Discrete physics using metrized chains

Abstract

Over the last fifty years, there have been numerous efforts to develop comprehensive discrete formulations of geometry and physics from first principles: from Whitney's geometric integration theory [33] to Harrison's theory of chainlets [16], including Regge calculus in general relativity [26, 34], Tonti's work on the mathematical structure of physical theories [30] and their discrete formulation [31], plus multifarious researches into so-called mimetic discretization methods [28], discrete exterior calculus [11, 12], and discrete differential geometry [2, 10]. All these approaches strive to tell apart the different mathematical structures---topological, differentiable, metrical---underpinning a physical theory, in order to make the relationships between them more transparent. While each component is reasonably well understood, computationally effective connections between them are not yet well established, leading to difficulties in combining and progressively refining geometric models and physics-based simulations. This paper proposes such a connection by introducing the concept of metrized chains, meant to establish a discrete metric structure on top of a discrete measure-theoretic structure embodied in the underlying notion of measured (real-valued) chains. These, in turn, are defined on a cell complex, a finite approximation to a manifold which abstracts only its topological properties. The algebraic-topological approach to circuit design and network analysis first proposed by Branin [7] was later extensively applied by Tonti to the study of the mathematical structure of physical theories [30]. (Co-)chains subsequently entered the field of physical modeling [4, 18, 24, 25, 31, 37], and were related to commonly-used discretization methods such as finite elements, finite differences, and finite volumes [1, 8, 21, 22]. Our modus operandi is characterized by the pivotal role we accord to the construction of a physically based inner product between chains. This leads us to criticize the emphasis placed on the choice of an appropriate dual mesh: in our opinion, the "good" dual mesh is but a red herring, in general.