PIER
 
Progress In Electromagnetics Research
ISSN: 1070-4698, E-ISSN: 1559-8985
Home | Search | Notification | Authors | Submission | PIERS Home | EM Academy
Home > Vol. 127 > pp. 93-120

INVERSE DESIGN OF DIELECTRIC MATERIALS BY TOPOLOGY OPTIMIZATION

By M. Otomori, J. Andkjaer, O. Sigmund, K. Izui, and S. Nishiwaki

Full Article PDF (1,617 KB)

Abstract:
The capabilities and operation of electromagnetic devices can be dramatically enhanced if artificial materials that provide certain prescribed properties can be designed and fabricated. This paper presents a systematic methodology for the design of dielectric materials with prescribed electric permittivity. A gradient-based topology optimization method is used to find the distribution of dielectric material for the unit cell of a periodic microstructure composed of one or two dielectric materials. The optimization problem is formulated as a problem to minimize the square of the difference between the effective permittivity and a prescribed value. The optimization algorithm uses the adjoint variable method (AVM) for the sensitivity analysis and the finite element method (FEM) for solving the equilibrium and adjoint equations, respectively. A Heaviside projection filter is used to obtain clear optimized configurations. Several design problems show that clear optimized unit cell configurations that provide the prescribed electric permittivity can be obtained for all the presented cases. These include the design of isotropic material, anisotropic material, anisotropic material with a non-zero off-diagonal terms, and anisotropic material with loss. The results show that the optimized values are in agreement with theoretical bounds, confirming that our method yields appropriate and useful solutions.

Citation:
M. Otomori, J. Andkjaer, O. Sigmund, K. Izui, and S. Nishiwaki, "Inverse design of dielectric materials by topology optimization," Progress In Electromagnetics Research, Vol. 127, 93-120, 2012.
doi:10.2528/PIER12020501
http://www.jpier.org/pier/pier.php?paper=12020501

References:
1. Bendsoe, M. P. and N. Kikuchi, "Generating optimal topologies in structural design using a homogenization method," Comput. Methods Appl. Mech. Engrg., Vol. 71, No. 2, 197-224, 1988.
doi:10.1016/0045-7825(88)90086-2

2. Yoo, J., N. Kikuchi, and J. L. Volakis, "Structural optimization in magnetic devices by the homogenization design method," IEEE T. Magn., Vol. 36, No. 3, 574-580, 2000.
doi:10.1109/20.846220

3. Nomura, T., S. Nishiwaki, K. Sato, and K. Hirayama, "Topology optimization for the design of periodic microstructures composed of electromagnetic materials," Finite Elem. Anal. Des., Vol. 45, No. 3, 210-226, 2009.
doi:10.1016/j.finel.2008.10.006

4. Nishiwaki, S., T. Nomura, S. Kinoshita, K. Izui, M. Yoshimura, K. Sato, and K. Hirayama, "Topology optimization for cross-section designs of electromagnetic waveguides targeting guiding characteristics," Finite Elem. Anal. Des., Vol. 45, No. 12, 944-957, 2009.
doi:10.1016/j.finel.2009.09.008

5. Yamasaki, S., T. Nomura, A. Kawamoto, K. Sato, and S. Nishiwaki, "A level set-based topology optimization method targeting metallic waveguide design problems ," Int. J. Numer. Meth. Eng., Vol. 87, No. 9, 844-868, 2011.
doi:10.1002/nme.3135

6. Borrvall, T. and J. Petersson, "Topology optimization of fluids in stoles flow," Int. J. Numer. Meth. Eng., Vol. 41, No. 1, 77-107, 2003.
doi:10.1002/fld.426

7. Sigmund, O. and J. S. Jensen, "Systematic design of phononic band-gap materials and structures by topology optimization," Phil. Trans. R. Soc. Lond. A, Vol. 361, 1001-1019, 2003.
doi:10.1098/rsta.2003.1177

8. Sigmund, O. and S. Torquato, "Design of materials with extreme thermal expansion using a three-phase topology optimization method ," J. Mech. Phys. Solids., Vol. 45, No. 6, 1037-1067, 1997.
doi:10.1016/S0022-5096(96)00114-7

9. Larsen, U. D., O. Sigmund, and S. Bouwstra, "Design and fabrication of compliant micromechanisms and structures with negative Poisson's ratio ," J. Microelectromech. S., Vol. 6, No. 2, 99-106, 1997.
doi:10.1109/84.585787

10. Sigmund, O., "Materials with prescribed constitutive parameters: an inverse homogenization problem," INT. J. Solids Struct., Vol. 31, No. 17, 2313-2329, 1994.
doi:10.1016/0020-7683(94)90154-6

11. Choi, J. S. and J. Yoo, "Design and application of layered composites with the prescribed magnetic permeability," Int. J. Numer. Meth. Eng., Vol. 82, No. 1, 1-25, 2010.

12. El-Kahlout, Y. and G. Kiziltas, "Inverse synthesis of electromagnetic materials using homogenization based topology optimization," Progress In Electromagnetics Research, Vol. 115, 343-380, 2011.

13. Sihvola, A., Electromagnetic Mixing Formulas and Applicataions, The Institution of Engineering and Technology, London, 1999.

14. Bensoussan, A., J. L. Lions, and G. Papanicolau, Asymptotic Analysis for Periodic Structures, North-Holland Publishing Company, Amsterdam, 1978.

15. Sanchez-Palencia, E., "Non-homogeneous media and vibration theory," Lecture Notes in Physics, Vol. 127, Springer-Verlag, Berlin, 1980.

16. Hashin, Z., "Analysis of composite materials --- A survey," J. Appl. Mech., Vol. 50, No. 3, 481-505, 1983.
doi:10.1115/1.3167081

17. Milton, G. W., The Theory of Composites, Cambridge University Press, Cambridge, 2001.

18. Sigmund, O., "On the usefulness of non-gradient approaches in topology optimization," Struct. Multidisc. Optim., Vol. 43, No. 5, 589-596, 2011.
doi:10.1007/s00158-011-0638-7

19. Bendsoe, M. P. and O. Sigmund, "Material interpolation schemes in topology optimization," Arch. Appl. Mech., Vol. 69, No. 9, 635-654, 1999.
doi:10.1007/s004190050248

20. Guest, J. K., J. H. Prévost, and T. Belytschko, "Achieving minimum length scale in topology optimization using nodal design variables and projection functions ," Int. J. Numer. Meth. Eng., Vol. 61, No. 2, 238-254, 2004.
doi:10.1002/nme.1064

21. Sigmund, O., "Morphology-based black and white filters for topology optimization," Struct. Multidisc. Optim., Vol. 33, No. 4-5, 401-424, 2007.
doi:10.1007/s00158-006-0087-x

22. Olesen, L. H., "A high-level programming-language implementation of topology optimization applied to steady-state Navier-Stokes flow," Int. J. Numer. Meth. Eng., Vol. 65, No. 7, 975-1001, 2006.
doi:10.1002/nme.1468

23. Milton, G. W., "Bounds on the complex permittivity of a two-component composite material," J. Appl. Phys., Vol. 52, No. 8, 5286-5293, 1981.
doi:10.1063/1.329385

24. Hashin, Z., "The elastic moduli of heterogeneous materials," J. Appl. Mech., Vol. 29, No. 1, 143-150, 1962.
doi:10.1115/1.3636446

25. Francfort, G. and F. Murat, "Homogenization and optimal bounds in linear elasticity," Archive for Rational Mechanics and Analysis, Vol. 94, No. 4, 307-334, 1986.
doi:10.1007/BF00280908

26. Lurie, K. A. and A. V. Cherkaev, "Optimization of properties of multicomponent isotropic composites," J. Optimiz. Theory App., Vol. 46, No. 4, 571-580, 1985.
doi:10.1007/BF00939160

27. Milton, G. W., "Modelling the Properties of Composites by Laminates,", Vol. 1, 150-174, Springer-Verlag, New York, 1985.

28. Norris, A. N., "A differential scheme for the effective moduli of composites," Mech. Mater., Vol. 4, No. 1, 1-16, 1985.
doi:10.1016/0167-6636(85)90002-X

29. Sigmund, O., "A new class of extremal composites," J. Mech. Phys. Solids, Vol. 48, No. 2, 397-428, 2000.
doi:10.1016/S0022-5096(99)00034-4

30. Grabovsky, Y. and R. V. Kohn, "Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. II: the Vigdergauz microstructure," J. Mech. Phys. Solids, Vol. 43, No. 6, 949-972, 1995.
doi:10.1016/0022-5096(95)00017-D

31. Vigdergauz, S., "Energy-minimizing inclusions in a planar elastic structure with macroisotropy," Struct. Optimization, Vol. 17, No. 2-3, 104-112, 1999.
doi:10.1007/BF01195935

32. Svanberg, K., "A class of globally convergent optimization methods based on conservative convex separable approximations," SIAM J Optimiz., Vol. 12, No. 2, 555-573, 2002.
doi:10.1137/S1052623499362822

33. Rayleigh, L., "On the influence of obstacles arranged in rectangular order upon the properties of a medium," Philos. Mag., Vol. 34, No. 211, 481-502, 1892.

34. Wallén, H., H. Kettunen, and A. Sihvola, "Composite near-field superlens design using mixing formulas and simulations," Metamaterials, Vol. 3, No. 3-4, 129-139, 2009.
doi:10.1016/j.metmat.2009.08.002


© Copyright 2014 EMW Publishing. All Rights Reserved