subdivision schemes for smooth contact surfaces …subdivision schemes for smooth contact surfaces...

To appear inInternational Journal of Numerical Methods in Engineering, September 2004

Subdivision schemes for smooth contact surfaces of arbitrary meshtopology in 3D

M. Stadler and G.A. Holzapfel�

Institute for Structural Analysis - Computational Biomechanics, Graz University of Technology,Schiesstattgasse 14B, A-8010 Graz, Austria

SUMMARY

This paper presents a strategy to parameterize contact surfaces of arbitrary mesh topology in 3D with at least��-continuity for both quadrilateral and triangular meshes. In the regular mesh domain, four quadrilaterals or sixtriangles meet in one node, even ��-continuity is attained. Therefore, we use subdivision surfaces, for which non-physical pressure jumps are avoided for contact interactions. They are usually present when the contact kinematicsis based on facet elements discretizing the interacting bodies. The properties of subdivision surfaces give rise tobasically four different implementation strategies. Each strategy has specific features and requires more or lessefforts for an implementation in a finite element program. One strategy is superior with respect to the othersin the sense that it does not use nodal degrees of freedom of the finite element mesh at the contact surface.Instead, it directly uses the degrees of freedom of the smooth surface. Thereby, remarkably, it does not requirean interpolation. We show how the proposed method can be used to parameterize adaptively refined meshes withhanging nodes. This is essential when dealing with finite element models whose geometry is generated by meansof subdivision techniques. Three numerical 3D problems demonstrate the improved accuracy, robustness andperformance of the proposed method over facet-based contact surfaces. In particular, the third problem, adoptedfrom biomechanics, shows the advantages when designing complex contact surfaces by means of subdivisiontechniques. Copyright c� 2003 John Wiley & Sons, Ltd.

KEY WORDS: Contact, smooth surface, subdivision schemes

1. Introduction

For the finite element simulation of contact problems various methods are used to incorporate contactconstraints in the variational formulation (see the pioneering work [1], and [2], [3], [4] among others).However, the application of these methods may cause sliding of nodes over edges of the master surfaceand may lead, consequently, to discontinuities in the distance function when the contact surface isdiscretized with facet elements. This is particularly the case when the surface to be discretized retains ahigh curvature throughout the simulation. Then, the assumptions for proofing uniqueness and convexityof the contact problem do not hold anymore [5]. The consequences may be oscillations of the contact

�Correspondence to: G.A. Holzapfel, Institute for Structural Analysis - Computational Biomechanics, Graz University ofTechnology, Schiesstattgasse 14B, 8010 Graz, Austria, [email protected]

SUBDIVISION SCHEMES FOR SMOOTH CONTACT 1

forces, non-realistic pressure jumps, contact cycling (i.e. consecutive changes of contact partnersduring a Newton iteration) and loss of quadratic convergence of the nonlinear solution scheme.

The goal of the present work is to avoid these types of problems by specifying a smooth interpolatingsurface on one of the contacting bodies. Several authors have demonstrated that for rigid bodies the useof ��-continuity of the contact surfaces is advantageous (see, for example, [6], [7] and [8]). Recentpapers ([9], [10]) claim that it would also be a desirable property for deformable bodies, partly dueto the fact that ��-continuity is a necessary condition for quadratic rate of convergence of nonlinearsolvers. This is especially the case if a contact problem involves dynamics – and thus acceleration ispresent, which may jump at the element interfaces – and if the interpolation is not ��-continuous [11].

Hence, also for deformable bodies a number of smoothing techniques were developed: they includecubic Hermite interpolation [12], [13], cubic Bezier spline [14], [13], Overhauser (or Catmull-Rom)spline [9], [10] and NURBS [15], [16]. These techniques were originally developed for 2D problems.Their bivariate extension to describe contact surfaces in 3D has the inherent disadvantage that it cannotrepresent surfaces with arbitrary mesh topology. One approach to circumvent this problem is the useof NURBS. Thereby, the surfaces are decomposed into a set of trimmed NURBS patches, i.e. the useof portions of the originally rectangular patches, which are finally assembled. However, this results incomputationally expensive algorithms which are prone to numerical errors and complex continuityconstraints. Hence, smoothing techniques for contact surfaces in the 3D domain represent a morechallenging task.

To the authors’ knowledge two different approaches address this issue. One is by PUSO andLAURSEN [17] who presented a technique using Gregory patches, which are applicable to bothstructured and unstructured meshes of quadrilaterals. The applicability to unstructured meshes wasshown for arbitrary valences. The valence of a vertex is the number of edges coming out of it. However,it only achieves ��-continuity resulting in a discontinuous surface metric at element boundaries.Geometrically this means that the direction of the tangent vector is continuous (not its length). Pusoand Laursen reconcile this by introducing a modified frictional formulation. The other approach isdocumented by KRSTULOVIC-OPARA et al. [18] and [19]. They employ quartic Bezier surfaces forthe interpolation through the nodes and the centroid of triangular elements. This approach leads toquasi-��-continuous surfaces, which are everywhere continuous except at the element nodes, whereno derivatives are continuous (��-continuous).

In the present work we use so-called subdivision surfaces for the interpolation. They originate fromthe Computer-Aided-Geometric-Design community, introduced in 1978 by CATMULL and CLARK

[20], and DOO and SABIN [21]. Both works generalize tensor product B-splines of bi-degree twoand three to arbitrary topologies by extending the refinement rules to irregular parts (valence � �� for quadrilateral-based meshes or � �� for triangular meshes) of the control polyhedron. Later, in��, LOOP [22] has generalized triangular Box splines [23] of total degree four to arbitrary triangularmeshes. These schemes have the desirable property that they admit a polynomial representation ofthe regular part of the mesh. Further developments produced schemes, which possess properties suchas interpolations without matrix inversion, true �� or ��-continuity, and the ability to interpolatemeshes of arbitrary topology as well as arbitrary genus (quadrilateral/triangle) with ��-continuity atthe quadrilateral/triangle interface. These features make this approach very attractive for smoothing 3Dcontact surfaces. For a general introduction to subdivision surfaces see [24].

An additional advantage of the proposed approach is that it allows to base the geometrical modelingand the finite element analysis on an identical representation paradigm. Hence, both the deformedand the undeformed geometries of the contact surfaces are described by the subdivision surface. Theimportance of such a unified framework for geometrical design and mechanical analysis is discussed

Copyright c� 2003 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2003; 0:0–0Prepared using nmeauth.cls

2 M. STADLER AND G.A. HOLZAPFEL

in [25] and [26] and is consequently exploited for shells in [27] and [28]. It yields a robust interfacebetween geometrical design and mechanical analysis, whereby a number of difficulties are avoided,inherent in the heterogeneous treatment of smooth surfaces of current software tools.

Although subdivision surfaces were developed more than �� years ago they were not applied untilrecently. One reason for that might be that there did not exist a solid mathematical foundation of theirconvergence and smoothness properties. This gap was closed most notably by the works [29], [30]and [31]. Due to the same reason, subdivision surfaces have only recently been applied in the FEcommunity. The beginning is marked by the seminal paper [27], which uses subdivision surfaces forthin-shell structures within the geometrically linear regime. Successively, subdivision surfaces werealso used for thin shells in the nonlinear regime (see [28] and [32]), hierarchical mesh refinement [33]and mesh generation (see [26],[34] and [35]). To the authors’ knowledge, the approach documented inthis paper is the first attempt to employ this class of surface description for contact problems.

The organization of the paper is as follows: Section 2 briefly reviews the governing equations and thevariational framework for frictional contact problems. In Section 3 we provide a detailed description offour different subdivision schemes, which are appropriate for the design of contact surfaces. In addition,we also discuss some aspects of smoothness and surface evaluation. Section 3 also serves to providean overview upon the existing literature regarding certain aspects of subdivision surfaces. With this inmind we have all necessary prerequisites to assemble the contact algorithm. Hence, Section 4 discussesfour different strategies for the implementation of the subdivision-based contact algorithm into a finiteelement program. In Section 5 we discuss how subdivision surfaces can be used to operate on two casesof �-adaptivity. In Section 6 we show three applications of the proposed smoothing technique, wherethe particular advantages will become clear. Finally, in Section 7 we provide concluding remarks. Theappendix gives some details of the necessary data structures for subdivision surfaces.

2. Contact formulation

We consider a contact problem of two bodies ��, � � �� . Each of them occupies the open set��

� with smooth boundaries �� , where �

� denotes the reference configuration of body � at(fixed) reference time � � �. During some closed time interval � � �� of interest the two bodiesundergo the motions expressed by the maps ��

��

� , where ��

� � �� denotes

the closure of the open sets �� . Note that the motions�� transform referential positions��

�� into

points ��

located at the current configuration � of body � at time �, where �� .Subsequently, we divide the current (smooth) boundaries �� of the bodies into the three regions��

, �� and �� so that �� , at which Dirichlet, von Neumann and contact boundary

conditions are prescribed. In addition, we consider a parametrization of the current boundary surface ��of body two. The parameter plane with region � � �

� is characterized by the convective coordinates� and �. Hence, a map ��

� gives the relation �� . For the geometrical situationsee Fig. 1.

2.1. Governing equations and normal contact of two bodies

Here we briefly review the field equations and the boundary conditions of body �. The displacementsolution �� must satisfy the momentum balance principles and the boundary conditions. Thus,



�c1

�c2

�1

g

n

x

�2

��1

�u1

�u2

��2

1

x2

x2N

�

�

�

��t( , )� �★

★

Figure 1. Definition of closest point projection (only the current configuration is shown).

following, for example, [36] and [37], strong forms read

��

(1)

where �� and �� denote the prescribed Cauchy traction and displacement vectors, respectively. Thesymmetric Cauchy stress tensor is characterized by ��, the body force defined per unit current volumeas ��, the current mass density is �, the Cauchy traction vector (force measured per unit surfacearea defined in the current configuration) is denoted by ��, while �� is the outward unit vector fieldperpendicular to the boundary ��, and �� denotes the divergence operator of �� with respect tothe spatial coordinates. The initial conditions are given by

��

� ��

��

��

� ��

(2)

where � is the unit vector and �� is the prescribed initial velocity field.By assuming that the two bodies �� come into contact we ask for the gap function �� as indicated

in Fig. 1. It is the normal projection of �� onto �� according to

��

��

�� (3)

where �� is considered to be a (slave) point, with �� , while �� on ��is the associated (master) point. The minimum distance problem is performed, in particular, by a local



Newton iteration of the form

��

��

��

��

��

�

� (4)

where the solution vector �� stands for the set �� . For subsequent use in Section (2.2) we notethat

�� (5)

and denote all geometric quantities ��, which are evaluated at �� , by ��.

2.1.1. Path integration for frictional contact Frictional contact is based on the knowledge of therelative tangential slip. If the frictional behavior is characterized by the local parametrization inducedby the FE triangulation, the frictional time integration becomes meaningless when the incremental slippath involves more than one surface element. An efficient method to avoid this problem is documentedin [38]. Therein, the slip path length is computed in terms of two positions and the local normal vectorsassociated with a slave point at the beginning and the end of a time increment. Based on this, anassumed slip path is generated. Geometrically, the assumed slip path can be viewed as a second-orderapproximation of the geodesic (i.e. a locally length-minimizing curve) defined by these two points.

Subsequently, we briefly review the basic ideas of this concept and adapt the algorithm, as proposedin [38]. Within the iterative solution scheme, we require to refer to quantities from the previous step.Therefore, subsequently, we denote quantities of the previous and current time steps as � �� and � ��,respectively. If all quantities within an equation refer to a single time step and if no ambiguity can arise,then the superscripts � �� and � �� are omitted for convenience. Time steps are used synonymous forload steps within the iterative solution scheme.

The assumed slip path is based on two positions which belong to different time steps. Therefore, thedistance between these steps can not be evaluated by standard methods. Hence, in order to compute thelength of the assumed slip path within a single time step, we require a mapping, �� say, of a geometricquantity �� in the current time step, back to the previous time step. Usually, a quantity is evaluatedby means of global variables (nodal displacement �), and local variables (�� from the same timeframe. However, the symbol �� means that it is evaluated with the global variable from the previoustime step and local variables from the current time step. Hence, for example, the map of the currentprojection point �� back to the previous time step has the form

�� (6)

where �� means that we compute a displacement vector by projecting the localcoordinates (�� ) of the current time step onto the nodal configuration (i.e. the finite element mesh)of the previous time step. The geometrical situation is illustrated in Fig. 2. Subsequently, the argumentsof the functions are omitted for simplicity.

Before proceeding it is necessary to define the assumed slip path, for which the orthonormal frames



Previous time frame

Current time frame

n

2��

★

n

2��

★

1��

n★

1��

2��

x2★+

x2★�

x2★+˜

˜

˜

˜

+

+

+

+

+

+

�

�

dm

assumed slip path

with length �

1��

Geodesic,

Figure 2. Mapping of the point of contact from the current time frame back to the previous time frame forcomputing the assumed slip path length � within a single time frame.

attached to �� and �� are required. Therefore, we define the quantities (see Fig. 2)

� �� ,

� � � �

�,

�� ,��

��,�� ,��

��.

��

(7)

Then, for the path parametrization we define the approximation to the geodesic as [38]

��

�

��

��

�

��

�� (8)

where �� are Hermite shape functions, defined in the domain � � �� as

��

� � � ��

�

� � � ��

��

� � ��

�

� � ��

�� (9)

The length of the assumed slip path, � say, can then be computed as the arc length of the geodesic

��

� �

��

��

�� (10)



which is evaluated numerically by, for example, Gauss quadrature. For later use, we define the direction�� of the increment of the total tangential slip, i.e.

�� with �� (11)

Therein, we have employed the operator � � � �� . The approximation to the geodesic and thecomputation of the length � of the assumed slip path is illustrated in Fig. 2.

2.2. Variational formulation

The goal of this section is to develop the finite element framework for frictional contact so that it cantake advantage of the introduced surface parametrization. We start with the virtual work contributiondue to contact. The contact constraint is enforced by the Karush-Kuhn-Tucker conditions, whichtransform the virtual work principle into an inequality variational principle. After regularization wearrive at equations, which describe the contact-related quantities necessary for the implementation intoa finite element program.

2.2.1. External virtual work – contact contribution The state of tangential contact is characterizedeither by stick (or adhesion) or slip, in the following indicated by �� or ��, respectively. Stickmeans that a node, which is in contact, does not move in the tangential direction, while for slip thenode moves. Note that this distinction is not required for normal contact. Consequently, it turns outthat it is convenient to decompose the contact-related variables into normal and tangential componentsaccording to, for example, [4]. For the contact contribution Æ�� to the external virtual work wemay write

Æ��

��

��Æ�� Æ �� (12)

whereby the Cauchy traction vector � is decomposed into a normal component ��, with �� (no adhesion), and a tangential component � � � � ��, while denote the tangential gap vectorto be defined below, and Æ �� is the variation of ��.

The conditions upon admissibility of �� regarding normal contact are summarized in the Karush-Kuhn-Tucker conditions, i.e.

�� on �� (13)

where �� , �� means contact (no contact means �� , �� ). The three relations (13)represent conditions for impenetrability, compressive normal interaction and complementarity of gapand contact pressure, respectively. Since they yield an inequality variational principle, a regularizationmethod is required. Here we employ the penalty method. For normal contact, it replaces the conditions(13)�–(13)� by the expression �� , where �� denotes the MACAULEY bracket, representingthe positive part of its argument and �� is the non-negative normal penalty parameter.

The frictional state, characterized either by stick (or adhesion) or slip, can be formally modeledby using the concept of elasto-plasticity theory (see [3] or [4] among others). Therein, plastic flow isgoverned by an evolution equation and a flow criterion. Since it is not known in advance if plasticflow occurs, a trial state is computed. This state is then checked if it satisfies the flow criterion or not.If yes, the trial state is accepted as an elastic state. If not, plastic flow occurs and the flow criterion issatisfied by means of a return-mapping algorithm [39]. For the related algorithm it is convenient to splitthe deformation into its elastic and plastic components. Hence, in an analogous manner, we additively



decompose the tangential slip into an (elastic) part � , responsible for stick or adhesion, and a(plastic) part � , responsible for slip. Thus,

� � � � � (14)

The ideal stick constraint � � � can then be introduced into the variational framework by a penaltyregularization according to

� � � � � (15)

where � denotes the non-negative tangential penalty parameter associated with tangential sliding. Thesliding process is modeled by an evolution law which is introduced through a slip potential � � � (theanalogue of the yield function in plasticity theories). From the maximum dissipation principle oneobtains the constitutive equation for the actual slip path

� � � ��

�� with � �� (16)

where we have introduced the unknown consistency parameter � (slip rate). Here, we particularize theslip potential with Coulomb’s law (equivalent to the plastic slip criterion) as

� � � � � �� (17)

where � denotes the coefficient of friction. For tangential contact, the Karush-Kuhn-Tucker conditionscan be specified as

� � �� (18)

Next we have to perform the algorithmic update of the tangential gap vector �� , which is performedby the unconditionally stable backward-Euler integration of the evolution equation and the elasticpredictor/plastic corrector concept [40]. In a first step, a trial traction solution, �� say, is thencomputed, which does, in general, not fulfill the slip criterion (18)�. The total slip � ��

�� is computed in terms of stick and slip parts according to (14). Then from (15) we may derive the trialtraction vector at the current time step according to

��

� � ��

�� (19)

where we have used the relation��

��

� � �� (20)

of the tangent traction vector �� of the previous time step. Note that the increment �� of the total tangential slip vector is given by its direction �� , defined in (11)�, and its length ��provided in (10). The slip potential (17) takes on the form ��

��. In a secondstep, the criterion (18)�, i.e. stick/slip, can be checked. For the case �� no friction takes place(pure stick), while for �� sliding occurs in the tangential direction. Hence, we may distinguishbetween stick and slip as follows:

� For the case of stick (i.e. ‘elastic’), the tangential traction vector �� is given by �� , asprovided in (19)�. Consequently, by using (19)� we find from (15) that

� � ��

� �� (21)

which defines the tangential stick vector.



� For the case of sliding (i.e. ‘plastic’), a return mapping to Coulomb’s friction cone has to beperformed. By using backward-Euler integration in order to approximate (16), we obtain

� � �� with �� (22)

where � � �� denotes the unknown incremental slip (plastic) parameter. Hence, with the useof (18) the incremental Karush-Kuhn-Tucker conditions read � � �, � �, �� , whichdetermine �. The traction vector is then

�� (23)

where we have introduced the definition �� . For a nonlinear slip potential,the return-mapping algorithm, required to determine the unknown incremental slip parameter�, leads to an iterative solution procedure. However, for the case of Coulomb’s law, � can bedetermined explicitly. By taking the norm of (23) we deduce from (17), by setting � � �, that

� ��

�

��

�� (24)

Finally, with the known �, the current traction vector �� and the tangential slip vector � � canbe obtained by substituting (24) into (23) and (22). Thus,

��

�� (25)

� � � � � ��

�

��

�� (26)

The components of the history vector, which are needed as input for the next load step, are the tangentialtraction vector �� , i.e. �� for stick and (25) for slip, and the current coordinates of the solution point,i.e. �� .

2.2.2. Aspects of the finite element implementation In the finite element context, we need to solvea nonlinear problem for which we employ Newton’s method. The MATHEMATICA package ACEGEN

[41] is used for the automatic derivation of the matrix formulae needed to describe the finite elementdiscretization of the smooth contact element.

We consider a smooth master surface, defined by a control polyhedron, which consists of � � �master nodes and an associated slave node. Hence, the total number of degrees of freedom of thesecontact partners is in three dimensions � � �� . The displacements of all involved nodes aregiven by the vector ��, � � �� . According to (12), the residual vector �� is then given as

��

� � ��

�� (27)

where we have omitted the superscripts �� and ��, because the equation is valid for both caseswhen the appropriate quantities are taken into account. Since ��

�� and � ��

�� , the implicit derivatives �� and �� are required. By application of thechain rule, they can be obtained from (5) as

��

��

��

��

��

��

��

��

��

��

� (28)



This allows to rewrite (27) as

��

� � �

��

��

��

��

� �

��

��

� � � � � �� (29)

where, for the frictionless case, we just have the first term ��. The corresponding tangentmatrix is obtained as

��

� � � �� (30)

which, for frictional slip, is a non-symmetric matrix, since Coulomb’s laws is non-associative. Theseexpressions are still independent of the smooth surface description, which we are going to particularizein the following section.

3. Subdivision schemes for contact problems

Until recently, NURBS were the defacto standard for computational geometry. They represent aversatile tool to model free-form surfaces. In addition, they possess the property of local support whichmeans that modification of a single node of the control polyhedron changes the surface only within abounded region. In regard to contact mechanics, this means that the element arrays (i.e. tangent matrixand the residual vector), which are associated with a particular finite element, are of limited size [16].However, NURBS are bivariate functions, and, therefore, they are not designed to directly representsurfaces of arbitrary topology. A possible work-around is to decompose the surface into (trimmed)NURBS patches. Unfortunately, this suffers from two drawbacks:

� Trimming two NURBS patches to match along their common boundary involves the complexcomputation of surface-to-surface intersection. These algorithms are, in general, computationallyexpensive and prone to numerical errors due to ill-conditioning.

� Complex and less intuitive continuity constraints across adjacent (trimmed) patches must beenforced throughout the deformation process.

These drawbacks have led to the development of subdivision surfaces (see [20] and [21]).Historically, they were derived from B-spline knot insertion rules to address the challenge of designingsmooth surfaces of arbitrary topology. We start with an arbitrary connectivity control polyhedron whichis described by control vertices and connecting edges in between. It forms a topological 2-manifold– possibly with boundary. The eventual surface (called the limit surface) is constructed through alimiting process of repeated refinement. The subdivision process consists of two parts, (i) a topologicalsplit rule describing how the existing control polyhedron is subdivided into additional polyhedrons,and (ii) some geometric rule with which the new control vertex positions are determined from theold positions. Most subdivision schemes use smoothing filters of finite support and local definitionwith constant coefficients depending only on the valence (i.e. the number of elements surrounding avertex) in the support of the filter. This leads to very efficient implementations. The valence of thevertices of the control polyhedron allows to divide the mesh into regular and irregular regions. Forquadrilateral-based schemes, discussed here, the valence required for regularity is � � �, for triangle-based schemes it is � � �. Other valences indicate irregular regions. Some schemes produce limitsurfaces with different properties in these two regions, as noted below.



(a) (b) (c) (d)

Figure 3. Catmull-Clark subdivision scheme in 2D.

To illustrate the process of applying a subdivision scheme, we consider a simple one-dimensionalexample and use the Catmull-Clark subdivision scheme [20]. We denote the vertices of a polygon,which is the result of � subdivision steps, by �� . Thereby, � is the index (starting from zero) of thevertex within one level (an overview of all indices used in this text is given in Appendix A.1). For 1D,for example, the topological split rule is simply described as an insertion of a new vertex between twoexisting vertices. The geometric rule for newly introduced vertices is

��

�

� �� (31)

while the new coordinates of existing vertices are computed as

��

�

� �� (32)

For existing terminal vertices, we use the rule

��

�

� ��

��

� �� (33)

for the left end or for the right end, respectively. Hence, the vertex positions of the refined polygon arecomputed as weighted averages of the unrefined polygon of the previous subdivision step. Figure 3(a)-(d) shows the repeated application of these rules in two dimensions. Within each step, the unrefinedpolygon of the previous step is shown by dashed lines. In the limiting process for � � �, we obtainsmooth curves, which are actually ��-continuous cubic splines. It can be observed that the resultingcurve in Fig. 3(d) is approximating the polygon in Fig. 3(a). For one-dimensional problems, subdivisionschemes do not offer an advantage over traditional approaches such as the methods discussed in theintroduction. Their strengths become apparent in the two-dimensional case and for problems witharbitrary-topology.

To illustrate this, we consider a coarse control polyhedron, as seen in Fig. 4(a). Its irregular vertices(of valence � � �) are shown as black dots. By repeated application of the subdivision scheme, thesurface converges to the smooth continuous limit surface, as shown in Fig. 4(c). For this example, thetwo-dimensional version of the Catmull-Clark [20] scheme was used, whose refinement rules will bedescribed later in Section 3.1. The reader who is interested in the topic of subdivision surfaces witha focus on geometrical modeling is referred to the standard introductory literature as mentioned inSection 1, and to [42].

There exist a number of subdivision schemes, which can be basically classified into two groups:



(a) (b) (c)

Figure 4. Example of the subdivision method using the Catmull-Clark scheme: control polyhedron (a),intermediate refinement (b), and limit surface (c). Irregular vertices are shown as black dots.

1. Interpolating schemes: Within each refinement step, the nodal positions of the coarser meshare not moved, while only the positions of the new vertices are computed. Therefore, vertexpositions of the initial mesh as well as all vertices generated during refinement lie on the limitsurface. Such schemes have been introduced for quadrilaterals by DYN et al. [43], KOBBELT etal. [44] and for triangles by ZORIN et al. [45].

2. Approximating schemes: These schemes compute new nodal positions for both the newlygenerated vertices and the vertices inherited from the coarser mesh. Therefore, the vertices ofthe control polyhedron do not lie on the limit surface. So, at each of subdivision, the existingvertices in the control polyhedron are moved closer to the limit surface. Approximating schemesare generally the favored schems for use in high-end animation. Candidates which fall into thiscategory were developed by CATMULL and CLARK [20] and LOOP [22].

Further criteria for classifications are (i) the type of the refinement rule (interpolating versusapproximating), (ii) the type of the generated mesh (triangular or quadrilateral), and (iii) the levelof smoothness of the limit surface for regular and irregular regions of the mesh (��, ��,...). We usethese criteria to characterize four schemes presented in the following section (see Fig. 5).

3.1. Refinement rules

In this text we discuss the Catmull-Clark (see CATMULL and CLARK [20]), Loop (see LOOP [22]),Kobbelt (see KOBBELT et al. [44]), and the modified Butterfly schemes (see ZORIN et al. [45], whichgoes back to the works by DYN et al. [43], [46]). Their properties as well as their so-called subdivisionmasks are summarized in Fig. 5. Among the many other schemes available in the literature we havechosen these, because they are well-established and simple to implement. Furthermore, their intrinsicproperties, when used to describe smooth contact surfaces, become more obvious. This might be notso clear with the more elaborate schemes. Figure 5 shows the subdivision masks for each schemeand presents the geometric (subdivision) rules for regular and irregular vertices in the first and secondrows. Thereby, the numbers as well as the symbols �� , �, , and !� , � � �� representthe variables associated with the weights, which are provided below. The third row in Fig. 5 provides



Regularvertex

1/41/4

1/4 1/4

1/161/16

3/8

1/161/16

3/8

��N��N

1/2 1/2

Catmull-Clark [20]

Crease/boundary

3/8 3/8

1/8

1/8

�

1��

1/8 3/4 1/8

Loop [22]

1/256 -9/256 -9/256 1/256

81/256

-9/256

-9/256

1/256 -9/256 -9/256 1/256

-9/256

-9/256

Kobbelt [44]

-1/16

9/16 9/16

-1/16

ModifiedButterfly [45], [46]

Continuity atregular vertex

Continuity atirregular vertex

Interpolating

Approximating

C 2

C 1

C 2 C 1 C 1

C 1 C 1 C 1

Mesh genus

a a

b

b

c c

cc

d d

s0

s1s2

s3

sN-1s

qIrregular

vertex

�

�

�

��

1/2 1/2

1/8 3/4 1/8

81/256

81/256 81/256

-1/16

9/16 9/16

-1/16

��N

��N

��N

��N

��N

��N

��N

��NN-2

1��

Figure 5. Subdivision schemes with associated subdivision masks and characteristic properties.

the rules for subdivision of a vertex on the boundary or at a crease (i.e. a part of the surface with��-continuity by design). The remaining rows summarize the specific properties of the subdivisionschemes regarding the continuity at regular and irregular vertices, the behavior of interpolation orapproximation of the control polygon and the mesh genus (i.e. quadrilateral or triangular).

For all of the four schemes except the Kobbelt scheme, the weights for the averaging process dependon the valence � of a particular vertex (see Fig. 5). These weights were determined such that ��-



Valence Weights� � � !� � �� !� � !� � �� !� � �� !� � !� � �� !� � �� !� � �� !�" �"�� !�" �"��

Table I. Values for the weights �� , � � �� , of the modified Butterfly scheme when using valences � �� ,[45].

continuity is achieved at irregular vertices. In summary, they are:

� Catmull-Clark scheme: In [20] it is suggested to use � �� and � �� .� Loop scheme: LOOP [22] proposed the relation

��

�

�

�

��

��

�

�!�"

�"

�

�� (34)

A simpler choice is reported in [23]: � �� for � � � and � �� for � � �.� Modified Butterfly scheme: If the edge, where a new node is created, connects two nodes of

valence � � �, then it is suggested to use � � ��, # � ��, $ � ��, � � �,see [46]. Therein, � can be chosen ‘suitably small’ or zero, resulting in the same rule set as theeight-point butterfly stencil introduced in [43]. If one node has a valence � �� , then the valuesfor !� , � � �� , are to be determined according to Table I, see [45]. The weight of thecenter node � � �� remains constant for any value of � .

3.2. Subdivision surface evaluation

Surface evaluation is the process of taking the control polyhedron, adding vertices, and braking facesinto more, smaller faces to find a better polygonal approximation of the limit surface. Basically thereexist three different approaches to evaluate subdivision surfaces: (i) recursive evaluation [20], [22],(ii) exact evaluation [47], [48], and (iii) pre-tabulated basis function composition [49]. In the presentwork we consider the first two approaches. The latter one is not suitable for the use within a contactalgorithm.

3.2.1. Recursive evaluation of subdivision surfaces Recursive evaluation, which is based on repeatedapplication of subdivision masks, as provided in Fig. 5, is the most direct implementation of thestandard definition of subdivision. For a curve this approach was briefly described in Section 3.However, within the nonlinear solution scheme of the contact problem we require the linearizationof geometric quantities based on the surface representation. Since recursive evaluation does notparametrization the surface with local coordinates � and �, the resulting smooth surface, which coversa finite element, must be treated by the contact algorithm as if standard facet elements were used.Hence, the normal vector, distance function and shape function derivatives are evaluated from thefacets, that were generated by the subdivision method. Although this approach is in principle possible,a recursively evaluated quantity leads to poor performance when an algebraic linearization is required.Therefore, an explicit form of the limit surface is of big advantage. This is only possible for the spline-based subdivision schemes such as Catmull-Clark and Loop, as shown next.



3.2.2. Exact evaluation of subdivision surfaces Within the regular domain of the mesh, theseschemes admit a simple evaluation of surface points since the control vertices of the Catmull-Clarkscheme coincide with those of rectangular B-splines and the control vertices of the Loop schemecoincide with those of triangular Box splines. However, at irregular nodes, these schemes can notbe applied. Until recently, the proper parametrization of subdivision surfaces in the vicinity of irregularvertices has been an unsolved problem. A solution, which is based on the eigendecomposition of thesubdivision matrix, is presented for (quadrilateral) Catmull-Clark surfaces in [47], and for (triangular)Loop surfaces in [48]. A modified version of this technique, which involves only one subdivisionstep, is documented in the recent work [27]. This is possible, because after one subdivision step theintegration point of interest lies within a regular mesh region. Since for contact problems the contactpoint may lie anywhere on the surface, this technique is not applicable for our purposes. Here webriefly review this technique and apply it to the Catmull-Clark surfaces. It differs only slightly fromthe Loop scheme, which is described in detail in [48].

We consider a quadrilateral patch (shaded region in Fig. 6(a)) with one irregular vertex (see node �),without loss of generality. This is due to the fact that after a single subdivision, even for a quadrilateralpatch with four irregular vertices, the geometric situation, as shown in Fig. 6(a), can always berecovered. The geometry is then defined by control vertices, ��

� �� say, whichare shown in Fig. 6(a) as black dots. Therein, % � �� denotes the number of control vertices thatinfluence the shape of the grey shaded region in Fig 6(a). Through the topological splitting process,new control vertices (small circles in Figs. 6(a) and (b)) can be generated, which allow to parameterizethree quarters �

, & � �� (see Fig. 6(b)) of the region, and to evaluate them as simple B-splines.These new control vertices are defined as

��

��

� �� (35)

with��

� �� (36)

where ' � %��. These control vertices for subdivision level � � � can be computed via an ' �%extended subdivision matrix � � as �� , where

� � �

��

��

�� (37)

Therein, �� is the common �� subdivision matrix for an irregular vertex of valence� . All submatrices are provided in Appendix A.2. To advance to any further level of subdivision� � �,a smaller % �% subdivision matrix

��

��

�(38)

can be used. Hence, for � � �, the control vertices for the B-splines can be obtained as

�� (39)

where we used �� . For example, if the origin in Fig. 6(c), denoted by a double circle,represents an irregular vertex, the evaluation of the surface at node ( requires three subdivisions.Hence, the prerequisites are obtained for defining the three surface regions, denoted as � ��

�,

& � �� , for any level of subdivision �, as

� ��

�� & � �� (40)



1

2

3 4 5

6

7

:(1)

:(6) :(2)

:(3)

:(4)

:(5)

:(8):(7)

(a)

�31

�21

�11

�32

�22

�12

A�

�

0 1

�

0

1

(b)

1

45

6

:(17) :(16) :(15) :(14) :(9)

:(8)

2

3

7

(1)

:(6):(7) :(2)

:(3)

:(4)

:(5)

:(10)

:(11)

:(12)

:(13)

�11

�21

�31

(c)

Figure 6. Parametrization of a Catmull-Clark subdivision surface near an irregular vertex of valence � :(a) numbering scheme for the set of control vertices ��, whereby ��

� � �� (black dots), whichdefines a surface region (grey shaded) near an irregular vertex, (b) numbering scheme for the control vertices ��,which define the B-spline surface regions ��

� , � � �� , and (c) partition of the unit square into an infinite familyof tiles (c). Therein, we used the shift-operator ��, defined as �� .

where �� (see Fig. 6(c)) and � � �. Therein, we have used the ��' ‘picking matrix’ ��,& � �� , which selects the control vertices from ��, associated with the &-th region, and � �� is the vector containing the �� cubic B-spline basis functions (see Appendix A.3). In addition we haveused the transformations

�� (41)

�� (42)

�� (43)

which map the tile � , & � �� , onto the unit square (see Fig. 6(c)). However, the evaluation, as

described in Eq. (40), is not very efficient, since it involves � � multiplications of the matrix ��,as seen in Eq. (39). As a possible remedy the subdivision is transformed into its eigenspace. Then, thesubdivision is equivalent to a simple scaling of its eigenvectors by their associated eigenvalues. Thistechnique allows to compute limit points and limit normals.



Subsequently, we aim to rewrite Eq. (40) in a way that utilizes the transformation of the subdivisionmatrix into its eigenspace in order to make the computation of this equation for large values of � moreefficient. Therefore, we consider the%�% subdivision matrix ��. For the Catmull-Clark and the Loopschemes it is non-defective for any valence, which means that the eigenvectors are complete. This isnot the case for several other schemes. The property that the eigenvectors are complete is importantfor the diagonalization of ��. Consequently, there exist always % linearly independent eigenvectors.The corresponding eigensystem of �� can be written as �� , where �� is the diagonal matrixcontaining the eigenvalues of ��, and �� is an invertible matrix whose columns are the correspondingright eigenvectors. The computation of the eigensystem is equivalent to the solution of the equation [47]

�� (44)

Since �� is invertible, we can rewrite �� as

�� (45)

which enables to rewrite Eq. (39) as

�� (46)

The last two matrices in Eq. (46) may be collected as #�� , representing the projectionof the % control vertices into the eigenspace of the subdivision matrix. Hence, with (46), Eq. (40) cannow be rewritten as

� �� #��

��

� � �� & � �� (47)

For simplicity, by collecting the terms in Eq. (47), which are independent of the control vertices andthe power �, we obtain

� �� #��

�� &�� & � �� (48)

where the definition� �� &��

��

� � �� (49)

has been introduced, and �� .Finally, we have an efficient technique for the evaluation of contact points on the interior of the

master surface. However, at the boundary of the subdivision surface, several control vertices aremissing. They are computed as a reflection of the existing nodes. For the reflecting plane, we use a planethat contains (i) the boundary edge and (ii) the surface normal vector at the boundary. For example, weconsider the grey shaded quadrilateral, as shown in Fig. 6(a), with valence � � �, and assume that itsright edge is part of the boundary. Hence, the vertices � �� and �� describe the boundary. To providean analytical description of the subdivision surface, we require the missing vertices with the indices�� and ��. Then, for example, vertex �� is computed as �� . A similartreatment applies to corners of the subdivision surface, where two reflecting planes are used. Note thatthis approach is different from the one we have used in the one-dimensional setting in Eq. (33). Thiscompletes our discussion of the exact evaluation of subdivision surfaces.

3.3. Surface continuity

Continuity is a major concern in contact mechanics. In a recent paper [16] we have shown that it mayinfluence the rate of convergence significantly. The analysis of the level of continuity of subdivision



schemes may cause problems, partly due to the fact that the limit surface can not be evaluated explicitlyfor all schemes. Although, Fourier methods were developed to analyze smoothness properties forregular mesh regions, these are not applicable at irregular mesh vertices. The breakthrough was recentlypublished in [50] and [31] to develop methods for proving the level of continuity. There exist schemes,which are ��-continuous for a regular mesh and �� at irregular vertices (Catmull-Clark and Loopschemes), schemes which are everywhere ��-continuous (Kobbelt and modified Butterfly schemes),and schemes which are ��-continuous for a regular mesh and ��-continuous at irregular vertices(Butterfly scheme). These properties are summarized in Fig. (5).

4. Implementation strategies

The properties of the various subdivision schemes suggest four reasonable implementations strategiesregarding the contact surface. In particular, the different approaches use (i) the interpolating schemesdirectly, (ii) the approximating schemes without interpolation, (iii) the approximating schemes withinterpolation, (iv) the approximating schemes by defining an interface finite continuum element, wherethe vertices of the control polyhedron form a part of its degrees of freedom. The specific merits of eachimplementation strategy is discussed in the following:

(i) Interpolating schemes: The Kobbelt and the modified Butterfly schemes can be used directlyfor interpolating the finite element nodes of quadrilateral or triangular genus, respectively. Theyhave ��-continuity everywhere. A disadvantage is that the only way to evaluate the surfaces isby application of the subdivision mask. Even by the use of automatic differentiation and codeoptimization techniques, such as proposed in [41], this may lead to long computation times forthe element arrays. However, a reasonable limit of accuracy may help to avoid this.

(ii) Approximating schemes without interpolation: Without any interpolation technique, the Loopand Catmull-Clark schemes can be used to approximate contact surfaces of quadrilateral ortriangular genus, respectively. Since they do not pass through the finite element nodes, meshrefinement changes the contact surface.

(iii) Approximating schemes with interpolation: As an extension to the method (ii), an interpolationtechnique as described in [51] can be used. If the method is local, as it is the case for the quasi-interpolation technique, then the element arrays are of limited size. Another method to avoid non-local element arrays, when interpolating, is to include the vertices of the control polyhedron inthe global system of equations and carry out the fitting procedure simultaneously to the solutionof the nonlinear system of equilibrium equations, as described in [16].

(iv) Define an interface finite continuum element with the control vertices as part of its degrees offreedom (composite contact element): Special finite continuum elements can be defined in orderto describe the outermost layer of elements forming the contact surface. Since they represent akind of interface between the continuum body and the contact surface, some of their degrees offreedom are the element nodes and some are the vertices of the control polyhedron. This leadsto an identical representation paradigm (see [25] and [26]), whereby both the deformed andthe undeformed geometries of the contact surfaces are described by the subdivision surface.It yields a robust interface between geometrical design and mechanical analysis, helping toavoid a number of difficulties inherent in the heterogeneous treatment of smooth surfaces ofcurrent software tools. Due to the property of local support of subdivision surfaces, only afew nodes of the control polyhedron need to be considered for the element arrays. This is



cn,1

cn,5

cn,9

cn,13

cn,14

cn,15

cn,16

cn,6

cn,7

cn,8

cn,12

cn,11

cn,10

cn,2

cn,3

cn,4

(P )5

P2

P3

(P )6

(P )7

(P )8

s( , )� �

P1

��

Figure 7. Hexahedral finite element with nodes �� , belonging to the finite element mesh, and nodes�� (shown in brackets), depending on the control polyhedron with nodes ��, � � �� . The

associated limit subdivision surface (partially cut away) is denoted by �� .

illustrated in Fig. 7 for a simple 8-noded hexahedral element, described by nodes ��, whichare associated with current coordinates ��, referential coordinates �� and nodal displacements�� . The nodes �� are part of the finite element mesh, while the nodes�� depend on the vertices of the control polyhedron denoted by ��, � � �� (with the associated displacements defined by ��, � � �� , where the superscript !indicates a displacement field of a node located at the control polyhedron). Hence, the geometry-related quantities �� , �� , and �� are evaluated via (40) (or (48)) as� ��

�, � � �� , & � �� , where �� , and � is

the level of subdivision. This method bypasses the need to interpolate the subdivision surfacethrough the finite element nodes, because they do not exist at the global element level. Hence thedisplacement-related variables of an 8-noded continuum element are described by the vector

��

�� (50)

Suppose that the material model is defined by some strain-energy function ), then the materialcontribution (indicated by the subscript �) to the element arrays are given by

�� )

�� (51)

��

��

��)

��

�� (52)

The contact related parts are provided through Eqs. (29) and (30). We call the elementa composite contact element, because, for a continuum element, this formulation is onlymeaningful when it also acts as a contact element.

In summary, the advantage of the interpolating scheme, i.e. (i), is its simplicity. It can easily beimplemented into an existing framework of a finite element system. The only ingredients requiredby the element is the mesh topology (i.e. element neighborhood information). However, the highestachievable level of continuity is �� for the regular and irregular domains of the mesh. Approaches



(ii)-(iv) have in common the salient feature of approximating subdivision schemes, i.e. they achieve��-continuity everywhere except at the irregular vertices, where the continuity is ��. In addition,they allow an explicit evaluation, as described in Section 3.2, which accounts for a short computationtime of the element arrays. Approach (ii) shares the simplicity with approach (i), however, it doesnot interpolate the finite element nodes. This difficulty is overcome in approach (iii) by means ofan interpolation technique. The first three approaches have in common that the contact constraint isapplied by an overlay element (see [3], [4]). This is different for the composite contact element, i.e.approach (iv), where the nodes�� , of the continuum element, which form a part of the contactsurface, do not appear as unknowns in the global element matrix, instead they depend on the nodes�� , of the control polyhedron. The composite contact element demands the largestimplementation effort when used within a standard finite element environment. However, it providesthe most attractive features regarding the contact algorithm.

Note that since Eq. (40) is linear in ��, the nodal displacements �� can efficiently be computedas

��

� ��

��

� ��

� � ��

� ��

��

�

��

�

� � ��

� �� & � �� (53)

Therein, we used right superscripts �� and �� to signify polyhedral coordinates associated withthe current and the reference configurations, respectively.

As a general remark, it should be noted that the valence � influences the number of entries in theelement arrays. However, this does not necessitate to derive the characteristic equations of the elementarrays for any value of � that might occur. Instead, it is sufficient to derive these equations for theregular case � � �. Subsequently, the element arrays can be supplemented by additional equations,which include the additional degrees of freedom for irregular cases � �� . This is illustrated as partof Section 5.

Regarding terminology, it should also be noted that the term ‘approximating subdivision scheme’refers to the fact that the limit surface approximates the vertices of the control polyhedron and does notpass through them. It has nothing to do with the approximation or interpolation of the finite elementnodes. In fact, it passes through the finite element nodes with arbitrary precision in method (iii) andexactly in method (iv).

5. Adaptivity

In this paper we only discuss �-adaptivity. Other approaches such as & or �-adaptivities do not changethe mesh topology and are therefore not of interest here. �-adaptivity can be classified into twoapproaches, one which does not generate hanging nodes and one which does. Clearly, the first approachleads to unstructured meshes with arbitrary topology. This case is naturally handled by subdivisionsurfaces. It remains to discuss the second case, which leads to structured meshes with hanging nodes.Both cases are illustrated in Fig. 8.

Adaptively refined meshes with hanging nodes are usually described by a hierarchical data structure.This has the advantage that neighborhood relations of elements can easily be generated, which areessential for the present contact algorithm. For more details about the associated data structures, the



(a) (b) (c)

Figure 8. Non-uniform (adaptive) refinement of a mesh shown in (a) with different strategies: (b) hierarchicallystructured with hanging nodes, and (c) unstructured without hanging nodes.

reader is referred to [52] and [53]. Here we will only consider refined meshes with no more than onehanging node per element edge, which is common practice.

The parametrization with non-spline-based (Kobbelt or modified Butterfly) subdivision schemes istrivial, since the additional finite element nodes of a refined mesh represent only information, whichdoes not need to be computed by the subdivision scheme. However, when parameterizing an adaptivelyrefined mesh with a spline-based subdivision scheme (Catmull-Clark or Loop), we encounter thesituation that at some mesh locations the set of control vertices ��, * � �� , required todescribe the B-spline, is not complete. This is discussed next and illustrated for two cases in Fig. 9.The element area to be parameterized, shown as the dark region , is adjacent to (a) coarser elementsand (b) finer elements. However, these missing support vertices (hollow circles in Fig. 9) can easilybe evaluated as a function of existing vertices (black dots in Fig. 9). It must be noted that the highestlevel of mesh refinement involved determines how the region must be partitioned so that it can beparameterized. This is obvious from Fig. 9(b), where has to be partitioned into four regions (theone currently parameterized is denoted by and indicated by a dashed line) to attain the same levelof subdivision as the highest level of its neighbors. If this procedure is omitted, important informationabout the surface, and, therefore, continuity, is lost.

To introduce the necessary dependencies between the missing nodes and the known �� , � �� %, we set up the vector ��, � � �� , which contains the index � of the unknown �� ,� � �� . Therein, � is the number of missing support vertices. To implement these additionalequations in the stiffness matrix, it is convenient to define a matrix ��, which describes thesedependencies. Thus, we may write

�� (54)

where �� is an ' � � matrix defined as

��

�� $ �� ,

� �%&'�(�"'.(55)

Hence, similar to hanging node constraints for finite element meshes, �� is supplemented by ��,which allows to eliminate the unknown degrees of freedom.



��

��

��

��

��

�� c1,16 c1,15 c1,14

c1,10

c1,3

c1,4 c1,5c1,11

c1,2 c1,1 c1,6c1,12

c1,9 c1,8 c1,7 c1,13

�

c1,4 c1,5 c1,11

c1,16

c1,3

c1,15

c1,2c1,1

c1,6c1,12

c1,9 c1,8 c1,7 c1,13

�

c1,14 c1,10

�

(a) (b)

c0,1c0,2

c0,3

c0,4 c0,5

c0,6

c0,7c0,8c0,9

c0,10

c0,11

c0,12

c0,13

c0,14c0,15c0,16

c0,1c0,2

c0,3 c0,4 c0,5

c0,6

c0,7c0,8c0,9

c0,10

c0,11

c0,12

c0,13

c0,14c0,15c0,16

Figure 9. Two cases for the parametrization of a structured mesh with hanging nodes and valence � � ,when using approximating subdivision schemes with implementation strategy (iv): the element surface to beparameterized is adjacent to (a) coarser elements and (b) refined elements (see Section 4). The control polyhedra

to parameterize the region are indicated by a checkerboard texture.

6. Numerical examples

Three representative numerical examples are selected to demonstrate the features of the proposedcontact algorithm. The examples aim to show the accuracy, which may be achieved with the presentedalgorithm. The first example, first published in [17], involves three concentric spherical shells, whichare in contact. It uses smooth surfaces with an unstructured mesh. The second example illustrates theuse of triangular composite contact elements by considering a cylindrical contactor sliding in a half-tube with friction. The distribution of the tangential traction along the path of the contactor is analyzedfor different contact surface types. The third example illustrates the proposed algorithm more from themodeling point of view. It aims to show the strengths when designing complex contact surfaces bymeans of subdivision techniques, and simulates the contact interaction between the facet joints of twolumbar vertebral bodies, which exhibits a research area by its own in the field of biomechanics. In orderto avoid confusion between vertebral facet joints and facet contact elements, we subsequently use theterm piecewise planar elements for facet contact elements.

6.1. Interference fit of concentric spheres

According to [17] we consider three concentric spherical shells, where the inner (I) and outer shells(O) are rigid, while the middle shell (M) is deformable (see Fig. 10(a)). For the discretization we usesmooth surfaces with an unstructured mesh. Shell (M) has an inner radius +� � ��, an outer radius+� � �� and consists of a material with the properties, � �� - � �� , whichare the Young’s modulus, Poisson’s ratio and the thermal expansion coefficient, respectively. The outer



Rigid shells

Deformable shell

(a) (b)

(O)

(M)

(I)

A

Figure 10. Three concentric spherical shells: (a) 3D geometry of the structure with shells partially removed, (b) 1/8section of the finite element mesh. The outer shell (O) and the inner shell (I) are rigid, while the middle shell (I)is deformable. The arrow indicates the direction of rotation of the inner shell (I). Symbol ‘A’ indicates an irregular

node.

surface of shell (M) is tied to shell (O), while the inner surface of (M) is in contact with shell (I). Thiscontact surface is characterized by a Coulomb frictional parameter � � ��.

In a first loading step, the temperature of shell (M) is raised up to 1000.0, in a second step, shell (I) isrotated by � Æ (as indicated in Fig. 10(b)) within 20 steps, while shell (O) is fixed in space. The momentis applied by a prescribed displacement on one node of shell (I). Two other nodes of shell (I) are fixedto define the axis of rotation. In [17], the analytical solution for the resulting moment is given by�� . We carried out simulations with 96, 384 and 864 elements for shell (M) (see Fig. 10(b)).In Fig. 11 the obtained frictional moments are plotted in logarithmic scale over the rotation angles andcompared for different meshes and contact surface types. Since only the sliding behavior is of interest,the transition from stick to slip of the finite element solution is not shown. Hence, we start directly fromthe slip behavior. For the different contact surface types, we used (i) standard piecewise planar contactelements (with ��-continuity), (ii) the Kobbelt subdivision scheme with implementation strategy (i)(with ��-continuity), and (iii) the Catmull-Clark subdivision scheme with implementation strategy (iv)(with ��-continuity in the regular mesh domain and ��-continuity in the irregular mesh domain). Inthis problem eight nodes, identified with symbol ‘A’ in Fig. 10(b)), define irregular mesh domains withvalence � � �. Clearly, the piecewise planar-based algorithm does not produce meaningful results,while the subdivision surface techniques quickly converge to the exact solution.

6.2. Cylindrical contactor sliding in a half-tube

This example illustrates the use of triangular composite contact elements. A similar version of theexample was published in [18]. We consider a half-tube, chosen as the master surface, and a cylindricalcontactor, chosen as the slave surface (see Fig. 12(a)). The inner and outer radii of the tube is�� and ��, respectively, while its length is ��. The dimension of the contactor is�� . For the purpose to generate an irregular mesh a hole with radius � � ��was placed at the lower edge of the half-tube. The contactor slides within the region indicated by thedashed line from one end to the other in the direction indicated by the arrow. Thereby, the contactorpasses regular (valence � � �) as well as irregular regions (valence � � � � � � � �). Irregular vertices,



0 10 20 30 40

109

1.5 109

�� 109

�� 109

5.0 109

�� 108

384

864

96

384

384

Num

ber

ofel

emen

ts

Surface type

Rotation angle ( )

Fri

ctio

nal m

omen

t (-

)

Piecewise planar

Kobbeltsubdivision surface

Catmull-Clarksubdivision surface

Figure 11. Frictional moment in logarithmic scale over the relative rotation angle of three concentric sphericalshells with different contact surface types.

which influence the surface within the contacting region, are shown as black dots. We used a coursemesh in order to make the differences between the used surface representations more apparent. Thehalf-tube/contactor problem was discretized with �� tetrahedral elements.

The half-tube is fixed at all (cutting) planes. The contactor has prescribed displacements at the uppersurface in order to move it along the path. In addition, a normal force of ��) is applied on theupper surface in order to press the contactor against the half-tube. Due to the (uneven) discretizedsurface, loading was applied in �� steps (i.e. �� prescribed displacements). Both bodies are simulatedas compressible hyperelastic neo-Hookean material described by the strain-energy function

) �$

� .� ��

/

� 0 �� (56)

with the material parameters $ � � ��)�� and / � ��)��. In (56) .� � *�� is thefirst invariant of the right Cauchy-Green tensor � � � �, where � is the deformation gradient and0 � �'%� � � is the local volume ratio. The frictional coefficient was chosen to be � � ��, and hencethe exact solution for the tangential traction vector is � � �). For the normal and the tangentialpenalty parameters we used the value �� .

The numerical simulations we carried out with three different surface representations, i.e. (i)piecewise planar elements, (ii) modified Butterfly, [46] and [45], and (iii) Loop subdivision surfaces[22]. We study the distribution of the tangential traction along the path length. Here we are onlyinterested in the distribution of � , since, for this problem, � is independent of the interpolationtechnique. We normalize � by dividing the numerical solution with the analytical solution. Theresults, as illustrated in Fig. 12(b), reveal that the modified Butterfly method already yields a betterapproximation and a significantly smoother tangential force distribution than the one obtained withpiecewise planar elements, which shows oscillations in the normalized tangential traction. The use ofthe Loop subdivision surface gives a further improvement.



Displacement along the path (mm)u

Norm

ali

zed

tan

gen

tial

tract

ion

(-)

1.00

1.05

1.10

0.90

0.95

1 4 72 3 5 60

Piecewise planar

Modified Butterflysubdivision surface

Loopsubdivision surface

Surface type

(b)(a)

u

Figure 12. Cylindrical contactor sliding in a half-tube: (a) Finite element mesh of the problem, irregular verticesare shown as black dots, (b) distribution of the normalized tangential traction along the path, i.e. the displacement

of the cylindrical contactor, computed with different contact surface types.

6.3. Contact interaction between two lumbar vertebral bodies

In this example the advantages of modeling complex surfaces with the subdivision technique shouldbecome more apparent. We discretize two lumbar vertebral bodies, in particular, we focus on the facetjoints, which form the posterior part of the vertebral body. Although this would not necessitate to modelthe whole body by subdivision surfaces, it has advantages when performing mesh refinement or whencontact develops in regions not known a priori.

Since these joints are known to be susceptible to degenerative processes and injuries, numerouspublications are devoted to study the underlying mechanisms. The issue was discussed, in particular,as a contact problem in [54], [55] and [56]. In a more general context, the problem is investigated forthe whole spine in [57] and [58], among others. All of them use piecewise planar contact surfaces.However, it turns out that the facet articular surfaces have a complex curvature [55], which maycontribute essentially to the obtained results. For example, in [57] it was shown that the relationshipbetween torsion moment and torsion angle is a function of the number of elements on the facetjoint surface. In addition, the geometry resulting from graded facetectomy (partial removal of thefacet joints) may be too complicated to be modeled reasonably by standard discretization methods.Subdivision surfaces allow to capture the associated geometry up to any desired level of accuracy. Thegoals of this example are twofold: (i) to illustrate the process of modeling such a complicated surfacewith the subdivision surface technique, and (ii) to assess how far the simplification of piecewise planar



Con

tact

stre

ss(N

/mm

)

Path length (mm)(starting from position ‘A’)

10

0

5

15

20

0 20105 15

Num

ber

of

elem

ents

along

the

pat

h

7

14

7

14

Surface type

2

Piecewise planar

Catmull-Clarksubdivision surface

(a) (b)

L2

L3

(c)

A

Figure 13. Contact interaction between lumbar vertebral bodies L2 and L3: (a) polygon proxy of a half of L2,(b) representation of both bodies by means of subdivision surfaces, and (c) contact stress distribution along thewhite dashed path (indicated in (b)) for different mesh densities and contact surface types. The ligaments and the

intervertebral disc are not shown.

contact surfaces is justified. In particular, we study the contact stress within the facet joint of the lumbarvertebral bodies L2 and L3 during physiologically normal loading.

Usually it may be difficult to represent a body of arbitrary topology with smooth surfaces. However,using subdivision surfaces, a whole vertebral body can be modeled by a single surface. Hence, it wasnot necessary to join individual surface patches as it would have been necessary with NURBS. Inparticular, for this example we have chosen the Catmull-Clark subdivision scheme and implementationstrategy (iv). Since the generated mesh contains hanging nodes, we had to use the technique describedin Sec. 5. We obtained generally ��-continuous surfaces, and only at irregular nodes the level of



ITL LF PLL ALL, CL, ISL, SSL1� +,-� �� 1� � �� $ +,-� ��

Table II. Material parameters for the ligaments, i.e. intertransverse ligament (ITL), ligamentum flavum (LF),posterior longitudinal ligament (PLL), anterior longitudinal ligament (ALL), capsular ligament (CL), interspinous

ligament (ISL) and supraspinous ligament (SSL).

continuity decreased to ��. The geometry was obtained from the Visible Human Project data set [59].The subdivision surface was designed using a proprietary software tool. In order to fill the surfacewith hexahedral finite elements, it is necessary to take some care when laying out the surface topology.The topology is governed by the so-called ‘polygon proxy’, which represents the coarsest controlpolyhedron of the model. Hence, the polygon proxy must be built purely from quadrilateral polygons,which is, in general, easy to obtain. The polygon proxy and the resulting subdivision surface forthe vertebral body are shown in Figs. 13(a) and 13(b), respectively. For the finite element mesh weperformed additional refinements as it was necessary for the numerical simulations.

The two vertebral bodies are connected with the following ligaments: intertransverse ligament(ITL), ligamentum flavum (LF), posterior longitudinal ligament (PLL), anterior longitudinal ligament(ALL), capsular ligament (CL), interspinous ligament (ISL) and supraspinous ligament (SSL). Theyare modeled as transversely isotropic materials, with an isotropic matrix material, defined by the strain-energy function (i.e. in analogy with [60], [61], see also [37])

) �� )� �� )� �� 2 0�� (57)

where)� �.��

$

� �.� �� (58)

represents the energy stored in the matrix material,

)� �.��

1��1�

�'./1� �.

� ��

(59)

represents the contribution from the collagen fibers, and

2 0� �/

� 0 �� (60)

is a scalar-valued function, with the property 2 �� , which is motivated mathematically. It servesas a penalty function enforcing the incompressibility constraint.

Therein, � � 0�� is the modified right Cauchy-Green tensor, �.� � *�� is the first invariantof �, and the pseudo invariant �.� � � � �, where � � � � � is a structural tensor with the vector� characterizing the direction of the fibers. The symbols $ � �, 1� � � and 1� � � are materialparameters, which were obtained by fitting the strain-energy function to the data from [62] (see thesummary provided in Table II). The value for / � � (considered as a positive penalty parameter) waschosen to be � ��+,- for all ligaments. For additional information about the model see [60]. For theintervertebral disc as well as for cortical and cancellous bone, the material parameters were adoptedfrom [63]. The articular cartilage was modeled as a �� thick layer of neo-Hookean material with



the parameters also adopted from [62]. To model the contact interaction, body L2 was chosen as theslave surface while L3 was the master surface. For the penalty parameters �� and � associated withcontact, the value �� was chosen, and the frictional coefficient � was taken to be ��.

To apply the load, the lower endplate of L3 was fixed, while the upper endplate of L2 was exposedto an overall force of ��). In Fig. 13(c) the contact stress distribution along the white dashed path(see Fig. 13(b)) is shown – the length measurement starts at position ‘A’.

As can be seen from Fig. 13(c), by representing the contact surface with only � elements and��-continuous subdivision surfaces, the distribution of the contact stress is much smoother than for��-continuous piecewise planar surfaces. To identify the quality of the solutions, a uniform meshrefinement was performed, using twice as many elements (i.e. �� elements along the path in Fig. 13(b)).It shows that the solution with only � subdivision surface elements is already a good approximation.The model using piecewise planar elements also seems to converge to the smooth solution, but it showsstrong discontinuities even for the refined mesh. Hence, for this type of problem, piecewise planarcontact surfaces may only yield reliable results with very high mesh densities, which is inefficient toperform.

7. Conclusion

Subdivision surfaces were used to parameterize contact surfaces of arbitrary mesh topology in 3Dwith at least ��-continuity everywhere, both for quadrilateral and triangular meshes. Therefore, forfrictional contact interaction a simple standard formulation, as described in [4], can be used.

We presented four different subdivision schemes, the Catmull-Clark [20], the Loop [22], the Kobbelt[44], and the modified Butterfly schemes [43], [46], [45]. The first two belong to the group ofspline-based subdivision schemes, which allow an analytical evaluation of the surface in the regularmesh domain. The other two are not spline-based and can, therefore, only be evaluated by recursiveapplication of the refinement rule, which is less efficient. We also reviewed how such surfaces canbe evaluated efficiently near irregular vertices (� �� for quadrilateral-based meshes or � �� fortriangular meshes), and on adaptively refined meshes with hanging nodes.

Four different implementation strategies of a contact finite element were presented. These strategiesdiffer in the implementation effort and in the quality of the surface representation (such as level ofcontinuity, interpolation versus approximation). The proposed implementation strategy (iv) is superiorwith respect to the strategies (i)-(iii) in the sense that it does not use nodal degrees of freedom ofthe finite element mesh at the contact surface. Instead, it directly uses the degrees of freedom of thesmooth surface, which makes an interpolation obsolete when dealing with spline-based subdivisionschemes. This leads to an identical representation paradigm. We call the resulting contact element acomposite contact element (as opposed to an overlay element), because of the fact that this formulationis only meaningful for a continuum element when it also acts as a contact element. The MATHEMATICA

package ACEGEN was used for the automatic derivation of the matrix formulae required to describethe finite element discretization of the smooth contact element.

When using the Catmull-Clark or Loop schemes, there is no additional numerical cost whencompared when other smoothing techniques that consider also all neighbor vertices of the currentelement in the element arrays. When compared to facet elements, the evaluation of the element arraysis more complex due to the larger number of variables involved. Those schemes, which allow only arecursive evaluation (i.e. the Kobbelt and the modified Butterfly schemes) make the numerical cost ofthe computation of surface derivatives dependent on the precision to be achieved.



Three numerical examples were used to demonstrate the special merits of the proposed method. Theyaimed to show the accuracy, which may be achieved with the presented algorithm. The results showa significant improvement over facet-based (i.e. piecewise planar) contact surfaces when comparedwith analytical solutions. In particular, the non-physical oscillations are avoided and the robustness issignificantly increased. Therefore, the additional effort for the evaluation of the element arrays pays offwell due to the better convergence behavior. The third example, adopted from biomechanics, showed,in addition, how easy it is to design complex contact surfaces by means of subdivision techniquesand how to use the contact surfaces directly in finite element analyses. To the authors’ knowledge,the detailed curvature of the facet articular joint has not been considered in simulations using sucha high level of accuracy before. The obtained high accuracy was clearly pointed out in comparisonwith piecewise planar contact surfaces. Hence, the presented approach has the potential to open up anarea which allows to efficiently simulate three-dimensional contact problems with complex geometryand topology in a numerically accurate manner. This is in virtue of the contact surface (including thefinite element mesh), which (i) can be modeled in a highly efficient way, (ii) has at least ��-continuityeverywhere, (iii) can deal with surface vertices of any valence, and (iv) does not need any interpolation.

Appendix A

A.1 Description of the used indices

Index Description� Level of subdivision� Valence

% � �� Number of control vertices for � � �' � % � � Number of control vertices for � � �& � �� Regular patch index

A.2 Subdivision matrices

With the given order of the control vertices, the �� subdivision matrix �� for theCatmull-Clark scheme, as introduced in Eqs. (37) and (38), is given by [20]

��

� �

� # $ # $ # � � � # $ # $ � � 3 3 � � � � � � � 3 34 4 4 4 � � � � � � � � �� 3 3 � 3 3 � � � � � � �4 � � 4 4 4 � � � � � � �...

.... . .

...� 3 � � � � � � � 3 3 � 34 4 � � � � � � � � � 4 4

�!!!!!!!!!!!�� (61)

Therein, � denotes the valence of the irregular vertex and � � � �� # � �� $ �� 3 � �� and 4 � ��. The submatrices �� and �� in �� (see Eq. (38))



follow from the B-spline knot insertion rules as [47]

��

� �

$ � � # � # � � ��3 � � 3 � � � � ��# � � $ # � # $ ��3 � � � � � � 3 ��3 � � � � 3 � � ��# $ # � # $ � � ��3 3 � � � � � � ��

�!!!!!!!!��

� �

$ # $ � # $ �� 3 3 � � � �� $ # $ � � �� 3 3 � � �� 3 3 �� $ # $� � � � � 3 3

�!!!!!!!!�� (62)

where � � �� # � �� $ � �� and �� . For the case � � �, there isno control vertex �� (it coincides with ��. Then, the components of the second column of �� mustbe replaced according to �� $� 3� �� $� 3� � � � � �� , and the last two columns in ��are missing. The submatrices �� and �� appearing in � � (see Eq. (37)) are [47]

��

� �

� � � � 4 � � �� 3 � �� 4 4 � �� 3 � 3 �� 4 4 �� 3 4 � � �� 4 � � � �� 3 � � � � �� 4 4 � � � ��

�!!!!!!!!!!!!��

� �

4 4 � � 4 � �3 � 3 � 3 � �� 4 4 � � � �� 3 � 3 � � �� 4 4 � � �3 3 � � � 3 �� 4 4 �� 3 � 3� � � � � 4 4

�!!!!!!!!!!!!�� (63)

where �� .

A.3 B-spline basis functions

The vector � �� in Eqs. (40) and (49) contains the 16 cubic B-spline basis functions. Thecomponent � of � �� is defined as

� �� (64)

where ‘0’ denotes the modulo operator returning the remainder of the division between the first andthe second argument, and ‘�’ denotes the integer division operator. The functions�� are the uniformcubic B-spline basis functions given by

��

� � �� ,

��

� � �� ,

��

� � � �� ,

��

��,

��

(65)

where � stands for either � or �.



ACKNOWLEDGEMENT

Special thanks go to Professor Rudolf Stollberger, Ph.D., from the Institute of Magnetic Resonance, Karl-Franzens-University Graz, to Margit Bauer, MD., from the Department of Obstetrics and Gynecology, Karl-Franzens-University Graz, and to Harald Bisail, M.Sc., from the Institute of Anatomy, Karl-Franzens-UniversityGraz, who assisted us in developing the second example and in generating the necessary data for the vertebralbodies. Financial support for this research was provided by the Austrian Science Foundation under START-AwardY74-TEC. This support is gratefully acknowledged.

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