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Continuation strategies that exclude stiff designs, shift target frequencies, or delay design robustness substantially increase the probability that sound transmission loss topology optimization converges to high-performing optima.

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arxiv 2509.24667 v2 pith:4IURHHPU submitted 2025-09-29 math.OC

Continuation strategies to mitigate convergence to low-performing local optima in topology optimization of sound transmission loss

Tom De Weer , Vanessa Cool , Elke Deckers This is my paper
classification math.OC
keywords optimaoptimizationconvergencedynamicfrequencieslocallosslow-performing
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Topology optimization is a computational design method that distributes material within a given space to optimize a performance measure. When the performance measure is dynamic, such as the sound transmission loss of a sandwich panel, the optimization landscape is full of sharp peaks and valleys. Optimizers frequently get trapped in 'mass-driven' designs: stiff structures that simply shift all vibration frequencies upward and behave like a single heavy plate. These designs are easy to find but perform poorly.

The authors quantify this failure on a benchmark sandwich panel problem. Using 20 random starting points for each of 16 target frequency ranges, they estimate the probability that the optimizer reaches a design at least 10 percent better than the mass law. Below 2500 Hz this probability is zero; above 4500 Hz it often drops below 50 percent.

They then implement and compare three strategy families. Exclusion strategies add a constraint that makes overly stiff designs infeasible. Frequency shift strategies first optimize at a higher target frequency, then gradually lower it to the desired range, transplanting high-performing solutions downward. Relaxation strategies delay the requirement that the design be robust to manufacturing variations, or replace the worst-case minmax objective with an aggregate one, before restoring the original formulation.

Monte Carlo results show all strategies improve the success probability in at least some frequency ranges, with different trade-offs. Frequency shifting is effective at low frequencies but doubles or triples computation time. Exclusion and relaxation are cheap and remove transition regions, but can bias designs toward compliance. The paper ends with practical guidance for choosing among these strategies.

Core claim

All investigated strategies demonstrate measurable benefits and trade-offs (abstract; Section 4.4). Concretely, the paper claims that frequency shift techniques can reduce the low-performing region at low frequencies until a physical limit, exclusion strategies can turn transition regions into high-performing regions, and relaxation variant R3 improves STL across the whole frequency range. If correct, these strategies provide a quantified way to raise the chance of a high-performing optimum without exhaustive reruns.

Load-bearing premise

The Monte Carlo estimate of PHP uses only N=20 random initial guesses per frequency range (Section 3.2, Appendix B), giving a standard error of about 11 percentage points at the 50 percent point. The paper classifies regions as low-performing, transition, or high-performing based on these noisy estimates, and compares strategies whose differences are sometimes smaller than this error. If these sampling errors are not representative, the ranking and the 'physical limit' conclusion (e.g., no high-performing optima below 500 Hz) could be wrong.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Axiom & Free-Parameter Ledger

7 free parameters · 5 axioms · 0 invented entities

The paper introduces no new physical entities. Its central claims rest on engineering modeling assumptions (periodicity, sub-wavelength acoustics, material interpolation) and on a set of hand-chosen strategy hyperparameters. The latter are experimental design choices rather than fitted model parameters, but they influence the reported success rates and are not explored in a sensitivity study.

free parameters (7)
  • Normalization constant C = 120
    Chosen in Sec 2.2.2 to scale the objective; affects gradient magnitudes but not the ranking of final optima.
  • High-performing threshold = 1.1 x STL_mass_law
    Defines success in Sec 3.2; the paper asserts insensitivity to this cutoff but does not show a sensitivity study.
  • Connectivity constraint limit mu_sw = 15
    Scales the allowed self-weight compliance in Sec 2.2.3; inherited from prior work, not tuned here.
  • Exclusion bounds J_min = -0.5, -0.05, adaptive to -0.05
    Hand-selected for variants E1-E3 in Sec 4.1; no systematic tuning study is provided.
  • Excess frequency steps for F1-F4 = omega* = 1000, 2000, 3000, 4000 Hz; step 100 Hz
    Hand-selected in Sec 4.2; the results depend on these choices, as shown by F2 degrading a baseline high-performing region.
  • Robustness steps Delta_eta = 0.02 to 0.1 in four steps
    Hand-selected for R2 and R3 in Sec 4.3; no sensitivity analysis is reported.
  • Mass law trigger criterion = 1.15 x STL_ml
    Used to start frequency continuation and robustness continuation in Secs 4.2 and 4.3; chosen ad hoc.
axioms (5)
  • domain assumption The sandwich panel is weakly periodic and the unit cell size is smaller than the acoustic wavelength, allowing exclusion of higher-order harmonics.
    Invoked in Sec 2 before the STL definition; limits the fidelity of the acoustic model to the sub-wavelength regime.
  • standard math Bloch-Floquet boundary conditions enforce infinite periodicity of the unit cell.
    Used in Sec 2.1 and Appendix A to reduce the infinite panel to one unit cell; standard in periodic medium analysis.
  • domain assumption The RAMP interpolation and artificial material properties (E_v, rho_v, etc.) adequately model intermediate densities.
    Appendix A defines the interpolation scheme; its accuracy for the considered frequency range is assumed from prior literature.
  • domain assumption The robust formulation with eroded, blueprint, and dilated designs is a valid proxy for manufacturability and design robustness.
    Sec 2.2.1 uses the three-design robust formulation from [32]; the paper relies on this to judge design quality but does not validate against manufactured parts.
  • standard math MMA with the stated continuation and move-limit strategies converges to a local optimum of the discretized problem.
    Sec 2.2.5 and Appendix A rely on the MMA solver's convergence properties; standard for topology optimization.

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Dynamic topology optimization problems often suffer from convergence to low-performing local optima. This typically results in stiff designs that do not exploit dynamical phenomena such as antiresonance and decoupling. To obtain better designs, researchers often repeat their optimizations with different initial guesses. However, such reruns are computationally expensive and the required number is unknown. To quantify this problem, random initial guesses are sampled and tested for different frequencies on two case studies: (1) dynamic compliance minimization of a reinforced cantilever, which exhibits poor optima for driving frequencies below the first natural frequency, and (2) sound transmission loss maximization of a sandwich panel, which additionally sees a strong tendency toward low-performing optima at high frequencies. To address this issue, the study first divides techniques to reduce the needed number of reruns into four categories: global optimization, exclusion, relaxation, and frequency shift methods. For the latter three, continuation strategies are proposed, illustrated, evaluated and compared on the sound transmission loss case, using Monte Carlo sampling to estimate success rates. All strategies show measurable benefits and trade-offs. To support broader applicability, the study concludes with practical guidelines for dealing with convergence to poor local optima in dynamic topology optimization.

Figures

Figures reproduced from arXiv: 2509.24667 by Elke Deckers, Tom De Weer, Vanessa Cool.

Figure 1
Figure 1. Figure 1: Sound transmission loss of a uniform plate (blue) and a sandwich panel connected with zero-mass spring (red). Both [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Problem setup. a) Infinite sandwich panel. b) Unit cell considered during optimization. c) Example of FE [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Baseline optimization run yielding a mass-driven optimum, for ∆ [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Baseline optimization run yielding a high-performing optimum, for ∆ [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Results of applying the Monte Carlo sampling methodology to the baseline implementation. Performances [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Worst- and best-performing designs of the low- and high-performing optima for the baseline implementation. [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Overview of the methodology’s results for the considered exclusion strategies, [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Overview of the methodology’s results for the four variants of the frequency shift strategy, F [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Optimization process overview of a run targeting ∆ [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Eroded, blueprint and dilated designs (right) and corresponding [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Monte Carlo sampling results for the relaxation strategy variants, R [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗

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Works this paper leans on

42 extracted references · 18 canonical work pages

  1. [1]

    M. P. Bendsøe, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, Computer Methods in Applied Mechanics and Engineering 71 (2) (1988) 197–224. doi:https://doi.org/10.1016/0045-7825(88) 90086-2. URL https://www.sciencedirect.com/science/article/pii/0045782588900862

    arXiv 1988

  2. [2]

    A. R. D ´ ıaaz, N. Kikuchi, Solutions to shape and topology eigenvalue optimization problems using a homogenization method, International Journal for Numerical Methods in Engineering 35 (7) (1992) 1487–1502. arXiv:https:// onlinelibrary.wiley.com/doi/pdf/10.1002/nme.1620350707, doi:https://doi.org/10.1002/nme.1620350707. URL https://onlinelibrary.wiley.co...

    work page doi:10.1002/nme.1620350707 1992

  3. [3]

    Lewis, M

    A. Lewis, M. Overton, Eigenvalue optimization, Acta Numerica 5 (1996) 149–190. doi:10.1017/S0962492900002646

    doi:10.1017/s0962492900002646 1996

  4. [4]

    H. S. Koh, G. H. Yoon, Large-scale topology optimization for dynamic problems using a repetitive substructuring approach, Structural and Multidisciplinary Optimization 67 (33) (2024). doi:https://doi.org/10.1007/s00158-024-03759-1

    work page doi:10.1007/s00158-024-03759-1 2024

  5. [5]

    M. Xiao, J. Ma, X. Gao, P. Breitkopf, B. Raghavan, W. Zhang, L. Cauvin, P. Villon, Primal–dual on-the-fly reduced- order modeling for large-scale transient dynamic topology optimization, Computer Methods in Applied Mechanics and Engineering 428 (2024) 117099. doi:https://doi.org/10.1016/j.cma.2024.117099. URL https://www.sciencedirect.com/science/article/...

    arXiv 2024

  6. [6]

    Q. Li, O. Sigmund, J. S. Jensen, N. Aage, Reduced-order methods for dynamic problems in topology optimization: A comparative study, Computer Methods in Applied Mechanics and Engineering 387 (2021) 114149. doi:https://doi.org/ 10.1016/j.cma.2021.114149. URL https://www.sciencedirect.com/science/article/pii/S0045782521004801

    arXiv 2021

  7. [7]

    V. Cool, O. Sigmund, N. Aage, F. Naets, E. Deckers, Vibroacoustic topology optimization for sound transmission minimization through sandwich structures, J. of Sound and Vibr. 568 (2024) 117959, https://doi.org/10.1016/j.jsv. 2023.117959

    arXiv 2024

  8. [8]

    Giraldo Guzman, C

    D. Giraldo Guzman, C. Lissenden, P. Shokouhi, M. Frecker, Topology optimization design of resonant structures based on antiresonance eigenfrequency matching informed by harmonic analysis, Journal of Mechanical Design 145 (10) (2023) 101704. doi:10.1115/1.4062882. URL https://doi.org/10.1115/1.4062882

    work page doi:10.1115/1.4062882 2023

  9. [9]

    Giannini, N

    D. Giannini, N. Aage, F. Braghin, Topology optimization of mems resonators with target eigenfrequencies and modes, European Journal of Mechanics - A/Solids 91 (2022) 104352.doi:https://doi.org/10.1016/j.euromechsol.2021.104352. URL https://www.sciencedirect.com/science/article/pii/S0997753821001182

    arXiv 2022

  10. [10]

    O. M. Silva, M. M. Neves, A. Lenzi, A critical analysis of using the dynamic compliance as objective function in topology optimization of one-material structures considering steady-state forced vibration problems, Journal of Sound and Vibration 444 (2019) 1–20. doi:https://doi.org/10.1016/j.jsv.2018.12.030. URL https://www.sciencedirect.com/science/articl...

    work page doi:10.1016/j.jsv.2018.12.030 2019

  11. [11]

    Andreassen, F

    E. Andreassen, F. Ferrari, O. Sigmund, A. R. Diaz, Frequency response as a surrogate eigenvalue problem in topology optimization, International Journal for Numerical Methods in Engineering 113 (8) (2018) 1214–1229. doi:https://doi. org/10.1002/nme.5563

    work page doi:10.1002/nme.5563 2018

  12. [12]

    Herrmann, O

    L. Herrmann, O. Sigmund, V. M. Li, C. Vogl, S. Kollmannsberger, On neural networks for generating better local optima in topology optimization, Structural and Multidisciplinary Optimization 67 (11) (NOV 2024). doi:10.1007/ s00158-024-03908-6

    2024

  13. [13]

    Assimi, A

    H. Assimi, A. Jamali, A hybrid algorithm coupling genetic programming and nelder-mead for topology and size optimization of trusses with static and dynamic constraints, Expert Systems with Applications 95 (2018) 127–141. doi:10.1016/j.eswa.2017.11.035

    work page doi:10.1016/j.eswa.2017.11.035 2018

  14. [14]

    Giannini, M

    D. Giannini, M. Schevenels, E. P. Reynders, Rotational and multimodal local resonators for broadband sound insulation of orthotropic metamaterial plates, Journal of Sound and Vibration 547 (2023) 117453. doi:https://doi.org/10.1016/ j.jsv.2022.117453. URL https://www.sciencedirect.com/science/article/pii/S0022460X22006368

    arXiv 2023

  15. [15]

    Zhang, J

    S. Zhang, J. A. Norato, Finding better local optima in topology optimization via tunneling, in: Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, ASME, Design Engn Div; ASME, Comp & Informat Engn Div, 2018, aSME International Design Engineering Technical Conferences (IDE...

    2018

  16. [16]

    P. Liu, X. Zhang, Y. Luo, Topological design of freely vibrating bi-material structures to achieve the maximum band gap centering at a specified frequency, Journal of Applied Mechanics-Transaction of the ASME 88 (8) (AUG 1 2021). doi:10.1115/1.4050621

    work page doi:10.1115/1.4050621 2021

  17. [17]

    Dalklint, R

    A. Dalklint, R. E. Christiansen, O. Sigmund, On the potential use of performance bounds based on semidefinite programs for topology optimization (2025)

    2025

  18. [18]

    V. Cool, O. Sigmund, N. Aage, A practical review on promoting connectivity in topology optimization, Struct. and Multidiscipl. Opt. 68 (73), https://doi.org/10.1007/s00158-025-04004-z (2025)

    work page doi:10.1007/s00158-025-04004-z 2025

  19. [19]

    C. B. Dilgen, N. Aage, Topology optimization of transient vibroacoustic problems for broadband filter design using cut elements, Finite Elements in Analysis and Design 234 (2024) 104123. doi:https://doi.org/10.1016/j.finel.2024. 104123. URL https://www.sciencedirect.com/science/article/pii/S0168874X24000179 25

    work page doi:10.1016/j.finel.2024 2024

  20. [20]

    Halkjær, O

    S. Halkjær, O. Sigmund, J. S. Jensen, Maximizing band gaps in plate structures, Structural and Multidisciplinary Optimization 32 (2006) 263–275. doi:https://doi.org/10.1007/s00158-006-0037-7

    work page doi:10.1007/s00158-006-0037-7 2006

  21. [21]

    R. Kohn, G. Allaire, Topology optimization and optimal shape design using homogenization, Kluwer, 1993, pp. 207–218

    1993

  22. [22]

    Sanders, J

    C. Sanders, J. Norato, T. Walsh, W. Aquino, An error-in-constitutive equations strategy for topology optimization for frequency-domain dynamics, Computer Methods in Applied Mechanics and Engineering 372 (DEC 1 2020). doi:10.1016/ j.cma.2020.113330

    arXiv 2020

  23. [23]

    X. Guo, G. Cheng, K. Yamazaki, A new approach for the solution of singular optima in truss topology optimization with stress and local buckling constraints, Structural and Multidisciplinary Optimization 22 (5) (2001) 364–372. doi: 10.1007/s00158-001-0156-0

    work page doi:10.1007/s00158-001-0156-0 2001

  24. [24]

    Olhoff, J

    N. Olhoff, J. Du, Generalized incremental frequency method for topological designof 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 54 (5, SI) (2016) 1113–1141. doi:10.1007/s00158-016-1574-3

    work page doi:10.1007/s00158-016-1574-3 2016

  25. [25]

    F. Meng, L. Meng, J. Wang, J. Zhu, B. P. Wang, S. Yuan, W. Zhang, Topology optimization for minimum dynamic compliance using an antiresonant frequency constraint, Structural and Multidisciplinary Optimization 67 (9) (SEP 2024). doi:10.1007/s00158-024-03878-9

    work page doi:10.1007/s00158-024-03878-9 2024

  26. [26]

    R. F. Boukadia, E. Deckers, C. Claeys, M. Ichchou, W. Desmet, A wave-based optimization framework for 1D and 2D periodic structures, Mech. Syst. and Signal Process. 139 (2020) 106603, https://doi.org/10.1016/j.ymssp.2019.106603

    arXiv 2020

  27. [27]

    F. Fahy, P. Gardonio, Sound and structural vibration: radiation, transmission and response, 2nd edition, Elsevier, UK, 2007

    2007

  28. [28]

    Bloch, ¨Uber die quantenmechanik der elektronen in kristallgittern, Z

    F. Bloch, ¨Uber die quantenmechanik der elektronen in kristallgittern, Z. f¨ ur Phys. 52 (7) (1929) 555–600, https://link. springer.com/article/10.1007/BF01339455

    doi:10.1007/bf01339455 1929

  29. [29]

    V. Cool, C. Claeys, H. Denayer, F. Naets, E. Deckers, Experimental validation of a topology optimized 2d sandwich panel for sound transmission minimization, Mechanical Systems and Signal Processing 211 (2024) 111197, https://doi.org/ 10.1016/j.ymssp.2024.111197

    arXiv 2024

  30. [30]

    de Melo Filho, L

    N. de Melo Filho, L. Van Belle, C. Claeys, E. Deckers, W. Desmet, Dynamic mass based sound transmission loss prediction of vibro-acoustic metamaterial double panels applied to the mass-air-mass resonance, J. Sound Vibr. 442 (2019) 28–44, https://doi.org/10.1016/j.jsv.2018.10.047

    work page doi:10.1016/j.jsv.2018.10.047 2019

  31. [31]

    Jensen, A simple method for coupled acoustic-mechanical analysis with application to gradient-based topology optimization, Struct

    J. Jensen, A simple method for coupled acoustic-mechanical analysis with application to gradient-based topology optimization, Struct. and Multidiscipl. Opt. 59 (5) (2019) 1567–1580, https://doi.org/10.1007/s00158-018-2147-4

    work page doi:10.1007/s00158-018-2147-4 2019

  32. [32]

    F. Wang, B. S. Lazarov, O. Sigmund, On projection methods, convergence and robust formulations in topology optimization, Struct. and Multidiscipl. Opt. 43 (6) (2011) 767–784, https://doi.org/10.1007/s00158-010-0602-y

    doi:10.1007/s00158-010-0602-y 2011

  33. [33]

    R. E. Christiansen, B. S. Lazarov, J. S. Jensen, O. Sigmund, Creating geometrically robust designs for highly sensitive problems using topology optimization, Struct. and Multidiscipl. Opt. 52 (4) (2015) 737–754, https://doi.org.10.1007/ s00158-015-1265-5

    2015

  34. [34]

    Svanberg, The method of moving asymptotes—a new method for structural optimization, Int

    K. Svanberg, The method of moving asymptotes—a new method for structural optimization, Int. J. for Num. Methods in Eng. 24 (2) (1987) 359–373, https://doi.org/10.1002/nme.1620240207

    doi:10.1002/nme.1620240207 1987

  35. [35]

    Hoffmann, Violin plot, https://www.mathworks.com/matlabcentral/fileexchange/45134-violin-plot , accessed: 2025-03-24

    H. Hoffmann, Violin plot, https://www.mathworks.com/matlabcentral/fileexchange/45134-violin-plot , accessed: 2025-03-24

    2025

  36. [36]

    Bourdin, Filters in topology optimization, Int

    B. Bourdin, Filters in topology optimization, Int. J. Num. methods Eng. 50 (9) (2001) 2143–2158, https://doi.org/10. 1002/nme.116

    2001

  37. [37]

    J. K. Guest, J. H. Pr´ evost, T. Belytschko, Achieving minimum length scale in topology optimization using nodal design variables and projection functions, Int. J. Num. Methods Eng. 61 (2) (2004) 238–254,https://doi.org/10.1002/nme.1064

    doi:10.1002/nme.1064 2004

  38. [38]

    Stolpe, K

    M. Stolpe, K. Svanberg, An alternative interpolation scheme for minimum compliance topology optimization, Struct. and Multidiscipl. Opt. 22 (2) (2001) 116–124, https://doi.org/10.1007/s001580100129

    doi:10.1007/s001580100129 2001

  39. [39]

    Hussein, M

    M. Hussein, M. Leamy, M. Ruzzene, Dynamics of phononic materials and structures: Historical origins, recent progress, and future outlook, Appl. Mech. Rev. 66 (4) (2014) 040802, https://doi.org/10.1115/1.4026911

    doi:10.1115/1.4026911 2014

  40. [40]

    Y. Yang, B. R. Mace, M. J. Kingan, Prediction of sound transmission through, and radiation from, panels using a wave and finite element method, J. Acoust. Soc. Am. 141 (4) (2017) 2452–2460, https://doi.org/10.1121/1.4977925

    work page doi:10.1121/1.4977925 2017

  41. [41]

    Y. Yang, B. Mace, M. Kingan, Vibroacoustic analysis of periodic structures using a wave and finite element method, J. of Sound and Vibr. 457 (2019) 333–353, https://doi.org/10.1016/j.jsv.2019.06.009

    work page doi:10.1016/j.jsv.2019.06.009 2019

  42. [42]

    K. E. Swartz, D. A. Tortorelli, D. A. White, K. A. James, Manufacturing and stiffness constraints for topology optimized periodic structures, Struct. and Multidiscipl. Opt. 65 (129), https://doi.org/10.1007/s00158-022-03222-z (2022). 26

    work page doi:10.1007/s00158-022-03222-z 2022