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arxiv: 2605.04735 · v1 · submitted 2026-05-06 · 💻 cs.CE · cs.NA· math.NA· math.OC

Recognition: unknown

Sequential topology optimization: SIMP initialization for level-set boundary refinement

2), (2) Faculty of Mechanical Engineering, Czech Academy of Sciences, Czech Republic, Czech Republic), Czech Technical University in Prague, Du\v{s}an Gabriel (1) ((1) Institute of Thermomechanics, J\'an Kopa\v{c}ka (1), Martin Isoz (1), Ond\v{r}ej Je\v{z}ek (1, Praha

Pith reviewed 2026-05-08 16:12 UTC · model grok-4.3

classification 💻 cs.CE cs.NAmath.NAmath.OC
keywords topology optimizationSIMPlevel-set methodsigned distance functionboundary refinementsequential optimizationthree-dimensional meshesmanufacturing-ready designs
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The pith

Initializing level-set topology optimization from a SIMP density field produces sharp manufacturable boundaries while reducing sensitivity to the starting design.

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

The paper establishes a sequential workflow that first uses density-based SIMP optimization to explore topologies efficiently, then converts the resulting density field into a signed distance function to initialize a level-set stage for boundary refinement. This addresses the complementary weaknesses of the two approaches: SIMP yields diffuse boundaries that need post-processing for manufacturing, while level-set methods maintain sharp interfaces but depend heavily on a good initial guess. A sympathetic reader would care because the combination could automate the generation of structures with clear, ready-to-manufacture interfaces without manual interpretation of gray regions. Validation on three-dimensional cantilever and MBB beam benchmarks shows compliance values comparable to standalone level-set runs, together with reported wall-clock speedups reaching 4.6 times on the cantilever case. The framework is released as open-source code to enable direct reproduction and extension.

Core claim

The SIMP-derived initialization mitigates sensitivity to the initial design in level-set optimization, and the level-set stage acts as optimization-driven post-processing that produces manufacturing-ready boundaries. The key step is an SDF-based geometry transfer formulated for three-dimensional meshes that converts the SIMP density distribution into an initial signed distance function for the subsequent level-set evolution.

What carries the argument

SDF-based geometry transfer that converts a SIMP density distribution into a signed distance function to initialize level-set boundary refinement.

If this is right

  • Level-set optimization can start from a topologically rich density field rather than an arbitrary initial guess, reducing the number of failed runs.
  • The final designs emerge with sharp material interfaces that require no additional thresholding or smoothing for manufacturing.
  • The overall procedure achieves compliance comparable to pure level-set optimization while delivering reported speedups of up to 4.6 times on the cantilever benchmark.
  • The same transfer step can be applied to other density-based methods that produce gray-scale fields, extending the post-processing benefit beyond SIMP.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may generalize to other physics problems where an initial diffuse solution can be sharpened into a crisp interface, such as certain fluid or thermal topology problems.
  • Because the level-set stage is treated as refinement rather than primary topology discovery, the method could be paired with faster but coarser density solvers to further reduce total compute time.
  • Open release of the code allows direct testing on user-defined three-dimensional geometries to measure whether the speedup and boundary sharpness hold outside the published cantilever and MBB cases.

Load-bearing premise

Converting the SIMP density distribution to a signed distance function preserves the essential topological features without introducing artifacts that the subsequent level-set optimization cannot correct.

What would settle it

Running the sequential method on the same three-dimensional cantilever and MBB benchmarks and obtaining either substantially higher final compliance than pure level-set optimization or visibly non-sharp boundaries after the level-set stage.

Figures

Figures reproduced from arXiv: 2605.04735 by 2), (2) Faculty of Mechanical Engineering, Czech Academy of Sciences, Czech Republic, Czech Republic), Czech Technical University in Prague, Du\v{s}an Gabriel (1) ((1) Institute of Thermomechanics, J\'an Kopa\v{c}ka (1), Martin Isoz (1), Ond\v{r}ej Je\v{z}ek (1, Praha.

Figure 1
Figure 1. Figure 1: Illustration of the sequential topology optimization methodology on a 2D beam, used here for visual clarity (the methodology and all validation studies are performed in 3D). (a) SIMP optimization result showing element-wise density distribution. (b) Elemental densities mapped to nodes. (c) Piecewise linear boundary extracted at the density threshold 𝜌𝑡 = 0.5. (d) SDF constructed on the finite element mesh … view at source ↗
Figure 2
Figure 2. Figure 2: Cantilever beam problem setup: fixed supports at four corner regions (blue) and load applied to a circular region at the opposite face (red). O. Jezek et al.: Preprint submitted to Elsevier Page 16 of 15 view at source ↗
Figure 3
Figure 3. Figure 3: Cantilever beam optimization results. Each pair shows the extracted SIMP geometry (left) and the level-set refined result (right). Cases (a)–(e): sequential SIMP→LS with varying convergence tolerance Δ𝜌max. Case (f): baseline from uniform porous initialization. Holes are marked to highlight topological differences between cases. Iteration [1] 0 25 50 75 100 125 150 C o m plia n c e [J] 25 30 35 40 45 Porou… view at source ↗
Figure 4
Figure 4. Figure 4: Cantilever beam level-set convergence histories: (a) Compliance and (b) volume fraction of extracted geometries. Sequential initializations (SIMP→LS) with varying Δ𝜌max are compared against the porous initialization baseline (dashed line). O. Jezek et al.: Preprint submitted to Elsevier Page 17 of 15 view at source ↗
Figure 5
Figure 5. Figure 5: MBB beam problem setup: sliding boundary condi￾tions (green) at the symmetry plane (𝑥 = 0) and bottom edge (𝑥 = 2.0); load (red) applied on a semicircular region of radius 0.1. O. Jezek et al.: Preprint submitted to Elsevier Page 18 of 15 view at source ↗
Figure 6
Figure 6. Figure 6: MBB beam optimization results. Each pair shows the extracted SIMP geometry (left) and the level-set refined result (right). Cases (a)–(e): sequential SIMP→LS with varying convergence tolerance Δ𝜌max. Case (f): baseline from uniform porous initialization. Holes are marked to highlight topological differences between cases. Iteration [1] 0 25 50 75 100 C o m plia n c e [J] 30 40 50 Porous Δ𝜌max = 0.5% Δ𝜌max … view at source ↗
Figure 7
Figure 7. Figure 7: MBB beam level-set convergence histories: (a) Compliance and (b) volume fraction of extracted geometries. Sequential initializations (SIMP→LS) with varying Δ𝜌max are compared against the porous initialization baseline (dashed line). O. Jezek et al.: Preprint submitted to Elsevier Page 19 of 15 view at source ↗
read the original abstract

Density-based topology optimization methods such as SIMP enable efficient topological exploration but produce diffuse material boundaries that require interpretation before manufacturing. Level-set methods maintain sharp interfaces but are sensitive to the initial design. This paper presents a sequential framework that addresses these complementary limitations through a signed distance function (SDF)-based geometry transfer, formulated for three-dimensional meshes. The SIMP density distribution is converted into an SDF that initializes subsequent level-set boundary refinement. From the level-set perspective, the SIMP-derived initialization mitigates sensitivity to the initial design. From the SIMP perspective, the level-set stage acts as optimization-driven post-processing that produces manufacturing-ready boundaries. Validation on three-dimensional cantilever and MBB benchmarks demonstrates compliance comparable to standalone level-set optimization, with up to 4.6x wall-clock speedup on the cantilever case. The full implementation is released under an open-source license to support reproducibility.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a sequential topology optimization framework that first performs SIMP density-based optimization, converts the resulting density field to a signed distance function (SDF) to initialize level-set optimization, and then refines the boundaries. This hybrid approach is claimed to combine SIMP's topological exploration with level-set's sharp interfaces, mitigating level-set sensitivity to initial designs while producing manufacturing-ready boundaries. Validation on 3D cantilever and MBB beam benchmarks reports compliance comparable to standalone level-set optimization, with up to 4.6x wall-clock speedup on the cantilever case, and the implementation is released open-source.

Significance. If the sensitivity-mitigation benefit is demonstrated, the method offers a practical route to robust 3D topology optimization with sharp boundaries and reduced overall cost. The open-source release is a clear strength that enables reproducibility and community extension of the SDF conversion step. However, the current evidence establishes only performance parity on standard benchmarks rather than the claimed robustness advantage.

major comments (2)
  1. [Validation section] Validation section (cantilever and MBB benchmarks): the central claim that 'the SIMP-derived initialization mitigates sensitivity to the initial design' lacks supporting experiments. The reported results show only that compliance is comparable to standalone level-set optimization (presumably from favorable initials), with no ablation studies using varied or poor initial designs to demonstrate degradation in the pure level-set case that the sequential method avoids. This directly undermines the stated benefit from the level-set perspective.
  2. [Method section] Method section (SDF conversion step): the assumption that converting the SIMP density distribution to an SDF 'preserves the essential topological features without introducing artifacts' is stated but not tested. No quantitative metrics or failure-case examples are provided to confirm that any conversion-induced issues are reliably corrected by the subsequent level-set evolution, which is load-bearing for the sequential framework's reliability.
minor comments (2)
  1. [Abstract] Abstract: the speedup factor (4.6x) and compliance comparisons are stated without error bars, convergence histories, or mesh-resolution sensitivity analysis, making it difficult to assess robustness of the performance claims.
  2. [Introduction] The manuscript would benefit from explicit comparison to other hybrid SIMP/level-set approaches in the literature to clarify the novelty of the SDF-based transfer.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below, agreeing that additional evidence would strengthen the claims on sensitivity mitigation and SDF conversion reliability. We will make the indicated revisions to the manuscript.

read point-by-point responses

  1. Referee: [Validation section] Validation section (cantilever and MBB benchmarks): the central claim that 'the SIMP-derived initialization mitigates sensitivity to the initial design' lacks supporting experiments. The reported results show only that compliance is comparable to standalone level-set optimization (presumably from favorable initials), with no ablation studies using varied or poor initial designs to demonstrate degradation in the pure level-set case that the sequential method avoids. This directly undermines the stated benefit from the level-set perspective.

    Authors: We agree that the manuscript would benefit from explicit ablation studies to directly demonstrate the sensitivity-mitigation benefit. The current results show performance parity and up to 4.6x speedup relative to standalone level-set optimization on standard benchmarks, which is consistent with the SIMP initialization providing a robust starting point that avoids poor local optima. To address the referee's point, we will add new experiments in the revised validation section using deliberately suboptimal initial designs for pure level-set optimization and compare outcomes with the sequential method. These will appear as additional figures and quantitative discussion. revision: yes

  2. Referee: [Method section] Method section (SDF conversion step): the assumption that converting the SIMP density distribution to an SDF 'preserves the essential topological features without introducing artifacts' is stated but not tested. No quantitative metrics or failure-case examples are provided to confirm that any conversion-induced issues are reliably corrected by the subsequent level-set evolution, which is load-bearing for the sequential framework's reliability.

    Authors: We acknowledge that the SDF conversion step is presented without quantitative validation of feature preservation or artifact analysis. In the revised manuscript we will add metrics such as the change in number of connected components and Euler characteristic before versus after conversion, together with boundary discrepancy measured by Hausdorff distance between the SIMP isosurface and the SDF zero level set. We will also discuss any observed artifacts in the cantilever and MBB cases and how level-set evolution corrects them, supported by the existing benchmark data. revision: yes

Circularity Check

0 steps flagged

No circularity in the algorithmic sequential framework

full rationale

The paper presents a procedural method: run SIMP, convert density field to SDF for level-set initialization, then refine boundaries. No equations, parameters, or results are defined in terms of themselves or prior outputs by construction. Claims about sensitivity mitigation and manufacturing-ready boundaries are supported by benchmark comparisons rather than self-referential definitions or fitted-input predictions. No load-bearing self-citations or imported uniqueness theorems appear in the provided text. The derivation chain is self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework inherits standard assumptions from density-based and level-set topology optimization without introducing new free parameters or invented physical entities beyond the algorithmic transfer step.

axioms (2)
  • domain assumption Finite-element discretization of the compliance minimization problem under volume constraint is a valid model of structural behavior.
    Invoked implicitly when both SIMP and level-set stages are run on the same mesh.
  • ad hoc to paper Signed distance function conversion from a density field yields a geometrically faithful initial interface for level-set evolution.
    Central to the sequential hand-off; no independent proof supplied in the abstract.

pith-pipeline@v0.9.0 · 5524 in / 1460 out tokens · 28429 ms · 2026-05-08T16:12:32.937275+00:00 · methodology

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Reference graph

Works this paper leans on

68 extracted references · 43 canonical work pages

  1. [1]

    Visualization

    Ahrens, J., Geveci, B., Law, C., 2005. ParaView: An end-user tool forlarge-datavisualization,in:TheVisualizationHandbook.Elsevier, pp. 717–731. doi:10.1016/B978-012387582-2/50038-1

    work page doi:10.1016/b978-012387582-2/50038-1 2005

  2. [2]

    doi:10.1016/bs.hna.2020.10.004 , author =

    Allaire, G., Dapogny, C., Jouve, F.o., 2021. Shape and topology optimization, in: Geometric Partial Differential Equations, Part II. Elsevier. volume 22 ofHandbook of Numerical Analysis, pp. 1–132. doi:10.1016/bs.hna.2020.10.004

    work page doi:10.1016/bs.hna.2020.10.004 2021

  3. [3]

    Structural optimization using topological and shape sensitivity via a level set method

    Allaire,G.,deGournay,F.,Jouve,F.,Toader,A.M.,2005. Structural optimization using topological and shape sensitivity via a level set method. Control and Cybernetics 34, 59–80. URL:http://eudml. org/doc/209353

    2005

  4. [4]

    2004 , issn =

    Allaire, G., Jouve, F., Toader, A.M., 2004. Structural optimization using sensitivity analysis and a level-set method. Journal of Compu- tational Physics 194, 363–393. doi:10.1016/j.jcp.2003.09.032

    work page doi:10.1016/j.jcp.2003.09.032 2004

  5. [5]

    Altair OptiStruct 2026 User Guide

    Altair Engineering Inc., 2026. Altair OptiStruct 2026 User Guide. Troy, MI. URL:https://2026.help.altair.com/2026/hwsolvers/os/ index.htm. accessed: 2026-01-09

    2026

  6. [6]

    Efficient topology optimization in MATLAB using 88 lines of code

    Andreassen,E.,Clausen,A.,Schevenels,M.,Lazarov,B.S.,Sigmund, O., 2011. Efficient topology optimization in MATLAB using 88 lines of code. Structural and Multidisciplinary Optimization 43, 1–

    2011

  7. [7]

    URL:https://doi.org/10.1007/s00158-010-0594-7, doi:10.1007/ s00158-010-0594-7

    work page doi:10.1007/s00158-010-0594-7

  8. [8]

    Ansys Mechanical 2025 R2: Topology Opti- mization

    ANSYS, Inc., 2025. Ansys Mechanical 2025 R2: Topology Opti- mization. Canonsburg, PA. URL:https://www.ansys.com/products/ structures/topology-optimization. accessed: 2026-01-09

    2025

  9. [9]

    Badia, F

    Badia, S., Verdugo, F., 2020. Gridap: An extensible finite element toolbox in julia. Journal of Open Source Software 5, 2520. URL: https://doi.org/10.21105/joss.02520, doi:10.21105/joss.02520

    work page doi:10.21105/joss.02520 2020

  10. [10]

    Hole seeding in level set topology optimization via density fields

    Barrera, J.L., Geiss, M.J., Maute, K., 2020. Hole seeding in level set topology optimization via density fields. Structural and Multidisciplinary Optimization 61, 1319–1343. doi:10.1007/ s00158-019-02480-8

    2020

  11. [11]

    Optimalshapedesignasamaterialdistribution problem

    Bendsøe,M.P.,1989. Optimalshapedesignasamaterialdistribution problem. Structural Optimization 1, 193–202. URL:https://doi. org/10.1007/BF01650949, doi:10.1007/BF01650949

    work page doi:10.1007/bf01650949 1989

  12. [12]

    Material interpolation schemes in topology optimization

    Bendsøe, M.P., Sigmund, O., 1999. Material interpolation schemes in topology optimization. Archive of Applied Mechanics 69, 635–

    1999

  13. [13]

    URL:https://doi.org/10.1007/s004190050248, doi:10.1007/ s004190050248

    work page doi:10.1007/s004190050248

  14. [14]

    (2003).Topology Optimization: Theory, Methods, and Applications

    Bendsøe, M.P., Sigmund, O., 2004. Topology Optimization: The- ory, Methods, and Applications. 2 ed., Springer, Berlin, Heidel- berg. URL:https://doi.org/10.1007/978-3-662-05086-6, doi:10. 1007/978-3-662-05086-6

    work page doi:10.1007/978-3-662-05086-6 2004

  15. [15]

    Multidimensional binary search trees used for associative searching

    Bentley, J.L., 1975. Multidimensional binary search trees used for associative searching. Communications of the ACM 18, 509–517. doi:10.1145/361002.361007

    work page doi:10.1145/361002.361007 1975

  16. [16]

    Bezanson, A

    Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B., 2017. Julia: A fresh approach to numerical computing. SIAM review 59, 65–98. URL:https://doi.org/10.1137/141000671

    work page doi:10.1137/141000671 2017

  17. [17]

    International Journal for Numerical Methods in Engineering , volume =

    Bourdin, B., 2001. Filters in topology optimization. International Journal for Numerical Methods in Engineering 50, 2143 – 2158. doi:10.1002/nme.116

    work page doi:10.1002/nme.116 2001

  18. [18]

    Guillard, C

    Bruns, T.E., Tortorelli, D.A., 2001. Topology optimization of non-linear elastic structures and compliant mechanisms. Com- puter Methods in Applied Mechanics and Engineering 190, 3443–3459. URL:https://www.sciencedirect.com/science/article/ pii/S0045782500002784, doi:https://doi.org/10.1016/S0045-7825(00) 00278-4

    work page doi:10.1016/s0045-7825(00 2001

  19. [19]

    Incorporating topolog- ical derivatives into level set methods

    Burger, M., Hackl, B., Ring, W., 2004. Incorporating topolog- ical derivatives into level set methods. Journal of Computa- tional Physics 194, 344–362. URL:https://www.sciencedirect. com/science/article/pii/S0021999103004868,doi:https://doi.org/10. 1016/j.jcp.2003.09.033

    2004

  20. [20]

    NearestNeighbors.jl: High performance nearest neighbor data structures and algorithms for Julia

    Carlsson, K., 2024. NearestNeighbors.jl: High performance nearest neighbor data structures and algorithms for Julia. URL:https:// github.com/KristofferC/NearestNeighbors.jl

    2024

  21. [21]

    SIMULIA Tosca Structure 2025 User Guide

    Dassault Systèmes, 2025. SIMULIA Tosca Structure 2025 User Guide. Vélizy-Villacoublay, France. URL:https://www.3ds.com/ products/simulia/tosca. accessed: 2026-01-09

    2025

  22. [22]

    A survey of structural and multidisciplinarycontinuumtopologyoptimization:post2000.Struc- tural and Multidisciplinary Optimization 49, 1–38

    Deaton, J.D., Grandhi, R.V., 2014. A survey of structural and multidisciplinarycontinuumtopologyoptimization:post2000.Struc- tural and Multidisciplinary Optimization 49, 1–38. doi:10.1007/ s00158-013-0956-z

    2014

  23. [23]

    Level- setmethodsforstructuraltopologyoptimization:areview

    vanDijk,N.P.,Maute,K.,Langelaar,M.,vanKeulen,F.,2013. Level- setmethodsforstructuraltopologyoptimization:areview. Structural andMultidisciplinaryOptimization48,437–472. URL:https://doi. org/10.1007/s00158-013-0912-y, doi:10.1007/s00158-013-0912-y

    work page doi:10.1007/s00158-013-0912-y 2013

  24. [24]

    An efficient method of triangulating equi- valued surfaces by using tetrahedral cells

    Doi, A., Koide, A., 1991. An efficient method of triangulating equi- valued surfaces by using tetrahedral cells. IEICE TRANSACTIONS on Information E74-D, 214–224

    1991

  25. [25]

    Real-Time Collision Detection

    Ericson, C., 2004. Real-Time Collision Detection. Morgan Kauf- mann, San Francisco

    2004

  26. [26]

    Velocityextensionforthelevel-setmethodand multipleeigenvaluesinshapeoptimization

    deGournay,F.,2006. Velocityextensionforthelevel-setmethodand multipleeigenvaluesinshapeoptimization. SIAMJournalonControl and Optimization 45, 343–367. doi:10.1137/050624108

    work page doi:10.1137/050624108 2006

  27. [27]

    Achieving minimum lengthscaleintopologyoptimizationusingnodaldesignvariablesand projectionfunctions

    Guest, J.K., Prévost, J.H., Belytschko, T., 2004. Achieving minimum lengthscaleintopologyoptimizationusingnodaldesignvariablesand projectionfunctions. InternationalJournalforNumericalMethodsin Engineering 61, 238–254. URL:https://onlinelibrary.wiley.com/ doi/abs/10.1002/nme.1064, doi:https://doi.org/10.1002/nme.1064

    work page doi:10.1002/nme.1064 2004

  28. [28]

    Density- based hole seeding in xfem level-set topology optimization of fluid problems

    Høghøj, L.C., Andreasen, C.S., Maute, K., 2025. Density- based hole seeding in xfem level-set topology optimization of fluid problems. Structural and Multidisciplinary Optimization 68,

    2025

  29. [29]

    URL:https://doi.org/10.1007/s00158-024-03956-y,doi:10.1007/ s00158-024-03956-y

    work page doi:10.1007/s00158-024-03956-y

  30. [30]

    On smooth or 0/1 designs of the fixed- mesh element-based topology optimization

    Huang, X., 2021. On smooth or 0/1 designs of the fixed- mesh element-based topology optimization. Advances in Engi- neering Software 151, 102942. URL:https://www.sciencedirect. com/science/article/pii/S0965997820309881,doi:https://doi.org/10. 1016/j.advengsoft.2020.102942

    work page arXiv 2021

  31. [31]

    Sequential topology optimization: SIMP initialization for level-set boundaryrefinement

    Ježek, O., Kopačka, J., Isoz, M., Gabriel, D., 2026a. Data for: “Sequential topology optimization: SIMP initialization for level-set boundaryrefinement”.Zenodo.doi:10.5281/zenodo.20024424.dataset, v1.0.0

    work page doi:10.5281/zenodo.20024424.dataset

  32. [32]

    Sequential topology optimization: SIMP initialization for level-set boundary re- finement.https://github.com/jezekon/2026-Jezek-SeqTopOpt

    Ježek, O., Kopačka, J., Isoz, M., Gabriel, D., 2026b. Sequential topology optimization: SIMP initialization for level-set boundary re- finement.https://github.com/jezekon/2026-Jezek-SeqTopOpt. Source O. Jezek et al.:Preprint submitted to ElsevierPage 14 of 15 Sequential topology optimization code repository, version 1.0.0

    2026

  33. [33]

    Smooth geometry extraction from simp topology optimization: Signed distance function approach with volume preservation

    Ježek, O., Kopačka, J., Isoz, M., Gabriel, D., Maršálek, P., Šo- tola, M., Halama, R., 2026. Smooth geometry extraction from simp topology optimization: Signed distance function approach with volume preservation. Advances in Engineering Software 212, 104071. URL:https://www.sciencedirect.com/science/article/pii/ S0965997825002091, doi:https://doi.org/10.1...

    work page doi:10.1016/j.advengsoft.2025 2026

  34. [34]

    Isosurfacestuffing:Fasttetrahedral meshes with good dihedral angles

    Labelle,F.,Shewchuk,J.R.,2007. Isosurfacestuffing:Fasttetrahedral meshes with good dihedral angles. ACM Transactions on Graphics 26, 57. doi:10.1145/1276377.1276448

    work page doi:10.1145/1276377.1276448 2007

  35. [35]

    Smoothing topology optimization results using pre-built lookup tables

    Li, Z., Lee, T.U., Yao, Y., Xie, Y.M., 2022. Smoothing topology optimization results using pre-built lookup tables. Advances in Engineering Software 173, 103204. URL:https: //www.sciencedirect.com/science/article/pii/S0965997822001090, doi:10.1016/j.advengsoft.2022.103204

    work page doi:10.1016/j.advengsoft.2022.103204 2022

  36. [36]

    Design of the multi-material structure using an MMC-SIMP sequential topology optimization method

    Li, Z., Xu, H., Zhang, S., Cui, J., Liu, X., 2025. Design of the multi-material structure using an MMC-SIMP sequential topology optimization method. PLOS ONE 20, e0321100. doi:10.1371/ journal.pone.0321100

    2025

  37. [37]

    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.,

  38. [38]

    Structural and Multidisciplinary Optimization 57, 2457–2483

    Currentandfuturetrendsintopologyoptimizationforadditive manufacturing. Structural and Multidisciplinary Optimization 57, 2457–2483. doi:10.1007/s00158-018-1994-3

    work page doi:10.1007/s00158-018-1994-3 1994

  39. [39]

    Lorensen,W.E.,Cline,H.E.,1987.Marchingcubes:Ahighresolution 3D surface construction algorithm, in: Proceedings of the 14th An- nual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH ’87), ACM. pp. 163–169. doi:10.1145/37401.37422

    work page doi:10.1145/37401.37422 1987

  40. [40]

    From topology optimization design to additive manufacturing: Today’s success and tomorrow’s roadmap

    Meng,L.,Zhang,W.,Quan,D.,Shi,G.,Tang,L.,Hou,Y.,Breitkopf, P., Zhu, J., Gao, T., 2020. From topology optimization design to additive manufacturing: Today’s success and tomorrow’s roadmap. Archives of Computational Methods in Engineering 27, 805–830. doi:10.1007/s11831-019-09331-1

    work page doi:10.1007/s11831-019-09331-1 2020

  41. [41]

    , series =

    Nocedal, J., Wright, S.J., 2006. Numerical Optimization. Springer Series in Operations Research and Financial Engineering. 2nd ed., Springer, New York. doi:10.1007/978-0-387-40065-5

    work page doi:10.1007/978-0-387-40065-5 2006

  42. [42]

    Osher, R

    Osher, S., Fedkiw, R., 2004. The Level Set Methods and Dynamic Implicit Surfaces. volume 57. doi:10.1115/1.1760520

    work page doi:10.1115/1.1760520 2004

  43. [43]

    Fronts propagating with curvature- dependent speed: Algorithms based on Hamilton-Jacobi formula- tions

    Osher, S., Sethian, J.A., 1988. Fronts propagating with curvature- dependent speed: Algorithms based on Hamilton-Jacobi formula- tions. Journal of Computational Physics 79, 12–49. doi:10.1016/ 0021-9991(88)90002-2

    1988

  44. [44]

    APDE- based fast local level set method

    Peng,D.,Merriman,B.,Osher,S.,Zhao,H.,Kang,M.,1999. APDE- based fast local level set method. Journal of Computational Physics 155, 410–438. doi:10.1006/jcph.1999.6345

    work page doi:10.1006/jcph.1999.6345 1999

  45. [45]

    Level-set topology optimization with PDE generated conformalmeshes

    Schmidt, M.R., Barrera, J.L., Mittal, K., Swartz, K.E., Tortorelli, D.A., 2024. Level-set topology optimization with PDE generated conformalmeshes. StructuralandMultidisciplinaryOptimization67,

    2024

  46. [46]

    doi:10.1007/s00158-024-03870-3

    work page doi:10.1007/s00158-024-03870-3

  47. [47]

    Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science

    Sethian, J.A., 1999. Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. 2nd ed., Cambridge Uni- versity Press, Cambridge

    1999

  48. [48]

    Tetgen, a delaunay-based quality tetrahedral mesh generator,

    Si, H., 2015. TetGen, a Delaunay-based quality tetrahedral mesh generator. ACM Transactions on Mathematical Software 41, 11:1– 11:36. doi:10.1145/2629697

    work page doi:10.1145/2629697 2015

  49. [49]

    On the design of compliant mechanisms using topology optimization

    Sigmund, O., 1997. On the design of compliant mechanisms using topology optimization. Mechanics of Structures and Machines 25, 493–524. URL:https://doi.org/10.1080/08905459708945415, doi:10. 1080/08905459708945415

    work page doi:10.1080/08905459708945415 1997

  50. [50]

    A 99 line topology optimization code written in matlab

    Sigmund, O., 2001. A 99 line topology optimization code written in matlab. Structural and Multidisciplinary Optimization 21, 120–

    2001

  51. [51]

    URL:https://doi.org/10.1007/s001580050176, doi:10.1007/ s001580050176

    work page doi:10.1007/s001580050176

  52. [52]

    Morphology-based black and white filters for topologyoptimization

    Sigmund, O., 2007. Morphology-based black and white filters for topologyoptimization. StructuralandMultidisciplinaryOptimization 33, 401–424. URL:https://doi.org/10.1007/s00158-006-0087-x, doi:10.1007/s00158-006-0087-x

    work page doi:10.1007/s00158-006-0087-x 2007

  53. [53]

    Topology optimization approaches: A comparative review,

    Sigmund, O., Maute, K., 2013. Topology optimization approaches: Acomparativereview. StructuralandMultidisciplinaryOptimization 48, 1031–1055. doi:10.1007/s00158-013-0978-6

    work page doi:10.1007/s00158-013-0978-6 2013

  54. [54]

    Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima

    Sigmund, O., Petersson, J., 1998. Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Structural Optimization 16, 68–75. doi:10.1007/BF01214002

    work page doi:10.1007/bf01214002 1998

  55. [55]

    Areviewofmethodsfor thegeometricpost-processingoftopologyoptimizedmodels

    Subedi,S.C.,Verma,C.S.,Suresh,K.,2020. Areviewofmethodsfor thegeometricpost-processingoftopologyoptimizedmodels. Journal of Computing and Information Science in Engineering 20, 060801. doi:10.1115/1.4047429

    work page doi:10.1115/1.4047429 2020

  56. [56]

    Geuzaine, J.-F

    Svanberg, K., 1987. The method of moving asymptotes—a new method for structural optimization. International Journal for Nu- merical Methods in Engineering 24, 359–373. doi:10.1002/nme. 1620240207

    work page doi:10.1002/nme 1987

  57. [57]

    Post-Processing of Topology Optimized Results:Amethodforretrievingsmoothandcrispgeometries

    Swierstra, M.K., 2017. Post-Processing of Topology Optimized Results:Amethodforretrievingsmoothandcrispgeometries. Master thesis. Delft University of Technology. URL:https://resolver. tudelft.nl/uuid:0db4da1e-07e5-433f-a261-3dc0cd6abfc0

    2017

  58. [58]

    Automated and accurate geometry extraction and shape optimization of 3d topology optimization results

    Swierstra, M.K., Gupta, D.K., Langelaar, M., 2020. Automated and accurate geometry extraction and shape optimization of 3d topology optimization results. ArXiv abs/2004.05448. URL:https://api. semanticscholar.org/CorpusID:215745139

    work page arXiv 2020

  59. [59]

    Explicit reconstruction and shape optimization of topology optimization re- sultswithmechanicalperformancepreservation

    Tang,Y.,Li,Y.,Xiang,X.,Luo,J.,Zhou,W.,Yao,W.,2026. Explicit reconstruction and shape optimization of topology optimization re- sultswithmechanicalperformancepreservation. ComputerModeling in Engineering & Sciences URL:http://www.techscience.com/CMES/ online/detail/26583, doi:10.32604/cmes.2026.079578

    work page doi:10.32604/cmes.2026.079578 2026

  60. [60]

    https://doi.org/10.1016/j

    Verdugo, F., Badia, S., 2022. The software design of gridap: A finite element package based on the julia JIT compiler. Computer Physics Communications 276, 108341. URL:https://doi.org/10.1016/j. cpc.2022.108341, doi:10.1016/j.cpc.2022.108341

    work page doi:10.1016/j 2022

  61. [61]

    A level set method for structuraltopologyoptimization

    Wang, M.Y., Wang, X., Guo, D., 2003. A level set method for structuraltopologyoptimization. ComputerMethodsinAppliedMe- chanics and Engineering 192, 227–246. doi:10.1016/S0045-7825(02) 00559-5

    work page doi:10.1016/s0045-7825(02 2003

  62. [62]

    Radial basis functions and level set method for structural topology optimization

    Wang, S., Wang, M., 2006. Radial basis functions and level set method for structural topology optimization. International Journal forNumericalMethodsinEngineering65,2060–2090. doi:10.1002/ nme.1536

    2006

  63. [63]

    GridapTopOpt.jl:ascalableJuliatoolboxforlevelset-basedtopology optimisation

    Wegert,Z.J.,Manyer,J.,Mallon,C.N.,Badia,S.,Challis,V.J.,2025a. GridapTopOpt.jl:ascalableJuliatoolboxforlevelset-basedtopology optimisation. Structural and Multidisciplinary Optimization 68,

  64. [64]

    URL:https://doi.org/10.1007/s00158-024-03927-3,doi:10.1007/ s00158-024-03927-3

    work page doi:10.1007/s00158-024-03927-3

  65. [65]

    Level-set topology optimisation with unfitted finite elements and automatic shape differentiation

    Wegert, Z.J., Manyer, J., Mallon, C.N., Badia, S., Challis, V.J., 2025b. Level-set topology optimisation with unfitted finite elements and automatic shape differentiation. Computer Methods in Applied MechanicsandEngineering445,118203. URL:https://doi.org/10. 1016/j.cma.2025.118203, doi:10.1016/j.cma.2025.118203

    work page doi:10.1016/j.cma.2025.118203 2025

  66. [66]

    A Hilbertian pro- jectionmethodforconstrainedlevelset-basedtopologyoptimisation

    Wegert, Z.J., Roberts, A.P., Challis, V.J., 2023. A Hilbertian pro- jectionmethodforconstrainedlevelset-basedtopologyoptimisation. Structural and Multidisciplinary Optimization 66, 204. doi:10.1007/ s00158-023-03663-0

    2023

  67. [67]

    A topology optimization method based on the level set method incorporating a fictitious interface energy

    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. URL:https://www.sciencedirect. com/science/article/pii/S0045782510001623,doi:https://doi.org/10. 1016/j.cma.2010.05.013

    2010

  68. [68]

    Thecocalgorithm,partii:Topological, geometrical and generalized shape optimization

    Zhou,M.,Rozvany,G.,1991. Thecocalgorithm,partii:Topological, geometrical and generalized shape optimization. Computer Methods in Applied Mechanics and Engineering 89, 309–336. URL:https: //www.sciencedirect.com/science/article/pii/0045782591900469, doi:https://doi.org/10.1016/0045-7825(91)90046-9. second World Congress on Computational Mechanics. O. Jezek...

    work page doi:10.1016/0045-7825(91)90046-9 1991