PIER B
 
Progress In Electromagnetics Research B
ISSN: 1937-6472
Home | Search | Notification | Authors | Submission | PIERS Home | EM Academy
Home > Vol. 45 > pp. 117-146

ISOGEOMETRIC SHAPE OPTIMIZATION FOR ELECTROMAGNETIC SCATTERING PROBLEMS

By D. M. Nguyen, A. Evgrafov, and J. Gravesen

Full Article PDF (1,195 KB)

Abstract:
We consider the benchmark problem of magnetic energy density enhancement in a small spatial region by varying the shape of two symmetric conducting scatterers. We view this problem as a prototype for a wide variety of geometric design problems in electromagnetic applications. Our approach for solving this problem is based on shape optimization and isogeometric analysis. One of the major difficulties we face to make these methods work together is the need to maintain a valid parametrization of the computational domain during the optimization. Our approach to generating a domain parametrization is based on minimizing a second order approximation to the Winslow functional in the vicinity of a reference parametrization. Furthermore, we enforce the validity of the parametrization by ensuring the non-negativity of the coefficients of a B-spline expansion of the Jacobian. The shape found by this approach outperforms earlier design computed using topology optimization by a factor of one billion.

Citation:
D. M. Nguyen, A. Evgrafov, and J. Gravesen, "Isogeometric shape optimization for electromagnetic scattering problems," Progress In Electromagnetics Research B, Vol. 45, 117-146, 2012.
doi:10.2528/PIERB12091308
http://www.jpier.org/pierb/pier.php?paper=12091308

References:
1. Ding, Y., "Shape optimization of structures: A literature survey," Computers & Structures, Vol. 24, No. 6, 985-1004, 1986.
doi:10.1016/0045-7949(86)90307-X

2. Delfour, M. and J. Zolesio, "Shapes and Geometries. Analysis, Differential Calculus, and Optimization, Advances in Design and Control,", Vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001.

3. Mohammadi, B. and O. Pironneau, Applied Shape Optimization for Fluids, Oxford University Press, 2001.

4. Braibant, V. and C. Fleury, "Shape optimal design using B-splines," Comput. Methods Appl. Mech. Engrg., Vol. 44, No. 3, 247-267, 1984.
doi:10.1016/0045-7825(84)90132-4

5. Olhoff, N., M. P. Bendsoe, and J. Rasmussen, "CAD-integrated structural topology and design optimization," Shape and Layout Optimization of Structural Systems and Optimality Criteria Methods, CISM Courses and Lectures, Vol. 325, 171-197, Springer, Vienna, 1992.

6. Nguyen, D. M., A. Evgrafov, A. R. Gersborg, and J. Gravesen, "Isogeometric shape optimization of vibrating membranes," Comput. Methods Appl. Mech. Engrg., Vol. 200, No. 13-16, 1343-1353, 2011.
doi:10.1016/j.cma.2010.12.015

7. Hughes, T., J. Cottrell, and Y. Bazilevs, "Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement," Comput. Methods Appl. Mech. Engrg., Vol. 194, No. 39-41, 4135-4195, 2005.
doi:10.1016/j.cma.2004.10.008

8. Wall, W., M. Frenzel, and C. Cyron, "Isogeometric structural shape optimization," Comput. Methods Appl. Mech. Engrg., Vol. 197, No. 33-40, 2976-2988, 2008.
doi:10.1016/j.cma.2008.01.025

9. Cho, S. and S. H. Ha, "Isogeometric shape design optimization: Exact geometry and enhanced sensitivity," Struct. Multidiscip. Optim., Vol. 38, No. 1, 53-70, 2009.
doi:10.1007/s00158-008-0266-z

10. Nagy, A., M. Abdalla, and Z. Gurdal, "Isogeometric sizing and shape optimisation of beam structures," Comput. Methods Appl. Mech. Engrg., Vol. 199, No. 17-20, 1216-1230, 2010.
doi:10.1016/j.cma.2009.12.010

11. Qian, X., "Full analytical sensitivities in nurbs based isogeometric shape optimization," Comput. Methods Appl. Mech. Engrg., Vol. 199, No. 29-32, 2059-2071, 2010.
doi:10.1016/j.cma.2010.03.005

12. Seo, Y. D., H. J. Kim, and S. K. Youn, "Shape optimization and its extension to topological design based on isogeometric analysis," International Journal of Solids and Structures, Vol. 47, No. 11--12, 1618-1640, 2010.
doi:10.1016/j.ijsolstr.2010.03.004

13. Cottrell, J. A., Y. Bazilevs, and T. J. R. Hughes, Isogeometric Analysis: Toward Integration of CAD and FEA, J. Wiley, West Sussex, 2009.

14. Knupp, P. and S. Steinberg, Fundamentals of Grid Generation, CRC Press, Boca Ranton, 1993.

15. Gravesen, J., A. Evgrafov, A. R. Gersborg, D. M. Nguyen, and P. N. Nielsen, "Isogeometric analysis and shape optimisation," Proceedings of NSCM-23: The 23rd Nordic Seminar on Computational Mechanics, Vol. 23, 14-17, 2010.

16. Aage, N., N. Mortensen, and O. Sigmund, "Topology optimization of metallic devices for microwave applications," International Journal for Numerical Methods in Engineering, Vol. 83, No. 2, 228-248, 2010.

17. Allaire, G., "Conception optimale de structures," Mathematiques et Applications, 58, Springer, 2007.

18. Bendse, M. and O. Sigmund, Topology Optimization. Theory, Methods and Applications, pringer-Verlag, Berlin, 2003.

19. Jin, J., The Finite Element Method in Electromagnetics, John Wiley & Sons, New York, 2002.

20. dams, R., Sobolev Spaces. Pure and Applied Mathematics, 65, Academic Press, New York, London, 1975.

21. O., R. Taylor, The Finite Element Method, Vol. 1, The Basis, 5th Ed., Butterworth-Heinemann, Oxford, 2000.

22. Gravesen, J., "Differential geometry and design of shape and motion,", Department of Mathematics, Technical University of Denmark, 2002, URL: http://www2.mat.dtu.dk/people/J.Gravesen/cagd.pdf.

23. Piegl, L. and W. Tiller, The NURBS Book (Monographs in Visual Communication), Springer-Verlag, Berlin, 1995.

24. De Boor, C. and G. Fix, "Spline approximation by quasiinter-polants," J. Approximation Theory, Vol. 8, 19-45, 1973.
doi:10.1016/0021-9045(73)90029-4

25., , MathWorks Inc., URL: http://www.mathworks.com.

26. Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1964.

27. Balanis, C. A., Advanced Engineering Electromagnetics, Wiley, New York, 2005.

28., , COMSOL Inc., URL: http://www.comsol.com.

29. Cheng, D. K., Fundamentals of Engineering Electromagnetics,, Addison-Wesley, Reading, Massachusetts, 1993.

30. Aage, N., Personal Communication, 2011.


© Copyright 2010 EMW Publishing. All Rights Reserved