Finite element‐based model order reduction of electromagnetic devices

Abstract

SUMMARYIn this paper an e$cient algorithm is presented for the development of compact and passive macro-models of electromagnetic devices through the systematic reduction of the order of discrete models for these devices obtained through the use of "nite elements. The proposed methodology is founded on a new "nite element formulation that casts Maxwell's curl equations in a state-space form. Such state-space representations are very compatible with a class of robust model order reduction techniques based on Krylov subspaces. However, the advantage of this compatibility appears to be hindered by the fact that the state matrix of the discretized Maxwellian system is of dimension almost twice that obtained from the "nite element approximation of the electromagnetic vector wave equation. It is shown in this paper that the apparent penalty in both memory and computation e$ciency can be avoided by a proper selection of the "nite element expansion functions used for the discretization of the electromagnetic "elds. More speci"cally, it is shown that the proper selection of expansion functions renders the state-space form of the Maxwellian system equivalent to the discrete problem obtained from the approximation of the vector wave equation using tangentially continuous vector "nite elements. This equivalence is then used to e!ect Krylov-based model order reduction directly from the "nite element approximation of the vector wave equation. In particular, a passive model order reduction algorithm is used for this purpose. The proposed reduced order macromodelling algorithm is demonstrated through its application to a variety of microwave passive components.

Model-order reduction techniques for linear electromagnetic problems - an overview

“…All these approaches share the property that the reduced-order model is obtained by projecting the state-space onto a Krylov subspace that has a lower rank, the projected problem being then efficiently solved. In [15] the authors propose a new FE formulation based on Maxwell's curl equations that enables the computation of the Generalised Impedance Matrix (GIM) of complex structures, including lossy media and unbounded regions. The GIM is expressed in a state-space form in the Laplace domain, the result being well suited for a processing by means of an Arnoldi-like MOR algorithm.…”

Parameterized Model Order Reduction Techniques for Finite Element Based Full Wave Analysis

“…The system in (8) exhibits linear dependency with respect to k and therefore the moments Qk can be computed efficiently using techniques based on the Krylov subspace such as the Arnoldi process [2]. Similarly, the Arnoldi process can also be used to calculate Q>1 .... Qi, for the case when the variation of the matrices G and C with respect to design parameters ( .)…”

“…This increase in the number of state variables is undesirable because the inverse of the system matrix in (6) is required to obtain the Krylov subspace, which is needed to generate the reduced system. However, by making use of the modified block Arnoldi approach proposed in [2], the generation of Krylov subspace can be formulated at the same computational cost of the solution of (5) at a single frequency.…”

“…This results in a significant overhead and reduced (1) where ko = W2t£o/Uo; eo and/,to are the permittivity and permeability for the free space; Ur ] and [er ] are the relative permeability and permittivity tensors, respectively and E is the electric field vector. For waveguide problems, on the conducting surfaces, the electric field satisfies the Dirichlet boundary condition as nxE=O (2) The boundary condition for the port i is given by [6] n x V x E + j/8n x (E x nh) = -2 j/jE c at S, (3) where f is the propagation constant of the dominant mode, S denotes the cross section of port i and Ei'c is the incident field. For rectangular waveguides, f is known analytically by(c2o o-(mT /a) 2-(nT /b)2)/, where a, b are the width and height of the waveguide; m, n represents the mode numbers of the transverse electric or magnetic waves propagating through a waveguide.…”

“…Equation (5) exhibits both a linear and quadratic dependence with respect to k whereas Krylov subspace methods require linear dependency of k in the discrete model. In order to use the Krylov subspace method and take advantage of the Arnoldi process [2], extra state variables are introduced to convert the form of (5) as (G+kC)X -B…”