Theoretical Developments and Simulation Tools for Discrete Geometric Computational Electromagnetics in the Time Domain

The original Finite Difference Time Domain (FDTD) method, devised by Yee in 1966, inspired a conspicuous amount of research in the field of numerical schemes for solving Maxwell's equations in the time domain, thanks to its simplicity and computational efficiency. The original algorithm, which computes the values of electric and magnetic fields on the points of two interlocked Cartesian orthogonal grids, has also been rewritten as a Finite Integration Technique (FIT) algorithm, where the computed quantities are the integrals of the field over geometric elements of the grids. Both formulations suffer from the so-called staircase approximation problem: when an interface between regions with discontinuous material properties is not flat, the expected convergence properties of the numerical solution are not guaranteed if an exaggeratedly fine grid is not used. In this regard, even recent improved techniques based on combined arithmetic and harmonic averaging techniques cannot achieve second order accuracy in time in the neighborhood of the interface. This problem is inherent to the Cartesian orthogonal discretization of the domain, as unstructured grids (tetrahedral or polyhedral) mesh generators avoid it with grids conformal to the discontinuities in material properties. Approaches that have had some degree of success in adapting the FDTD algorithm to unstructured grids include schemes based on the Finite Element method (FEM), on the Cell Method and, more recently, formulations based on the Discontinuous Galerkin (DG) approach. Yet, consistency issues of discontinuous methods question their accuracy, since these methods do not explicitly force tangential continuity of the fields across mesh element interfaces, weakening the local fulfillment of physical conservation laws (charge conservation in particular). On the other hand, classical FEM formulations, which do not share this drawback, trade their geometric flexibility with an implicit time-stepping scheme, i.e. the computation includes solving a linear system of algebraic equations at each time-step. This severely limits the scalability of the algorithm. Recently, a technique has been introduced by Codecasa et al., based on a Discrete Geometric Approach (DGA) which instead yields an explicit, consistent and conditionally stable algorithm on tetrahedral grids. Due to the promising features of this approach, a thorough analysis of its performance and accuracy is in order, since neither have been widely tested yet. This work addresses the issue and shows that the latter approach compares favorably with equal order FEM approaches on unstructured grids. An important drawback of the DGA approach is that it was originally formulated for strictly dielectric materials. The way to overcome this limitation is unfortunately not obvious. The present work addresses this issue and solves it without sacrificing any property of the original algorithm. Furthermore, although the properties of the material operators in the original formulation show that the resulting scheme is conditionally stable, a Courant--Friedrich--Lewy (CFL) condition equivalent to the one of the original FDTD algorithm is not given. This is also dealt with in the bulk of this thesis and a sufficient condition for the stability of this algorithm is given with proof. Finally a practical toolbox for time domain electromagnetic simulations, tentatively named TetFIT and resulting from the coding efforts of the author is presented, with preliminary results on its performance when running on Graphical Processing Units (GPUs).The original Finite Difference Time Domain (FDTD) method, devised by Yee in 1966, inspired a conspicuous amount of research in the field of numerical schemes for solving Maxwell's equations in the time domain, thanks to its simplicity and computational efficiency. The original algorithm, which computes the values of electric and magnetic fields on the points of two interlocked Cartesian orthogonal grids, has also been rewritten as a Finite Integration Technique (FIT) algorithm, where the computed quantities are the integrals of the field over geometric elements of the grids. Both formulations suffer from the so-called staircase approximation problem: when an interface between regions with discontinuous material properties is not flat, the expected convergence properties of the numerical solution are not guaranteed if an exaggeratedly fine grid is not used. In this regard, even recent improved techniques based on combined arithmetic and harmonic averaging techniques cannot achieve second order accuracy in time in the neighborhood of the interface. This problem is inherent to the Cartesian orthogonal discretization of the domain, as unstructured grids (tetrahedral or polyhedral) mesh generators avoid it with grids conformal to the discontinuities in material properties. Approaches that have had some degree of success in adapting the FDTD algorithm to unstructured grids include schemes based on the Finite Element method (FEM), on the Cell Method and, more recently, formulations based on the Discontinuous Galerkin (DG) approach. Yet, consistency issues of discontinuous methods question their accuracy, since these methods do not explicitly force tangential continuity of the fields across mesh element interfaces, weakening the local fulfillment of physical conservation laws (charge conservation in particular). On the other hand, classical FEM formulations, which do not share this drawback, trade their geometric flexibility with an implicit time-stepping scheme, i.e. the computation includes solving a linear system of algebraic equations at each time-step. This severely limits the scalability of the algorithm. Recently, a technique has been introduced by Codecasa et al., based on a Discrete Geometric Approach (DGA) which instead yields an explicit, consistent and conditionally stable algorithm on tetrahedral grids. Due to the promising features of this approach, a thorough analysis of its performance and accuracy is in order, since neither have been widely tested yet. This work addresses the issue and shows that the latter approach compares favorably with equal order FEM approaches on unstructured grids. An important drawback of the DGA approach is that it was originally formulated for strictly dielectric materials. The way to overcome this limitation is unfortunately not obvious. The present work addresses this issue and solves it without sacrificing any property of the original algorithm. Furthermore, although the properties of the material operators in the original formulation show that the resulting scheme is conditionally stable, a Courant--Friedrich--Lewy (CFL) condition equivalent to the one of the original FDTD algorithm is not given. This is also dealt with in the bulk of this thesis and a sufficient condition for the stability of this algorithm is given with proof. Finally a practical toolbox for time domain electromagnetic simulations, tentatively named TetFIT and resulting from the coding efforts of the author is presented, with preliminary results on its performance when running on Graphical Processing Units (GPUs)

An arbitrary-order Cell Method with block-diagonal mass-matrices for the time-dependent 2D Maxwell equations

We introduce a new numerical method for the time-dependent Maxwell equations on unstructured meshes in two space dimensions. This relies on the introduction of a new mesh, which is the barycentric-dual cellular complex of the starting simplicial mesh, and on approximating two unknown fields with integral quantities on geometric entities of the two dual complexes. A careful choice of basis-functions yields cheaply invertible block-diagonal system matrices for the discrete time-stepping scheme. The main novelty of the present contribution lies in incorporating arbitrary polynomial degree in the approximating functional spaces, defined through a new reference cell. The presented method, albeit a kind of Discontinuous Galerkin approach, requires neither the introduction of user-tuned penalty parameters for the tangential jump of the fields, nor numerical dissipation to achieve stability. In fact an exact electromagnetic energy conservation law for the semi-discrete scheme is proved and it is shown on several numerical tests that the resulting algorithm provides spurious-free solutions with the expected order of convergence.Comment: 34 pages, 14 figures, submitte

Lean cohomology computation for electromagnetic modeling

Solving eddy current problems formulated by using a magnetic scalar potential in the insulator requires a topological pre-processing to find the so-called first cohomology basis of the insulating region, which may be very time-consuming for challenging industrially driven problems. The physics-inspired D\u142otko-Specogna (DS) algorithm was shown to be superior to alternatives in performing such a topological pre-processing. Yet, the DS algorithm is particularly fast when it produces as output not a regular cohomology basis but a so-called lazy one, which contains the regular one but it keeps also some additional redundant elements. Having a regular basis may be advantageous over the lazy basis if a technique to produce it would take about the same time as the computation of a lazy basis. In the literature, such a technique is missing. This paper covers this gap by introducing modifications to the DS algorithm to compute a regular basis of the first cohomology group in practically the same time as the generation of a lazy cohomology basis. The speedup of this modified DS algorithm with respect to the best alternative reaches more than two orders of magnitudes on challenging benchmark problems. This demonstrates the potential impact of the proposed contribution in the low-frequency computational electromagnetics community and beyond. \ua9 2017 IEEE

Computation of Relative 1-Cohomology Generators From a 1-Homology Basis for Eddy Currents Boundary Integral Formulations

Efficient boundary integral formulations based on stream functions for solving eddy current problems in thin conductors, which are modeled by the orientable combinatorial two-manifold with boundary, need generators of the first relative cohomology group to make the problem well defined. The state-of-the-art technique is to compute directly the relative cohomology generators with a combinatorial algorithm having linear worst-case complexity. In this paper, we propose to compute the relative cohomology generators from the homology generators, introducing a novel and general algorithm whose running time is again linear in the worst case. The advantage is that one may use an off-the-shelf software to compute the homology generators and implement only a simple and cheap procedure to obtain the required relative cohomology generators. Although the presented applications relate to ac power systems, the proposed technique is of general interest, and may be used for other applications in computational science and engineering. \ua9 2016 IEE

Arbitrary order spline representation of cohomology generators for isogeometric analysis of eddy current problems

The eddy current problem has many relevant practical applications in science, ranging from non-destructive testing to magnetic confinement of plasma in fusion reactors. It arises when electrical conductors are immersed in an external time-varying magnetic field operating at frequencies for which electromagnetic wave propagation effects can be neglected. Popular formulations of the eddy current problem either use the magnetic vector potential or the magnetic scalar potential. They have individual advantages and disadvantages. One challenge is related to differential geometry: Scalar potential based formulations run into trouble when conductors are present in non-trivial topology, as approximation spaces must be then augmented with generators of the first cohomology group of the non-conducting domain. For all existing algorithms based on lowest order methods it is assumed that the extension of the graph-based algorithms to high-order approximations requires hierarchical bases for the curl-conforming discrete spaces. However, building on insight on de Rham complexes approximation with splines, we will show in the present submission that the hierarchical basis condition is not necessary. Algorithms based on spanning tree techniques can instead be adapted to work on an underlying hexahedral mesh arising from isomorphisms between spline spaces of differential forms and de Rham complexes on an auxiliary control mesh

GPU Accelerated Time-Domain Discrete Geometric Approach Method for Maxwell's Equations on Tetrahedral Grids

A recently introduced time-domain method for the numerical solution of Maxwell's equations on unstructured grids is reformulated in a novel way, with the aim of implementation on graphical processing units (GPUs). Numerical tests show that the GPU implementation of the resulting scheme yields correct results, while also offering an order of magnitude in speedup and still preserving all of the main properties of the original finite-difference time-domain algorithm. © 2017 IEEE

T-Ω formulation with higher order hierarchical basis functions for non simply connected conductors

This paper extends the T-\u3a9 formulation for eddy currents based on higher order hierarchical basis functions so that it can deal with conductors of arbitrary topology. To this aim we supplement the classical hierarchical basis functions with non-local basis functions spanning the first de Rham cohomology group of the insulating region. Such non-local basis functions may be efficiently found in negligible time with the recently introduced DS algorithm

High order geometric methods with splines: an analysis of discrete Hodge--star operators

A new kind of spline geometric method approach is presented. Its main ingredient is the use of well established spline spaces forming a discrete de Rham complex to construct a primal sequence $\{X^k_h\}^n_{k=0}$, starting from splines of degree $p$, and a dual sequence $\{\tilde{X}^k_h\}_{k=0}^n$, starting from splines of degree $p-1$. By imposing homogeneous boundary conditions to the spaces of the primal sequence, the two sequences can be isomorphically mapped into one another. Within this setup, many familiar second order partial differential equations can be finally accommodated by explicitly constructing appropriate discrete versions of constitutive relations, called Hodge--star operators. Several alternatives based on both global and local projection operators between spline spaces will be proposed. The appeal of the approach with respect to similar published methods is twofold: firstly, it exhibits high order convergence. Secondly, it does not rely on the geometric realization of any (topologically) dual mesh. Several numerical examples in various space dimensions will be employed to validate the central ideas of the proposed approach and compare its features with the standard Galerkin approach in Isogeometric Analysis.Comment: Updated with revised version: 25 pages, 8 figures, 4 table