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Variational formulations of transport phenomena on combinatorial meshes

Kiprian Berbatov, Andrey P. Jivkov

TL;DR

This work introduces Combinatorial Mesh Calculus (CMC), a variational framework for transport phenomena formulated directly on cell complexes that encode multi-dimensional microstructures. By deriving primal and mixed weak formulations and mapping them to discrete combinatorial meshes via Forman subdivisions, the method preserves conservation laws on curved and irregular polytopal meshes without requiring smooth embeddings or circumcentric duals. A diagonal stiffness-like matrix in the mixed formulation enables efficient solution strategies, and the framework is validated on 2D and 3D domains with exact continuum solutions, demonstrating accurate handling of diffusion-driven transport across complex topologies. The results suggest significant potential for structure-aware materials modelling, including polycrystals, composites, and porous media, with open-source code to enable broader adoption and further development.

Abstract

We develop primal and mixed variational formulations of transport phenomena on cell complexes with simple polytope connectivity. This framework addresses materials with internal structures comprising components of different topological dimensions, where cells of each dimension may possess distinct physical properties. The approach, which we call Combinatorial Mesh Calculus (CMC), extends Forman's combinatorial differential forms, previously used to formulate strong conservation laws. CMC operates directly on meshes without requiring smooth embeddings, using discrete analogues of the exterior derivative, Hodge star, and co-differential operators. Our mixed formulation achieves computational efficiency through diagonal stiffness matrix A, which admits direct inversion and enable efficient solution strategies. CMC differs from Discrete Exterior Calculus, which requires circumcentric duality and well-centred meshes, and from Finite Element Exterior Calculus, which constructs polynomial spaces on smooth domains. Our framework applies to general cell complexes, including curved cells and irregular meshes without geometric quality constraints. The mathematical development proceeds in parallel between the smooth and discrete settings, establishing correspondences between continuous and discrete operators. Initial boundary value problems are formulated for mass diffusion, heat conduction, charge transport, and fluid flow through porous media. Numerical examples on regular and irregular meshes in two and three dimensions demonstrate agreement with analytical solutions. The framework enables modelling of transport in materials where microstructural topology influences macroscopic behaviour, with applications to polycrystalline materials, composites, and porous media.

Variational formulations of transport phenomena on combinatorial meshes

TL;DR

This work introduces Combinatorial Mesh Calculus (CMC), a variational framework for transport phenomena formulated directly on cell complexes that encode multi-dimensional microstructures. By deriving primal and mixed weak formulations and mapping them to discrete combinatorial meshes via Forman subdivisions, the method preserves conservation laws on curved and irregular polytopal meshes without requiring smooth embeddings or circumcentric duals. A diagonal stiffness-like matrix in the mixed formulation enables efficient solution strategies, and the framework is validated on 2D and 3D domains with exact continuum solutions, demonstrating accurate handling of diffusion-driven transport across complex topologies. The results suggest significant potential for structure-aware materials modelling, including polycrystals, composites, and porous media, with open-source code to enable broader adoption and further development.

Abstract

We develop primal and mixed variational formulations of transport phenomena on cell complexes with simple polytope connectivity. This framework addresses materials with internal structures comprising components of different topological dimensions, where cells of each dimension may possess distinct physical properties. The approach, which we call Combinatorial Mesh Calculus (CMC), extends Forman's combinatorial differential forms, previously used to formulate strong conservation laws. CMC operates directly on meshes without requiring smooth embeddings, using discrete analogues of the exterior derivative, Hodge star, and co-differential operators. Our mixed formulation achieves computational efficiency through diagonal stiffness matrix A, which admits direct inversion and enable efficient solution strategies. CMC differs from Discrete Exterior Calculus, which requires circumcentric duality and well-centred meshes, and from Finite Element Exterior Calculus, which constructs polynomial spaces on smooth domains. Our framework applies to general cell complexes, including curved cells and irregular meshes without geometric quality constraints. The mathematical development proceeds in parallel between the smooth and discrete settings, establishing correspondences between continuous and discrete operators. Initial boundary value problems are formulated for mass diffusion, heat conduction, charge transport, and fluid flow through porous media. Numerical examples on regular and irregular meshes in two and three dimensions demonstrate agreement with analytical solutions. The framework enables modelling of transport in materials where microstructural topology influences macroscopic behaviour, with applications to polycrystalline materials, composites, and porous media.
Paper Structure (25 sections, 13 theorems, 215 equations, 7 figures, 8 tables)

This paper contains 25 sections, 13 theorems, 215 equations, 7 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Overview and contributions
  4. Style and notations
  5. Overview of exterior calculus
  6. Topological operations
  7. Metric operations
  8. Exterior calculus variational formulations
  9. Primal weak formulations of transport phenomena
  10. Mixed weak formulations of transport phenomena
  11. Domains for physical quantities
  12. Combinatorial mesh calculus (CMC)
  13. Cell complexes (meshes)
  14. Topological operations
  15. Metric operations
  16. ...and 10 more sections

Key Result

Corollary 2.27

Let $M$ be an oriented smooth manifold with boundary, $D = \dim M$. An important consequence of the graded Leibniz rule, eq:exterior_calculus/leibniz, and the Stokes-Cartan theorem, eq:exterior_calculus/stokes_cartan, is the integration by parts formula: for any $p \in \{0, ..., D - 1\},\ \omega \in

Figures (7)

  • Figure 1: Rainbow colour scheme used for visualisation of potentials and flow rates
  • Figure 2: Regular curvilinear polar mesh on a disk and its Forman subdivision
  • Figure 3: Solutions for diffusion with quadratic potential on a disk with a regular polar mesh $\mathcal{K}$
  • Figure 4: Projections of a regular mesh on a hemisphere and its Forman subdivision
  • Figure 5: Projections of solutions for diffusion with quadratic potential on a hemisphere with a spherical mesh
  • ...and 2 more figures

Theorems & Definitions (98)

  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Remark 2.9
  • Definition 2.10
  • Definition 2.11
  • ...and 88 more