Subdivision Exterior Calculus for Geometry Processing Fernando de Goes Mathieu Desbrun Mark Meyer Tony DeRose Pixar Animation Studios Caltech Pixar Animation Studios Pixar Animation Studios Pixar Technical Memo #16-01 April, 2016 DEC SEC Figure 1: Subdivision Exterior Calculus (SEC). We introduce a new technique to perform geometry processing applications on subdivision surfaces by extending Discrete Exterior Calculus (DEC) from the polygonal to the subdivision setting. With the preassemble of a few operators on the control mesh, SEC outperforms DEC in terms of numerics with only minor computational overhead. For instance, while the spectral conformal parameterization [Mullen et al. 2008] of the control mesh of the mannequin head (left) results in large quasi-conformal distortion (mean = 1:784, max = 9:4) after subdivision (middle), simply substituting our SEC operators for the original DEC operators significantly reduces distortion (mean=1:005, max=3:0) (right). Parameterizations, shown at level 1 for clarity, exhibit substantial differences. Abstract Among the various polygonal mesh techniques, Discrete Exterior Calculus (DEC) [Desbrun et al. 2008] is a coordinate-free formal- This paper introduces a new computational method to solve differ- ism for solving scalar and vector valued differential equations. In ential equations on subdivision surfaces. Our approach adapts the particular, it reproduces, rather than merely approximates, essen- numerical framework of Discrete Exterior Calculus (DEC) from tial properties of the differential setting such as Stokes’ theorem. the polygonal to the subdivision setting by exploiting the refin- Given that the control mesh of a subdivision surface is a polygonal ability of subdivision basis functions. The resulting Subdivision mesh, applying existing DEC methods directly to the control mesh Exterior Calculus (SEC) provides significant improvements in ac- may seem tempting. However, this approach ignores the geometry curacy compared to existing polygonal techniques, while offering of the limit surface, thus introducing a significant loss of accuracy exact finite-dimensional analogs of continuum structural identities in the discretization process (Fig.1). A customary workaround is such as Stokes’ theorem and Helmholtz-Hodge decomposition. We to perform computations on a denser polygonal mesh generated by demonstrate the versatility and efficiency of SEC on common ge- a finite number of subdivision steps to improve accuracy, but this ometry processing tasks including parameterization, geodesic dis- solution dramatically increases the number of degrees of freedom, tance computation, and vector field design. computation time, and memory footprint. Furthermore, the mis- match between the geometry representation and the basis functions Keywords: Subdivision surfaces, discrete exterior calculus, dis- used for solving the differential equations weakens the convergence crete differential geometry, geometry processing. rate of the numerics. This issue is broadly recognized in the finite Concepts: •Mathematics of computing ! Discretization; Com- element literature and has led to the development of isoparameteric putations in finite fields; and isogeometric methods [Hughes et al. 2005]. Yet most of these recent techniques are neither adapted to subdivision surfaces nor do they preserve structural properties of the smooth theory. 1 Introduction To overcome these limitations, we introduce in this paper a sys- tematic derivation of discrete differential operators for subdivision Subdivision surfaces have become the de facto geometry represen- surfaces through a technique we call Subdivision Exterior Calculus tation in the entertainment industry [Loop et al. 2013]. By recur- (SEC). This variant of DEC can be easily retrofitted to existing im- sively refining a control mesh using linear combinations of ver- plementations of a variety of geometry processing algorithms, and tices, subdivision provides an effective tool for modeling, anima- yields an effective numerical framework that retains core proper- tion, and rendering of smooth surfaces of arbitrary topology [Zorin ties of smooth differential operators defined on the limit subdivi- and Schroder¨ 2000; Warren and Weimer 2001]. In spite of this sion surface, while offering the accuracy expected from the use of prominence, little attention has been paid to numerically solving higher-order basis functions. differential equations on subdivision surfaces. This is in sharp con- trast to a large body of work in geometry processing that developed discrete differential operators for polygonal meshes [Botsch et al. 1.1 Related Work 2010] serving as the foundations for several applications ranging from parameterization to fluid simulation [Crane et al. 2013a]. Our contributions relate to a number of research efforts in computer graphics, computational physics, and discrete differential geometry, which all seek numerical solutions of partial differential equations. Finite element methods. The finite element methodology has long tion were altered from regular grids to Catmull-Clark subdivision been regarded as the foundational tool to numerically solve varia- surfaces. In [Lounsbery et al. 1997], the authors showed how tional problems. Using test and trial function spaces of finite di- wavelets could be constructed from subdivision surfaces. Grin- mensions, a differential equation is discretized by characterizing its spun et al. [2002] extended adaptive finite elements using sub- solution in a so-called weak, or integral, form. These computa- division basis functions for the simulation of thin shells, while tional methods are commonly based on a polygonal approximation Thomaszewski et al. [2006] computed bending forces on the limit of the domain geometry and basis functions over each polygonal el- surface of a Loop scheme. The work of Riffnaller-Schiefer et ement. Increased accuracy is obtained either through h-refinement, al. [2015] also simulated thin shells via NURBS-compatible sub- where the geometry approximation (and thus the solution space) division surfaces. These methods have used quadratures based on is refined using more polygons, or p-refinement, where the geo- exact evaluation [Stam 1998] at selected limit surface locations to metric discretization is kept intact but higher-order basis functions approximate differential operators, similar to the approach favored are used [Babuska and Suri 1994]. However, any mismatch be- in the IGA literature, but none offer a structure-preserving calcu- tween the basis functions describing the geometry and the ones lus that can be directly used for increasing accuracy of scalar and defining the solution space on the geometry hampers the analysis vector computations. and convergence of these approaches, an issue coined a “variational crime” [Strang and Fix 1973]. Smooth Whitney forms. Our work is closely related to the con- struction of Whitney forms through subdivision of vertices, edges, Isogeometric analysis. The concept of isogeometric analysis and faces introduced in [Wang et al. 2006]. However, the authors (IGA) was introduced in [Hughes et al. 2005] to mitigate this mis- exploited smooth differential forms for visualization purposes only, match in computer-aided design. Typically, a B-spline representa- reverting to DEC operators on the control mesh for solving dif- tion is used both for modeling the geometry of a physical domain ferential equations. Instead, we show that the refinability of these and for defining function spaces needed for the weak formulation higher-order Whitney forms can be leveraged to design accurate employed by the finite element methodology (see [Cirak et al. 2002] and structure-preserving discrete differential operators for subdivi- for an early extension to subdivision surfaces). A primary advan- sion surfaces. tage of IGA methods is that they are geometrically exact no matter how coarse the discretization is: the coarsest level of discretiza- 1.2 Contributions tion entirely defines the geometry, while refinements of the func- tion space can be performed to better capture the solution of a dif- In this paper, we adapt DEC, the coordinate-free language for cal- ferential equation without affecting the shape of the domain. How- culus on polygonal meshes, to subdivision surfaces. This is done by ever, exact computation of integrals appearing in weak formulations extending the work of Wang et al. [2006] that unified subdivision (such as stiffness or mass matrices) is not possible in general, so with the exterior derivative operator. Here we complete the pic- quadrature rules are invoked to approximate them. In particular, ture by unifying subdivision with the inner product of differential adaptive quadrature strategies are required to resolve the integrands forms. The resulting Subdivision Exterior Calculus (SEC) allows in regions of high curvature and near extraordinary vertices [Baren- for computations on the control mesh with significantly improved drecht 2013; Nguyen et al. 2014;J uttler¨ et al. 2016]. accuracy compared to DEC as it reduces discretization errors, while retaining core properties of the smooth theory. In particular, we Structure-preserving computing. Alongside the developments prove that our SEC-based differential operators satisfy the prop- mentioned above, computational approaches involving “compat- erties recognized in [Auchmann and Kurz 2006; Wardetzky et al. ible” or “mimetic” discretizations have been shown to improve 2007]. Finally, we apply the SEC framework to archetypal geome- numerics