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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.

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.

This paper presents a fast numerical method for computing certified far-field patterns of phased antenna arrays over broad frequency bands as well as wide ranges of steering and look angles. The proposed scheme combines finite-element analysis, dual-corrected model-order reduction, and empirical interpolation. To assure the reliability of the results, improved a posteriori error bounds for the radiated power and directive gain are derived. Both the reduced-order model and the error-bounds algorithm feature offline-online decomposition. A real-world example is provided to demonstrate the efficiency and accuracy of the suggested approach.

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...

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.

This paper presents a fast certified numerical method for computing radiation patterns of electrically large antenna arrays over broad frequency bands and wide ranges of look angles. The suggested scheme combines finite-element analysis with empirical interpolation and employs a sub-domain approach on the Huygens surface to reduce computational costs in the offline part of the algorithm. To assure the reliability of the numerical results, an accurate and efficiently computable a posteriori error bound for the radiation patterns is proposed. A real-world example is presented to demonstrate the efficiency and accuracy of the suggested approach.

Purpose – The goal is to derive a numerical method for computing parametric reduced-order models (PROMs) from finite-element (FE) models of microwave structures that feature geometrical parameters. Design/methodology/approach – First, a parameter-dependent FE mesh is constructed by a topology-preserving mesh-morphing algorithm. Then, multivariate polynomial interpolation is employed to achieve explicit geometrical parameterization of all FE matrices. Finally, a PROM based on parameter-dependent projection matrices is constructed by means of interpolation and state transformation techniques. Findings – The resulting PROMs are of low dimension and fast to evaluate. Moreover, the method features high rates of convergence, and the number of FE solutions required for constructing the PROM is small. The accuracy of the PROM is only limited by that of the underlying FE model and can be controlled by varying the PROM dimension. Research limitations/implications – Since the method uses topology-preserving mesh-mor...

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.

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.

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.

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.

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...

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.

PurposeThe purpose of this paper is to compare competing adaptive strategies for fast frequency sweeps for driven and waveguide‐mode problems and give recommendations for practical implementations.Design/methodology/approachThe paper first summarizes the theory of adaptive strategies for multi‐point (MP) sweeps and then evaluates the efficiency of such methods by means of numerical examples.FindingsThe authors' numerical tests give clear evidence for exponential convergence. In the driven case, highly resonant structures lead to pronounced pre‐asymptotic regions, followed by almost immediate convergence. Bisection and greedy point‐placement methods behave similarly. Incremental indicators are trivial to implement and perform similarly well as residual‐based methods.Research limitations/implicationsWhile the underlying reduction methods can be extended to any kind of affine parameter‐dependence, the numerical tests of this paper are for polynomial parameter‐dependence only.Practical implicationsThe present paper describes self‐adaptive point‐placement methods and termination criteria to make MP frequency sweeps more efficient and fully automatic.Originality/valueThe paper provides a self‐adaptive strategy that is efficient and easy to implement. Moreover, it demonstrates that exponential convergence rates can be reached in practice.

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.

Summayr form only given. Starting from an initial mesh, discretizations of higher resolution can be constructed very easily by partitioning existing finite elements (FEs) recursively (Rude, U., 1993). Besides guaranteeing a lower bound on mesh quality, the resulting nested structure adds a high degree of regularity to the FE discretization, which greatly simplifies the application of fast geometrical multigrid solvers. At the same time, efficient error control requires the mesh size to adjust locally according to the spatial behavior of the electromagnetic fields and thus implies the need for strongly non-uniform mesh refinement. Our approach allows finite elements of unequal refinement levels to touch. As a result, the FE hierarchy is always perfectly nested. We focus on the computational efficiency of the generalized hanging variables framework. We first prove that, for the whole set of permissible constraints, the corresponding sequences of finite element spaces remain perfectly nested. We then propose one specific method which is easy to implement and which combines low memory requirements and rapid numerical convergence. Numerical examples are given to validate our findings.