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In this paper, we introduce a discretization scheme for the Yang–Mills equations in the two-dimensional case using a framework based on discrete exterior calculus. Within this framework, we define discrete versions of the exterior covariant derivative operator and its adjoint, which capture essential geometric features similar to their continuous counterparts. Our focus is on discrete models defined on a combinatorial torus, where the discrete Yang–Mills equations are presented in the form of both a system of difference equations and a matrix form. [ABSTRACT FROM AUTHOR]

Fractional vector calculus is the building block of the fractional partial differential equations that model non-local or long-range phenomena, e.g., anomalous diffusion, fractional electromagnetism, and fractional advection-dispersion. In this work, we reformulate a type of fractional vector calculus that uses Caputo fractional partial derivatives and discretize this reformulation using discrete exterior calculus on a cubical complex in the structure-preserving way, meaning that the continuous-level properties curl α grad α = 0 and div α curl α = 0 hold exactly on the discrete level. We discuss important properties of our fractional discrete exterior derivatives and verify their second-order convergence in the root mean square error numerically. Our proposed discretization has the potential to provide accurate and stable numerical solutions to fractional partial differential equations and exactly preserve fundamental physics laws on the discrete level regardless of the mesh size. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a discretization scheme for the Yang–Mills equations in the two-dimensional case using a framework based on discrete exterior calculus. Within this framework, we define discrete versions of the exterior covariant derivative operator and its adjoint, which capture essential geometric features similar to their continuous counterparts. Our focus is on discrete models defined on a combinatorial torus, where the discrete Yang–Mills equations are presented in the form of both a system of difference equations and a matrix form.

Computational modeling is a key resource to gather insight into physical systems in modern scientific research and engineering. While access to large amount of data has fueled the use of machine learning to recover physical models from experiments and increase the accuracy of physical simulations, purely data-driven models have limited generalization and interpretability. To overcome these limitations, we propose a framework that combines symbolic regression (SR) and discrete exterior calculus (DEC) for the automated discovery of physical models starting from experimental data. Since these models consist of mathematical expressions, they are interpretable and amenable to analysis, and the use of a natural, general-purpose discrete mathematical language for physics favors generalization with limited input data. Importantly, DEC provides building blocks for the discrete analog of field theories, which are beyond the state-of-the-art applications of SR to physical problems. Further, we show that DEC allows to implement a strongly-typed SR procedure that guarantees the mathematical consistency of the recovered models and reduces the search space of symbolic expressions. Finally, we prove the effectiveness of our methodology by re-discovering three models of continuum physics from synthetic experimental data: Poisson equation, the Euler’s elastica and the equations of linear elasticity. Thanks to their general-purpose nature, the methods developed in this paper may be applied to diverse contexts of physical modeling.

We consider new degrees of freedom for higher order differential forms on cubical meshes. The approach is inspired by the idea of Rapetti and Bossavit to define higher order Whitney forms and their degrees of freedom using small simplices. We show that higher order differential forms on cubical meshes can be defined analogously using small cubes and prove that these small cubes yield unisolvent degrees of freedom. Importantly, this approach is compatible with discrete exterior calculus and expands the framework to cover higher order methods on cubical meshes, complementing the earlier strategy based on simplices. [ABSTRACT FROM AUTHOR]

The physical models and numerical methodologies of Computational Fluid Dynamics (CFD) are historically linked to the concept of continuous medium and to analysis where continuity, derivation and integration are defined as limits at a point. The first consequence is the need to extend these notions in a multidimensional space by establishing a global inertial frame of reference in order to project the variables there. In recent decades, the emergence of methodologies based on differential geometry or exterior calculus has changed the point of view by starting with the creation of entangled polygonal and polyhedral structures where the variables are located. Mimetic methods and Discrete Exterior Calculus, notably, have intrinsic conservation properties which make them very efficient for solving fluid dynamics equations. The natural extension of this discrete vision relates to the derivation of the equations of mechanics by abandoning the notion of continuous medium. The Galilean frame of reference is replaced by a local frame of reference composed of an oriented segment where the acceleration of the material medium or of a particle is defined. The extension to a higher dimensional space is carried from cause to effect, from one local structure to another. The conservation of acceleration over a segment and the Helmholtz–Hodge decomposition are two essential principles adopted for the derivation of a discrete law of motion. As the fields covered by CFD are increasingly broad, it is natural to return to the deeper meaning of physical phenomena to try a new research or new path which would preserve the properties of current formulations.

The discrete exterior calculus (DEC) is very promising, though not yet widely used, discretization method for photonic crystal (PC) waveguides. It can be seen as a generalization of the finite difference time domain (FDTD) method. The DEC enables efficient time evolution by construction and fits well for nonhomogeneous computational domains and obstacles of curved surfaces. These properties are typically present in applications of PC waveguides that are constructed as periodic structures of inhomogeneities in a computational domain. We present a two‐dimensional DEC discretization for PC waveguides and demonstrate it with a selection of numerical experiments typical in the application area. We also make a numerical comparison of the method with the FDTD method that is a mainstream method for simulating PC structures. Numerical results demonstrate the advantages of the DEC method. [ABSTRACT FROM AUTHOR]

In this paper, we apply the exact controllability concept for time-harmonic electromagnetic scattering. The problem is presented in terms of the differential forms, and the discrete exterior calculus is utilized for spatial discretization. Accordingly, the physical properties of the problem are maintained. Despite we consider time-harmonic problems, we concentrate on transient wave equations treated by the exact controllability approach. Essentially, we use a controlled variation of the asymptotic approach with periodic constraints, in which the time-dependent equation is simulated in time, until the time-harmonic solution is reached. [ABSTRACT FROM AUTHOR]

New high-performance computing (HPC) software designed for massively parallel computers with high-speed interconnects is presented to accelerate research into geometric formulations of solid mechanics based on discrete exterior calculus (DEC). DEC is a relatively new and entirely discrete approach being developed to model non-smooth material processes, for which continuum descriptions fail. Until now, progress has been slowed by limited HPC software. The tool presented herein integrates the DEC library ParaGEMS into the well-established parallel finite-element (FE) code ParaFEM, leveraging ParaFEM’s diverse IO routines, optimised solvers, and interfaces to third-party libraries. This is accomplished by interpreting FE elements, or their subdivision, as independent DEC simplicial complexes. The element-wise contribution to the global system matrix is then replaced with the DEC formalism, superimposing contributions from the dual mesh at element boundaries. The integrated tool is validated using five miniApps for scalar diffusion and linear elasticity on synthetic microstructures with emerging discontinuities, showing the performance for both continuum and discrete problems. Profiling indicates DEC calculations have excellent scaling and the solver achieves approximately 80% parallel efficiency using naïve partitioning on ∼8000 cores with >135 million unknowns. The tool is now being used to develop DEC formulations of more complex phenomena, such as material nonlinearity and fracture.

We introduce a framework for representing face-based directional fields of an arbitrary piecewise-polynomial order. Our framework is based on a primal-dual decomposition of fields, where the exact component of a field is the gradient of piecewise-polynomial conforming function, and the coexact component is defined as the adjoint of a dimensionally-consistent discrete curl operator. Our novel formulation sidesteps the difficult problem of constructing high-order non-conforming function spaces, and makes it simple to harness the flexibility of higher-order finite elements for directional-field processing. Our representation is structure-preserving, and draws on principles from finite-element exterior calculus. We demonstrate its benefits for applications such as Helmholtz-Hodge decomposition, smooth PolyVector fields, the vector heat method, and seamless parameterization. [ABSTRACT FROM AUTHOR]

We present a systematic way to implement higher order Whitney forms in numerical methods based on discrete exterior calculus. Given a simplicial mesh, we first refine the mesh into smaller simplices which can be used to define higher order Whitney forms. Cochains on this refined mesh can then be interpolated using higher order Whitney forms. Hence, when the refined mesh is used with methods based on discrete exterior calculus, the solution can be expressed as a higher order Whitney form. We present algorithms for the three required steps: refining the mesh, solving the coefficients of the interpolant, and evaluating the interpolant at a given point. With our algorithms, the order of the Whitney forms one wishes to use can be given as a parameter so that the same code covers all orders, which is a significant improvement on previous implementations. Our algorithms are applicable with all methods in which the degrees of freedom are integrals over mesh simplices — that is, when the solution is a cochain on a simplicial mesh. They can also be used when one simply wishes to approximate differential forms in finite-dimensional spaces. Numerical examples validate the generality of our algorithms. [ABSTRACT FROM AUTHOR]

We discuss a discretization of the de Rham–Hodge theory in the two-dimensional case based on a discrete exterior calculus framework. We present discrete analogues of the Hodge–Dirac and Laplace operators in which key geometric aspects of the continuum counterpart are captured. We provide and prove a discrete version of the Hodge decomposition theorem. The goal of this work is to develop a satisfactory discrete model of the de Rham–Hodge theory on manifolds that are homeomorphic to the torus. Special attention has been paid to discrete models on a combinatorial torus. In this particular case, we also define and calculate the cohomology groups. [ABSTRACT FROM AUTHOR]

The focus of this paper is to propose a novel computational approach for the solution of large-scale flow-based topology optimization problems using a graphics processing unit (GPU). A marker-and-cell method is first used to discretize a fluid flow design domain. This is followed by a finite difference method to solve the Stokes equations for steady-state incompressible fluid flow. An adjoint method is then employed to conduct design sensitivity analysis for the optimization. We use a generalized minimal residual method as the base solver for the linear system and develop an efficient geometric multigrid preconditioner on GPU in a matrix-free form. We simplify the treatment of different boundary conditions with improved accuracy based on the theory of discrete exterior calculus. Numerical results utilizing different resolutions are presented and highlight a nearly linear computational time scalability. Consequently, intricate branching flow structures may be automatically and efficiently discovered at high resolutions. Our approach is capable of solving indefinite problems (i.e., one forward solution of the Stokes equations) with over 7 million elements in three dimensions (3D) and over 16 million elements in two dimensions (2D) within two minutes using a single desktop computer. Furthermore, all numerical experiments reported in this paper are performed on a single NVIDIA Quadro RTX 8000 graphics card. We subsequently compare the optimized flow structures obtained using the newly proposed method with those obtained by commercial finite element software in an established optimization loop and find the optimized structures from both methods to be in good agreement. To highlight the advantage of GPU acceleration, a quantitative run-time comparison study with the commercial finite element software is performed. Our implementation is shown to solve fluid flow problems with orders of magnitude higher resolution using only a fraction of the computational time. [ABSTRACT FROM AUTHOR]

We are concerned with a special class of discretizations of general linear transmission problems stated in the calculus of differential forms and posed on R n . In the spirit of domain decomposition, we partition R n = Ω ∪ Γ ∪ Ω + , Ω a bounded Lipschitz polyhedron, Γ : = ∂ Ω , and Ω + unbounded. In Ω , we employ a mesh-based discrete co-chain model for differential forms, which includes schemes like finite element exterior calculus and discrete exterior calculus. In Ω + , we rely on a meshless Trefftz–Galerkin approach, i.e., we use special solutions of the homogeneous PDE as trial and test functions. Our key contribution is a unified way to couple the different discretizations across Γ . Based on the theory of discrete Hodge operators, we derive the resulting linear system of equations. As a concrete application, we discuss an eddy-current problem in frequency domain, for which we also give numerical results. [ABSTRACT FROM AUTHOR]

While the Discrete Exterior Calculus (DEC) discretization of the diffusive term of the Transport Equation is well understood, the DEC discretization of the convective term, as well as its stabilization, is an ongoing area of research. In this paper, we propose a local discretization for this term based on DEC and geometric arguments, considering the particle velocity field prescribed at the vertices of the primal mesh. This formulation is similar to that of the Finite Element Method with linear interpolation functions (FEML) and can be stabilized using known stabilization techniques, such as Artificial Diffusion. Using this feature, numerical tests are carried out on simple stationary and transient problems with domains discretized with coarse and fine simplicial meshes to show numerical convergence.

We discuss a discretization of the de Rham–Hodge theory in the two-dimensional case based on a discrete exterior calculus framework. We present discrete analogues of the Hodge–Dirac and Laplace operators in which key geometric aspects of the continuum counterpart are captured. We provide and prove a discrete version of the Hodge decomposition theorem. The goal of this work is to develop a satisfactory discrete model of the de Rham–Hodge theory on manifolds that are homeomorphic to the torus. Special attention has been paid to discrete models on a combinatorial torus. In this particular case, we also define and calculate the cohomology groups.

Abstract Some of the most important geometric integrators for both ordinary and partial differential equations are reviewed and illustrated with examples in mechanics. The class of Hamiltonian differential systems is recalled and its symplectic structure is highlighted. The associated natural geometric integrators, known as symplectic integrators, are then presented. In particular, their ability to numerically reproduce first integrals with a bounded error over a long time interval is shown. The extension to partial differential Hamiltonian systems and to multisymplectic integrators is presented afterwards. Next, the class of Lagrangian systems is described. It is highlighted that the variational structure carries both the dynamics (Euler–Lagrange equations) and the conservation laws (Nœther’s theorem). Integrators preserving the variational structure are constructed by mimicking the calculus of variation at the discrete level. We show that this approach leads to numerical schemes which preserve exactly the energy of the system. After that, the Lie group of local symmetries of partial differential equations is recalled. A construction of Lie-symmetry-preserving numerical scheme is then exposed. This is done via the moving frame method. Applications to Burgers equation are shown. The last part is devoted to the Discrete Exterior Calculus, which is a structure-preserving integrator based on differential geometry and exterior calculus. The efficiency of the approach is demonstrated on fluid flow problems with a passive scalar advection.

A geometric method for analysis of elastic and elastic-plastic solids is proposed. It involves operators on naturally discrete domains representing a material's microstructure, rather than the classical discretisation of domains for solving continuum boundary value problems. Discrete microstructures are considered as general cell complexes, which are circumcentre-dual to simplicial cell complexes. The proposed method uses the separation of the total deformation energy into volumetric and distortional parts as a base for introducing elastoplastic material behaviour. Volumetric parts are obtained directly from volume changes of dual cells, and the distortional parts are calculated from the distance changes between primal and dual nodes. First, it is demonstrated that the method can accurately reproduce the elastic behaviour of solids with Poisson's ratios in the thermodynamically admissible range from -0.99 to 0.49. Further verification of the method is demonstrated by excellent agreement between analytical results and simulations of the elastic deformation of a beam subjected to a vertical displacement. Second, the Prandtl operator approach is used to simulate the behaviour of the solid during cyclic loading, considering its elastoplastic material properties. Results from simulations of cyclic behaviour during alternating and variable load histories are compared to expected macroscopic behaviour as further support to the applicability of the method to elastic-plastic problems. [ABSTRACT FROM AUTHOR]

We present a compendium of Hodge decompositions of vector fields on tetrahedral meshes embedded in the 3D Euclidean space. After describing the foundations of the Hodge decomposition in the continuous setting, we describe how to implement a five-component orthogonal decomposition that generically splits, for a variety of boundary conditions, any given discrete vector field expressed as discrete differential forms into two potential fields, as well as three additional harmonic components that arise from the topology or boundary of the domain. The resulting decomposition is proper and mimetic, in the sense that the theoretical dualities on the kernel spaces of vector Laplacians valid in the continuous case (including correspondences to cohomology and homology groups) are exactly preserved in the discrete realm. Such a decomposition only involves simple linear algebra with symmetric matrices, and can thus serve as a basic computational tool for vector field analysis in graphics, electromagnetics, fluid dynamics and elasticity. [ABSTRACT FROM AUTHOR]

We present a new hybrid discrete exterior calculus (DEC) and finite difference (FD) method to simulate fully three-dimensional Boussinesq convection in spherical shells subject to internal heating and basal heating, relevant in the planetary and stellar phenomenon. We employ DEC to compute the surface spherical flows, taking advantage of its unique features including coordinate system independence to preserve the spherical geometry, while we discretize the radial direction using FD method. The grid employed for this novel method is free of problems like the coordinate singularity, grid non-convergence near the poles, and the overlap regions. We have developed a parallel in-house code using the PETSc framework to verify the hybrid DEC-FD formulation and demonstrate convergence. We have performed a series of numerical tests which include quantification of the critical Rayleigh numbers for spherical shells characterized by aspect ratios ranging from 0.2 to 0.8, simulation of robust convective patterns in addition to stationary giant spiral roll covering all the spherical surface in moderately thin shells near the weakly nonlinear regime, and the quantification of Nusselt and Reynolds numbers for basally heated spherical shells. • Boussinesq convection in spherical shells. • Hybrid discrete exterior calculus and finite difference (DEC-FD) discretization. • Surface and radial operators are approximated by DEC and FD, respectively. • No coordinate singularity and grid non-convergence near poles, and overlap regions. • A plethora of robust convective patterns and resolving high wavenumber features. [ABSTRACT FROM AUTHOR]

Highlights • We presented a new finite-element time-domain Maxwell solver for body-of-revolution geometries. • The solver enables the use of a Cartesian 2-D code with no modifications necessary except to accommodate anisotropic media. • The proposed solver does not require any modifications on the basis functions adjacent to the symmetry axis. • The new Maxwell solver is illustrated in a number of problems involving electromagnetic fields in cylindrical structures. Abstract We present a finite-element time-domain (FETD) Maxwell solver for the analysis of body-of-revolution (BOR) geometries based on discrete exterior calculus (DEC) of differential forms and transformation optics (TO) concepts. We explore TO principles to map the original 3-D BOR problem to a 2-D one in the meridian ρz -plane based on a Cartesian coordinate system where the cylindrical metric is fully embedded into the constitutive properties of an effective inhomogeneous and anisotropic medium that fills the domain. The proposed solver uses a (TE ϕ , TM ϕ) field decomposition and an appropriate set of DEC-based basis functions on an irregular grid discretizing the meridian plane. A symplectic time discretization based on a leap-frog scheme is applied to obtain the full-discrete marching-on-time algorithm. We validate the algorithm by comparing the numerical results against analytical solutions for resonant fields in cylindrical cavities and against pseudo-analytical solutions for fields radiated by cylindrically symmetric antennas in layered media. We also illustrate the application of the algorithm for a particle-in-cell (PIC) simulation of beam-wave interactions inside a high-power backward-wave oscillator. [ABSTRACT FROM AUTHOR]

For analysis of electromagnetic fields, differential forms are useful mathematical tools. Discrete exterior calculus (DEC) is a finite-difference-type framework of discretization of the discrete differential forms. Although several numerical estimations of errors of DEC have been reported, a tangible theoretical error bound is not established. One reason of this is that there are two norms that have been commonly used for the discrete differential forms. In this paper, we show that on any quasi-uniform mesh, the two norms of the discrete differential forms in DEC are equivalent, and moreover, the order of accuracy of the discrete differential forms are independent of the choice of the norm. As an application, it is shown that the accuracy of the discrete Hodge operator of DEC is of the second-order if any quasi-uniform Delaunay mesh is adopted. [ABSTRACT FROM AUTHOR]

We present a discrete exterior calculus (DEC) based discretization scheme for incompressible two-phase flows. Our physically-compatible exterior calculus discretization of single phase flow is extended to simulate immiscible two-phase flows with discontinuous changes in fluid properties such as density and viscosity across the interface. The two-phase incompressible Navier-Stokes equations and conservative phase field equation for interface capturing are first transformed into the exterior calculus framework. The discrete counterpart of these smooth equations is obtained by substituting with discrete differential forms and discrete exterior calculus operators. We prove the boundedness of the method for the forward Euler and predictor-corrector time integration schemes in the DEC framework. With a proper choice of two free parameters, the scheme remains phase field bounded without the requirement of any ad hoc mass redistribution. We verify the scheme against several standard test cases (for interface capturing) comprising not only the flat domains but also the curved domains, leveraging the advantage that DEC operators are independent of the coordinate system. The results show excellent properties of boundedness, mass conservation and convergence. Moreover, we demonstrate the ability of the scheme to handle large density and viscosity ratios as well as surface tension in the simulation of various two phase flow physical phenomena on flat or curved surfaces. [ABSTRACT FROM AUTHOR]

We introduce a general class of second-order boundary value problems unifying application areas such as acoustics, electromagnetism, elastodynamics, quantum mechanics, and so on, into a single framework. This also enables us to solve wave propagation problems very efficiently with a single software system. The solution method precisely follows the conservation laws in finite-dimensional systems, whereas the constitutive relations are imposed approximately. We employ discrete exterior calculus for the spatial discretization, use natural crystal structures for three-dimensional meshing, and derive a "discrete Hodge" adapted to harmonic wave. The numerical experiments indicate that the cumulative pollution error can be practically eliminated in the case of harmonic wave problems. The restrictions following from the CFL condition can be bypassed with a local time-stepping scheme. The computational savings are at least one order of magnitude. [ABSTRACT FROM AUTHOR]

In this paper, we perform a numerical analysis in frequency domain for various electromagnetic problems based on discrete exterior calculus (DEC) with an arbitrary 2-D triangular or 3-D tetrahedral mesh. We formulate the governing equations in terms of DEC for 3-D and 2-D inhomogeneous structures, and also show that the charge continuity relation is naturally satisfied. Then we introduce a general construction for signed dual volume to incorporate material information and take into account the case when circumcenters fall outside triangles or tetrahedrons, which may lead to negative dual volume without Delaunay triangulation. Then we examine the boundary terms induced by the dual mesh and provide a systematical treatment of various boundary conditions, including perfect magnetic conductor (PMC), perfect electric conductor (PEC), Dirichlet, periodic, and absorbing boundary conditions (ABC) within this method. An excellent agreement is achieved through the numerical calculation of several problems, including homogeneous waveguides, microstructured fibers, photonic crystals, scattering by a 2-D PEC, and resonant cavities. [ABSTRACT FROM AUTHOR]

Discrete exterior calculus (DEC) is a numerical method for solving partial differential equations on meshes with applications in computer graphics, numerics, and physical simulations. It discretizes PDEs in a way such that important integral theorems hold exactly instead of being approximated. The drawback of the method is that it has strong requirements on the mesh. While the narrow range of admissible meshes mentioned in the original work could be widened, current methods still exclude an essential category of meshes, i.e., meshes with concyclic triangle pairs. Such meshes are common, as many synthetic meshes, e.g., triangulations of CAD models, handcrafted 3D models, and even results of surface meshing algorithms, contain such triangle pairs. Our paper describes an approach that allows us to use meshes with arbitrary triangulations that may contain concyclic triangle pairs by defining DEC operators for concyclic polygons. • Lifting one of the last mesh requirements for using DEC. • Allows for using circumcentric duality in meshes with degenerate dual edges. • The method only requires minor changes in existing DEC codes. [ABSTRACT FROM AUTHOR]

The collocated Lebedev grid has previously been proposed as an alternative to the Yee grid for electromagnetic finite-difference time-domain (FDTD) simulations. While it performs better in anisotropic media, it performs poorly in isotropic media because it is equivalent to four overlapping, uncoupled Yee grids. We propose to couple the four Yee grids and fix the Lebedev method using discrete exterior calculus (DEC) with higher-order Hodge duals. We find that higher-order Hodge duals do improve the performance of the Lebedev grid, but they also improve the Yee grid by a similar amount. The effectiveness of coupling overlapping Yee grids with a higher-order Hodge dual is thus questionable. However, the theoretical foundations developed to derive these methods may be of interest in other problems. [ABSTRACT FROM AUTHOR]

The Laplace operator plays a fundamental role in geometry processing. Several discrete versions have been proposed for 3D meshes and point clouds, among others. We define here a discrete Laplace operator for temporally coherent mesh sequences, which allows to process mesh animations in a simple yet efficient way. This operator is a discretisation of the Laplace-Beltrami operator using Discrete Exterior Calculus on CW complexes embedded in a four-dimensional space. A parameter is introduced to tune the influence of the motion with respect to the geometry. This enables straightforward generalisation of existing Laplacian static mesh processing works to mesh sequences. An application to spacetime editing is provided as example. [ABSTRACT FROM AUTHOR]

We model meteoroids entering the Earth׳s atmosphere as objects surrounded by non-magnetized plasma, and consider efficient numerical simulation of radar reflections from meteors in the time domain. Instead of the widely used finite difference time domain method (FDTD), we use more generalized finite differences by applying the discrete exterior calculus (DEC) and non-uniform leapfrog-style time discretization. The computational domain is presented by convex polyhedral elements. The convergence of the time integration is accelerated by the exact controllability method. The numerical experiments show that our code is efficiently parallelized. The DEC approach is compared to the volume integral equation (VIE) method by numerical experiments. The result is that both methods are competitive in modelling non-magnetized plasma scattering. For demonstrating the simulation capabilities of the DEC approach, we present numerical experiments of radar reflections and vary parameters in a wide range. [ABSTRACT FROM AUTHOR]

This paper introduces a new computational method to solve differential equations on subdivision surfaces. Our approach adapts the numerical framework of Discrete Exterior Calculus (DEC) from the polygonal to the subdivision setting by exploiting the refin-ability of subdivision basis functions. The resulting Subdivision Exterior Calculus (SEC) provides significant improvements in accuracy compared to existing polygonal techniques, while offering exact finite-dimensional analogs of continuum structural identities such as Stokes' theorem and Helmholtz-Hodge decomposition. We demonstrate the versatility and efficiency of SEC on common geometry processing tasks including parameterization, geodesic distance computation, and vector field design. [ABSTRACT FROM AUTHOR]

We consider the computationally efficient time integration of Maxwell's equations using discrete exterior calculus (DEC) as the computational framework. With the theory of DEC, we associate the degrees of freedom of the electric and magnetic fields with primal and dual mesh structures, respectively. We concentrate on mesh constructions that imitate the geometry of the close packing in crystal lattices that is typical of elemental metals and intermetallic compounds. This class of computational grids has not been used previously in electromagnetics. For the simulation of wave propagation driven by time-harmonic source terms, we provide an optimized Hodge operator and a novel time discretization scheme with nonuniform time step size. The numerical experiments show a significant improvement in accuracy and a decrease in computing time compared to simulations with well-known variants of the finite difference time domain method. [ABSTRACT FROM AUTHOR]

We derive a numerical method for Darcy flow, and also for Poisson’s equation in mixed (first order) form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is one of its discretizations on simplicial complexes such as triangle and tetrahedral meshes. DEC is a coordinate invariant discretization, in that it does not depend on the embedding of the simplices or the whole mesh. We start by rewriting the governing equations of Darcy flow using the language of exterior calculus. This yields a formulation in terms of flux differential form and pressure. The numerical method is then derived by using the framework provided by DEC for discretizing differential forms and operators that act on forms. We also develop a discretization for a spatially dependent Hodge star that varies with the permeability of the medium. This also allows us to address discontinuous permeability. The matrix representation for our discrete non-homogeneous Hodge star is diagonal, with positive diagonal entries. The resulting linear system of equations for flux and pressure are saddle type, with a diagonal matrix as the top left block. The performance of the proposed numerical method is illustrated on many standard test problems. These include patch tests in two and three dimensions, comparison with analytically known solutions in two dimensions, layered medium with alternating permeability values, and a test with a change in permeability along the flow direction. We also show numerical evidence of convergence of the flux and the pressure. A convergence experiment is included for Darcy flow on a surface. A short introduction to the relevant parts of smooth and discrete exterior calculus is included in this article. We also include a discussion of the boundary condition in terms of exterior calculus. [ABSTRACT FROM PUBLISHER]

Abstract: In this study, we perform numerical computations of electromagnetic fields including applications such as light scattering. We present two methods for discretizing the computational domain. One is the discrete-dipole approximation (DDA), which is a well-known technique in the context of light scattering. The other approach is the discrete exterior calculus (DEC) providing the properties and the calculus of differential forms in a natural way at the discretization stage. Numerical experiments show that the DEC provides a promising alternative for solving the general Maxwell equations. [Copyright &y& Elsevier]

• A parameterized exterior calculus is introduced for learning physics. • Analysis provides guarantees for learning well-posed nonlinear elliptic operators. • Exact treatment of conservation and characterization of nontrivial null-spaces. • Novel training strategy guarantees exact physics independent of available data. As traditional machine learning tools are increasingly applied to science and engineering applications, physics-informed methods have emerged as effective tools for endowing inferences with properties essential for physical realizability. While promising, these methods generally enforce physics weakly via penalization. To enforce physics strongly, we turn to the exterior calculus framework underpinning combinatorial Hodge theory and physics-compatible discretization of partial differential equations (PDEs). Historically, these two fields have remained largely distinct, as graphs are strictly topological objects lacking the metric information fundamental to PDE discretization. We present an approach where this missing metric information may be learned from data, using graphs as coarse-grained mesh surrogates that inherit desirable conservation and exact sequence structure from the combinatorial Hodge theory. The resulting data-driven exterior calculus (DDEC) may be used to extract structure-preserving surrogate models with mathematical guarantees of well-posedness. The approach admits a PDE-constrained optimization training strategy which guarantees machine-learned models enforce physics to machine precision, even for poorly trained models or small data regimes. We provide analysis of the method for a class of models designed to reproduce nonlinear perturbations of elliptic problems and provide examples of learning H (d i v) / H (c u r l) systems representative of subsurface flows and electromagnetics. [ABSTRACT FROM AUTHOR]

A direct formulation of linear elasticity of cell complexes based on discrete exterior calculus is presented. The primary unknowns are displacements, represented by a primal vector-valued 0 -cochain. Displacement differences and internal forces are represented by a primal vector-valued 1 -cochain and a 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 Poisson's 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. Excellent 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. • Formulation of linear elasticity using discrete exterior calculus (DEC). • Derivation of new discrete sharp musical isomorphism. • Derivation of non-local and non-diagonal discrete Hodge star for linear elasticity. • Numerical validation using standard mechanical problems with analytic solutions. [ABSTRACT FROM AUTHOR]

Abstract: An n-simplex is said to be n-well-centered if its circumcenter lies in its interior. We introduce several other geometric conditions and an algebraic condition that can be used to determine whether a simplex is n-well-centered. These conditions, together with some other observations, are used to describe restrictions on the local combinatorial structure of simplicial meshes in which every simplex is well-centered. In particular, it is shown that in a 3-well-centered (2-well-centered) tetrahedral mesh there are at least 7 (9) edges incident to each interior vertex, and these bounds are sharp. Moreover, it is shown that, in stark contrast to the 2-dimensional analog, where there are exactly two vertex links that prevent a well-centered triangle mesh in , there are infinitely many vertex links that prohibit a well-centered tetrahedral mesh in . [Copyright &y& Elsevier]

Abstract: In this paper, we present a geometric discretization scheme for incompressible linearized elasticity. We use ideas from discrete exterior calculus (DEC) to write the action for a discretized elastic body modeled by a simplicial complex. After characterizing the configuration manifold of volume-preserving discrete deformations, we use Hamilton’s principle on this configuration manifold. The discrete Euler–Lagrange equations are obtained without using Lagrange multipliers. The main difference between our approach and the mixed finite element formulations is that we simultaneously use three different discrete spaces for the displacement field. We explicitly derive the governing equations for the two-dimensional case, where the discrete spaces for the displacement field are constructed by polynomials over primal meshes for incompressibility constraint, polynomials over dual meshes for the kinetic energy, and polynomials over support volumes for the elastic energy, and the discrete space of the pressure field is constructed by polynomials over primal meshes. We test the efficiency and robustness of this geometric scheme using some numerical examples. In particular, we do not see any volume locking and/or checkerboarding of pressure in our numerical examples. This suggests that our choice of discrete solution spaces is compatible. [Copyright &y& Elsevier]

The evolution of a viscous thin film on a curved geometry is numerically approximated based on the natural time discretization of the underlying gradient flow. This discretization leads to a variational problem to be solved at each time step, which reflects the balance between the decay of the free (gravitational and surface) energy and the viscous dissipation. Both dissipation and energy are derived from a lubrication approximation for a small ratio between the characteristic film height and the characteristic length scale of the surface. The dissipation is formulated in terms of a corresponding flux field, whereas the energy primarily depends on the fluid volume per unit surface, which is a conserved quantity. These two degrees of freedom are coupled by the underlying transport equation. Hence, one is naturally led to a PDE-constrained optimization problem, where the variational time stepping problem has to be solved under the constraint described by the transport equation. For the space discretization a discrete exterior calculus approach is investigated. Various applications demonstrate the qualitative and quantitative behavior of one- and two-dimensional thin films on curved geometries. [ABSTRACT FROM AUTHOR]

Abstract: We define signed dual volumes at all dimensions for circumcentric dual meshes. We show that for pairwise Delaunay triangulations with mild boundary assumptions these signed dual volumes are positive. This allows the use of such Delaunay meshes for Discrete Exterior Calculus (DEC) because the discrete Hodge star operator can now be correctly defined for such meshes. This operator is crucial for DEC and is a diagonal matrix with the ratio of primal and dual volumes along the diagonal. A correct definition requires that all entries be positive. DEC is a framework for numerically solving differential equations on meshes and for geometry processing tasks and has had considerable impact in computer graphics and scientific computing. Our result allows the use of DEC with a much larger class of meshes than was previously considered possible. [Copyright &y& Elsevier]

This article describes the algorithms, features, and implementation of PyDEC, a Python library for computations related to the discretization of exterior calculus. PyDEC facilitates inquiry into both physical problems on manifolds as well as purely topological problems on abstract complexes. We describe efficient algorithms for constructing the operators and objects that arise in discrete exterior calculus, lowest-order finite element exterior calculus, and in related topological problems. Our algorithms are formulated in terms of high-level matrix operations which extend to arbitrary dimension. As a result, our implementations map well to the facilities of numerical libraries such as NumPy and SciPy. The availability of such libraries makes Python suitable for prototyping numerical methods. We demonstrate how PyDEC is used to solve physical and topological problems through several concise examples. [ABSTRACT FROM AUTHOR]