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The emergence of long-range order at low temperatures in atomistic systems with continuous symmetry is a fundamental, yet poorly understood phenomenon in Physics. To address this challenge we study a discrete microscopic model for an elastic crystal with dislocations in three dimensions, previously introduced by Ariza and Ortiz. The model is rich enough to support some realistic features of three-dimensional dislocation theory, most notably grains and the Read-Shockley law for grain boundaries, which we rigorously derive in a simple, explicit, geometry. We analyze the model at positive temperatures, in terms of a Gibbs distribution with energy function given by the Ariza-Ortiz Hamiltonian plus a contribution from the dislocation cores. Our main result is that the model exhibits long range positional order at low temperatures. The proof is based on the tools of discrete exterior calculus, together with cluster expansion techniques., 54 pages, 8 figures. Final version accepted for publication on JEMS. Compared to version v2: minor corrections and new latex format

In this paper we discuss the geometry and mechanics of discrete manifolds—so called simplicial complexes that are essentially triangulated regions—motivated by the works on discrete exterior calculus and differential geometry in computer graphics. We show that the classical ideas related to the geometry of continuous manifolds, such as metric, curvature, and affine connections take on a much simpler and more intuitive aspect when discussed in the context of such discrete triangulated manifolds. We introduce the notion of a dual mesh to describe “dual” variables (technically differential forms). We use the dual mesh to show that the geometry of the defects such as microcracks, dislocations and incoherent boundaries as well as balance laws and constitutive relations can be introduced directly, without the need for “discretization” from a continuum. This opens up the possibility of direct simulations of these bodies without the need for a continuous counterpart. We end by demonstrating how the second law of thermodynamics can be used in such a situation including discrete non-local systems.

We present a geometric particle-in-cell (PIC) algorithm on unstructured meshes for studying electrostatic perturbations with frequency lower than electron gyrofrequency in magnetized plasmas. In this method, ions are treated as fully kinetic particles and electrons are described by the adiabatic response. The PIC method is derived from a discrete variational principle on unstructured meshes. To preserve the geometric structure of the system, the discrete variational principle requires that the electric field is interpolated using Whitney 1-forms, the charge is deposited using Whitney 0-forms and the electric field is computed by discrete exterior calculus. The algorithm has been applied to study the ion Bernstein wave (IBW) in two-dimensional magnetized plasmas. The simulated dispersion relations of the IBW in a rectangular region agree well with theoretical results. In a two-dimensional circular region with fixed boundary condition, the spectrum and eigenmode structures of the IBW are obtained from simulations. We compare the energy conservation property of the geometric PIC algorithm derived from the discrete variational principle with that of previous PIC methods on unstructured meshes. The comparison shows that the new PIC algorithm significantly improves the energy conservation property.

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.

Discrete mechanics is presented as an alternative to the equations of fluid mechanics, in particular to the Navier-Stokes equation. The derivation of the discrete equation of motion is built from the intuitions of Galileo, the principles of Galilean equivalence and relativity. Other more recent concepts such as the equivalence between mass and energy and the Helmholtz-Hodge decomposition complete the formal framework used to write a fundamental law of motion such as the conservation of accelerations, the intrinsic acceleration of the material medium, and the sum of the accelerations applied to it. The two scalar and vector potentials of the acceleration resulting from the decomposition into two contributions, to curl-free and to divergence-free, represent the energies per unit of mass of compression and shear. The solutions obtained by the incompressible Navier-Stokes equation and the discrete equation of motion are the same, with constant physical properties. This new formulation of the equation of motion makes it possible to significantly modify the treatment of surface discontinuities, thanks to the intrinsic properties established from the outset for a discrete geometrical description directly linked to the decomposition of acceleration. The treatment of the jump conditions of density, viscosity and capillary pressure is explained in order to understand the two-phase flows. The choice of the examples retained, mainly of the exact solutions of the continuous equations, serves to show that the treatment of the conditions of jumps does not affect the precision of the method of resolution., Comment: 34 pages, 29 figures

Explicit structure-preserving geometric particle-in-cell (PIC) algorithm in curvilinear orthogonal coordinate systems is developed. The work reported represents a further development of the structure-preserving geometric PIC algorithm achieving the goal of practical applications in magnetic fusion research. The algorithm is constructed by discretizing the field theory for the system of charged particles and electromagnetic field using Whitney forms, discrete exterior calculus, and explicit non-canonical symplectic integration. In addition to the truncated infinitely dimensional symplectic structure, the algorithm preserves exactly many important physical symmetries and conservation laws, such as local energy conservation, gauge symmetry and the corresponding local charge conservation. As a result, the algorithm possesses the long-term accuracy and fidelity required for first-principles-based simulations of the multiscale tokamak physics. The algorithm has been implemented in the SymPIC code, which is designed for high-efficiency massively-parallel PIC simulations in modern clusters. The code has been applied to carry out whole-device 6D kinetic simulation studies of tokamak physics. A self-consistent kinetic steady state for fusion plasma in the tokamak geometry is numerically found with a predominately diagonal and anisotropic pressure tensor. The state also admits a steady-state sub-sonic ion flow in the range of 10 km s−1, agreeing with experimental observations and analytical calculations Kinetic ballooning instability in the self-consistent kinetic steady state is simulated. It is shown that high-n ballooning modes have larger growth rates than low-n global modes, and in the nonlinear phase the modes saturate approximately in 5 ion transit times at the 2% level by the E × B flow generated by the instability. These results are consistent with early and recent electromagnetic gyrokinetic simulations.

This article introduces a new and general construction of discrete Hodge operator in the context of Discrete Exterior Calculus (DEC). This discrete Hodge operator permits to circumvent the well-centeredness limitation on the mesh with the popular diagonal Hodge. It allows a dual mesh based on any interior point, such as the incenter or the barycenter. It opens the way towards mesh-optimized discrete Hodge operators. In the particular case of a well-centered triangulation, it reduces to the diagonal Hodge if the dual mesh is circumcentric. Based on an analytical development, this discrete Hodge does not make use of Whitney forms, and is exact on piecewise constant forms, whichever interior point is chosen for the construction of the dual mesh. Numerical tests oriented to the resolution of fluid mechanics problems and thermal transfer are carried out. Convergence on various types of mesh is investigated. Flat and non-flat domains are considered.

A conservative primitive variable discrete exterior calculus (DEC) discretization of the Navier-Stokes equations is performed. An existing DEC method (Mohamed, M. S., Hirani, A. N., Samtaney, R. (2016). Discrete exterior calculus discretization of incompressible Navier-Stokes equations over surface simplicial meshes. Journal of Computational Physics, 312, 175-191) is modified to this end, and is extended to include the energy-preserving time integration and the Coriolis force to enhance its applicability to investigate the late time behavior of flows on rotating surfaces, i.e., that of the planetary flows. The simulation experiments show second order accuracy of the scheme for the structured-triangular meshes, and first order accuracy for the otherwise unstructured meshes. The method exhibits second order kinetic energy relative error convergence rate with mesh size for inviscid flows. The test case of flow on a rotating sphere demonstrates that the method preserves the stationary state, and conserves the inviscid invariants over an extended period of time.

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.

In this paper, the time domain Maxwell’s equations are solved using the discrete exterior calculus (DEC) formalism in the two-dimensional space. To truncate the computational domain, the complex frequency-shifted perfectly matched layer (CFS-PML) concept is applied to create a reflectionless artificial layer. The paper presents a new numerical procedure to easily implement the CFS-PML with curved inner boundary. In order to numerically realize the PML, in a simplicial mesh, this paper proposes to utilize the nearest neighbor algorithm to associate point sets to boundary points. The distance from points to the boundary curve defines the attenuation function inside the PML. The performance of the approach is assessed by measuring the reflection error for three numerical experiments.

Maxwell's equations are discretized on a Face-Centered Cubic (FCC) lattice instead of a simple cubic as an alternative to the standard Yee method for improvements in numerical dispersion characteristics and grid isotropy of the method. Explicit update equations and numerical dispersion expressions, and the stability criteria are derived. Also, several tools available to the standard Yee method such as PEC/PMC boundary conditions, absorbing boundary conditions, and scattered field formulation are extended to this method as well. A comparison between the FCC and the Yee formulations is made, showing that the FCC method exhibits better dispersion compared to its Yee counterpart. Simulations are provided to demonstrate both the accuracy and grid isotropy improvement of the method.

Discrete exterior calculus (DEC) is a structure-preserving numerical framework for partial differential equations solution, particularly suitable for simplicial meshes. A longstanding and widespread assumption has been that DEC requires special (Delaunay) triangulations, which complicated the mesh generation process especially on curved surfaces. This paper presents numerical evidences demonstrating that this restriction is unnecessary. Convergence experiments are carried out for various physical problems using both Delaunay and non-Delaunay triangulations. Signed diagonal definition for the key DEC operator (Hodge star) is adopted. The errors converge as expected for all considered meshes and experiments. This relieves the DEC paradigm from unnecessary triangulation limitation.

Discrete exterior calculus (DEC) offers a coordinate–free discretization of exterior calculus especially suited for computations on curved spaces. In this work, we present an extended version of DEC on surface meshes formed by general polygons that bypasses the need for combinatorial subdivision and does not involve any dual mesh. At its core, our approach introduces a new polygonal wedge product that is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule. Based on the discrete wedge product, we then derive a novel primal–to–primal Hodge star operator. Combining these three ‘basic operators’ we then define new discrete versions of the contraction operator and Lie derivative, codifferential and Laplace operator. We discuss the numerical convergence of each one of these proposed operators and compare them to existing DEC methods. Finally, we show simple applications of our operators on Helmholtz–Hodge decomposition, Laplacian surface fairing, and Lie advection of functions and vector fields on meshes formed by general polygons.

Crystal defects have microstructure, and this microstructure should be related to the microstructure of the original crystal. Hence each type of crystals may have similar defects due to the same failure mechanism originated from the same microstructure, if they are under the same loading conditions. In this work, we propose a multiscale crystal defect dynamics (MCDD) model that models defects by considering its intrinsic microstructure derived from the microstructure or material genome of the original perfect crystal. The main novelties of present work are: (1) the discrete exterior calculus and algebraic topology theory are used to construct a scale-up (coarse-grained) dual lattice model for crystal defects, which may represent all possible defect modes inside a crystal; (2) a higher order Cauchy–Born rule (up to the fourth order) is adopted to construct atomistic-informed constitutive relations for various defect process zones, and (3) an hierarchical strain gradient theory based finite element formulation is developed to support an hierarchical multiscale cohesive (process) zone model for various defects in a unified formulation. The efficiency of MCDD computational algorithm allows us to simulate dynamic defect evolution at large scale while taking into account atomistic interaction. The MCDD model has been validated by comparing of the results of MCDD simulations with that of molecular dynamics (MD) in the cases of nanoindentation and uniaxial tension. Numerical simulations have shown that MCDD model can predict dislocation nucleation induced instability and inelastic deformation, and thus it may provide an alternative solution to study crystal plasticity.

Our aim in this paper is to provide a theory of discrete Riemann surfaces based on quadrilateral cellular decompositions of Riemann surfaces together with their complex structure encoded by complex weights. Previous work, in particular of Mercat, mainly focused on real weights corresponding to quadrilateral cells having orthogonal diagonals. We discuss discrete coverings, discrete exterior calculus, and discrete Abelian integrals. Our presentation includes several new notions and results such as branched coverings of discrete Riemann surfaces, the discrete Riemann-Hurwitz Formula, double poles of discrete one-forms and double values of discrete meromorphic functions that enter the discrete Riemann-Roch Theorem, and a discrete Abel-Jacobi map., Comment: 35 pages, 9 figures. Minor corrections, new Figures 3, 5, 6, and 7

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.

Numerous tasks in imaging and vision can be formulated as variational problems over vector-valued maps. We approach the relaxation and convexification of such vectorial variational problems via a lifting to the space of currents. To that end, we recall that functionals with polyconvex Lagrangians can be reparametrized as convex one-homogeneous functionals on the graph of the function. This leads to an equivalent shape optimization problem over oriented surfaces in the product space of domain and codomain. A convex formulation is then obtained by relaxing the search space from oriented surfaces to more general currents. We propose a discretization of the resulting infinite-dimensional optimization problem using Whitney forms, which also generalizes recent "sublabel-accurate" multilabeling approaches., Oral presentation at CVPR 2019

A discretization of the Convection-Diffusion equation is developed based on Discrete Exterior Calculus (DEC). While DEC discretization of the diffusive term in the equation is well understood, the convective part (with non-constant convective flow) had not been DEC discretized. In this study, we develop such discretization of the convective term using geometric arguments. We can discretize the convective term for both compressible and incompressible flow. Moreover, since the Finite Element Method with linear interpolation functions (FEML) and DEC local matrix formulations are similar, this numerical scheme is well suited for parallel computing. Using this feature, numerical tests are carried out on simple domains with coarse and fine meshes to compare DEC and FEML and show numerical convergence for stationary problems.

International audience; 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 inte-grators 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 (Noether'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.

The de Rham-Hodge theory is a landmark of the 20$^\text{th}$ Century's mathematics and has had a great impact on mathematics, physics, computer science, and engineering. This work introduces an evolutionary de Rham-Hodge method to provide a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds constructed from a filtration, which induces a family of evolutionary de Rham complexes. While the present method can be easily applied to close manifolds, the emphasis is given to more challenging compact manifolds with 2-manifold boundaries, which require appropriate analysis and treatment of boundary conditions on differential forms to maintain proper topological properties. Three sets of unique evolutionary Hodge Laplacian operators are proposed to generate three sets of topology-preserving singular spectra, for which the multiplicities of zero eigenvalues correspond to exactly the persistent Betti numbers of dimensions 0, 1, and 2. Additionally, three sets of non-zero eigenvalues further reveal both topological persistence and geometric progression during the manifold evolution. Extensive numerical experiments are carried out via the discrete exterior calculus to demonstrate the utility and usefulness of the proposed method for data representation and shape analysis., 28 pages, 15 figures

We investigate the performance of various discrete Hodge star operators for discrete exterior calculus (DEC) using circumcentric and barycentric dual meshes. The performance is evaluated through the DEC solution of Darcy and incompressible Navier-Stokes flows over surfaces. While the circumcentric Hodge operators may be favorable due to their diagonal structure, the barycentric (geometric) and the Galerkin Hodge operators have the advantage of admitting arbitrary simplicial meshes. Numerical experiments reveal that the barycentric and the Galerkin Hodge operators retain the numerical convergence order attained through the circumcentric (diagonal) Hodge operators. Furthermore, when the barycentric or the Galerkin Hodge operators are employed, a super-convergence behavior is observed for the incompressible flow solution over unstructured simplicial surface meshes generated by successive subdivision of coarser meshes. Insofar as the computational cost is concerned, the Darcy flow solutions exhibit a moderate increase in the solution time when using the barycentric or the Galerkin Hodge operators due to a modest decrease in the linear system sparsity. On the other hand, for the incompressible flow simulations, both the solution time and the linear system sparsity do not change for either the circumcentric or the barycentric and the Galerkin Hodge operators. Various definitions for discrete Hodge star operators are investigated.Darcy and incompressible Navier-Stokes flows are used for numerical experiments.Barycentric Hodge star reproduces the convergence rate of the circumcentric Hodge.The difference in the computational cost is generally minor.

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.

We model meteoroids entering the Earth[U+05F3]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. © 2016 Elsevier Ltd.

A conservative discretization of incompressible Navier-Stokes equations is developed based on discrete exterior calculus (DEC). A distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product. The governing equations are first rewritten using the exterior calculus notation, replacing vector calculus differential operators by the exterior derivative, Hodge star and wedge product operators. The discretization is then carried out by substituting with the corresponding discrete operators based on the DEC framework. Numerical experiments for flows over surfaces reveal a second order accuracy for the developed scheme when using structured-triangular meshes, and first order accuracy for otherwise unstructured meshes. By construction, the method is conservative in that both mass and vorticity are conserved up to machine precision. The relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step., 35 pages, 8 figures

We revisit the theory of Discrete Exterior Calculus (DEC) in 2D for general triangulations, relying only on Vector Calculus and Matrix Algebra. We present DEC numerical solutions of the Poisson equation and compare them against those found using the Finite Element Method with linear elements (FEML)., We made some corrections, included one more example and added several figures

The shallow water equations provide a useful analogue of the fully compressible Euler equations since they have similar characteristics: conservation laws, inertia-gravity and Rossby waves, and a (quasi-) balanced state. In order to obtain realistic simulation results, it is desirable that numerical models have discrete analogues of these properties. Two prototypical examples of such schemes are the 1981 Arakawa and Lamb (AL81) C-grid total energy and potential enstrophy conserving scheme, and the 2007 Salmon (S07) Z-grid total energy and potential enstrophy conserving scheme. Unfortunately, the AL81 scheme is restricted to logically square, orthogonal grids, and the S07 scheme is restricted to uniform square grids. The current work extends the AL81 scheme to arbitrary non-orthogonal polygonal grids and the S07 scheme to arbitrary orthogonal spherical polygonal grids in a manner that allows for both total energy and potential enstrophy conservation, by combining Hamiltonian methods (work done by Salmon, Gassmann, Dubos, and others) and discrete exterior calculus (Thuburn, Cotter, Dubos, Ringler, Skamarock, Klemp, and others). Detailed results of the schemes applied to standard test cases are deferred to part 2 of this series of papers.

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.

A class of variational schemes for the hydrodynamic-electrodynamic model of lossless free-electron gas in a quasineutral background is developed for high-quality simulations of surface plasmon polaritons. The Lagrangian density of lossless free-electron gas with a self-consistent electromagnetic field is established, and the dynamical equations with the associated constraints are obtained via a variational principle. Based on discrete exterior calculus, the action functional of this system is discretized and minimized to obtain the discrete dynamics. Newton-Raphson iteration and the biconjugate gradient stabilized method are equipped as a hybrid nonlinear-linear algebraic solver. Instead of discretizing the partial differential equations, the variational schemes have better numerical properties in secular simulations, as they preserve the discrete Lagrangian symplectic structure, gauge symmetry, and general energy-momentum density. Two numerical experiments were performed. The numerical results reproduce characteristic dispersion relations of bulk plasmons and surface plasmon polaritons, and the numerical errors of conserved quantities in all experiments are bounded by a small value after long-term simulations., Comment: 9 figures

We present a novel linear subdivision scheme for face-based tangent directional fields on triangle meshes. Our subdivision scheme is based on a novel coordinate-free representation of directional fields as halfedge-based scalar quantities, bridging the mixed finite-element representation with discrete exterior calculus. By commuting with differential operators, our subdivision is structure preserving: it reproduces curl-free fields precisely and reproduces divergence-free fields in the weak sense. Moreover, our subdivision scheme directly extends to directional fields with several vectors per face by working on the branched covering space. Finally, we demonstrate how our scheme can be applied to directional-field design, advection, and robust earth mover’s distance computation, for efficient and robust computation.

We consider a numerical approach for the incompressible surface Navier-Stokes equation on surfaces with arbitrary genus $g(\mathcal{S})$. The approach is based on a reformulation of the equation in Cartesian coordinates of the embedding $\mathbb{R}^3$, penalization of the normal component, a Chorin projection method and discretization in space by surface finite elements for each component. The approach thus requires only standard ingredients which most finite element implementations can offer. We compare computational results with discrete exterior calculus (DEC) simulations on a torus and demonstrate the interplay of the flow field with the topology by showing realizations of the Poincar\'e-Hopf theorem on $n$-tori.

We discuss the discretization of Maxwell-Vlasov equations based on a discrete exterior calculus framework, which provides a natural factorization of the discrete field equations into topological (metric-free) and metric-dependent parts. This enables a gain in geometrical flexibility when dealing with general grids and also the ab initio, exact preservation of conservation laws through discrete analogues. In particular, we describe a particle-in-cell (PIC) implementation of time-dependent discrete Maxwell-Vlasov equations, whereby the electromagnetic field are discretized using Whitney forms and coupled to particle dynamics by means of a gather-scatter scheme that yields exact charge-conservation on general grids. Numerical examples of PIC simulations such as vacuum diode and backward-wave oscillator are used to illustrate the approach.

We present a new finite-element time-domain (FETD) solver for analysis of axisymmetric devices based on discrete exterior calculus (DEC) and transformation optics (TO) concepts. The proposed FETD solver decomposes the fields into TEϕ and TMϕ modes, which are expanded by using appropriate set of (vector or scalar) basis functions. Utilizing DEC, trigono-metric orthogonality, and a leap-frog time-integrator, we obtain energy-conserving fully discrete Maxwell's equations. We explore TO principles to map the original problem from a cylindrical system to an equivalent problem on a Cartesian mesh embedded on an effective (artificial) inhomogeneous medium with radial variation. The new FETD solver is illustrated for the efficient solution of a backward-wave oscillator (BWO) encompassing a slow-wave waveguide with sinusoidal corrugations.

We discuss a new electromagnetic particle-in-cell algorithm for the simulation of Maxwell-Vlasov equations on unstructured grids. The use of discrete exterior calculus and differential forms of various degrees enables numerical charge conservation from first principles, down to the numerical precision floor. In addition, energy conservation is obtained via a symplectic field update. The algorithm is illustrated for the modeling of high-power microwave devices based on Cerenkov radiation driven by relativistic plasma beams.

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 3D and 2D inhomogeneous structures, and also show that the charge continuity relation is naturally satisfied. Then we introduce effective signed dual volume to incorporate material information into Hodge star operators 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 demonstrate the implementation 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., 41 pages with 19 figures. Briefly presented in IEEE APS 2016 meeting

We present the discrete exterior calculus (DEC) to solve discrete partial differential equations on discrete objects such as cell complexes. To cope with advection-dominated problems, we introduce a novel stabilization technique to the DEC. To this end, we use the fact that the DEC coincides in special situations with known discretization schemes such as finite volumes or finite differences. Thus, we can carry over well-established upwind stabilization methods introduced for these classical schemes to the DEC. This leads in particular to a stable discretization of the Lie-derivative. We present the numerical features of this new discretization technique and study its numerical properties for simple model problems and for advection-diffusion processes on simple surfaces.

We consider a numerical approach for the incompressible surface Navier-Stokes equation. The approach is based on the covariant form and uses discrete exterior calculus (DEC) in space and a semi-implicit discretization in time. The discretization is described in detail and related to finite difference schemes on staggered grids in flat space for which we demonstrate second order convergence. We compare computational results with a vorticity-stream function approach for surfaces with genus \(g(\mathcal{S}) = 0\) and demonstrate the interplay between topology, geometry and flow properties. Our discretization also allows to handle harmonic vector fields, which we demonstrate on a torus.

A staggered discretization of the incompressible Navier–Stokes equations is presented for polyhedral non orthogonal nonsmooth meshes admitting a barycentric dual mesh. The discretization is constructed by using concepts of discrete exterior calculus. The method strictly conserves mass, momentum and energy in the absence of viscosity.

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.

A discrete diagonal Hodge star operator for pairwise Delaunay triangulations was described in Hirani et al. (2013). This note fixes errors in two figures and some indexing mistakes in the original paper. One consequence of this fix is that discrete exterior calculus may be applicable to a wider class of meshes.

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., Comment: Corrected error in Figure 1 (columns 3 and 4) and Figure 6 and a formula error in Section 2. All mathematical statements (theorems and lemmas) are unchanged. The previous arXiv version v3 (minus the Appendix) appeared in the journal Computer-Aided Design

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. Graphical abstractDisplay Omitted HighlightsWe define a discrete Laplace operator for temporally coherent mesh sequences.A parameter controls the influence of the motion and w.r.t. the geometryApplication to spacetime editing is provided as example.

We apply high-quality discretizations to simulate electromagnetic microwaves. Instead of the vector field presentations, we focus on differential forms and discretize the model in the spatial domain using the discrete exterior calculus. At the discrete level, both the Hodge operators and the time discretization are optimized for time-harmonic simulations. Non-uniform spatial and temporal discretization are applied in problems in which the wavelength is highly-variable and geometry contains sub-wavelength structures. peerReviewed

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 R^2, there are infinitely many vertex links that prohibit a well-centered tetrahedral mesh in R^3., Approximately 30 pages. Contains 25 figures. Some figures include multiple graphics

Current mesh reduction techniques, while numerous, all primarily reduce mesh size by successive element deletion (e.g. edge collapses) with the goal of geometric and topological feature preservation. The choice of geometric error used to guide the reduction process is chosen independent of the function the end user aims to calculate, analyze, or adaptively refine. In this paper, we argue that such a decoupling of structure from function modeling is often unwise as small changes in geometry may cause large changes in the associated function. A stable approach to mesh decimation, therefore, ought to be guided primarily by an analysis of functional sensitivity, a property dependent on both the particular application and the equations used for computation (e.g. integrals, derivatives, or integral/partial differential equations). We present a methodology to elucidate the geometric sensitivity of functionals via two major functional discretization techniques: Galerkin finite element and discrete exterior calculus. A number of examples are given to illustrate the methodology and provide numerical examples to further substantiate our choices., 6 pages, to appear in proceedings of the SIAM-ACM Joint Conference on Geometric and Physical Modeling, 2009

This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for the continuous problem. After a brief introduction to finite element methods, the discretization methods we consider, we develop an abstract Hilbert space framework for analyzing stability and convergence. In this framework, the differential complex is represented by a complex of Hilbert spaces and stability is obtained by transferring Hodge theoretic structures from the continuous level to the discrete. We show stable discretization is achieved if the finite element spaces satisfy two hypotheses: they form a subcomplex and there exists a bounded cochain projection from the full complex to the subcomplex. Next, we consider the most canonical example of the abstract theory, in which the Hilbert complex is the de Rham complex of a domain in Euclidean space. We use the Koszul complex to construct two families of finite element differential forms, show that these can be arranged in subcomplexes of the de Rham complex in numerous ways, and for each construct a bounded cochain projection. The abstract theory therefore applies to give the stability and convergence of finite element approximations of the Hodge Laplacian. Other applications are considered as well, especially to the equations of elasticity. Background material is included to make the presentation self-contained for a variety of readers., Comment: 74 pages, 8 figures; reorganized introductory material, added additional example and references; final version accepted by Bulletin of the AMS, added references to codes and adjusted some diagrams. Bulletin of the American Mathematical Society, to appear 2010

The inclusion of source terms in discrete gravity is a long-standing problem. Providing a consistent coupling of source to the lattice in Regge Calculus (RC) yields a robust unstructured spacetime mesh applicable to both numerical relativity and quantum gravity. RC provides a particularly insightful approach to this problem with its purely geometric representation of spacetime. The simplicial building blocks of RC enable us to represent all matter and fields in a coordinate-free manner. We provide an interpretation of RC as a discrete exterior calculus framework into which non-gravitational fields naturally couple with the simplicial lattice. Using this approach we obtain a consistent mapping of the continuum action for non-gravitational fields to the Regge lattice. In this paper we apply this framework to scalar, vector and tensor fields. In particular we reconstruct the lattice action for (1) the scalar field, (2) Maxwell field tensor and (3) Dirac particles. The straightforward application of our discretization techniques to these three fields demonstrates a universal implementation of coupling source to the lattice in Regge calculus., Comment: 10 pages, no figures, Latex, fixed typos and minor corrections.