Department of Mathematics. Department of Computer Science. University of J&m, Jaffna. Sri Lanka. R-Eaoale. Dept. of Engineermg. Hamy Mudd College.
IEEE TRANSACTIONS ON MAGNETICS, VOL. 27. NO.5, SEPTEMBER 1991
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TWO RE$UISITE TOOLS IN THE OPTIMAL DESIGN OF ELECTROMAGNETIC DEVICES Brfrlvme 8ubrrm.nl.m 8. a.n.lhn.th.n Department of Computer Science Department of Mathematics University of J&m, Jaffna. Sri Lanka.
R-Eaoale ABSTRACT
Dept. of Engineermg. H a m y Mudd College Claranont, CA 91711, USA. removed.
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The requisite methodology for performing optimal design the sythesis of devices from specified performance standards is now in place.
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What remains to be done, is the building of software that is reliable and would be transparent to the user. If this could be accomplished. then these sophisticated design methods would be quickly accepted in industry.
This paper presents two of the requisite tools for such automatic
II. ME3H GENERATION Presently finite element mesh generators are available that allow the user to define the mesh a s ob ects that are drawn within a window that is the boundary o/ the solution domain 15.61. Thereafter. these mesh generators use some scheme, ofkm the Delaunay method of mesh optimlzation(7j. to mesh the region. As noted however. the fact that these generatom require user input in the form of d r a m makes them unsuited to the task optimal des@ without modification.
implementation: 1. A parametrized mesh generator that allows the design iterations to proceed without interruption and ii. An opti&zation algorithm-that takes care of different object function shapes, using different optimization algorithms, including the principle of tunneling.
L THE STATE OF THE ART IN OPTIMAL DESIGN We are now beginning to make headway in our movement from solving the classic direct analysis problem to solving the more relevant indirect inverse problem 11-41. The essence of the method involves the definition of a n object function which is " k e d with respect to the parameters that describe the device under design. Thus we start with a design, analyze and obtain its performance and then, inverse problem methodology tells us how we must change the parameters to move towards our object which is the satisfaction of some performance criterion expressed through the object function. Iteratively. thus, we move progressively towards the perfe-ct synthesis. These sophisticated methods of device synthesis are now increasingly available in research laboratories. When scientific publications report results. the results are essentially obtained by the utpert. the goal being to establish the method of synthesis, rather than to write easy to use software. The emphasis is not on creating a design tool whose methods are 'black-boxed' to the design engineer, who, if he is to ever use these tools, would be interested only in the end result.
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Thus, once a design parameter is changed in the research laboratory. the research scientist would model the new device defined by the new set of parameters using a new flnite element mesh and try to get the next set of new values. Similarly. if the parameters do not converge by one method of optimization, the scientist would stop and start with another optimizauon method. However. it is a necessarv measure of the success of these methods that they should be accepted in routine design synthesis by engineers. This in turn implies that once the parameters are changed. analysis should proceed without user input so that the untutored user might use these methods easily. Similarly. when one algorithm for minimization of the object function fails. another should be used automatically. The success of these methods, then is a matter of making these methods work with little user intervention. This paper describes the implementation of a n optimization program using: 1.
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A parametric mesh generator that treats a two dimensional finite element mesh a s a collection of objects described through parameters and generates the mesh from this information. Thus once the initial estimates of the parameters of the device are specified. the program automatically generates the mesh, solves the problem and can call the optimization algorithm. The new set of iterated parameters from the optimization algorithm. are used to generate the mesh for the new devoce description. An optimization package that employs a wide range of optimlzatmn methods. Starting from the simplest, if and when one method fails. another is called in. As a result. the onus on the user to choose a n alternative " f z a t i o n method is
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Filun1:PMmetrlcDaui~of.~ In this paper, as shown in Fig. la, the polygonal objects are defIned within a rectangular frame through parameters. Then is no serious limit on the number of polygons or number of polygon sides. mmpt Each polygon those imposed in the code for dimensioning purp-. is identified by two real numbers (the permeabiltty and current in magnetostatic problems). an integer n describing the number of sides it has. an integer m that specifies the number of nodes required on each edge of the polygon, a pair of real numbers gMng the coordinates of the first node and then. n-2 pairs of real numbers giving the coordinate increments to the following node. These parameters then completely describe the polygons. Once this is done, then using existing methodology using Voronoi tessellations 171. the meshing may be accomplished. Circles and c w e d objects are modeled using polygons. Convex polygons are modeled using more than one concave polygon with restraints on the parameters of one. However. one of the weaknesses of inverse problem methodology is that it takes a long while to reach the optimum and this is one of the chief impediments to its full acceptance. It is therefore necessary to do all one can to save time. It was found that with
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many, many noded polygons, the Voronoi approach slows the mesh generation. To overcome this. the approach we have adopted is to mesh the interior of each polygon (by computing an interior node to each polygonal node as shown in Fig. lb) and then connect the node to the two closest nodes of the exterior rectangle wherever it is warranted. Thereafter, we use the Delaunay scheme to optimize this set of connections a s described in 151 and proceed to add points to the mesh until a certain triangle quality as given by obtuse angles and specifled by the polygons is realized. Figures 1 and 2 represent 2 positions of a polygonal current source that is being identified through matching the finite element solution with measurements. The first is the original position and the second is the position and shape after 7 iterations. III. OPTIMIZATION METHODS
IILl General Concept In inverse problems, a scalar function with continuous first and second derivatives is minimized to build the best product; the scalar function is usually defined a s a function of a physical quantity. Electromagnetic systems are defined by geometric parameters and physical parameters. These parameters and the physical quantities are related to each other by electromagnetic equations. In minimizing the object function with respect to the design/descriptive parameters of interest, one has a range of choices as to what algorithm is to be used: simplex. steepest descent, conjugate gradients, etc. Each is specifically useful in a situation, and all take one to the " m u m closest to where one is.
IEEE TRANSACTIONS ON MAGNETICS, VOL. 27, NO.5, SE€TEMBER 1991 break down because the function is not smooth on account of our using dmerent meshes. When the function is not smooth, the
gradient methods break down, particularly close to a m u m . But because of tunneling it is not sufficient to stop close to the minimum. Otherwise we would be unable to escape the local minimum that lies close to it after the function has been modified. As and when these gradient methods break-down, we switch to a robust but slow simplex method and do reach the real local minimum before resorting to tunneling. In the following we describe the implemetational details of this
approach. In general, we consider the minimization of a scalar function F b ) , where F has continuous first and second derlvatlves, and A= A(x) is a physical quantity. If Xn is a global minimizer of F(A(x)) then F(A(xi+l))
