Search results for "Discrete exterior calculus"

Controlled time integration for the numerical simulation of meteor radar reflections

2016

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 …

Using Wave Propagation Simulations and Convolutional Neural Networks to Retrieve Thin Film Thickness from Hyperspectral Images

2021

Ill-posed inversion problems are one of the major challenges when there is a need to combine measurements with the theory and numerical model. In this study, we demonstrate the use of wave propagation simulations to train a convolutional neural network (CNN) for retrieving sub-wavelength thickness profiles of thin film coatings from hyperspectral images. The simulations are produced by solving numerically one-dimensional wave equation with a method based on Discrete Exterior Calculus (DEC). This approach provides a powerful tool to produce large sets of training data for the neural network. CNN was verified by simulated verification sets and measured reflectance spectra, both of which showe…

High-quality discretizations for microwave simulations

2016

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

New degrees of freedom for differential forms on cubical meshes

2022

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. Significantly, 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.

Discrete exterior calculus for photonic crystal waveguides

2022

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. …

Systematic implementation of higher order Whitney forms in methods based on discrete exterior calculus

2022

AbstractWe 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…

On a numerical solution of the Maxwell equations by discrete exterior calculus

2014

Generalized wave propagation problems and discrete exterior calculus

2018

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 cumulat…

Efficient Time Integration of Maxwell's Equations with Generalized Finite Differences

2015

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 discretizati…