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A new serendipity function space for hexahedral H(curl)-conforming finite elements, the so-called mixed-order serendipity space, is proposed. In the case of arbitrary parallelepiped meshes, the resulting asymptotic rate of convergence of the H(curl)-norm error is exponential in the base of the mesh size and in terms of the finite element order. Compared to tensor product spaces, the number of unknowns is much smaller, whereas the rate of convergence remains the same. While the proposed serendipity space is not suitable for elements of general shape, it allows the construction of hierarchical basis functions that are compatible with some of the costly but versatile tensor product spaces. This property permits mixing different finite element spaces within one mesh, without affecting conformity.For the general curvilinear case, this paper introduces an iso-serendipity finite element for curvilinear hexahedra, which employs the proposed serendipity space and basis for representing the fields and H1 serendipity basis for representing the geometry. As the main contribution, this element achieves the same convergence rate for both curvilinear and parallelepiped meshes, by means of a special yet simple mesh refinement technique. In contrast to isogeometric methods, it only requires certain interpolation points on curvilinear boundaries rather than the entire geometry mapping. All results are supported by mathematical proofs and validated experimentally via numerical examples.

A numerical method for optimizing the frequency response of linear time-invariant microwave structures at the fields level is presented. It is based on the finite-element method in the frequency-domain, a quasi-Newton optimizer, and the reduced-basis method. The reduced-order model is constructed adaptively, in the course of the optimization process, and its validity is restricted to some neighborhood of the trajectory drawn by the optimizer in parameter space. The computational complexity of the method is independent of the number of design variables. The present study is restricted to material parameters which lead to affine parameterization. Numerical results demonstrate that the number of finite-element evaluations, which are computationally expensive, is greatly reduced, even in the pre-asymptotic region of the optimization process. Moreover, it is shown that computing the gradient of the cost function analytically provides significant savings in computer runtime over finite-difference approximations.

In this paper, compact time-domain (TD) models featuring materials with frequency-dependent electromagnetic (EM) properties are derived. The considered frequency-dependent material models include multiterm Debye and Lorentz models for the electric permittivity and the magnetic permeability and a multiterm Drude model for the electric conductivity. The TD models are based on finite-element systems in the frequency domain (FD). To render the model compact and computationally efficient, the dimension of the FD system is compressed with the help of projection-based model-order reduction. In contrast to older approaches, the TD transformation is performed on the FD model itself rather than on the transfer function. The result is a state-space representation, which may either be solved by custom time integrators or imported into commercial circuit simulators. The advantages of the new approach include provable passivity of the FD model, provable causality of the TD model, and the ability to reconstruct the transient EM fields.

This paper presents a frequency-sweep technique based on model-order reduction and finite elements, for the broadband analysis of structures fed by waveguides (WGs) possessing frequency-dependent modal field patterns. Standard order reduction requires the matrices and right-hand sides (RHSs) to exhibit affine frequency parameterization. This precondition is violated when the transverse fields of the WG modes vary with frequency. The proposed solution involves two steps. First, a reduced-order model (ROM) for the WG is constructed. It enables the accurate yet inexpensive computation of propagation characteristics. Second, order reduction is applied to the driven problem, wherein the reduced WG model is utilized to construct affine approximations to the matrices and RHSs. Since this process requires operations on reduced-order matrices only, it is computationally cheap and enables offline/online decomposition. Both impedance and scattering formulations are considered. For the latter, an alternative to the transfinite element method is proposed, which does not employ modal field patterns as shape functions. It avoids interior resonances and computes scattering parameters more efficiently when only a limited set of excitations is of interest. The resulting algebraic system is of somewhat larger dimension but easier to assemble. Its simple structure greatly facilitates the construction of the ROM.

The finite-element time-domain simulation of nonlinear eddy-current problems requires the iterative solution of a large, sparse system of equations at every time-step. Model-order reduction is a powerful tool for reducing the computational effort for this task. In this paper, an adaptive order-reduction methodology with error control is proposed. In contrast to previous approaches, it treats the nonlinearity without simplification, by rewriting the original equations as a quadratic–bilinear system.

A coupled finite-element boundary-element method for solving parametric models of eddy-current problems is proposed. Affine approximation by the empirical interpolation method makes the numerical model accessible to projection-based parametric model-order reduction. The resulting low-dimensional system provides high evaluation speed at an accuracy comparable with that of the underlying discretization method.

A new method for calculating the geometric sensitivities of curvilinear finite elements is presented. Approximating the relevant metric tensors by hierarchical orthogonal polynomials enables the sensitivity matrices to be integrated analytically. The resulting numerical method is based on pre-calculated universal matrices and achieves significant savings in computer runtime over conventional techniques based on numerical integration. Moreover, there exists a representation limit for the geometry, i.e., the degree of basis functions fully determines a critical order of the geometry expansion, beyond which the derivatives of the finite-element matrices will remain constant. To validate the suggested approach, a numerical example is presented.

Model-order reduction provides an efficient way of computing frequency sweeps for finite-element models, because the dimension of the reduced-order system depends on the complexity of the frequency response rather than the size of the original model. For electrically large domains, however, the applicability of such methods is unclear because the system response may be very complicated. This paper provides a numerical study of the effects of bandwidth, electrical size, and scan angle on the size and convergence of the ROM, by considering linear antenna arrays. A mathematical model is proposed and validated against numerical experiments.

Purpose – The purpose of this paper is to determine the broadband frequency response of the impedance matrix of wireless power transfer (WPT) systems comprising litz wire coils. Design/methodology/approach – A finite-element (FE)-based method is proposed which treats the microstructure of litz wires by an auxiliary cell problem. In the macroscopic model, litz wires are represented by a block with a homogeneous, artificial material whose properties are derived from the cell problem. As the frequency characteristics of the material closely resemble a Debye relaxation, it is possible to convert the macroscopic model to polynomial form, which enables the application of model reduction techniques of moment-matching type. Findings – FE-based model-order reduction using litz wire homogenization provides an efficient approach to the broadband analysis of WPT systems. The error of the reduced-order model (ROM) is comparable to that of the underlying original model and can be controlled by varying the ROM dimension...

Methods of parametric order reduction are very appealing for solving parameter-dependent models at the fields level, because they provide fast simulations and low systematic error. This paper presents a self-adaptive framework for computing reduced-order models featuring affine and non-affine parameters. It is based on a hypercube partitioning of the domain of non-affine parameters and employs non-uniform grid refinement, controlled by a suitable error indicator. Compared with state-of-the-art entire-domain methods, the proposed sub-domain approach achieves significant improvements in memory consumption and computer run time.

The finite-element formulation proposed in this paper is for the frequency domain and covers the entire range from static/stationary fields to wave propagation. It does not involve any frequency-dependent thresholds, leads to complex symmetric system matrices and unique solutions, and applies to the most general structures, comprising both lossy and lossless regions. Compared with non-stabilized methods, e.g., the electric field formulation, the price to be paid is one extra scalar potential in the lossy region. Numerical examples demonstrate that the proposed method is well-suited for direct and iterative solvers, and remains stable in the static case.

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.

Model-order reduction provides an appealing approach to solving parametric large-scale models stemming from spatial discretization methods. The high-dimensional model at the fields-level is replaced by a surrogate model that is fast to evaluate, at a controllable level of error. This paper presents an interpolation-based order-reduction method for systems with non-affine parameters. The main novelty is the construction of the parameter-dependent projection matrix. For a given error level, the suggested approach reduces the number of instantiations of the fields-level model compared to state-of-the-art methods.

A fast and accurate simulation framework for characterizing inductive power transfer systems with respect to coil alignment and frequency is presented. It combines the finite-element method with homogenization techniques and employs parametric model-order reduction. The reduced-order models feature low systematic errors and allow for thousands of evaluations per second. The proposed method is capable of handling litz wire coils.

In the lossless case, immittance formulations of the finite element method lead to real-valued system matrices. This reduces memory consumption and solution times. However, their solutions may get corrupted by interior resonances. The scattering formulation, in contrast, is inherently stable, but the resulting matrices are always complex-valued. This article presents a model-order reduction technique for lossless microwave structures that combines the low computational costs of the impedance formulation with the high robustness of the scattering approach. It extends the authors' previous work to structures fed by inhomogeneous waveguides.

This paper presents an improved finite-element formulation for axially uniform electromagnetic waveguides. It allows for both dielectric and conduction losses and covers the entire range from optics down to the static limit. Propagation coefficients of small magnitude, particularly those of transmission line modes in the low-frequency regime, are computed much more accurately than with previous approaches.

Since typical finite-element systems are of high dimension, the analysis of parameter-dependent microwave structures over broad frequency bands tends to be very time-consuming. This issue is addressed by parametric order reduction, which provides a systematic methodology for constructing surrogate models that are cheap to evaluate and feature low and controllable levels of error. This article presents an order reduction technique for finite-element models that depends on the operating frequency and features explicit and implicit parameters for material properties and shape, respectively. It uses polynomial interpolation to resolve implicit parameter dependencies and employs parameter-dependent bases defined on sub-domains of parameter space. The resulting reduced-order models are of very small dimension and preserve the structure and frequency dependency of the original finite-element model. Numerical results demonstrate that the proposed method reduces solution times by several orders of magnitud...

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.

Numerical simulation methods at the fields level, such as the finite-element method, are highly accurate but computationally expensive. In the context of mathematical optimization, this implies that the cost function, which needs to be evaluated a large number of times, is very expensive to compute. To overcome this shortcoming, this paper proposes to employ parametric reduced-order models for computing the cost function and, if required, its gradients. They introduce low systematic error, require little memory, and allow for hundreds to thousands of model evaluations per second. The great utility of the suggested approach in both deterministic and stochastic optimization methods is demonstrated by a numerical example, featuring 11 geometric parameters.

Purpose – The article aims to present an efficient numerical method for computing the far-fields of phased antenna arrays over broad frequency bands as well as wide ranges of steering and look angles. Design/methodology/approach – The suggested approach combines finite-element analysis, projection-based model-order reduction, and empirical interpolation. Findings – The reduced-order models are highly accurate but significantly smaller than the underlying finite-element models. Thus, they enable a highly efficient numerical far-field computation of phased antenna arrays. The frequency-slicing greedy method proposed in this paper greatly reduces the computational costs for constructing the reduced-order models, compared to state-of-the-art methods. Research limitations/implications – The frequency-slicing greedy method is intended for use with matrix factorization methods. It is not applicable when the underlying finite-element system is solved by iterative methods. Practical implications – In contrast to conventional finite-element models of phased antenna arrays, reduced-order models are very cheap to evaluate. Hence, they provide an enabling technology for computing radiation patterns over broad frequency bands and wide ranges of steering angles. Originality/value – The paper presents a two-step model-order reduction method for efficiently computing the far-field patterns of phased antenna arrays. The suggested frequency-slicing greedy method constructs the reduced-order models in a systematic fashion and improves computing times, compared to existing methods.

Projection-based model-order reduction is a powerful methodology for solving parameter-dependent linear systems of equations. Adaptive multi-point methods commonly employ a greedy strategy for expansion point placement: The location where some error measure is maximum is selected. This requires evaluating an error indicator on a dense sampling of the parameter domain at each iteration of the model generation phase. To reduce runtimes, a hierarchical refinement strategy that reuses information from previous steps is proposed.

This contribution describes a methodology for computing compact time-domain models of electromagnetic devices containing Lorentz materials from finite-element systems in the frequency domain. The procedure starts with projection-based model-order reduction, to downsize system dimension. The resulting reduced-order model is transformed to the time-domain and leads to a state-space representation. The proposed method is mathematically proved to preserve important system properties including passivity, causality, and reciprocity, and allows for reconstructing the transient electromagnetic fields.

A fast frequency sweep method for finite-element models of linear time-invariant structures is presented, which is valid from the static/stationary limit up to the optical range. The key ingredients are a low-frequency stable finite-element formulation and a projection-based method of model-order reduction. The dimension of the reduced model is governed by the complexity of the system dynamics only; it does not depend on the size of the finite-element matrix. A numerical example is presented to demonstrate the efficiency of the suggested approach.

A method for constructing time-domain models including Drude materials from finite-element data in the frequency-domain is presented. To produce a model that is compact, projection-based order-reduction techniques are applied. The time-domain model is based on a state space representation and can be proven to be passive and causal. Moreover, the suggested approach provides an efficient way for reconstructing the transient electromagnetic fields.

When Courant prepared the text of his 1942 address to the American Mathematical Society for publication, he added a two-page Appendix to illustrate how the variational methods first described by Lord Rayleigh could be put to wider use in potential theory. Choosing piecewise-linear approximants on a set of triangles which he called elements, he dashed off a couple of two-dimensional examples and the finite element method was born. … Finite element activity in electrical engineering began in earnest about 1968-1969. A paper on waveguide analysis was published in Alta Frequenza in early 1969, giving the details of a finite element formulation of the classical hollow waveguide problem. It was followed by a rapid succession of papers on magnetic fields in saturable materials, dielectric loaded waveguides, and other well-known boundary value problems of electromagnetics. … In the decade of the eighties, finite element methods spread quickly. In several technical areas, they assumed a dominant role in field problems. P.P. Silvester, San Miniato (PI), Italy, 1992 Early in the nineties the International Workshop on Finite Elements for Microwave Engineering started. This volume contains the history of the Workshop and the Proceedings of the 13th edition, Florence (Italy), 2016 . The 14th Workshop will be in Cartagena (Colombia), 2018.

Established order reduction methods for nonlinear descriptor systems project the nonlinear system onto a linear submanifold. This may lead to reduced models of large scale which provide little computational speed-up. To overcome this deficiency, the proposed method replaces the construction of a global projection matrix by an interpolation of locally reduced models. The present paper gives the underlying theory and demonstrates the accuracy and efficiency of the suggested approach by a finite element model of an eddy current problem.

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.

The goal of this paper is to compute k-β diagrams of photonic crystals, accurately and fast. For this purpose, we propose a multipoint model-order reduction scheme for the modal analysis of periodic structures, based on the finite element method. Our numerical example demonstrates that the new approach is significantly faster than conventional finite-element solutions, while error levels are very similar. The proposed method allows for an adaptive choice of the expansion points for the reduced model.

Parametric model-order reduction is a very powerful methodology for analyzing large-scale systems with multiple parameters. This paper extends the theory of moment-matching single-point methods to linear systems parameterized by multivariate polynomials. We propose a new algorithm that exhibits high numerical robustness and short runtimes, allows for direction-dependent model orders, and is easy to parallelize. To demonstrate the accuracy and efficiency of the suggested approach, we present the response surfaces of two microwave finite-element models, featuring the operating frequency and material properties as parameters.

Single-point methods of moment-matching type provide a most effective means for realizing fast frequency sweeps. The impedance formulation of the finite element method is very well-suited for such techniques, but it is susceptible to interior resonances, especially in the case of lossless microwave structures. This deficiency can be resolved by imposing transparent boundary conditions at the waveguide ports, but the resulting matrices exhibit nonpolynomial frequency-dependency unless all waveguide modes are of the transverse electric and magnetic type. In consequence, single-point methods cannot be applied without approximations. This article proposes a fast frequency sweep technique that handles not only transverse electric and magnetic modes but also transverse electric and transverse magnetic modes without any approximations. Moreover, it retains the efficiency of the impedance formulation but prevents interior resonances from occurring. The theoretical background of and numerical evidence for...

PurposeThe purpose of this paper is to enable fast finite element (FE) analysis of electromagnetic structures with multiple geometric design variables.Design/methodology/approachThe proposed methodology combines multi‐variable model‐order reduction with mesh perturbation techniques and polynomial interpolation of parameter‐dependent FE matrices.FindingsThe resulting reduced‐order models are of comparable accuracy as but much smaller size than the original FE systems and preserve important system properties such as reciprocity.Research limitations/implicationsThe method is limited to mesh variations that are obtained from a nominal discretization by continuous deformation. Topological changes in the mesh are not permissible.Practical implicationsIn contrast to the underlying FE models, the resulting reduced‐order systems are very cheap to analyze. Possible applications include parametric libraries, design optimization, and real‐time control.Originality/valueThe paper extends the scope of moment‐matching order‐reduction techniques to a class of non‐polynomial systems arising from FE models with geometric parameters.

The Jacobi-Davidson method is very attractive for the large-scale finite element simulation of electromagnetic cavity problems, because it just requires approximate solutions to some auxiliary systems of linear equations, which are easily obtained by iterative solvers. Moreover, rates of convergence are quadratic to cubic. In consequence, the time spent within the pre-asymptotic region at the early stages of the iteration may have strong impact on overall performance, particularly when the number of eigenvalues sought is small. Therefore, we propose a startup procedure that utilizes p hierarchical finite element spaces to reduce the cost of bridging the pre-asymptotic region.

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.

This paper deals with time-harmonic electromagnetic fields only. It presents finite-element formulations of improved robustness and low-frequency behavior, fast hp multi-level solution techniques, and order-reduction methods that handle parameter-dependent systems at low computational cost.

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 st...

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.

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.

We present an advanced geometric multigrid strategy for finite-element (FE) meshes of nonuniform refinement levels. The proposed method exploits the fact that levels of high resolution usually extend over small subdomains only. By restricting the smoothing operations at the finer levels to the corresponding partial meshes, both memory consumption and operation count are kept at a minimum. Our computer implementation is based on a hierarchy of perfectly nested FE spaces and employs hanging variables to interface between regions of unequal refinement levels. To validate the effectiveness of the suggested approach, several numerical examples are given

This paper presents a self-adaptive reduced-basis method for driven microwave problems which incorporates the frequency-dependent mode patterns and dispersive propagation coefficients of inhomogeneous waveguides. A reduced-order model provides a low-dimensional subspace for approximating the relevant waveguide modes over the considered frequency range. That information is exploited to establish affine parameter-dependence in the three-dimensional driven model, which hereby becomes accessible to projection-based model-order reduction.