Efficient FEM-Based EM Optimization Technique Using Combined Lagrangian Method With Newton’s Method

Abstract

Gradient-based optimization algorithms are popularly used in electromagnetic (EM)-based design optimizations. Among the gradient-based optimization algorithms, Newton’s method is not practically applicable to EM optimization because it is very time consuming to obtain the Hessian matrix containing the second-order derivatives of the EM responses with respect to the geometrical variables. This article addresses this situation and proposes an efficient gradient-based EM optimization technique using the combined Lagrangian method with Newton’s method. EM optimizations can be reformulated into constrained optimizations when the finite element method (FEM) is applied to perform EM simulations. In this article, we propose to elevate the Lagrangian method (i.e., a popular constrained optimization method) to EM optimization. By using the Lagrangian method to perform the EM optimization, the Hessian matrix can be obtained efficiently without the time-consuming evaluations of second-order derivatives of the EM responses with respect to the geometrical variables. With the efficiently calculated the Hessian matrix, Newton’s method can be applied. We derive new formulations of Newton’s method specifically for the EM optimization with the Lagrangian method. The proposed EM optimization using the combined Lagrangian method with Newton’s method can converge faster than direct EM optimizations with other gradient-based optimization methods. The proposed technique is demonstrated by two EM optimization examples of microwave components.