IsoGeometric Analysis with Subdivision Surfaces

Pieter J. BarendrechtEindhoven University of Technology

January 19, 2013

Abstract

In this introductory project we have applied IsoGeometric Analysis (IGA) on geometries modeled withCatmull-Clark subdivision surfaces. The fundamental concept of subdivision surfaces is the repeated subdi-vision of the control net, which basically represents a rough draft of an envisioned geometry. In the limitthis control net converges to a surface referred to as the limit surface, which can be completely described by(piecewise) basis functions. This means that subdivision surfaces can be used in conjunction with IGA.

Using a combination of Blender and Matlab, we have created a workflow that allows us to successivelyimport coarsely modeled objects, subdivide them, calculate their limit surface and eventually apply analysis(e.g. elasticity analysis) on them. Although this environment is rather static and still requires some manualintervention, it shows that the two topics go hand-in-hand very well.

This report discusses the absolute basics of both univariate and bivariate subdivision, along with a fewrelated topics. Using the concept of the limit surface, the connection to IGA is made. The combination ofthese two areas of research is ultimately illustrated with an example.

1. Introduction

Subdivision surfaces are a powerful modelingtechnique, much used in animated movies andgames. However, it has been virtually absent inComputer Aided Engineering (CAE) for a long time.One of the reasons might be that it can be ratherdifficult to model a geometry exactly as intended(e.g. design an object from a blueprint). Anothercause might be the difficulty in evaluating points onthe resulting surface, called the limit surface.

Nowadays, these inconveniences are practicallyresolved. There are extensive possibilities regardingshape control [14], NURBS-compatible subdivisionsurfaces [6] (NURBS [17] are still the standard inCAD), and stable methods to evaluate any pointon the limit surface, whether it is regular, extraor-dinary [34], a point near the boundary or near acrease [14]. This means that subdivision surfacescan now be combined with analysis. A relativelynew Finite Element (FEA) technique, IsoGeometricAnalysis (IGA) [11], seems the right candidate forthis conjunction [5].

The intention of this introductory project was tobriefly study a popular subdivision scheme and sub-sequently use it for analysis. It has, however, got-ten slightly out of hand, resulting in an interactiveMatlab environment to experiment with differentschemes, shape control and other related topics likereverse subdivision. The topic is of such interestthat the research described in this report will becontinued and extended in my MSc thesis. There-

fore, we have chosen to only focus on the basics ofa single scheme for this report – Catmull-Clark [8].

In this text, it is assumed that the reader has abasic knowledge of CAGD (e.g. Bezier and B-Splinecurves and standard knot-insertion/refinement algo-rithms). If not, [31] and [18] are excellent referencesto catch up. Good books discussing FEA and IGAare [3], [28] and [19]. Regarding the topic of subdi-vision – there are many well-written papers and sur-veys [24, 7, 29], but unfortunately not many books.Both [1] and [30] have been regularly consulted forthis project.

To conclude this section, an overview of the con-tents. We will start more or less from scratch withunivariate subdivision. The main concepts will beintroduced, and are extended to the bivariate case inthe subsequent section. Following up is the most im-portant part, discussing the limit surface. Since thelimit surface forms the connection between subdivi-sion and analysis, it is only logical to look at thistopic in a little more detail. Next is an overviewof IGA, followed by its application to subdivisionsurfaces. We conclude with some thoughts on theimplementation and a look ahead regarding futureresearch.

2. Subdivision curves

Let us begin with an arbitrary curve C(t) =PTA(t) where P is a vector of control points, andA(t) a vector of basis functions. For many reasons itcould be desirable to describe the exact same curve

in a different way as D(t) = QTB(t). For instance,a change of basis could be useful (e.g. using theBernstein basis instead of the B-spline basis whenapplying Bezier decomposition), or a different set ofcontrol points (e.g. for a better control of the curvewhen using it to model an object), where each newcontrol point is a linear combination of the old con-trol points.

Note that when a different set of basis func-tions is chosen, a different set of control points mustbe used as well, and vice versa. This can be eas-ily seen when writing the new control points asa linear combination of the old ones: Q = SP ,where S is the matrix containing the linear combi-nations. It follows that QT = PTST , and thereforethe curve D(t) can be written as PTSTB(t). SinceD(t) = C(t) by definition, it results in the expres-sion A(t) = STB(t).

In the case of binary subdivision (whether it isunivariate or bivariate), we’re interested in findinga two-scale relation for the original basis functions,and use this relation to compute the new set of con-trol points. For special cases, the matrix S relatingQ to P can be found directly, but in the generalcase we have to resort to the two-scale relation (alsocalled scaling or refinement relation).

We start by taking a look at both these ap-proaches, and briefly touch the topic of convergence.The subsequent section will expand the concepts tosurfaces.

The graphical method to find the new set of con-trol points is due to the Lane-Riesenfeld algorithm[22], which extends the de Rham-Chaikin algorithm[9] to arbitrary degree B-splines. To illustrate theapproach, the cubic case is depicted in Figure 1.

P2 = (2, 3, 4) P3 = (3, 4, 5)

P4 = (4, 5, 6)

P1 = (1, 2, 3) P5 = (5, 6, 7)

Q2 Q4

Q3

Q5Q1

Q6

Q7

Fig. 1 Lane-Riesenfeld for d = 3. Each controlpoint has an associated label, containing the param-eters for the blossom. Labels for Qi are ommitted.

Note that the mathematical interpretation ofthe Lane-Riesenfeld algorithm is midpoint knot-refinement, which can be observed when making use

of the blossom (i.e. the polar form of the curve)[27, 25].

The line linearly interpolating the control pointsis called the control polygon, which is the key con-cept of subdivision. Note that the control polygonof the new set of control points already looks a lotmore like the actual B-spline curve than the old con-trol polygon. If we were to apply the same methodrepeatedly, the control polygon would (in the limit)converge to the actual curve – the limit curve. Thismeans that instead of drawing the actual curve, wecould just take its initial control polygon, subdivideit a few times, and display the result instead. Whilethis is not that useful for curves, it certainly is forsurfaces. It is used a lot in the animation and gam-ing industry (e.g. the power of the computer beingused determines the smoothness of the objects in agame). As of yet, it is not widely used in CAD.

In matrix notation, the new control points Qi

can be obtained from the old control points Pi bymultiplying them with a subdivision matrix S. Forthe cubic case (d = 3) this results in

...

Q1

Q2

Q3

Q4

Q5

Q6

Q7

...

=

1

8

· · · · · · ·· 4 4 0 0 0 ·· 1 6 1 0 0 ·· 0 4 4 0 0 ·· 0 1 6 1 0 ·· 0 0 4 4 0 ·· 0 0 1 6 1 ·· 0 0 0 4 4 ·· · · · · · ·

...

P1

P2

P3

P4

P5

...

Note that this matrix S directly follows from the ap-plication of the Lane-Riesenfeld algorithm in Figure1 (see also Figure 2).

0 1 2 3 4 5 6 7 80

0.5

1A1(t) A2(t) A3(t) A4(t) A5(t)

0 1 2 3 4 5 6 7 80

0.5

1B2 B4 B6B3 B5 B7B1

Fig. 2 Unrefined (top) and refined (bottom) B-Spline basis. Ai(t) are the accompanying basis func-tions for Pi, likewise Bi(t) for Qi.

When studying the dependencies of the new con-trol points, it can be observed that there are in facttwo different cases – there are two different stencils.A stencil [30] illustrates how much a new controlpoint depends on which old control points. The onesfor the example are 1/8 [4 4] and 1/8 [1 6 1]. Thesestencils represent an affine combination (in this andmost other cases even a convex combination) – for

convenience the factor 1/8 could therefore be omit-ted.

The more general approach is finding the mask.A mask [30] is the opposite of a stencil – it showshow much an old control point contributes to whichnew control points. For splines, the two-scale rela-tion for the original basis functions can be used tofind the mask. Using the geometric definition of boxsplines (including uniform B-splines), the refinementrelation is readily obtained (see Figure 2 and 3).

Fig. 3 Graphical representation of the two-scalerelations for d = 1 (using the projection of a two-dimensional box) and d = 3. The associated masksare 1/2 (1 2 1) and 1/8 (1 4 6 4 1).

It appears that the new basis functions are scaledand shifted versions of the old ones (which is basi-cally the definition of a two-scale relation) – for ex-ample, A3(t) equals 1/8(B2(t) + 4B3(t) + 6B4(t) +4B5(t) +B6(t)), which is equivalent to 1/8(A3(2t) +4A3(2t−1)+6A3(2t−2)+4A3(2t−3)+A3(2t−4)).See Figure 2 and 3. Note that the coefficients add upto 2, because the mask corresponds to a binary sub-division scheme. Again, the factor 1/8 could there-fore be omitted. A geometrical interpretation of acoefficient in a mask is simply the number of sub-cubes projected onto the same part of the support.

A different approach for finding the two-scale re-lation for B-splines relies on the Fourier transfor-mation [26]. Other methods of obtaining the maskinclude the z-transform (i.e. using the generatingpolynomial of a scheme) [23]. A really intuitive ap-proach is the arrow method [15]. Combined withmask convolution, this is a very powerful technique.

It is straightforward to obtain the stencils froma mask (the sequences of every other coefficient ina mask constitute the stencils), so for [1 4 6 4 1] we

have(

1 4 6 4 1)

= [1 6 1] and(

1 4 6 4 1)

=

[4 4]. The stencils can then be used to construct thesubdivision matrix S.

It is important to note that in the literature theterm subdivision matrix is used for (at least) threedifferent matrices. The first one is the bi-infinite oneas illustrated above. Additionally, because the ba-sis functions of these subdivision schemes have local

support, only a small part of the matrix is enough toderive the properties of the scheme (i.e. the smooth-ness or continuity). For this kind of analysis, thecurve in the neighborhood of a point of interest hasto be known. For the cubic case, this leads to the5× 5 matrix

1

8

1 6 1 0 0

0 4 4 0 0

0 1 6 1 0

0 0 4 4 0

0 0 1 6 1

which is used to obtain the control points defininga cubic segment left and right of the point of inter-est, in other words its two-ring neighborhood. Whenonly interested in the evaluation of the scheme ata certain point, an even smaller matrix suffices. Inthe cubic case, this is the 3× 3 matrix

1

8

4 4 0

1 6 1

0 4 4

which corresponds to the one-ring neighborhood ofthe point of interest.

To conclude this section, let us think about whatwill happen when the same matrix (either the 5×5 orthe 3× 3 one) is repeatedly (i.e. n times) applied torespectively 5 or 3 control points: R = SnP . Insteadof iteratively applying S for a number of n times, wecould try to compute Sn and use the resulting ma-trix to find R directly. Using the eigendecompositionof S = V ΛV −1, it follows that Sn = V ΛnV −1. Ob-taining the eigenstructure for these small matricesis relatively easy, but for larger ones we will needspecial techniques, as explained in the next section.

Looking at the expression R = SnP for the 3×3matrix, this reduces (for n→∞) to R1

R2

R3

=1

6

1 4 1

1 4 1

1 4 1

P1

P2

P3

so any three consecutive control points of the con-trol polygon of a uniform cubic B-spline will con-verge to a single point. Rewriting this point to12 ( 1

3P1 + 23P2)+ 1

2 ( 23P2 + 1

3P3) and using blossoming,it is not difficult to see that this point indeed lies onthe actual B-spline curve.

3. Subdivision surfaces

The first bivariate subdivision schemes werethose of Doo-Sabin [16] and Catmull-Clark [8]. Thefoundation of these schemes is constituted by thetensor product of the Lane-Riesenfeld algorithm forrespectively d = 2 and d = 3 (note that this isnot historically correct, as [22] was published sev-eral years later than [16] and [8]). To illustrate this,

let us take the tensor products of the two stencilsfor the d = 3 case:[

4 4]⊗[

4 4]

=

[16 16

16 16

][

4 4]⊗[

1 6 1]

=

[4 24 4

4 24 4

][

1 6 1]⊗[

4 4]

=

4 4

24 24

4 4

[

1 6 1]⊗[

1 6 1]

=

1 6 1

6 36 6

1 6 1

These stencils also follow from the mask by takingevery other coefficient on every other row, as illus-trated for the second stencil:

1 4 6 4 1

4 16 24 16 4

6 24 36 24 6

4 16 24 16 4

1 4 6 4 1

Note that there are in fact three unique stencils –the second and thirds ones are rotational variants.These three stencils are illustrated in Figure 4.

The first one describes the centroid of a quadri-lateral face – the resulting new vertex is called aface point. The second stencil creates an edge point,whereas the third one updates the position of an oldvertex, resulting in a vertex point.

f f

ff

d

dd

d

e e a

c c

cc

b

b

b

b

Fig. 4 The three stencils, resulting in a new facepoint, a new edge point and an updated vertex point.

To avoid any confusion, the coefficients d, e andf are just the values from the stencils:

d =4

64=

1

16, e =

24

64=

3

8, and f =

16

64=

1

4.

The same goes for a, b and c – we will come back tothis later.

Similarly as for the univariate case, repeatedlyapplying the subdivision matrix (i.e. the one corre-sponding to the one-ring neighborhood) to a set of3-by-3 neighboring control points results in conver-gence to a single point. This can be observed whenS is constructed from the stencils and subsequentlyused to compute Sn. Every row of S consists of ex-actly one stencil, with its coefficients moved to the

right places (see Figure 4).

S =1

64

36 6 6 6 6 1 1 1 1

24 24 4 0 4 4 0 0 4

24 4 24 4 0 4 4 0 0

24 0 4 24 4 0 4 4 0

24 4 0 4 24 0 0 4 4

16 16 16 0 0 16 0 0 0

16 0 16 16 0 0 16 0 0

16 0 0 16 16 0 0 16 0

16 16 0 0 16 0 0 0 16

Again, we can use the eigendecomposition to findSn for n → ∞. Because of the rotational symme-try of the scheme, S is actually (apart from the firstrow and column) a block-circulant matrix contain-ing 2×2 blocks of size 4×4. Using the 4×4 DiscreteFourier Transformation (DFT) matrix F4 and its in-verse F−1

4 , the eigenvalues of each 4×4 block M areeasy to find: F4MF−1

4 = Λ [12]. Note that F−14 is

in fact the conjugate transpose of F4, i.e. F ∗4 . As

an example, let us look at the first 4× 4 block, i.e.

M =1

64

24 4 0 4

4 24 4 0

0 4 24 4

4 0 4 24

.

The 4× 4 DFT matrix F4 is defined as

F4 =1

2

1 1 1 1

1 −i −1 i

1 −1 1 −1

1 i −1 −i

which leads us to the (diagonal) matrix of eigenval-ues Λ

F4MF ∗4 =

12

0 0 0

0 38

0 0

0 0 14

0

0 0 0 38

.

Calculating the eigenvalues of all the blocks at oncecan be done using the matrix I2⊗F4, where I2 is the2 × 2 identity matrix. Finally, pre-multiplying theresulting matrix by a permutation matrix W andpost-multiplying it by W−1 it is possible to obtaina block-diagonal matrix. This matrix is still simi-lar (i.e. it has the same eigenvalues) to the origi-nal block-circulant matrix. The eigenvalues of theblock-diagonal matrix are readily computed, sincethey are the union of the eigenvalues of the individ-ual blocks. More details on this process (i.e. involv-ing the first row and column of S and finding theeigenvectors) can be found in [12, 34]. In the nextsection we will take a closer look at the limit surface.

In order to create a subdivision scheme that canbe used on a non-regular mesh (containing general

n-gons and extraordinary or irregular vertices), thesubdivision rules need to be generalized. From Fig-ure 4 it follows that the application of the first sten-cil results in the average value (the face point) ofthe 4 vertices defining the quadrilateral. An obvi-ous choice is to generalize this to the average valueof the n vertices defining an n-gon. An edge pointcan then be defined as the average value of the ad-jacent two new face points and the two old verticesconstituting the edge.

We continue with the third stencil. This oneseems rather problematic, considering that we haveto allow both n-gons and extraordinary vertices atthe same time. However, studying the results afterone application of the first and second stencil, it iseasy to see that all n-gons are quadrangulated. Fig-ure 5 shows the result of this primal quadrisection.

Fig. 5 Quadrisection of a few different n-gons.Note that the valence of the new face point is n.

Therefore, assuming that all faces are quads, wecan focus on just the extraordinary points. We canrecognize three types of vertices in the regular stencil(see Figure 4), so it makes sense to continue workingwith the three different coefficients a, b and c in thegeneral case (see Figure 6).

a

b

b

b

b

c

cc

c

c

b

b

Fig. 6 Catmull-Clark stencil for an extraordinaryvertex.

The optimal values for these coefficients dependon the valence of the central point. There havebeen many suggestions for these coefficients (mainlyby using DFT to obtain the eigenstructure of thesubdivision matrix of the two-ring neighborhood) –Catmull and Clark just generalized them from thegeometrical interpretation of the third stencil [8],resulting in an intuitive scheme:

Vnew =n− 3

nVold +

1

n2

∑F +

2

n2

∑M (1)

where F are the face points and M the midpointsof the incident edges of the vertex Vold. This results

in the following expressions for the coefficients:

an = 1− 7

4n, bn =

3

2n2and cn =

1

4n2.

Since it is sufficient to only consider quadrilaterals,the coefficients d, e and f are the same as before.Note that the relation in (1) is often written as

Vnew =n− 2

nVold +

1

n2

∑F +

1

n2

∑P

where P are the vertices at the end of the incidentedges of Vold. Additionally, it can also be written as

Vnew =n− 3

nVold −

1

n2

∑F +

4

n2

∑E

where E are the edge points of the incident edges ofVold – it is quite suitable for implementation.

Using the first form in (1), it is particularly easyto apply reverse Catmull-Clark subdivision (that is,on a mesh that is the result of a regular Catmull-Clark subdivision), by isolating Vold:

Vold =1

n− 3

(nVnew −

1

n

∑F − 2

n

∑M

)The strategy is to find all original vertices withn 6= 3, which can only be done if it is known whichcurrent vertices are the updated vertices with respectto the coarser mesh (which can be taken care ofby ordering the newly calculated face points, edgepoints and vertex points in a logical way). Finally,using the vertices already computed, the ones cor-responding to n = 3 can be found. Note that thismethod might fail if there are too many (or evenexclusively) old vertices with valence 3.

To conclude, it should be mentioned that thereare tools available that allow the user to shape theresulting geometry by adding sharpness to the edgesand vertices [14]. See Figure 7 for an example.The rules for these features derive from special sten-cils for boundary edges and vertices, which will bebriefly discussed in the next section.

Fig. 7 Left: Coarse model of a chess piece. Mid-dle: Regular Catmull-Clark subdivision. Right:Catmull-Clark subdivision using selective sharpness.Created in MATLAB, rendered with Blender 2.6.

4. The limit surface

As implicitly mentioned in the previous section,the limit surface corresponding to the regular facesin the control net consists of uniform bi-cubic B-Spline patches. However, we would also like to eval-uate points on patches corresponding to non-regularfaces in the control net – the faces containing an ex-traordinary vertex. Therefore, the first question iswhether the subdivision matrices representing theone-ring neighborhoods of these extraordinary ver-tices have the same convergence behavior as for theregular vertices.

For a non-defective subdivision matrix, a nec-essary and sufficient condition for convergence to asingle point is a simple (i.e. with multiplicity 1)dominant eigenvalue having value 1. When all en-tries of the matrix are non-negative, this is not dif-ficult to check. Because the rows of the subdivisionmatrix represent affine combinations (i.e. each rowsums up to unity), S has an eigenvector v1 consistingof all ones, accompanied by an eigenvalue λ1 hav-ing value 1. Gershgorin’s circle theorem provides anupper bound of value 1 for all eigenvalues, whereasthe Perron-Frobenius theorem for non-negative ir-reducible matrices guarantees a simple dominanteigenvalue. The one-ring neighborhood matrix S isirreducible because the entries of its first row andcolumn are all positive, resulting in a strongly con-nected associated digraph.

Since the (2v+1)×(2v+1) subdivision matrix Sfor Catmull-Clark is non-defective for all valences v,we know that S has 2v+1 linear independent eigen-vectors. Saving the right eigenvectors vi as columnsin V , and the corresponding eigenvalues λi in a diag-onal matrix Λ, we can write SV = V Λ. Because thecolumns of V are linearly independent, V −1 existsand thus we can write V −1S = ΛV −1, which meansthat every row of V −1 is in fact a left eigenvectorwi.

Repeatedly applying S to the vertices constitut-ing the one-ring neighborhood of an extraordinaryvertex leads to Q = SnP , where P and Q are bothvectors of 2v + 1 control points. This means thatwe can write P as a vector in the space spannedby the 2v + 1 eigenvectors vi, so P =

∑vihi where

hi are row vectors containing the coefficients (re-call that each control point in P has an x, y andz component). In matrix form this can be writ-ten as P = V H, and therefore the coefficientscan be computed as H = V −1P . This results inQ = SnP = Sn

∑vihi =

∑Snvihi =

∑λni vihi.

When n → ∞, only λn1 = 1 remains, along with itsaccompanying eigenvector v1 consisting of all ones.This means that the point of convergence is sim-ply h1, which is the product of the first row of V −1

(i.e. the left eigenvector w1) and the original controlpoints P .

Using Stam’s approach [34], we assume that eachface in the control net has at most one extraordinaryvertex (which is easily satisfied by applying Catmull-Clark subdivision once or at most twice). A patchcorresponding to a face with one extraordinary ver-tex is defined by K = 2v + 1 + 7 control points P0,where v is the valence of the extraordinary vertex.This is illustrated Figure 8 for v = 5.

Fig. 8 The K control points for the case v = 5.The extraordinary vertex of the face is highlighted ingreen.

Now when we apply our subdivision scheme tothese K control points, we end up with M = K+9 =2v + 1 + 7 + 9 new ones. They describe the exactsame patch, but looking a little closer, we observethat the old patch has been subdivided into foursubpatches. Of these subpatches, three are regular– which means that they can be evaluated! – andone is non-regular, i.e. again corresponding to aface with one extraordinary point. See Figure 9.

Fig. 9 Left: The M new control points for the casev = 5 – the non-regular face is highlighted. The col-ors of the control points correspond to the matricesbelow. Right: The 16 control points B2,1 defining aregular subpatch highlighted in red.

The non-regular subpatch is of course again de-fined by K control points P1, calculated as P1 =AP0, whereas the M control points P1 defining allfour subpatches (constituting one patch of the pre-vious level) are obtained via P1 = AP0. Both A andA are so-called extended subdivision matrices:

A =

(S 0

S11 S12

)A =

S 0

S11 S12

S21 S22

The matrices Sij are comprised of regular stencils(i.e. corresponding to the case v = 4) [34]. It fol-lows that Pn = AnP0 and Pn = AAn−1P0.

Repeating the step above results in a tiling ofthe parameter domain Ω = [0, 1] × [0, 1]. Each tileΩn

k corresponds to a subdivision level n and a sectork as depicted below in Figure 10.

ΩΩ1

2

Ω23 Ω2

2

Ω21

Ω13

Ω11

Fig. 10 The unit square Ω decomposed into tilesΩn

k , where k ∈ 1, 2, 3 and n ∈ N+.

The 16 control points Bk,n for a subpatch at leveln in sector k ∈ 1, 2, 3 are obtained from the Mcontrol points in Pn by using a (16 × M) pickingmatrix Xk, i.e. Bk,n = XkPn. The subpatch corre-sponding to the tile Ωn

k is therefore described by

sk,n(s, t) = BTk,nb(s, t) = PT

n XTk b(s, t) (2)

where b(s, t) are the regular bivariate cubic B-Splines. Of course we don’t compute Pn by re-peatedly applying A and then A – these controlpoints are computed directly from the original con-trol points P0. We like to call this approach n stepsof virtual subdivision. Using A = V ΛV −1 we canwrite

Pn = AV Λn−1V −1P0 (3)

which subsequently leads to the expression

sk,n(s, t) = PT0 Λn−1

(XkAV

)Tb(s, t). (4)

where P0 = V −1P0, i.e. the coefficients for writingP0 as a vector in the space spanned by eigenvectors(see earlier in this section).

A point (u, v) ∈ Ω on an extraordinary patchis thus computed as follows. First, the maximumof u and v is determined as m = max(u, v). Ifm = 0, we return the point of convergence h1 asdescribed above. Otherwise, the virtual subdivisionlevel n is determined by n = ceil(− log2(m)). Inorder to determine k ∈ 1, 2, 3, we use n to scale(u, v) by a factor 2n−1. Together, n and k deter-mine the tile Ωn

k corresponding to a certain sub-patch. Since the B-Spline basis functions are definedon Ω, we have to map Ωn

k to Ω in order to obtainthe right point of evaluation (s, t). Take for example(u, v) = (.35, .20), which results in n = 2. Scalingby 2n−1 = 2 results in (u, v) = (.70, .40), so k = 1.The point (s, t) then follows from a simple transfor-mation, in this case (s, t) = (2u− 1, 2v) = (.40, .80).See Figure 11.

Fig. 11 The point (u, v) = (.35, .20) trans-formed to (u, v) = (.70, .40) transformed to (s, t) =(.40, .80).

At first sight this might seem as lot of requiredcomputations. However, since most of these stepsdon’t depend on the value of n (i.e. the numberof virtual subdivisions) or the set of control pointsP0, they can be pre-calculated. For each valencev we could therefore compute the K eigenvalues(λ1, λ2, . . . , λK), the matrix V −1 and the matrixDk = (XkAV )T for k ∈ 1, 2, 3, and save them toa file. Note that the dimensions of Dk are (K× 16).A more concise notation to express a point on thepatch associated with tile Ωn

k is therefore

sk,n(s, t) = PT0 Λn−1Dkb(s, t). (5)

In practice it means that this data has to be loadedfrom a file before evaluating a mesh (the controlpolygon). Next, for each non-regular face in themesh, described by the control points P0, the co-efficients P0 have to be determined only once. Anypoint (u, v) ∈ Ω on the associated patch can then betransformed to the correct point of evaluation (s, t)and evaluated using the relation in (5). Although itis possible to use (5) to evaluate patches associatedwith regular faces in the mesh, it is slightly faster todirectly use the B-Spline basis functions b(u, v) forthis case.

At this point, we can evaluate any point of apatch associated with an internal face of the mesh.However, a mesh does not have to be a closed man-ifold – there could be a boundary. Furthermore,the boundary faces could be either regular or non-regular. Because our geometry of interest could bemodeled in a way without any non-regular bound-ary faces (see Figure 12), we have not studied theevaluation of such patches in full detail. In short,the main difficulty lies in the fact that the asso-ciated subdivision matrices are often defective (i.e.the eigenvectors are not linearly independent). Thismeans that S is not diagonalizable as V ΛV −1, whichcomplicates the calculation of Sn. Fortunately thereis another kind of matrix decomposition called theJordan normal form, which can be used to computeSn. The subdivision matrix can then be written asS = WJW−1, where J is a block-diagonal matrix(containing m blocks where m is the number of lin-early independent eigenvectors) and W the matrixcontaining the generalized eigenvectors as columns.

Note that the same approach can be used for inter-nal edges with sharp edges or vertices – see [33] formore details.

Fig. 12 The types and locations of extraordinaryvertices (highlighted) might depend on the modelingmethod. The coarse meshes are depicted on the left,followed by two iterations of Catmull-Clark subdivi-sion.

The boundary faces of a mesh require specialstencils for the calculation of both edge-points andvertex-points (face points don’t pose a problem). Anintuitive way to look at this problem is to decom-pose the mesh into its boundary curve and its innerpart, see Figure 13.

Fig. 13 Simple geometry decomposed into itsboundary curve and inner part, followed by two it-erations of Catmull-Clark.

It is then easy to see that we can use the univari-ate stencils for the boundary – after all, it is a curve.This means that an edge point on the boundary iscalculated as the mean value of its two neighbor-ing vertices (i.e. using the stencil [4 4]), whereas avertex point is calculated using the stencil [1 6 1].Strictly speaking, there is one additional concern –application of the regular stencil for an edge-pointcomprising two boundary faces could lead to foldsin the resulting surface (e.g. if the mesh contains

a concave section). In [4], an improved stencil issuggested.

In case of a regular boundary face, which is de-fined by 12 control points (see Figure 18), we cancompute 20 new control points using a 20× 12 sub-division matrix E. These 20 points can then beused to evaluate two subpatches. The control pointsdefining a single subpatch are obtained by using the16× 20 picking matrix XL or XR. See Figure 14.

Fig. 14 Application of the matrix E results in 20new control points, defining two subpatches.

Note that these two subpatches only constitute1/2 of the original patch, whereas for the internalcase, 3/4 of the original patch could be evaluated af-ter applying the matrix A. This leads to a differenttiling of the domain Ω, as is depicted in Figure 15.

Fig. 15 Tiling of the domain Ω for the regularboundary case compared to the internal case. Notethat the latter has a constant number of three sub-patches per level.

It is clear that in this case the number of sub-patches at level n is not constant, but 2n. In orderto compute the control points defining a subpatchΩn

k , we therefore have to use two different 12 × 12matrices EL and ER. The subdivision matrix EL

computes the 12 left-most control points, whereasER computes the 12 right-most ones. See Figure16.

Fig. 16 Application of the subdivision matricesEL and ER to the original control points.

It follows that the control points defining a groupof two subpatches at level n are found by repeat-edly applying EL and ER in an alternating fashion,followed by one application of E. The alternatingpattern can be derived from the binary tree associ-ated with the tiling depicted in Figure 15. A left

branch in this tree corresponds to the application ofAL, a right branch to the application of AR. Notethat each leaf in the tree corresponds to a set of 12control points, i.e. the situation depicted in Figure14. As example, the control points Pn=5 definingthe group of two subpatches highlighted in Figure17 are found by P5 = EERELELERP0 (comparethis to the relation in (3)). Subsequently applyingXL or XR results in the control points defining asingle subpatch Ωn

k .

Fig. 17 The sequence of EL and ER follows fromthe path in the associated binary tree.

Regrettably, this approach is rather inefficient.Using the eigendecomposition of EL and ER is ofno use, since they appear in an arbitrary alternatingpattern. In addition, the case of a corner face is evenmore complicated! The tiling for this case reveals arecursive relation for #n, the number of subpatchesat level n. Starting with #1 = 1, the relation reads#n = 3 + 2#n−1. The resulting associated tree is acombination of a binary and a ternary tree.

Fortunately there is a much easier method forevaluating points on patches associated with aboundary or a corner face. This technique, which isbriefly mentioned in [21, 33], makes use of so calledphantom control points. For the regular boundarycase, the idea is to combine the original 12 controlpoints with 4 phantom points, such that the reg-ular patch s(u, v) corresponding to these 16 con-trol points is precisely the same as the boundarypatch. The key for finding these phantom pointsis to equate the boundary curve of the boundarypatch to the isoparameter curve s(u, 0) of the regularpatch. This results in the relation Pi = 2Pi+4−Pi+8

for i ∈ 1, 2, 3, 4, as illustrated in Figure 18. If acoordinate-free approach [13] is used, this relationshould be rewritten to Pi = Pi+4 + (Pi+4 − Pi+8).

An equivalent way of deriving these phantompoints is by equating the results from the stencilsapplied on the boundary, i.e. the results from thebivariate stencils should equal the results from theunivariate stencils. As example, the bivariate edge-point stencil [4 4; 24 24; 4 4] applied to P1, P2, P5,P6, P9 and P10 should have the same result as theunivariate edge-point stencil [4 4] applied to P5 andP6. See Figure 18.

P1 P2 P3 P4

P5

P9

P13

P6 P7 P8

P10 P11 P12

P14 P15 P16

P1

P5

P9

P13

P2 P3 P4

P6 P7 P8

P10 P11 P12

P14 P15 P16

Fig. 18 A boundary face and a double boundary(corner) face. The boundary is indicated by the thickline.

A similar approach can be followed for a cor-ner face, but only when we assume that the cornervertex P6 is marked as sharp. This means that itsposition is fixed, resulting in the interpolation of thispoint. The phantom points P2, P3 and P4 can thenbe calculated from Pi = 2Pi+4−Pi+8 and the phan-tom points P5, P9 and P13 from Pi = 2Pi+1 − Pi+2.Finally, the phantom point P1 follows from equatingthe point s(0, 0) of the regular patch to the controlpoint P6, resulting in P1 = 4P7+4P10+P11−8P6, orP1 = P11 +4(P7−P6)+4(P10−P6) for a coordinate-free approach. See Figure 18.

To conclude, note that we are in fact usingonly 12 basis functions for the evaluation of regu-lar boundary patches, and 9 basis functions for thecorner patches. In a later section we will look atthese basis functions in some more detail.

5. IsoGeometric Analysis in a nutshell

IsoGeometric Analysis (IGA) is a relatively newform of FEA that is currently receiving a lot of in-terest in the academic world. The concepts of thetechnique were first discussed in [20], and later inthe book [11]. IGA ultimately attempts to unite theworlds of CAGD and FEA – the similarities betweenthe two are abundant, but the different terminologystill causes a lot of confusion.

The central notion of IGA are basis functions.As we have seen, geometries can be modeled usingbasis functions (either directly by using NURBS, orindirectly by using subdivision surfaces). The keyobservation is that IGA uses the same basis func-tions for both the geometry and the analysis. Thisimplies that we don’t need an expensive mesh gen-erator anymore, since the mesh now follows fromthe geometry. Note that although the isoparametricconcept (which is often used in FEA) might soundsimilar, it is fundamentally different.

Since a geometry modeled with Catmull-Clarksurfaces is C2 continuous at regular parts and C1

continuous near non-regular parts, the basis func-tions are suitable to use for analysis. Note that sub-division surfaces have been combined with analysisbefore, for instance in [10].

With regard to the analysis, the basis functionsspan the trial space – the approximation to the so-lution of the differential equation to be solved isan element of this space. Because of the continu-ity across the element boundaries, the approxima-tion will have a higher continuity than is the case inclassical FEA (which commonly uses the Lagrangebasis functions).

The employment of IGA in conjunction withsubdivision surfaces is demonstrated in the next sec-tion through the use of an example.

6. IGA and subdivision surfaces

We start this section with a recap of the differ-ent basis functions. Figure 19 illustrates the regularuniform cubic B-Splines Br(u), along with the basisfunctions Bb(u) for the boundary case.

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 10

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 10

.1

.2

.3

.4

.5

.6

.7

u → u →

Fig. 19 The basis functions Br(u) and Bb(u). No-tice that both sets of basis functions form a partitionof unity, i.e.

∑Br(u) = 1 and

∑Bb(u) = 1.

The latter basis functions can be obtained as fol-lows. It is given that a single column of three con-trol points P (based on Figure 18), accompanied bythree unknown basis functions Bb(u), describe a cer-tain curve PTBb(u). Furthermore, we know that thefour control points Q, accompanied by the knownbasis functions Br(u), describe the exact same curveQTBr(u) – note that the first control point Q1 is aphantom point. The matrix C, such that Q = CP ,is readily derived as

C =

2 −1 0

1 0 0

0 1 0

0 0 1

, CT =

2 1 0 0

−1 0 1 0

0 0 0 1

As explained in the section on subdivision curves,we can then use CT to relate the basis functionsBb(u) to Br(u) as Bb(u) = CTBr(u).

The bivariate basis functions now follow as setout below:

Regular patchesThe regular bivariate cubic B-Splines, definedby the tensor product Brr(u, v) = Br(u) ⊗Br(v).

Boundary patchesThe tensor product Brb(u, v) = Br(u)⊗Bb(v).Note that this operation is not commutative!

Corner patchesThe tensor product Bbb(u, v) = Bb(u)⊗Bb(v).Note that the corner vertex will be interpo-lated.

Non-regular patchesRewriting the equations (2) and (3) assk,n(s, t) = PT

0 V−T Λn−1V T ATXT

k b(s, t), itfollows that these basis functions – restrictedto a tile Ωn

k – are described by Bnr(u, v)∣∣Ωn

k

=

V −T Λn−1V T ATXTk b(s, t).

Switching to Finite Element Analysis, we can nowset up the abstract Bubnov-Galerkin setting (G):

(G)

Find uh ∈ Sh such that

B(wh, uh) = l(wh) ∀wh ∈ V h

where V h represents our discrete test space andSh the discrete trial space constructed from V h asSh = vh + qh

∣∣vh ∈ V h, where the lifting functionqh satisfies the eventual Dirichlet boundary condi-tions. Note that the operators B(wh, vh) and l(wh)are, respectively, the bilinear form and the linearform.

In our case the space V h is spanned by thebasis functions Bi(u, v) discussed above, so V h =span (Bi(u, v)).

Let us move on to a specific problem. Our ge-ometry or interest is based on Figure 12, and is cho-sen because it contains all four types of patches de-scribed above. It represents a block of material in-cluding some imperfections (e.g. grains) – see Figure20.

Fig. 20 The original geometry followed by two it-erations of Catmull-Clark. The color green corre-sponds to regular patches, yellow to extraordinarypatches, orange to boundary patches and red to cor-ner patches.

The initial mesh clearly contains faces with morethan one extraordinary vertex, so we apply an itera-tion of Catmull-Clark subdivision. After this singleiteration the conditions for evaluating the limit sur-face are satisfied, see Figure 21.

Fig. 21 The limit surface after one and two iter-ations of the Catmull-Clark algorithm.

For our analysis we consider isotropic linear elas-tic behavior:

∇ · σ = 0 on Ω

σ = 2µε+ λ tr(ε) I

ε = ∇su

u = (Fm − I)x + u on Γ

where µ and λ are the Lame parameters using ν = .3and E = 1. Fm is the macroscopic deformation gra-dient, defined as

Fm =

(1 .05

.05 1

)which means that we shear the block along the x = ydiagonal. Finally, u represents a microscopic fluctu-ation which is applied periodically on both the leftand right sides and the top and bottom sides – afterall our geometry is just a small block of materialthat is eventually repeated in all directions, thusconstituting a larger unit.

Note that the description of the model can bewritten in the abstract form mentioned above (seefor example [19]).

After solving for the displacement u, we cancompute the strains ε and stresses σ. The stressfield in the x-direction, σxx, is plotted in Figure 22.

σxx

.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

Fig. 22 Plot of σxx for the coarse case. A vectorversion is available online (click on the plot).

To conclude, we apply the same analysis on aglobally refinement geometry. Since the test space(and therefore the trial space) is now richer, we ob-tain an improved description of the stress field – seeFigure 23.

Fig. 23 Plot of σxx for the globally refined case.A vector version is available online.

7. Implementation

We conclude this report with a short overviewof the workflow. The first step in the process wasthe implementation of the Catmull-Clark scheme.Although there are several good implementationsavailable (see for example CGAL and OpenMesh),we chose to implement the scheme in Matlab our-selves, inter alia for a better comprehension of thescheme. After a rather inefficient attempt using aface-vertex mesh, we switched to a half-edge mesh.The half-edge data structure provides easy access toneighboring components.

There are several ways to implement subdivision.An existing mesh can either be modified (by addingcomponents to the mesh) or completely rewritteninto a new mesh. Moreover, subdivision schemes canbe split into refinement operators and smoothing op-erators, leading to a so-called factored approach [36].The other method is to apply each stencil (represent-ing both operators) at once. Our main implementa-tion uses the rewriting method by directly applyingthe stencils. In addition, due to some overlap withanother project, the schemes were also implementedusing stellar operators (see [2] for the accompanyingreport). It uses the modifying method, and by thenature of the stellar operators [35], the splitting ap-proach.

Next up was the implementation of an importscript for .OBJ files, exported from Blender.Since .OBJ uses a face-vertex format to store amesh, we had to write a script to convert it to ahalf-edge mesh. Additionally we added an export

script for .OBJ to be able to render our results inBlender.

Over time the implementation in Matlab wasextended to an interactive environment, in order tobe able to indicate the sharpness of both verticesand edges. We also took a look at other subdivisionschemes (e.g. Doo-Sabin, Loop, Kobbelt’s

√3 and

Velho’s 4−8 scheme) which are not discussed in thisreport. Functionality for reverse Catmull-Clark wasadded later.

An interesting topic for future research is study-ing the difference between CPU and GPU im-plementations (either using OpenGL, OpenCL orCUDA).

8. Conclusion

After a succinct discussion of subdivisionschemes, the limit surface and IGA, the exampledemonstrated that the two topics go together quitewell. Only a more dynamic and generally applicableworkflow can provide a decisive answer, but theseearly signs certainly look promising. In the conclu-sion of my MSc thesis I hope to give a somewhatmore extensive review.

Intriguing future research topics include the ex-tension of the discussed concepts to general volu-metric subdivision methods (e.g. box spline solids).Taking a detailed look at convergence rates andcomparing them to other numerical methods mightalso be of interest.

Additionally, implementing adaptive mesh re-finement should be very useful. Triangular schemesare probably the most suitable for this purpose, al-though the extension to volumes might be slightlyproblematic (subdivision of a tetrahedron is notcompletely trivial).

Another option would be to implement an al-ternative method to Stam’s approach like the onepresented in [32] and subsequently compare the twomethods (e.g. in the extend of efficiency). An ad-vantage of this alternative method is that it worksfor any subdivision scheme, not just for polynomialones. A possible drawback might be the fact thatthe limit surface can only be evaluated at points ona predefined grid.

Some possible future directions with regard tothe implementation were already mentioned in theprevious section. In addition it might be practicalto make use of the Python interface of Blender,which provides the possibility to access the meshdirectly. The recent switch to BMesh makes thisidea even more appealing.

Acknowledgements

First and foremost I would like to thank my su-pervisor Clemens Verhoosel for his close involve-ment in this project. Additionally my gratitude

goes to Gokturk Kuru for the many fruitful dis-cussions we had. Finally a word of thanks to theInkscape and Blender communities and the peo-ple at Math.StackExchange!

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