Reduced basis approximations in microwave filters and diplexers: Inf-sup constant behavior

Abstract

In this work, a model order reduction technique for fast frequency sweep in microwave filters and diplexers via the Reduced Basis Method is detailed. The Finite Element Method is used to solve the time-harmonic Maxwell's equations in these resonant circuit problems. Taking into account the electromagnetic field varies smoothly as a function of the frequency parameter, a low dimension system parameter manifold is identified. Thus, the original frequency-dependent Finite Element problem can be approximated by a model of reduced size using a Reduced Basis approximation. Based on the field frequency behavior in these resonant structures, the Reduced Basis space where the approximation holds is identified and characterized. Finally, special emphasis is placed in the behavior of the infsup constant used to provide an a posteriori error estimator, and its consequences in the Reduced Basis approximation are highlighted. Index Terms-Error analysis, finite element methods, Galerkin method, reduced basis methods, reduced order systems.

A Compact Basis for Reliable Fast Frequency Sweep via the Reduced-Basis Method

“…Eigenvalues co 2 n have finite multiplicity; they are infinitely many and accumulate at infinity. If e n solves (27), then its complex conjugate also does. As a result, the eigenfunctions e n are real-valued functions and, in particular, e n can be replaced by e" in (24) and (26).…”

“…First of all, as shown in Section II, let us point out that the solution to Maxwell's equations in the whole electromagnetic spectrum requires indeed the knowledge of an infinite number of orthogonal field solutions, {Fo,e"\n e N}, which result in a complete solution basis [see (28)]. Recall that Fo is the solution to problem (15) and [e"\n e N} are the eigenfunctions in the eigenproblem (27). As discussed in Section II, all these solutions are orthogonal with respect to the inner product (-,•)//,£• Now, our interest is focused on a specific frequency band B and we can proceed as follows.…”

“…By the same token, the ROM needs to be certified and an a posteriori error analysis is developed. Different error estimators have been proposed [17], [22], [23], [25], [26] and some of them are based on the inf-sup constant in the frequency-dependent variational problem [27]. However, whenever a resonance is hit in the frequency band of interest, which is normally the case in microwave circuits, since most devices involve discontinuities that often result in resonances, these inf-sup constant-based error estimators are no longer valid.…”

Fast A Posteriori State Error Estimation for Reliable Frequency Sweeping in Microwave Circuits via the Reduced-Basis Method

“…This is the case for inf-sup constant-based error estimators [18,19,35,37]. The residual norm divided by this costly inf-sup constant [14] bounds the state error. As already stated in [12], keeping the inf-sup constant in the denominator of the error estimator causes potential risk for many problems with small inf-sup constants.…”

“…As already stated in [12], keeping the inf-sup constant in the denominator of the error estimator causes potential risk for many problems with small inf-sup constants. This is quite common in microwave circuits, where resonances show up in the frequency band of analysis, dropping the inf-sup constant down to zero [10,14]. Different strategies for a posteriori error estimation should be considered, reducing its computational cost to the same order of the ROM, if possible.…”