Volker Hill

A stabilized multilevel vector finite-element solver for time-harmonic electromagnetic waves

An enhanced finite-element method (FEM) for the vector wave equation is presented. For improved speed and stability ranging from microwave frequencies down to the static limit, we propose a multilevel solver that uses a tree-gauged formulation on the coarsest mesh and a partially gauged scheme for the iterative cycle. Moreover, we have generalized the concept of hanging nodes to higher order H(curl)-conforming tetrahedral elements. The combination of hierarchical basis functions and the hanging variables framework yields great flexibility in placing degrees of freedom and provides a very attractive alternative to remeshing in an hp-adaptive context.

Multi-parameter polynomial order reduction of linear finite element models

In this paper we present a numerically stable method for the model order reduction of finite element (FE) approximations to passive microwave structures parameterized by polynomials in several variables. The proposed method is a projection-based approach using Krylov subspaces and extends the works of Gunupudi etal. (P. Gunupudi, R. Khazaka and M. Nakhla, Analysis of transmission line circuits using multidimensional model reduction techniques, IEEE Trans. Adv. Packaging 25 (2002), pp. 174–180) and Slone etal. (R.D. Slone, R. Lee and J.-F. Lee, Broadband model order reduction of polynomial matrix equations using single-point well-conditioned asymptotic waveform evaluation: derivations and theory, Int. J. Numer. Meth. Eng. 58 (2003), pp. 2325–2342). First, we present the multivariate Krylov space of higher order associated with a parameter-dependent right-hand-side vector and derive a general recursion for generating its basis. Next, we propose an advanced algorithm to compute such basis in a numerically stable way. Finally, we apply the Krylov basis to construct a reduced order model of the moment-matching type. The resulting single-point method requires one matrix factorization only. Numerical examples demonstrate the efficiency and reliability of our approach.

Finite-element waveguide solvers revisited

A general analysis of potential- or field-based finite element solvers for axially uniform electromagnetic waveguides is presented. Explicit equations for the set of permissible gauge transformations are derived, which are later utilized to identify side effects that may affect the reliability of present approaches. We show that such limitations can be eliminated by an enhanced formulation employing a decomposition of the transverse components of the vector potential together with a particular choice of gauge. Numerical results support our findings.

Multivariate finite element model order reduction for permittivity or permeability estimation

An efficient finite element method for the analysis of passive microwave components over wide ranges of frequencies and material parameters is proposed. First, a parameterized equation system of low order is generated, which is then solved for specific parameter values. Since the evaluation of the reduced order system is very fast, the proposed method is very well-suited as a forward solver for permittivity or permeability estimation

Finite element basis functions for nested meshes of nonuniform refinement level

We propose a systematic methodology for the construction of hanging variables to connect finite elements of unequal refinement levels within a nested tetrahedral mesh. While conventional refinement schemes introduce irregular elements at such interfaces which must be removed when the mesh is further refined, the suggested approach keeps the discretization perfectly nested. Thanks to enhanced regularity, mesh-based methods such as refinement algorithms or intergrid transfer operators for use in multigrid solvers can be implemented in a much simpler fashion. This paper covers H/sup 1/ and H(curl) basis functions for triangular or tetrahedral elements.

Efficient Implementation of Non-Uniform Refinement Levels in a Geometric Multigrid Finite Element Method for Electromagnetic Waves

We propose an efficient geometric multigrid (GMG) strategy for finite element (FE) meshes of non-uniform refinement levels. The suggested method exploits that levels of high resolution usually extend over small sub-domains only. By restricting all fine level smoothing operations to the corresponding partial meshes, memory consumption and operation count are kept at a minimum

Ordnungsreduktion linearer zeitinvarianter Finite-Elemente-Modelle mit multivariater polynomieller Parametrierung (Model Order Reduction of Linear Finite Element Models Parameterized by Polynomials in Several Variables)

Abstract Der vorliegende Beitrag stellt ein Verfahren zur Ordnungsreduktion linearer Gleichungssysteme vor, die durch Polynome in mehreren Variablen parametriert sind und aus der Finite-Elemente-Methode hervorgehen. Der vorgeschlagene Ansatz beruht auf multivariaten Krylov-Unterräumen und erweitert bestehende Verfahren in zweierlei Hinsicht: Erstens beinhaltet er einen neuen Algorithmus zur Berechnung einer stabilen Basis für das reduzierte System, und zweitens verallgemeinert er das Konzept der Krylov-Unterräume höherer Ordnung auf Mehrparametersysteme.

A Jacobi-Davidson Method for the hp Multilevel Analysis of Cavity Modes

Abstract This article presents a variant of the Jacobi–Davidson (JD) method for the modal analysis of electromagnetic (EM) cavities. The suggested algorithm exploits the hp multilevel (ML) structure of the underlying finite element (FE) space to solve the JD correction equation at low computational complexity and to impose constraints that prevent the occurrence of nonphysical solutions. The proposed method is suitable for large-scale models and covers structures with dielectric and/or magnetic losses.

Non-Uniform Refinement and Preconditioning of hp Multi-Level Finite Element Methods

This paper presents an efficient finite element method for the time-harmonic Maxwell equations. It is based on unstructured tetrahedral meshes and uses a multi-level solver that sweeps through a nested family of finite element spaces. To handle non-uniform refinement levels efficiently, we employ a subdomain method that applies smoothing operations to the corresponding partial meshes only.

Multilevel Analysis of Electromagnetic Resonators Using Hierarchical p-Type Finite Elements

We propose an efficient eigenmode solver for electromagnetic cavities based on the Jacobi-Davidson (JD) method and hierarchical finite elements (FE) of high order. The JD algorithm enables us to exploit the multilevel structure of the FE space at various stages of the solution process. As a result, the computational complexity of the suggested method is very low, which is particularly useful for large-scale projects