Isogeometric analysis of minimal surfaces on the basis of extended Catmull--Clark ...lsec.cc.ac.cn/~chench/papers/CMAME_1.pdf · 2018. 5. 11. · Catmull–Clark subdivision Qing

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Comput. Methods Appl. Mech. Engrg. 337 (2018) 128–149www.elsevier.com/locate/cma

Isogeometric analysis of minimal surfaces on the basis of extendedCatmull–Clark subdivision

Qing Pana,∗, Timon Rabczukb, Chong Chenc, Guoliang Xuc, Kejia Pand

a Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), Hunan NormalUniversity, Changsha, 410081, China

b Institute of Structural Mechanics, Bauhaus Universität-Weimar, Weimar, 99423, Germanyc LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

d School of Mathematics and Statistics, Central South University, Changsha, 410083, China

Received 12 June 2017; received in revised form 6 January 2018; accepted 27 March 2018Available online 3 April 2018

Highlights

• Demonstrate the discretization workflow of isogeometric analysis based on extended Catmull–Clark subdivision (IGA-CC)approach which can be naturally integrated into the frame work of standard finite element method (FEM).

• Establish the inverse inequalities and the approximation properties for the limit form of extended Catmull–Clark subdivisionwhich are similar to those for FEM.

• Present the detailed convergence study for the minimal surface models discretized by the fashion of IGA-CC approach.• Numerical tests are carried out with comparison to classical FEM based on the linear elements.

Abstract

We study the application of Isogeometric Analysis based on extended Catmull–Clark subdivision approach for the minimalsurface models on planar domains. Subdivision approaches are compatible with NURBS as the standard of CAD systems whichare capable of the refinability of B-spline techniques. The exactness of the physical domain of interest is fixed patchwise by thecoarsest quadrilateral mesh and maintained through refinement. By performing extended Catmull–Clark subdivision, the controlmesh can be repeatedly refined, and the geometry is described as an infinite set of bicubic splines while maintaining its originalexactness. The finite element space is spanned by the limit form of extended Catmull–Clark subdivision, which possesses C1

smoothness and the flexibility of mesh topology. In this work we establish the approximation properties and inverse inequalities forthis space which are similar to the ones of classical finite elements. The approximation estimates for the minimal surface models

∗ Corresponding author.E-mail address: [email protected] (Q. Pan).

https://doi.org/10.1016/j.cma.2018.03.0400045-7825/ c⃝ 2018 Elsevier B.V. All rights reserved.

http://crossmark.crossref.org/dialog/?doi=10.1016/j.cma.2018.03.040&domain=pdfhttp://www.elsevier.com/locate/cmahttps://doi.org/10.1016/j.cma.2018.03.040http://www.elsevier.com/locate/cmamailto:[email protected]://doi.org/10.1016/j.cma.2018.03.040

Q. Pan et al. / Comput. Methods Appl. Mech. Engrg. 337 (2018) 128–149 129

are developed with the aid of the H1-norm convergence property of its linearization model. The performance of numerical tests isconsistent with the theoretical results. We also compare these numerical calculations with classical linear finite element methods.c⃝ 2018 Elsevier B.V. All rights reserved.

Keywords: Isogeometric analysis; Extended Catmull–Clark subdivision; Error estimation; Minimal surfaces

1. Introduction

Today the system of Computer Aided Design (CAD) mostly uses the boundary structure (B-rep) to describe thegeometries by spline basis functions and often Non-Uniform Rational B-Splines (NURBS) of different polynomialorder. Numerical simulations based on finite element analysis (FEA) are performed on the desired objects which aremostly represented by Lagrange polynomials. The incompatible mathematical representations make the communica-tion between CAD and numerical simulations based on FEA very challenging. This challenge today is addressed byexpensive and time-consuming human intervention.

Isogeometric analysis (IGA), introduced by Hughes et al. [1,2] aims to bridge the gap between CAD and FEA.IGA uses NURBS instead of polynomials to represent the geometry and construct an approximate numerical solutionof a finite element discretization. Significantly this avoids a reapproximation of the geometry. Moreover, we canuse h-refinement by knot insertion, and p-refinement by order elevation to improve the simulation accuracy withoutchanging the geometry. Significant effort was devoted to the development of splines that allows local refinement. Oneof the most frequently used approaches in CAD is based on T-Splines [3,4], which however lead to flat. Unstructuredmeshes are difficult to refine while insuring linear independence, particularly in 3D. From the analytical point of view,hierarchical bases are preferred. Several constructions of such spaces have been proposed, the most common of whichare (Truncated) Hierarchical B-Splines (THB) [5,6], Locally Refined (LR)-Splines [7,8] and Polynomial/RationalSplines over Hierarchical T-Meshes (PHT/RHT splines) [9–13]. Different from the other constructions, the PHT/RHTsplines involve refinement at element level (rather than splitting the basis functions as in THB and LR splines), whileusing a quad/oct-tree structure for the resulting mesh at all refinement levels. The linear independence of the basis isalso guaranteed by the construction for PHT splines, at the cost of lower continuity (maximum C1 for cubic splinesinstead of C2 for the other splines). A framework of computation is reused in IGA on a set of three-dimensionalmodels with similar semantic features in [14].

Surface subdivision provides a simple and efficient recursive refinement to construct smooth surfaces from arbitrarymeshes [15–17]. Many subdivision technologies have been widely adopted in computer graphics applications forcomplexity of the models. Constructing surfaces through subdivision elegantly addresses some issues faced bycomputer graphics and CAD practitioners. More importantly, they need to handle control meshes of arbitrary topology,while maintaining surface smoothness and visual quality automatically. Subdivision surfaces easily admit multi-resolution extensions, thus enabling efficient hierarchical representations of complex surfaces. The most popularsubdivision schemes extend splines, thus maintaining continuity with previously used representations and inheritingsome of the appealing qualities of splines. Another important advantage of subdivision surfaces is that simple localmodifications of subdivision rules make it possible to introduce surface features of many different types.

Since subdivision algorithms can be used to define basis on arbitrary mesh domains, they become a naturalcandidate for higher-order finite element calculations in engineering applications, such as shell problems [18]. Naturalrefinement structure of subdivision surfaces leads to adaptive hierarchal finite element constructions [19]. Subdivision-based mesh generation for FEA has been explored in [20]. Mixed finite element methods based on subdivisionwere used to construct high-order smooth surfaces with specified boundary conditions [21]. Truncated hierarchicalCatmull–Clark subdivision [22] was developed to support local refinement for arbitrary topology mesh. The useof subdivision surfaces as a common foundation for modeling, simulation, and design in a unified framework wasproposed in [23]. A bound on the distance between a Catmull–Clark subdivision surface patch and its limit facein terms of the maximum norm of the second order differences was derived in [24]. A framework of realizing theintegration of CAD and boundary element analysis was presented based on subdivision methods [25,26].

Subdivision surfaces are compatible with NURBS as the standard of CAD systems which are capable of therefinability of B-spline techniques. There recently have been a few works on the application of subdivision methods

130 Q. Pan et al. / Comput. Methods Appl. Mech. Engrg. 337 (2018) 128–149

in IGA. Volumetric IGA based on Catmull–Clark solids was investigated in [27]. For the IGA methods over complexphysical domain, Powell–Sabin splines were used as IGA tools for advection–diffusion–reaction problems [28]. Thebivariate splines in the rational Bernstein–Bézier form over the triangulation were applied in [29]. A reproducingkernel triangular B-spline-based finite element method was proposed to solve PDEs [30]. Collocated isogeometricboundary element methods and unstructured analysis-suitable T-spline surfaces was coupled for linear elastostaticproblems in [31]. A new generalized surface and IGA elements have the vertices of the irregular quad mesh throughcomplementing bi-3 splines by bi-4 splines near irregularities in the mesh layout was presented in [32]. A frameworkfor geometric design and IGA on unstructured quadrilateral meshes was proposed in [33]. A new type of Hermitebases for bicubic spline defined over a rectangular mesh with arbitrary topology was investigated in the framework ofIGA [34] .

Minimal surfaces have several desirable properties. Firstly, minimal surfaces have the least surface area, whichmakes them to be widely used in large scale and light roof constructions. Secondly, minimal surfaces have separableproperty, i.e., any sub-patch, no matter how small, cut from a minimal surface still has the least area of all surfacesub-patches with the same boundary. Thirdly, minimal surfaces have balanced surface tension, which stabilizes thewhole construction since the tension is in equilibrium at each point on a roof, as on a soap film. Finally, there are noumbilicus points on a minimal surface, hence no water could stay on the minimal surface roof. Architecture inspiredfrom minimal surfaces embodies the union of economy and beauty. Scientists and engineers have anticipated thenanotechnology applications of minimal surfaces in areas of molecular engineering and materials science.

In this paper we study the isogeometric analysis based on extended Catmull–Clark subdivision (IGA-CC) approachfor the minimal surface models on planar domains. The exact geometry is fixed at the coarsest level of the quadrilateraldiscretization with any topological structure, which is thought of as the initial control mesh of the subdivision. Byperforming extended Catmull–Clark subdivision, the mesh can be refined while maintaining the original exactnessof the geometry. The solutions of equations and the geometry share the same bicubic splines as the basis functions.It means that the analysis we perform on the actual geometry is not shared by standard finite element method(FEM). We need introduce our former work [35] where we established the approximation properties for the extendedCatmull–Clark surface subdivision function, and performed three Poisson’s equations with the Dirichlet boundarycondition as numerical tests. It should be noted that the interpolation error estimation for the limit function spaceof the extended Catmull–Clark subdivision derived in [35] is significant for the starting of the error analysis aboutthe minimal surface models. Moreover, the adaptive numerical methods of IGA-CC and some related optimizationtechnique will be also used in this paper.

The main contributions of this paper include

1. Establish the inverse inequalities and the approximation properties for the limit form of extended Catmull–Clarksubdivision which are similar to those for FEM.

2. Present the detailed convergence study for the minimal surface models discretized by the fashion of IGA-CCapproach.

3. Numerical tests are carried out with comparison to classical FEM based on the linear elements (FEM-Linear).

This paper is organized as follows: Section 2 describes the function space defined by the limit form of extendedCatmull–Clark subdivision in which inverse inequalities and approximation properties are established. Section 3presents the detailed convergence study for the minimal surface models discretized by the fashion of IGA-CCapproach. Section 4 gives the numerical computation by means of classical Newton method. In Section 5 we performthree numerical examples to test our theoretical results, and all numerical examples are carried out by comparisonwith FEM-Linear. The paper is finished by a short conclusion in Section 6.

2. Approximation properties of extended Catmull–Clark subdivision function space

Throughout the analysis, we use the classical Sobolev spaces W k,p(Ω ), for k a positive integer, and 1 ≤ p ≤ ∞,endowed with the usual norm ∥ ·∥k,p and seminorm |·|k,p. For the classical Hilbert spaces H k(Ω ), ∥ ·∥k and |·|k denotetheir norms and seminorms respectively. Let Ω ⊂ R2 with x = [x, y]T ∈ Ω be the physical domain of interest withthe boundary ∂Ω . The spaces of functions on Ω with k-order continuous derivatives is denoted as Ck(Ω ).


Fig. 1. (a) A regular patch over the shaded quadrilateral with its neighboring 16 control vertices. (b) Local parametric transformation between twoadjacent patches τ and τ ′.

2.1. Extended Catmull–Clark subdivision

Subdivision schemes can generate smooth surfaces via a limit procedure of an iterative refinement process startingfrom an initial control mesh of the limit surface. Several schemes of subdivision for generating smooth surfaces havebeen proposed. Subdivision schemes where the vertex positions of the coarse mesh are fixed, and only the newlyadded vertex positions need to be computed are named as interpolation (see [36,37]), while others are approximation(see [15,38]).

The original Catmull–Clark subdivision scheme was designed to generalize uniform B-spline knot insertion tomeshes with arbitrary topology which is applicable only for closed surfaces. Its extension [39] supplements thesubdivision rules near boundaries, so it can overcome some problems, such as lack of smoothness at extraordinaryboundary vertices and folds near concave corners, and allow the generation of surfaces with prescribed normals bothon the boundaries and the interior sharp edges. The control vertices of the refined meshes are generated from thecontrol vertices of the previous step by a portfolio of weight coefficients. Finally, this sequence of meshes convergesto a limit surface composed of an unlimited number of surface patches.

Each quadrilateral of the control mesh, regarded as the parametric domain, corresponds to a quadrilateral patchof the surface. If all control vertices of the patch have the valence 6 and none of its two-ring neighbor vertices is aboundary vertex, the resulting surface patch is called regular. It can be exactly described by a bicubic B-spline with16 control vertices xi :

x(ξ, η) =16∑

i=1

Bi (ξ, η)xi , (2.1)

where (ξ, η) are the barycentric coordinates of the unit square T = {[ξ, η]T ∈ R2 : 0 ≤ ξ ≤ 1, 0 ≤ η ≤ 1} (seeFig. 1(a)). If a patch is irregular, i.e., at least one of its control vertices has the valence other than 6 and none of itstwo-ring neighbor vertices is a boundary vertex, the resulting surface patch is not a bicubic B-spline. For the evaluationof irregular patches, we use the fast scheme proposed by Stam [16]. In this strategy the mesh needs to be subdividedrepeatedly until the parameter values of interest are interior to a regular patch.

2.2. Finite element function space

The IGA framework adopts the uniform representation for the geometric computational domain and the numericalsimulation. In this paper, the generalized bicubic B-splines are utilized for geometrical domain modeling and theformulation of isoparametric finite elements, which can be suitable for quadrilateral meshes of arbitrary topology andany shaped boundaries.

Let us introduce some notations which will be used in the following representation. Denote Ω as the limit surfaceof extended Catmull–Clark subdivision. We describe Ω with an initial control mesh M0 and let x0i be its i th controlvertex. The subsequent finer meshes Mk, k = 1, 2, . . . , can be achieved through repeatedly applying extendedCatmull–Clark subdivision where we denote xki be the i th control vertex. The limit of the subdivision process generates


a smooth surface which converges at extraordinary vertices. The limit position of each vertex can be found explicitly,which is described as Lemma 2.1 (see [39] for details). We use Ωh to denote the discretized representation of thelimit form Ω where we denote x̂i as its i th control vertex and h is the maximal edge length. The discretized formΩh =

⋃iα=1τα , τ̊α

⋂τ̊β = ∅ for α ̸= β, where τ̊α is the interior of the patch τα . The domain of each patch τα on

the discretization Ωh can always be locally represented as an explicit bicubic B-spline (2.1). The boundaries of thedomain Ω are represented as the cubic B-spline curves which are preserved as the subdivision proceeds. It meansthat the given boundary curves are interpolated. Therefore Catmull–Clark subdivision elements can exactly representgeometries in the same way which is consistent with the concept of isogeometric strategy.

Lemma 2.1. Let xki be a vertex of the control mesh Mk with the valence n. Mark its 1-ring adjacent edgepoints withsubscript e and 1-ring facepoints with subscript f , then all these vertices converge to a single position

x̂i :=n

n + 5xki +

4n(n + 5)

n∑j=1

xke j +1

n(n + 5)

n∑j=1

xkf j , k = 0, 1, 2 · · · , (2.2)

as the subdivision step k goes to infinity.

It means that we can evaluate the limit position of the surface at any finite subdivision level k and at any vertexxki ∈ Ω k, k = 1, 2, . . . , by averaging the vertex and its neighbors according to the subdivision schemes.

Next let us describe the behavior of the tangent plane at extraordinary vertex x00 with the valence n. Its 1-ringadjacent edgepoints are contained in the set E0 = {E01, . . . , E

0n} where E0jmodn shares an edge with E

0( j+1)modn . In this

way, E0 is cyclically ordered. Its 1-ring facepoints are contained in the set F0 = {F01, . . . , F0n} where F0jmodn and

F0( j+1)modn locate two adjacent faces respectively. In this way, F0 is also cyclically ordered. The surface has a well

defined tangent plane at x00. An explicit formulation of the plane is given as Lemma 2.2.

Lemma 2.2. Let T be a periodic function whose i th component is the vector from x00 to E0i ,

Ti =

⎛⎝ n∑j=1

l1e j E0j

⎞⎠ r1ei +⎛⎝ n∑

j=1

l2e j E0j

⎞⎠ r2ei +⎛⎝ n∑

j=1

l1f j F0j

⎞⎠ r1fi +⎛⎝ n∑

j=1

l2f j F0j

⎞⎠ r2fi , (2.3)where the coefficients

l1e j = 4 sin jθn, l2e j = 4 cos jθn, r

1ei =

1σ

sin iθn, r2ei =1σ

cos iθn,

l1f j =sin jθn + sin( j + 1)θn

4λ − 1, l2f j =

cos jθn + cos( j + 1)θn4λ − 1

,

r1fi =sin iθn + sin(i + 1)θn

σ (4λ − 1), r2fi =

cosiθn + cos(i + 1)θnσ (4λ − 1)

,

and λ = 5/16 + 1/16(cos θn + cos θn/2√

9 + cos 2θn), σ = n(

2 + 1+cosθn(4λ−1)2

), with θn = 2π/n.

Since the coefficients r1ei , r2ei , r

1fi

and r2fi are constants, the vectors Ti lie in a plane. Note that T is a function of E0

and F0 which does not depend on x00. A surface normal at any extraordinary vertex can be found by evaluating (2.3)at any j and j + 1, and taking the cross product of the resulting vectors. The well defined curvature functions exist atordinary vertices of the mesh, however the same second order effects at extraordinary vertices cannot be assured.

We propose to use extended Catmull–Clark subdivision functions as the basis functions of IGA-CC approach. Thefinite function space is defined by the limit of Catmull–Clark subdivision which is denoted as Vh(Ω ). We use it fordescribing the computation domain and performing the numerical simulation to arrive at a unified discretization ofour problem. For each control vertex x̂i , i = 1, . . . , m, including boundary control vertices, of the control mesh Ωh ,we associate it with a basis function φi , where φi is defined by the limit of the extended Catmull–Clark subdivisionfor the zero control values everywhere except at x̂i where it is one. Hence the support of φi is local and it covers the2-ring neighborhood of vertex x̂i . The control polygon Ωh , as a piecewise linear surface, is served as the definitiondomain of the basis function φi . The mapping from Ωh to φi is defined by a dual subdivision process. More precisely,when the extended Catmull–Clark subdivision scheme is applied to the control function values recursively, the linear


subdivision scheme (each quadrilateral is partitioned into four equal-sized sub-quadrilaterals) is applied to the controlmesh correspondingly. The limit of the former is φi and that of later is Ωh itself. The basis functions share someproperties with the well known box spline basis. These properties are important in our finite element method. Let usdescribe them as follows.

1. Positivity. The weights of the extended subdivision rules are positive. Hence the basis function φi is nonnegativeeverywhere and positive around x̂i .

2. Locality. It is known that the limit value at a control vertex is a linear combination of the one-ring neighborvalues. Hence, the limit value is zero at a control vertex if the control values on the one-ring neighbor controlvertices are zeros. Therefore, the support of the basis function is within the two-ring neighborhood.

3. Partition of Unity. Since all the subdivision rules have the properties that the weights are summed to one.Therefore, if we choose all the control values as one, the control values after one subdivision step are still one.This implies that

∑mi=1φi = 1. This property is called partition of unity.

4. Interpolatory Properties at the Boundary. The extended subdivision rules on the boundary do not involve theinterior control points. Hence the basis functions for the interior control points are zero at the boundary. Thismeans that the given boundary curves are interpolated.

5. Linear Independency. As a set of basis functions, {φi }mi=1 must be linearly independent. This fact can be derivedfrom the result on the solvability of the following interpolation: For the given function values {ui }m1 , find thecontrol function values {vi }m1 such that

m∑j=1

v jφ j (x̂i ) = ui , i = 1, . . . , m, (2.4)

where ui = u(x̂i ) is the i th interpolation function value, v j is the j th control function value, φ j is the j th basisfunction. The interpolation problem (2.4) always has a unique solution (see [35] for the proof).

Based on the existence and uniqueness of the interpolation problem (2.4), we derived the following interpolationerror estimate (see [35] for the proof).

Lemma 2.3. There exists an interpolation function u I ∈ Vh(Ω ) such that

∥u − u I ∥s ≤ Ch2−s∥u∥2, s = 0, 1, ∀u ∈ H 20 (Ω ), (2.5)

where the constant C is independent of h.

2.3. Inverse inequalities for the limit form of Catmull–Clark subdivision

In this section we will prove the general inverse inequalities for the limit form of extended Catmull–Clarksubdivision which is described as the following Theorem 2.1.

Theorem 2.1. Suppose Ωh be the discretized representation of the limit form Ω of the extended Catmull–Clarksubdivision where we denote h is the maximal edge length. Denote Vh(Ω ) as the finite function space defined by thelimit of Catmull–Clark subdivision. Let s, t be integers with s ≥ t , and p, q be integers with 1 ≤ p, q ≤ ∞. Wehave the inverse inequalities on the patch τ ∈ Ωh

|u|s,q,τ ≤ Cht−s+2(1/q−1/p)τ |u|t,p,τ , u ∈ Vh(Ω ). (2.6)

Suppose the mesh Ωh is quasi-uniform, we further have the inverse inequalities on the domain Ωh

(∑τ∈Ωh

|u|qs,q,τ )1q ≤

⎧⎪⎪⎨⎪⎪⎩Cht−s+2(1/q−1/p)(

∑τ∈Ωh

|u|pt,p,τ )1p , q > p,

Cht−s(∑τ∈Ωh

|u|pt,p,τ )1p , q ≤ p,

(2.7)

where the constant C is independent of p, q, τ and h.


Proof. Notice that the limit position of each vertex on the mesh Ωh can be found explicitly in Lemma 2.1, and weproved the existence, uniqueness and solvability for its corresponding interpolant in our former work [35]. We willuse the classical parametric transformation method. Let M be a rectangular parametric domain of points. We assumethat G is smooth invertible such that

Ω = G(M), M = G−1(Ω ).

It provides a parameterization for the limit form Ω where each patch τ ∈ Ω is mapped onto a unit square T ∈ M.We denote by hτ the diameter of the element τ ∈ Ωh , for u ∈ W s,p(Ω ), û = u ◦ G−1. By Lemma 3 in [40], we can

obtain

|u|s,q,τ ≤ C |det ∇G|1/q0,∞,T |∇G

−1|s0,∞,τ |û|s,q,T ,

and

|û|t,p,T ≤ C |det ∇G−1|1/p0,∞,τ |∇G|

t0,∞,T |u|t,p,τ ,

where a constant C is independent of p and q. Joining the above two bounds, it completes the proof of (2.6).Denote h = max{hτ |τ ∈ Ωh}, we obtain

(∑τ∈Ωh

|u|qs,q,τ )1/q

≤ Cht−s+2(1/q−1/p)(∑τ∈Ωh

|u|qt,p,τ )1/q .

When p < q , it follows

(∑τ∈Ωh

|u|qs,q,τ )1/q

≤ Cht−s+2(1/q−1/p)

⎛⎝(∑τ∈Ωh

|u|q−pt,p,τ )1/(q−p)

∑τ∈Ωh

|u|pt,p,τ

⎞⎠1/q≤ Cht−s+2(1/q−1/p)(

∑τ∈Ωh

|u|qt,p,τ )1/q .

When p ≥ q , with (1 − q/p) + q/p = 1, we have by the Hölder inequality

(∑τ∈Ωh

|u|qs,q,τ )1/q

≤ Cht−s+2(1/q−1/p)

⎡⎣(∑τ

1

)1−q/p⎛⎝∑τ∈Ωh

|u|q·p/qt,p,τ

⎞⎠q/p⎤⎦1/q≤ Cht−s+2(1/q−1/p) · Ch−2(1/q−1/p)(

∑τ∈Ωh

|u|qt,p,τ )1/q

≤ Cht−s(∑τ∈Ωh

|u|qt,p,τ )1/q .

Finally (2.7) is proved.

2.4. Approximation of the limit form of Catmull–Clark subdivision

We adopt the method of the orthogonal polynomial expansion to study the approximation properties of extendedCatmull–Clark subdivision. It is necessary to introduce the Legendre polynomial sequence L in the unit intervalT = [−1, 1],

L0 = 1, L1 = t, L2 = (3t2 − 1)/2, . . . , L j =1

2 j j !∂

jt (t

2− 1) j , t ∈ T,

where the orthogonal properties ⟨L i , L j ⟩ = 0 when i ̸= j , and ⟨L j , L j ⟩ = 22 j+1 . It is well known that j degreepolynomial L j is orthogonal to any j − 1 degree polynomial Pj−1. Integrate L polynomials to get another sequenceof L polynomials

L0 = 1, L1 = t, L2 = (t2 − 1)/2, . . . ,L j+1 =1

2 j j !∂

j−1t (t

2− 1) j , t ∈ T,

where the quasi-orthogonal properties ⟨Li ,L j ⟩ ̸= 0 when i − j = 0 or ±2, or else ⟨Li ,L j ⟩ = 0.


For any function f ∈ W k+1,p(T ), its orthogonal description can be described as the following Definition 2.1(see [41] for details).

Definition 2.1. For any function f ∈ W k+1,p(T ), T = [−1, 1], there exists a k degree polynomial

fk(t) =k∑

i=0

biLi (t), t ∈ T,

where the coefficients

b0 =f (1) + f (−1)

2, b1 =

f (1) − f (−1)2

, bi =2i − 1

2

∫ 1−1

f ′(t)L i−1(t)dt, i = 1, 2, . . . .

The local coordinate transformation for the limit form x̂(x̂, ŷ) of extended Catmull–Clark subdivision will bedescribed as follows. Let

x̂ = x̂(ξ, η), ŷ = ŷ(ξ, η), (ξ, η) ∈ T,

which is an invertible transformation from the physical domain Ω into the parametric domain T = [−1, 1] × [−1, 1],i.e., the determinant of Jacobian matrix |J | =

⏐⏐⏐ ∂(x̂,ŷ)∂(ξ,η) ⏐⏐⏐ ̸= 0. With Definition 2.1, the orthogonal expansion of theparametric transformation for x̂(ξ, η) can be written as an infinite series form (the same for ŷ(ξ, η))

x̂(ξ, η) =∞∑

i, j=0

bi jLi (ξ )L j (η),

where the coefficients bi j can be obtained by using the integration by parts with i times in ξ -direction and/or j timesin η-direction

b00 = (x̂1 + x̂2 + x̂3 + x̂4)/4,

bi0 = ci

∫ 1−1

(x̂ξ (ξ, 1) + x̂ξ (ξ, −1))L i−1(ξ )dξ, bi1 = ci

∫ 1−1

(x̂ξ (ξ, 1) − x̂ξ (ξ, −1))L i−1(ξ )dξ, i ≥ 1,

b0 j = c j

∫ 1−1

(x̂η(1, η) + x̂η(−1, η))L j−1(η)dη, b1 j = c j

∫ 1−1

(x̂η(1, η) − x̂η(−1, η))L j−1(η)dη, j ≥ 1,

bi j = ci j

∫∫T

x̂ξη(ξ, η)L i−1(ξ )L j−1(η)dξdη, i, j ≥ 1,

(2.8)

where ci , c j and ci j are constants. With the limit position of x̂ and its tangent derivatives x̂ξ and x̂η described inLemmas 2.1 and 2.2, the coefficients b00, bi0, bi1, b0 j and b1 j can be determined. Therefore, we have the approx-imation remainder

∑∞

i, j=1bi jLi (ξ )L j (η) for the limit form of extended Catmull–Clark subdivision. The followingapproximation order can be derived (see Fig. 1(b))

2xξ = 2b10 + O(h2) = x̂2 − x̂4 + O(h2) = l2 cos β + O(h2),

and

2xη = l1 cos α + O(h2), 2yξ = l2 sin β + O(h2), 2yη = l1 sin α + O(h2).

The method of the orthogonal polynomial expansion described above lets us study some approximation propertiesfor the limit form of extended Catmull–Clark subdivision, which will help us to further research the finite elementapproximation on the space supported by extended Catmull–Clark subdivision basis functions.

3. Convergence analysis for the minimal surface problem of IGA-CC approach

Let Du = (D1u, D2u) with D1u = ux , D2u = u y , and partial derivatives Dαu = Dαu

Dxα1 Dyα2 , α = α1 + α2,α, α1, α2 ≥ 0. Consider the minimal surface problem:{

−Div(ai (Du)) = 0, in Ω ,u = ϕ, on ∂Ω , (3.1)


where Div is the divergence operator, and ai (Du) = DDi u (1 + |Du|2)1/2 = (1 + |Du|2)−1/2 Di u, i = 1, 2. The weak

formulation of (3.1) is: find u ∈ H 1ϕ (Ω ) such that⎧⎪⎨⎪⎩A(u, v) =∫Ω

2∑i=1

ai (Du)Divdxdy = 0, ∀v ∈ H 10 (Ω ),

u = ϕ, on ∂Ω .

(3.2)

The finite element space Shϕ (Ω ) = {v|v ∈ Vh(Ω ), v|∂Ω = ϕ} where Vh(Ω ) is defined by the limit form of extendedCatmull–Clark subdivision as described above. Find the finite element solutions uh ∈ Shϕ (Ω ) of (3.2) which satisfy

A(uh, v) = 0, ∀v ∈ Sh0 (Ω ). (3.3)

Using (3.2) and (3.3), the error e = u − uh satisfies

A(u − uh, v) = 0, ∀v ∈ Sh0 (Ω ). (3.4)

The convergence results of the error u − uh are represented in Theorem 3.3 which will be proved in Section 3.2,however the proof is not trivial. Let us describe the main framework of the proof. Firstly consider the correspondinglinear elliptic problem for the minimal surface models. We will construct an orthogonal polynomial expansion u Iwhich is super close to its finite element solutions ũh based on the method of Section 2.4. Here we obtain theH 1-norm convergence property of ũh − u I as Theorem 3.1 which will be discussed in the following Section 3.1.Secondly, combining Theorem 3.1 with the interpolant estimate Lemma 2.3, we achieve the super convergence resultfor the finite element solutions uh of problem (3.3) approached by the finite element solutions ũh of its linear model(3.7), which is represented as Theorem 3.2 and discussed in Section 3.2. Finally, we can naturally get the final resultTheorem 3.3.

3.1. H 1-norm convergence property of the linear elliptic problem

Consider the linear elliptic problem with zero boundary condition{−Div(Du) = f, in Ω ,u = 0, on ∂Ω . (3.5)

Define the function space S0 = {u|u ∈ H 30 (Ω )}. The weak formulation of (3.5) is: find u ∈ S0 such that

Ã(u, v) =∫Ω

2∑i=1

Di u Divdxdy =∫Ω

f vdxdy, ∀v ∈ S0. (3.6)

The finite element approximation of (3.6) is: find ũh ∈ Sh0 (Ω ) such that

Ã(ũh, v) = ( f, v), ∀v ∈ Sh0 (Ω ). (3.7)

With (3.6) and (3.7), the error ẽ = u − ũh satisfies

Ã(u − ũh, v) = 0, ∀v ∈ Sh0 (Ω ). (3.8)

Theorem 3.1. Let u and ũh be the solutions of problems (3.6) and (3.7) respectively, and u I be the interpolantapproximation of u. There exists a constant C such that

∥ũh − u I ∥1 ≤ Ch2∥u∥3. (3.9)

The proof of Theorem 3.1 involves the analysis on the patch τ of the domain Ωh . With the chain rule uξ =ux xξ + u y yξ , uη = ux xη + u y yη, we have

∇u =[

uxu y

]= [J ]−1

[uξuη

], [J ]−1 =

1|J |

[yη −yξ

−xη xξ

].


The bilinear form Ã(u, v) of (3.6) is rewritten as

Ã(u, v) =∫Ω

(a11uξvξ + a12uξvη + a21uηvξ + a22uηvη)dξdη,

where the coefficients

a11 = (y2η + x2η)/|J |, a12 = a21 = (−yη yξ − xηxξ )/|J |, a22 = (y

2ξ + x

2ξ )/|J |.

Proof. Using the method of the orthogonal polynomial expansion in Section 2.4, we have

u =∞∑

i, j=0

bi jLi (ξ )L j (η).

With Lemmas 2.1–2.3, the remainder R = u − u I can be represented as

R = u − u I =∞∑

i, j=1

bi jLi (ξ )L j (η). (3.10)

Recalling (3.8), and decomposing the error ẽ = u − ũh = R − θ̃ with θ̃ = ũh − u I , we get

Ã(θ̃ , v) = Ã(R, v) =∑

τ

∫τ

(Rxvx + Ryvy)dxdy

=

∑τ

∫T

(a11 Rξvξ + a12 Rξvη + a21 Rηvξ + a22 Rηvη)dξdη

= I1 + I2 + I3 + I4.

(3.11)

Denote Pk as the set of any k degree polynomial. We firstly discuss the terms I1 and I4. Since vξ ∈ P0(ξ ) × P1(η),we know that L i (ξ )⊥vξ when i ≥ 1, and L j (η)⊥v when j > 3. Thus∫

TRξvξ dξdη =

3∑j=2

b1 j

∫T

L0(ξ )L j (η)vξ dξdη, (3.12)

where with 1 + j ≥ 3, the coefficient

|b1 j | ≤ c∫

T|∂3u|dξdη ≤ ch

∫τ

|D3u|dxdy ≤ ch∥u∥3,τ .

Then summing up all patches τ of the domain Ωh , there is

|I1| = |∑

τ

∫T

a11 Rξvξ dξdη| ≤ Ch2∥u∥3∥v∥1.

It is similar to obtain the estimate |I4| ≤ Ch2∥u∥3∥v∥1.Next discuss the remaining terms I2 and I3. Since vη ∈ P1(ξ ) × P0(η), we know that L j (η)⊥vη when j > 2, and

L′i (ξ ) = L i−1(ξ )⊥vη when i > 2. Thus∫T

a12 Rξvηdξdη =∫

Ta12b12L0(ξ )L2(η)vηdξdη +

∫T

a12b2L′2(ξ )vηdξdη, (3.13)

where

b2 = c∫ 1

−1uξ (ξ, η)L1(ξ )dξ.

Considering the first term of (3.13), we get∫T

a12b12L0(ξ )L2(η)vηdξdη ≤ ch2∥u∥3,τ∥v∥1,τ , (3.14)


for the inequality |b12| ≤ ch∫τ|D3u|dxdy. With regard to the second term of (3.13), by use of the integration by parts,

we have∫T

a12b2L′2(ξ )vηdξdη = −∫

Ta12b2L2(ξ )vξηdξdη

=

∫T

a12b′2L2(ξ )vξ dξdη −∫

∂Ta12b2L2(ξ )vξ dξ.

(3.15)

We obtain∫T

a12b′2L2(ξ )vξ dξdη ≤ ch2∥u∥3,τ∥v∥1,τ , (3.16)

for the inequality |b′2| ≤ c∫ 1−1|∂

2ξ ∂ηu(ξ, η)|dξ ≤ ch

2∥u∥3,τ . Considering the linear integration of (3.15), the

coefficients b2 and a12 on the common edge between two adjacent patches τ and τ ′ respectively have the saltus

[b2] = c∫

τ+τ ′[∂2ξ u(ξ, η)]dξ = ch∥u∥2,τ , [a12] = O(h),

which is necessary in the performance of patch merging. Considering v = 0 on the boundary ∂Ω , sum up all of thelinear integration inside the domain Ωh and on the boundary ∂Ωh⏐⏐⏐⏐⏐∑

τ

∫τ+τ ′

[a12b2]L2(ξ )vξ dξ

⏐⏐⏐⏐⏐ = ch2∥u∥3∥v∥1. (3.17)Based on formula (3.14), (3.16) and (3.17), summing up all patches τ of the domain Ωh , we obtain

|I2| = |∑

τ

∫T

a12 Rξvηdξdη| ≤ Ch2∥u∥3∥v∥1.

Similarly we get the estimate |I3| ≤ Ch2∥u∥3∥v∥1. Thus

| Ã(θ, v)| = | Ã(R, v)| ≤ |I1| + |I2| + |I3| + |I4| ≤ Ch2∥u∥3∥v∥1. (3.18)

We choose v = ũh − u I in (3.18), then the proof of this theorem is finished.

3.2. Convergence properties of the minimal surface problem

We represent the solvability of (3.2) as follows. Assume that there is an appropriate smooth isolated solutionu ∈ H 1ϕ (Ω ), and ∥u∥1,∞ = M is bounded. It means that there exists a domain

Bε(u) = {u′ ∈ W 1,∞(Ω ), u′|∂Ω = ϕ, ∥u − u′∥1,∞ < ε}, ε > 0,

which does not contain other solutions of (3.2).Introduce the following bilinear form

au(w, v) =∫Ω

2∑i, j=1

ai j (Du)D jwDivdxdy, (3.19)

where ai j (Du) =∂ai (Du)∂ D j u

, ai jk(Du) =∂2ai (Du)

∂ D j u∂ Dk u, i, j, k = 1, 2. Suppose for ∀ w ∈ Bε(u), there is a ν > 0 such that

the bilinear form

aw(v, v) ≥ ν ∥v∥21, v ∈ H10 (Ω ). (3.20)

Denote ut = v + t(u − v), t ∈ [0, 1], with u1 = u, u0 = v. By use of the Newton–Leibniz formula, we get for∀ u, v ∈ Bε(u),

A(u, u − v) − A(v, u − v) =∫Ω

∫ 10

2∑i, j=1

ai j (D(ut ))D j (u − v)Di (u − v)dtdxdy ≥ ν∥u − v∥21.


Denote the error e = u − uh , and let ut = uh + t(u − uh) = uh + te, t ∈ [0, 1]. Suppose u, uh ∈ Bε(u), thenut ∈ Bε(u). Recalling (3.4), we use the Taylor expansion to get

0 = A(u, v) − A(uh, v) =∫ 1

0

ddt

A(ut , v)

=

∫Ω

2∑i, j=1

ai j (Du)D j eDivdxdy +∫Ω

∫ 10

2∑i, j,k=1

ai jk(Dut )D j eDkeDivdtdxdy,(3.21)

then by the Hölder inequality

|au(e, v)| ≤ C ∥e∥21,4 ∥v∥1,2, ∀v ∈ Sh0 (Ω ). (3.22)

We introduce the quadratic inequality which is described as the following Lemma 3.4.

Lemma 3.4. If y ≥ 0 satisfies the quadratic inequality y ≤ a + by2, a > 0, b > 0, when 4ab ≤ 1, y continuouslyincreases from zero to the solution y ≤ 2a (see [41] for details).

By the aid of the auxiliary linear elliptic projector ũh ∈ Shϕ (Ω ) for the corresponding linear problem (3.5), weobtain the convergence result as Theorem 3.2.

Theorem 3.2. Let uh and ũh be the solutions of problems (3.3) and (3.7) respectively. There exists a constant C suchthat

∥ũh − uh∥1 ≤ Ch2. (3.23)

Proof. An auxiliary projector ũh ∈ Sh0 (Ω ) is constructed such that it satisfies

au(u − ũh, v) = 0, ∀v ∈ Sh0 (Ω ).

Decompose the error e = u − ũh + (ũh − uh) = ẽ + θ where θ = ũh − uh . Recalling (3.22), we get

|au(θ, v)| = |au(e, v)| ≤ C ∥e∥21,4 ∥v∥1,2, v ∈ Sh0 (Ω ). (3.24)

Choosing v = θ , then using (3.20) and Theorem 2.1, we have

∥θ∥1,4 ≤ ch−1/2∥θ∥1 ≤ ch−1/2 ∥e∥21,4 . (3.25)

Combining Lemma 2.3, Theorem 2.1 and Theorem 3.1 leads to

∥ẽ∥1,4 ≤ ∥u − u I ∥1,4 + ∥u I − ũh∥1,4 ≤ c1h∥u∥2,4 + c2h2−1/2∥u∥3.

Then we obtain

∥e∥1,4 ≤ ∥ẽ∥1,4 + ∥θ∥1,4 ≤ Ch + C2h−1/2 ∥e∥21,4 .

Consider Lemma 3.4 if h is appropriately small such that 4CC2h−1/2 < 1, it follows that ∥e∥1,4 ≤ 2C1h. Recalling(3.25), we finish the proof of Theorem 3.2.

The boundedness estimation is also achieved∥u − uh∥1,∞ ≤ ∥ẽ∥1,∞ + ∥θ∥1,∞

≤ C3h−1/2∥ẽ∥1,4 + C4h−1∥θ∥1 ≤ Ch < ε,

if h is appropriately small. It guarantees that uh ∈ Bε(u).We then have the H 1-norm and H 0-norm error estimates described as Theorem 3.3.

Theorem 3.3. Suppose Ω be the convex domain. Let u and ũh be the solutions of problems (3.2) and (3.3) respectively.We have

∥u − uh∥s = O(h2−s), s = 0, 1. (3.26)


Proof. Using Lemma 2.3 with ũh = u I , and Theorem 3.2, we easily get

∥u − uh∥1 ≤ ∥u − ũh∥1 + ∥ũh − uh∥1= ∥u − u I ∥1 + ∥ũh − uh∥1 = O(h).

Then we directly have the H 0-norm estimate using the Aubin–Niestche argument

∥u − uh∥0 = O(h2).

4. Discretization performance of IGA-CC approach

In this section we aim at finding the finite element solution uh ∈ Sh0 (Ω ) satisfying

A(uh, v) =∫Ω

(1√

1 + |Duh |2∇uh, ∇v

)dxdy = 0, v ∈ Sh0 (Ω ).

Suppose φi be a basis function in the space Sh0 (Ω ) which corresponds to the control vertex x̂i , i = 1, 2, . . . , m, of themesh Ωh . Assume that x̂1, x̂2, . . . , x̂n are the interior control vertices, and x̂n+1, x̂n+2, . . . , x̂m are the boundary controlvertices. Then the finite element solutions uh can be represented as

uh =n∑

j=1

φ j uhj +m∑

j=n+1

φ j uhj , (4.1)

where the values of uh are fixed at the boundary control vertices x̂n+1, x̂n+2, . . . , x̂m , and the ones are to be determinedat the interior control vertices x̂1, x̂2, . . . , x̂n . Choosing the test functions v = φi , i = 1, . . . , n, then we get a systemof nonlinear equations

gi (uh1, uh2, . . . , uhn) = A(uh, φi ) = 0, i = 1, 2, . . . , n.

Suppose we have known the approximate values U kh = (ukh1, u

kh2, . . . , u

khn) at the kth step, we can obtain the values

U k+1h = (uk+1h1 , u

k+1h2 , . . . , u

k+1hn ) at the (k + 1)-th step by using the classical Newton method

U k+1h = Ukh − (G

′(U kh ))−1G(U kh ), (4.2)

where G = [g1, g2, . . . , gn]T . Actually we change (4.2) into the following system{U k+1h = U

kh + ∆U

kh ,

K∆U kh = b,(4.3)

where ∆U kh = Uk+1h − U

kh . The coefficient matrix and the right-hand side of the second equation in (4.3) are

respectively

K = G ′(U kh ) = [g′

1(Ukh ), g

′

2(Ukh ), . . . , g

′

n(Ukh )]

T∈ Rn×n,

b = G(U kh ) = [g1(Ukh ), g2(U

kh ), . . . , gn(U

kh )]

T∈ Rn×1.

Substitute (4.1) into K and b, then the elements in K and b are defined as follows:

gi (U kh ) =∫Ω

m∑j=1

ukh j

⎛⎝(1 + ( m∑j=1

ukh j∂φ j

∂x)2 + (

m∑j=1

ukh j∂φ j

∂y)2)− 12

∇φ j , ∇φi

⎞⎠ dxdy,g′i (U

kh ) =

Dgi (U )DU

⏐⏐⏐⏐U=Ukh

=

∫Ω

m∑j=1

⎛⎝(1 + ( m∑j=1

ukh j∂φ j

∂x)2 + (

m∑j=1

ukh j∂φ j

∂y)2)− 12

∇φ j , ∇φi

⎞⎠ dxdy. (4.4)The discretization scheme of IGA-CC approach for the second equation of (4.3) is similar to the framework of

standard FEM. The elements gi and g′i in the formula (4.4) are related to Uk , therefore they need to be recomputed at

each step k. However we can precompute the related basis functions φi and their derivatives on each patch of the meshbecause they are independent of the values of uh . We describe the computation method in the following Section 4.1.

The integrations for computing the matrix elements are computed by a 12-point Gaussian quadrature rule. That is,each quadrilateral is subdivided into four sub-quadrilaterals and a 3-point Gaussian quadrature rule is employed on


Table 1The most subdivision times for four Gauss–Legendre knots of three types of patches.

gi (ξi , ηi ) Interior Sub-boundary Boundary

g1 (0.2113249, 0.2113249) 3 3 4g2 (0.2113249, 0.7886751) 1 1 3g3 (0.7886751, 0.2113249) 1 1 3g4 (0.7886751, 0.7886751) 1 1 2

The second column lists the parameter value (ξi , ηi ) of the Gaussian knot gi over the unit square. The third, fourth and fifth columns respectivelygive us the most subdivision times for the four Gaussian knots of the three types of patches.

each of the sub-quadrilaterals. The 3-point Gaussian quadrature rule has error bound O(h3), where h is the maximaledge length. The fast evaluation scheme by Stam [16] suitable for interior patches is considered for the implementationof our method. Since good initial values U 0h are necessary for the Newton method, here we achieve them via the methodin [42].

4.1. Precomputing the basis functions

The computation of Catmull–Clark subdivision basis functions and their derivatives is not intuitive because therequired two rings of neighbors around each patch have arbitrary topological structure, and the additional individualgeometric data are contained in the subdivision schemes around the boundaries. The basis functions correspondingto the interior control vertices are zero at the boundaries because the boundary rules do not involve the interiorcontrol vertices. We classify the control mesh into interior patches, sub-boundary and boundary patches. The patchescontaining boundary vertices are named as boundary patches, the ones adjacent to boundary patches are called sub-boundary patches, and all of the others are called interior patches. We use the following scheme to treat these threetypes of patches.

Interior patch. Apply Stam’s algorithm for this case (see [16]);Sub-boundary patch. Subdivide a sub-boundary patch once will result in four interior quadrilaterals which can

be evaluated using the above method for the interior patch;Boundary patch. Subdivide a boundary patch repeatedly till its sub-patches belong to the sub-boundary case, then

use the above method to evaluate them.With the above description, Stam’s fast evaluation scheme [16] is always implemented which is suitable for interior

patches with only one extraordinary vertex. Therefore, it is necessary to first subdivide once each patch of the initialmesh. The evaluation of basis functions over their support elements uses general Gaussian integration which onlyneeds a few subdivision steps in order to bring Gaussian quadrature knots into a bicubic B-spline patch. In ourimplementation, we need to estimate the subdivision times in advance for the parameter value (ξi , ηi ) of any Gaussianquadrature knot gi , then adaptively carry out the evaluation. In this work we consider four Gaussian integrationformula, the most subdivision times for the three types of patches after one necessary initial subdivision are shown inTable 1.

Note that, for a given control mesh, the number of actual control vertices does not increase although we implementonce initial Catmull–Clark subdivision for the efficient use of Stam’s fast evaluation schemes, and the number ofvalues to be solved stays the same. The integration over each initial patch is the sum of the values on all knots of itsfour sub-patches with the same number of Gauss–Legendre knots.

All basis functions and their derivatives for each patch are pre-computed and stored in a data structure beforesolving equations. For the case of interior patches, the valences of their four control vertices uniquely determine theassociated basis functions. We merge interior patches into several categories according to the list of valences of theircontrol vertices, so the patches with the same list of valences share the same set of basis functions. It greatly reducesour computation cost and storage. The remaining sub-boundary and boundary patches have their uniquely associatedbasis functions because their individual geometric information embodies the involved boundary subdivision rules.

5. Examples

In this section, we present three numerical experiments for the minimal surface models on planar geometricdomains. The numerical solution is operated on the limit representation of extended Catmull–Clark subdivision,


Fig. 2. (a) Scherk surface. (b) Helicoid surface. (c) Catenoid surface. Here the vertical coordinate represents the values of u, and the horizontalcoordinate plane represents the xy-plane.

therefore the integration evaluation of the Gauss–Legendre knots is conducted on the quadrilateral meshes of the limitform of the subdivision. Three minimal surface models considered here are Scherk, Helicoid and Catenoid minimalsurfaces (see [43]). Scherk minimal surface is represented as u = ln cos ycos x , Helicoid one is represented as u = arctan

yx ,

and Catenoid one is represented as u = ln(√

x2 + y2 +√

x2 + y2 − 1), which are plotted in Fig. 2(a), (b) and (c)respectively.

Consider the minimal surface models on three different planar geometries. The first geometry is a square

Ω1 := {(x, y)| |x | ≤ 1, |y| ≤ 1},

where the minimal surface model is a Scherk surface. The second one is an L-shape

Ω2 := {(x, y)| (1 ≤ x ≤ 3, 1 ≤ y ≤ 3) \ (2 < x ≤ 3, 2 < y ≤ 3)},

where the minimal surface model is a Helicoid surface. The third one is a circular disk with a central hole

Ω3 := {(x, y)| (√

x2 + y2 ≤ 4) \ (|x | < 1.2, |y| < 1.2)},

where the minimal surface model is a Catenoid surface.The three geometries Ω1,Ω2 and Ω3 are respectively demonstrated in Fig. 3, Fig. 4 and Fig. 5. For each of the three

geometries, four almost uniform meshes at four different density levels are constructed as shown from (a) to (b), (c)and (d) in the first row of the three figures. Once refinement is added from (a) to (b), (b) to (c) and (c) to (d) so thatthe number of quadrilaterals on the refined meshes increases four times and their sizes approximately decrease byhalf. To show that extended Catmull–Clark subdivision scheme does not require structured meshes and can supportthe same meshes with any topological structure as standard finite elements, the valences of the control vertices in thethree geometries are in the range of 3 to 7. In this section, we apply FEM-Linear to solve the same three examples,and compare their accuracy, convergence and computational cost.

5.1. Accuracy and convergence

In this section we compare the accuracy between IGA-CC and FEM-Linear approaches. In Figs. 3–5, the errordistribution u − uh is shown in the second row and the third row which correspond to the models with four densitylevels of the first row. The second row shows the results from FEM-Linear, and the third one shows the results fromIGA-CC. The error range for both methods decreases with the mesh refinement procedure going on. For the samecontrol mesh, the error span produced from FEM-Linear is bigger than that from IGA-CC, and the error fluctuationfrom the former is also bigger than the latter.

We compute H 0-norm errors ∥u − uh∥0 of both approaches and depict them in Figs. 6–8. It is obvious that theirH 0-norm errors decrease with mesh refinement proceeding. The error of IGA-CC is smaller than that of FEM-Linearfor the same models. Based on the numerical error comparison, we can observe that IGA-CC converges faster thanFEM-Linear. The figures also suggest that their convergence order is around order two.


Fig. 3. A square Ω1. (a), (b), (c) and (d) are four control meshes where one time refinement is implemented from (a) to (b), (b) to (c) and (c) to (d).The corresponding distribution of the error u − uh resulting from FEM-Linear and IGA-CC is respectively shown in (a′), (b′), (c′) and (d′) of thesecond row, and (a′′), (b′′), (c′′) and (d′′) of the third row.

5.2. Computational cost

In this section we compare the computational cost between IGA-CC and FEM-Linear. For the models (a), (b), (c)and (d) of Figs. 3–5, we list the corresponding time cost in Tables 2–4. The first column shows the number of controlvertices and patches of the meshes. The second and the third columns list the time cost (in seconds) of computing thebasis functions and their derivatives because they can be pre-computed and saved in a data structure. The computationof FEM-Linear for the same control meshes is faster because it is unnecessary for us to compute the derivatives ofthe linear basis functions. The time cost does not increase four times after each refinement step for IGA-CC strategy.As we mentioned in Section 4.1, most of interior patches share the same set of basis functions which depend only onthe valence number of their control vertices. With the mesh refinement going on, the increasing rate for the numberof interior patches is much faster than the other sub-boundary and boundary patches, then a large number of interiorpatches are merged into the same categories which reduces the computation expense.

The systems generated from IGA-CC and FEM-Linear are highly sparse. Hence a stable iterative method for theirsolutions is desirable. We adopt GMRES iterative solver for the second equation of (4.3) where the threshold valuefor controlling the iteration-stopping is 6.0 × 10−8. We list their time cost in the fourth and the fifth columns (inseconds) where the termination condition is specified as ∥U k+1 − U k∥∞ < δ, and δ is a given threshold value. Thecomputation time of FEM-Linear is faster than that of IGA-CC. We know that the number of non-zero elements ofthe linear system generated from IGA-CC is larger than that from FEM-linear. Moreover, we depict the H 0-norm


Fig. 4. An L-shape Ω2. (a), (b), (c) and (d) are four control meshes where one time refinement is implemented from (a) to (b), (b) to (c) and (c) to(d). The corresponding distribution of the error u − uh resulting from FEM-Linear and IGA-CC is respectively shown in (a′), (b′), (c′) and (d′) ofthe second row, and (a′′), (b′′), (c′′) and (d′′) of the third row.

Table 2Data of the examples in Fig. 3.

Vertices/patches Basis func.(s) Solving linear sys.(s)

FEM-linear IGA-CC FEM-linear IGA-CC

125/104 0.02 0.12 0.02 0.034457/416 0.05 0.22 0.08 0.201745/1664 0.10 0.56 0.42 0.746817/6656 0.24 1.02 1.62 3.56

errors versus the total time including computing basis functions, assembling and solving linear systems in Figs. 9, 10,11. The conclusion is that the method of IGA-CC has better accuracy with more time consuming than FEM-Linear.The proposed method is implemented by C++ in Linux system running on a PC with 2.4 GHz Q6600 Intel CPU anddouble precision arithmetic operation.

6. Conclusions

In this work, we have presented the discretization framework of IGA-CC approach taking the minimal surfaceproblems on planar domains as the models. We have also given the detailed convergence results for these models,where inverse inequalities for the limit form of extended Catmull–Clark subdivision and its approximation propertieswere established. Notably, these theoretical results are essential for other equation models based on IGA-CC approach.


Fig. 5. A circular disk with a central hole Ω3. (a), (b), (c) and (d) are four control meshes where one time refinement is implemented from (a) to(b), (b) to (c) and (c) to (d). The corresponding distribution of the error u − uh resulting from FEM-Linear and IGA-CC is respectively shown in(a′), (b′), (c′) and (d′) of the second row, and (a′′), (b′′), (c′′) and (d′′) of the third row.

Fig. 6. Domain Ω1. Comparison of the convergence rate of the errors versus the subdivision times between FEM-Linear and IGA-CC. Here thenumbers 0, 1, 2 and 3 on the x-axis correspond to the models of Fig. 3(a), (b), (c) and (d) respectively, and e on the y-axis is the H0-norm error.







97/75 0.01 0.09 0.014 0.016340/300 0.03 0.16 0.056 0.111285/1200 0.05 0.32 0.24 0.464969/4800 0.16 0.74 1.28 2.48

Furthermore, we have applied three different geometrical interests, and performed numerical tests which corroboratethe theoretical results. The numerical examples have been performed by comparison with the methods of standardFEM-Linear.





382/328 0.03 0.23 0.06 0.161420/1312 0.06 0.50 0.32 0.545464/5248 0.21 0.99 1.72 3.1621424/20992 0.83 1.96 3.88 9.15

Fig. 9. Domain Ω1. Comparison of the convergence rate of the errors versus the total time complexity between FEM-Linear and IGA-CC. Herex-axis represents the time cost (in seconds) and e on the y-axis is the H0-norm error. ∗ symbols correspond to the models of Fig. 3 (a), (b), (c) and(d) respectively.

Fig. 10. Domain Ω2. Comparison of the convergence rate of the errors versus the total time complexity between FEM-Linear and IGA-CC. Herex-axis represents the time cost (in seconds) and e on the y-axis is the H0-norm error. ∗ symbols correspond to the models of Fig. 4(a), (b), (c) and(d) respectively.

Acknowledgments

We would like to thank Prof. Chuanmiao Chen from the College of Mathematics and Computer Science atHunan Normal University, for his helpful discussions and comments. Qing Pan is supported by National NaturalScience Foundation of China (NSFC) (No. 11671130), Scientific Research Fund of Hunan Provincial EducationDepartment (No. 15A110) and Hunan Provincial Natural Science Foundation of China (No. 2018JJ2248). Chong


Fig. 11. Domain Ω3.Comparison of the convergence rate of the errors versus the total time complexity between FEM-Linear and IGA-CC. Herex-axis represents the time cost (in seconds) and e on the y-axis is the H0-norm error. ∗ symbols correspond to the models of Fig. 5(a), (b), (c) and(d) respectively.

Chen is supported by National Natural Science Foundation of China (NSFC) (No. 11301520). Kejia Pan is supportedby National Natural Science Foundation of China (NSFC) (No. 41474103), the Excellent Youth Foundation of HunanProvince of China (No. 2018JJ1042) and the Innovation-Driven Project of Central South Univeristy (No. 2018CX042).

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Isogeometric analysis of minimal surfaces on the basis of extended Catmull–Clark subdivisionIntroductionApproximation properties of extended Catmull–Clark subdivision function spaceExtended Catmull–Clark subdivisionFinite element function spaceInverse inequalities for the limit form of Catmull–Clark subdivisionApproximation of the limit form of Catmull–Clark subdivision

Convergence analysis for the minimal surface problem of IGA-CC approachH1-norm convergence property of the linear elliptic problemConvergence properties of the minimal surface problem

Discretization performance of IGA-CC approachPrecomputing the basis functions

ExamplesAccuracy and convergenceComputational cost

ConclusionsAcknowledgmentsReferences

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Isogeometric analysis of minimal surfaces on the basis of extended Catmull--Clark ...lsec.cc.ac.cn/~chench/papers/CMAME_1.pdf · 2018. 5. 11. · Catmull–Clark subdivision Qing