Ortwin Farle

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.

An Adaptive Multi-Point Fast Frequency Sweep for Large-Scale Finite Element Models

Single- and multi-point model order reduction methods constitute two complementary approaches to the fast finite element analysis of passive electromagnetic structures over wide frequency bands. This paper presents an adaptive point placement strategy for multi-point methods. Numerical experiments show that, for a given approximation error limit, the proposed algorithm produces reduced-order systems of lower dimension than single point methods. Hence, solution times for the reduced-order model are improved, which is important when large numbers of function evaluation are required, such as in parametric libraries. For large-scale problems, which are not accessible to direct solvers, even the computer runtimes for generating the reduced-order systems are superior to those of single-point methods.

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.

A New Method for Accurate and Efficient Residual Computation in Adaptive Model-Order Reduction

Projection-based model-order reduction is a powerful methodology for solving parameter-dependent linear systems of equations. The efficient computation of the residual norm is of paramount importance in adaptive model reduction schemes because it is heavily used in error indicators and a posteriori error bounds. These guide the adaptive selection of expansion points in multi-point methods and serve as stopping criteria for subspace enrichment. This paper demonstrates that the standard algorithm for fast residual norm computation leads to premature stagnation, and it presents a new approach of improved accuracy.

A Model Order Reduction Method for the Finite-Element Simulation of Inhomogeneous Waveguides

A finite-element-based model order reduction method for the broadband analysis of the dominant modes of transversally inhomogeneous waveguides is presented. We show that the sub-space projections used in conventional multipoint methods may result in spurious modes in the reduced-order model and propose an improved formulation to overcome this problem. The suggested method achieves error levels comparable to those of the underlying finite-element method, while overall solution times are significantly smaller.

Passivity preserving parametric model-order reduction for non-affine parameters

Parametric model-order reduction (pMOR) has become a well-established technology for analysing large-scale systems with multiple parameters. However, the treatment of non-affine parameters is still posing significant challenges, because projection-based order-reduction methods cannot be applied directly. A common remedy is to establish affine parameter-dependencies approximately, but present extraction methods do not take important system properties, such as passivity, into account. This article proposes a new order-reduction approach that preserves passivity, reciprocity and causality and applies to a wide class of linear time-invariant (LTI) systems. We present the theory of the suggested method and demonstrate its practical usefulness by numerical examples taken from computational electromagnetics.

A Posteriori Error Bounds for Krylov-Based Fast Frequency Sweeps of Finite-Element Systems

Projection-based model reduction is a well-established methodology for computing fast frequency sweeps of finite-element (FE) approximations to passive microwave structures. This contribution presents a novel provable error bound for moment-matching reduced-order models of lossless systems. It improves over existing methods by increasing the accuracy of the estimate and by reducing numerical costs. Numerical studies demonstrate the benefits of the suggested approach.