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IGA Solver

6È¸

ënóÆðó§Æ

IGA Solver–Xuefeng Zhu ënóÆ

SN0

1 IGA for Linear Elastic Problems

2 Isogeometric Shell Analysis

3 Trimmed Isogeometric Analysis

4 B++ Splines with Applications to IGA


kÚAÛ©Û¦)ìuÐ§

1 1943c§CourantuL1¦^n/«õª¼ê5

¦)Û=¯KØ©"

2 1956 cÅÑúiTurner§Clough§Martin ÚTopp 3©ÛÅ

(XÚïÄlÑ\!ù!n/üfÝLª

3 1960 cClough 3?n²¡5¯K§1gJÑ¿¦^/k

0(finite element method)¶¡"

4 1967 cZienkiewicz ÚCheung Ñ1k'k©Û

;Í"

5 1970 c±§km©A^u?n5ÚC/¯

K"Í¶ÆöXTed BelytschkoÚThomas Hughes.

6 2005c§Thomas HughesÇJÑAÛ©Û.


5åÆ§


¥mÃ¯K

e¡±¥mÃ²¡AC¯K~§505

¯KAÛ©Û

Ù)Û)µ

σr ,r (r , θ) =Tx

2(1− R2

r2 ) +Tx

2(1− 4

R2

r2 + 3R4

r4 )cos2θ (1)

σθ,θ(r , θ) =Tx

2(1 +

R2

r2 )− Tx

2(1− 3

R4

r4 )cos2θ (2)

σr ,θ(r , θ) =Tx

2(1 + 2

R2

r2 − 3R4

r4 )sin2θ (3)


5¯KAÛ©ÛÄÚ½

5¯KAÛ©ÛÄÚ½8B±eÚ½µ

(AÛëêz

ü©Û-üf

N©Û-of

\Dirichlet>.^Ú1Ö>.^

¦) £|

?n


I(AÛëêz-Volumetric parameterizations

Jessica Zhang, Gang Xu, Xin Li, Na Lei et al.


I(AÛëêz-Surface parameterizations

Xianfeng Gu, Xin Li, Jessica Zhang, Gang Xu et al.


I(AÛëêz-²¡¯K


IIü©Û


IIü©Û

ÀJ £¼ê(ë)

S =n∑i

Ri (ξ, η)Pi

u =n∑i

Ri (ξ, η)di


IIü©Û

∏=

∫Ω

12εσdΩ−

∫Ω uT fdΩ−

∫∂Ω uT TdΓ

þã¼ª=zzüþ¼ªÚµ∏=

∑e∏e =

∑e(∫Ωe

12ε

TeσedΩ−

∫Ωe

ueT fedΩ−

∫∂Ωe

uTe TdΓ)

Ù¥

σe =

σxx

σyy

τxy

= Dεe εe =

εxx

εyy

γxy

= Lue

ue =

u

v

L =

∂∂x 0

0 ∂∂y

∂∂y

∂∂x


IIü©Û

bneq´,ü':ê8

ue =

u

v

=

R1 0 · · · Ri 0 · · · Rneq 0

0 R1 · · · 0 Ri · · · 0 Rneq

dx1

dy1...

dxi

dyi

...

dxneq

dyneq


IIü©Û-2

∏e =∑

e dTe∫Ωe

12 BT DBdΩdT

e−deT ∫

ΩRT fedΩ−dT

e∫∂Ωe

RT Td∂Ω

³U¼égdÝ¦

∂∏∂d

= 0

éuzü¼aq¦Xeª

∂∏∂d

=∑

e

(

∫Ωe

12

BT DBdΩdTe −

∫Ω

RT fedΩ−∫∂Ωe

RT TdS) = 0

üfÝ§ª

Kede = fe

Ù¥ke´üfÝÝ§de´ü!:gdÝ§fe´ü1Öþ"


IIü©Û

K eij =

∫Ωe

BTi DBjdΩe =

∫Ωe

D0

∂Ri∂x 0 ∂Ri

∂y

0 ∂Ri∂y

∂Ri∂x

1 ν 0

ν 1 0

0 0 1-ν2

∂Rj∂x 0

0 ∂Rj∂y

∂Rj∂y

∂Rj∂x

∂Ri

∂ξ

∂Ri

∂η

=

∂x∂ξ

∂y∂ξ

∂x∂η

∂y∂η

∂Ri

∂x∂Ri

∂y

= J

∂Ri

∂x∂Ri

∂y


IIü©Û


IIIN©Û


IIIN©Û

N²ï§µ

Kd = f

ª¥§kNfÝÝ¶dN!: £þ¶fN1

Öþ"

§¦)3Ú\>.^c§N²ï§´ÛÉ§=

N§Ø)"3Ú\>.^§§)"


III\ £>.^Ú1Ö

NURBSØ÷vKronecker delta5§ÏdÃ\ £>.

^§ÀÀÇ)ûù¯K"¦ò £>.':=z

>.:§KN:gdÝÚ>.:gdÝ±ïáéXµ

d = Td∗

KKd = fCµ

TT KTd∗ = TT f


IV?n(J


SN0






Isogeometric Shell Analysis



~^NnØ)Reissner-MindlinnØ

Úkirchhoff-LovenØ"ö¦%CAÛC1ëY§ cö

KØI"



NAÛ¼êµ

x(ξ, η, ς) = xN(ξ, η) + ςh(ξ, η)

2f(ξ, η)

âëg§ÑN £¼ê

u

v

w

=∑

i

Ni (ξ, η)

ui

vi

wi

+12ςhi

[V1i −V2i

] αi

βi

þ/ª:



u

v

w

= Nae; ae =

ae

i...

aej

with aei =

ui

vi

wi

αi

βi

(4)

N =

N1 N2 · · · N9

ae =

u1 v1 w1 ϕ1 ψ1 · · · u9 v9 w9 ϕ9 ψ9

(5)

Ù¥

Ni =

Ni (ξ, η) 0 0 ζ

2 hi l1iNi (ξ, η) − ζ2 hi l2iNi (ξ, η)

0 Ni (ξ, η) 0 ζ2 him1iNi (ξ, η) − ζ2 him2iNi (ξ, η)

0 0 Ni (ξ, η) ζ2 hin1iNi (ξ, η) − ζ2 hin2iNi (ξ, η)

(6)



ÛIÚÛÜIm'Xµ

X = θX′

Ù¥

X =[

x y z]T

X′ =[

x ′ y ′ z ′]T

θ =[

V1 V2 V3

]=

l1 l2 l3

m1 m2 m3

n1 n2 n3



ÛÜIXeÛÜAåÛÜACµ

ε′ =[εx ′ εy ′ γx ′y ′ γy ′z′ γz′x ′

]T

=[

∂u′

∂x ′∂v ′

∂y ′∂u′

∂y ′ + ∂v ′

∂x ′∂v ′

∂z′ + ∂w ′

∂y ′∂u′

∂z′ + ∂w ′

∂x ′

]T

σ′ =[σx ′ σy ′ τx ′y ′ τy ′z′ τz′x ′

]T= Dε′

Ù¥

D =E

1− ν2

1 ν 0 0 0

ν 1 0 0 0

0 0 1−ν2 0 0

0 0 0 1−ν2k 0

0 0 0 0 1−ν2k



ÛIXÚÛÜIX £ê'Xµ∂u′

∂x ′∂v ′

∂x ′∂w ′

∂x ′

∂u′

∂y ′∂v ′

∂y ′∂w ′

∂y ′

∂u′

∂z′∂v ′

∂z′∂w ′

∂z′

= θT

∂u∂x

∂v∂x

∂w∂x

∂u∂y

∂v∂y

∂w∂y

∂u∂z

∂v∂z

∂w∂z

θÙ¥

θ =[

V1 V2 V3

]=

l1 l2 l3

m1 m2 m3

n1 n2 n3



ÛÚÛÜêfm'Xµ



ÛACL«

ε = Lgug = Lgug = LgNae

Ù¥



ÛIXÚÛÜIXAC'Xµ

ε′ = Qgεg = QgLgug = QgLgNae = Bae

zNURBSfÝL«µ

Ke =

∫V e

Hdxdydz =

∫ 1

0

∫ 1

0

∫ 1

0BT DB |J|dξdηdζ



gÌïuçôge(AÛ©Û£)182N¤


SN0






Trimmed Isogeometric Analysis


SN0

1 IGA for linear elastic problems





Immersed methods and non-body fitted methods

IGA-based immersed methods include immersed IGA, FCM,

XIGA et al.

Traditional non-body fitted methods include such as IBM,

XFEM, EFGM, RBF-based meshless method et al.

Issue IIt is difficult for these methods to strongly impose Dirichlet boundary

conditions because they do not satisfy Kronecker delta property.


Incorporating boundary collocation points or boundary curves/surfaces

Issue IIAnother disadvantage of immersed methods is that they are

difficult to incorporate boundary collocation points or boundary

curves/surfaces as boundary representation.


Analysis-suitable volume parameterization is still a big challenge

Issue IIIVolume parameterization for isogeometric analysis is still a big

challenge, especially for the CSG models with complex topology

structures. Stress functions are usually discontinuous!


B++ spline method

B++ Splines = Boundary Plus Plus Splines

Advantages of B++ splinesStrongly impose Dirichlet boundary conditions

Incorporate boundary collocation points as the boundary

representation.

Say bye to multi-blocks and stress functions are continuous.


B++ splines

Its idea is replacing the control points of the background

spline mesh using the boundary collocation points or the

boundary spline curves/surfaces.


B++ spline basis functions

B++ spline basis functions satisfy the Kronecker delta property,

build the partition of unity. They are also linearly independent.


Mutually orthogonal components

By BPB = PC , we can rewrite PPPB, in terms of two mutually

orthogonal components, one in the row space of B and the other in

the null space of B, that is

PB = BTΦΦΦ + Z = BTΦΦΦ + RΨΨΨ (7)

where

BR = 0 (8)

Combining these equations, we get

PB = BT (BBT )−1PC + RPE (9)


Hierarchical B++ spline representation of a trimmed spline surface

Thus, S is reformulated

S = NAPA + NBBT (BBT )−1PC + NBRPE (10)

For simplicity

S = NAPA + NCPC + NEPE (11)

PA Control points with complete basis functions.

PC Boundary collocation points or the control points of trim

curves.

PE Enriched control points.

NA Complete basis functions.

NC Basis functions of the boundary collocation points or the

control points of trim curves.

NE Basis functions of enriched control points.IGA Solver–Xuefeng Zhu ënóÆ

T

The matrix T is

T = [T1 T2] =[

BT (BBT )−1 R]. (12)

where T1 = BT (BBT )−1 and T2 = R.

Advantages

Linear independence. Yes

Kronecker delta property. Yes

Partition of unity. Yes.


Applications-Structural analysis


Applications-XIGA


Applications-Topology optimization


Applications-Onestep inverse isogeometric analysis


Processing and potential applications on B++ splines

In processing:

Fluid-solid interactions using B++ splines.

Topology optimization using B++ spline-based isogeometric

analysis.

B++ spline-based XIGA.

Potential applications:

Simulations of failure and crack using B++ spline-based XIGA.

Multigrid method using B++ splines

Hierarchical B++ splines with applications to adaptive iso-

geometric analysis.


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