mapped b-spline basis functions for shape design and isogeometric analysis over an arbitrary...

Comput. Methods Appl. Mech. Engrg. 269 (2014) 87–107

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Comput. Methods Appl. Mech. Engrg.

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Mapped B-spline basis functions for shape designand isogeometric analysis over an arbitrary parameterization

0045-7825/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cma.2013.10.023

⇑ Corresponding author. Tel.: +852 34429548; fax: +852 34420172.E-mail addresses: [email protected] (X. Yuan), [email protected] (W. Ma).

Xiaoyun Yuan, Weiyin Ma ⇑Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Kowloon, Hong Kong

a r t i c l e i n f o a b s t r a c t

Article history:Received 26 June 2013Received in revised form 18 October 2013Accepted 23 October 2013Available online 1 November 2013

Keywords:B-splinesMapped basis functionsGravity center methodRe-parameterizationArbitrary topologyIsogeometric analysis

It is well-known that B-spline surfaces are defined by a regular array of control vertices. Incase of models with arbitrary topology, it is extremely difficult to maintain continuity con-ditions among neighboring surfaces. The scenario is also true for applications in isogeomet-ric analysis (IGA). This paper presents a novel method for shape design and isogeometricanalysis from a quadrilateral control mesh of arbitrary topology using mapped B-splinebasis functions. Based on an arbitrary input quadrilateral control mesh, a global parameter-ization of the final surface is first defined through a Gravity Center Method (GCM). A re-parameterization method is then applied to map a B-spline basis function to others thatare explicitly defined and are tailored to each of the control vertices which can be eitherregular or extraordinary ones. The final surface is defined by all the input control verticeswith their corresponding mapped basis functions. For practical implementation, the sur-face can be evaluated patch by patch. Depending on the order of the B-spline basis functionused for mapping to others, the global continuity of the resulting surface, including atextraordinary points, can be arbitrary higher order. In the present paper, a uniform cubicB-spline basis function is used and the resulting surface is globally C2 continuous. Severalnumerical examples are provided to demonstrate the proposed method for both shapedesign and isogeometric analysis.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

B-spline curves and surfaces are widely used in the CAD and graphics communities. Their rational representation, knownas Non-Uniform Rational B-splines (NURBS), has been the de facto industrial standard over the past decades. B-splines pro-vide a convenient set of basis functions for a control mesh of regular topology. They have been widely used in shape design,surface reconstruction, shape deformation and animation, image processing, biomedical applications, and recently on iso-geometric analysis (IGA) [6,14,18,26,28]. However, similar to other tensor-product schemes, B-splines suffer from a severelimitation in modeling shapes of arbitrary topology. It is extremely difficult to maintain continuity conditions among neigh-boring surfaces with spline-based modeling.

Since the concept of IGA was introduced by [18], IGA based on B-splines has been widely discussed. Many researchershave applied B-splines and NURBS as the basis for IGA applications such as fluid mechanics [5,3], structural mechanics[20], thermal analysis [1], shape optimization [27] and so on. Other splines have also been applied for IGA, such as T-splines[4,11], hierarchical B-splines [33], PHT-splines (polynomial splines over T-meshes) [31,34], adaptive or locally refined splines[12,15], and generalized B-spline [23]. Most of these schemes provide convenient basis functions with efficient local refine-ments. They provide promising solutions for IGA and attracted a lot of attention in recent years. However, all these schemes

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88 X. Yuan, W. Ma / Comput. Methods Appl. Mech. Engrg. 269 (2014) 87–107

are still built upon rectangular parametric domains and it is not so convenient for efficient design of models with extraor-dinary patches or cells. In connection with the proposed scheme using mapped basis functions of this paper, one may alsofind a similar approach in [24,25] for parametric modeling from a control mesh of arbitrary topology. Their approach also useB-spline basis functions and the resulting surfaces can also be arbitrary higher order at extraordinary points. However, themethodologies in defining a global parameterization and re-parameterization are different. The Gravity Center Method(GCM) proposed in this paper produces a smoother global parameterization compared with that in [24,25]. The re-param-eterization method using Lagrangian interpolation formula proposed in this paper is also intuitive and easy to implement. Toour best knowledge, their approach has not been applied to isogeometric analysis.

In connection with IGA for models with arbitrary topology, very little work has been reported so far. One of the alternativesolutions is subdivision-based modeling. Subdivision-based IGA has been addressed in [7,22]. However, the continuity con-dition at extraordinary points is G1 only, which poses a limitation for IGA applications. Some other researchers also discussedthe treatment of irregular cases. The basic idea is to convert unstructured mesh to IGA suitable models based on commonlyused basis functions [13,35–37]. Recently, Jeong et al. [19] presented some mapping techniques similar to the method ofauxiliary mapping [2] in FEM for isogeometric analysis containing singularities. It works with proper selection of control ver-tices, but the continuity at extraordinary points was not discussed. Scott et al. [30] also provided some other direct treat-ments for IGA on irregular meshes by building special analysis-suitable T-splines. The continuity of the resulting model ishowever G1 as well at extraordinary points.

This paper presents a novel method for shape design and isogeometric analysis using a quadrilateral control mesh of arbi-trary topology. Based on the input control mesh, a smooth global parameterization is first defined using a Gravity CenterMethod (GCM). A B-spline basis function is further defined locally centered at the knot coordinates corresponding to a par-ticular control vertex. Depending on the shape and orientation of the local parametric domain, the basis function is furtherscaled and aligned with the local parametric space. A re-parameterization technique is finally applied for easy evaluation ofthe basis functions corresponding to the parametric domain of current interest. The method can be directly applied to iso-geometric analysis. The proposed method possesses some very nice properties:

� The space spanned by the mapped B-spline basis functions is an extension of tensor-product B-splines over an arbitraryparameterization.� The continuity conditions of the resulting surfaces is determined by the mapped basis functions, which can be arbitrary

higher over the entire surface, including at extraordinary points. For uniform cubic B-spline basis functions used in thispaper, global C2 continuity is obtained.� The above property with higher-order continuity at extraordinary points is quite important for IGA applications, and it

ensures that the continuity of the entire solution space is also smooth, which leads more accurate simulation results com-pared with other approaches in the presence of extraordinary corner patches.� The proposed method can be further extended to other basis functions as well as to non-quadrilateral meshes.

In the following, some preliminaries on spline-based modeling are highlighted in Section 2. Section 3 presents the methodfor producing mapped basis functions, including smooth global parameterization using a Gravity Center Method (GCM), ba-sis function generation, re-parameterization and evaluation. Section 4 describes the method for shape design on an arbitraryquadrilateral parameterization. Further applications of the mapped basis for isogeometric analysis are discussed in Section 5.Numerical examples on both shape design and isogeometric analysis of linear elasticity problems using the mapped basisfunctions can also be found in Section 4 and Section 5, respectively. Some general conclusions are summarized in Section 6.

2. Background knowledge on spline-based modeling

A B-spline curve of order k (or degree d ¼ k� 1) is defined by [9,10,14,26]:

PðnÞ ¼Xn

i¼1

viBi;kðnÞ for nk 6 n 6 nnþ1; ð1Þ

where PðnÞ is a point in R3 on the B-spline curve, vi 2 R3 for i ¼ 1;2; . . . ;n are a set of n control points, and Bi;kðnÞ fori ¼ 1;2; . . . ;n are the k-th order normalized B-spline basis functions that are uniquely defined by its order k, number of con-trol points n and a set of nþ k knots fnignþk

i¼1 using the following recursive equation

Bi;1ðnÞ ¼1; for ni 6 n < niþ1;

0; otherwise;

�ð2Þ

and

Bi;kðnÞ ¼n� ni

niþk�1 � niBi;k�1ðnÞ þ

niþk � nniþk � niþ1

Biþ1;k�1ðnÞ; ð3Þ

for i ¼ 1;2; . . . ;n. Tensor-product surface and volume are defined in a similar way as,

Fig. 1.functio

X. Yuan, W. Ma / Comput. Methods Appl. Mech. Engrg. 269 (2014) 87–107 89

Sðn;gÞ ¼X

i;j

vijBijðn;gÞ; ð4Þ

Vðn;g; fÞ ¼Xi;j;l

vijlBijlðn;g; fÞ; ð5Þ

for nkn6 n < nnnþ1, gkg

6 g < gngþ1 and fkf6 f < fnfþ1, where Bijðn;gÞ and Bijlðn;g; fÞ are the usual blending basis functions.

The knots can be a set of any non-decreasing numbers, such as uniform knots or a general set of non-uniform knots.One can also use multiple knots both internally within the definition domain or at the two curve ends or surface and cellboundaries. Fig. 1(a) illustrates one periodical B-spline basis function with order k ¼ 4, i.e., defined with global uniformknots. Fig. 1(b) shows a tensor-product blending basis function from two B-spline basis functions on the parametric 2Dspace. Fig. 2(a) illustrates the topological structure of a set of regular control points for a bi-cubic B-spline surface.Fig. 2(b) provides the corresponding mesh formed by the collocation of all the knots in parametric space, we call it aparametric mesh. The green area shown in Fig. 2(a) illustrates the final geometry corresponding to the green area ofparametric domain shown in Fig. 2(b). Based on the correspondence between the input control mesh and the finalparametric mesh, we develop the following scheme for defining mapped basis functions.

3. Mapped basis functions over an arbitrary parameterization

As it can be seen from Fig. 2(b), each of the basis functions for k ¼ 4 is centered at a knot position on the 2D parametricplane. We can therefore simply use the center knot position to represent the corresponding basis function. If a basis functionpositioned at the origin in the parametric space is given as Bðn;gÞ, any other basis functions can be defined through a simplere-parameterization as Bðn� ni;g� giÞ, where ni and gi are the center knot coordinates. For regular cases when k ¼ 4 or othereven order, the topological structure of the parametric mesh coincides with that of the control mesh, while for regular caseswith an odd order, such as a quadratic surface with order k ¼ 3, the topological structure of the parametric mesh is a dualmesh to that of the control mesh.

In case of a control mesh of arbitrary topology, we have the same scenario, which leads to the proposed method for pro-ducing mapped basis functions over an arbitrary parameterization. The method is composed of four major tasks as follows.

� Construction of a smooth global parameterization: Given a control mesh of arbitrary topology, we first construct a smoothparametric mesh having exactly the same topological structure as that of the given control mesh. The resulting paramet-ric mesh will then be used as a global parametrization for defining the final surface.� Generation of mapped basis functions: We will then assign a basis function to each of the knot positions, i.e. at each of the

vertices of the parametric mesh. In the present paper, we use a tensor-product cubic B-spline basis function that are posi-tioned at the knot positions on the global parametric space with proper scaling and alignment as if there is an underlyingregular parametric mesh at the neighborhood of the corresponding knot position. One may however use any other basisfunctions for assignment and mapping.

Illustration of cubic B-spline basis functions: (a) a cubic B-spline basis function with order k ¼ 4; (b) a bi-cubic tensor-product B-spline basisn with local support regions.

Fig. 2. A control mesh with regular parameterization: (a) a regular control mesh; and (b) the corresponding parametric mesh of (a).


� Re-parameterization of mapped basis functions: A re-parameterization method is further used to crop the correspondingpieces of basis functions for later evaluation of a surface patch.� Evaluation of mapped basis functions: Given a surface patch for evaluation, all influential basis functions are then collected

and evaluated based on the above mentioned mapping or transformations.

We illustrate each of the above tasks in the following subsections.

3.1. Construction of a smooth global parameterization

Given an arbitrary control mesh, we first project it to a proper projection plane for initialization. The projection plane willbe used as the global parametric plane and it should be chosen such that there will be a one to one mapping from the originalcontrol mesh to the projected mesh with no overlapping. Other surfaces that can be later unfolded to a plane can also be usedfor projection. One may use, for instance, a cylindrical surface for projection and then unfold the cylindrical surface with theprojected mesh onto a plane for parameterization.

With the projected mesh, we further generate the parametric mesh using some smoothing algorithms. We call thewhole smoothing procedure a Gravity Center Method (GCM) as the main algorithm operates upon gravity centers ofthe quadrangles on the mesh. We also apply special treatments in handling boundary and corner vertices. The proposedmethod works with a general quadrilateral mesh of arbitrary topology. In the literature, one may also find many otheralgorithms for mesh smoothing [16,17,32] that might also be adapted for producing a smooth parametric mesh with spe-cial boundary treatments. As the proposed smoothing method is a kind of iterative computation, for each level of inter-active smoothing, there are three types of algorithms for inner vertices, boundary vertices and corder vertices,respectively. In the following, we take a projected control mesh of a landscape model as an example to illustrate theGCM smoothing algorithm.

(1) All inner vertices vI ’s like A in Fig. 3(a) is updated as the gravity center of its one-ring quadrangles incident to the cor-responding vertex, such as the blue colored polygons for updating vertex A. The new position of point A will be moved fromthe solid red dot to the hollow red dot defined below

vI ¼Pn

i¼1Ai � ciPni¼1Ai

ð6Þ

where, n is the number of surrounding quadrangles; ci is the gravity center of each surrounding quadrangle that is also de-fined using the above equation from its two splitting triangles; Ai is the area of each surrounding quadrangle.

(2) Boundary vertices vB, such as B in Fig. 3(a), are computed starting from an inner vertex v2I , such as F in Fig. 3(a), vectort pointing towards outside while keeping the same angle from lines EF and FG, with magnitude being the average length ofall edges of the old mesh. The new boundary point is thus defined by the following equation

vB ¼ v2I þ t ð7Þ

where, v2I is the updated inner vertex connected to the target vertex by an edge; t is the vector computed as mentionedabove. Point vB for C is similar, but one of the points for computing t needs to be based on a vertex like H that may havenot been updated at this time yet. In this case the old vertex H is used.

Fig. 3. Illustration of the Gravity Center Method (GCM): (a) the projection of the input control mesh to x–y plane; (b) the resulting parametric mesh afterone GCM smoothing iteration; (c) the resulting parametric mesh after four GCM smoothing iterations; and (d) the final parametric mesh.


(3) The corner vertex vC like D is computed as the missing point of a parallelogram formed by K; L and M in Fig. 3(a)

vC ¼ v1B þ v2B � v3I ð8Þ

where, v1B ¼ K and v2B ¼ M are updated boundary vertices connected with the target vertex by an edge in one quadrangle;v3I ¼ L is the updated vertex opposite to the target vertex of the corresponding corner quadrangle.


The above completes one iteration for computing all updated vertices. Fig. 3(b) illustrates the resulting mesh after onelevel of the above smoothing iterations; Fig. 3(c) shows the resulting mesh after four levels of smoothing iterations.

The process is continued until a smooth stable mesh is obtained. There are many termination criterion and the conditionwe use is

Di ¼ei

u � eil

�ei� ei�1

u � ei�1l

�ei�1

�� < e ð9Þ

where, eiu is the maximum edge length of the new mesh; ei

l is the minimum edge length of the new mesh; �ei is the averageedge length of the new mesh; ei�1

u is the maximum edge length of the old mesh; ei�1l is the minimum edge length of the new

mesh; �ei�1 is the average edge length of the old mesh; e is a small tolerance and we often assign e ¼ 10�4.Fig. 3(d) provides the final parametric mesh when it is considered as a stable mesh. The total number of GCM iterations

for the stable mesh is 21 in this case. It can be seen that the resulting mesh is well organized and is smooth. The lengths of allthe edges are almost the same, so are the areas of the quadrangles. This is the parametric mesh we want.

Upon final termination, the mesh is updated again by dividing the coordinates of all vertices by the average edge �ei. Thelengths of all the edges of the final parametric mesh are thus around 1.

3.2. Generation of mapped basis functions

With our present scheme, all mapped basis functions come from a bi-cubic uniform B-spline basis functions. Visually, theproposed procedure for generating mapped basis functions is equivalent to that for individually assigning a basis function toeach of the knot positions on the parametric mesh, i.e., to each of the control vertices due to the one-to-one mapping be-tween the control mesh and the parametric mesh. Each of the mapped basis functions is obtained through three basic oper-ations, namely translation, rotation and scaling. Here the illustration is based on a knot position vector v on the parametricmesh of Fig. 4(a) marked in dark blue, which is the same parametric mesh shown in Fig. 3(d).

The first operation is translation. A bi-cubic uniform B-spline basis function is translated to the position centered at thesaid knot position v, leaving the function covering the pink area shown in Fig. 4(b).

We then apply a rotation operation to change the orientation of the basis function to align with the mesh locallyaround the knot position. To do so, a primary axis direction based on whom the rotation would apply should be decided.We try to find the most symmetric axis of the mesh locally around the knot position. For the dark blue knot in Fig. 4(c),there are four incident edges shown as the four dark blue lines. A quantitative value is computed for each edge for eval-uating its suitability as a primary axis. To compute the quantitative value for each of the incident edges, we find the topo-logical symmetry point for every outer point, such as the small blue vertex shown in Fig. 4(c). We further calculate thedistance between the outer point and its topological symmetry point after fold them to one side. If there is no topologicalsymmetry point, the distance from the corresponding point to the axis is computed. The quantitative value is thencomputed as the sum of the distances. The primary axis direction for that knot position is then the edge direction corre-sponding to the least quantitative value of the above sum of symmetric differences. Fig. 5 shows the distances in greenlines that are used to compute the quantitative value, i.e. the sum of all the distances, for edge 1 in Fig. 4(c). For this darkblue knot position v, the primary axis direction is computed to be edge 3 defined by knot positions from v to w as shownin Fig. 4(d). A rotation operation can then be applied to align the translated basis function shown in 4(b) to that as shownin 4(d).

Finally, in order to let the basis function to better fit the local region at the respective knot position, a scaling operation isapplied such that the first coverage of the basis function would just cover the four edges incident to the knot position, nomore and no less. As the average edge length of the parametric mesh is about 1 (see Section 3.1), the largest values of xand y as shown in Fig. 4(e) are then the scales for x- and y-directions, respectively, for the scaling operation.

After the scaling operation, the basis function more or less covers two rings of local quadrangles. Note that the scalingoperation doesn’t affect the magnitude of the basis function. Fig. 4(f) shows the final shape of the basis function at the darkblue knot position.

To assign mapped basis functions to all knot positions or control vertices of the parametric or control mesh, one may ap-ply a similar procedure as discussed above to each of the knot positions. For a regular mesh, the resulting parametric meshand the mapped basis functions should be adequate for surface evaluation. The resulting surface would be exactly the sameas that of uniform bi-cubic B-spline surface and Catmull–Clark subdivision. For an irregular mesh, however, a further re-parameterization step is needed in order to conveniently extract the required piece of basis function, such as that of themapped basis function shown in Fig. 4(f), for each of the surface patches.

It should be noted that during the so called process for generating the mapped basis functions discussed in this section,we only need to determine the set of parameters, namely v for translation, w (and v) for rotation, and x and y for scaling, foreach of the knot positions or control points. We call the collection of these parameters the parameterization of the respectivecontrol point, denoted by P ¼ fv;w; x; yg, from which a unique local parametric space for the respective mapped basis func-tion of v can be defined. Actual evaluation will be discussed in Section 3.4 and is performed using the respective mappingoperations discussed in this and the following sub-sections.

Fig. 4. Generation of a mapped basis function at knot position v on the global parametric mesh: (a) the global parametric mesh resulting from the GCMmethod; (b) translation of the basis function for v; (c) identification of the primary axis (edge #3) for the corresponding basis function of v; (d) rotation ofthe basis function for v; (e) scaling of the basis function for v; (f) the final shape of the basis function for v.

Fig. 5. A quantitative evaluation of edge 1 shown in Fig. 4(c) for determining the orientation of the final mapped basis function: light blue representing theoriginal mesh; dark blue line for the axis of edge 1; purple mesh for the mirror of the original mesh about the axis of edge 1; and green lines between blackdots showing the distances contributing to the sum for the quantitative evaluation. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)


3.3. Re-parameterization of mapped basis functions

The idea of re-parameterization is similar to the treatment of trimmed surfaces for spline-based meshfree method [21].Schmidt et al. [29] also applied a kind of re-parameterization for the evaluation of trimmed NURBS surfaces for isogeometric


analysis. In applications of trimmed surface rendering, re-parameterization is also often used for easy rendering and otherapplications, such as tool path generation.

The aim of our re-parameterization is similar to the above work and we wish to transform each of the irregular patches asshown in Fig. 6(b) into a square parametric domain within ½0;1�2. The re-parameterization is realized by applying theLagrangian interpolation formula with simple adjustment.

Given an arbitrary quadrilateral patch with knots coordinates ðn1;g1Þ; ðn2;g2Þ; ðn3;g3Þ; ðn4;g4Þ on the global parametricspace ðn;gÞ, we wish to re-parameterize the patch as a local patch ðs; tÞ within ½0;1�2 as shown in Fig. 6(a). Fig. 6(b) showsthe parametric relation between ðn;gÞ and ðs; tÞ. Similar to the isoparametric element in Finite Element Analysis (FEA), apply-ing the adjusted Lagrangian interpolation formula, the re-parameterization of the patch can be performed using the follow-ing equation

Fig. 6.parame

n ¼ P1n1 þ P2n2 þ P3n3 þ P4n4

g ¼ P1g1 þ P2g2 þ P3g3 þ P4g4

�ð10Þ

where,

Re-parameterization of mapped basis functions: (a) local parametric space ðs; tÞ of a surface patch; (b) global parametric space ðn;gÞ with thetric domain of a surface patch; (c) local parametric space ðu; vÞ of a mapped basis function with the parametric domain of a surface patch.


P1 ¼ ð1� sÞð1� tÞP2 ¼ sð1� tÞP3 ¼ st

P4 ¼ ð1� sÞt:

8>>><>>>:

ð11Þ

We have then,

n ¼ stðn1 � n2 þ n3 � n4Þ þ sðn2 � n1Þ þ tðn4 � n1Þ þ n1

g ¼ stðg1 � g2 þ g3 � g4Þ þ sðg2 � g1Þ þ tðg4 � g1Þ þ g1

�ð12Þ

where, s 2 ½0;1�; t 2 ½0;1�. Following the above re-parameterization, a basis function Bðn;gÞ can be re-parameterized to Bðs; tÞ,for s 2 ½0;1� and t 2 ½0;1�, on the patch.

3.4. Evaluation of mapped basis functions

Given a surface patch Sðs; tÞ for ðs; tÞ 2 ð0;1Þ2 for evaluation, we first need to identify all non-zero mapped basis functionswhose support reaches the respective patch. We apply an initial check using the center point c of the respective patch Sðs; tÞin the global parametric space developed in Section 3.1, i.e., with c having coordinates of the global parametric space.

� If c is far away from the origin of the local parametric space of a mapped basis function defined in Section 3.2 byP ¼ fv;w; x; yg, that basis function would have no effect to the final evaluation of the surface patch centered at c. Weset a criterion that if the distance between c and v in the global parametric space is larger than a threshold value 3:5,the corresponding basis function of v will be ignored. In case of a regular mesh, the threshold value is 2.5.� If a basis function whose support reaches in the patch domain for evaluation or can not be ignored following the above

check, we will then perform evaluation using the method discussed in this sub-section.

The evaluation of a basis function Bvðs; tÞ for ðs; tÞ 2 ð0;1Þ2, is performed through two steps of parametric mapping definedin Sections 3.3 and 3.2, respectively. As shown in Fig. 6, the re-parameterization method of Section 3.3 is first applied to mapthe parametric coordinates of the surface patch in ðs; tÞ as shown in Fig. 6(a) to the global parametric space in ðn;gÞ as Bvðn;gÞas shown in Fig. 6(b). The parametric coordinates in ðn;gÞ are then further mapped to the local coordinate space ðu;vÞ of therespective mapped basis function of v, shown as the pink area in Fig. 6(c). It should be noticed that Bðu;vÞ is actually a uni-form bi-cubic B-spline basis function centered at the origin of ðu;vÞ, which is the same for all the knot positions or controlpoints that can be efficiently evaluated using explicit polynomial functions.

In case of a quadrilateral mesh of arbitrary topology, a normalization operation is also applied to the resulting basis func-tions, which is defined as follows

Nvðsj; tjÞ ¼Bvðsj; tjÞPni¼1Biðsj; tjÞ

; ð13Þ

where Nvðsj; tjÞ stands for the mapped basis function at v after normalization, Bvðsj; tjÞ stands for the mapped basis functionat v before normalization, and Biðsj; tjÞ’s, for i ¼ 1;2; �;n, represent the n non-zero or effective mapped basis functions of thesurface patch with local parametric coordinates ðsj; tjÞ for evaluation. Basis functions Bvðn;gÞ in the global parametric spaceðn;gÞ can also be normalized as Nvðn;gÞ in the same way. It is worth mentioning that the normalization is not required forregular meshes, The normalization process can also be saved for regions far away from extraordinary points of an irregularmesh, as the parametric mesh for that area approaches to regular after GCM processing.

4. Mapped basis functions for shape design

4.1. Highlights of the methodology

Following the method discussed in Section 3, we compute the resulting surface patch by patch. For the j-th patch, e.g., thelocal surface is defined as

Sjðsj; tjÞ ¼Xn

i¼1

viNiðsj; tjÞ ¼Xn

i¼1

viNiðn;gÞ ¼ Sjðn;gÞ; ð14Þ

where, sj 2 ½0;1�; tj 2 ½0;1�. For all patches needed to be computed in Fig. 3(b), the local parametric spaces (systems) can beseen in Fig. 7. The origin for each local parametric space is almost randomly assigned by the control point with the smallestindex number of the vertices of that patch. After the re-parameterization and patch-wise evaluation, the surface model forthe control mesh shown in Fig. 3(a) is illustrated in Fig. 8(a). The respective local parametric directions in the physical spacecan be inferred from Fig. 7.

Fig. 7. Illustration of a global parameterization with local parametric spaces.

Fig. 8. A landscape surface produced by mapped B-spline basis functions and in comparison with Catmull–Clark subdivision: (a) resulting surface usingmapped basis functions with its initial control vertices; (b) resulting surface using Catmull–Clark subdivision with its initial control vertices; (c) resultingsurface using mapped basis functions with reflection lines display; (d) resulting surface using Catmull–Clark subdivision with reflection lines display.


Fig. 9. Illustration of a flower model with two surfaces: (a) initial control meshes of the petal and stamen surfaces of a flower model; (b)-(c) globalparameterization resulting from the GCM method for the outer petal surface and inner stamen surface.


Notice that as we are using mapped B-spline basis functions, each of the basis functions is compactly supported as that ofthe original B-spline basis functions.

The continuity of the proposed scheme: The re-parameterization method can be seen as a kind of cropping approach.Because we always define the model back in Eq. (4), as long as the basis functions Niðn;gÞ are continuous, the geometry of thefinal surface must be continuous as well. In other words, the continuity of the final surface will be exactly the same as that ofthe individual mapped basis functions whose continuity remains the same as that of the original B-spline basis function. Asuniform bi-cubic B-spline basis functions with C2 continuity are applied in this paper, the continuity of the resulting geom-etry is also C2 continuous with respect to the global parameterization n and g.

Without going to further details, we highlight the main properties of mapped B-spline basis functions as follows:

� Generalization: The space spanned by the mapped B-spline basis functions is an extension of tensor-product B-splinesover an arbitrary parameterization.� Continuity: The continuity conditions of the resulting surfaces is determined by the mapped basis functions, which can be

arbitrary higher order over the entire surface, including at extraordinary points. For cubic B-splines used in this paper,global C2 continuity is obtained.� Partition of unity: The mapped B-spline basis functions require a normalization step for patches close to an extraordinary

point in order to maintain the property of partition of unity.� Extension: The proposed method can be extended to other basis functions as well as to non-quadrilateral meshes.

The above property with higher-order continuity at extraordinary points is quite important for IGA applications, and itensures that the continuity of the entire solution space is also smooth, which leads more accurate simulation result com-pared with other approaches in the presence of extraordinary corner patches.

4.2. Examples and further discussions

In the following, we demonstrate the use of the mapped B-spline basis functions for shape design using three exam-ples. One of the surfaces is the landscape model used in the previous section. The other two surfaces are a flower surfacemodel and a face mask model. The original B-spline basis function used for mapping to others for all these examples dis-cussed in this section is an uniform bi-cubic B-spline basis function that is C2 continuous. It is therefore expected that theresulting final geometry should be C2 as well with respect to n and g, while the solution in the physical space should besmooth.

Example 1. A landscape modelFor the landscape model, Fig. 3(a) shows the initial control mesh. The resulting global parametric mesh after GCM

parameterization is illustrated in Fig. 3(d). Fig. 6(b) also shows global parameterization together with the local parametricspace for each of the surface patches. The resulting surface with control vertices is illustrated in Fig. 8(a). Fig. 8(c) shows thereflection analysis, i.e., rendering with highlight lines, of the surface of Fig. 8(a). The reflection lines for the model producedby mapped basis functions in Fig. 8(c) vary smoothly even at extraordinary points, confirming that the continuity of the

Fig. 10. A flower model produced by mapped B-spline basis functions in comparison with Catmull–Clark subdivision: (a) resulting surfaces with initialcontrol vertices using mapped basis functions; (b) resulting surfaces with initial control vertices using Catmull–Clark subdivision; (c), (e) and (g) resultingsurfaces using mapped basis functions in reflection lines display; (d), (f) and (h) resulting surfaces using Catmull–Clark subdivision in reflection linesdisplay; (c) and (d) illustration of both of the petal and stamen surfaces; (e) and (f) illustration of the petal surface only; (g) and (h) a scaled up view of thesurfaces near the central extraordinary point.


resulting surface is C2. For comparison, we also show the resulting surface produced by Catmull–Clark subdivision and its thereflection lines in Fig. 8(b) and (d). It is easy to see that there are obvious disturbance with the reflection lines atextraordinary points from Fig. 8(d), especially for regions close to the upper two extraordinary points of valence 3 and 6compared with similar regions in Fig. 8(c). Viewing the whole picture, the reflection lines are less harmonious in Fig. 8(d)than that produced by mapped basis functions in Fig. 8(c).

Example 2. A flower modelThis example illustrates a flower model defined by two surfaces, namely an outer petal surface and an inner stamen

surface. We also compare the results with that of Catmull–Clark surfaces. The initial control meshes are illustrated inFig. 9(a). The resulting global parameterizations generated by GCM of the two surfaces are shown in Fig. 9(b) and (c). Withthese two parametric meshes, the flower surfaces are produced using the proposed mapped basis functions. For comparison,we also produce the surfaces using Catmull–Clark subdivision. Fig. 10(a) shows the flower surfaces with control mesh in


russetish color produced from the proposed mapped B-spline basis functions. Fig. 10(b) shows the flower surfaces withcontrol mesh in pink color produced from Catmull–Clark subdivision. Fig. 10 illustrate the resulting surfaces with reflectionanalysis produced from the proposed mapped B-spline basis functions. From Catmull–Clark subdivision, the resultingsurfaces with reflection lines display are shown in Fig. 10. Scaled up local views of the surfaces near the central extraordinarypoints are shown in Fig. 10(g) and (h).

As the two extraordinary points are at the center and all the rest of the mesh is regular, there is no big difference betweenthe two sets of resulting surfaces for this example. Nevertheless, comparing the two sets of resulting surfaces with reflectionlines shown in Fig. 10, especially the center region of the scaled up views, the ones on the left side from mapped basisfunctions in Fig. 10 are still smoother than that on the right side from Catmull–Clark subdivision shown in Fig. 10.

Example 3. A face mask modelA face mask control mesh is shown in Fig. 11(a). There are in total 23 extraordinary points in the input mesh. The

parametric mesh by GCM is shown in Fig. 11(b). The resulting surface from mapped B-spline basis functions with GCM globalparameterization and the corresponding reflection lines rendering are illustrated in Fig. 12(a) and (c). The resulting surfacefrom Catmull–Clark subdivision with relevant reflection analysis can be seen in Fig. 12(b) and (d). Fig. 12(e) and (f) alsoprovide scaled up views of the two reflection lines rendering from the proposed mapped basis functions and Catmull–Clarksubdivision, respectively. While the continuity of the resulting surface at extraordinary points from Catmull–Clarksubdivision is G1, the continuity of the resulting surface from the mapped basis functions are C2. The surface quality can beviewed by the reflection lines on the two sides of the nose in the scaled up views. Some sharp features appear in Catmull–Clark subdivision, which indicates its G1 property. Nevertheless, as the mesh of this example is much denser than that ofExamples 1 and 2, variations among adjacent vertices are small. As a result, there is no big global difference between thereflection line displays resulting from mapped basis functions and Catmull–Clark subdivision, respectively, including at mostof the extraordinary areas.

To summarize, the plots of the reflection lines in Examples 1–3, in comparison with the results obtained by applying theCatmull–Clark subdivision scheme, show the good behavior of the proposed method. In Example 2, the only remarkabledifference between the reflection lines of the two methods are essentially around the extraordinary vertices, in view of theregularity of the mesh elsewhere. Much more difference can be appreciated for the landscape model of Example 1. Even ifExample 3 presents more extraordinary vertices than Example 2, the difference between the illustrations showing the reflectionlines of the two methods are again very well localized and almost equal elsewhere. All these examples indicate that theproposed method produces resulting surfaces with better behaviors than that of Catmull–Clark subdivisions in regions withextraordinary points, while it produces the same surfaces as that of Catmull–Clark subdivision in regular regions.

5. Mapped basis functions for isogeometric analysis

5.1. Highlights of the methodology

Isogeometric analysis (IGA) is widely studied over the last decade using various different basis functions and for differentapplications. The following highlights the basic methodology of IGA for stress and strain analysis under elastic deformation.The physical properties in the solution space is defined over the same set of modeling basis functions as

uðn;g; fÞ ¼Xn

i¼1

uiNiðn;g; fÞ ð15Þ

where u ¼ ½ux; uy;uz�T stands for the main variables of the solution space. Let

Fig. 11. A face mask model and its global parameterization: (a) initial control mesh; (b) GCM global parameterization.

Fig. 12. A face mask model produced by mapped B-spline basis functions and Catmull–Clark subdivision: (a)-(b) resulting surfaces; (c)-(d) reflection linesdisplay; (e)-(f) scaled-up reflection lines display; (a), (c) and (e) results from mapped B-spline basis functions; (b), (d) and (f) results from Catmull–Clarksubdivision.


d ¼ ½u1x;u1y;u1z; . . . ;unx;uny; unz�T ð16Þ

Eq. (15) can then be further written as

u ¼ N � d: ð17Þ

Following the theory of linear elasticity, we have also the following equation for strain definition

� ¼

�x

�y

�z

cxy

cxz

cyz

2666666664

3777777775¼

@@x 0 00 @

@y 0

0 0 @@z

@@y

@@x 0

@@z 0 @

@x

0 @@z

@@y

26666666664

37777777775

u ¼ ½@�u ¼ D � d ð18Þ

Fig. 13. Isogeometric analysis of a model with regular domains: (a) initial control mesh with degree d ¼ 2 (or order k ¼ 3); (b) control mesh after degreeelevation to d ¼ 3 (or order k ¼ 4); (c) the resulting geometry (illustration from degree d ¼ 2); (d) IGA results with d ¼ 2; and (e) IGA results with d ¼ 3 (ork ¼ 4) after degree elevation or p-refinement.


Fig. 14. Definition of a spanner model: (a) initial control mesh; (b) global parameterization resulting from the GCM method.


where

D ¼ ½@�N: ð19Þ

The corresponding stress is further defined as

r ¼ E�; ð20Þ

in which

E ¼ 2Gð1� 2mÞ

1� m m m 0 0 0m 1� m m 0 0 0m m 1� m 0 0 00 0 0 1�2m

2 0 00 0 0 0 1�2m

2 00 0 0 0 0 1�2m

2

2666666664

3777777775: ð21Þ

where,

G ¼ E2ð1þ mÞ ð22Þ

is the shear modulus with E and m being the Young’s modulus and Poisson’s ratio, respectively. The strain energy of a linearelastic body is defined as

due ¼ ddT �Z flþ1

fl

Z gkþ1

gk

Z njþ1

nj

DT EDjJjdndgdf � d; ð23Þ

The work produced by external loading is defined as

dwe ¼ ddTr � fr ¼ ddT �

Xme

j¼1

½Nð�jÞ�T ½fj��

; ð24Þ

in which ½fj�, for j ¼ 1;2; � � � ;me, stands for external loading applied to the elastic body at positions ð�jÞ ¼ ð�nj; �gj;�fjÞ, respec-tively. Following a virtual work principle due ¼ dwe, we obtain
Z flþ1
fl

Z gkþ1

gk

Z njþ1

nj

DT EDjJjdndgdf � d ¼Xme

j¼1


; ð25Þ

or

k � d ¼ f; ð26Þ

Fig. 15. Isogeometric analysis of a spanner model: (a) loading and boundary conditions (( for loading applied in the analysis, 4 for constraints applied inrespective directions); (b) Y stress computed based on mapped B-spline basis functions with GCM parameterization: max 24.296 MPa and min�89.393 MPa (Max displacement is 1.883e-004 m).


where

k ¼Z flþ1

fl

Z gkþ1

gk

Z njþ1

nj

DT EDjJjdndgdf ð27Þ

is the stiffness matrix of the underlying elastic cell and

f ¼Xme

j¼1


ð28Þ

is the loading acting upon the elastic cell. When the stiffness matrices k for all individual cells are calculated, the stiffnessmatrix for the entire elastic body can be further assembled. It is then able to work out d for further evaluation of displace-ment and other physical space parameters. Finally, the displacement, strain and stress are obtained from Eqs. (17), (18) and(20), respectively.

In case of a 2D simplified problem, the stiffness matrix becomes

k ¼Z gkþ1

gk

Z njþ1

nj

DT EDjJjdndg ð29Þ

where,

E ¼

E1�m2

mE1�m2 0

mE1�m2

E1�m2 0

0 0 E2ð1þmÞ

2664

3775: ð30Þ

In irregular case, as re-parameterization is applied, the stiffness matrix becomes

k ¼Z 1

0

Z 1

0DT EDjJ1jjJ2jdsdt ð31Þ

where,

J1 ¼@x@n

@y@n

@x@g

@y@g

" #; J2 ¼

@n@s

@g@s

@n@t

@g@t

" #: ð32Þ

Fig. 16. Y stress of a spanner model computed by Ansys for comparison: max 32.1 MPa and min �86.0 MPa (Max displacement is 1.35e-004 m).

Fig. 17. Definition of a bottle opener model: (a) initial control mesh; (b) global parameterization resulting from the GCM method.


5.2. IGA examples and further discussions

One of the important features of the proposed scheme using mapped basis functions is that both the geometry and thesolution in physical space can be arbitrary higher order depending on the basis functions used for mapping to others. Thisproperty is most important to isogeometric analysis in the presence of extraordinary points. Since most other schemes canonly achieve G1 continuity at such positions, the domain must be re-parameterized to avoid such singularities, otherwise,discontinuity may occur.

IGA Example 1: IGA of a model with regular topology.This example is obtained using regular B-spline basis functions. It shows that to obtain a smooth solution, the basis func-

tion used in the solution space should have sufficiently higher order. The geometry of this model comes from Fig. 2.15 onpage 34 in [8]. The initial control mesh can be seen in Fig. 13(a). The coordinates of the control points are: v1;1;v1;2;v1;3,v2;1;v2;2;v2;3, v3;1;v3;2;v3;3, v4;1;v4;2;v4;3 ¼ ð0;0Þ; ð�1; 0Þ; ð�2;0Þ, ð0;1Þ; ð�1;2Þ; ð�2;2Þ, ð1;1:5Þ; ð1;4Þ; ð1;5Þ,ð3;1:5Þ; ð3;4Þ; ð3;5Þ (unit: cm). Knots for ’’n’’ and ’’g’’ are given by N ¼ f0;0;0;0:5;1;1;1g and R ¼ f0;0;0;1;1;1gwith a com-mon order k ¼ 3. The total loading applied in this example for IGA is F ¼ 2� 10N ¼ 20N with boundary conditions as shownin Fig. 13(c). The model material is assumed to be carbon steel with Young’s modulus E ¼ 200 GPa, Possion’s ratio m ¼ 0:28,and shearing modulus G ¼ 78:13 GPa.

The computation result using B-splines of degree d ¼ 2 (order k ¼ 3) is shown in Fig. 13(d). In this figure, it can be seenthat the von Mises stress distribution curves are not smooth along patch boundary in the middle. This is because the con-tinuity of the basis function is only C1, and the continuity of the solution is dropped to C0. We further apply a p-refinement byelevating the degree from 2 to 3 (or order from 3 to 4). Knots for ’’n’’ and ’’g’’ after the refinement become:N ¼ f0;0;0;0;0:5;1;1;1;1g and R ¼ f0;0;0;0;1;1;1;1g and the resulting control mesh is illustrated in Fig. 13(b). The


new result with degree d ¼ 3 (order k ¼ 4) is shown in Fig. 13(e) which is then smooth. Thus, the order k should be at least 4for IGA in order to obtain a smooth solution.

In the following, we further present two examples in applying mapped B-spline basis functions for isogeometric anal-ysis of 2D models. The original B-spline basis function used for mapping to others for all the examples discussed in thissection is again an uniform bi-cubic B-spline basis function that is C2 continuous. It is therefore expected that the resultingfinal geometry should be C2 with respect to n and g, while the solution in the physical space should be smooth. Theseexamples are computed in a C++ programming environment and, similar to the previous example, the same material ofcarbon steel is used with Young’s modulus E ¼ 200GPa, Poisson’s ratio m ¼ 0:28 and shearing modulus G ¼ 78:13GPa.Gaussian quadrature is used for matrix integration, in which the number of points used for integration is five. For the visu-alization of the simulation result, the color hue is linearly interpolated from deep red (highest stress) to purple (loweststress).

In this example, a spanner model is analyzed using isogeometric analysis based on mapped cubic B-spline basis functions.There are two extraordinary points of valence 5 in the control mesh as shown in Fig. 14(a). The height of the model is200 mm and its thickness is 10 mm. The parametric mesh generated from GCM is shown in Fig. 14(b). The spanner modelbuilt with the GCM parameterization and boundary conditions can be seen in Fig. 15(a). For isogeoemtric analysis, the span-ner is fixed at two positions and a total of 80 N loading is applied on one side of the handle as shown in Fig. 15(a). The result-ing Y stress is computed as shown in Fig. 15(b). With the same feature vertices and boundary condition, a simulation is alsoperformed using a traditional FEA software named Ansys. The number of elements used in Ansys is 532. The result of Y stressis shown in Fig. 16.

IGA Example 2: IGA of a spanner model.As seen from Figs. 15,16, the distributions of the Y-stresses from mapped basis functions and that from Ansys are similar.

The resulting maximum stresses are also similar, while the maximum computed displacements are agreed to the same mag-nitude of 10�4. The positions of the maximum stresses can also be found from the respective figure, which are actually ap-peared at different positions. The maximum stress produced from Ansys appeared at positions of the constraints (Fig. 19),which is likely caused by stress concentration, while the maximum stress produced using mapped basis functions appearedon the back side of the handle (Fig. 15(b)), which is reasonable considering the loading and constraints applied for analysis.But the main distributions and range of the stresses are consistent. The stress with our simulation mainly ranges from�21 MPa to 18 MPa, while in Ansys, it is from �20.4 MPa to 19.0 MPa.

IGA Example 3: IGA of a dolphin-shape bottle opener.As shown in Fig. 17(a), this model has one extraordinary point of valence 6. The global parametric mesh generated from

GCM is shown in Fig. 17(b). The length, height and thickness of the model is 110 mm, 89 mm and 2 mm, respectively. Anexternal loading of 33.9 N in total is applied on the handle and two positions are fully constrained as shown in Fig. 18(a).The result of the von Mises stress is shown in Fig. 18(b). With the same feature vertices and boundary conditions, a

Fig. 18. Isogeometric analysis of a bottle opener model: (a) loading and boundary conditions (( for loading applied in the analysis, 4 for constraintsapplied in respective directions); (b) the von Mises stress computed using mapped B-spline basis functions with GCM parameterization: max 37.634 MPaand min 0.116 MPa (Max displacement is 1.06484e-005 m).

Fig. 19. The von Mises stress of a bottle opener model computed by Ansys for comparison: max 37.0 MPa (Max displacement is 1.02e-005 m).


simulation is also performed using traditional FEA in Ansys software. The number of elements used in Ansys is 869. The vonMises stress distribution is shown in Fig. 19. Both the stress distribution and the maximum stresses are basically consistentfrom both of the methods for this example.

6. Conclusions

In this article, a novel method using mapped B-spline basis functions are proposed for both shape design and isogeomet-ric analysis over an arbitrary quadrilateral parameterization. Given an arbitrary quadrilateral input mesh, the method startsfrom the computation of a smooth global parameterization using a Gravity Center Method (GCM). A uniform bi-cubic B-spline basis function is then used to map to other mapped basis functions at knot positions with appropriate translation,rotation and scaling corresponding to the respective control vertex. For evaluation, a re-parameterization method is also pro-posed to map an irregular parametric domain in the global parametric space to a unit square in the local parametric space ofa mapped basis function.

The proposed method using mapped basis functions has a number of nice properties as summarized in Section 4. Thespace spanned by the mapped B-spline basis functions is an extension of tensor-product B-splines over an arbitrary param-eterization. The continuity conditions of the resulting surfaces is determined by the mapped basis functions, which can bearbitrary higher order, which is most important for applications in isogeometric analysis. For cubic B-spline basis functionsused in this paper, a global C2 continuity is obtained. The proposed method can also be extended to other basis functions aswell as to non-quadrilateral meshes. The mapped B-spline basis functions require a normalization step for patches close toan extraordinary point in order to maintain the property of partition of unity.

The mapped B-spline basis functions are also further applied for isogeometric analysis in Section 5. One of the main attraction inapplying mapped basis functions for IGA applications is that the proposed scheme can be arbitrary higher order. Some examplesare presented to demonstrate the application of mapped basis functions in both shape design and isogeometric analysis.

While we think that the resulting mapped basis functions are linearly independent if they are defined over the smoothglobal parametric domain, a rigorous proof is required as future work. The refinement properties of the proposed basis func-tions should also be further studied. We also expect to extend the proposed scheme to other basis functions, such as that forinterpolatory modeling schemes. Further algorithms for non-quadrilateral meshes and volumetric models should also beinvestigated.

Acknowledgements

This work presented in this paper is partially supported by Research Grants Council of Hong Kong SAR (CityU 118512) andCity University of Hong Kong (Strategic Research Grant No. 7002735). The highlight or zebra lines illustrated in Figs. 8, 10and 12 are produced using a software named Rhinoceros. For comparison, Figs. 16 and 19 are produced using a softwarenamed Ansys for finite element analysis.

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