Finite element methods for Maxwell's equations.

Numerical Methods for Scientists and Engineers

In this paper, a reduced-basis-approximation-based model-order reduction for fast and reliable frequency sweep in the time-harmonic Maxwell's equations is detailed. Contrary to what one may expect by observing the frequency response of different microwave circuits, the electromagnetic field within these devices does not drastically vary as frequency changes in a band of interest. Thus, instead of using computationally inefficient, large dimension, numerical approximations such as finite- or boundary-element methods for each frequency in the band, the point in here is to approximate the dynamics of the electromagnetic field itself as frequency changes. A much lower dimension, reduced-basis approximation sorts this problem out. Not only rapid frequency evaluation of the reduced-order model is carried out within this approach, but also special emphasis is placed on a fast determination of the error measure for each frequency in the band of interest. This certifies the accurate response of the reduced-order model. The same scheme allows us, in an offline stage, to adaptively select the basis functions in the reduced-basis approximation and automatically select the model-order reduction process whenever a preestablished accuracy is required throughout the band of interest. Finally, real-life applications will illustrate the capabilities of this approach.

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Reliable Fast Frequency Sweep for Microwave Devices via the Reduced-Basis Method

The reduced basis method (RBM) generates low- order models for the solution of parametrized partial differential equations to allow for efficient evaluation in many-query and real-time contexts. We show the theoretical framework in which the RBM is applied to Maxwell's equations and present numerical results for model reduction in frequency domain. Using rigorous error estimators, the RBM achieves low-order models under variation of material parameters and geometry. The RBM reduces model order by a factor of 50 to 100 and reduces compute time by a factor of 200 and more for numerical experiments using standard circuit elements.

Fast Evaluation of Time–Harmonic Maxwell's Equations Using the Reduced Basis Method

In this paper, an efficient computational framework for the full-wave design by optimization of complex microwave passive devices, such as antennas, filters, and multiplexers, is described. The framework consists of a computational engine, a 3D object modeler, and a graphical user interface. The computational engine, which is based on a finite element method with curvilinear higher-order tetrahedral elements, is coupled with built-in or external gradient-based optimization procedures. For speed, a model order reduction technique is used and the gradient computation is achieved by perturbation with geometry deformation, processed on the level of the individual mesh nodes. To maximize performance, the framework is targeted to multicore CPU architectures and its extended version can also use multiple GPUs. To illustrate the accuracy and high efficiency of the framework, we provide examples of simulations of a dielectric resonator antenna and full-wave design by optimization of two diplexers involving tens of unknowns, and show that the design can be completed within the duration of a few simulations using industry-standard FEM solvers. The accuracy of the design is confirmed by measurements.

An Efficient Framework For Fast Computer Aided Design of Microwave Circuits Based on the Higher-Order 3D Finite-Element Method

In this paper, a domain decomposition approach for design purposes is proposed. The analysis domain is divided into subdomains according to the arbitrarily shaped parts that should be modified in the synthesis process. A full-wave matrix-valued transfer function describes each decomposition subdomain, namely, an admittance-type matrix. Field continuity between subdomains is directly enforced by an admittance matrix connection. This methodology makes it possible to analyze only those parts of the analysis domain that are supposed to evolve in order to satisfy the design specifications. Furthermore, several modifications in the shape of the components are allowed as a consequence of the easy matrix connection process, where the consideration of different admittance-type matrices or absence of them gives rise to distinct geometric structures. A model order reduction technique is also considered for fast frequency sweeping. Finally, several numerical examples illustrate the capabilities of the proposed procedure, as well as its accuracy.

Microwave Circuit Design by Means of Direct Decomposition in the Finite-Element Method

In this paper, an efficient method for calculating TE and TM modes in closed waveguides made up of cylindrical perfect electric multiconductors and homogeneous isotropic media is presented. An ad hoc analytical formulation of the problem is carried out, and special emphasis is placed on its resolution. As a result, the determination of the cutoff wavenumbers is achieved using a robust and efficient algorithm. An adaptive approximation scheme is also proposed to ensure an acceptable precision and an efficient computation without missing any mode. Finally, several comparisons demonstrate the accuracy of the proposed method

V. de la Rubia

Papers

Reliable Fast Frequency Sweep for Microwave Devices via the Reduced-Basis Method

Microwave Circuit Design by Means of Direct Decomposition in the Finite-Element Method

An efficient method for determining TE and TM modes in closed waveguides made up of N cylindrical conductors