I944 zyxwvutsrqponmlkjih IEEE TRANSACTIONS ON MAGNETICS. zyxwvu VOL. 31. NO 3. MAY zyxw 1995 Nonlinear Shape Design Sensitivity Analysis of Magnetostatic Problems using Boundary Element Method zyx Chang Seop Koh and Osama A. Mohammed Dept. of Electrical & Computer Engineering, Florida International University, Miami, Florida 33199, USA Song-yop Hahn Dept. of Electrical Engineering, Seoul National University, Seoul 15 1-742, KOREA zyxwv Abstract-A design sensitivity analysis is extended to cover the nonlinear magnetostatic problems. The formula is derived using material derivative as boundary integrals of the state and adjoint variables over the deformable boundary. A solution scheme for nonlinear state and adjoint equations using boundary element method is also presented to get accurate sensitivity coefficients. And a normalizing scheme for the pointwise sensitivity coefficients are proposed to avoid a zig-zag shapes during optimization. I. INTRODUCTION In practical engineering, synthesizing the best engineering solution to a given physical design problem is required. Several methods have been developed for this purpose such as deterministic method using design sensitivity analysis[ 1,2,3, 4,5], evolution strategy[6], simulated airnealingl71, and artificial intelligence[8]. Among them the deterniinistic method using design sensitivity analysis has proved to give a proper design within a reasonable number of iterations for problems that have complex shape with relatively good initial design and many design variables over 30. In the method, an accurate computation of the design sensitivity coefficients is very important to increase the computational efficiency. There are two methods for the bensitivity computation: the discrete approach based on discretized system equation and continuum approach based on matenal derivative. However, the latter is considered to give more accurate sensitivity coefficients than the former because less numerical errors are involved. In the continuum approach, the design sensitivity formula should be expressed as boundary integrals of the variables over the deformable boundary. This is because the design velocity vector can not be uniquely defined inside the domain. Hence, in this approach, the boundary element is preferred to finite element method because the method is considered to give more accurate solutions on the boundary than the finite element method. In this paper, the analytic design sensitivity formula for nonlinear magnetostatic problems is derived based on the formula for linear problems[ 1,2]. The pointwise sensitivity coefficients are suggested to be normalized to avoid a zig- zag shape during optimization. Manuscript received on July 6, 1994 11. NONLINEAR DESIGN SENSITIVITY ANALYSIS A. zyxwvu Design sensitivify formula Let us consider the interface problem between nonlinear and linear materials. The nonlinear magnetic material such as iron is R1 and the linear one such as air is 02 where the exciting current or permanent magnet exists. And the interface between the two materials is y. The outward unit normal vector n is defined on the interface from R1 to R2. The region of interest may be assumed belong to the linear region (usually air region) for convenience. In the design of electromagnetic system, the objective function that represents the system performance such as distribution of the force, torque or flus density quantitatively may be written as : where zyxwvu f is a differential function and numeric subscripts denote the region and mp is a characteristic function through which the region of interest can be indicated. The augmented objective function, F1, is, at first, defined as the difference between the objective function and the variational form of the governing equation of the system. And the shape design sensitivity is derived from the material derivative of the augmented objective function as in linear problems[ 1,2]. Considering the weighting functions A,, A, and their material derivatives belong to the admissible set of variation K, and m =0, zyxwv J=O on y and using the material derivative formula, t i e variation of the augmented objective function can be represented as follows:[ 1,2,9]: -- 4 = IQ, { zyxwv fA, * 22 + fB . B( ) }jl/p dv -Jnl { v,B(~ zyxwv )*B(A,) +~xB(A,) .B(~,)B(A~ ).B(A, )WV -In, V*B(k2).B(;?Z)d\4*, J,.(VA, .V)dv +Jq { v, +J*,{V~B(VA,.V).B(A,)+ v,B(A~)*B(vA, .~)}dv + Jnl { 2,@( AI ) * B( VAl. v) B (if, ). B( zyx 3 ) )dv - J,, { ~ B ( A I *B(A, ) - V~B( 4 ). B(A, ))vnds -J*,{~A* ‘(VA2.V)+fB.B(VA,’Y))nr,h, B(VAl *V)*B(Al) + yB(A,)-B(VA, .V)}h (2) 0018-9464195$04.00 0 1995 lEEE