2 Singularity particle shape functions 2007.pdf

8/10/2019 2 Singularity particle shape functions 2007.pdf

1/23

Comput Mech (2007) 41:135157

DOI 10.1007/s00466-007-0174-x

O R I G I N A L PA P E R

The reproducing singularity particle shape functions for problemscontaining singularities

Hae-Soo Oh Jae Woo Jeong June G. Kim

Received: 23 October 2006 / Accepted: 13 March 2007 / Published online: 11 April 2007

Springer-Verlag 2007

Abstract In this paper, we construct particle shape func-

tions that reproduce singular functions as well as polynomialfunctions. We also construct piecewise polynomial convolu-

tion partition of unity functions by taking the convolution of

the scaled conical window function with the characteristic

functions of quadrangular patches (we provide the computer

code for this construction). We demonstrate that the reprodu-

cing singular particle shape functions yield highly accurate

numerical solutions for the singularity problems with crack

singularity or a jump boundary data singularity.

Keywords Reproducing polynomial particle shape

functions

Reproducing singularity particle shape

functionsPatch-wise uniformly spaced particlesInterpolation error estimateThe convolution partition ofunity function

1 Introduction

The finite element method (FEM) has been widely used to

solve many important science and engineering problems.

H.-S. Oh was supported in part by funds provided by the University of

North Carolina at Charlotte. J. G. Kim was supported in part by the

Research Grant of the Kangwon National University. J. G. Kim is aVisiting Professor of the University of North Carolina at Charlotte.

H.-S. Oh (B)J. W. JeongDepartment of Mathematics and Statistics,

University of North Carolina at Charlotte,

Charlotte, NC 28223, USA

e-mail: [email protected]

J. G. Kim

Department of Mathematics,

Kangwon National University,

Chunchon 200-701, South Korea

However, the conventional FEM has several obstacles, such

as mesh refinement and constructing smooth global basisfunctions. Recently, several generalized finite element

methods (GEFM) that circumvent the obstacles of the

conventional FEM were introduced. Among many GFEMs

that use meshes minimally or do not use meshes at all [ 13],

those methods related to this paper areElement Free Galerkin

Method (EFGM) [1,7,8,10,1315], h-p Cloud Method

[6], Partition of Unity Finite Element Method (PUFEM) [17,

24,25], and Reproducing Kernel Element Method (RKEM)

[1012].

This paper is a continuation of the previous paper [22] that

is closely related to those element free methods: RKPM and

RKEM. The Reproducing Kernel Particle Method (RKPM)

[7,8,10,1315] is a meshfree method that yields highly accu-

rate approximation to smooth functions by using the repro-

ducing kernel particle (RKP) shape functionsthat canexactly

interpolate the polynomials of a fixed degree. The RKP shape

functions can be constructed to be smooth up to any desired

order by selecting smooth window functions.

However, the RKP shape functions constructed by using

specific window functions with compact supports are gene-

rally fractionalfunctions with complicated denominators that

aresolutions of thesystem of algebraicequations.Thus,these

RKP shape functions have the following problems:

(1) They do notsatisfy theKronecker delta property;hence,

it has difficulties in dealing with Dirichlet boundary

conditions.

(2) Accuracy is compromised in numerical integrations for

these complex fractional shape functions.

In order to alleviate these obstacles, in [21], we construc-

ted piecewise polynomial Cr-reproducing polynomial

1 3


2/23

136 Comput Mech (2007) 41:135157

particle (RPP) shape functions associated with uniformly (or

non-uniformly) distributed particles, that satisfy the Krone-

cker delta property, for any integer r 0, and any desi-red reproducing order. Furthermore, in [22], by transforming

these piecewise polynomial RPP shape functions via bilinear

mappings, we construct piecewise polynomial particle shape

functions, associated with patch-wise uniformly (or non-

uniformly) distributed particles in a polygonal domain, thathave the property of polynomial reproducing of a reduced

order. However, elliptic boundary value problems on non-

convex domains (especially, cracked domains) contain sin-

gularities. Moreover, it is well known that the polynomial

shape functions poorly approximate the singular functions.

Thus, in this paper, to deal with singularity problems in the

framework of meshfree particle methods, using RPP shape

functions of order 2N and special mappings similar to

[4,9,18,19], we construct special particle shape functions

that reproduce three kinds of singularities:

(1) r

0.5

+l

cos(0.5+l), r0.5

+l

sin(0.5+l), l =0, . . . ,N1;(2) r0.5, r1.5, r0.5 cos(1.5 );(3) r0.5 sin(1.5 ),

and the complete polynomialsxk1yk2 , 0k1+ k2 N.The main ideas of the construction for reproducing singu-

larity particle(RSP) shape functions are as follows:

(1) Let Qs be a rectangular reference patch constructedby tensor product of reproducing polynomial particle

(RPP) shape functions on[1, 1]as shown in [21].(2) Suppose a polygonal domain contains a point singu-

larity at the origin(0, 0)of the rectangular coordinatesystem. We properly select a rectangular subdomainQscontaining the singularity on which the influence of the

singularity is dominant. Next, R:= \ Q s , wherethe singularity effects are tolerable, is divided into qua-

drangular patches Qj , j = 2, . . . ,NQ , as shown in[21].

(3) Leti j , i j , be RPP shape functions on Qs .Then,we construct special mappings Ts : Qs Qs suchthati j T1s , i j , reproduce the singular functionsarising in the crack singularity.

(4) Through the multiplication of the convolution partition

of unity toi jT1s ,we obtain RSP shape functionson the singular zone Q s with compact supports.

The resulting closed form particle shape functions

satisfy the Kronecker delta property except at few par-

ticles around the singular zone Q s . However, it can be

made so that the Kronecker delta property is satisfied

at all particle along the boundary of the domain.

The paper is organized in the following manner. Section2

introduces the notations and definitions used in this paper.

Section3reviews the construction of the (flat-top) convolu-

tion partition of unity functions that was given in [22]. We

also present those theorems proved in the previous paper

[22] for the purpose of using them for the construction of

reproducing singularity particle shape functions. Section 4

proves three main theorems and tree different mappings that

construct the reproducing singular particle shapefunctions as

well as reproducing polynomial particle shape functions. InSect.5,by combing the (flat-top) piecewise polynomial par-

tition of unity functions, and the singular particle shape func-

tions,that were constructed in Sects. 3 and 4, respectively, we

construct reproducing singular particle shape functions with

compact supports. In Sect.6,by comparing the interpolation

errors in the L2-norm and the H1-semi norm, we demons-

trate the effectiveness of the proposed reproducing singular

particle (RSP) shape functions in dealing with the crack sin-

gularity as well as the jump boundary data singularity. The

concluding remarks are stated in Sect.7. In Appendix, we

present the diagram of intersection of patches and the sup-

port of a scaled window function that are essential in theconstruction of the convolution partition of unity functions.

2 RKP shape functions and RPP shape functions

Throughout this paper,, Zd are multi-indices andx=(1x,2 x, . . . , dx),xj = (1xj , 2xj , . . . , dxj ) denote pointsin Rd. However, if there is no confusion, we also use the

conventional notation for the points in Rd or Zd as x=(x1,x2, . . . ,xd)and= (1, 2, . . . , d).We also use thefollowing notations:

(xxj ) :=(1x 1xj )1 . . . (dx dxj )d,

xu:= ||u

x11 xdd

, || :=1+ 2+ + d.

Let be a domain in Rd. For any non-negative integer m ,

Cm () denotes the space of all functions such that toge-

ther with all their derivatives D of orders

|

| m, are

continuous on .The support of is defined by

supp= {x : (x)=0}.

In the following, a function Cm () is said to be aCm - function.

We also use the usual Sobolev space denoted by Hk().

Foru Hk(),the norm and the semi-norm, respectively,

1 3


3/23

Comput Mech (2007) 41:135157 137

are

uk,=||k

| u|2d x

1/2

, and

|u

|k,

= ||=k | u

|2d x

1/2

.

A weight function (or window function) is a non-negative

continuous function with compact support and is denoted

byw(x).Two typical window functions are as follows: For

x R,

(a) Conical:

w(x)

= (1 x2)l , |x| 1,0, |x|> 1,

(1)

which is aCl1-function.(b) Gaussian:

w(x)=

(e1/1x2 ) if|x|< 1,0 if|x| 1, (2)

which is an infinitely smooth function.

InRd, the weight function w(x) can be constructed from a

one-dimensional weight function either as w (x)= w(x)or as w(x)= di=1 w(xi ), where x= (x1, . . . ,xd) andx2 = x21+ +x2d.In this paper, we use the latter for ahigher dimensional window function.

Let beafiniteindexsetand denotes a bounded domain

in Rd. Let{xj : j } be a set of a finite number ofuniformly or non-uniformly spaced points in Rd, that are

called particles.

Definition 2.1 Let k be a non-negative integer. Then the

functions j

(x) corresponding to the particles xj

, j

are called the RPP shape functions with the reproducing pro-

perty of orderk(or simply, of reproducing order k) if and

only if it satisfies the following condition:j

(xj ) j (x)=x , forxRd and for 0 || k.

(3)

By applying a similar argument to [2,7], one can easily

prove the following: the condition (3) for the RPP shape

functions is equivalent toj

(xxj ) j (x)=0||, for 0 || kand x Rd.

(4)

For the construction of RPP shape functions, we do not

use any specific window functions[21] in one-dimensional

case. For higher dimensional RPP shape functions, we use

unbounded RPP shape functions multiplied by the convolu-

tion partition of unity functions.

On the other hand, the construction of RKP shape function

is directly related to the window functions as follows:

The RKP shape function, associated with the particle

xj , j , is constructed by

j (x)=w(xxj )

0||k(xxj ) b (x) (5)

whereb (x)are chosen so that (3) is satisfied and w(x)is a

window function. This gives rise to a linear system in b (x),namely0||k

m+ (x)b (x)= 0|| for 0 || k, (6)

where 0||is the Kronecker delta, and

m (x)=jZ

w(xxj )(xxj ) .

For one-dimensional case, this system can be written as

M(x) [

b0(x), b1(x) , . . . , bk(x)]

T

= [1, 0, . . . , 0

]T,

where

M(x)=j

w(xxj )

1

(xxj )1(xxj )2

...

(xxj )k

[1, (xxj )1, . . . , (xxj )k].The coefficient matrixM(x) of the linear system (6) is called

the moment matrix.

3 The convolution (flat-top) partition of unity shape

function and the construction non-uniformly

distributed particles on quadrangular patches

In this section, we briefly review the construction of the flat-

top partition of unity shape functions and the construction of

patch-wise non-uniformly distributed particles. The detailed

descriptions and proofs can be found in [22].

1 3


4/23

138 Comput Mech (2007) 41:135157

3.1 The construction of the convolution partition of unity

shape function

For brevity, we denote the coordinates of points ofR2 by

x=(x,y)or = (, ).Definition 3.1 The convolution of functions f(x) andg(x),

which is denoted by( f g)(x), is defined by(f g)(x)=

R2

f(y)g(xy )d y.

The characteristic function of E R2, denoted by E, isdefined by

E(x)=

1 x E,0 otherwise.

The (flat-top) convolution partition of unity(PU) functions

are constructed as follows:

1. Suppose is a polygonal domain and

= {x: dist(x, )},x=(x,y)

which is called the -framed (the outer rectangle in

Fig. 1 bounded by thedotted line).Now is partitioned

into large bounded quadrangles (may not be rectangles),

Qk, k=1, . . . , nsuch that

Qk = Qk,

= n

k=1 Qk ,int(Qk) int(Ql)= , fork=l .

Here, for k= 1, . . . , n, the subset int(Qk) denote theinterior ofQ k.Then we have

nk=1

int(Qk)=1, a.e. on . (7)

Fig. 1 Diagram of , , for = 0.1 and the patches Qk,Qk,k=1, 2, 3, 4

Actually, Qk is enlarged from Qk by only along theboundary part Q k ( Qk= Qk ifQ k =. Itis elaborated in Sect.6.2)

2. Consider the scaled conical window function, defined

by

w (,) =

A(1

(

)2)l (1

(

)2)l if

||

and

||,0 otherwise,

(8)

where

A1 =

(1 (

)2)l d

(1 (

)2)l d,

andl is an integer with 3l


5/23

Comput Mech (2007) 41:135157 139

Fig. 2 Graphs of the

convolution PU functions

k(x,y), k=1, 2, 3, 4,on the .In the labels of four figures,

Ek stands for Qk w ,fork=1, . . . , 4

X

-1 -0.5 0 0.5

1

Y

0

1

0

0.2

0.4

0.6

0.8

1

YX

Z

E

*1

X

-1 -0.5 0 0.5

1

Y0

0.51

0

0.5

1

XY

Z

*E2

X-1 -0.5

0 0.5 1

Y

0

0.5

1

0

0.5

1

Y

X

Z

*E3

X

-1 -0.50

0.51

Y

0 0.5

1

ZZ

Z Z

0

0.2

0.4

0.6

0.8

1

Y

Z

X

*E4

Let us note that the integral domain P(x,y) is one of a tri-

angle, a quadrangle, a pentagon, and a hexagon, as shown in

Appendix A.

In other words, k isan integral of the polynomial w over

a polygon P(x,y)that is bounded by linear functions. Hence,

it is a piecewise polynomial whose support is the -framed

Qk(i.e., the set of all points that are within-distance fromQ

k). Specifically, this flat-top bubble function is as follows:

k(x,y)=

1, if[B + (x,y)] Qk;0, if[B + (x,y)] R2\Qk;r(x,y) >0,if[B + (x,y)] Qk is

a proper subset ofB + (x,y).(11)

The piecewise polynomialr(x,y)can be obtained in a clo-

sed form function; however, it is complicated except when

Q is a rectangle. Thus, r(x,y) can be determined numeri-

cally by using the Gaussian quadrature that can yield the

exact integral. Fortran code for r(x,y) can be found in our

previous paper [22]. In[20], we prove the decomposition of

supp k into subsets on which the convolution PU functionis a polynomial.

It is worth noting that the convolution partition of unity

shape function k(x,y) is as smooth as the window function

w (, ),since

x [Qk w] = Qk [w (,)],

where=(1, 2)denotes a multi index andx= x1 x2 .The graphs of the convolution partition of unity func-

tions k(x,y), k= 1, 2, 3, 4 for the quadrangles Qk, k=

1, 2, 3, 4 in Fig.1are shown in Fig.2. These figures and the

graphs of derivatives of kcan also be found in [22]. Let us

note that in Fig.2, we have 4

k=1 k

(x,y)=1, for all(x,y) ,

but not 1 for(x,y)

\.

3.2 The construction of reproducing polynomial particle

shape functions associated with patch-wise

non-uniformly distributed particles

In order to show that bilinear mappings preserve the pro-

perty of reproducing polynomials, we adopt the following

notations.

1. Supposebis a fixed positive real number and the piece-wise polynomial reference particle shape functions have

the polynomial reproducing property of order 2N:

2Nj=0

jj ( )= , for 0 2Nfor all[0,b].

(12)

Throughout this paper, we use the following notations:

h=b/2N, j= h j, j= 0, 1, . . . , 2N,

1 3


6/23

140 Comput Mech (2007) 41:135157

Fig. 3 Mapping from the reference patch to physical patches

j ( ) = the particle shape functions constructed inAppendix B (where j are not uniformly spaced) or,

the Lagrange interpolating polynomials corresponding

to the nodesj .

Taking tensor product of one dimensional shape func-

tions, the reference patch is

Q= [0,b] [0,b](see, Fig.3)

and the reference particle shape functions have the

extended-polynomial reproducing property as follows:

(j1,j2)2N

k1j1

k2j2

E xtj1 ( )E xtj2

()=k1 k2 ,

for 0k1, k22N, for all(, )R2. (13)

Then, we observe the following.

(a) It is important to note that the polynomial repro-

ducing property holds not only for (,)Q, butalso for(, )Q(the complement ofQ).

(b) If E xtj , j=0, 1, . . . , 2Nare the piecewise poly-nomials shown in appendix, those piecewise poly-

nomials which are not zero on[0, h] (or[h,b])are extended to(, h)(or [h, )). Since thesepiecewise polynomial particle shape functions are

global polynomials on (, h] (or[h, )) and

satisfy the polynomial reproducing property ofreproducing order 2Nfor all [0, h], the exten-ded shape functions also satisfy the polynomial

reproducing property of reproducing order 2Nfor

all (, 0]([b, )). Abusing notations, theshape functions, that are extended to the outside

of[0,b], are also denoted byj .2. LetQ kbe a quadrangular patch whose four vertices are

(xi ,yi ), i= 1, 2, 3, 4.

Then, a bijective mapping Tk: Q Qk(see, Fig.3)is defined by

(x,y)=Tk(,),

where

x= x1b2

(b )(b ) + x2b2

()(b ) + x3b2

()()

+x4b2

(b )(),

y= y1b2

(b )(b ) + y2b2

()(b ) + y3b2

()()

+y4b2

(b )().

Let

i j (x,y)=i j (T1k (x,y)),

where

i j (,)= Ex ti ( ) Ex tj ().Then the transformed particle shape functions have the poly-

nomial reproducing property with reduced reproducing order

N(one half of the original reproducing order), as stated in

the following lemma, which was proved in [22].

Let2Nbe the index set{i j: 1i, j2N+ 1}.Lemma 3.1 Suppose the reproducing property (13) holds

for

0

(k1

+k2)

2N.

Then the transformed particle shape functions have the fol-

lowing polynomial reproducing propertyi j2N

x1i y

2j

i j (x,y)=x1y2 ,

for0(1+ 2) N, (x,y)R2.

3.3 RPP shape functions with compact support

The supports of the extended piecewise polynomial RPP

shape functions, i j , constructed in theSect. 3.2, are unboun-

ded. Thus, we need to make these particle shape functionswith small compact support by capping i jwith the convolu-tion partition of unity functions kconstructed in Sect.3.1.

Let us define the particle shape functions by

i j (x,y):= [i j k](x,y).Then the reduced particle shape functions, i j (x,y) become

a piecewise polynomial RPP shape functions with polyno-

mial reproducing property of order N with supp(i j ) supp( k), for somek. It was proved in[22] that the capped

1 3


7/23

Comput Mech (2007) 41:135157 141

particle shape functions also have the reproducing polyno-

mial property of order N. That is,i j2N

xk1i y

k2j i j (x,y)=xk1yk2 ,

for 0(k1+ k2)N, (x,y) .

4 Reproducing singularity particle shape functions that

reproduce polynomials and singular functions

Suppose a polygonal domain is non convex at a point at

which the internal angle is / , where is a real number

with 0< 1, we use=1/m for the singular patch mapping T.

Then the inverse mapping is

T1 (x,y)=(r cos(), r sin()), (17)Let=1/,then their Jacobians are

J(T )= r1

cos( 1) sin( 1) sin( 1) cos( 1)

, (18)

J(T)1 = (1/r1)

cos(1) sin(1)sin(1) cos(1)

, (19)

|J(T )| = 2r2(1). (20)Next, we construct the particles and the polynomial repro-

ducing particle shape functions on a reference patch, Q=[0,a] [0,b]. Leti= ( a2N)(i 1), i= 1, 2, . . . , 2N+ 1j= ( b2N)(j 1), j=1, 2, . . . , 2N+ 1.

Let fi ( ) and g j () be the Lagrange interpolating poly-

nomials corresponding to the nodes i , andj , respectively.

The Lagrange interpolating polynomials is simple, but it is

not the best choice because the supports are whole interval.

Thus, forsmaller supports of shape functions, it is recommen-

ded to use those particle shape functions shown in appendix

B for fi andgj .

Let us consider the tensor product of fi ( ) andgj () defi-

ned by

i j (,)= fi ( ) gj ()for the reproducing polynomial shape functions correspon-

ding to the particles (i , j ) with reproducing polynomial

property of order 2N.

To generate the singular particle shape functions that can

deal with the crack singularity,

1. we choose the conformal patch mapping with=1/2.Then,

z=T(w)=w 2, w=T1 (z)=

z.

2. we choose a singular zone Qs . For example, Qs =[0.5, 0.5] [0, 0.5] is the rectangle A B C D aroundthe crack tip(0, 0)as shown in Fig.5.

It is worth notingthat ourconstruction forRSP shape func-

tions is not restricted to the crack singularity. For example,

the patch mapping

z=T (w)=w 3, =1/3,can generate RSP shape functions to deal with the re-entrant

corner singularity,r2/3 cos(2/3).

Remark 4.1 There are no particular rules to choose the size

of the singular zone Qs . One can choose the singular zone

1 3


8/23

142 Comput Mech (2007) 41:135157

so that the pollution effect of the crack singularity can be

tolerable on the outside of this zone.

For,i = 1, 2, . . . , 2N+ 1, and j= 1, 2, . . . , 2N+ 1,we denote the particles and the corresponding singular shape

functions as follows:

1. Let

T(i , j )=(xi j ,yi j ), T1 (ri j , i j )=(i , j ),(ri j ,i j )= the polar coordinates of(i , j ),(ri j , i j )= the polar coordinates ofT (i , j ).

Then

T1 :

i =ri j cos(i j )j= ri j sin(i j ) ;

T:

xi j= r1/

i j cos(i j /)yi j= r1/i j sin(i j /).

(21)

2. The singular shape functions corresponding to the par-

ticles(xi j ,yi j )are

i j (x,y)=(E xti j T1 )(x,y),

where

i j (,)

= fE xti ( )

gE xtj ().

Henceforth, we simply writeE xti j byi j .Let 2N= {kl:1k, l2N+ 1}.Then we haveLemma 4.1 (1) For i j, kl2N,we have

i j (k, l )=i kjl .

(2) For any point (,) in the w-plane (and hence in

T1 (Qs )), the reference global polynomial shape func-tions, i j (,),(i, j ) 2N, have the reproducing

polynomial property of order2N :i j2N

k1i

k2j

i j (,)=k1 k2 ,

for0k1+ k22N. (22)

Let us note that the mapped particles in the singular zone

are not uniformly distributed (see, Fig. 4). Moreover, the

corresponding singular shape functions i j

(x,y) do not have

compact supports.

Theorem 4.1 Suppose the RPP shape functions (Lagrange

interpolants or the piecewise polynomials in Appendix B)

i j (, ), for i j 2Ncorresponding to the particles (i , j ), i j 2Nsatisfy the

following relation:

i j2N

k1

i

k2

j

i j(,)

=k1 k2 , for0

k

1+k

22N.

(23)

Let

i j (x,y)=i j (T1 (r, ) )=i j (r cos(), r sin()).Then, we have

(1) i j satisfy the Kronecker delta property.(2) For = 1/2, the particle shape functions i j , i j

2N,reproduce the complete polynomial of order N:

xk1yk2 , 0k1+ k2 N. (24)

(3) For any , the particle shape functionsi j , i j 2N,reproduce the following singular functions:

r(2k+1) cos((2k+1)), r(2k+1) sin((2k+1)),for k=0, . . . , (N 1). (25)

Henceforth, because of the property (3) of Theorem4.1,

the shape functioni j= i j T1 are said to be the RSPshape functions. However,the supports ofi jare unbounded,and hence they are not yet the required RSP shape functions.

As stated in Sect.3.3,after multiplying the convolution PU

function s, the adjusted shape functions i j := s i jwith compact supports will be actually called the RSP shape

functions.

ProofBy the conformal mapping (16), the Eq. (23) is trans-

formed to(i,j )2N

ri jcos(i j )

k1 ri jsin(i j )

k2i j (r

cos(),

r sin())= [r cos()]k1[r sin()]k2 ,for 0

k1

+k2

2N. (26)

In what follows, we denote the right-hand side of Eq. (26) by

P(k1, k2; r, )=

r cos()k1 r sin()k2

=r(k1+k2) [cos()]k1 [sin()]k2 .[A] Suppose k1 +k2is an even integer and =1/2, we havethe following cases:

(A-0) Ifk1+k2= 0, there is only one case (k1, k2)=(0, 0).

P(k1, k2; r, /2)=1.

1 3


9/23

Comput Mech (2007) 41:135157 143

(A-2) If k1+ k2 = 2, there are three case (k1, k2) =(2, 0), (1, 1), (0, 2).

P(2, 0; r, /2)=rcos2

2

=r 1 + cos

2= 1

2(x+ r),

P(1, 1; r, /2)=rcos 2

sin

2= 1

2rsin = 1

2y,

P(0, 2; r, /2)=rsin2 2

=r1 cos 2

= 12

(rx)

(A-4) Ifk1+ k2= 4, there are five cases (k1, k2)=(4, 0),(3, 1), (2, 2), (1, 3), (0, 4).

P(3, 1; r, /2)=(r1/2 cos( /2))2(r1/2 cos( /2))(r1/2 sin( /2))

= 12

(x+ r) 12

y= 14

(x y+yr),P(1, 3; r, /2)= P(1, 1; r, /2) P(1, 2; r, /2)

= 14

(x y+yr),

P(2, 2; r, /2)= P(1, 1; r, /2) P(1, 1; r, /2)= 14

y2,

P(4, 0; r, /2)=r2 cos4 2

=r2 14

(1 + 2 cos + cos2 )

= 14

(r2 + 2r x+x2),

P(0, 4; r, /2)=r2 sin4 2

= 14

(r2 2r x+x2)

From these relations, we can generate polynomials of degree

2 as follows:

1

= P(0, 0

;r, /2),

x = P(2, 0; r, /2) P (0, 2; r, /2),y =2 P(1, 1; r, /2),y2 =4 P(2, 2; r, /2)x y=2 P(3, 1; r, /2) 2 P(1, 3; r, /2),x2 =[ P(2, 0; r, /2) P (0, 2; r, /2)]2

= P(4, 0; r, /2) 2 P(2, 2; r, /2)+P(0, 4; r, /2).

(27)

(A-6) Ifk1 + k2=6, there are the following seven relations:

P(4, 2; r, /2)= P(2, 0; r, /2) P(2, 2; r, /2)

= 1

8(x y2

+r y2),

P(2, 4; r, /2)= P(0, 2; r, /2) P(2, 2; r, /2)= 1

8(r y2 x y2),

P(5, 1; r, /2)= P(1, 1; r, /2) P(4, 0; r, /2)= 1

8(2x2y+y 3 + 2r x y),

P(1, 5; r, /2)= P(1, 1; r, /2) P(0, 4; r, /2)= 1

8(2x2y+y 3 + 2r x y),

P(3, 3; r, /2)= P(1, 1; r, /2) P(2, 2; r, /2)= 1

8(y3),

P(6, 0; r, /2)= P(2, 0; r, /2)3

= 18

(x+ r)3,P(0, 6

;r, /2)

= P(2, 0

;r, /2) P(2, 2

;r, /2)

= 18

(rx)3,Using all of above cases, we obtain all of the monomials of

degree 3:

x y2 =4( P(4, 2; r, /2) P (2, 4; r, /2)),y3 =8 P(3, 3; r, /2),x3 = P(6, 0; r, /2) P (0, 6; r, /2)

3( P(4, 2; r, /2) P (2, 4; r, /2)),x2y=2(P(5, 1; r, /2) P (1, 5; r, /2))

4 P(3, 3; r, /2)Inductively, we can combine all cases up to k

1 +k

2=2N, to

show that the RSP shape functionsi j ( T1 ), i j 2N,reproduce the complete polynomials of order N:

xk1yk2 , for 0k1+ k2 N. (28)[B] Next, let us consider the cases when k1+k2 is an oddinteger. Supposek1+ k2=2k 1, k=1, 2, . . . ,N .

Using the identity cos(2k 1)+i sin(2k 1) =(ei)2k1 =(cos + isin )2k1,we obtain

cos(2k 1) =k1l=0

(1)l

2k 12l

cos2k12l sin2l ,

sin(2k1) = kl=1

(1)l1

2k 12l 1

cos2k2l sin2l1 .

Thus, fork=1, 2, . . . ,N,we have the following relations:r(2k1) cos (2k 1)

=k1l=0

(1)l

2k 12l

r(2k1) cos2k12l sin2l ,

=k1l=0

(1)l

2k 12l

P(2k 1 2l, 2l; r,).

r(2k

1) sin(2k

1)

=k

l=1(1)l1

2k 12l 1

r(2k1) cos2k2l sin2l1

=k

l=1(1)l1

2k12l1

r(2k1) P(2k 2l, 2l 1; r,).

Specifically, whenk1+ k2=1, we haver cos()= P(1, 0; r,),r sin()= P(0, 1; r,).

1 3


10/23

144 Comput Mech (2007) 41:135157

Whenk1+ k2=3, we have

r3 cos(3 )=1

l=0(1)l

3

2l

P(3 2l, 2l; r, )

= P(3, 0; r, ) 3P(1, 2; r,),

r3 sin(3 )

=

2

l=1

(

1)l1 3

2l 1P(32l, 2l 1; r, )=3 P(2, 1; r, ) P (0, 3; r,).

Whenk1+ k2=5, we have

r5 cos(5 )=2

l=0(1)l

5

2l

P(5 2l, 2l; r, )

= P(5, 0; r, ) 10P(3, 2; r, )+5 P(1, 4; r,),

r5 sin(5 )=3

l=1(1)l+1

5

2l 1

P(62l, 2l 1; r, )

=5 P(4, 1; r, ) 10 P(2, 3; r, )+P(0, 5; r,).

Taking=1/2 in Theorem4.1,we have the following

Corollary 4.1 Suppose the RSP shape functions on the

z-plane areconstructed by the conformal mapping, T1 (z)=z1/2,as follows:

i j (x,y)=(i j T1 )(x,y)=i j (r1/2 cos( /2),r1/2 sin( /2)).

Then, we have

(1) The shape functions, i j (x,y), i j 2N, have theKronecker delta property:

i j (xkl ,ykl )= kli j .

(2) The shape functions, i j (x,y), i j 2N, have thepolynomial reproducing property of order N:

xk1yk2 , for0

k1

+k2

N. (29)

(3) The shape functions, i j (x,y), i j 2N,reproducethe singular functions:

r1/2+l cos(1/2 + l) , r1/2+l sin(1/2 + l) ,l=0, . . . , (N 1). (30)

Remark 4.2 The transformed relation (26) reproduces poly-

nomials when the reproducing orders are even numbers (that

is,(k1+k2) is 0, 2, 4, . . . , 2N), whereas the same relation

reproduces singular shape functions when the reproducing

orders are add numbers (that is, (k1+ k2) is 1, 3, 5, . . . ,2N 1).

Corollary 4.2 Suppose the intensity of singularity is a ratio-

nal number=n/m with1< n < m. The RSP shape func-tions constructed via the singular patch mapping defined by

T (,)=(r1/ cos((1/) ),r1/ sin((1/))), (31)

where=1/m, generate the singular functions:

rn/m+l cos(n/m+ l) , rn/m+l sin(n/m+ l) ,l=0, 1, 2, . . . (32)

ProofWithout loss of generality, we assume that

= 2/3

(the re-entrant corner singularity). In order to generate thesingular functions:

r2/3+l cos(2/3 + l), r2/3+l sin(2/3 + l) ,l=0, 1, 2, . . . (33)

we can consider the following patch mapping:

T (,)=(r1/ cos((1/) ),r1/ sin((1/))), (34)

where=1/3. Observing that2=2/3, 5=(2/3 + 1), 8=(2/3 + 2),

Theorem4.1implies that the RSP shape functions generate

the required singular functions

Corollary 4.3 Suppose u(x,y) is a linear combination of

the singular functions, listed in(30), and the complete poly-

nomials, listed in (29). Then theRSP shapefunctions i j , i j2Nexactly interpolate u(x,y). That is, for all (x,y)

R

2,

u(x,y)=2N+1

i=1

2N+1j=1

u(xi j ,yi j )i j (x,y).

Proof It is sufficient to prove this claim when u (x,y) has

two terms. For example, suppose

u(x,y)=C1r1/2 sin( /2) + C2x y.

1 3


11/23

Comput Mech (2007) 41:135157 145

From the relation (27), we have

u(x,y)=C1 P(0, 1; r, /2) + C2[2P(3, 1; r, /2)2 P(1, 3; r, /2)]

=

i j2NC1

r

1/2

i j sin(i j /2)1

i j (T (x,y))

+C2 2 r1/2i j cos(i j /2)3 r1/2i j sin(i j /2)12

r

1/2

i j cos(i j /2)

1 r

1/2

i j sin(i j /2)

3i j (T (x,y))

=

i j2NC1

r

1/2

i j sin(i j /2)i j (T (x,y))

+C2

(xi jyi j+yi j ri j ) + (xi jyi j+yi j ri j )

i j (T (x,y))=

i j2NC1

r1/2

i j cos(i j /2)+ C2[xi jyi j ]

i j (T (x,y))=

i j2N

u(xi j ,yi j )(x,y).

In the following three subsections, without loss of gene-

rality, we use 25 particles (that is, N= 2) in the referencepatch Q.

4.2 Particle shape functions that generate the singular

functions

r,

rcos(3/2)

Consider the following domains

Dcos= {(r, ):0 < r 1/2, 2/3 };D2= {(r, ):0 < r 1/2, 0 /3};D3= (, log(1/2)]

0,

2

;

Qcos= (0, 1/

2] [0, 1]; Q0= [1, 1] [1, 1].Now we define the following bijective mappings:

f12(r, )=(r, 23 ) : Dcos D2f23(r, )=(log(r), 32 ) : D2 D3f

34(

x,y)

=(e

x/2, cos(

y))

:D

3 Q

cos(,)=(22 1, 2 1):QcosQ0

(35)

Define a singular mapping from Dcos onto Qcos by

T1cos (r, )=

r, cos

3

2( 2/3)

, (36)

which is the composition of f12, f23, and f34. Then the patch

mapping from a reference patch Qcos onto Dcos is

Tcos (,)=

2,2

3+2

3cos1()

. (37)

Theorem 4.2 Leti (), 1 j 5, be the reference par-ticle shape function defined by (12),i j (,)= i ( )j ()and suppose

5i,j=1

k1i

k2j i j (,)=k1 k2 ,

for0k1, k24, ( , )Qcos . (38)Then the particle shapes

[cos]i j (i j ) T1cos , i=1, . . . , 5, j= 1, . . . , 5,satisfy the Kronecker delta property and generate the singu-

lar functions

r1/2, r1/2 cos(3 /2), r3/2.

ProofIt follows from (36)and (38)that

5

i,j=1

Tcos (k1i

k2j )(i j T1cos (r, )=(r1/2)k1 (cos 3/2)k2 ,

for 0k1, k24.By selecting proper combinations ofk1and k2, we obtain the

the required singular functions. Remark 4.3 The tensor product shape functions,i j= ij , 1i , j 5,has stronger reproducing polynomial pro-perty with serendipity reproducing order than the order res-

tricted by 0k1+ k24.Corollary 4.4 The particlescorresponding to the RSP shape

functions [cos]i j are all in the domain Dcos, and hence theseRSP shape functions exactly interpolate a linear combinationof those singular functions, a1r

1/2 +a2r1/2 cos((3/2) )+a3r

3/2,on the domain Dcos .

4.3 Particle shape functions that generate the singular

functions

r,

rsin(3 /2)

Let

Dsi n= {(r, ):0 < r 1/2, /3 };D2= {(r, ):0 < r 1/2, 2/30};D3=(, log(1/2)] [0, ];Qsi n=(0, 1/2] [1, 1].

Then we have the following bijective mappings:

g12(r, )=(r, /3) : Dsi n D2g23(r, )=(log(r), 32 ) : D2 D3g34( x,y)=(e x/2, cos( y)):D3Qsi n(,)=(22 1, ) : Qsi nQ0

(39)

Define a singular patch mapping from Dsi n onto Qsi n byT1si n (r, )=(

r, cos((3/2)( /3)), (40)

1 3


12/23

146 Comput Mech (2007) 41:135157

which is the composition ofg12, g23, andg34. Then the patch

mapping from a reference patch Qsi n onto Dsi n is

Tsi n(,)=(2, /3 + (2/3) cos1()). (41)

By the similar methods to the proof of Theorem 4.3, we

have the following:

Theorem 4.3 Supposei j , i=1, . . . , 5, j= 1, . . . , 5, res-pectively, are the reproducing polynomial particle shape

functions associated the particles(i , j )Qsi n such that5

i,j=1

k1i

k2j

i j (,)=k1 k2 , for0k1, k24. (42)

Then the particle shapes

[si n]i j (i j ) T1si n , i= 1, . . . , 5, j= 1, . . . , 5,

satisfy the Kronecker delta property and generate the singu-

lar functions

r1/2, r1/2 sin(3/2), r3/2.

Corollary 4.5 The particlescorresponding to the RSP shape

functions [si n]i j are all in the domain Dsi n, and hence theseRSP shape functions exactly interpolate a linear combination

of those singular functions, b1r1/2 +b2r1/2 sin((3/2) )+

b3r3/2,on the domain Dsi n .

4.4 Interpolation error of the Crack singularity associated

with RSP shape functions

Let us consider the following three linear combinations of

those singular functions that are generated in the previous

three subsections:

U (r, ) =r0.5 cos(0.5 ) + r1.5 cos(1.5 )+r0.5 sin(0.5 ) + r1.5 sin(1.5 ),

Ucos (r, )=r0.5 + r0.5 cos(1.5 ),Usi n(r, )=r0.5 + r0.5 sin(1.5 ).

(43)

Then, It follows from Corollary 4.3 through Corollary 4.5

that the RSP shape functions{i j (T11/2 ): 1 i, j 5},{i j (T1cos ):1i, j5} and{i j (T1si n ): 1 i, j 5},exactly interpolateU(r, ) , Ucos (r, ) ,and Usi n(r, ) ,res-pectively.

Let D= {(r, ): r < 0.5, 0 } and Q =[0, 1/2] [0, 1/2].Then the conformal mapping T1 :DQ with=1/2 is defined by

T1 (r, )=(r0.5 cos(0.5 ), r0.5 cos(0.5)).

For the purpose of computing the derivatives of the inter-

polation errors, we compute the Jacobian of the mappings,

T1 , T1cos , T1

si n,respectively, as follows:

J(T1 )=

/x/x

/y /y

= r1

cos( 1) sin( 1) sin( 1) cos( 1)

,

J(T1cos )=

(1/2)r0.5 cos (3/2) sin( 32

)(sin )/r

(1/2)r0.5 sin (3/2) sin( 32

)(cos )/r

,

J(T1si n )=

(1/2)r0.5 cos (3/2) cos( 32 )(sin )/r(1/2)r0.5 sin (3/2) cos( 3

2 )(cos )/r

.

Now, the gradient ofi j (T1cos )is as follows: i j (T1cos (r,))/x

i j (T1cos (r,))/y

= J(T1cos ) J()

i (s)j (t)/ s

i (s)j (t)/ t

, (44)

where: Qcos Q0 is defined by(35), and i (t) is theLagrange interpolation polynomial associated with the node

i in[1, 1]. The gradients ofi j (T1 ) and i j (T1si n )can be computed in a similar manner.

Let us note that, as r 0, the test singular functions(43)approach zero, whereas their derivatives become infinitely

large. However, numerical tests show that for each case, the

absolute maximum interpolation errors are virtually zero as

shown in Table1. From Table1,we observe the followings:

1. we compute the maximum interpolation errors by eva-

luating the interpolation errors at the following61 points:

(r, ) , =k(/60), k=0, 1, . . . , 60,

for each of the ten layers: r= 0.5, 0.2, 0.1, 0.1E1,. . . , 0.1E7.

2. Rerr stands for the sum of the maximum ofDx(error)and the maximum ofDy (error).

3. Since U/xat only one point(r, /2)is very small,

the relative error is about four digits larger than all the

other numbers even though the absolute error is vir-

tually zero. Thus, in estimating the maximum relativeDx(error) of the first case, the relative error at the point

(r, /2)was excluded.

4. The errors for the second case is lager than the other

two cases. The error would be as good as that of the

first case if calculating error were restricted on Dcos={(r, ): 0 < r 5, 3/2}. Let us note thatthe particles corresponding to the RSP shape functions

i j (T1

cos (r, ) )are in the domain Dcos that is one third

of the upper half disk with radius r=0.5 (see, Fig.4).

1 3


13/23

Comput Mech (2007) 41:135157 147

Table 1 Max relative errors along the half circleSr= {(r, ):0 } for the radiir= 0.5, 0.2, 0.1, 0.1E1, 0.1E2, 0.1E3, 0.1E4,. . . ,0.1E7

Maximum relative errors

S. type: r1/2 cos( /2), r1/2 sin( /2) r1/2, r1/2 cos(3 /2) r1/2, r1/2 sin(3 /2)

Layers(r): Rerr-1 Rerr-1 Rerr-2 Rerr-2 Rerr-3 Rerr-3

0.5E+0 8.31E16 3.98E14 9.23E14 7.55E13 2.12E15 9.83E150.2E+0 5.51E16 1.11E13 8.08E14 3.12E13 2.52E15 9.32E150.1E+0 8.69E16 4.74E14 5.25E14 2.42E13 4.75E15 7.44E150.1E1 2.75E15 6.96E14 6.77E14 6.17E13 6.89E15 1.17E140.1E2 3.42E15 4.78E14 1.10E13 1.16E12 1.46E14 1.90E140.1E3 4.95E15 3.36E14 1.12E13 1.77E12 1.13E14 3.17E140.1E4 8.34E15 5.87E14 9.86E14 1.84E12 1.24E14 3.90E140.1E5 2.18E14 2.01E14 1.01E13 1.33E12 2.66E14 2.49E140.1E6 6.97E14 4.42E14 9.91E14 2.18E12 6.39E14 2.42E130.1E7 2.26E13 3.32E14 1.55E13 1.91E12 1.28E13 4.37E13Rerr indicates the sum ofx-derivative and y -derivative of errors in the maximum norm. Rerr stands for the relative errors in the maximumnorm

Remark 4.4 Under stress boundary condition alongthe crack

front, the solution vector u(r, )= (ur, u) of the linearelasticity equations around the crack tip can be expressed as

follows ([23], p. 290):

ur= 1

4

r2

1/2{[(2k 1) cos( /2) cos(3 /2)]KI

[(2k 1) sin( /2) 3 sin(3 /2)]KI I} + o(r1/2)u

= 1

4 r

21/2 {[(2k+ 1) sin( /2) + sin(3 /2)]KI

[(2k+ 1) cos( /2) 3cos(3/2)]KI I} + o(r1/2),where k=34for plane strain, =E/(2(1+)),andKIand KI Iare the opening mode and the sliding mode stress

intensity factors, respectively.

We have seen that those singular functions appeared in

the singular displacement vector functions can be exactly

interpolated by using one of the RSP shape functions

{i j (T11/2 (r, ) ): 1 i, j 5},{i j (T1cos (r, ) ): 1i, j 5} and{i j (T1si n (r, ) ): 1 i, j 5}. Thus, themeshfree method incorporated with these RSP shape func-

tions around the crack tip could yield highly accurate stressanalysis.

From the RSP shape function constructed in this section,

we observe the followings:

1. Our RSP shape functions are different from adopting

the enriched base functionr0.5 incorporated with a cut-

off function in the classical FEM [5]. Assuming that

the boundary is along the line= , the five particles

along the-axis in the reference rectangles Qcos ,Qsi n(the -axis in the reference rectangle Q ) are sent to theboundary partQ s of the patchQ s= [0.5, 0.5][0, 0.5]. Since the RSP shape functions satisfy the Kro-necker delta property at these particles, it is easy to

handle the Dirichlet boundary condition.

However, it is important to note that, if the reference

patches are Q= [0,H1] [0,H1],Qcos= [0,H2]

[0, 1

],

Qcos

= [0,H3

] [1, 1

], then all three types

RSP shape functions concurrently satisfy the Kroneckerdelta property at the five particles on the negativex-axis

(= ) only when H1= H2= H3= H. We usedH= 1/2 for Fig.4.

2. One can use 16 particles on the reference patches Qinstead of 25 particles, however, the five particles on the

boundary handles the boundary condition better than the

four particles does.

3. The particles corresponding to the RSP shape functions

constructed in Sects.4.1(for the case I singularity), 4.2

(for the case II singularity), 4.3 (for the case III singu-

larity), respectively, are plotted in Fig. 4. If

Q were

[0, 0.5] [0, 0.5],the particles (diamonds in Fig.4)would have been inside Q= [0.5, 0.5] [0, 0.5]. Inthat case, the accuracy along the layerr= 0.5 drops byone digit, whereas the accuracy along the layer

r= 0.1E7 is slightly improved. Moreover, in this case,the Kronecker delta property is not satisfied for the RSP

shape functions for the case II and the case III sin-

gularities, unless the reference patches for these cases

are also adjusted to Qcos= [0, 0.5] [0, 1],Qcos=[0, 0.5] [1, 1].

1 3


14/23

148 Comput Mech (2007) 41:135157

X

Y

-0.4 -0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

Particles for Case I Singularity

X

Y

-0.4 -0.2 0 0.2 0.40

0.2

0.4

0.6

Particles for Case II Singularity

X

Y

-0.4 -0.2 0 0.2 0.40

0.2

0.4

Paricles for Case III singularity

Fig. 4 Particles corresponding to the RSP shape functions for the Case I singularity (U ), the Case II singularity(Ucos ), and the Case III

singularity(Usi n ), respectively

5 The construction of RSP and RPP global shape

functions with compact supports

Without loss of generality, we assume that all reference

patches,Qk,Q ,Qcos ,Qsi n , are equipped with the standardRPP shape functions of reproducing order 4.

In Sect.3, using the bilinear patch mapping Tk: QQkwe constructed the RPP shape functions that reproduce

the complete polynomial ofm (m= 4 if Qk is rectangle,m=2 ifQ kis a quadrangle):

x1y2 , 01+ 2m , for all(x,y)R2.

InSect. 4, through the singular patch mapping T :QQ , = 1/2,we constructed the RSP shape functions thatreproduce the complete polynomials

x1y2 , 01+ 22, for all(x,y)R2,

as well as the singular functions:

r0.5+k cos((0.5+k)), r0.5+k sin((0.5+k)), fork=0, 1.

Using the singular patch mappings Tcos and Tsi n , we

constructed the RSP shape functions that reproduce the sin-

gular functions:

r0.5, r1.5, r0.5 cos(1.5 ), r0.5 sin(1.5 ).

From now on, the singular patch mappings T , Tcos , Tsi nare denoted byTs and the bilinear patch mapping is denoted

byTk.

Let us note that

thesupportsof these particle shape functionsare unboun-ded;

the mapped particles by the bilinear patch mappingTkare overlapping along the common edges of patches that

do not contain singularities.

[A]We reduce these particle shape functions to the func-

tions with compact supports by multiplying the convolution

partition of unity functions k= w Qk constructed in

1 3


15/23

Comput Mech (2007) 41:135157 149

Sect.3.1. For(x,y) and fori j 4, we defineki j (x,y)= [(i,j) T1k (x,y)] [ k(x,y)],skl (x,y)= [(k,l) T1s (x,y)] [ s (x,y)].Then supp(ki j ) and supp(

skl

) are compact subsets{(x,y):dist((x,y),Qk }. Here Qk is enlarged by only alongthe boundary part Q k (see, Sect.6.2for details).

[B] (Global numbering of particles) Consider the follo-

wing mapped particles obtained by the patch mappings

Tk(i , j ), k=1, 2, 3, . . . , n Q , 1i, j5,Ts (i , j ), 1i , j5(s stands for the index of the patch

mappings for the singularities).

[B1: Numbering among RPP shape functions] To each of

these local particle numbers Tk(i , j )corresponding to the

RPP shape functions, we assign one global particle number.

If several mapped particles associated with RPP shape func-

tions share one point in common, we assign the same global

number to thses particles and these RPP shape function is the

sum of the associated RPP shape functions.

Suppose, for example,

Tk1 (1, 1), Tk2 (2, 2), Tk3 (3, 3), Tk4 (4, 4),

are the same point on the common edge of two patches or the

common vertex of several patches. Then the global particle

number and the global RPP shape functions are determined

as follows:

I(x,y)=ki j (x,y)if one mapped particle corresponds to one

point in ,

I(x,y)=ki j (x,y) + k

i j (x,y),

if two mapped particles corresponds to one

point in an edge ,

I(x,y)=k1i1j1 (x,y) + k2i2j2

(x,y) + k3i3j3 (x,y)+k4i4j4 (x,y),

if several mapped particles(e.g. four)correspond

to one point at a vertex .

[B2:Numbering among RSP shape functions] To each local

particle number Ts (i , j ) corresponding to the RSP shape

function, we assign a different global particle number even

though several local particles fall on the same point in .

For example, since T (r, /2)= Tcos (r, 0)= Tsi n(r, 0)=(r2, ) , three different RSP shape functions are correspon-

ded to each of five points, ([k/(42)]2, ) , k=0, 1, 2, 3, 4.Thus, at least three different global particle numbers are

assigned to each of these five points. The reason for this

assignment is that three different types of RSP shape func-

tions corresponding to the particles T (i , j ), Tcos (i , j ),

Tsi n(i , j ) should be used within the same rectangular patch

to deal with the crack singularity.

[B3:Numbering among RSP and RPP shape functions whose

corresponding particles are the same point] If one local par-

ticle corresponds to an RSP shape function as well as an

RPP shape function, then this point has two different global

particle numbers.

Let = R S denotes the set of indices of theglobally numbered particles, where R is the index set for

the global RPP shape functions and S is the index set forthe global RSP shape functions. Then we have the following

theorem.

Theorem 5.1 1. I,I R, are the reproducing poly-nomial particle shape functions of order at least2 for

all (x,y) that are not in the patches containingsingularities.

2. For I R, I has a compact support and satisfythe Kronecker delta property at all boundary particles

as well as the inside particles, except for the singular

particles that correspond to the RSP shape functions.

3. If all RSP shape functions (of three different types)constructed in Sect. 4 are used, thevector space spanned

by I,I S, include the singular functions r0.5+lsin(0.5+l) , r0.5+l cos(0.5+l) , l =0, 1; r0.5, r1.5, r0.5sin(1.5) , r0.5 sin(1.5), multiplied by the convolution

PU function, which is 1 around the crack tip.

4. If the window functionw Cl (R2), I Cl () forall I ,except at the singularity points.

Proof (1) If(x,y) is an interior point of Qk1 that is the -

distance away from Qk1 , then the convolution PU function

k1becomes 1 at (x,y). Hence, I,I

satisfy the

reproducing polynomial (as well as singular) shape function

property at(x,y).

On the other hand, if(x,y)is inside of the 2-band along

Qk1 (hence 0 < k1

(x,y) < 1), then this point is also

inside the 2-band along Qk2 , where Qk2 is a patch adja-

cent to the patch Qk1 . Thus, if dist((x,y), Qk1 ) anddist((x,y), Qk2 ),thenI

x1I y

2I I(x,y)=

i j4

x1i j y

2i j i j (x,y) k1 (x,y)

+ kl4

x1kl y

2kl

kl (x,y) k2 (x,y)

= x1y2 k1 (x,y) + k2 (x,y)= x1y2 .

(2) Suppose

Tk1 (i1 , j1 )=Tk2 (i2 , j12)=Tk3 (i3 , j3 )=(xI,yI),

theniljl T1kl (xI,yI)=iljl (il , jl )=1, for l=1, 2, 3.Hence,

1 3


16/23

150 Comput Mech (2007) 41:135157

I(xI,yI)=

3l=1

iljl T1kl kl

(xI,yI)

=

3l=1

kl

(xI,yI)=1.

(3) and (4) are obvious.

6 Interpolation error associated with RSP shape

functions

For brevity, we consider the case when u(x,y) contains only

the type I singularity stated in Sect. 4.1.

The interpolation ofu(x,y) associated with the combined

RSP shape functions and the RPP shape functions is defined

by

Iu=I

u(xI,yI)I(x,y)

=NQk=1

st

u(kqst)[kst(x,y)]

+

st

u(s qst)[s st(x,y)]

=NQk=1

st

u(kqst)(st T1k ) k

+

st

u(s qst)(st T1 ) s

,

where

s

qst= T(s

s ,

s

t), = 1/2, stand for the particles(diamonds in Fig. 5) corresponding to the RSP shape func-tions. The rectangular patch Qs = [0.5, 0.5] [0, 0.5]contains a singularity in Fig.5.

X

Y

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

bc

d a e

f

ghij

k

l

Decomposition of the support of Convolution PU functionon which it is a polynomial

BC

D A

Q

QQQ

Q 1

234

5

m

no

p

Fig. 5 Qs= [0.5, 0.5][, 0.5] is an enlarged rectangle suchthatQs=Q s . Then the intersection of [the support of s= Qsw ]and is theouter dash-dot rectangle (eg ji ). Therectangle[supp s ] is dividedinto six rectangles on which s is a polynomial.Thediamonds indicates the particles generated by the singular patch

mapping T (w)=w 2, =1/2

A local interpolation ofu on the patch Q l is defined by

Il u=

st

u(Tl (s , t))(st T1l (x,y)).

Lemma 6.1 Suppose xp Ll=1[supp l]. If the local inter-polation error of a function u(x,y) at the point xpis denoted

by

(Il u u)(xp),then the global interpolation error of u at xp is the sum of

the local errors multiplied by the convolution PU functions

as follows:

Ll=1

l (xp) (Il u u)(xp).

Proof SinceL

l=1 l (xp)=1,we have

Ll=1

l (xp) (Il u u)(xp)

=L

l=1 l (xp) Il u(xp)

Ll=1

l (xp)

u(xp)

=

Ll=1

l(xp) Il u(xp)

u(xp),

which is the global interpolation error at the point xp .

6.1 Interpolation error in energy norm

Theformulas for computation of interpolation error in energy

norm can be found in [22]. We briefly describe parts of the

formulas for our numerical example. The squared L2()-

norm of the interpolation error is computed by the following

mater patch approach.

Iu u20=N pi=1

Qi

st

u(i qst)[st T1i ] i

+ ki st u(kqst)

[ st

T

1k ]

ku

2

=N pi=1

Q

st

u(i qst)[st] i Ti

+

ki

st

u(kqst)[i j T1k Ti ] k Ti

u Ti2

|J(Ti )|.

1 3


17/23

Comput Mech (2007) 41:135157 151

Here i= {k: [supp k] Q i= }(the index set ofthe neighboring patches around Qi ). Letx:= ( x, y )T,and Ji j denotes the (i, j )- component of the inverse of theJacobian ofTi . Then, the squaredH1-semi norm of the inter-

polation error is defined by

x(Iu u)20=

x(Iu u)

2

+

y(Iu u)

2dx dy,

where each term of this integral can be computed on the

reference patch, for example

x(Iu u)

2dx dy

=N pi=1

Qi

x

st

u(i qst)[st T1i ] i

+ x

ki

st

u(kqst)[st T1k ] k

u

x

2

=N pi=1

Q

st

u(i qst)[(J11,J12) st] ( i Ti )

+st x

i Ti+

ki

st

u(kqst)

x

st T1k

Ti k Ti+ [st T1k Ti ]

x

k Ti

ux

Ti2

|J(Ti )|.

Moreover, for effective evaluation of these integrals, we

observe the following:

1. x

[stT1k ]Tiand y [stT1k ]Tican be computedby the following chain rules:

x[st T1k ] Ti= x[st T1k ] (Tk T1k ) Ti=

J(Tk)1 st

(T1k Ti )

= [J(Tk)1 (T1k Ti )][st (T1k Ti )]. (45)

2. An explicit form of the inverse function T1k is notavailable in general. Thus, (T1k Ti ) is evaluated byNewtons method, that yield the desired inverse coor-

dinates in two or three iterations because Tk is bilinear

mapping. For k= s, the inverse of the singular patchmappingTs is available in an explicit form in Sect. 4.

6.2 Numerical example

We explain the procedures of constructing RSP shape func-

tions as well as RPP shape functions in conjunction with

Fig. 5. We assume that our test problem contains a jump

boundary data singularity at(0, 0)(see, Fig.5).

(1) Mappings for patch-wise non-uniformly spaced par-

ticles.

T: Qs [0, 1/

2] [0, 1/2] (conformalmapping:T (w)=w 2, =1/2)T

k:

[1, 1

][1, 1

] Q

k, k

=1, . . . , 5 (bilinear

mapping),

where Q 1, . . . ,Q5 are rectangles in Fig.5.

(2) With=0.1, the convolution partition of unity shapefunctions are constructed by k= w Qk, k=1, , 5,where Q kare rectangles whose vertices areas follows:

Qs :(0.5, ),(0.5, 0.5),(0.5, 0.5),(0.5, )Q1 : (0.5, ),(1 + , ),(1 + , 0.5),(0.5, 0.5),Q2: (0.5, 0.5), (1+, 0.5), (1+, 1+), (0.5, 1+), Q5 : (1 , ),(0.5, ),(0.5, 0.5),(1 , 0.5).Let us note that the enlarged quadrangle Qk contains

Qk so that the convolution PU shape function k

becomes 1 alongQ k. We also note thatQk=Qk fork=1, . . . , 5 ands .

In this section, the interpolation errors on the patch Qkare estimated in the following two norms: the L 2-norm and

the H1-semi norm, respectively, defined by

error|QkH0=

Qk

(Iu u)2d x d y

1/2

,

|error|Qk|H1=

Qk

(Iu u)2d x d y

1/2

.

In the computational perspectives, we observe the follo-

wings:

1. For the particle shape functions corresponding to the

particles in the patch Q s (Fig.5), we use the conformal

1 3


18/23

152 Comput Mech (2007) 41:135157

mappingT, =1/2. Thus, the components Ji j of theinverse matrix and the determinant|J(Ts )|are those in(19)and (20).

2. If xp T[Qs] [ \Q s],thenT1 (xp)is an outsidepoint ofQs , and hence E xti j (T1 (xp)) is very large,whenever dist(T1 (xp),Qs ) is large and the polyno-mial degree of

i j is high.

Thus, it is recommended to choose Qs so that it can

be as close to T (Qs )as possible, especially when thereference RPP shape functions of high order (>4) are

employed.

3. At the particles in the outside of Qs , the RSP shape

functions I,I , do not satisfy the Kroneckerdelta property. In particular, K(T(1/

2, 1/

2))=

0, for all particle shape functions Kthat correspond to

the mapped particles in Q s (diamonds inQ s ) because

T(1/

2, 1/

2) is not in the support of K (the

-neighborhood of Qs ). However, that does not mean

the amplitude of the global RSP shape function

J cor-responding to the particle T(1/2, 1/2) is zero inthe interpolation approximations as well as the finite

element approximations.

We are about to show the effectiveness of the RSP shape

functions in dealing with problems containingthe jump boun-

dary data singularity.

Let us consider the benchmark problem, known as the

Motz Problem ([16], p. 335):

u=0 in (shown in Fig.5),

u=

500 along the vertical linex=10 along the negativex-axis,

u

n=0 along the non negativex-axis.

Thenu has a jump boundary data at(0, 0). It is known that

around the singularity point(0, 0), the solution of the Motz

problem is dominated by

l=0

blrl+1/2 cos(l+ 1/2) ,

where the coefficients bl are quickly decreasing to zero as

l .

Example 1 Let us compute the interpolation error of a sin-

gular function with jump-boundary data singularity

u(r, )=0.5r1/2 cos( /2) + 0.4r1/2 sin( /2) + 0.2r3/2 cos(3 /2) + 0.1r3/2 sin(3 /2), (46)

associated with the RSP shape functions corresponding to

the particles shown as the 25 diamonds in Fig. 5,and the

RPP shape functions corresponding to the 81-particles in the

patchesQ 1, . . . ,Q5.

Without loss of generality, we will show the interpola-

tion error only on the patch Qs= [0.5, 0.5] [0, 0.5],that contains the singularity. For this purpose, we decompose

[supp s ] Qs , into the following sixrectangles (see, Fig. 5):

Qs0=rectangle(abcd)Qs1=rectangle(a Amb)Qs2=rectangle(bm Bn )Qs3=rectangle(cbno)Qs4=rectangle(pcoC)Qs5=rectangle(Ddc p)

Let us note that, on each of these rectangles, the convolution

PU function s is a polynomial. In particular, s is1onQ s0.

The RSP columns of Table2are computed as follows:

1. By Corollary 4.3 and Table1, we have

1i,j5

u(T (si,

sj ))(i j (T

1 (x,y) u(x,y)=0,

for all(x,y)Qs . (47)

Moreover, k(x,y)=0,k=1, . . . , 5, for all(x,y)

Qs0.Thus, we have(Iu

u)

|Qs0

=0.(see, the first

row of RSP columns)2. From Lemma 6.1 and Eq. (47), we have the followings:

(a) (Iu u)|Qs1=

1i,j5

[u(T (si, sj ))(i j (T1 (x,y))u(x,y)] s

+

1i,j9[u(T1(i , j ))(i j (T11 (x,y)) u(x,y)] 1

=

1i,j9[u(T1(i , j ))(i j (T11 (x,y)) u(x,y)] 1

(b) (Iu u)|Qs2=

1i,j5

[u(T(si , sj ))(i j (T1 (x,y)) u(x,y)] s

+3

k=1

1i,j9u(Tk(i , j ))(i j (T

1k (x,y))u(x,y)

k

=3

k=1

1i,j9u(Tk(i , j ))(i j (T

1k

(x,y))u(x,y) k.

1 3


19/23

Comput Mech (2007) 41:135157 153

Table 2 Interpolation error inL 2-norm and H1-semi norm with respect to RPP shape functions as well as the RSP shape functions

Jump boundary data singularity

Norm Error inL 2-norm Error in H1-semi norm

Basis RPP RSP RPP RSP

(Iu

u)on Q s0 5.18E

3 1.52E

16 2.36E

1 5.86E

16

(Iu u)on Q s1 7.89E3 1.91E9 3.18E1 8.04E8(Iu u)on Q s2 5.60E5 2.70E7 3.38E3 9.52E6(Iu u)on Q s3 4.96E5 8.43E8 2.92E3 3.80E6(Iu u)on Q s4 5.62E5 2.26E7 3.39E3 8.01E6(Iu u)on Q s5 7.73E3 2.08E9 3.14E1 9.01E8(Iu u)on Q s 1.22E2 3.62E7 3.20E0 1.30E5We use 25 RSP shape functions for RSP columns, whereas 81 RPP shape functions (polynomial reproducing order=8) for RPP columns

Fig. 6 Upper leftThe graph of

the exactu(x)containing thejump data singularity on the

singular zone Q s .Upper right

The graph of the interpolation

function ofu associated with the

RSP shape functions on Q s .

Two figures agrees up five digits

or more at almost all mesh

points for graphs and hence two

are numerically exactly the

same.CenterThe graph of the

interpolation ofu associated

with the RPP shape functions on

Qs , that shows a big difference

from the graph of the exact

function

X

-0.2

0

0.2

Y

0

0.2

0.4

0.6

Z

0

0.2

0.4

Y

X

Z

Jump Boundary Data Singularity

X

-0.2

0

0.2 Y0

0.2

0.4

0.6

Z

0

0.2

0.4

Y

Z

X

Interpolation of

jump boundary data singularity

by RSP shape ft

X0 Y

0

0.5

Z

0

0.5

Y

X

Z

Interpolation of Jump Data Singularity

associated with RPP shape functions

We use 25 RSP shape functions onQsand81 RPP shape

functions (polynomial reproducing order = 8) on each

Qk, k= 1, . . . , 5 for the numerical results on RSPcolumns. On the other hand, for the interpolation errors

on the RPP columns we used the 81 RPP shape func-

tions on all six patches. If the 25 RPP shape functions

were used for all six patches, the accuracy would be

dropped by one half.

3. OnQ sk, k1, the interpolation ofuis mixed with res-pect to RSP shape functions and the RPP shape function

of order 8. The complete polynomials of degree 8 that

are generated by RPP shape functions are not able to

1 3


20/23

154 Comput Mech (2007) 41:135157

Fig. 7 The Interpolation

functions ofu associated with

the capped RSP shape functions

Leftand with the capped RPP

shape functionsright,

respectively

X

-0.4-0.2

00.2

0.4

Y

0

0.5

Z

0

0.2

0.4

Y

ZZ

X

X

-0.4-0.2

00.2

0.4

Y0

0.5

Z

0

Y

X

Interpolation of Jump Data Singularity

associated with RSP shape ft

multiplied by Flat-Top convolution PU

Interpolation of Data Singularity

associated with RPP shape ft

multiplied by flat-Top convolution PU

exactly interpolate the singular functions. That is why

the interpolation errors on Qsk, k= 1, . . . , 5 are notzero in Table2.

In Figs.6and7,we show that

1. In Fig. 6, the exact u(r, ) with jump boundary data

singularity agrees with the interpolation ofu associated

with RSP shape function on Q s0.Thus, two figures are

the same. However, the graph of the interpolation ofu

associated with RPP shape functions (lower center graph

of Fig.6) is quite different from the graph of the exact

u(upper left figure).

2. In Fig. 7, the interpolations ofu (r, ) associated with

capped RSP shape function (that is, multiplied by the

convolution PU function) is compared with the inter-polations ofu (r, )associated with capped RPP shape

functions over the singular zone[supp s ]. From thisfigure, one can see that approximability of RSP shape

functions is far better than that of RPP shape functions

in dealing with singularities.

Example 2 Let us consider the following function that

contains the cracksingularity: w(r, )=w1(r, )+w2(r, )+w3(r, ) ,where

w1(r, )=0.5r1/2 cos( /2) + 0.4r1/2 sin( /2)

+0.2r3/2 cos(3 /2) + 0.1r3/2 sin(3 /2)

w2(r, )=r0.5 + r1.5 + r0.5 cos(1.5 )

w3(r, )=r0.5 + r1.5 + r0.5 sin(1.5 )From Corollaries 4.3, 4.4, 4.5, we know that (Iwj wj )|Qs ,j=1, 2, 3 should be similar to the errors in Table2. Howe-ver, Lemma 6.1 does not hold for (Iww)|Qs (the interpola-tion error associated with the combined RSP shape functions

of three different types) because the RSP shape functions

related to type I singularity do not have the Kronecker delta

property with respect to the particle associated with type II

singularity, and so on.

However, as stated in Remark 4.3, the finite element

approximation with respect to combined RSP shape func-

tions should be as good as the interpolation approximation

of Table2for Example1.

7 Concluding remarks

The singularities of the crack on homogeneous materials are

the monotone singularity of type:x ,x (logx)l .Ontheother

hand, the singularities of the cracks in the composite mate-

rials are the oscillating singularity of type: x cos logx,

where 0 < < 1 is called an intensity of singularity and a

small real number is called the oscillating factor.

In this paper, we constructed three different types of RSP

shape functions that can reproduce singular functions captu-

ring the almost all monotone singularities.

More work should be done for constructions of particle

shape functions that can handle the oscillating singularities,

arising at the cracks of the composite materials.

The proof of error estimate for an interpolation associated

with RPP shape function was given in [22]. An error estimate

of an interpolation associated with the RSP shape functions

will be reported in a forthcoming paper.

There are three different types of three-dimensional

domain singularities: vertex singularity, edge singularity and

vertex-edge combined singularity. We expect that it is not

very difficult to extend the two-dimensional construction

of reproducing singularity particle shape functions to the

three-dimensional cases. However, it may not be easy to

extend the convolution partition of unity shape functions to

the three-dimensional cases. These three dimensional issues

will be discussed in the forthcoming paper.

1 3


21/23

Comput Mech (2007) 41:135157 155

A Diagram of intersecting polygons and their vertices in

the convolution PU functions

B C1 particle shape functions of reproducing order 4

whose supports are subsets of[0, n]

The basic C1-particle shape function with support[3, 3]and reproducing property of order 4 for uniformly distributed

particles is uniquely determined as follows [21]:

([3,3];1;4)(x)=

1120

x(x+ 2)(x+ 3)2(x+ 7) x [3, 2] 1

24(x+ 1)(x+ 2)(x3 + 6x2 3x 24) x [2, 1]

112

(x+ 1)(x4 + 2x3 15x2 12x+ 12) x [1, 0] 1

12(x 1)(x4 2x3 15x2 + 12x+ 12) x [0, 1]

124

(x 2)(x 1)(x3 6x2 3x+ 24) x [1, 2] 1

120(x 7)(x 3)2(x 2)x x [2, 3]

As mentioned above, since (x0)|[0,3], (x1)|[0,3], (x2)|[0,3],(x3)|[0,3], (x4)|[0,3], and (x5)|[0,3] are polynomials, thatpartof these piecewise polynomial particle shape functions can

be globally extended. Note thatn should be greater than or

equal to 12.

1 3


22/23

156 Comput Mech (2007) 41:135157

1. The extended particle shape function corresponding to the particlex0=0 is the following:

(x0)(x)=

15400

(x 5)(x 4)(x 3)2 (x 2)(x 1)(8x+ 15) x [0, 3] (, 0]0 x(, 3]


(x1)(x)=

11080

(x 5)(x 4)(x 3) (x2)x(8x217x36) x [0, 3] (, 0]([3,3];1;4)(x 1) x [3, 4]0 x(, 4]


(x2)(x)

=

1540

(x5)(x4)(x3) (x1)x(8x225x27) x [0, 3] (, 0](

[3,3

];1

;4)(x

2) x

[3, 5

]0 x(, 5]


(x3)(x)=

1540

(x 5)(x 4)(x 2) (x1)x(8x2 33x18) x [0, 3] (, 0]([3,3];1;4)(x 3) x [3, 6]0 x (, 6]


(x4)(x)=

11080 (x 5)(x 3)(x 2) (x1)x(8x241x9) x [0, 3] (, 0]([3,3];1;4)(x 4) x [3, 7]0 x (, 7]


(x5)(x)=

15400

(x 4)(x 3) (x2)(x1)x2(8x49) x [0, 3] (, 0]([3,3];1;4)(x 5) x [3, 8]0 x(, 8]

7. For particlexj (j=6, 7, . . . , n 6)

(xj )(x)=([3,3];1;4)(x j )

8. For particlexj (j=n 5, n 4, n 3, n 2, n 1, n)

(xj )(x)=(xnj )((x n))

1 3


23/23

Comput Mech (2007) 41:135157 157

References

1. Atluri S, Shen S (2002) The meshless method. Tech Science Press

2. Babuska I, Banerjee U, Osborn JE (2003) Survey of meshless

and generalized finite element methods: a unified appraoch. Acta

Numer, pp. 1-125. Cambridge Press, Cambridge

3. Babuska I, Banerjee U, Osborn JE (2004) On the approximabi-

lity and the selection of particle shape functions. Numer math

96:6016404. Babuka I, Oh H-S (1990) The p-version of the finite element

method for domains with corners and for infinite domains. Numer

Methods PDEs 6:371392

5. Babuka I, Rosenzzweig MR (1972) A finie element scheme for

domains with corners. Numer Math 20:371392

6. Duarte CA, Oden JT (1996) Anhpadaptive method using clouds.

Comput Methods Appl Mech Eng 139:237262

7. Han W, Meng X (2001) Error alnalysis of reproducing kernel par-

ticle method. Comput Methods Appl Mech Eng 190:61576181

8. Han W, Meng X (2002) On a Meshfree method for singular pro-

blems. CMES(Tech Science Press) 3:6576

9. KimH, LeeSJ,Oh H-S(2003) Numerical methodsanderroranaly-

sis forone diemensionalellipticproblems containing singularities.

Numer Methods PDEs 19:399420

10. Li S, Liu WK (2004) Meshfree particle methods. Springer,Heidelberg

11. Li S, Lu H, Han W, Liu WK, Simkins DC Jr (2004) Reprodu-

cing kernel element method: part II. Globally conformingIm /Cn

hierarchies. Comput Methods Appl Mech Eng 193:953987

12. Liu WK, Han W, Lu H, Li S, Cao J (2004) Reproducing kernel

element method: part I. Theoretical formulation. Comput Methods

Appl Mech Eng 193:933951

13. Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle

methods. Int J Numer Methods Fluids 20:10811106

14. Liu WK, Liu S, Jun S, Li Adee J, Belytschko T (1995) Reprodu-

cing kernel particle methods for structural dynamics. Int J Numer

Methods Eng 38:16551679

15. Liu WK, Li S, Belytschko T (1997) Moving least square reprodu-

cing kernel method. Part I: Methodology and convergence. Com-

put Methods Appl Mech Eng 143:422453

16. Lucas TR, Oh H-S (1993) The method of auxiliary mapping for

the finite element solutions of elliptic problems containing singu-

larities. J Comput Phys 108:327342

17. MelenkJM, Babuka I (1996) Thepartition of unityfinite element

method: theory and application.Comput Methods Appl MechEng

139:239314

18. Oh H-S, Babuka I (1995) The method of auxiliary mapping for

the finite element solutions of plane elasticityproblems containing

singularities. J Comput Phys 121:193-212

19. Oh H-S, Kim H, Lee S-J (2001) The numerical methods for oscil-

lating singularities in elliptic boundary value problems. J Comput

Phys 170:742763

20. Oh H-S, Kim JG (2006) The piecewise polynomial partition of

unity shape functions for the generalized finite element methods

(revision process)

21. Oh H-S, KimJG, Jeong JW (2005) TheClosed Form Reproducing

polynomial particleshapefunctions formeshfree particlemethods.

Comput Methods Appl Mech Eng (to appear)

22. Oh H-S, Kim JG, Jeong JW (2006) The smooth piecewise

polynomial particle shape functions corresponding to patch-wise

non-uniformly spaced particles for meshfree particles methods.

Comput Mech (to appear)

23. Parton VZ, Perli PI (1984) Mathematical methods of the theroy of

elasticity, I. MIR Publisherd, Moscow

24. Stroubolis T, Copps K, Babuska I (2001) Generalized finite ele-

ment method. Comput Methods Appl Mech Eng 190:40814193

25. Stroubolis T, Zhang L, Babuska I (2003) Generalized finite ele-

ment method using mesh-based handbooks: application to pro-

blems in domains with many voids. Comput Methods Appl Mech

Eng 192:31093161

Documents

2 Singularity particle shape functions 2007.pdf