A boundary conformal Discontinuous Galerkin method for electromagnetic field problems on Cartesian grids

The thesis presents a novel numerical method based on the high order Discontinuous Galerkin (DG) method for three dimensional electrostatic and electro-quasistatic field problems where materials are of very complex shape and may move over time. A well-known example is water droplets oscillating on the surface of high voltage power transmission line insulators. The electric field at the surface of the insulator causes the oscillation of the water droplets. The oscillation, in turn, triggers partial discharges which have damaging effects on the polymer insulation layers of high voltage insulators. The simulation of such phenomena is highly complex from an electromagnetic point of view. Most numerical methods which are applied to such field problems use conforming meshes where the elements are fitted exactly to the material geometry. This implies that the element interfaces conform to the material boundaries or material interfaces. In general, the generation of conforming meshes is computationally expensive. Furthermore, when dealing with materials that move over time, conforming meshes need to be adapted to the changing material geometry at each point in time. To avoid the often computationally costly generation and adaption step of conforming meshes, the numerical method proposed in this thesis operates on a single fixed structured Cartesian mesh. First, field problems with non-moving materials are considered. To obtain accurate simulation results on field problems with complex-shaped materials, an additional approach, namely the cut-cell discretization approach, is applied. The cut-cell discretization approach subdivides the elements at material boundaries or interfaces into smaller sub-elements which are referred to as cut-cells. The approach is embedded into the Discontinuous Galerkin (DG) method for standard Cartesian meshes since the DG method allows for high order approximations and offers a great flexibility for additional approaches. Since the mesh is not fitted to the material geometry, geometrically small cut-cells might emerge. Therefore, two supplementary approaches, the adaptive approximation order method and the cell merging method are proposed which enable an accurate approximation even on geometrically small cut-cells. Furthermore, a DG hybridization is presented which lowers the number of degrees of freedom in domains where the high number of DG degrees of freedom is not necessary to obtain accurate results. The numerical method comprising all above mentioned approaches is labelled as boundary conformal DG (BCDG) method. In a second step, the BCDG method is extended to field problems where materials move over time. We refer to this approach as extension of the BCDG (EBCDG) method. The EBCDG method adapts to the moving materials by recalculating only the cut-cells at each point in time while the underlying Cartesian grid is kept fixed. Therefore, no computationally expensive mesh adaption or mesh generation steps are needed. The BCDG and the EBCDG method are applied to numerical examples of electrostatic (ES) and electro-quasistatic (EQS) field problems. First, numerical results of the BCDG method on a verification example of a cylindrical capacitor filled with two dielectric layers are shown. A convergence study and a comparison study illustrate the high accuracy of the BCDG method with respect to the number of degrees of freedom. Finally, the EBCDG method is applied to an example of a water droplet oscillating artificially on the insulation layer of a high voltage insulator. A convergence study demonstrates that even on a coarse mesh a high resolution of the potential and electric field solution can be achieved

Numerical simulation of deformation of a droplet in a stationary electric field using DG

Numerical simulation of deformation of a droplet in a stationary electric field is performed in the present research. The droplet is suspended in another immiscible fluid with the same density and viscosity but a different dielectric property (permittivity). By applying the electric field, the fluids are polarized that gives rise to mechanical forces and deformation. A two-way coupling occurs because of the forces exerted from the electric field on the droplet and the deformation of the droplet which changes the geometry for the electric field calculations. The droplet continues to deform until a force balance between the electric force, pressure and the surface tension is achieved and the droplet becomes a spheroid. An electromechanical approach is adopted to solve the above mentioned problem, which includes solving the governing equations of both the electric and fluid fields, computing the coupling forces and capturing the movement of the interface of the droplet and the surrounding fluid. A one-fluid approach is followed, which enables us to solve one set of the governing equations for both the droplet and the surrounding fluid. The interface is represented as the zero iso-value of a level set function and an advection equation is solved to find the movement of the interface. A diffuse interface model is used to regularize the jump in the fluid and electric properties. The governing equations of the electric and fluid fields and the level set advection equation are discretized using the Discontinuous Galerkin Finite Element method (DG) in the BoSSS code for solving conservation laws. The electric field is computed from the electric potential by considering the electrostatic equations. To find the electric potential, a Laplace equation is solved which has a jump in the permittivity at the interface. The Laplace equation is discretized using the interior penalty method (IP) which we modified for the case of high jumps in the permittivity. Assuming that the fluids are linear dielectric materials, the electric force is the dielectrophoretic force which is computed from the Kortweg-Helmholtz formula. This force is added as a body force to the incompressible Navier-Stokes equations, which are the governing equations for the fluid flow. Considering that there is no jump in the fluid properties, a single phase solver of the Navier-Stokes equations including the surface tension at the interface is developed. The surface tension force is added as a body force to the Navier-Stokes equations using the continuum surface force model (CSF). This model is known for producing a spurious velocity field. To decrease the spurious velocities, the surface tension term is calculated by using high degree polynomials for a precise calculation of the normal vector and curvature. To solve the incompressible Navier-Stokes equations using the DG method, a projection scheme with a consistent Neumann pressure boundary condition is employed and the same polynomial order for the velocity and pressure (equal-order method) is applied. Using the above-mentioned pressure boundary condition leads to an optimal convergence rate of k + 1 in the L2-norm for the pressure, which is not reported from other DG solvers. However, using the DG method, we have observed that discontinuities in the solutions at the cell boundaries can affect the solution accuracy and even cause a numerical instability. These accuracy and stability issues occur when the derivatives of the solution are computed. Therefore a flux-based method for calculation of the derivatives of the flow variables was adopted. As the results showed considerably improved accuracy and stability characteristics, we used the proposed method also in solving the above mentioned coupled problem

Wideband Impedance Boundary Conditions for FE/DG Methods for Solving Maxwell Equations in Time Domain

In this work, dispersive surface impedance boundary conditions are applied to Discontinuous Galerkin Method (DG-FEM) in the time and frequency domains, on a wide frequency band. Three different kinds of surface impedance boundary conditions are considered, namely Standard Impedance Boundary Condition (SIBC) for modeling smooth conductor surfaces with high conductivity, Corrugated Surface Boundary Condition (CSBC) for modeling corrugated conducting surfaces, and Impedance Transmission Boundary Condition (ITBC) for modeling electrically thin conductive sheets. Two different schemes for modeling dispersive surface impedance boundary conditions on a wide frequency band are presented, one in the frequency domain, and another in the time domain. In the frequency domain, a procedure for solving a complex nonlinear eigenvalue problem (EVP) arising from applying the dispersive impedance boundary conditions to the discrete Maxwell’s equations, is presented. The procedure is based on fixed point iteration, and it enables to solve for the nonlinear EVP as a linear EVP, and therefore to simplify the computational task significantly. In the time domain scheme, the dispersive boundary conditions are first approximated in the frequency domain as series of rational functions, and then transformed into the time domain by means of Laplace transform. The time stepping schemes for time domain simulations are obtained by means of Recursive Convolution (RC) and Auxiliary Differential Equation (ADE) methods. The frequency domain scheme, as well as the time domain scheme, are verified and validated by investigating the Q factors and the fundamental frequencies of different resonant structures. Numerical examples are given, and convergence studies are performed. The results are compared with the analytical results, as well as results obtained by commercial softwares. The developed schemes appear to be computationally efficient, and the accuracy very high, already with coarse meshes and low basis function orders

Conformal electromagnetic wave propagation using primal mimetic finite elements

Elektromagnetische Wellenausbreitung bildet die physikalische Grundlage für unzählige Anwendungen in verschiedenen Bereichen der heutigen Welt. Um räumliche Szenarien zu modellieren, muss der kontinuierliche Raum in geeigneter Weise in ein Rechengebiet umgewandelt werden. Üblich diskretisierte Modelle – welche auf verschiedenen Größen beruhen – berücksichtigen die Beziehungen zwischen Feldvariablen mittels Relationen, welche durch partielle Differentialgleichungen repräsentiert werden. Um mathematische Beziehungen zwischen abhängigen Variablen in zweckdienlicher Art nachzubilden, schaffen hyperkomplexe Zahlensysteme ein passendes alternatives Rahmenwerk. Dieser Ansatz bezweckt das Einbinden bestimmter Systemeigenschaften und umfasst zusätzlich zur Modellierung von Feldproblemen, bei denen alle Variablen vorkommen, auch vereinfachte Modelle. Um eine wettbewerbsfähige Alternative zur üblichen numerischen Behandlung elektromagnetischer Felder in beobachtungsorientierter Weise darzubieten, wird das elektrische und magnetische Feld elektromagnetischer Wellenfelder als eine zusammengefasste Feldgröße, eingebettet im Funktionenraum, verstanden. Dieses Vorgehen ist intuitiv, da beide Felder in der Elektrodynamik gemeinsam auftreten und direkt messbar sind. Der Schwerpunkt dieser Arbeit ist in zwei Ziele untergliedert. Auf der einen Seite wird ein umformuliertes Maxwell-System in einer metrikfreien Umgebung mittels dem sogenannten „bikomplexen Ansatz“ umfassend untersucht. Auf der anderen Seite wird eine mögliche numerische Implementierung hinsichtlich der Finite-Elemente-Methode auf modernem Wege durch Nutzung der diskreten äußeren Analysis mit Fokus auf Genauigkeitsbelange bewertet. Hinsichtlich der numerischen Genauigkeitsbewertung wird demonstriert, dass der vorgelegte Ansatz grundsätzlich eine höhere Exaktheit zeigt, wenn man ihn mit Formulierungen vergleicht, welche auf der Helmholtz-Gleichung beruhen. Diese Dissertation trägt eine generalisierte hyperkomplexe alternative Darstellung von gewöhnlichen elektrodynamischen Ausdrucksweisen zum Themengebiet der Wellenausbreitung bei. Durch die Nutzung einer direkten Formulierung des elektrischen Feldes in Verbindung mit dem magnetischen Feld wird die Rechengenauigkeit von Randwertproblemen erhöht. Um diese Genauigkeitserhöhung zu erreichen, wird eine geeignete Erweiterung der de Rham-Kohomologie unterbreitet.Electromagnetic wave propagation provides the physical basis for countless applications in various subjects of today’s world. In order to model spatial scenarios, the continuous space must be converted to an appropriate computational domain. Ordinarily discretized models – which are based on distinct quantities – consider the connection between field variables by relations which are represented by partial differential equations. To reproduce mathematical relationships between dependent variables in a convenient manner, hypercomplex number systems build a suitable alternative framework. This approach aims to incorporate certain system properties and covers, in addition to the modeling of field problems where all variables are present, also simplified models. To provide a competitive alternative to the ordinary numerical handling of electromagnetic fields in an observation-based way, the electric and magnetic field of electromagnetic wave fields is understood as only one combined field variable embedded in the function space. This procedure is intuitive since both fields occur together in electrodynamics and are directly measureable. The focus of this thesis is twofold. On the one side, a reformulated Maxwell system is broadly investigated in a metric-free environment by the use of the so-called ”bicomplex approach”. On the other side, a possible numerical implementation concerning the Finite Element Method is evaluated in a modern way by the use of discrete exterior calculus with focus on accuracy matters. Regarding the numerical accuracy evaluation, it is demonstrated that the presented approach yields a higher exactness in general when comparing it to formulations which are based on the Helmholtz equation. This thesis contributes generalized hypercomplex alternative representations of ordinary electrodynamic expressions to the topic of wave propagation. By the use of a direct formulation of the electric field in conjunction with the magnetic field, the computational accuracy of boundary value problems is improved. In order to achieve this increase of accuracy, an appropriate enhancement of the de Rham cohomology is proposed

Modeling EMI Resulting from a Signal Via Transition Through Power/Ground Layers

Signal transitioning through layers on vias are very common in multi-layer printed circuit board (PCB) design. For a signal via transitioning through the internal power and ground planes, the return current must switch from one reference plane to another reference plane. The discontinuity of the return current at the via excites the power and ground planes, and results in noise on the power bus that can lead to signal integrity, as well as EMI problems. Numerical methods, such as the finite-difference time-domain (FDTD), Moment of Methods (MoM), and partial element equivalent circuit (PEEC) method, were employed herein to study this problem. The modeled results are supported by measurements. In addition, a common EMI mitigation approach of adding a decoupling capacitor was investigated with the FDTD method

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described