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In this work, a multiplicative regularization scheme is applied to the two-dimensional EIT inversion. In the cost functional, the data misfit is multiplied by a weighted L2-norm-based regularization factor. Gauss-Newton method is used to minimize the cost functional iteratively. In the implementation of the multiplicative regularization scheme, the gradient and divergence operators need to be approximated on triangular meshes. For this purpose, discrete exterior calculus (DEC) theory is applied to rigorously formulate these operators. Numerical examples show a good reconstruction and anti-noise performance of the multiplicative regularization scheme.

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.

A novel computational electromagnetic method developed with discrete exterior calculus (DEC) (M. Desbrun et al., arXiv:math/0508341) on simplicial mesh is presented. As is well known, differential forms can been used to recast Maxwell's equations in a more succinct fashion, which completely separate metric-free and metric-dependent components. Instead of dealing with vectorial field as finite difference method and finite element method (FEM), DEC studies electric and magnetic fields in discrete differential forms, i.e. their integrals over simplicial structures. In DEC, electric and magnetic fields are represented as cochains, which are vectors with finite length. Differential operators, such as curl and divergence, are replaced by discrete exterior derivative, which are highly sparse matrices only containing geometrical connection information with elements 0, 1 or −1. From Yee grid, the necessity and importance of a dual mesh or dual grid has been realized. In DEC, the dual mesh is constructed by connecting circumcenters of simplicial structures (triangles in 2D and tetrahedrons in 3D). With this circumcenter dual or Voronoi dual, the discrete Hodge star operators, which map from primal cochains to dual cochains, can be constructed as diagonal matrices. Effective dual volume is also introduced to incorporate material information into Hodge star operators. Then constitutive relations in an inhomogeneous medium can be described with these Hodge star operators. The case when circumcenters fall outside triangles or tetrahedrons is also considered and treated appropriately. The implementation of various boundary conditions is also illustrated.

In this article, we present the development of Java library implementing Discrete Exterior Calculus (DEC) constructs using the Processing IDE rendering environment to analyze WaveFront .obj objects. The library contains common DEC structures and operators found in other implementations and provides local iterators to explore the topological relations that DEC theory relies on, along with classes able to render both vector and scalar fields using the Processing IDE capabilities.

A self-contained electromagnetic theory is developed on a simplicial lattice. Instead of dealing with vectorial field, discrete exterior calculus (DEC) studies the discrete differential forms of electric and magnetic fields. Circumcenter dual is adopted to achieve diagonality and simplicity of Hodge star operators. In this paper, Gauss' theorem and Stokes' theorem are shown to be satisfied inherently. Many other electromagnetic theorems, like reciprocity theorem, can be derived on this simplicial lattice consistently with an appropriate definition of wedge product between forms. The preservation of these theorems guarantees that this treatment of Maxwell's equations will not lead to spurious solutions.

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

A generalized modal analysis with discrete exterior calculus for 2-D or 3-D electromagnetic system is presented. For an arbitrarily shaped inhomogeneous system, discrete exterior calculus provides a simple way to formulate the problem based on a triangular or tetrahedral discretization. In this paper, circumcenter dual is adopted to achieve diagonal discrete Hodge star operators. We also consider signed dual volumes for all dimensions to keep the correctness and accuracy of discrete Hodge star operators. Traveling modes of inhomogeneous waveguides and resonant cavity modes are numerically calculated to validate this method.

This paper addresses the issue of structure-preserving discretization of open distributed-parameter systems with Hamiltonian dynamics. Employing the formalism of discrete exterior calculus, we introduce simplicial Dirac structures as discrete analogues of the Stokes-Dirac structure and demonstrate that they provide a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. This approach of discrete differential geometry, rather than discretizing the partial differential equations, allows to first discretize the underlying Stokes-Dirac structure and then to impose the corresponding finite-dimensional port-Hamiltonian dynamics. In this manner, we preserve a number of important topological and geometrical properties of the system.

The paper demonstrates how discrete exterior calculus (DEC) tools may be useful in computer vision and graphics. A variational approach provides a link with mechanics. Our development of DEC includes discrete differential forms, discrete vector fields and the operators acting on these. This development of a discrete calculus, when combined with the methods of discrete mechanics and other recent work is likely to have promising applications in a field like computer vision which offers such a rich variety of challenging variational problems to be solved computationally. As a specific example we consider the problem of template matching and show how numerical methods derived from a discrete exterior calculus are starting to play an important role in solving the equations of averaged template matching. We also show some example applications using variational problems from computer graphics and mechanics to demonstrate that formulating the problem discretely and using discrete methods for solution can lead to efficient algorithms.

A highly accurate FDTD formulation was developed on an overlapped cubic grid that greatly reduces numerical dispersion errors. However, errors in in the FDTD method arise not only from numerical dispersion, but from geometrical modelling as well. Although representing a significant progress in addressing the numerical dispersion problem, it is still confined to a cubic grid with the subsequent "stair-casing" geometric approximations that it entails. The material presented represents a fundamentally new paradigm for finite-difference methods which hopes to address both issues of numerical dispersion and geometrical modelling. It involves a rigorous mathematical framework based on concepts from topology and differential geometry. Particularly, it involves a construction of a discrete analog to the calculus of differential forms. It should be noted that the use of differential forms, and their lattice counterparts, is well known within the field of algebraic topology. However, the original contribution here lies in the introduction of a metric onto the lattice. It is with the metric that the adjoint exterior derivative may be defined, which is required for most physical systems not the least of importance being Maxwell's equations.