arXiv subjects

Pieter D. Boom

Publications and source records attributed to Pieter D. Boom.

6 recordsLinked to original sources

arXiv2026

Incidence-based Combinatorial Geometry on Cell Complexes

Combinatorial Mesh Calculus (CMC) formulates conservation laws directly on cell complexes using combinatorial differential forms and their cochain representations. This note develops an incidence-based geometric extension of that framework for directional quantities and local geometric structure on explicit cellular organisation. The central construction is a family of local fibres generated intrinsically by the incidence structure of the complex. Each vertex carries a vector space spanned by its incident edge directions, providing a combinatorial analogue of a tangent space whose dimension reflects local topology. These fibres define local coefficient spaces for incidence-based vector-, covector- and endomorphism-valued cochains. Directed transport maps compare states attached to neighbouring fibres, while a canonical solder form relates fibre directions to the underlying cell-complex structure. Together they give rise to finite combinatorial analogues of transport, torsion, curvature and metric structure. Evaluation cup products provide pairings between kinematic and force-like quantities, while bundle Hodge operators induced by the fibre metric relate vector- and covector-valued cochains, and weighted covariant incidence operators define degree-raising transport-corrected operations on incidence cochains. The framework does not assume a smooth manifold, fixed-rank bundle, cellular sheaf or local system. Instead, geometric structure is generated directly from the organisation represented by the cell complex. The resulting theory establishes a combinatorial scaffold for geometry on explicit cellular organisation and identifies the principal mathematical questions required for its further development, including admissible transport classes, metric compatibility, Cartan-type structure equations and locality-dependent algebraic structures.

math.GM↗

arXiv2022

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

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.

math-ph↗

arXiv2021

Parallelized Discrete Exterior Calculus for Three-Dimensional Elliptic Problems

A formulation of elliptic boundary value problems is used to develop the first discrete exterior calculus (DEC) library for massively parallel computations with 3D domains. This can be used for steady-state analysis of any physical process driven by the gradient of a scalar quantity, e.g. temperature, concentration, pressure or electric potential, and is easily extendable to transient analysis. In addition to offering this library to the community, we demonstrate one important benefit from the DEC formulation: effortless introduction of strong heterogeneities and discontinuities. These are typical for real materials, but challenging for widely used domain discretization schemes, such as finite elements. Specifically, we demonstrate the efficiency of the method for calculating the evolution of thermal conductivity of a solid with a growing crack population. Future development of the library will deal with transient problems, and more importantly with processes driven by gradients of vector quantities.

cs.MS↗

arXiv2021

A Geometric Formulation of Linear Elasticity Based on Discrete Exterior Calculus

A direct formulation of linear elasticity of cell complexes based on discrete exterior calculus is presented. The primary unknown are displacements, represented by primal vector-valued 0-cochain. Displacement differences and internal forces are represented by primal vector-valued 1-cochain and dual vector-valued 2-cochain, respectively. The macroscopic constitutive relation is enforced at primal 0-cells with the help of musical isomorphisms mapping cochains to smooth fields and vice versa. The balance of linear momentum is established at primal 0-cells. The governing equations are solved as a Laplace equation with a non-local and non-diagonal material Hodge star. Numerical simulations of several classical problems with analytic solutions are presented to validate the formulation. Good agreement with known solutions is obtained. The formulation provides a method to calculate the relations between displacement differences and internal forces for any lattice structure, when the structure is required to follow a prescribed macroscopic elastic behaviour. This is also the first and critical step in developing formulations for dissipative processes in cell complexes.

math-ph↗

arXiv2016

Runge-Kutta Characterization of the Generalized Summation-by-Parts Approach in Time

This article extends the theory of dual-consistent summation-by-parts (SBP) and generalized SBP (GSBP) time-marching methods by showing that they are implicit Runge-Kutta schemes. Through this connection, the accuracy theory for the pointwise solution, as well as the solution projected to the end of each time step, is extended for nonlinear problems. Furthermore, it is shown that these minimum guaranteed order results can be superseded by leveraging the full nonlinear order conditions of Runge-Kutta methods. The connection to Runge-Kutta methods is also exploited to derive conditions under which SBP and GSBP time-marching methods associated with dense norms are nonlinearly stable. A few known and novel Runge-Kutta methods with associated GSBP operators are presented. The novel methods, all of which are L-stable and algebraically-stable, include a four-stage seventh-order fully-implicit method, a three-stage third-order diagonally-implicit method, and a fourth-order four-stage diagonally-implicit method.

math.NA↗

arXiv2016

High-Order Implicit Time-Marching Methods Based on Generalized Summation-By-Parts Operators

This article extends the theory of classical finite-difference summation-by-parts (FD-SBP) time-marching methods to the generalized summation-by-parts (GSBP) framework. Dual-consistent GSBP time-marching methods are shown to retain: A and L-stability, as well as superconvergence of integral functionals when integrated with the quadrature associated with the discretization. This also implies that the solution approximated at the end of each time step is superconvergent. In addition GSBP time-marching methods constructed with a diagonal norm are BN-stable. This article also formalizes the connection between FD-SBP/GSBP time-marching methods and implicit Runge-Kutta methods. Through this connection, the minimum accuracy of the solution approximated at the end of a time step is extended for nonlinear problems. It is also exploited to derive conditions under which nonlinearly stable GSBP time-marching methods can be constructed. The GSBP approach to time marching can simplify the construction of high-order fully-implicit Runge-Kutta methods with a particular set of properties favourable for stiff initial value problems, such as L-stability. It can facilitate the analysis of fully discrete approximations to PDEs and is amenable to to multi-dimensional spcae-time discretizations, in which case the explicit connection to Runge-Kutta methods is often lost. A few examples of known and novel Runge-Kutta methods associated with GSBP operators are presented. The novel methods, all of which are L-stable and BN-stable, include a four-stage seventh-order fully-implicit method, a three-stage third-order diagonally-implicit method, and a fourth-order four-stage diagonally-implicit method. The relative efficiency of the schemes is investigated and compared with a few popular non-GSBP Runge-Kutta methods.

math.NA↗