Free-form Reflector Design Using Differential Evolution Algorithm

NURBS surface representation, combined with Differential Evolution (DE), enables us to perform automated non-imaging reflector design. The overall result is a simple automated nonimaging reflector design technique and only a little data such as desired illuminance distribution and searching limits are needed. Merit functions specific to non-imaging reflector design are presented. Using different merit functions, the generated illuminance distribution can be uniform as well as concentrated. DE is performed to obtain the reflector that generates the desired illuminance distribution. The photometric distributions are calculated through Monte-Carlo ray tracing and the illuminance value is used to calculate the merit function value. The validity of the proposed approach is demonstrated by optimization examples. Almost the same uniform illuminance distribution can be obtained using the algorithm proposed in this paper as that obtained by edge-raymethod. A concentrated illuminance distribution can also be generated using the algorithm proposed. Introduction Non-imaging reflectors are widely used in illumination and solar energy systems.[1,2] In these applications, the nonimaging reflector is designed to convert the lambertian source into a beam with a divergence of half-angle θ without loss of throughput. One direct design approach is using algebraic methods, among which tailored edge ray approach has made great success. [3-5] In this approach, the two-dimensional reflector is determined by the reflector equations that are deduced from edge-ray method and conservation of energy. For three dimensions reflector design, it is common to solve the problem by reducing to two dimensions (2D), then the 3D reflector can be generated by means of rotational or translational symmetry. The second approach is indirect and it uses optimization-based design techniques.[6-8] Free form surfaces e.g. Bézier surface and NURBS surface are appropriate to represent the shape of a reflector in this automated design approach. Since Holland laid down the basic principles of genetic algorithms, [9]the genetic algorithm has been proven to be valuable both in imaging and non-imaging optics.[10,11] The Differential Evolution(DE) is a variation of the genetic algorithm and was proposed by R.Storn and K.Price. [12] Algorithm that combines the global search capability of the DE with the flexibility of the shape adjustment of free form surface has been used in the second approach of reflector design. S.Doyle used cubic Bézier curve to represent the profile of a reflector and paraboloid and ellipse reflector can be obtained using his algorithm.[8] For the design of the non-imaging reflector, we developed an algorithm to perform an automated free form reflector generation. The optimization examples show that the proposed approach is effective and robust. Design Method NURBS Curve. NURBS has become the de facto industry standard for the representation, design and data exchange of geometric information processed by computers. The excellent mathematical and algorithmic properties, combined with successful industrial applications, have contributed to the enormous popularity of NURBS. The mathematical expression of a NURBS curve is as follows: Key Engineering Materials Vols. 364-366 (2008) pp 138-142 online at http://www.scientific.net © (2008) Trans Tech Publications, Switzerland Online available since 2007/Dec/06 All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 130.203.133.34-15/04/08,15:59:54)