Methods of topological optimization in problems of increasing vibration resistance of plate aviation structures


Аuthors

Ryzhova E. S.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: RyzhovaES@mai.ru

Abstract

This article provides a comprehensive review of modern topological optimization methods aimed at enhancing the vibration resistance of plate-type aviation structures. The study focuses on the application of topological optimization techniques to improve the dynamic performance of thin-walled components such as panels, skins, and hatches, which are subjected to dynamic loads from acoustic pressure, engine vibrations, and aeroelastic effects. The primary optimization objectives include maximizing natural frequencies—particularly the fundamental frequency—and minimizing dynamic compliance under harmonic excitation, thereby shifting resonance zones and reducing the risk of fatigue failure.  
Key methodological approaches are examined, including the density-based method (SIMP), the level-set method, and evolutionary techniques, within the context of eigenvalue problems and dynamic response minimization. The influence of boundary conditions—such as clamped versus simply supported edges—on the optimal topology is emphasized, demonstrating that constraint conditions significantly affect both the dynamic behavior and the resulting reinforcement layout. The integration of manufacturing constraints, particularly those related to additive manufacturing, is also discussed as a critical factor in the design of realizable optimized structures.  
A practical case study is presented, illustrating the topological synthesis of a reinforcing frame for a rectangular plate under volume constraints. Using the SIMP method, the evolution of the frame topology is analyzed for volume fractions ranging from 5% to 30%. The results show a transition from simple arched supports to complex, branched lattice structures as the volume allowance increases, highlighting the non-linear relationship between material usage and frequency improvement.  
Despite progress, several challenges remain, such as high computational costs, the need for nonlinear and damping-inclusive models, and the difficulty of translating density-based results into manufacturable CAD models. Future research directions include the development of multiphysics optimization frameworks, digital twin integration, and advanced AI-driven generative design methodologies.  
In conclusion, topological optimization proves to be a powerful tool for improving the vibration resistance of plate structures, offering significant gains in specific stiffness and dynamic performance. The continued integration of additive manufacturing constraints and intelligent algorithms is expected to further enhance the applicability and efficiency of these methods in aerospace engineering and beyond.

Keywords:

topology optimization, plate structures, natural frequency, dynamic compliance

References

  1. Bolotin V.V. Dinamicheskaya ustoichivost’ uprugikh system [Dynamic stability of elastic systems], Moscow, Gostekhizdat, 1956. 600 p.
  2. Zhuravlev V.F., Klimov D.M. Prikladnye metody v teorii kolebanii [Applied methods in the theory of oscillations]. Moscow, Nauka, 1988, 328 p.
  3. Bendsøe M.P., Sigmund O. Topology optimization: theory, methods and applications, Berlin, Springer, 2003, 370 p.
  4. Rozvany G.I.N. A critical review of established methods of structural topology optimization. Structural and Multidisciplinary Optimization, 2009, vol. 37, no. 3. pp. 217–237. DOI 10.1007/s00158-007-0217-0.
  5. Sigmund O. A 99 line topology optimization code written in Matlab. Structural and Multidisciplinary Optimization, 2001, vol. 21, no. 2, pp. 120–127. DOI 10.1007/s001580050176.
  6. Osher S., Sethian J. A. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton – Jacobi formulations. Journal of Computational Physics, 1988, vol. 79, no. 1. pp. 12–49. DOI 10.1016/0021-9991(88)90002-2.
  7. Debroy T., Wei H.L., Zuback J.S., Mukherjee T., Elmer J.W., Milewski J.O., Beese A.M., Wilson-Heid A., Amitava De, Zhang W. Additive manufacturing of metallic components — Process, structure and properties. Progress in Materials Science, 2018, vol. 92, pp. 112–224. DOI 10.1016/j.pmatsci.2017.10.001.
  8. Jankovics D., Gohari H., Tayefeh M., Barari A. Developing topology optimization with additive manufacturing constraints in ANSYS®. IFAC-PapersOnLine, 2018, vol. 51, iss. 11, pp. 1359–1364. DOI 10.1016/j.ifacol.2018.08.340.
  9. Olhoff N., Du J. Generalized incremental frequency method for topological design of continuum structures for minimum dynamic compliance subject to forced vibration at a prescribed low or high value of the excitation frequency. Structural and Multidisciplinary Optimization, 2016, vol. 54, pp. 1113–1141. DOI 10.1007/s00158-016-1574-3.
  10. Niu B., He X., Shan Y., Zhang P. On objective functions of minimizing the vibration response of continuum structures subjected to external harmonic excitation. Structural and Multidisciplinary Optimization, 2018, vol. 57, pp. 2291–2307. DOI 10.1007/s00158-017-1859-1.
  11. Leissa A.W. Vibration of plates: scientific and technical information division. Office of Technology Utilization National Aeronautics and Space Administration. Washington, D.C., 1969. vii, 353 p. NASA SP-160 
  12. Reddy J.N. Theory and analysis of elastic plates and shells, 2nd ed., Boca Raton, CRC Press, 2007.
  13. Díaaz A.R., Kikuchi N. Solutions to shape and topology eigenvalue optimization problems using a homogenization method. International Journal for Numerical Methods in Engineering, 1992, vol. 35, no. 7, pp. 1487–1502. DOI 10.1002/nme.1620350707.
  14. Ma Z.-D., Kikuchi N., Cheng H.-C. Topological design for vibrating structures. Computer Methods in Applied Mechanics and Engineering, 1995, vol. 121, no. ¼, pp. 259–280. DOI 10.1016/0045-7825(94)00714-X.
  15. Du J., Olhoff N. Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps. Structural and Multidisciplinary Optimization, 2007, vol. 34, no. 2, pp. 91–110. DOI 10.1007/s00158-007-0101-y.
  16. Timoshenko S.P. Kolebaniya v inzhenernom dele [Vibration problems in ingineering], Moscow, Nauka, 1967. 444 p.
  17. Gorman D.J. Free vibration analysis of rectangular plates with symmetrically distributed point supports along the edges. Journal of Sound and Vibration, 1980, vol. 73, no. 4, pp. 563–574. DOI 10.1016/0022-460X(80)90668-9.
  18. Pedersen N.L. Maximization of eigenvalues using topology optimization. Structural and Multidisciplinary Optimization, 2000, vol. 20, no. 1, pp. 2–11. DOI 10.1007/s001580050130.
  19. Fu Y., Kennedy G. Quasi-Newton corrections for compliance and natural frequency topology optimization problems. Structural and Multidisciplinary Optimization, 2023, vol. 66, art. 176. DOI 10.1007/s00158-023-03630-9.
  20. Bendsøe M. P. Optimal shape design as a material distribution problem. Structural Optimization, 1989, vol. 1, no. 4. P. 193–202. DOI 10.1007/BF01650949.
  21. Hu S., Manansala B., Fitzer U., Hohlfeld D., Bechtold T.Two-Phase approach for fast topology optimization of multi-resonant MEMS involving model order reduction. Micromachines, 2025, vol. 16, no. 4, art. 401. DOI 10.3390/mi16040401.
  22. Lazarov B.S., Sigmund O. Filters in topology optimization based on Helmholtz‐type differential equations. International Journal for Numerical Methods in Engineering, 2011, vol. 86, no. 6, pp. 765–781. DOI 10.1002/nme.3072.
  23. Wang M.Y., Wang X., Guo D. A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 2003, vol. 192, no. 1/2. pp. 227–246. DOI 10.1016/S0045-7825(02)00559-5.
  24. Allaire G., Jouve F., Toader A.-M. Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics, 2004, vol. 194, no. 1. pp. 363–393. DOI 10.1016/j.jcp.2003.09.032.
  25. Querin O.M., Steven G.P., Xie Y.M. Evolutionary structural optimisation (ESO) using a bidirectional algorithm. Engineering Computations, 1998, vol. 15, no. 8. pp. 1031–1048. DOI 10.1108/02644409810244129.
  26. Goodarzimehr V., Fanaie N., Talatahari S. Geometric and size optimization of structures under natural frequency constraints using improved material generation algorithm. International Journal of Optimization in Civil Engineering, 2025, vol. 15, no. 1. pp. 15–37.
  27. Leissa A.W. The free vibration of rectangular plates. Journal of Sound and Vibration, 1973, vol. 31, no. 3. pp. 257–293.
  28. Bathe K.J. Finite element procedures, 2nd ed., Upper Saddle River, Prentice Hall, 2014, 1037 p.
  29. Liu K., Bai Y., Yao S., Luan S. Topology optimization of shell-infill structures for natural frequencies. Engineering Computations, 2022, vol. 39, no. 5, pp. 1821–1846. DOI 10.1108/EC-03-2022-0135.
  30. Langelaar M. Topology optimization of 3D self-supporting structures for additive manufacturing. Additive Manufacturing, 2016, vol. 12, pp. 60–70. DOI 10.1016/j.addma.2016.06.010.
  31. Gaynor A.T., Guest J.K. Topology optimization considering overhang constraints: Eliminating sacrificial support material in additive manufacturing through design. Structural and Multidisciplinary Optimization, 2016, vol. 54, no. 5, pp. 1157–1172. DOI 10.1007/s00158-016-1550-y.
  32. Liu J., Gaynor A.T., Chen S., Kang Z., Suresh K., Takezawa A., Li L., Kato J., Tang J., Wang C.C.L., Cheng L., Liang X., To A.C. Current and future trends in topology optimization for additive manufacturing. Structural and Multidisciplinary Optimization, 2018, vol. 57, no. 6, pp. 2457–2483. DOI 10.1007/s00158-018-1994-3.
  33. Gibson L.J., Ashby M.F. Cellular solids: structure and properties, 2nd ed., Cambridge, Cambridge University Press, 1997, 510 p.
  34. Shevtsova V.S., Shevtsova М.S. Vestnik Yuzhnogo nauchnogo tsentra, 2013, vol. 9, no. 1, pp. 8–16.
  35. Xu S., Wang M., Zhou C., Zhou Y., Wan S., Wang B. Topology optimization for cyclic periodic structures with frequency objectives of nodal diameter modes, Engineering Optimization, 2024, pp. 1–24. DOI 10.1080/0305215X.2024.2314661.
  36. Siqueira L.O., Azevêdo A.S.C., Silva E.C.N., Picelli R. Topology optimization of rotating structures considering turbulent fluid–structure interaction problems and natural frequency constraints. Structural and Multidisciplinary Optimization, 2025, vol. 68, art. 90. DOI 10.1007/s00158-025-04017-8.
  37. Honshuku Y., Isakari H. A topology optimisation of acoustic devices based on the frequency response estimation with the Padé approximation. Applied Mathematical Modelling, 2022, vol. 110, pp. 819–840. DOI 10.1016/j.apm.2022.06.020.
  38. Giannone G., Ahmed F. Diffusing the Optimal Topology: A Generative Optimization Approach. Proceedings of the ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Volume 3A: 49th Design Automation Conference (DAC). Boston, Massachusetts, USA, August 20–23, 2023, paper no. V03AT03A012. DOI 10.1115/DETC2023-116595.
  39. Qu Y., Zhou Y., Luo Y. Structural topology optimization for frequency response problems using adaptive second-order Arnoldi method. Mathematics, 2025, vol. 13, no. 10, art. 1583. DOI 10.3390/math13101583.
  40. Sun J., Cai Z. Topology optimization for eigenfrequencies of a flexible multibody system. Multibody System Dynamics, 2025, vol. 64, pp. 307–330. DOI 10.1007/s11044-024-10018-0.
  41. Min S., Kikuchi N., Park Y.C., Kim S., Chang S. Optimal topology design of structures under dynamic loads. Structural Optimization, 1999, vol. 17, no. 2/3. pp. 208–218. DOI 10.1007/BF01195945.
  42. Ramasamy M., Slíva A., Govindaraj P., Nag A. Topology optimization and testing of connecting rod based on static and dynamic analyses. Applied Sciences, 2025, vol. 15, no. 4, art. 2081. DOI 10.3390/app15042081.


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