Topology optimization of conductors in electrical circuit

  • Katsuya NomuraEmail author
  • Shintaro Yamasaki
  • Kentaro Yaji
  • Hiroki Bo
  • Atsuhiro Takahashi
  • Takashi Kojima
  • Kikuo Fujita
Research Paper


This study proposes a topology optimization method for realizing a free-form design of conductors in electrical circuits. Conductors in a circuit must connect components, such as voltage sources, resistors, capacitors, and inductors, according to the given circuit diagram. The shape of conductors has a strong effect on the high-frequency performance of a circuit due to parasitic circuit elements such as parasitic inductance and capacitance. In this study, we apply topology optimization to the design of such conductors to minimize parasitic effects with maximum flexibility of shape manipulation. However, when the distribution of conductors is repeatedly updated in topology optimization, disconnections and connections of conductors that cause open and short circuits, respectively, may occur. To prevent this, a method that uses fictitious electric current and electric field calculations is proposed. Disallowed disconnections are prevented by limiting the maximum value of the fictitious current density in conductors where a current is induced. This concept is based on the fact that an electric current becomes concentrated in a thin conductor before disconnection occurs. Disallowed connections are prevented by limiting the maximum value of the fictitious electric field strength around conductors where a voltage is applied. This is based on the fact that the electric field in a parallel plate capacitor is inversely proportional to the distance between the plates. These limitations are aggregated as a single constraint using the Kreisselmeier-Steinhauser function in the formulation of optimization problems. This constraint prevents only disallowed disconnections and connections, but does not prevent allowed topology changes. The effectiveness of the constraint is confirmed using simple examples, and an actual design problem involving conductors in electromagnetic interference filters is used to verify that the proposed constraint can be utilized for conductor optimization.


Topology optimization Geometric constraint Conductor Electrical circuit Electromagnetic interference filter 



  1. Aage N, Egede Johansen V (2017) Topology optimization of microwave waveguide filters. Int J Numer Methods Eng 112(3):283–300MathSciNetGoogle Scholar
  2. Aage N, Mortensen N, Sigmund O (2010) Topology optimization of metallic devices for microwave applications. Int J Numer Methods Eng 83(2):228–248MathSciNetzbMATHGoogle Scholar
  3. Alexandrov O, Santosa F (2005) A topology-preserving level set method for shape optimization. J Comput Phys 204(1):121–130MathSciNetzbMATHGoogle Scholar
  4. Allaire G, Dapogny C, Frey P (2011) Topology and geometry optimization of elastic structures by exact deformation of simplicial mesh. Comptes Rendus Mathematique 349(17-18):999–1003MathSciNetzbMATHGoogle Scholar
  5. Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224MathSciNetzbMATHGoogle Scholar
  6. Bendsøe MP, Sigmund O (2004) Topology optimization: theory, methods, and applications. Springer, BerlinzbMATHGoogle Scholar
  7. Chen S, Wang MY, Liu AQ (2008) Shape feature control in structural topology optimization. Comput Aided Des 40(9):951–962Google Scholar
  8. Choi JS, Izui K, Nishiwaki S, Kawamoto A, Nomura T (2012) Rotor pole design of ipm motors for a sinusoidal air-gap flux density distribution. Struct Multidiscip Optim 46(3):445–455Google Scholar
  9. Christiansen AN, Nobel-jørgensen M, Aage N, Sigmund O, Bærentzen JA (2014) Topology optimization using an explicit interface representation. Struct Multidiscip Optim 49(3):387–399MathSciNetGoogle Scholar
  10. Erentok A, Sigmund O (2008) Three-dimensional topology optimized electrically-small conformal antenna. In: 2008 IEEE antennas and propagation society international symposium. IEEEGoogle Scholar
  11. Erentok A, Sigmund O (2011) Topology optimization of sub-wavelength antennas. IEEE Trans Antennas Propag 59(1):58–69Google Scholar
  12. Frickey DA (1994) Conversions between s, z, y, h, abcd, and t parameters which are valid for complex source and load impedances. IEEE Trans Microwave Theory Tech 42(2):205–211Google Scholar
  13. Guo X, Zhang W, Zhong W (2014) Explicit feature control in structural topology optimization via level set method. Comput Methods Appl Mech Eng 272:354–378MathSciNetzbMATHGoogle Scholar
  14. Ha SH, Cho S (2008) Level set based topological shape optimization of geometrically nonlinear structures using unstructured mesh. Comput Struct 86(13-14):1447–1455Google Scholar
  15. Haber RB, Jog CS, Bendsøe MP (1996) A new approach to variable-topology shape design using a constraint on perimeter. Structural Optimization 11(1–2):1–12Google Scholar
  16. Han X, Xu C, Prince JL (2003) A topology preserving level set method for geometric deformable models. IEEE Trans Pattern Anal Mach Intell 25(6):755–768Google Scholar
  17. Hassan E, Wadbro E, Berggren M (2014) Topology optimization of metallic antennas. IEEE Trans Antennas Propag 62(5):2488–2500MathSciNetzbMATHGoogle Scholar
  18. Hayt WH, Kemmerly JE, Durbin SM (1986) Engineering circuit analysis. McGraw-Hill, New YorkGoogle Scholar
  19. Jensen JS, Sigmund O (2004) Systematic design of photonic crystal structures using topology optimization: low-loss waveguide bends. Appl Phys Lett 84(12):2022–2024Google Scholar
  20. Kawamoto A, Matsumori T, Yamasaki S, Nomura T, Kondoh T, Nishiwaki S (2011) Heaviside projection based topology optimization by a PDE-filtered scalar function. Struct Multidiscip Optim 44(1):19–24zbMATHGoogle Scholar
  21. Kreisselmeier G, Steinhauser R (1979) Systematic control design by optimizing a vector performance index. IFAC Proceedings Volumes 12(7):113–117zbMATHGoogle Scholar
  22. Kreissl S, Maute K (2012) Levelset based fluid topology optimization using the extended finite element method. Struct Multidiscip Optim 46(3):311–326MathSciNetzbMATHGoogle Scholar
  23. Kurokawa K (1965) Power waves and the scattering matrix. IEEE Trans Microwave Theory Tech 13(2):194–202Google Scholar
  24. Liu S, Wang Q, Gao R (2014) A topology optimization method for design of small GPR antennas. Struct Multidiscip Optim 50(6):1165–1174Google Scholar
  25. Luo J, Luo Z, Chen S, Tong L, Wang MY (2008) A new level set method for systematic design of hinge-free compliant mechanisms. Comput Methods Appl Mech Eng 198(2):318–331zbMATHGoogle Scholar
  26. McRae DS (2000) r-refinement grid adaptation algorithms and issues. Comput Methods Appl Mech Eng 189 (4):1161–1182zbMATHGoogle Scholar
  27. Nomura T, Sato K, Taguchi K, Kashiwa T, Nishiwaki S (2007) Structural topology optimization for the design of broadband dielectric resonator antennas using the finite difference time domain technique. Int J Numer Methods Eng 71(11):1261–1296zbMATHGoogle Scholar
  28. Paul CR (2006) Introduction to electromagnetic compatibility. Wiley, New YorkGoogle Scholar
  29. Sato Y, Yamada T, Izui K, Nishiwaki S (2017) Manufacturability evaluation for molded parts using fictitious physical models, and its application in topology optimization. Int J Adv Manuf Technol 92(1–4):1391–1409Google Scholar
  30. Tsuji Y, Hirayama K, Nomura T, Sato K, Nishiwaki S (2006) Design of optical circuit devices based on topology optimization. IEEE Photon Technol Lett 18(7):850–852Google Scholar
  31. Van Miegroet L, Duysinx P (2007) Stress concentration minimization of 2d filets using X-FEM and level set description. Struct Multidiscip Optim 33(4-5):425–438Google Scholar
  32. Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767–784zbMATHGoogle Scholar
  33. Wang S, Wang MY (2006) A moving superimposed finite element method for structural topology optimization. Int J Numer Methods Eng 65(11):1892–1922MathSciNetzbMATHGoogle Scholar
  34. Wang S, Lee FC, Chen DY, Odendaal WG (2004) Effects of parasitic parameters on EMI filter performance. IEEE Trans on Power Electron 19(3):869–877Google Scholar
  35. Wei P, Wang MY, Xing X (2010) A study on X-FEM in continuum structural optimization using a level set model. Comput Aided Des 42(8):708–719Google Scholar
  36. Wrenn GA (1989) An indirect method for numerical optimization using the Kreisselmeir-Steinhauser function. NASA Contractor Report (4220)Google Scholar
  37. Xia Q, Shi T, Liu S, Wang MY (2012) A level set solution to the stress-based structural shape and topology optimization. Comput Struct 90:55–64Google Scholar
  38. Yamada T, Watanabe H, Fujii G, Matsumoto T (2013) Topology optimization for a dielectric optical cloak based on an exact level set approach. IEEE Trans Magn 49(5):2073–2076Google Scholar
  39. Yamasaki S, Nomura T, Kawamoto A, Sato K, Nishiwaki S (2011) A level set-based topology optimization method targeting metallic waveguide design problems. Int J Numer Methods Eng 87(9):844–868MathSciNetzbMATHGoogle Scholar
  40. Yamasaki S, Yamada T, Matsumoto T (2013) An immersed boundary element method for level-set based topology optimization. Int J Numer Methods Eng 93(9):960–988MathSciNetzbMATHGoogle Scholar
  41. Yamasaki S, Kawamoto A, Nomura T, Fujita K (2015) A consistent grayscale-free topology optimization method using the level-set method and zero-level boundary tracking mesh. Int J Numer Methods Eng 101(10):744–773MathSciNetzbMATHGoogle Scholar
  42. Yamasaki S, Kawamoto A, Saito A, Kuroishi M, Fujita K (2017a) Grayscale-free topology optimization for electromagnetic design problem of in-vehicle reactor. Struct Multidiscip Optim 55(3):1079–1090MathSciNetGoogle Scholar
  43. Yamasaki S, Yamanaka S, Fujita K (2017b) Three-dimensional grayscale-free topology optimization using a level-set based r-refinement method. Int J Numer Methods Eng 112(10):1402–1438MathSciNetGoogle Scholar
  44. Yoo J, Kikuchi N, Volakis JL (2000) Structural optimization in magnetic devices by the homogenization design method. IEEE Trans Magn 36(3):574–580Google Scholar
  45. Zhang W, Zhong W, Guo X (2014) An explicit length scale control approach in SIMP-based topology optimization. Comput Methods Appl Mech Eng 282:71–86MathSciNetzbMATHGoogle Scholar
  46. Zhou S, Li W, Li Q (2010) Level-set based topology optimization for electromagnetic dipole antenna design. J Comput Phys 229(19):6915–6930MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Toyota Central R&D Labs., Inc.NagakuteJapan
  2. 2.Department of Mechanical EngineeringOsaka UniversitySuitaJapan

Personalised recommendations