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arxiv: 2509.09236 · v3 · submitted 2025-09-11 · 🧮 math.NA · cs.CE· cs.NA· math.OC

Isogeometric Topology Optimization Based on Topological Derivatives

Guilherme Henrique Teixeira , Nepomuk Krenn , Peter Gangl , Benjamin Marussig This is my paper

Pith reviewed 2026-05-18 18:04 UTC · model grok-4.3

classification 🧮 math.NA cs.CEcs.NAmath.OC
keywords topology optimizationisogeometric analysistopological derivativeslevel-set methodimmersed boundarystructural designnumerical methods
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The pith

A level-set method inside an immersed isogeometric framework lets topology optimization change shapes without remeshing or predefined holes.

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

The paper shows that topological derivatives can drive the evolution of a level-set function while the underlying structure is represented and solved on an immersed isogeometric mesh. Because the mesh stays fixed, geometry updates happen by simply adjusting the level-set values rather than regenerating elements. The authors test the effect of polynomial degree on both the level-set representation and the physical solution field. They find that higher-degree functions raise the accuracy of the displacement or stress solution, yet linear functions are enough to represent the level-set itself. Two benchmark problems illustrate that the resulting designs match expected optima while preserving the no-remeshing property.

Core claim

The combination of a level-set representation, an immersed isogeometric discretization, and topological derivatives produces optimized structures through successive geometry updates that require neither remeshing nor the insertion of initial holes. Higher-degree basis functions improve the accuracy of the approximated solution fields, while linear basis functions suffice for the level-set description.

What carries the argument

An immersed isogeometric discretization paired with a level-set function that is updated by topological derivatives.

If this is right

  • Topological changes can be introduced directly through the level-set evolution without seeding initial voids.
  • Higher polynomial degrees in the physical solution field raise the fidelity of stress or displacement predictions during optimization.
  • Linear polynomials remain adequate for representing the level-set, keeping the description of the design boundary simple.
  • The fixed background mesh removes the need for repeated mesh generation steps that usually accompany topology changes.

Where Pith is reading between the lines

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

  • The same fixed-mesh strategy could extend to problems that require frequent topology changes, such as crack propagation or multi-phase flow.
  • Because linear level-set functions suffice, the computational overhead of the design variable representation stays low even when the physics solver uses higher order.
  • If the immersed quadrature remains stable under large level-set motions, the method may scale to three-dimensional industrial geometries without mesh adaptation.

Load-bearing premise

The immersed isogeometric solution stays accurate enough for topological derivatives to be evaluated reliably as the level-set interface moves, without extra stabilization or quadrature changes.

What would settle it

A side-by-side comparison on a standard compliance-minimization benchmark in which the final compliance or volume fraction obtained without remeshing deviates measurably from the result of an equivalent remeshed reference computation.

Figures

Figures reproduced from arXiv: 2509.09236 by Benjamin Marussig, Guilherme Henrique Teixeira, Nepomuk Krenn, Peter Gangl.

Figure 1
Figure 1. Figure 1: Different types of optimization: a) Parameter Optimization; b) Shape Optimization; c) Topology [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Representation of the domain problem: a) Domain [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Distribution of the Greville abscissae on the elements for different polynomial degrees and basis [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Type identification of the elements for assembling of the material property [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Approaches to treat the cut elements 10 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Cantilever problem 0 10 20 30 8 10 12 Iteration J Cost Function 0 10 20 30 8 10 12 Iteration J Cost Function 0 10 20 30 0 50 100 Iteration θ Angle 0 10 20 30 0 50 100 Iteration θ Angle 0 10 20 30 0.4 0.6 0.8 1 Iteration Ai/A0 Area 0 10 20 30 0.4 0.6 0.8 1 Iteration A i/A 0 Area p = 1 (d = 1); p = 2 (d = 2); p = 3 (d = 3); p = 4 (d = 4); . p = 2 (d = 1); p = 3 (d = 1); p = 4 (d = 1) [PITH_FULL_IMAGE:figure… view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the evolution of the cost function, angle, and area for different polynomial [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Final shape for different basis functions degree for approximating the solution: a) Level-set [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Curved cantilever problem 0 100 200 4 6 8 10 12 Iteration J Cost Function 0 100 200 4 6 8 10 12 Iteration J Cost Function 0 100 200 0 50 100 150 Iteration θ Angle 0 100 200 0 50 100 150 Iteration θ Angle 0 100 200 0.2 0.4 0.6 0.8 1 Iteration Ai/A0 Area 0 100 200 0.2 0.4 0.6 0.8 1 Iteration A i/A 0 Area p = 1 (d = 1); p = 2 (d = 2); p = 3 (d = 3); p = 4 (d = 4); . p = 2 (d = 1); p = 3 (d = 1); p = 4 (d = 1)… view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the evolution of the cost function, angle, and area for different polynomial [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Final shape for different basis functions degree for approximating the solution: a) Level-set [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
read the original abstract

Topology optimization is a valuable tool in engineering, facilitating the design of optimized structures. However, topological changes often require a remeshing step, which can become challenging. In this work, we propose an isogeometric approach to topology optimization driven by topological derivatives. The combination of a level-set method together with an immersed isogeometric framework allows seamless geometry updates without the necessity of remeshing. At the same time, topological derivatives provide topological modifications without the need to define initial holes [7]. We investigate the influence of higher-degree basis functions in both the level-set representation and the approximation of the solution. Two numerical examples demonstrate the proposed approach, showing that employing higher-degree basis functions for approximating the solution improves accuracy, while linear basis functions remain sufficient for the level-set function representation.

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 manuscript proposes an isogeometric topology optimization framework that combines topological derivatives with a level-set representation inside an immersed isogeometric discretization. The central claim is that this combination permits seamless geometry updates on a fixed background mesh without remeshing, while topological derivatives eliminate the need to prescribe initial holes. The authors further investigate the effect of polynomial degree, asserting that higher-degree bases improve accuracy for the state solution whereas linear bases suffice for the level-set function. Two numerical examples are presented to support these statements.

Significance. If the immersed discretization is shown to remain accurate without auxiliary quadrature or stabilization machinery, the approach would offer a practical route to topology optimization that exploits the exact geometry representation of IGA while avoiding remeshing costs. The explicit comparison of basis degrees for the level-set versus the solution field supplies actionable guidance for similar immersed methods.

major comments (2)
  1. [§3] §3 (Immersed discretization and quadrature): the claim that geometry updates occur “without the necessity of remeshing” rests on the assumption that standard tensor-product quadrature remains accurate on cut elements when the level-set evolves and topological derivatives are evaluated. The manuscript must specify the quadrature rule employed on cut cells and, if unmodified Gauss quadrature is used, provide evidence (e.g., convergence tables or comparison with adaptive subdivision) that integration error does not degrade the reported accuracy gains.
  2. [Numerical examples] Numerical examples (presumably §4): the abstract states that the examples demonstrate improved accuracy with higher-order solution bases, yet no quantitative error measures, compliance histories, or mesh-convergence studies are referenced. Without these data it is impossible to assess whether the observed improvement is attributable to the higher-degree bases or to other implementation choices.
minor comments (2)
  1. [§2] Notation for the topological derivative and the level-set evolution equation should be introduced with explicit references to the cited work [7] so that readers can trace the precise formulas employed.
  2. [Figures] Figure captions for the two numerical examples should state the polynomial degrees used for the solution and level-set fields in each run, together with the number of degrees of freedom.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed review. The comments help clarify important aspects of the immersed discretization and the presentation of numerical results. We address each major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [§3] §3 (Immersed discretization and quadrature): the claim that geometry updates occur “without the necessity of remeshing” rests on the assumption that standard tensor-product quadrature remains accurate on cut elements when the level-set evolves and topological derivatives are evaluated. The manuscript must specify the quadrature rule employed on cut cells and, if unmodified Gauss quadrature is used, provide evidence (e.g., convergence tables or comparison with adaptive subdivision) that integration error does not degrade the reported accuracy gains.

    Authors: We agree that explicit details on quadrature for cut elements are necessary to support the claim of remeshing-free updates. Section 3 describes the immersed isogeometric framework with a fixed background mesh and level-set-driven geometry updates, but does not fully specify the quadrature procedure on intersected elements. We will revise §3 to state that standard tensor-product Gauss quadrature is applied to the physical portion of each cut element (determined by the level-set), without additional stabilization or adaptive subdivision. To address the integration-error concern, we will add a short convergence study in the revised manuscript comparing results obtained with the current quadrature against a reference adaptive quadrature on the same meshes, confirming that integration errors remain subordinate to discretization errors for the polynomial degrees considered. revision: yes

  2. Referee: [Numerical examples] Numerical examples (presumably §4): the abstract states that the examples demonstrate improved accuracy with higher-order solution bases, yet no quantitative error measures, compliance histories, or mesh-convergence studies are referenced. Without these data it is impossible to assess whether the observed improvement is attributable to the higher-degree bases or to other implementation choices.

    Authors: We acknowledge that the current presentation relies primarily on visual comparison of final designs and qualitative statements about accuracy. Section 4 contains two numerical examples that illustrate the effect of polynomial degree on the state solution, but quantitative supporting data (error norms, compliance histories, and systematic mesh-convergence tables) are not tabulated or explicitly referenced in the text. We will expand §4 to include (i) compliance histories for each polynomial degree, (ii) relative error measures with respect to a reference solution, and (iii) mesh-convergence studies that isolate the contribution of the solution-space degree while keeping the level-set representation linear. These additions will allow readers to directly attribute the observed improvements to the higher-order bases. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation combines established external methods

full rationale

The paper presents a combination of level-set methods, immersed isogeometric analysis, and topological derivatives to enable topology optimization without remeshing. It explicitly cites [7] for the topological derivative approach that avoids initial holes and relies on standard IGA machinery for the discretization. No equations or steps reduce a claimed prediction or uniqueness result to a fitted parameter, self-defined quantity, or load-bearing self-citation chain; the numerical examples serve as empirical demonstration rather than circular validation. The central claims therefore remain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract does not introduce new free parameters, invented entities, or ad-hoc axioms beyond reliance on standard isogeometric analysis, level-set evolution, and the existence of topological derivatives as referenced in [7].

axioms (1)
  • domain assumption Topological derivatives exist and can be evaluated for the linear elasticity problems considered.
    The method depends on the availability of topological derivatives to drive topology changes, as cited from reference [7].

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Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    Allaire and F

    G. Allaire and F. Jouve. Coupling the level set method and the topological gradient in structural optimization, [in:]Proceedings IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials, pp. 3-12. Dordrecht, 2006. doi: https://doi.org/10.1007/1-4020-4752-5 1

    work page doi:10.1007/1-4020-4752-5 2006

  2. [2]

    Allaire, F

    G. Allaire, F. Jouve, and A. Toader. A level-set method for shape optimization.C. R. Acad. Sci. Paris, Ser. I,334:1125–1130, 2002. doi: https://doi.org/10.1016/S1631-073X(02)02412-3

    work page doi:10.1016/s1631-073x(02)02412-3 2002

  3. [3]

    Discretization of Dirac delta functions in level set methods.Journal of Computational Physics, 207(1):28–51, July 2005.doi:10.1016/j.jcp

    G. Allaire, F. Jouve, and A. Toader. Structural optimization using sensitivity analysis and a level-set method.Journal of Computational Physics,194:363–393, 2 2004. doi: https://doi.org/10.1016/j.jcp. 2003.09.032

    work page doi:10.1016/j.jcp 2004

  4. [4]

    Aminzadeh and S

    M. Aminzadeh and S. M. Tavakkoli. A parameter space approach for isogeometrical level set topology optimization.International Journal for Numerical Methods in Engineering,123:3485–3506, 8 2022. doi: https://doi.org/10.1002/nme.6976

    work page doi:10.1002/nme.6976 2022

  5. [5]

    Aminzadeh and S

    M. Aminzadeh and S. M. Tavakkoli. Multiscale topology optimization of structures by using iso- geometrical level set approach.Finite Elements in Analysis and Design,235, 8 2024. doi: https: //doi.org/10.1016/j.finel.2024.104167

    work page doi:10.1016/j.finel.2024.104167 2024

  6. [6]

    S. Amstutz. Sensitivity analysis with respect to a local perturbation of the material property.Asymptotic Analysis,49(1-2):87–108, 2006. doi: https://doi.org/10.3233/ASY-2006-778

    work page doi:10.3233/asy-2006-778 2006

  7. [7]

    Amstutz and H

    S. Amstutz and H. Andr¨a. A new algorithm for topology optimization using a level-set method.Journal of Computational Physics,216:573–588, 8 2006. doi: https://doi.org/10.1016/j.jcp.2005.12.015. 16

    work page doi:10.1016/j.jcp.2005.12.015 2006

  8. [8]

    M. P. Bendsøe and N. Kikuchi. Generating optimal topologies in structural design using a homoge- nization method.Computer Methods in Applied Mechanics and Engineering,71:197–224, 1988. doi: https://doi.org/10.1016/0045-7825(88)90086-2

    work page doi:10.1016/0045-7825(88)90086-2 1988

  9. [9]

    Burger, B

    M. Burger, B. Hackl, and W. Ring. Incorporating topological derivatives into level set methods.Journal of Computational Physics,194:344–362, 2 2004. doi: https://doi.org/10.1016/j.jcp.2003.09.033

    work page doi:10.1016/j.jcp.2003.09.033 2004

  10. [10]

    de Prenter, C

    F. de Prenter, C. V . Verhoosel, E. H. van Brummelen, M. G. Larson, and S. Badia. Stability and conditioning of immersed finite element methods: Analysis and remedies.Archives of Computational Methods in Engineering,30:3617–3656, 7 2023. doi: https://doi.org/10.1007/s11831-023-09913-0

    work page doi:10.1007/s11831-023-09913-0 2023

  11. [11]

    Ded `e, M

    L. Ded `e, M. J. Borden, and T. J.R. Hughes. Isogeometric analysis for topology optimization with a phase field model.Archives of Computational Methods in Engineering,19:427–465, 9 2012. doi: https://doi.org/10.1007/s11831-012-9075-z

    work page doi:10.1007/s11831-012-9075-z 2012

  12. [12]

    P. Gangl. A multi-material topology optimization algorithm based on the topological derivative.Com- puter Methods in Applied Mechanics and Engineering,366, 7 2020. doi: https://doi.org/10.1016/j. cma.2020.113090

    work page doi:10.1016/j 2020

  13. [13]

    Gangl and K

    P. Gangl and K. Sturm. A simplified derivation technique of topological derivatives for quasi-linear transmission problems.ESAIM - Control, Optimisation and Calculus of Variations,26, 2020. doi: https://doi.org/10.1051/cocv/2020035

    work page doi:10.1051/cocv/2020035 2020

  14. [14]

    Gangl, T

    P. Gangl, T. Komann, N. Krenn, and S. Ulbrich. Robust topology optimization of electric machines using topological derivatives.Arxiv, 4 2025. URLhttp://arxiv.org/abs/2504.05070

    work page arXiv 2025

  15. [15]

    Hughes, J

    T.J.R. Hughes, J. A. Cottrell, and Y . Bazilevs. Isogeometric analysis: Cad, finite elements, nurbs, exact geometry and mesh refinement.Computer Methods in Applied Mechanics and Engineering, 194:4135–4195, 10 2005. doi: https://doi.org/10.1016/j.cma.2004.10.008

    work page doi:10.1016/j.cma.2004.10.008 2005

  16. [16]

    H. A. Jahangiry and S. M. Tavakkoli. An isogeometrical approach to structural level set topology optimization.Computer Methods in Applied Mechanics and Engineering,319:240–257, 6 2017. doi: https://doi.org/10.1016/j.cma.2017.02.005

    work page doi:10.1016/j.cma.2017.02.005 2017

  17. [17]

    Khatibinia, M

    M. Khatibinia, M. Khatibinia, and M. Roodsarabi. Structural topology optimization based on hy- brid of piecewise constant level set method and isogeometric analysis.International Journal of Op- timization in Civil Engineering,10:493–512, 2020. URLhttps://www.researchgate.net/ publication/342924423. 17

    work page arXiv 2020

  18. [18]

    N. Krenn. Multi-material topology optimization subject to pointwise stress constraints for Additive Manufacturing. Master’s thesis, Graz University of Technology, 2021

    work page 2021

  19. [19]

    B. Ma, J. Zheng, G. Lei, J. Zhu, P. Jin, and Y . Guo. Topology optimization of ferromagnetic com- ponents in electrical machines.IEEE Transactions on Energy Conversion,35:786–798, 6 2020. doi: https://doi.org/10.1109/TEC.2019.2960519

    work page doi:10.1109/tec.2019.2960519 2020

  20. [20]

    Osher and J

    S. Osher and J. A. Sethian. Fronts propagating with curvature-dependent speed: Algorithms based on hamilton-jacobi formulations.Journal of Computational Physics,79:12–49, 1988. doi: https: //doi.org/10.1016/0021-9991(88)90002-2

    work page doi:10.1016/0021-9991(88)90002-2 1988

  21. [21]

    Roodsarabi, M

    M. Roodsarabi, M. Khatibinia, and S. R. Sarafrazi. Hybrid of topological derivative-based level set method and isogeometric analysis for structural topology optimization.Steel and Composite Struc- tures,21:1389–1410, 8 2016. doi: https://doi.org/10.12989/scs.2016.21.6.1389

    work page doi:10.12989/scs.2016.21.6.1389 2016

  22. [22]

    R. I. Saye. High-order quadrature methods for implicitly defined surfaces and volumes in hyperrect- angles.SIAM Journal on Scientific Computing,37:A993–A1019, 2015. doi: https://doi.org/10.1137/ 140966290

    work page 2015

  23. [23]

    R. I. Saye. High-order quadrature on multi-component domains implicitly defined by multivariate polynomials.Journal of Computational Physics,448, 1 2022. doi: 10.1016/j.jcp.2021.110720

    work page doi:10.1016/j.jcp.2021.110720 2022

  24. [24]

    Schillinger and M

    D. Schillinger and M. Ruess. The finite cell method: A review in the context of higher-order structural analysis of cad and image-based geometric models.Archives of Computational Methods in Engineer- ing,22:391–455, 7 2015. doi: https://doi.org/10.1007/s11831-014-9115-y

    work page doi:10.1007/s11831-014-9115-y 2015

  25. [25]

    Shojaee, M

    S. Shojaee, M. Mohamadian, and N. Valizadeh. Composition of isogeometric analysis with level set method for structural topology optimization.International Journal of Optimization in Civil Engineer- ing,2:47–70, 2012. URLhttps://www.researchgate.net/publication/259593893

    work page arXiv 2012

  26. [26]

    Sigmund and K

    O. Sigmund and K. Maute. Topology optimization approaches: A comparative review.Struc- tural and Multidisciplinary Optimization,48:1031–1055, 12 2013. doi: https://doi.org/10.1007/ s00158-013-0978-6

    work page 2013

  27. [27]

    Sigmund and J

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

    work page

  28. [28]

    doi: https://doi.org/10.1007/BF01214002. 18

    work page doi:10.1007/bf01214002

  29. [29]

    G. H. Teixeira, M. Loibl, and B. Marussig. Comparison of integration methods for cut elements. ArXiv, 1 2025. doi: https://doi.org/10.23967/eccomas.2024.098. URLhttp://arxiv.org/abs/ 2501.03854

    work page doi:10.23967/eccomas.2024.098 2025

  30. [30]

    Toprak, M

    T. Toprak, M. Loibl, G. H. Teixeira, I. Shiskina, C. Miao, J. Kiendl, B. Marussig, and F. Kummer. Employing continuous integration inspired workflows for benchmarking of scientific software – a use case on numerical cut cell quadrature.ArXiv, 4 2025. URLhttps://arxiv.org/abs/2503. 17192

    work page 2025

  31. [31]

    Verzicco

    R. Verzicco. Immersed boundary methods: Historical perspective and future outlook.Annual Review of Fluid Mechanics,11:39, 2023. doi: https://doi.org/10.1146/annurev-fluid-120720

    work page doi:10.1146/annurev-fluid-120720 2023

  32. [32]

    V ´azquez

    R. V ´azquez. A new design for the implementation of isogeometric analysis in octave and matlab: Geopdes 3.0.Computers and Mathematics with Applications,72:523–554, 8 2016. doi: https://doi. org/10.1016/j.camwa.2016.05.010

    work page doi:10.1016/j.camwa.2016.05.010 2016

  33. [33]

    M. Y . Wang, X. Wang, and D. Guo. A level set method for structural topology optimization.Computer Methods in Applied Mechanics and Engineering,192:227–246, 2003. doi: https://doi.org/10.1016/ S0045-7825(02)00559-5

    work page 2003

  34. [34]

    X. S. Wang, L. T. Zhang, and W. K. Liu. On computational issues of immersed finite element methods. Journal of Computational Physics,228:2535–2551, 4 2009. doi: https://doi.org/10.1016/j.jcp.2008. 12.012

    work page doi:10.1016/j.jcp.2008 2009

  35. [35]

    Wiesheu, T

    M. Wiesheu, T. Komann, M. Merkel, S. Sch ¨ops, S. Ulbrich, and I. Cortes Garcia. Combined pa- rameter and shape optimization of electric machines with isogeometric analysis.Optimization and Engineering,26:1011–1038, 2024. doi: https://doi.org/10.1007/s11081-024-09925-0. 19

    work page doi:10.1007/s11081-024-09925-0 2024