Study on the Clear Boundary Determination from Results of the Phase Field Design Method

  • Cheol Woong Kim
  • Hong Kyoung Seong
  • Jeonghoon YooEmail author
Regular Paper


This study proposes a systematic process to clarify the boundary of optimum shapes obtained from the phase field design method. Topology optimization generally suffers from unclear boundary definition due to the gray scale area. The phase field design method, one of topology optimization design methods, has benefits not only to avoid the re-initialization process in traditional level set method-based structural design but also to define high-resolution boundary compared with other topological design schemes. However, the diffusion region generated near boundaries of the resultant structure from the phase field design method still causes gray scale area. To clarify boundaries of the resultant optimal shape, the appropriate value of the diffusion coefficient is determined, and successive application of the adaptive mesh refinement is performed. The proposed process was applied to two numerical examples for MBB beam design and three-dimensional nano-antenna design and the validity of the process to define the clear structural boundary is confirmed.


Clear boundary definition Phase field design method Diffusion coefficient Adaptive mesh refinement Nano antenna design 



The authors greatly acknowledge the support from the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MEST) (NRF-2016R1A2B4008501) and also supported by Human Resources Program in Energy Technology R&D Program of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) granted financial resource from the Ministry of Trade, Industry & Energy, Korea (No. 20184030201940).


  1. 1.
    Bendsøe, M. P. (1989). Optimal shape design as a material distribution problem. Structural Optimization, 1, 193–202.CrossRefGoogle Scholar
  2. 2.
    Sigmund, O., & Peterson, J. (1998). Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Structural and Multidisciplinary Optimization, 16, 68–75.CrossRefGoogle Scholar
  3. 3.
    Sigmund, O. (2007). Morphology-based black and white filters for topology optimization. Structural and Multidisciplinary Optimization, 33, 401–424.CrossRefGoogle Scholar
  4. 4.
    Bendsøe, M. P., & Sigmund, O. (2003). Topology optimization: Theory, methods and applications. Berlin: Springer.zbMATHGoogle Scholar
  5. 5.
    Diaz, A. R., & Sigmund, O. (1995). Checkerboard patterns in layout optimization. Structural Optimization, 10, 40–45.CrossRefGoogle Scholar
  6. 6.
    Osher, S. J., & Santosa, F. (2001). Level set method for optimization problems involving geometry and constraints: I. Frequencies of a two-density inhomogeneous drum. Journal of Computational Physics, 171, 272–288.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Wang, M. Y., Wang, X., & Guo, D. (2003). A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 192, 227–246.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Allaire, G., Jouve, F., & Toader, A. (2004). Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics, 194, 363–393.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Yamada, T., Izui, K., Nishiwaki, S., & Takezawa, A. (2010). A topology optimization method based on the level set method incorporating a fictitious interface energy. Computer Methods in Applied Mechanics and Engineering, 199, 2876–2891.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Takezawa, A., Nisiwaki, S., & Kitamura, M. (2010). Shape and topology optimization based on the phase field method and sensitivity analysis. Journal of Computational Physics, 229(7), 2697–2718.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Allen, S. M., & Cahn, J. W. (1979). A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metallurgica, 27(6), 1085–1095.CrossRefGoogle Scholar
  12. 12.
    Choi, J. S., Yamada, T., Izui, K., Nishiwaki, S., & Yoo, J. (2011). Topology optimization using a reaction-diffusion equation. Computer Methods in Applied Mechanics and Engineering, 200(29–32), 2407–2420.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lim, H., Yoo, J., & Choi, J. S. (2014). Topological nano-aperture configuration by structural optimization based on the phase field method. Structural and Multidisciplinary Optimization, 49, 209–224.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Choi, J. S., Izui, K., Nishiwaki, S., Kawamoto, A., & Nomura, T. (2012). Rotor pole design of IPM motors for a sinusoidal air-gap flux density distribution. Structural and Multidisciplinary Optimization, 46(3), 445–455.CrossRefGoogle Scholar
  15. 15.
    Yamada, T., Izui, K., & Nishiwaki, S. (2011). A level set-based topology optimization method for maximizing thermal diffusivity in problems including design-dependent effects. Journal of Mechanical Design, 133, 031011.CrossRefGoogle Scholar
  16. 16.
    Kim, H., Kim, C., Seong, H., & Yoo, J. (2015). Structural optimization of a magnetic actuator with simultaneous consideration of thermal and magnetic performances. IEEE Transactions on Magnetics. Scholar
  17. 17.
    Lim, D., Shin, D., Shin, H., Kim, K., & Yoo, J. (2014). A systematic approach to enhance off-axis directional electromagnetic wave by two-dimensional structure design. Optics Express, 22(6), 6511–6518.CrossRefGoogle Scholar
  18. 18.
    Borel, P. I., Frandsen, L. H., Harpøth, A., Kristensen, M., Jensen, J. S., & Sigmund, O. (2005). Topology optimised broadband photonic crystal Y-splitter. Electrons Letters, 41(2), 69–71.CrossRefGoogle Scholar
  19. 19.
    Verfurth, R. (1994). A posteriori error estimation and adaptive mesh-refinement techniques. Journal of Computational and Applied Mathemactics, 50(1-3), 67–83.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Boettinger, W., Warren, J., Beckermann, C., & Karma, A. (2002). Phase-field simulation of solidification. Annual Review of Materials Research, 32, 163–194.CrossRefGoogle Scholar
  21. 21.
    Kim, J. (2012). Phase-field models for multi-component fluid flows. Communications in Computational Physics, 12(3), 613–661.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Schmitz, G., Böttger, B., Eiken, J., Apel, M., Viardin, A., Carré, A., et al. (2010). Phase-field based simulation of microstructure evolution in technical alloy grades. International Journal of Advances in Engineering Sciences and Applied Mathematics, 2(4), 126–139.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Bourdin, B., & Chambolle, A. (2000). Optimisation topologique de structures soumises à des forces de pression. In Actes du 32ème Congrès National d’Analyse Numérique.Google Scholar
  24. 24.
    Qin, R., & Bhadeshia, H. (2013). Phase field method. Materials Science and Technology, 26(7), 803–811.CrossRefGoogle Scholar
  25. 25.
    Wheeler, A. (1998). Cahn–Hoffman ξ-vector and its relation to diffuse interface models of phase transitions. Journal of Statistical Physics, 95(5–6), 1245–1280.MathSciNetzbMATHGoogle Scholar
  26. 26.
    Frey, H., Witt, S., Felderer, K., & Guckenberger, R. (2004). High-resolution imaging of single fluorescent molecules with the optical near-field of a metal tip. Physical Review Letters, 93(20), 200801.CrossRefGoogle Scholar
  27. 27.
    Sundaramurphy, A., Schuck, P., Conley, N., Fromm, D., Kino, G., & Moerner, W. (2006). Toward nanometer-scale optical photolithography: Utilizing the near-field of bowtie optical nanoantennas. Nano Letters, 6(3), 355–360.CrossRefGoogle Scholar
  28. 28.
    Garcia-Parajo, M. (2008). Optical antennas focus in on biology. Nature Photonics, 2, 201–203.CrossRefGoogle Scholar
  29. 29.
    Andkjær, J., Nishiwaki, S., Nomura, T., & Sigmund, O. (2010). Topology optimization of grating couplers for the efficient excitation of surface plasmons. Journal of the Optical Society of America B, 27(9), 1828–1832.CrossRefGoogle Scholar

Copyright information

© Korean Society for Precision Engineering 2019

Authors and Affiliations

  1. 1.Graduate School of Mechanical EngineeringYonsei UniversitySeoulSouth Korea
  2. 2.School of Mechanical EngineeringYonsei UniversitySeoulSouth Korea

Personalised recommendations