NURBS-based parametric mesh-free methods

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Comput. Methods Appl. Mech. Engrg. 197 (2008) 1541–1567

NURBS-based parametric mesh-free methods

Amit Shaw, D. Roy *

Structures Lab, Department of Civil Engineering, Indian Institute of Science, Bangalore 560012, India

Received 7 December 2006; received in revised form 28 November 2007; accepted 29 November 2007Available online 8 December 2007

Abstract

A mesh-free error reproducing kernel method (ERKM) has recently been proposed by [A. Shaw, D. Roy, A NURBS-based errorreproducing kernel method with applications in solid mechanics, Comput. Mech. 40(1) (2007) 127–148]. The ERKM is based on an ini-tial approximation of the target function by non-uniform-rational-B-splines (NURBS) followed by reproduction of the error via a familyof non-NURBS basis functions. However, specifications of the window supports of non-NURBS basis functions remain a tricky issue.Moreover, NURBS in higher dimensions (>1) is generally defined over rectangular (cuboidal in 3D) grid structures and thus, in manyproblems of practical interest, the geometric complexity of the domain would prevent making use of NURBS in the ERKM. Presently,we develop a parametric reformulation of the ERKM purely using NURBS. This reformulation allows the method to be applicable tonon-rectangular (non-cuboidal in 3D) physical domains in two or still higher dimensions without a need to explicitly specify the supportsize of the window. A key feature of this development is a geometric map that provides a local bijection between the physical domain andrectangular (cuboidal) parametric domain. The shape functions and their derivatives are then constructed over the parametric domain sothat polynomial reproduction and interpolation properties are satisfied over the physical domain with the geometric map being simul-taneously preserved. A couple of new schemes are also proposed to empower the shape functions with the interpolation property that inturn enables a precise imposition of Dirichlet boundary conditions. We illustrate the parametric mesh-free method in the context ofstrong/weak solutions of a few linear and non-linear boundary and initial value problems of engineering interest.� 2007 Elsevier B.V. All rights reserved.

Keywords: Parametric ERKM; NURBS; Geometric map; Error reproducing kernels; Mesh-free methods; Interpolations

1. Introduction

Difficulties in the solutions of available mathematicalmodels of many physical as well as engineering systemsarise not only owing to intricate forms of non-linearity,but also because of complicated domain boundaries as wellas involved boundary conditions. Given the scarcity ofanalytical solutions in most cases of practical interest andthe versatility of computational procedures in handlingsuch problems, research efforts continue to focus onimproving such numerical schemes. The finite elementmethod (FEM), which continues to remain the most popu-lar, constitutes an element-wise application of the varia-

0045-7825/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2007.11.024

* Corresponding author. Tel.: +91 8022933129; fax: +91 8023600404.E-mail address: [email protected] (D. Roy).

tional or Galerkin method, where the interpolating basisand test functions are restricted to elements whose unionin turn obtains the domain of interest. Discretization ofthe domain into elements is a key feature of the FEM. Ele-ment-wise approximations of field variables not only pro-vide a relief from the search of globally admissiblefunctions, it also introduces versatility in approximatingcomplex geometries with the accuracy of approximationgenerally increasing with decreasing element sizes. Eventhen, thanks to the use of elements in the FEM, there aresome numerical as well as computational aspects thatdemand scrutiny.

1. Two basic steps in any engineering analysis are: (a)geometry modeling with generation of mesh throughCAD and (b) computation of the response. The

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1542 A. Shaw, D. Roy / Comput. Methods Appl. Mech. Engrg. 197 (2008) 1541–1567

discretized geometry used by the numerical solver is gen-erally a crude approximation to the CAD geometry.Domain approximation is thus an important source oferror in the FEM. Moreover, during adaptive meshrefinements, interactions between the CAD and numer-ical solver make the FE mesh generation more costly.

2. When the highest order derivative in the PDE is morethan 2 (as in beams, plates and shells), higher order con-tinuity is generally required at the inter-element bound-ary. However the enforcement of even C1 continuityacross inter-element boundaries is not a very trivial task.

3. In the FEM, the domain (often a closed subset of thecontinuum) is assumed to be connected. Therefore it isdifficult to model fracture of the material body into anumber of pieces. In the same vein, discontinuity (asin a crack propagation problem), and non-differentiabil-ity of the response are difficult to treat.

Such limitations of the FEM may be ascribed to the dis-cretization of the domain into elements. Alternatively themesh-free methods, unlike classical forms of the FEM,do not require a mesh generation for discretizing compli-cated geometries and have evoked considerable interestamong researchers. However, whilst dispensing with thediscretization through elements, mesh-free shape functionsare of more complex nature than polynomial-based shapefunctions used with the FEM. Amongst the availablemesh-free methods, mention may be made of the diffuseelement method (DEM) [29], the element free Galerkinmethod (EFG) [3,26], the partition of unity method(PUM) [2,28], the h-p Clouds [12], the reproducing kernelparticle methods (RKPM) [22,23], the moving least-squarereproducing kernel method (MLSRK) [24,7,8], the mesh-less local boundary integral equation method (LBIE) [40],the smooth particle hydrodynamics method [13,38], themeshless local Petrov–Galerkin method (MLPG) [1], thepoint interpolation method [20], the boundary point inter-polation methods [14], and the reproducing kernel elementmethod [19,21,25,37].

Among the methods noted above, the moving least-square (MLS) approximation and the RKPM appear tobe the most popular. They are essentially based on discreteand approximated forms of the integral kernel representa-tion of a function with the kernel (window) function, /a(x)being finite-valued and compactly supported. /a(x), essen-tially a functional representation of the Dirac delta mea-sure, is such that

1. /a(x) is compactly supported,2.R

X /aðx� yÞdy ¼ 1,3. /a(x) monotonically decreases away from its maximum,4. /aðxÞ !

a!0dðxÞ.

Although the MLS/RKPM have been advanced andsuccessfully applied to many problems involving solids,structures, acoustics, fluids, etc. its implementationinvolves some numerical difficulties as enumerated below.

1. Choosing the support size (a) of the window function isa tricky issue. In principle, a should not be so small as toresult in a singular moment matrix and should not be solarge as to include a large number of points within itssupport, thereby leading to an exaggerated smoothnessin the approximation. Generally the user needs to opti-mize the size of the window function based on somenumerical experiments. A mesh-free approximation withautomatically and uniquely obtainable support size (fora given nodal distribution) has so far been elusive.

2. Approximations of non-differentiable functions posechallenges.

In an effort to address some of these difficulties, [34] haverecently proposed a NURBS (non-uniform-rational-B-spline)-based error reproducing kernel method (ERKM)wherein the target function is first approximated via NURBSbasis functions. Then the error (remainder) function, result-ing from the NURBS approximation of the target function,is reproduced (within a finite-dimensional polynomial space)through a non-NURBS family of basis functions and addedto the NURBS approximation of the target function. This isunlike the regular RKPM [22,23] that uses a (multiplicative)correction term to the kernel function and obtains the samethrough polynomial reproduction. The use of NURBS toconstruct one of the pair of families of basis functions helpsbring in the local support, variation diminishing and convexhull properties. While [34] have demonstrated that theERKM is significantly less sensitive to window sizes unlikethe MLS or the RKPM, the non-NURBS error reproducingpart of the shape functions still needs a window size to besupplied by the user. However it would be more desirableto entirely bypass the issue of user-supplied support sizes.Unfortunately, since NURBS in higher dimensions (>1) isconstructed through tensor products of one-dimensionalfunctions, it is generally defined over rectangular (cuboidalin 3D) grid structures. Given that many problems of practi-cal interest have geometrically complex domain boundaries;procedures to extend the ERKM to such cases must beexplored.

The present work is largely motivated by the concept ofcoupling geometry and approximations via NURBS, whichhas been introduced by Hughes and his co-workers via theso-called ‘‘isogeometric” analysis [16]. In the last work, theemphasis has been on modeling of the ‘‘exact” CAD geom-etry through the non-uniform-rational-B-splines (NURBS)basis functions followed by the discretization of the modelusing ‘‘NURBS elements”. Finally the isoparametric con-cept is invoked to define the field variables, i.e. the fieldvariables are approximated via the same NURBS basisfunctions. Modeling and mesh generation processes in theFEM generally consume the major portion of the totalcomputational time. Moreover, in most engineering analy-ses, mesh is generated from the CAD data and hence a con-tinuous interaction with the CAD is needed during themesh refinement (h-adaptive) process. However in ‘‘isogeo-metric” analysis, once the ‘‘exact” CAD geometry is

A. Shaw, D. Roy / Comput. Methods Appl. Mech. Engrg. 197 (2008) 1541–1567 1543

obtained and the ‘‘geometric data” (basis functions, con-trol points, etc.) is stored, any subsequent refinement maybe done in the solver itself with the known ‘‘geometricdata” without actually having any interaction with theCAD system. Finally the isoparametric concept is invokedto define and approximate the field variables via the sameNURBS basis functions. A mathematical analysis of the‘‘isogeometric” approach is given in [4]. The ‘‘isogeomet-ric” analysis approach is applied to many problems involv-ing structural vibration [10], incompressible flows [5], fluidstructure interaction [6,39], etc.

Following the concept of ‘‘isogeometric” analysis, aparametric mesh-free method is proposed in the presentstudy. The first step in the parametric formulation is todefine a parametric space �X ¼ ½0; 1�n (where n denotes thephysical dimension of the domain X) such that there is alocal bijection F : �X! X. The transformation F is con-structed through NURBS. Once the geometric map F isavailable, any subsequent refinement is done using F. Thenthe shape function and its derivatives are so constructedover the parametric space �X that polynomial reproductionand interpolation properties get satisfied over X, which isgenerally non-rectangular (non-cuboidal in 3D), and thegeometric map F is simultaneously preserved. Given thatNURBS are constructed only over the parametric space�X, the issue of domain complexity is effectively tackled.Moreover, since the parametric mesh-free method usesonly NURBS in the derivation of shape functions, the issueof choosing a support size is no longer relevant. We alsopropose a couple of additional modifications, viz. a pointinverse scheme and another via Kriging, to seamlesslybring in the interpolation property of the parametric shapefunctions. The proposed parametric formulation is thennumerically illustrated for strong and weak solutions of afew boundary and initial value problems with quite arbi-trary domain geometries. Some of these results are alsocompared with those available in the literature. These illus-trations help bring into focus the relative numerical advan-tages of the new method.

All bold faced letters in this paper indicate vectors or ten-sor quantities. While a � b denotes the standard tensor prod-uct of two vectors, a � b denotes the dot product and a � b thecross product of a and b. The rest of the paper is organized asfollows. In the first part of Section 2, construction of thegeometric map via NURBS is reviewed. The parametricmesh-free method is then developed and interpolation issuesdiscussed in the second part of Section 2. Several issuesrelated to the numerical implementation of the parametricmesh-free method are considered in Section 3. A few testcases are provided in Section 4 to explore the efficacy ofthe proposed formulation in dealing with arbitrary domaingeometries. Conclusions are drawn in Section 5.

2. Parametric mesh-free method

As already noted, the motivation behind developing theparametric mesh-free method is that tensor products of

NURBS (or B-splines) generally require an n-dimensionalhypercube. In order to realize the objective of dealing witha domain X with irregular geometry, first a parametricspace �X ¼ ½0; 1�n is defined such that a local bijectionF : �X! X is possible. Then the parametric space �X is dis-cretized and projected onto X via F. Finally shape func-tions are constructed over �X such that the reproduction(and interpolation) within any polynomial space Pp = Pp

(X) is achieved and the geometric map F : �X! X is simul-taneously preserved. Needless to say, the geometric trans-formation F plays a very crucial role in the parametricformulation. The construction of the geometric map is con-sidered in the next subsection.

2.1. Construction of the geometric map via NURBS

In any numerical technique, especially the FEM, amajor portion of the analysis time is consumed in modelingand discretization of the domain geometry. Geometriccomplexity makes the discretization process even morecostly. Therefore, it is always desirable to have a less costlyrepresentation of a complicated domain with the addedrequirement that, whenever required, the original geometrycan be reproduced almost exactly. NURBS is a mathemat-ical tool [31] that provides a parametric representation ofboth analytic and free-form surfaces. Moreover, it is usedfor design and data exchange of geometric informationprocessed by the computer. In this section we will brieflyreview the parametric representation of geometry and theirdiscretization via NURBS and this parametric representa-tion will be exploited in Section 2.2 for the constructionof parametric mesh-free shape functions.

NURBS is generally defined by its order, control points,weights associated with control points and knot vectors[31]. For example, in one-dimension, a NURBS curve isdefined by

CðnÞ ¼XNCP

i¼1

Rgi;pðnÞPi; n 2 �X ¼ ½a; b� � R: ð1Þ

Rgi;pðnÞ denotes the NURBS basis function and is defined as

Rgi;pðnÞ ¼

N gi;pðnÞwiPNCP

k¼1 N gk;pðnÞwk

: ð2Þ

fPigNCP

i¼1 are the control points, fwigNCP

i¼1 are the correspond-ing weights, and N g

i;pðnÞ are the B-spline basis functionswith degree pdefined over a knot vector (non-decreasing,uniform or non-uniform) N ¼ fn1 ¼ a; n2; . . . ; nNCPþp;nNCPþpþ1 ¼ bg. In Eqs. (1) and (2) the superscript ‘g’ impliesthat the basis functions are used for geometric mapping.

The objective is now to achieve a NURBS-based param-eterization of the original geometry (by control points,weights and knot vectors) such that, for any subsequentdiscretization (required for approximating the field vari-able), it suffices to work with the parametric domain andnot with the complicated geometry of the physical domain.Once discretization is accomplished over the parametric


space �X, it may be projected onto the physical spaceX ¼ fCðnÞ : n 2 �Xg via Eq. (1). For further elucidation,consider the NURBS approximation of a unit circle. B-spline basis functions (Fig. 2) are constructed over the knotvector N = {0,0,0,.25,.25,.5,.5,.75,.75,1.,1.,1} with p = 2and number of control points NCP = 9. Control pointsand corresponding weights are shown in Fig. 1.

NURBS approximation (as projected on the physicalspace X) of any point �x 2 �X ¼ ½0; 1� is given by

x ¼ ðx; yÞ ¼X9

i¼1

Rgi;pð�xÞPi: ð3Þ

Let the parametric space �X be discretized by NP pointsff�xigNP

i¼1g � �X. Then the corresponding discretization ofthe physical space ffxigNP

i¼1g � X may be obtained via Eq.(3) as xi ¼ ðxi; yiÞ ¼

P9i¼1Rg

i;pð�xiÞP i. Fig. 3 shows the discret-ization of the parametric space and its projection onto thephysical space.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 1. Quadratic B-spline basis functions defined over the knot vectorN = {0,0,0,.25,.25,.5,.5,.75,.75,1.,1.,1}.

3P4P

5P

6P7P

Fig. 2. Control points and associ

By way of an example in the higher dimension, considerthe NURBS approximation of a torus. Control points andthe corresponding weights are given in [16]. B-spline basisfunctions are constructed over the knot span N� } 2�X ¼ ½0; 1�2 with p = 2 and NCP = 9 � 9 = 81. Projectionof any point �x 2 �X onto the physical space X � R3 maybe written as

x ¼ ðx; y; zÞ ¼X9

j¼1

X9

i¼1

Rgij;pð�xÞPij; ð4Þ

where

Rgij;pð�xÞ ¼ Rg

ij;pð�x; �yÞ ¼N g

i;pð�xÞN gj;pð�yÞwijP9

i¼1

P9j¼1N g

i;pð�xÞN g

j;pð�yÞwij

: ð5Þ

Now the parametric space �X is first discretized by a set ofgrid points ff�xigNP

i¼1g � �X (Fig. 4) and the correspondingdiscretization ffxigNP

i¼1g � X of the physical space (Fig. 5)is obtained via Eq. (5).

2.1.1. Uniformity index

Normally we choose a uniform particle distribution inthe parametric domain. However, once it gets mapped tothe physical domain through F, the resulting discretizationin X will generally be non-uniform (see Fig. 6). Accordinglywe define an index, referred to as the ‘‘uniformity index”

(UI), as

UI ¼ hmin

hmax

;

where hmin and hmax are respectively the minimum andmaximum step size (distance between two adjacent nodes)in the physical domain. It is obvious that the UI = 1 repre-sents the ideal situation (in that the geometric map does notintroduce a possible ill-conditioning of the discretized sys-tem matrices). Sometimes the geometric map may lead to anodal distribution with very low uniformity index. This

i Pi wi

1 (1,0) 1

2 (1,1) 1 2

3 (0,1) 1

4 (-1,1) 1 2

5 (-1,0) 1

6 (-1,-1) 1 2

7 (0,-1) 1

8 (1,-1) 1 2

9 (1,0) 1

1 9,P P

2P

8P

ated weights for a unit circle.

0 -4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4

ix

ix

Fig. 3. (a) Discretization of the parametric space �X ¼ ½0; 1� for a unit circle and (b) projection onto the physical space X � R2 via Eq. (3).

a b

Fig. 4. (a) Parametric space for the toroidal surface and (b) control net the for toroidal surface.

Fig. 5. (a) Discretization of the parametric space �X ¼ ½0; 1�2 for the toroidal surface and (b) projection onto the physical space X � R3 via Eq. (4).


could, in turn, lead to numerical corruption and affect thesolution of the governing PDE-s. However numericalexperiments, reported later in Section 4.1, reveal that theparametric mesh-free method works very well for a reason-ably large range of the uniformity index.

Having discussed the use of the geometric map for aparametric discretization of the domain, the stage is nowset to couple this geometric information with polynomialreproduction and interpolation properties while construct-ing mesh-free shape functions. This is considered in thenext subsection.

2.2. Construction of parametric shape functions

Let the physical domain X � Rn be defined via thediscretization:

X ¼ x : x ¼XNCP

k¼1

Rgk;qð�xÞP k

( ); ð6Þ

where Rgk;qð�xÞ is the NURBS basis function of degree q de-

fined over a parametric space �X ¼ ½0; 1�n and Pk the corre-sponding control points. Image a point �x 2 �X in X may be

Fig. 6. (a) Uniform discretization of the parametric domain and (b) corresponding discretization (dense near P3 and coarse near P1) of the physicaldomain. Control points are shown by blue dots; UI = 5.3 � 10�2.


obtained through the local bijection F : �X! X. Now wediscretize the parametric domain �X by grid pointsff�xigNP

i¼1g � �X and denote ffxigNPi¼1g to be the corresponding

image in X (obtained by F). We need to construct the para-metric shape function over �X so that any functionu(x) 2 C(X) can be approximated as

uaðxÞ ¼XNP

i¼1

WiðxÞui; ð7Þ

where ui , u(xi) Here Wi(x) is the parametric shape func-tion. Its derivation must be consistent with the reproducing(and interpolating) requirements over X while simulta-neously maintaining the transformation F : �X! X. Thereproducing condition may be written as

XNP

i¼1

WiðxÞHðx� xiÞ ¼ Hð0Þ; ð8Þ

where H(x) = {xa}jaj6p is the set of monomial basis func-tions and a is the multi-index notation. From Eq. (6), wemay write

x� xi ¼XNCP

k¼1

Rgk;qð�xÞP k �

XNCP

k¼1

Rgk;qð�xiÞP k; ð9Þ

) Hðx� xiÞ ¼ HXNCP

k¼1

½Rgk;qð�xÞ � Rg

k;qð�xiÞ�P k

!

¼ Hð�xi; �xÞ: ð10Þ

Now Wi(x) may be written as

WiðxÞ ¼ HTð�xi; �xÞbðxÞRi;pð�xÞ; ð11Þ

where Ri;pð�xÞ is the NURBS basis functions (with constantweights) defined over �X with discretized grid pointsff�xigNP

i¼1g � �X and b(x) is the vector of unknown coeffi-cients. Using Eqs. (8) and (11), we may write

)XNP

i¼1

Hð�xi; �xÞbðxÞRi;pð�xÞHð�xi; �xÞ ¼ Hð0Þ; ð12Þ

)MðxÞbðxÞ ¼ Hð0Þ; ð13Þ) bðxÞ ¼M�1ðxÞHð0Þ: ð14Þ

Here MðxÞ is the parametric moment matrix given by

MðxÞ ¼XNP

i¼1

Hð�xi; �xÞHTð�xi; �xÞRi;pð�xÞ: ð15Þ

Using Eqs. (11), (12) and (15), the parametric mesh-freeshape function Wi(x) may be expressed as

WiðxÞ ¼ HTð0ÞM�1ðxÞHð�xi; �xÞRi;pð�xÞ: ð16Þ

Note that NURBS in Eq. (16) are constructed only overthe parametric space �X. Therefore the issue of domaincomplexity is effectively bypassed. Moreover no separatewindow function with user-specifiable support size is nowrequired for the construction of improved parametricmesh-free shape function (via Eq. 16). This is quite unlikemany other mesh-free approximation schemes includingthe MLS and the RKPM.

2.3. Construction of derivatives of parametric shape

functions

A stable and numerically accurate scheme for derivativecalculation is proposed in [34]. It is based on the premisethat bth derivatives of mesh-free basis functions exactlyreproduce bth derivatives of an arbitrary element of thespace Pp of polynomials of degree p P jbj. Using the sameprinciple, consistency relation for the derivatives may bewritten as

XNP

i¼1

WðbÞi ð�xÞHðx� xiÞ ¼ ð�1ÞjbjH ðbÞð0Þ; 8jbj 6 p; ð17Þ

Node Knot knot outside the parametric space

Fig. 7. Selection of knot vector over the parametric space.


where WðbÞi ð�xÞ , DbWið�xÞ be another family of mesh-freebasis functions, which exactly reproduce bth derivativesof elements in the space Pp for p P jbj. Now WðbÞi ð�xÞ maybe written as

WðbÞi ðxÞ ¼ H Tð�xi; �xÞbbðxÞRi;pð�xÞ; ð18Þ

where bbðxÞ ¼ fbbaðxÞgjaj6p is the vector of unknown coeffi-

cients for derivative reproduction. Substituting Eq. (18)into Eq. (17), one may readily obtain the final form ofWðbÞi ðxÞ as

WðbÞi ð�xÞ ¼ ð�1ÞjbjHTðbÞð0ÞM�1ðxÞHð�xi; �xÞRi;pð�xÞ: ð19Þ

The parametric shape functions obtained from Eq. (16) arenot interpolating. A special treatment is therefore requiredto impose the essential (in particular Dirichlet) boundaryconditions precisely. Towards realizing this objective, acouple of extended forms of the parametric formulationwith nodal interpolation property are discussed in the nextsection. However, before we start outlining these exten-sions, an important issue related to the regularity of theparticle distribution must be addressed.

2.4. Regularity of particle distribution

2.4.1. Non-singularity of moment matrix

Following Proposition 3.5 in [15], the necessary condi-tion for the moment matrix MðxÞ to be invertible at anypoint x 2 X is that x must at least be covered bydim(Pp) = (p + n)!/p!n! basis functions. A higher dimen-sional NURBS basis function is constructed by takingthe tensor product of NURBS basis functions in one-dimension. Thanks to its local support property, any�x 2 �X n Nn is exactly covered by p + 1 and its n-dimen-sional counterpart is exactly covered by (p + 1)n NURBSbasis function. Here Nn � Rn is the knot space in n-dimen-sion. Now, using the relations

ðp þ nÞ!p!n!

¼Yn

j¼1

ðp þ jÞj

< ðp þ 1Þn; 8n > 1 ð20aÞ

and

ðp þ 1Þ!p!

¼ ðp þ 1Þ; n ¼ 1: ð20bÞ

Therefore, the moment matrix MðxÞ, obtained via Eq. (15),is always invertible for all �x 2 �X n Nn. However this is notthe case when �x 2 Nn. Knots are usually covered by pn basisfunctions (assuming there is no multiplicity of knots) andthus at the knot positions one may not always haveðpþnÞ!

p!n!6 pn. This may lead to a singular moment matrix at

any knot position. This difficulty may, in principle, be re-moved by taking X and Nn such that X \ Nn = /. Howeverthis may not work always. A case in point arises during thefunctional evaluation at a Gauss point that lies in Nn – ascenario that is encountered when a weak form of themesh-free method is being implemented. As an alternative,we suggest the usage of NURBS with degree p + 1 for con-

structing the parametric shape functions of pth order con-sistency. Accordingly Eqs. (16) and (19) may be recast as

WiðxÞ ¼ HTð0ÞM�1ðxÞHð�xi; �xÞRi;pþ1ð�xÞ; ð21ÞWðbÞi ðxÞ ¼ ð�1ÞjbjHTðbÞð0ÞM�1ðxÞHð�xi; �xÞRi;pþ1ð�xÞ: ð22Þ

2.4.2. Selection of knots

The appropriate selection of knot vectors (Nn) remainsan issue in the construction of the shape functions. Uni-form knot distribution over the parametric space appearsto be a natural choice. However, this may not always workif the particle distribution does not follow a uniform gridstructure. Our numerical experiments, only a few of whichare reported in Section 4, suggest that the selection of coin-cident nodes and knot points works as well or better thanmost other choices (Fig. 7).

2.5. Interpolation issues

The parametric mesh-free shape functions, derived inSection 2.2, are not interpolating. One may obtain theinterpolating property (at a given node) by using multiple(p times) knots at that point. However, use of multipleknots reduces the continuity of the basis functions and thusemploying this strategy in the approximation of smoothfunctions entails a loss of accuracy. In other words,repeated knots should be sparingly used only to capturediscontinuities and sudden jumps in the approximation(see Section 4.3). The objective of this section is to outlinea couple of interpolating versions of the parametric formu-lation and numerically explore their relative advantages.

2.5.1. The point inverse methodLet the domain of interest X � Rn be discretized by a

set of grid points X ¼ ffxigNPi¼1g � X. The function

u(x) 2 C(X) may then be approximated as

uaðxÞ ¼XNP

i¼1

WiðxÞui ¼ fWiðxÞgNPi¼1 �U ; ð23Þ

where U ¼ fuigNPi¼1 is the set of the discretized function val-

ues. Since the parametric shape function does not recoverthe nodal values (i.e. the associated basis functions, whileforming a partition of unity, do not satisfy the Kroneckerdelta property), one generally has ua(xk) 6¼ u(xk) for almostall xk 2 X. Satisfying Eq. (23) at xk 2 X "k, we get

Ua ¼ WU ; ð24Þ) U ¼ UUa; ð25Þ


where Ua ¼ fuaðxiÞgNPi¼1. W is a positive definite matrix gi-

ven by Wij = Wi(xj) and U = W�1. Substituting Eq. (25) intoEq. (23), we have

uaðxÞ ¼ fWiðxÞgNPi¼1 �UUa; ð26Þ

) uaðxÞ ¼XNP

i¼1

WiðxÞXNP

j¼1

Uijuaj ; ð27Þ

) uaðxÞ ¼XNP

j¼1

XNP

i¼1

UjiWiðxÞ" #

uaj ; ð28Þ

) uaðxÞ ¼XNP

j¼1

Winj ðxÞua

j : ð29Þ

In the way, we derive the modified basis as

Wini ðxÞ ¼

XNP

j¼1

UijWjðxÞ: ð30Þ

As shown below, the modified basis Wini ðxÞ is endowed with

both reproduction and nodal interpolation properties.

Reproduction property:

The parametric mesh-free basis function Wi(x) satisfiesthe reproduction condition and thus we may write

xa ¼XNP

i¼1

WiðxÞxai ; jaj 6 p: ð31Þ

Satisfying Eq. (31) xk 2 X "k, we have

Xa ¼ WXa ) X a ¼ W�1Xa ) Xa ¼ UXa; ð32Þwhere Xa ¼ ffxa

i gNPi¼1g. Now from Eq. (30), we may write

XNP

i¼1

Wini ðxÞxa

i ¼XNP

i¼1

XNP

j¼1

UijWjðxÞxai

¼XNP

j¼1

WjðxÞXNP

i¼1

Uijxai ¼

XNP

j¼1

WjðxÞxaj

¼ xa ½from Eqs: ð30Þ; ð31Þ and ð31Þ�

)XNP

j¼1

Winj ðxÞxa

j ¼ xa: ð33Þ

Interpolation property:

Following Eq. (30) Wini ðxkÞ may be written as

Wini ðxkÞ ¼

XNP

j¼1

UijWjðxkÞXNP

j¼1

W�1ij WjðxkÞ ¼ dik: ð34Þ

The interpolation scheme proposed in this section isstraightforward. However due to the use of the inversetransformation U in Eq. (30), bandwidth of a discretizedsystem via the interpolating shape function is larger thanthat obtainable through the basis without the nodal inter-polation property. This increases the size of the influencedomain and thus may even affect the accuracy of the solu-tion, especially when effects of non-linearity are localized.

With this in mind, a localized Kriging-based interpolationscheme is discussed in the next subsection.

2.5.2. Interpolation via KrigingAn effective way of introducing the interpolation prop-

erty within a given set of basis functions via Kriging [17]is given in [11,18,36]. In the Kriging-based interpolationscheme, the basis functions are obtained by satisfying twoconditions: (i) unbiasedness and (ii) minimum variance,which lead to the following set of n + dim(Pp) linear alge-braic equations [36]:

Xn

j¼1

E½uðxiÞuðxjÞ�Wjðx0Þ þXjaj6p

laH a

¼ E½uðx0ÞuðxiÞ�; i 2 ½1; n�; ð35aÞ

Xn

j¼1

Wjðx0ÞH a ¼ djaj;0; jaj 6 p; ð35bÞ

where Wj(x0) is the Kriging-based interpolating shape func-tion, n ¼ cardfxi � X :j x0 � xi j6 ax0

g, ax0is the dilation

parameter at x0, l:¼{la}a6p is the vector of Lagrange mul-tipliers and Ha is the ath component of H(x � xi). In Eq.(35), covariance E[u(xi)u(xj)] is generally replaced by asemi-variogram c (xi,xj):¼c(h) (with a slight abuse of nota-tions), which is defined as cðhÞ ¼ 1

2E½fðuðxiÞ � uðxjÞg2� with

h = jxi � xjj being the separation distance between thepoints. There are several possible choices for the semi-vari-ogram [30].

In the numerical implementation of Eq. (35), the majordifficulty is an appropriate selection of the dilation param-eter ax0

(similar to the support size of the window functionin many mesh-free methods). In order to render this selec-tion irrelevant in the parametric mesh-free formulation, theNURBS basis function is once more used to construct thesemi-variogram model. Thus replacing c(h) by the NURBSbasis, we may write

cðxi; xjÞ ¼ Ri;pð�xiÞ � Rj;pð�xiÞ; ð36aÞcðx0; xiÞ ¼ Ri;pð�x0Þ � Ri;pð�xiÞ: ð36bÞ

Substituting Eq. (36) into (35), the latter may be written inthe matrix form as

R P

PT 0

" #W

l

( )¼

c

Hð0Þ

( ); ð37Þ

where W ¼ fWiðx0Þgni¼1 and R, P and c are given by

R ¼

Ri;pð�x1Þ � Ri;pð�x1Þ � � � Ri;pð�x1Þ � Ri;pð�xnÞ

� � � � � � � � �

Ri;pð�xnÞ � Ri;pð�x1Þ � � � Ri;pð�xnÞ � Ri;pð�xnÞ

2664

3775;ð38aÞ


P ¼1 � � � Hp

� � � � � � � � �1 � � � Hp

264

375; ð38bÞ

c ¼

Ri;pð�x0Þ � Ri;pð�x1Þ

..

.

Ri;pð�x0Þ � Ri;pð�xnÞ

8>><>>:

9>>=>>;: ð38cÞ

Following a similar approach, bth derivative of the para-metric mesh-free shape function may be obtained via thefollowing set of equations.

R P

PT 0

" #WðbÞ

lðbÞ

( )¼

cðbÞ

H ðbÞð0Þ

( )ð39Þ

with WðbÞ ¼ fWðbÞi ðx0Þgni¼1 and c(b) given by

cðbÞ ¼

RðbÞi ð�x0Þ � RðbÞi ð�x1Þ

..

.

RðbÞi ð�x0Þ � RðbÞi ð�xnÞ

8>>><>>>:

9>>>=>>>;: ð40Þ

Before we go in for the numerical illustrations, a fewimportant properties of parametric shape functions arenoted below.

1. If any one of the interpolation schemes proposed aboveare employed, Wi(x) would have the Kronecker deltaproperty, i.e. Wi(xj) 2 dij.

2. In the parametric formulation, the basis functions are soconstructed over a parametric space ð½0; 1�n � RnÞ thatthey not only possess the reproduction and interpolationproperties over the problem domain but also serve as ageometric map between the parametric space and theproblem domain. In the process, any Lipschitz andsmooth boundary gets mapped (point wise) to that cor-responding to the parametric space with the same accu-racy with which the field variables are approximated.

B2

B1

H

px

Fig. 8. Plane strain problem – g

3. There is no need to explicitly specify the support size ofthe window function. Due to the local support propertyof NURBS, the parametric basis function adjusts thesupport size automatically depending on the order ofconsistency.

4. Any desired order of continuity (including C0) is achiev-able by appropriately repeating the knots at a selectedpoint over the knot span and hence the parametricmesh-free method can even well approximate functionswith discontinuous derivatives. Moreover, thanks tothe variation diminishing property of NURBS, it hasadvantages in representing sharp layers without muchof numerical instability. The property is further demon-strated in Section 4.1.

3. A step-by-step implementation

Towards a detailed numerical implementation of theproposed method, consider a plane strain problem asshown in Fig. 8. Since the physical domain X is non-rectan-gular (non-cuboidal), the tensor product NURBS basisfunctions over X cannot be constructed. However it is pos-sible to map X to a parametric space �X and construct mesh-free shape functions over �X so as to approximate any func-tion over X. As noted in Section 2, the construction ofparametric basis functions involves three main steps:

� definition of the geometric map� discretization of the parametric space and transforma-

tion of discretization over the physical domain� construction of basis functions.

3.1. Definition of the geometric map

Consider a parametric space �X ¼ ½0; 1�2. Rgi;1ð�xÞ, i 2 [1,4],

denotes the ith NURBS basis function with fwi ¼ 1g4i¼1

defined over the knot span N � � with N = {0,0,1,1}and � = {0,0,1,1}. (Pxi,Pyi) is the coordinate of the ithcontrol point (see Fig. 9). Now the image (in X) of anypoint ð�x; �yÞ 2 �X may be written as

Parameter Value

1B 1 m

2B 4 m

H 4 m

E 25 GPa

ν 0.3

xp 10 kN/m

eometry and material data.

1 (0,0)P = 2 (4,0)P =

4 (1, 4)P =3 (0, 4)P =

Ω

3 4 1ξ ξ= =

Ω

1 2 0ξ ξ= =1 2 0η η= =

1 2 1η η= =

ba

Fig. 9. (a) Parametric space �X ¼ ð0; 1Þ2 for the plane strain problem in Fig. 8 and (b) corresponding control points.


x ¼X4

i¼1

Rgi;1ð�xÞP xi y ¼

X4

i¼1

Rgi;1ð�xÞP yi ð41Þ

with

Rg1;1ð�xÞ ¼ ð1� nÞð1� gÞ; ð42Þ

Rg2;1ð�xÞ ¼ nð1� gÞ; ð43Þ

Rg3;1ð�xÞ ¼ ð1� nÞg; ð44Þ

Rg4;1ð�xÞ ¼ ng: ð45Þ

Eqs. (41)–(45) provide the required geometric transforma-tion F.

3.2. Domain discretization and construction of shape

functions

Once the geometric map is defined, the physical domainX may be discretized via two main steps; (a) discretizationof the parametric space �X and (b) transformation of thediscretized parametric space via Eq. (51). As an illustra-tion, let the parametric space be discretized into two ele-ments as shown in Fig. 10. Let x1 and x2 be theprojections of �x1 and �x2, respectively, on X. FollowingEq. (41), x1 and x2 may be obtained as

1 2 0η η= =

1 2 1η η= =

3 4 1ξ ξ= =1 2 0ξ ξ= =

Ω

1 (0.5,0)=x

2 (0.5,1)=xa b

Fig. 10. (a) Discretization of the parametric space by 6 nodes (two elements) an

x1 ¼X4

i¼1

Rgi;1ð�x1ÞP i ¼ :5ð0; 0Þ þ :5ð4; 0Þ þ 0ð0; 4Þ þ 0ð1; 4Þ

¼ ð2; 0Þ;

x2 ¼X4

i¼1

Rgi;1ð�x1ÞP i ¼ 0ð0; 0Þ þ 0ð4; 0Þ þ 0:5ð0; 4Þ þ 0:5ð1; 4Þ

¼ ð0:5; 4Þ:

Similarly any discretization of the parametric space may betransformed over the physical domain using the geometricmap given in Eq. (41). Fig. 11 shows the discretization ofthe domain with NP = 15 � 15 points.

Once the parametric domain �X is discretized by gridpoints ff�xigNP

i¼1g and the corresponding discretizationffxigNP

i¼1g of physical domain X is obtained, we may con-struct the parametric shape function and its derivativesvia Eqs. (21) and (22).

3.3. Numerical integration

In a mesh-free method, the physical domain is first dis-cretized by a set of points (or particles) followed by theconstruction of shape functions about these particles. If

Ω

2 (0.5, 4)=x

1 (2,0)=x1P

3P

2P

4P

d (b) projection of the discretized parametric space onto the physical space.

ba

dc

Fig. 11. (a) Discretization of the parametric space by 15 � 15 nodes and (b) corresponding discretization of the physical space and (c) integration cells inparametric space and (d) integration cells in physical space.


the governing differential equations need to be solved in theweak sense, then a background mesh must be used for per-forming numerical integrations that naturally arise duringthe weak formulation. Integration is generally done overeach background cell (element) by a quadrature technique(Gauss quadrature, for instance). However, choosing thebackground mesh is not a very trivial task. Moreover foran arbitrary geometry, the need for a background meshbrings back the difficulty of mesh generation as in theFEM. In contrast, no separate background mesh over thephysical domain is required in the present parametric for-mulation. Integration is done in each cell over the paramet-ric space. This may lead to a larger number of backgroundcells than is actually used in other mesh-free formulations(like the element-free Galerkin (EFG) method). In theEFG and other mesh-free methods as applied to obtainweak solutions, we need to use a higher order quadraturerule in each cell. However in the present parametric formu-lation, it is observed through numerical experiments (someof which are reported here) that the second order Gaussquadrature rule is adequate for a large class of problems.Therefore, the number of floating point operationsrequired in the parametric mesh-free method and any othermethod are comparable. The stiffness matrix and nodalload vector may be computed as

K2NP�2NP ¼XN cell

m¼1

ZXm

BT2NP�2D2�2B2�2NP dX; ð46Þ

f2NP�1 ¼XN cell

m¼1

ZoXm

NT2NP�2p2�2 dðoXÞ; ð47Þ

where K is the stiffness matrix, f the nodal load vector, Dthe constitutive matrix, B the strain displacement matrix,N the shape function matrix and p the externally appliedtraction. Since shape functions and their derivatives are di-rectly obtained based on the polynomial reproduction overthe physical domain X, computing Jocobians is not re-quired (unlike the isoparametric formulation in the FEM)for transformation of derivatives from the parametricspace �X to the problem domain X.

3.4. Essential boundary conditions

Presently the interpolation property is achieved at theboundary points via the Kriging-based interpolationscheme. Distribution of rx is shown in Fig. 12.

The numerical implementation of the parametric formu-lation has been systematically demonstrated in this sectionin order to facilitate a better understanding. For further

Fig. 12. Distribution of horizontal stress rx.


numerical exploration, the proposed method is applied tosome linear and non-linear boundary and initial valueproblems with arbitrary domain geometry in Section 4.However, towards the completeness of the discussion, somelimitations and scope for future developments of thepresent parametric formulation are addressed in thefollowing.

a

a b

b

Fig. 13. A non-simply connected domain.

3.5. Limitations and scope for future developments

The key step in the parametric formulation (Section 2.2)is the coupling of geometric map F : �X! X with the repro-duction (and interpolation) property. The construction ofthe geometric map (as illustrated in Section 2.1) is essen-tially based on a parametric representation (by knot vec-tors, control points and associated weights) of theoriginal (physical) geometry via NURBS. It is assumedthat F is bijective (one-to-one and onto). However thereare many situations (especially for non-simply connecteddomains) where one may not have a bijective geometrictransformation and thus the inverse map F �1 : X! �Xmay not exist. For further illustration, consider a non-sim-ply connected domain shown in Fig. 13. Fig. 14 shows theparametric space and the corresponding NURBS model(control net in this case). fP ig10

i¼1 is the image set offP ig10

i¼1 obtained via the transformation F. We note that,even as P 1 (and P 5) and P 6 (and P 10) are two distinct pointsin the parametric space �X, their images P1 (and P5) and P6

(and P10) coincide in the physical space X. Therefore thegeometric map F : �X! X is not one-to-one (see Fig. 15).

One more example of a circular cylinder where the geomet-ric map is not one-to-one is shown in Fig. 16.

We have discussed above a few cases where the pro-posed parametric formulation may not work. However insuch cases we may decompose the domain into an FEM-like collection of simply connected sub-domains (elements)and still apply the parametric formulation over each sub-domain without affecting the continuity across the inter-domain boundary. The authors are in the process of devel-oping such a modification to deal with very complex andnon-simply connected domains. Details of this develop-ment will be addressed elsewhere.

{ }0,0,.3,0.5,.7,1,1Ξ =

1 5,P P

2P 3P

4P

6 10,P P

7P 8P

9P6P 7P 8P 9P 10P

1P 2P 3P 4P 5P

1A 2A 3A 4A1A

2A

3A

4A

{}

0,0,

1,1

ϒ=

ba

Fig. 14. (a) Parametric space and (b) corresponding control net.

Fig. 15. A non-simply connected domain where the geometric map is not one-to-one: (a) discretization of the parametric space and (b) projection of thediscretized parametric space on the physical space.

ba

Fig. 16. A circular cylinder where the geometric map is not one-to-one: (a) discretization of the parametric space and (b) projection of the discretizedparametric space on the physical space.


4. Numerical examples

The present section provides additional numericalexplorations of the proposed method for strong and weaksolutions of a few linear and non-linear boundary valueproblems (BVP-s) of interest in applied mechanics. In allthe examples involving weak formulations, second orderGauss quadrature is used for numerical integrations.

Details of the geometric data (e.g. knot vectors, controlpoints and the associated weights) for examples 4, 5 and6 are given in Appendix I.

4.1. Example 1: Two-dimensional Poisson’s equation

We recall from Section 2.1 the parametric discretization,even if uniform, is transformed via the geometric map to a

i Pi wi

1 (0,0) 1

2 (1 2,0) 1

3 (1,0) 1

4 (0,1 2) 1

5 (1 4,1 4) 1

6 (2 2,0)− (1 1 2) / 2+

7 (0,1) 1

8 (0,2 2)− (1 1 2) / 2+

9 ( 2 1, 2 1) 2− − 1P P2 P P

P

P

P

P

P

i Pi wi

1 (0, 1)− (1 1 2) / 2+2 ( 2 1, 1)− − 13 (1,1 2)− 14 (1,0) (1 1 2) / 2+5 (1 2, 1)− − 16 (0,1 2)− 17 ( 2 1,0)− 18 (1, 2 1)− 19 ( 1,1 2)− − 110 (1 2,0)− 111 (0, 2 1)− 112 ( 2 1,1)− 113 ( 1,0)− (1 1 2) / 2+14 ( 1, 2 1)− − 115 (1 2,1)− 116 (0,1) (1 1 2) / 2+

-1 -0.5 0 0.5 1

P16

P5 P2P1

P9

P10 P7

P3

P14

P6

P11P8

P13

P15 P12

P4

a

b

Fig. 17. (a) and (b) The two geometries used in example 1. Control points and weights are shown in the adjoining table.


non-uniform particle distribution in the physical domain.The purpose of this section is to demonstrate the perfor-mance of the parametric mesh-free method for particle

b

a

Fig. 18. Discretization of the domain shown in Fig. 17a: (a) discretizationcorresponding distribution of the physical space; UI = 2.9 � 10�3.

distributions with very low ‘‘uniformity index”. We con-sider Poisson’s equation over two different geometries(Fig. 17):

of the parametric space with 441 uniformly distributed nodes and (b)

b

a

Fig. 19. Discretization of the domain shown in Fig. 17b: (a) discretization of the parametric space with 441 uniformly distributed nodes and (b)corresponding distribution of the physical space; UI = 1.6 � 10�1.


o2u

ox2þ o

2uoy2¼ 4 in X ð48aÞ

with

u ¼ �u on oX: ð48bÞ

The exact solution to Eq. (48) is

uðx; yÞ ¼ x2 þ y2: ð49Þ

First, a uniform discretization of the parametric domainwith NP = 21 � 21 = 401 nodes is considered. Correspond-ing discretizations of the physical domain for both thecases are shown in Figs. 18 and 19. Shape functions withlinear consistency (p = 1) are constructed over X. For pur-poses of comparisons with the RKPM, a variable supportsize (depending on the nodal density) is used to constructthe RKPM shape functions. Relative L2 and H1 errornorms via the parametric mesh-free method and theRKPM are shown in Table 1.

In the above examples, L2 and H1 error norms via theparametric mesh-free method and the RKPM are compa-rable (see Table 1. In order to demonstrate the perfor-mance of the parametric mesh-free method for very low

Table 1aRelative L2 and H1 error norms for the domain shown in Fig. 17a(discretization shown in Fig. 18)

L2 norm H1 norm

RKPM 8.7 � 10�3 3.0 � 10�1

Parametric mesh-free method 6.0 � 10�3 7.3 � 10�2

Table 1bRelative L2 and H1 error norm for the domain shown in Fig. 17b(discretization shown in Fig. 19)

L2 norm H1 norm

RKPM 7.9 � 10�3 6.5 � 10�2

Parametric mesh-free method 6.3 � 10�3 5.3 � 10�2

values of the ‘‘uniformity index”, �X is next discretized intoNP = 401 ‘‘randomly distributed” nodes (generated by uni-formly distributed pseudo-random numbers in [0,1]). Thusthe parametric discretizations do not presently follow auniform grid structure (see Figs. 20 and 21). Relative L2

and H1 error norms through the parametric mesh-freemethod and the RKPM are shown in Table 2. It is clearthat the RKPM may suffer a very significant loss of accu-racy for such non-uniform particle distributions. The para-metric mesh-free method, on the other hand, continues toperform far more accurately.

4.2. Example 2: Kirchhoff plate under uniformly distributed

load

A simply supported equilateral triangular Kirchhoffplate under uniform load is now considered. This problemis used by [25,37] in order to demonstrate the performanceof the reproducing kernel element method (RKEM). Thegoverning equations may be written as

r4wðxÞ ¼ qD

x 2 X ð50aÞ

subject to boundary conditions:

wðxÞ ¼ 0 on oX; ð50bÞ

where $4w = wxxxx + 2wxxyy + w,yyyy and D ¼ Eh3

12ð1�m2Þ.In the above equations, l, E and h, respectively, denote

Poisson’s ratio, modulus of elasticity and thickness of theplate. Moreover, w(x) is the transverse displacement func-tion and q a uniformly distributed lateral load. The exactsolution of Eq. (50) is given by

wðx; yÞ ¼ p64aD

x3 � 3y2x� aðx2 þ y2Þ þ 4

27a3

� �

� 4

9a2 � x2 � y2

� �; ð51Þ

b

a

Fig. 20. Discretization of the domain shown in Fig. 17a: (a) discretization of the parametric space with 441 distributed nodes and (b) correspondingdistribution of the physical space; UI = 6.5 � 10�5.

b

a

Fig. 21. Discretization of the domain shown in Fig. 17b: (a) discretization of the parametric space with 441 distributed nodes and (b) correspondingdistribution of the physical space; UI = 7.3 � 10�3.

Table 2aRelative L2 and H1 error norms for the domain shown in Fig. 17a(discretization shown in Fig. 20)

L2 Norm H1 Norm

RKPM 1.9 � 10�1 6.78Parametric mesh-free method 8.2 � 10�2 9.8 � 10�2

Table 2bRelative L2 and H1 error norms for the domain shown in Fig. 17b(discretization shown in Fig. 21)

L2 Norm H1 Norm

RKPM 11.35 45.74Parametric mesh-free method 2.6 � 10�2 6.8 � 10�2


where a is the height of the triangle (see Fig. 22a). Thedomain X is discretized via NP = 5 � 5 = 25 and NP =9 � 10 = 90 nodes as shown in Fig. 22b and c respectively.The reason for choosing such nodal discretizations is tokeep the total degree of freedoms nearly the same as usedin the RKEM so as to facilitate a fair comparison. Shapefunctions with quadratic consistency (p = 2) are con-structed over X. The logarithms of L2 and H1 error norms

via the parametric mesh-free method and the RKEM fordifferent discretizations are shown in Table 3. It is of rele-vance to take note of the fact that RKEM-based numericalexperiments are reportedly performed with 64 (sometimes576) quadrature points per element in order to achieve‘‘good” numerical accuracy. Parametric mesh-free method,on the other hand, performs much better with just2 � 2 = 4 quadrature points per integration cell. Thismay be verified using Table 3.

0 L0

2h/3

h

3 4{0, 0, 0, 0, , , , , , }PN L L L Lξ ξ ξΞ =

℘=h 3

{-4,-3

,-2,1

,2,5

,6,7

}

,...

h/3

Fig. 23. A typical space–time block �X with NP = 51 and Nt = 4.

cba

Fig. 22. (a) Triangular plate and (b) discretization with NP = 5 � 5 = 25 nodes and (c) discretization with NP = 9 � 10 = 90 nodes.

Table 3Comparison of L2 and H1 error norms

No. ofnodes

log (L2

errornorm)

log (H1

errornorm)

RKEM (values are digitized fromFig. 9a in [25] and Fig. 17b in [37])

28 �5.5 �4.391 �6.3 �5.3

Parametric mesh-free method 25 �8.65 �7.2990 �9.36 �8.21


4.3. Example 3: Inviscid burgers equation

In the first example, a one-dimensional Burgers equationis considered. Here time evolution leads to a breakdown ofthe continuous solution and the appearance of a shock.The quasi-linear form of the equation may be written as

ouotþ u

ouox¼ 0: ð52Þ

A general form of the initial distribution is given by

uðx; 0Þ ¼ gðxÞ; �1 < x <1; t ¼ 0: ð53Þ

Eq. (52) is presently solved for an initially sinusoidal waveprofile given by

gðxÞ ¼uðx; 0Þ ¼ sin px

L ; 0 6 x 6 L;

0; x < 0 and x > L:

�ð54Þ

This example is used to demonstrate how the improvedNURBS-based mesh-free basis functions are useful inapproximating solutions of PDE-s that develop shocks orhave discontinuities. It is worthwhile to note that the pres-ent method works purely within a continuous frameworkand, unlike the extended finite element formulations [9],does not employ any enrichment based on step functionsto capture discontinuities.

First, a space–time approach (where time is also treatedas another independent dimension, just as the spatialdimensions, during the mesh-free discretization) for dis-cretizing Eq. (52) is considered. In order to avoid a largesystem matrix, Eq. (52) is solved incrementally over theith space–time block �Xi ¼ ½0; L� � ½ti; ti þ h�. The computed

solution over �Xi is used to obtain initial and boundary con-ditions for the solution over �Xiþ1. Now the parametricspace–time block �X ¼ ½0; L� � ½0; h� is discretized byNP(=51) � Nt(=4) (Fig. 23) nodes where NP and Nt

respectively denote the number of nodes in x and t direc-tions. Towards the construction of NURBS basis functions(with p = 3), two knot vectors N ¼ f0; 0; 0; 0; n5; n6; . . .nNP

; L; L; L; Lg and } ¼ t3f�4;�3;�2; 1; 2; 5; 6; 7g are

defined over �X. We emphasize that the knot vector N alongx-axis is open (first and last knots are repeated p + 1 times).The reason for this particular choice will be elaboratedupon shortly.

Fig. 24 shows the distribution of the wave profile inspace and time. We observe that, as the profile evolves intime, a shock appears at t = 0.5 s. It is readily verifiablefrom the figure that the improved NURBS-based mesh-freemethod is able to capture this shock quite accurately via apurely (locally) C0 approximation. Next, Eq. (52) is solvedvia a semi-discrete approach. First the domain is spatiallydiscretized by NP = 51 particles. One-dimensional basisfunctions are constructed with the knot vector N. Theresulting (semi-discrete) ODE-s are numerically integratedin time through an explicit Euler method. Snapshotsof the wave profiles at different time instants (t = 0.1 s,

Fig. 24. Temporal evolution of the wave profile of an initially sinusoidal distribution obtained via the new ERKM (defined over �X as shown in Fig. 7) withL = 1 and h = .05 s.


t = 0.3 s and t = 0.5 s) are plotted in Fig. 24. It is reported[9] that the semi-discrete approach behaves very poorly incapturing shocks/discontinuities and the resulting oscilla-tions in the vicinity of the shock/discontinuity. Curiously,as suggested by Fig. 25, a semi-discrete approach via theproposed method (without any discontinuous enrichment)appears to capture the shock reasonably accurately. How-ever, as compared with the space–time strategy, relativelyhigher order consistency (p � 1 P 3) over the spatialdomain is required in the semi-discrete approach in orderto obtain comparably accurate results.

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u(x)

t=0.1

t=0

Fig. 25. Snapshots of wave profiles of the initially sinusoidal distribution v

4.3.1. Effect of knot vectors

Recall the property of NURBS that any order of conti-nuity at a given location can be achieved simply by usingmultiple knots at that location. The ability of the improvedNURBS-based parametric mesh-free method to approxi-mate non-differentiable (or even discontinuous) functionsaccurately is a direct consequence of this multiplicity prop-erty of NURBS. Towards a numerical demonstration of therole of multiple knots, Eq. (52) is again considered. How-ever in this case, the new basis functions are constructedthrough a closed knot vector N ¼ f0; n2; n3; . . . nNPþp; Lg

0.5 0.6 0.7 0.8 0.9 1

x

t=0.3

t=0.5

ia the new ERKM-based semi-discrete approach, NP = 51, p � 1 = 3.


along x. Figs. 26 and 27, respectively, show the evolutionsof wave in time, as obtained via space–time and semi-dis-crete approaches. At the position of shock, the function isnot (apparently) differentiable. However, in the absence ofmultiple knots, the approximation is forced to be differen-

Fig. 26. Evolution of the wave profile of an initially sinusoidal distribution withh = .05 s.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

x

u(x)

t=0.4t=0.5

Fig. 27. Wave profile of an initially sinusoidal distribution at different time insemi-discrete approach with NP = 51, p � 1 = 3.

Γ1

Γ5Γ6

Γ7

Γ8

a

w

h

Fig. 28. Steady-state

tiable (undesirably smooth) even at or around the spatiallocation of the shock. In other words, accuracy of the sim-ulation is seriously compromised in the vicinity of theshock. This phenomenon is clearly reflected in Figs. 26and 27.

time, approximated via the new ERKM with closed knots with L = 1 and

0.6 0.7 0.8 0.90.7

0.75

0.8

0.85

0.9

0.95

1

1.05

x

u(x)

t=0.4

t=0.5

stant, approximated via the new ERKM-based (with closed knot vector)

Γ2

Γ3

Γ4

b

heat conduction.

Table 4Parameter values for example 4

Parameter Value

kx 1ky 1a 10b 2w 2h 0.5


4.4. Example 4: Steady-state heat conduction

In the second example, the steady heat conduction in atwo-dimensional domain X as shown in Fig. 28 is consid-ered. The governing equation in the strong form is

kxo2Tox2þ ky

o2Tox2¼ 0 in X: ð55Þ

{0,0,0Ξ =

5P1P

2P

6P

{}

0,0,

1,1

℘=

b

a

Fig. 29. (a) Parametric space for the heat conduction problem given in Fig

a

b

Fig. 30. (a) Discretization of the parametric space �X ¼ ½0; 1�2 for the heat condX � R2 (see Appendix I for details).

The following boundary conditions are presently adopted:T = 100 on C1 and T = 300 on C4, C5 and C6

Here k denotes the thermal conductivity of the materialand T the temperature. Parameter values used are given inTable 4.

Fig. 29 shows the parametric space and the correspond-ing control net. The parametric space is discretized byNP = 21 � 21 points as shown in Fig. 30a. Fig. 30b showsthe corresponding discretization of the physical domain.Parametric basis functions are constructed with p = 2.For brevity, we skip the details of the weak formulationand report the resulting temperature distribution in Fig. 31.

4.5. Example 5: Wrinkling of an isotropic circular membrane

As a third example, an annular membrane with an outerradius R = 300 mm and an inner radius r = 50 mm is con-

}.2,0.2,1,1

3P

4P7P

8P

. 10 and (b) corresponding control points (see Appendix I for details).

uction problem given in Fig. 28 and (b) Projection onto the physical space

Fig. 32. A circular membrane with an anti-clockwise rotation of 0.92 � atthe hub.

Fig. 31. Temperature distribution.


sidered. An anti-clockwise rotation of 0.92 � is applied asindicated in Fig. 32. The following properties are consid-ered in the example [27]: ED = 1022 kgf/cm, m =0.267(1 kgf = 9.81 N) where E, m and D are, respectively,the Young’s modulus, Poisson’s ratio and thickness ofthe membrane.

P1

P4

{}

0,0,

1,1

℘=

{ }0,0,0,1,1,1Ξ =

ba

Fig. 33. (a) Parametric space for the circular membrane and (b

The strong form of the equilibrium (Cauchy’s) equationsmay be written as

r � r ¼ 0; ð56Þ

where $� is the divergence operator and r is the symmetricCauchy stress tensor. Since the weak formulation has pres-ently to be carried out strictly within a tension field theory,a brief account of this formulation is presented in the fol-lowing. The weighted residual form of the equilibriumEq. (56) over the closed domain X may be written asZ

Xðr � rÞ � hdX ¼ 0: ð57Þ

Here h is a kinematically consistent (but otherwise arbi-trary) vector function. Integrating Eq. (57) by parts andapplying Gauss’ theorem, we arrive atZ

X½r : ðrhÞT�dX ¼

ZoX

nA � ½rh�dðoXÞ: ð58Þ

Transforming to the undeformed (initial) configuration,Eq. (58) may be recast in the final form as [32]:Z

X0

p : ðr0hÞT dV 0 ¼Z

oX00

f 0 � hdðoX0Þ; ð59Þ

where p = S�1(Jr) is the first Piola–Kirchhoff (non-sym-metric) stress tensor; S is the deformation gradient withoutconsidering variations in the membrane thickness, f0 is theforce on the deformed surface transformed to the initial

P2

P3P6

P5

) corresponding control points (see Appendix I for details).

b

a

Fig. 34. (a) Discretization of the parametric space �X ¼ ½0; 1�2 for the circular membrane and (b) projection on the physical space X � R2 (see Appendix Ifor details).

Fig. 35. Orientation of wrinkles (direction of maximum principal stressn1).

Fig. 36. Distribution of b1.


surface and $0 is the gradient operator with respect to theinitial configuration. To account for the wrinkled and slackzones, the wrinkling algorithm proposed by [33,35] hasbeen used. First the Cauchy stress tensor is obtained as

r ¼ ð1=JÞFHðEÞFT: ð60Þ

Fdenotes the deformation gradient tensor; J = det(F) theJacobian of the deformation gradient tensor; E =0.5(FTF � I) the Green–Lagrange strain tensor and H atensor function of E (describing the material constitutiverelation). Once the Cauchy stress tensor is obtained, amaterial point is checked for possible wrinkling/slackingbased on the principal stresses and principal strains[32]. Real stresses in wrinkled region are determined bymodifying the deformation tensor in the constitutive equa-tion as

Jr ¼ �FHð�EÞ�FT ð61aÞ

with

�F ¼ ½I þ b1n1 � n1�F and �E ¼ 0:5ð�FT �F � IÞ: ð62bÞ

Here b1 is a spatially continuous wrinkliness parameter (see[32,33,35]. Now b1 and the membrane thickness D aredetermined using the following coupled conditions:

(a) there is no stress along the wrinkling direction, i.e.:n1 � (Jr)n1 = 0 and

(b) the membrane is in a state of plane stress, i.e.:n3 � (Jr)n3 = 0.

Once the modified stress tensor is computed via Eq.(61a), one obtains the finally projected form of the responseequations as

KlkðUÞduk ¼ fl � �f : ð63Þ

Here Klk is the (l,k)th 3 � 3 sub-matrix of the tangent stiff-ness matrix, U = {uk} the discretized displacement vectorand fl � �f 2 R3 the residual nodal force on node l.

Eq. (63) has presently been solved by the parametricmesh-free method. Because of axial symmetry, only one

Fig. 37. Square footing under vertical load.

1P

2P

3P

7P

8P

4P

6P

9P 10P

11P12P

13P14P

15P16P

5P{}

0,0,

0.2,

0.5,

1,1

{}

0,0,1,1

℘=

{}

0, 0,1,1=

ba

Fig. 38. (a) Parametric space for the square footing and (b) corresponding control points (see Appendix I for details).

a

b

Fig. 39. (a) Discretization of the parametric space �X ¼ ½0; 1�3 for the square footing and (b) projection on the physical space X � R2 (see Appendix I fordetails).


Fig. 40. Distribution of ryy in MPa.

Parameter Value

R 25

r 1

E 25 GPa

ν 0.3

P 10 kN

P

R

A

A

rRSection A-A

Fig. 41. Cantilever ring subjected to vertical load.


quadrant of the plate is modeled. Fig. 33 shows the para-metric space and the corresponding control net. The basisfunctions are constructed with p = 3 and NP = 25 �15 points. Discretization of the parametric space and its

P1 – P16

P17 –P32

P33 –P48

P49 – P64

P128 – P144

Fig. 42. Position of control points fo

transformation on to the physical domain are shown inFig. 34.

The orientation of wrinkles and the distribution of b1

are shown in Figs. 35 and 36. Experimental results on the

P65 – P80

P81 – P96

P97 –P112

P113 –P128

r circular ring shown in Fig. 41.

Fig. 43. Contour plot for vertical displacements.

Fig. 44. Contour plot for rzz.


wrinkling of a stretched circular membrane under in-planetorsion are reported in [27]. Considering the same problem,the orientation of wrinkles, shown in Fig. 31, is observed tobe in very good agreement with the experimental results(see, e.g. Fig. 17 in [27]).

4.6. Example 6: A square footing under vertical load

As the next example, a square footing subjected to a ver-tical load (as shown in Fig. 37) is considered. Note that,unlike the previous cases, this is a spatially three-dimen-

sional problem. Fig. 38 shows the parametric space andcorresponding control points. The parametric space is dis-cretized by 11 � 11 � 11 points as shown in Fig. 39. Distri-bution of ryy is shown in Fig. 40.

4.7. Example 7: A cantilever ring subjected to vertical load

As the last example, a cantilever ring subjected to verti-cal concentrated load at the free end (Fig. 41) is considered.Control points are shown in Fig. 42. Distribution of verti-cal displacement and stresses are shown in Figs. 43 and 44,respectively.

Table I.1Control points and weights for example 1

i Pi wi

1 (0,2) 12 (0,0) 13 (10,0) 14 (10,2) 15 (4,2) 16 (4,1.5) 17 (6,1.5) 18 (6,2) 1


i Pi wi

1 (0,50) 12 ð50=

ffiffiffi2p

; 50=ffiffiffi2pÞ 1=

ffiffiffi2p

3 (50,0) 14 (0,300) 15 ð300=

ffiffiffi2p

; 300=ffiffiffi2pÞ 1=

ffiffiffi2p

6 (300,0) 1


i Pi wi

1 (0,0,0) 12 (4,0,0) 13 (0,4,0) 14 (4,4,0) 15 (0,0,0.3) 16 (4,0,0.3) 17 (0,4,0.3) 1


5. Concluding remarks

A NURBS-based parametric mesh-free method isproposed. Employing NURBS to construct the shapefunctions potentially enables a superior numerical perfor-mance (thanks to the variation diminishing property ofNURBS) and also addresses the problematic issue of anon-unique support size of the window function. Giventhat NURBS basis functions over higher dimensions aregenerally constructed through tensor products of vectorsof one-dimensional basis functions, the proposed para-metric method resolves the problem of how to constructsuch shape functions in non-rectangular (non-cuboidal)domains over two or still higher dimensions. As in theparametric formulation of the FEM, the key to the pres-ent approach is a geometric map that provides a localbijection between the physical and parametric domains,the latter being always rectangular (cuboidal). Once thegeometric map is available, any subsequent refinement isdone using this map. The shape function and its deriva-tives are constructed over the parametric domain suchthat reproduction and interpolation properties aresatisfied over the physical domain while simultaneouslypreserving the geometric map. A couple of novel interpo-lation schemes, viz. a point inverse scheme and a Kriging-based scheme, are also proposed to empower the shapefunctions with the interpolation property without jeopar-dizing the targeted smoothness in the approximation.However, given the higher computational overhead withthe point inverse algorithm, the Kriging-based interpola-tion logic appears to be a better choice enforcing Dirichletboundary conditions. The numerical implementation andperformance of the method are illustrated with a few lin-ear and non-linear boundary and initial value problems.

While the NURBS-based parametric mesh-free methodis applicable to a reasonably large class of problems withcomplicated geometry and those with highly non-uniformnodal distributions, the method still lacks the universalitythat the FEM enjoys. Indeed for many non-simply con-nected domains, a unique inverse of the geometric mapmay not exist and the parametric mesh-free method failsto be applicable to such problems. A way out is to decom-pose the domain into a collection of simply connected sub-domains and then apply the parametric formulation overeach sub-domain while making sure that the continuityacross the inter-domain boundaries remains unaffected.Such a non-trivial modification to deal with more complexand non-simply connected domains is being reportedseparately.

8 (4,4,0.3) 19 (1.75,1.75,0.5) 1

10 (2.25,1.75,0.5) 111 (2.25,2.25,0.5) 112 (1.75,2.25,0.5) 113 (1.75,1.75,2) 114 (2.25,1.75,2) 115 (2.25,2.25,2) 116 (1.75,2.25,2) 1

Acknowledgement

Financial grant from the Vikram Sarabhai Space Centre(VSSC) of the Indian Space Research Organization in con-ducting this research is gratefully acknowledged.

Appendix I. Geometric data for NURBS model used in

Section 5

I.1. Example 2: Steady-state of heat conduction

Bi-linear tensor product B-spline basis functions areconstructed over N � � = {0,0,.2,.2,1,1} � {0,0,1,1} 2[0,1]2. Control points and associated weights are given inTable I.1.

I.2. Example 3: Wrinkling of a circular membrane with a

central hub

B-spline basis functions are constructed by taking thetensor product of one quadratic B-spline defined overN = {0,0,.2,.2,1,1} 2 [0,1] and a linear B-spline defined over


� = {0,0,1,1} 2 [0,1]. Control points and associatedweights are given in Table I.2.

I.3. Example 4: Square footing under vertical load

Tri-linear B-spline basis functions are constructed overthe knot span N � � � R = {0,0,1,1} � {0,0,1,1} � {0,0,.2,.2,1,1} 2 [0,1]3. Control points and the associated weightsare given in Table I.3.

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NURBS-based parametric mesh-free methods

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