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Diffusion in multi-dimensional solids using Forman's combinatorial differential forms

Kiprian Berbatov, Pieter D. Boom, Andrew L. Hazel, Andrey P. Jivkov

TL;DR

This work develops an intrinsic, combinatorial framework for diffusion on discrete polyhedral meshes by extending Forman's combinatorial differential forms to include a metric on cochains and a discrete inner product. It introduces the Forman subdivision $K$ to realize forms as cochains, defines adjoint coboundary, Laplacian, and Hodge star, and derives a diffusion operator $Δ_0^α = δ^*_1 ∘ α ∘ δ_0$ that can assign different diffusivities to cells of different dimensions. Numerical demonstrations on regular and irregular meshes show structure-induced deviations from continuum diffusion and reveal percolation-like increases in effective diffusivity in composites with graphene plates or CNTs. The framework enables intrinsic modeling of heat, mass, and charge transport in heterogeneous materials and paves the way for intrinsic vector problems, such as solid deformation, on discrete spaces.

Abstract

The formulation of combinatorial differential forms, proposed by Forman for analysis of topological properties of discrete complexes, is extended by defining the operators required for analysis of physical processes dependent on scalar variables. The resulting description is intrinsic, different from the approach known as Discrete Exterior Calculus, because it does not assume the existence of smooth vector fields and forms extrinsic to the discrete complex. In addition, the proposed formulation provides a significant new modelling capability: physical processes may be set to operate differently on cells with different dimensions within a complex. An application of the new method to the heat/diffusion equation is presented to demonstrate how it captures the effect of changing properties of microstructural elements on the macroscopic behavior. The proposed method is applicable to a range of physical problems, including heat, mass and charge diffusion, and flow through porous media.

Diffusion in multi-dimensional solids using Forman's combinatorial differential forms

TL;DR

This work develops an intrinsic, combinatorial framework for diffusion on discrete polyhedral meshes by extending Forman's combinatorial differential forms to include a metric on cochains and a discrete inner product. It introduces the Forman subdivision to realize forms as cochains, defines adjoint coboundary, Laplacian, and Hodge star, and derives a diffusion operator that can assign different diffusivities to cells of different dimensions. Numerical demonstrations on regular and irregular meshes show structure-induced deviations from continuum diffusion and reveal percolation-like increases in effective diffusivity in composites with graphene plates or CNTs. The framework enables intrinsic modeling of heat, mass, and charge transport in heterogeneous materials and paves the way for intrinsic vector problems, such as solid deformation, on discrete spaces.

Abstract

The formulation of combinatorial differential forms, proposed by Forman for analysis of topological properties of discrete complexes, is extended by defining the operators required for analysis of physical processes dependent on scalar variables. The resulting description is intrinsic, different from the approach known as Discrete Exterior Calculus, because it does not assume the existence of smooth vector fields and forms extrinsic to the discrete complex. In addition, the proposed formulation provides a significant new modelling capability: physical processes may be set to operate differently on cells with different dimensions within a complex. An application of the new method to the heat/diffusion equation is presented to demonstrate how it captures the effect of changing properties of microstructural elements on the macroscopic behavior. The proposed method is applicable to a range of physical problems, including heat, mass and charge diffusion, and flow through porous media.
Paper Structure (38 sections, 10 theorems, 83 equations, 9 figures, 2 tables)

This paper contains 38 sections, 10 theorems, 83 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Continuum problem
  3. Overview of the paper
  4. Key notions from \ref{['sec:polytope_mesh']}
  5. Key notions from \ref{['sec:orientation']}
  6. Key notions from \ref{['sec:topology']}
  7. Key notions from \ref{['sec:geometry']}
  8. Key notions from \ref{['sec:hodge']}
  9. Notation and conventions
  10. Topological operations on a mesh.
  11. Discrete differential forms
  12. Forms as cochains on the Forman subdivision
  13. Quasi-cubical cup product of cochains and wedge product of forms
  14. Discrete metric operations
  15. General inner product
  16. ...and 23 more sections

Key Result

Theorem 2.16

$\smile$ satisfies the following properties (see arnold2012discrete):

Figures (9)

  • Figure 1: Triangulation (a) and its Forman subdivision (b)
  • Figure 2: The Forman subdivision of: (a) cube; (b) tetrahedron; (c) hexahedron; (d) square pyramid
  • Figure 3: Examples of: (a) cup product; (b) Hodge star
  • Figure 4: Examples of: (a) inner product; (b) Laplacian
  • Figure 5: Regular and irregular meshes (extended complex not shown).
  • ...and 4 more figures

Theorems & Definitions (134)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Example 2.8
  • Claim 2.9
  • proof
  • ...and 124 more