Gradient-based adjoint-variable optimization of broadband microstrip antennas with mixed-order prism macroelements

Abstract

This paper describes a new finite element method (FEM)-based adjoint-variable approach to design and optimization of planar microwave circuits using sensitivity analysis, with respect to small variations of their design parameters. The implementation is based on a specially developed class of mixed-order prismatic macroelements, suitable for an efficient analysis of planar microwave structures. Their use, combined with a simple probe feed model, not only reduces the overall computational effort but also facilitates a straightforward derivation of port parameter sensitivities. Explicit expressions are given for the adjoint solution and S-parameter sensitivity and the process is verified through the design and optimization of two slotted broadband microstrip patch antennas.

Introduction

The design and optimization of microwave circuits and antennas is often performed by means of evolutionary strategies, like genetic algorithms or similar techniques [1]. Several variations of this particular class of algorithms have been widely applied to optimization of various RF and microwave circuits, especially planar ones [2], [3]. Such approaches have come up with very good results, since they perform an extensive search over the set of possible solutions. Their popularity and wide applicability are also justified by simplicity and ease of implementation, particularly due to the fact that the electromagnetic analysis and the optimization algorithm are fully separated, which facilitates the use of commercial tools and existing libraries. This, however, comes at the price of increased computational cost, mainly in terms of design time. A highly promising alternative seems to be the application of gradient-based optimization techniques, especially when design parameters are real valued. Although such techniques have been established in lumped-circuit sensitivity analysis and used in waveguide optimization problems [4], [5], their application is relatively limited in the area of microwave design. Recently, there seems to be a reconsideration of such techniques, since there is plenty of evidence that they may be more robust and efficient in terms of computational time. In such a framework, the variation of the response of a microwave device with respect to its geometrical or material parameters, known as design sensitivity, is of great importance for design optimization. In an optimization problem, the design sensitivity is expressed by the gradient of a response (objective) function, which is chosen to describe the structure's performance, according to a specified criterion, e.g. bandwidth, gain, compactness, etc. Since, the objective function and its gradient need to be repeatedly evaluated, a fast EM solver and a convenient calculation of sensitivities are crucial for such an algorithm to be effective.

The use of the finite element method (FEM), is a likely candidate as an EM solver, since it is easily adaptable to geometric features of arbitrary shape, while it also offers the possibility of a rigorous sensitivity analysis. However, it is essential to keep the number of degrees of freedom low or speedup the computation by other means, otherwise the optimization process may be computationally cumbersome. In this paper, a new mixed-order prism vector macroelement, suitable for the analysis of planar microwave devices is described and utilized. These special kind of elements, having different degrees of approximation along different directions, fit particularly well into the framework of planar structures, since the implementation is based on the fact that the geometry of planar circuits is essentially two dimensional, and the main field variation is observed toward the direction that is vertical to the planar interfaces (Fig. 1). The “mixed-order” terminology, used herein, implies our requirement for arbitrary orders of variation toward different directions. The employment of such elements not only makes possible the use of two-dimensional mesh generators and a simple extrusion to get the three-dimensional structure, but also leads to a remarkable reduction of the number of degrees of freedom. By applying the A–V formulation [6] and convergence-optimized schemes [7] for a third-order macroelement, additional numerical convergence acceleration is achieved.

As for the sensitivity analysis, the most efficient way to obtain the design sensitivities is the adjoint-variable method (AVM) [8]. The adjoint-based sensitivity formula requires the calculation of the derivatives of the FEM system matrix with respect to the design parameters and the solution of the adjoint problem. Generally, the derivatives can be approximated using finite differences [9], but in FEM analysis this approach is disadvantageous, since it requires additional simulations for each design variable. It may be even prohibitive if the perturbation of design variable nodes does not preserve the mesh topology. In the case of FEM, however, an explicit analytical relation between the entries of the assembled system matrix and the coordinates of the grid nodes can be established, thus enabling the analytical calculation of derivatives. This was applied in [10], [11] for tetrahedral edge elements and high-order vector elements, respectively, and a similar approach in the case of nodal elements was used in [12]. In [10], it was also shown that under certain assumptions, no additional adjoint simulation is needed and the required adjoint solution is directly obtained from the original problem solution. An extension of this point is described in [13], where the relation between original and adjoint vectors for network parameters is also derived.

Here, the development of a third-order prism vector element is discussed first. The AVM is briefly outlined and the adjoint-based sensitivity formula is, then, presented. The employment of the probe feed model results in a simple expression for the adjoint solution and the system matrix derivatives are calculated, element by element, during the assembly procedure. To confirm our approach, a line-search scheme for bandwidth optimization of two slotted patch antennas is included.