Discrete Differential Forms for Computational Modeling Mathieu Desbrun Eva Kanso∗ Yiying Tongy Applied Geometry Lab Caltechz 1 Motivation coordinate-independent nature of quantities written in tensorial no- tation: local and global invariants are hard to notice by just staring The emergence of computers as an essential tool in scientific re- at the indices. On the other hand, invariants are easily discovered search has shaken the very foundations of differential modeling. when expressed as differential forms by invoking either Stokes’ Indeed, the deeply-rooted abstraction of smoothness, or differentia- theorem, the Poincare´ lemma, or by applying exterior differentia- bility, seems to inherently clash with a computer’s ability of storing tion. Note also that the exterior derivative of differential forms— only finite sets of numbers. While there has been a series of com- the antisymmetric part of derivatives—is one of the most important putational techniques that proposed discretizations of differential parts of differentiation, since it is invariant under coordinate system equations, the geometric structures they are simulating are often change. In fact, Sharpe states in [Sharpe 1997] that every differ- lost in the process. ential equation may be expressed in term of the exterior derivative of differential forms. As a consequence, several recent initiatives 1.1 The Role of Geometry in Science have been aimed at formulating physical laws in terms of differen- tial forms. For recent work along these lines, the reader is invited Geometry is the study of space and of the properties of shapes in to refer to [Burke 1985; Abraham et al. 1988; Lovelock and Rund space. Dating back to Euclid, models of our surroundings have 1993; Flanders 1990; Morita 2001; Carroll 2003; Frankel 2004] for been formulated using simple, geometric descriptions, formalizing books offering a theoretical treatment of various physical theories apparent symmetries and experimental invariants. Consequently, using differential forms. geometry is at the foundation of many current physical theories: general relativity, electromagnetism (E&M), gauge theory as well 1.3 Differential vs. Discrete Modeling as solid and fluid mechanics all have strong underlying geometri- cal structures. Einstein’s theory for instance states that gravitational We have seen that a large amount of our scientific knowledge relies field strength is directly proportional to the curvature of space-time. on a deeply-rooted differential (i.e., smooth) comprehension of the In other words, the physics of relativity is directly modelled by the world. This abstraction of differentiability allows researchers to shape of our 4-dimensional world, just as the behavior of soap bub- model complex physical systems via concise equations. With the bles is modeled by their shapes. Differential geometry is thus, de sudden advent of the digital age, it was therefore only natural to facto, the mother tongue of numerous physical and mathematical resort to computations based on such differential equations. theories. However, since digital computers can only manipulate finite sets Unfortunately, the inherent geometric nature of such theories is of- of numbers, their capabilities seem to clash with the basic founda- ten obstructed by their formulation in vectorial or tensorial nota- tions of differential modeling. In order to overcome this hurdle, a tions: the traditional use of a coordinate system, in which the defin- first set of computational techniques (e.g., finite difference or par- ing equations are expressed, often obscures the underlying struc- ticle methods) focused on satisfying the continuous equations at a tures by an overwhelming usage of indices. Moreover, such com- discrete set of spatial and temporal samples. Unfortunately, focus- plex expressions entangle the topological and geometrical content ing on accurately discretizing the local laws often fails to respect of the model. important global structures and invariants. Later methods such as Finite Elements (FEM), drawing from developments in the calculus 1.2 Geometry-based Exterior Calculus of variations, remedied this inadequacy to some extent by satisfying local conservation laws on average and preserving some important The geometric nature of these models is best expressed and elu- invariants. Coupled with a finer ability to deal with arbitrary bound- cidated through the use of the Exterior Calculus of Differential aries, FEM became the de facto computational tool for engineers. Forms, first introduced by Cartan [Cartan 1945]. This geometry- Even with significant advances in error control, convergence, and based calculus was further developed and refined over the twentieth stability of these finite approximations, the underlying structures of century to become the foundation of modern differential geometry. the simulated continuous systems are often destroyed: a moving The calculus of exterior forms allows one to express differential rigid body may gain or loose momentum; or a cavity may exhibit and integral equations on smooth and curved spaces in a consis- fictitious eigenmodes in an electromagnetism (E&M) simulation. tent manner, while revealing the geometrical invariants at play. For Such examples illustrate some of the loss of fidelity that can fol- example, the classical operations of gradient, divergence, and curl low from a standard discretization process, failing to preserve some as well as the theorems of Green, Gauss and Stokes can all be ex- fundamental geometric and topological structures of the underlying pressed concisely in terms of differential forms and an operator on continuous models. these forms called the exterior derivative—hinting at the generality The cultural gap between theoretical and applied science commu- of this approach. nities may be partially responsible for the current lack of proper Compared to classical tensorial calculus, this exterior calculus has discrete, computational modeling that could mirror and leverage several advantages. First, it is often difficult to recognize the the rich developments of its differential counterpart. In particu- lar, it is striking that the calculus of differential forms has not yet ∗Now at the University of Southern California. had an impact on the mainstream computational fields, despite ex- yNow at Michigan State University. cellent initial results in E&M [Bossavit 1998] or Lagrangian me- zE-mail: [email protected] chanics [Marsden and West 2001]. It should also be noticed that some basic tools necessary for the definition of a discrete calculus many of the standard tools used in differential geometry have dis- already exist, probably initiated by Poincare´ when he defined his crete combinatorial analogs, the discrete versions of forms or man- cell decomposition of smooth manifolds. The study of the structure ifolds will be formally identical to (and should partake of the same of ordered sets or simplices now belongs to the well-studied branch properties as) the continuum models. Additionally, such an ap- of mathematics known as Combinatorial Differential Topology and proach should clearly maintain the separation of the topological Geometry, which is still an active area of research (see, e.g., [For- (metric-independent) and geometrical (metric-dependent) compo- man 2003] and [Bjorner¨ and Welker 1995] and references therein). nents of the quantities involved, keeping the geometric picture (i.e., intrinsic structure) intact. 1.4 Calculus ex Geometrica A discrete differential modeling approach to computations will also Given the overwhelming geometric nature of the most fundamental be often much simpler to define and develop than its continuous and successful calculus of these last few centuries, it seems relevant counterpart. For example, the discrete notion of a differential form to approach computations from a geometric standpoint. will be implemented simply as values on mesh elements. Likewise, the discrete notion of orientation will be more straightforward than One of the key insights that percolated down from the theory of its continuous counterpart: while the differential definition of ori- differential forms is rather simple and intuitive: one needs to recog- entation uses the notion of equivalence class of atlases determined nize that different physical quantities have different properties, and by the sign of the Jacobian, the orientation of a mesh edge will be must be treated accordingly. Fluid mechanics or electromagnetism, one of two directions; a triangle will be oriented clockwise or coun- for instance, make heavy use of line integrals, as well as surface terclockwise; a volume will have a direction as a right-handed helix and volume integrals; even physical measurements are performed or a left-handed one; no notion of atlas (a collection of consistent as specific local integrations or averages (think flux for magnetic coordinate charts on a manifold) will be required. field, or current for electricity, or pressure for atoms’ collisions). Pointwise evaluations or approximations for such quantities are not the appropriate discrete analogs, since the defining geometric prop- erties of their physical meaning cannot be enforced naturally. In- stead, one should store and manipulate those quantities at their geometrically-meaningful location: in other words, we should con- sider values on vertices, edges, faces, and tetrahedra as proper dis- crete versions of respectively pointwise functions, line integrals, surface integrals, and volume integrals: only then will we be