2040 E E E TRANSACTIONS ON MAGNETICS, VOL. 29, NO. 2, MARCH 1993 zyx RF Scattering and Radiation by Using a Decoupled Helmholtz Equation Approach John D'Angelo General Electric Corporate Research and Development Isaak D. Mayergoyz Electrical EngineeringDepartment University of Maryland zyxwvuts Abstract-A new finite element formulation for the solution of 3D RF scattering and radiation problems is presented here. This formulation is based on the solution of a set of decoupled Helmholtz equations for the Cartesian components of the field vectors. This results in a robust, computer efficient method which eliminates previous difficulties associated with "curl-curl" type partial differential equations. Although presented here in the frequency domain, the method is easily extendable to the time domain. I. INTRODUCTION The solution of 3D RF scattering and radiation problems by "fiite" methods (finite difference, f"iite elements, and finite volumes) can be approached in the space domain by using either two coupled first order partial differential equations (PDEs) zyxwvutsrqp [ 1,2] or a single second order equation [3,4]. In the time domain, the equations can be solved by using either time stepping methods or psuedo-time stepping methods zyxwvuts [5]; while in the frequency domain, direct or iterative matrix solution techniques are utilized. In this paper, a robust and computer efficient second-order, frequency domain f ite element method is described. Second order electromagnetic PDEs have definite advantages over first order equations. These advantages are: reduced number of unknowns, eliminating the need of constructing an "intertwined" staggered mesh, and better definition of material interfaces. In our paper [6], we have discussed a new approach to the formulatibn of 3D electromagnetic scattering problems. This approach is based on the replacement of "zero-divergence'' equations for field vectors by additional and equivalent boundary conditions. This results in boundary value problems for which Cartesian components of the field vectors are completely decoupled as far as the PDE equations are concerned coupling is realized only through the boundary conditions. This decoupling substantially reduces the computer memory and computational requirements over methods which couple field components throughout the solution region. Methods presented in [3,4] also produce decoupled equations by a different approach of using the "curl-curl'' formulation and applied gauge conditions. While this formulation and [3,4] produce the same vector Helmholtz equations in homogeneous material regions, the formulations differ at material interfaces and perfect conductors. The outlined approach for our formulation has been discussed in [6] for the electric field only. In this paper, we discuss this approach for the magnetic field formulation and illustrate its effectiveness on some numerical examples. A special effort is made to demonstrate this approach by solving "large problems", that is, problems in which the physical dimensions zyxwvutsrq are much larger than a wavelength. II. FORMULATION Consider an electromagnetic scattering of an incident field zyxwvutsr 2 by a perfect electric conductor with the boundary S (see figure 1). By using Maxwell theory, it can be shown that the above problem can be reduced to the following boundary value problem for the scattered field H. Find the solution of equation vxvxG-a2u=o zyxwvutsrqp inv- (a2=w2po&o) (1) subject zyxwvutsrqp to the boundary conditions Fig. 1 Solution region It is also assumed that zyxwv k satisfies the outgoing radiation conditions. It is easy to see that equation (1) is equivalent to the following two equations: v2e+ n2e = 0 (3) divH = o (4) Thus accordingg the boundary value problem (1)-(2), three Cartesian components of H satisfy four independent partial differential equations (3)-(4) and two independent boundary conditions (1)-(2). Next, we show that the "zero-di_vergenc_e" equation (4) can be replaced by the boundary condition ?.H = -?.Ho. For the sake of technical simplicity. we consider here the case of piece-wise flat boundaries (multi-faceted boundaries). It is asserted that the boundary value problem (1)-(2) is equivalent to the following boundary value problem: find the solution to the equation v2e+ Re = 0 subject to the boundary conditions: %, - aH,, JHT2 - JH,O? av m, av av (7) H,, = -H," (8) where: 21 and 22 are orthogonal to one another and tangential to S and H" is the incident magnetic field. It is easy to see that the boundary conditions, (6) and (7) follow from (2) and (8) while from (6). (7) and (8) one can get boundary condition (2). To prove the equivalence of where: 5; is a unit vector of outward normal to S, while i = fl 0018-9464/93$03.00 0 1993 IEEE