Parametrized Local Reduced-Order Models With Compressed Projection Basis for Fast Parameter-Dependent Finite-Element Analysis

This paper proposes an automated parametric local model-order reduction scheme for the expedited design of microwave devices using the full-wave finite-element method (FEM). The approach proposed here results in parameterized reduced-order models (ROMs) that account for the geometry and material variation in the selected subregion of the structure. In each subregion, a parameter-dependent projection basis is generated by concatenating several local bases that correspond to different parameter values, yielding a small, dense ROM. The process is automated and uses an adaptive scheme guided by a local goal-oriented error estimator to select points in the parameter space at which a local basis needs to be computed. A two-stage basis compression technique is also proposed to remove the redundancy from the projection basis and yields compact macromodels. Numerical examples, including FE analysis of a fifth-order filter with seven geometric variables as parameters, demonstrate that the approach provides a significant reduction in computational time while preserving the accuracy of the simulations.

I. Introduction

Numerical simulations are used in almost all areas of engineering as a tool for modeling physical phenomena. In the past few decades, they have become indispensable tools in the process of designing and optimizing microwave devices. One of the most popular numerical techniques used in computational electromagnetics (CEM) is the finite-element method (FEM), which allows for full-wave solutions of Maxwell’s equations. It is especially advisable to use FEM when the computational domain contains complex geometries or inhomogeneous or anisotropic materials. However, the analysis of such problems with FEM usually leads to large systems of equations with as many as millions of unknowns of the form

$$\begin{equation} \mathbf {A}(f) \mathbf {x}(f) = \mathbf {b}(f) \end{equation}$$ where is a frequency-dependent sparse system matrix, is an excitation vector, and is a solution vector. For a large number of unknowns, the solution of (1) may take many hours or even days. The problem of long simulation times is exacerbated if the analysis involves repeated changes in the material or geometrical parameters of the structure—as for example in optimization or parametric studies. The coefficient in the system matrix becomes parameter-dependent and computations take much longer, since the system matrix needs to be updated each time a design variable or a parameter is modified; (1) thus needs to be solved anew many times in the specified frequency bandwidth.