [B1 B2 B3 B4]

=

r sin

2,

r cos

2,

r sin

2sin(),

r cos

2sin()

B

r = 0

uh(X) =IS

NI(X) uI+

ISc(X)NI(X) H(fI(X)) aI

+

ISt(X)NI(X)

K

BK(X) bKI

St

B

a b c d a

p


121/147

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 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 01 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 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 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 01 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 10 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 01 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 10 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 01 1 1 1 1 11 1 1 1 1 11 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 aa crackbcd

A+

A

r+ r

r+ =A+

A+ + A, r =

A

A+ + A

a b c d a b

KuuIJ KuaIJ K

ubIJK

KauIJ KaaIJ K

abIJK

KbuIJK KbaIJK K

bbIJK

uJaJ

bJK

=

fextIfextIfextIK

K d = fext

K d = {u a b}T fext =

fu fa fb

T fb =

fb1 fb2 fb3 fb4

fuI =

NI b d +

t

NI t d

faI =

NI (H((X)) H((XI))) b d+

tNI (H((X)) H((XI))) t d


122/147

fblI =

NI

BlI(X) BlI(XI)

b d+

t

NI

BlI(X) BlI(XI)

t d

K =

BT C B d

B

BuI = NI,X 0

0 NI,Y

NI,Y NI,X

BaI =

NI,X (H((X)) H((XI))) 00 NI,Y (H((X)) H((XI)))NI,Y (H((X)) H((XI))) NI,X (H((X)) H((XI)))

BblI |l=1,2,3,4 =

NI

BlK(X) BlK(XI),X

0

0

NI

BlK(X) BlK(XI),Y

NI

BlK(X) BlK(XI)

,Y

NI

BlK(X) BlK(XI)

,X

NI B

lK(X)

,i

= NI,i BlK(X) + NI B

lK(X),i

Bl,i = Bl,r r,i + B

l, ,i

r

, i

Bl

,r

Bl

,

B1,r =sin(/2)

2

2B1, =

2cos(/2)

2

B2,r =cos(/2)

2

2B2, =

2sin(/2)

2

B3,r =sin(/2) sin()

2

2B3, =

r

cos(/2) sin()

2+ sin(/2) cos()

B4,r =

cos(/2) sin()

2

2B4, =

r

sin(/2) sin()

2+ cos(/2) cos()


123/147

X

Y

X

Y

r

r,X = cos() ,X = sin/rr,Y = sin() ,Y = cos/r

B1,X =sin(/2)

2

2B1,Y =

cos(/2)

2

2

B2,X =cos(/2)

2

2B2,Y =

sin(/2)

2

2

B3,X =

sin(3/2) sin()

22B3,Y =

sin(/2) + sin(3/2) cos()

22B4,X =

cos(3/2) sin()

2

2B4,Y =

cos(/2) + cos(3/2) cos()

2

2

B,X = B,X cos() + B,Y sin()

B,Y = B,X sin() + B,Y cos()


124/147

Branching discontinuityIntersecting discontinuity

1(x) = 01(x) = 0

2(x) = 02(x) = 0

S1c

1(X) = 0 S2c 2(X) = 0

S3c = S1c

S2c

S1t

S2t

uh(X) =IS(X)

NI(X) uI+

IS1c(X)NI(X) H(1(X)) a

(1)I

+

IS2c(X)NI(X) H(2(X)) a

(2)I

+ IS3c(X)

NI(X) H(1(X)) H(2(X)) a(3)

I

+

IS1t (X)NI(X)

K

B(1)K (X) b

(1)KI

+

IS2t (X)NI(X)

K

B(2)K (X) b

(2)KI


125/147

(1 < 0, 2 < 0) (1 >0, 2 > 0) (1 > 0, 2 < 0) (1 > 0, 2 < 0) (1 > 0, 2 >0) (1 < 0, 2 < 0) 1 X

1(X) =

01(X),

02(X1)

02(X) > 0

02(X), 02(X1)

02(X) < 0

0

uh(X) =IS(X)

NI(X) uI+ncn=1

ISc(X)

NI(X) H((n)I (X)) a

(n)I

+

mtm=1

ISt(X)

NI(X)K

B(m)K (X) b

(m)KI

nc mt


126/147

0 P b = X 0 \ c0

u(X, t) = u(X, t) on u0

n0 P(X, t) = t0(X, t) on t0

n0 P(X, t) = 0 on c0 if not in contact

t+0t = t0t = 0, t

+0N = t0N on c0 if in contact

[[uN]] 0 on c0

[[n P]] = 0 on c0

t0N = n P n t0t

[[uN]] = u+ n+ = u n 0

n+ = n

0

( u)T : P d0

0

u b d0

t0

u t0 d0 +

c0

[[uN]] d0 0

C1

C0


127/147

KII KII

c

vc

=KI2r

fIh(, vc) +KII

2rfIIh (, vc)

fIh fIIh

vc c c

c =KcI2r

KcI

KIsinc + KII (3cosc 1) = 0


128/147

c = 2arctan

KI

K2I + 8K

2II

4KII

mdiag =m

nnodes

1

mes(el)

el

2 del

el m mes()el nnodes

M

lumped

II = J MconsistentIJ , orMlumpedII = m

MconsistentIIJ

MconsistentIJ

t tc = 2/max

uh(X) = N1 u1 + N1 1 a1 + N2 u2 + N2 2a2

lumped =

m1 0 0 00 m2 0 00 0 m3 00 0 0 m4

max det(K M) K M


129/147

mi Ehkin = 0.5uT Mlumped u

Ekin = 0.5

elv2 d

u a

Ehkin = 0.5

m1 u21 + m2 u

22

= 0.5 u

2(m1 + m2)

Ekin = 0.5 m u2

= Ehkin m1 = m2 = 0.5 m m

u = a1(x) u

Eh

kin = 0.5 m3 a21 + m4 a22 = 0.5 a2 (m3 + m4)

Ekin = 0.5 a2

el

21 del

m3 m4

m3 = m4 =m

2 mes()el

el

21 del

l

N1(x) = 1 xl

N2(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

max= l

3E

lumpedFE = A l

1/2 0

0 1/2

tlumpedc,FE = l

E

=

3tc,FE


130/147

s

s

uh(x) = N1(x) u1 + N1(x) S(x s) a1+ N2(x) u2 + N2(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 + 2/3s31/6 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 11 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

2tlumpedc,FE


131/147

crack crack

crack

effective crack length

a) b)

c) d)

1 1 2

34


132/147

Sc

2sin(/2)

Sc

a b

a n0 = b n0 = 0 n0

a b


133/147

crack

T(X)

a 0T= 0T a = 0 in 0b

0T

= 0T

b

= 0 in 0

t+ v = 0


134/147

r = 2 + 2 = arctan(/)

= = 0


135/147


136/147


137/147

P

I

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(rJ; h

)

h

3


138/147

c,ext

c,ext

strong embedded elements


139/147

embedded elements


140/147

c,ext P

a) b) c)


141/147

a) b)

uh(X) =IS

NI(X) uJ+M(e)s (X) [[u

(e)I (X)]]

u u

(e)

M(e)s

M(e)s (X) =

0 (e) / SH

(e)s (e) (e) S

(e) =N+e

I=1N+I (X)

Hs S N+e

(e)

+0

h(X) =IS

(0NI(X) uI)S(e) [[u(e)I (X)]]

S+

(e)s

k

[[u

(e)I (X)]] n

S

S (e)s /k


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(e)s

(e)s =

1 X Ske0 X / Ske

k

K

(e)uu K

(e)uu

K(e)

uuK

(e)

uu u(e)

[[u(e)

I]] =

FextI0

K(e)uu =

0

BT C B d0

K(e)uu =

0

BT C B d0

K(e)uu =

0

BT C B d0

K(e)uu =

0

BT C B d0

C

B

(e) =

(e)

x 0

0 (e)

y(e)

y(e)

x

n(e) =

nx 00 nyny nx

B

B = B

B = B [[u(e)I ]]

[[u(e)I ]] =

K

(e)uu

1K

(e)uu u

(e)

[[u(e)I ]]

K u = f

K = Kuu Kuu K1uu Kuu


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S

+

S

interelement separation methods


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lecture xfem meshfree

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