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arxiv: 2605.11994 · v2 · submitted 2026-05-12 · 🧮 math.NA · cs.NA

The SiMPL Method for Multi-Material Topology Optimization

Peter Gangl , Brendan Keith , Dohyun Kim , Boyan S. Lazarov , Thomas M. Surowiec This is my paper

Pith reviewed 2026-05-15 05:30 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords multi-material topology optimizationBregman divergencemirror descentpolytopal constraintsdensity-based optimizationstructural designmagnetic flux optimizationArmijo line search
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The pith

The SiMPL method penalizes designs around the previous iterate with a Bregman divergence to enforce strict point-wise polytopal constraints in multi-material topology optimization.

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

The paper presents the SiMPL method for density-based multi-material topology optimization. It integrates mirror descent with point-wise constraints that require each spatial point to occupy exactly one vertex of a convex polytope representing distinct material states. A Bregman divergence penalizes the space around the prior iterate, smoothing the landscape so that new iterates stay strictly feasible with respect to these local rules. Global constraints such as mass limits are then recovered by solving a small dual problem. The resulting algorithm is simple to code and pairs naturally with an Armijo line search, as shown on structural problems with isotropic and anisotropic materials and on magnetic flux design in motors.

Core claim

The framework generates a descending sequence of iterates by penalizing the design space around the previous iterate with a generalized distance function tailored to the convex geometry of the n-dimensional polytope. This distance function, called a Bregman divergence, smooths the optimization landscape, ensuring that each iterate strictly satisfies the point-wise constraints. Subsequently, global constraints can be enforced easily by solving a small, finite-dimensional dual problem. The method is simple to implement and demonstrates robustness and efficiency when combined with an Armijo-type line search algorithm.

What carries the argument

The Bregman divergence tailored to the n-dimensional polytope, which acts as a generalized distance penalty that smooths the landscape around the previous design iterate and enforces strict satisfaction of point-wise material-state constraints.

If this is right

  • Each iterate satisfies the point-wise constraints exactly, eliminating the need for post-processing projections.
  • Global constraints such as total mass are recovered by solving one small finite-dimensional dual problem after the local update.
  • The same framework applies without change to both isotropic and anisotropic material models.
  • The algorithm extends directly to magnetic flux optimization problems in electric motors.
  • Implementation reduces to standard mirror-descent steps plus the dual solve and an Armijo line search.

Where Pith is reading between the lines

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

  • The method may reduce the need for specialized projection operators that other multi-material schemes require.
  • Because the penalty is defined directly on the polytope geometry, the approach could transfer to any optimization problem whose feasible set is a product of polytopes.
  • Efficiency on large meshes would allow routine treatment of problems with five or more candidate materials.
  • Replacing the Bregman divergence with other divergences adapted to the same polytope could yield families of related algorithms.

Load-bearing premise

The chosen Bregman divergence, when paired with an Armijo line search, produces iterates that remain strictly feasible with respect to the point-wise polytopal constraints for the material models and problem geometries considered.

What would settle it

A run on an anisotropic-material structural problem in which at least one spatial point in an iterate violates a polytopal vertex constraint after the line search step.

Figures

Figures reproduced from arXiv: 2605.11994 by Boyan S. Lazarov, Brendan Keith, Dohyun Kim, Peter Gangl, Thomas M. Surowiec.

Figure 1
Figure 1. Figure 1: Polytopes parametrized using ∇R∗ (ψ); cf. Theorem 3.6. (a) A hexagon and (b) a bipyramid with an octagonal base. Colors indicate the magnitude of the latent variable |ψ|. This relationship allows us to define a latent variable ψ ∈ L∞(Ω; R n) parameterizing the design variable via η = ∇R∗ (ψ). In turn, the update rule (3.3) becomes ψ k+1/2 = ψ k − αk∇F(η k (3.4a) ), ψ k+1 = ψ k+1/2 − W⊤µ k (3.4b) , where µ … view at source ↗
Figure 2
Figure 2. Figure 2: Optimized designs for compliance minimization using isotropic materials with [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Convergence history for (a) compliance F(η k ), (b) gap residual resk, and (c) step size αk for 2D compliance minimization with isotropic materials. of h = 2−8 . The indicator space is discretized using piecewise constant elements, while both the filtered indicator space and the displacement space are approximated using continuous piecewise bilinear elements. The optimal design for each interpolation schem… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Optimized design for compliance minimization using orthotropic material [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Optimization of electric motor: (a) Cross-section of the sector of an electric [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
read the original abstract

We introduce an efficient and scalable method for density-based multi-material topology optimization, integrating classical mirror descent techniques with point-wise polytopal design constraints. Such constraints arise naturally in this class of problems, wherein the vertices of convex polytopes correspond to distinct design states, only one of which should be occupied at each point in space. The framework generates a descending sequence of iterates by penalizing the design space around the previous iterate with a generalized distance function tailored to the convex geometry of the $n$-dimensional polytope. This distance function, called a Bregman divergence, smooths the optimization landscape, ensuring that each iterate strictly satisfies the point-wise constraints. Subsequently, global constraints (e.g., bounds on the structural mass) can be enforced easily by solving a small, finite-dimensional dual problem. The resulting method is simple to implement and demonstrates robustness and efficiency when combined with an Armijo-type line search algorithm. We validate the method in structural design problems involving the optimal arrangement of both isotropic and anisotropic materials, as well as magnetic flux optimization in electric motors.

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 introduces the SiMPL method for density-based multi-material topology optimization. It combines mirror descent with a Bregman divergence tailored to the convex geometry of n-dimensional polytopes to generate a descending sequence of iterates that strictly satisfy point-wise design constraints (vertices corresponding to pure material states). Global constraints such as mass bounds are then enforced by solving a small finite-dimensional dual problem. The method is paired with an Armijo-type line search, claimed to be simple to implement, robust, and efficient, and is validated on structural optimization problems with isotropic and anisotropic materials as well as magnetic flux optimization in electric motors.

Significance. If the central feasibility claim holds, the work would provide a useful contribution to multi-material topology optimization by offering a constraint-preserving update that avoids explicit projections or heavy penalties. The integration of mirror descent theory with polytopal sets is a natural extension and could generalize to other problems with simplex-like constraints. Validation on both isotropic/anisotropic stiffness and electromagnetic examples demonstrates practical applicability, and the absence of free parameters in the core update is a strength.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (method derivation): The load-bearing claim that the tailored Bregman divergence plus Armijo line search automatically produces iterates strictly inside the relative interior of the polytope is asserted but not derived. No explicit form of the divergence (e.g., which entropy or barrier on the simplex) is given, nor is there an argument showing that the line-search acceptance criterion preserves strict feasibility rather than merely descent; this leaves open the possibility that components can approach zero or that the property fails for anisotropic tensors.
  2. [§4] §4 (numerical results): No convergence rates, a priori error bounds, or analysis of the dual solver's conditioning are provided, even though the abstract emphasizes efficiency and scalability. The reported robustness on example problems is useful but insufficient to support the broader claims without quantitative rates or failure-mode analysis.
minor comments (2)
  1. [§2] Notation for the polytopal vertices and material interpolation could be introduced earlier and made consistent across sections to aid readability.
  2. [Figures] Figure captions should explicitly state the number of materials and the polytope dimension used in each example.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help clarify the presentation of the SiMPL method. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and additional material.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (method derivation): The load-bearing claim that the tailored Bregman divergence plus Armijo line search automatically produces iterates strictly inside the relative interior of the polytope is asserted but not derived. No explicit form of the divergence (e.g., which entropy or barrier on the simplex) is given, nor is there an argument showing that the line-search acceptance criterion preserves strict feasibility rather than merely descent; this leaves open the possibility that components can approach zero or that the property fails for anisotropic tensors.

    Authors: We agree that the derivation of strict feasibility preservation requires more explicit detail. The Bregman divergence is constructed from a strictly convex barrier function on the polytope (negative entropy for the simplex, or its direct analogue for general polytopes via affine transformation of coordinates). This divergence diverges to +∞ as any coordinate approaches the boundary of the relative interior. The mirror-descent update therefore remains in the relative interior whenever the previous iterate does, because the objective of the proximal subproblem tends to infinity at the boundary. The Armijo line search is performed along the line segment connecting the current interior point to the candidate update; any accepted step stays in the open relative interior. The property is independent of the material tensors (isotropic or anisotropic), since the pointwise constraints act only on the design variables. We will add the explicit functional form of the divergence and a concise proof of interior-point preservation to §3. revision: yes

  2. Referee: [§4] §4 (numerical results): No convergence rates, a priori error bounds, or analysis of the dual solver's conditioning are provided, even though the abstract emphasizes efficiency and scalability. The reported robustness on example problems is useful but insufficient to support the broader claims without quantitative rates or failure-mode analysis.

    Authors: We acknowledge that the manuscript currently provides only empirical robustness on the chosen examples and lacks both theoretical rates and conditioning analysis. Full a priori convergence rates are difficult to obtain for this non-convex setting and are not claimed in the paper; however, we will augment §4 with quantitative numerical evidence, including semi-log plots of objective decrease versus iteration count for all reported examples and tables of iteration numbers and CPU times. For the dual problem (a small, low-dimensional convex program with one or two equality constraints), we will add a short paragraph describing its solution via a safeguarded Newton method and noting that the Hessian remains well-conditioned in practice because the number of dual variables is fixed and small. A brief failure-mode discussion will be included, observing that the line search and dual solver succeeded on all tested instances (including anisotropic and electromagnetic cases) and identifying the only observed sensitivity (very tight global mass bounds). These additions will strengthen the efficiency and scalability claims with concrete data. revision: partial

Circularity Check

0 steps flagged

No circularity: SiMPL applies established mirror descent with geometry-tailored Bregman divergence to polytopal constraints

full rationale

The derivation chain rests on standard mirror descent theory (Bregman divergence chosen to the polytope geometry) plus an Armijo line search to produce feasible iterates. The abstract states that the divergence 'smooths the optimization landscape, ensuring that each iterate strictly satisfies the point-wise constraints,' but this follows from general properties of the method rather than any self-definition, fitted input renamed as prediction, or self-citation load-bearing step. No equations reduce the claimed strict feasibility to a tautology or to parameters fitted from the target result itself. The subsequent dual solve for global constraints is a standard finite-dimensional step independent of the local update. The paper is therefore self-contained against external benchmarks of mirror descent on convex sets.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The approach assumes standard properties of convex polytopes and Bregman divergences from mirror descent literature; no free parameters are explicitly fitted in the abstract description, and the new element is the tailored application rather than new entities.

axioms (2)
  • domain assumption Vertices of convex polytopes correspond to distinct design states with only one occupied at each spatial point
    Stated directly in the abstract as arising naturally in multi-material problems.
  • domain assumption Bregman divergence generated by a suitable convex function smooths the landscape while preserving strict feasibility
    Central to the claim that each iterate satisfies point-wise constraints.
invented entities (1)
  • SiMPL method no independent evidence
    purpose: Efficient enforcement of point-wise polytopal constraints in multi-material topology optimization
    The overall algorithm is the novel contribution introduced in the paper.

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Reference graph

Works this paper leans on

64 extracted references · 64 canonical work pages

  1. [1]

    2025 , eprint =

    A novel multi-thickness topology optimization method for balancing structural performance and manufacturability , author =. 2025 , eprint =

    work page 2025

  2. [2]

    1970 , publisher =

    Convex Analysis , author =. 1970 , publisher =

    work page 1970

  3. [3]

    Mirror descent and nonlinear projected subgradient methods for convex optimization , journal =

    Beck, Amir and Teboulle, Marc , year =. Mirror descent and nonlinear projected subgradient methods for convex optimization , volume =. Operations Research Letters , publisher =. doi:10.1016/s0167-6377(02)00231-6 , number =

    work page doi:10.1016/s0167-6377(02)00231-6

  4. [4]

    Larsen, S. D. and Sigmund, O. and Groen, J. P. , journal =. Optimal truss and frame design from projected homogenization-based topology optimization , year =. doi:10.1007/s00158-018-1948-9 , refid =

    work page doi:10.1007/s00158-018-1948-9 1948

  5. [5]

    On approaches for avoiding low-stiffness regions in variable thickness sheet and homogenization-based topology optimization , year =

    Giele, Reinier and Groen, Jeroen and Aage, Niels and Andreasen, Casper Schousboe and Sigmund, Ole , journal =. On approaches for avoiding low-stiffness regions in variable thickness sheet and homogenization-based topology optimization , year =. doi:10.1007/s00158-021-02933-z , refid =

    work page doi:10.1007/s00158-021-02933-z

  6. [6]

    Multi-material topology optimization using ordered

    Zuo, Wenjie and Saitou, Kazuhiro , journal =. Multi-material topology optimization using ordered. 2017 , issn =. doi:10.1007/s00158-016-1513-3 , refid =

    work page doi:10.1007/s00158-016-1513-3 2017

  7. [7]

    and Lund, E

    Stegmann, J. and Lund, E. , journal =. Discrete material optimization of general composite shell structures , year =. doi:https://doi.org/10.1002/nme.1259 , keywords =

    work page doi:10.1002/nme.1259

  8. [8]

    On structural optimization of composite shell structures using a discrete constitutive parametrization , year =

    Lund, Erik and Stegmann, Jan , journal =. On structural optimization of composite shell structures using a discrete constitutive parametrization , year =. doi:https://doi.org/10.1002/we.132 , keywords =

    work page doi:10.1002/we.132

  9. [9]

    Material interpolation schemes for unified topology and multi-material optimization , year =

    Hvejsel, Christian Frier and Lund, Erik , journal =. Material interpolation schemes for unified topology and multi-material optimization , year =. doi:10.1007/s00158-011-0625-z , refid =

    work page doi:10.1007/s00158-011-0625-z

  10. [10]

    doi:10.48550/arXiv.2107.08011 , title =

    Antonakopoulos, Kimon and Mertikopoulos, Panayotis , booktitle =. doi:10.48550/arXiv.2107.08011 , title =

    work page doi:10.48550/arxiv.2107.08011

  11. [11]

    doi:10.1142/9789812815750_0009 , year =

    Nonlinearity and Functional Analysis: Lectures on Nonlinear Problems in Mathematical Analysis , author =. doi:10.1142/9789812815750_0009 , year =

    work page doi:10.1142/9789812815750_0009

  12. [12]

    Bendsøe and Noboru Kikuchi

    Generating optimal topologies in structural design using a homogenization method , journal =. 1988 , issn =. doi:10.1016/0045-7825(88)90086-2 , author =

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

  13. [13]

    and Bolte, J\'

    Bauschke, Heinz H. and Bolte, J\'. A Descent Lemma Beyond. Mathematics of Operations Research , volume =. 2017 , doi =

    work page 2017

  14. [14]

    B. S. Lazarov and O. Sigmund , title =. Numerical Methods in Engineering , volume =. 2011 , pages =

    work page 2011

  15. [15]

    Lazarov and Mattias Schevenels , keywords =

    Oded Amir and Ole Sigmund and Boyan S. Lazarov and Mattias Schevenels , keywords =. Efficient reanalysis techniques for robust topology optimization , journal =. 2012 , doi =

    work page 2012

  16. [16]

    and Svanberg, K

    Stolpe, M. and Svanberg, K. , date =. An alternative interpolation scheme for minimum compliance topology optimization , volume =. Structural and Multidisciplinary Optimization , number =. doi:10.1007/s001580100129 , id =

    work page doi:10.1007/s001580100129

  17. [17]

    Filters in topology optimization.International Journal for Numerical Methods in Engineering, 50(9):2143–2158, 2001

    Bourdin, Blaise , title =. International Journal for Numerical Methods in Engineering , volume =. doi:10.1002/nme.116 , year =

    work page doi:10.1002/nme.116

  18. [18]

    A 99 line topology optimization code written in Matlab.Structural and Multidisciplinary Optimization, 21(2):120–127, 2001

    Sigmund, O. , date =. A 99 line topology optimization code written in. Structural and Multidisciplinary Optimization , number =. doi:10.1007/s001580050176 , id =

    work page doi:10.1007/s001580050176

  19. [19]

    On Equivalence Between Optimality Criteria and Projected Gradient Methods with Application to Topology Optimization Problem , volume =

    Ananiev, Sergey , date =. On Equivalence Between Optimality Criteria and Projected Gradient Methods with Application to Topology Optimization Problem , volume =. Multibody System Dynamics , number =. doi:10.1007/s11044-005-2530-y , id =

    work page doi:10.1007/s11044-005-2530-y

  20. [20]

    , date =

    Zillober, C. , date =. A globally convergent version of the method of moving asymptotes , volume =. Structural optimization , number =. doi:10.1007/BF01743509 , id =

    work page doi:10.1007/bf01743509

  21. [21]

    Topology Optimization , doi =

    Bends. Topology Optimization , doi =. 2004 , address =

    work page 2004

  22. [22]

    Svanberg, ‘The method of moving asymptotes—a new method for structural optimization’, Numerical Meth En- gineering, vol

    Svanberg, Krister , title =. International Journal for Numerical Methods in Engineering , volume =. doi:10.1002/nme.1620240207 , year =

    work page doi:10.1002/nme.1620240207

  23. [23]

    2023 , eprint =

    Proximal Galerkin: A structure-preserving finite element method for pointwise bound constraints , author =. 2023 , eprint =

    work page 2023

  24. [24]

    , title = "

    BARZILAI, JONATHAN and BORWEIN, JONATHAN M. , title = ". IMA Journal of Numerical Analysis , volume =. 1988 , month =

    work page 1988

  25. [25]

    A simplified view of first order methods for optimization , volume =

    Teboulle, Marc , date =. A simplified view of first order methods for optimization , volume =. Mathematical Programming , number =. doi:10.1007/s10107-018-1284-2 , id =

    work page doi:10.1007/s10107-018-1284-2

  26. [26]

    Krister Svanberg , title =

    work page

  27. [27]

    Topology optimization using

    Aage, Niels and Andreassen, Erik and Lazarov, Boyan Stefanov , date-added =. Topology optimization using. Structural and Multidisciplinary Optimization , number =. doi:10.1007/s00158-014-1157-0 , id =

    work page doi:10.1007/s00158-014-1157-0

  28. [28]

    One-shot procedures for efficient minimum compliance topology optimization , volume =

    Amir, Oded , date-added =. One-shot procedures for efficient minimum compliance topology optimization , volume =. Structural and Multidisciplinary Optimization , number =. doi:10.1007/s00158-024-03763-5 , id =

    work page doi:10.1007/s00158-024-03763-5

  29. [29]

    Topology optimization of multi-story buildings under fully non-stationary stochastic seismic ground motion , volume =

    Angelucci, Giulia and Quaranta, Giuseppe and Mollaioli, Fabrizio , date =. Topology optimization of multi-story buildings under fully non-stationary stochastic seismic ground motion , volume =. Structural and Multidisciplinary Optimization , number =. doi:10.1007/s00158-022-03319-5 , id =

    work page doi:10.1007/s00158-022-03319-5

  30. [30]

    Bridge topology optimisation with stress, displacement and frequency constraints , doi =

    Guan, Hong and Chen, Yin-Jung and Loo, Yew-Chaye and Xie, Yi-Min and Steven, Grant P , journal =. Bridge topology optimisation with stress, displacement and frequency constraints , doi =. 2003 , publisher =

    work page 2003

  31. [31]

    High-performance finite elements with

    Julian Andrej and Nabil Atallah and Jan-Phillip B\". High-performance finite elements with. 2024 , eprint =

    work page 2024

  32. [32]

    2020 , month =

    Modular Finite Element Methods , author =. 2020 , month =. doi:10.11578/dc.20200303.5 , howpublished =

    work page doi:10.11578/dc.20200303.5 2020

  33. [33]

    INFORMS Journal on Optimization , volume =

    Lu, Haihao , title =. INFORMS Journal on Optimization , volume =. 2019 , doi =

    work page 2019

  34. [34]

    2004 , issn =

    Structural optimization using sensitivity analysis and a level-set method , journal =. 2004 , issn =. doi:10.1016/j.jcp.2003.09.032 , author =

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

  35. [35]

    On the (non-)optimality of Michell structures , volume =

    Sigmund, Ole and Aage, Niels and Andreassen, Erik , year =. On the (non-)optimality of Michell structures , volume =. Structural and Multidisciplinary Optimization , publisher =. doi:10.1007/s00158-016-1420-7 , number =

    work page doi:10.1007/s00158-016-1420-7

  36. [36]

    , howpublished =

    Kim, D. , howpublished =

    work page

  37. [37]

    Lazarov and Thomas M

    Brendan Keith and Dohyun Kim and Boyan S. Lazarov and Thomas M. Surowiec , year =. Analysis of the. SIAM Journal on Optimization , number =

    work page

  38. [38]

    Lazarov and Thomas M

    Dohyun Kim and Boyan S. Lazarov and Thomas M. Surowiec and Brendan Keith , year =. A simple introduction to the. Structural and Multidisciplinary Optimization , volume =

    work page

  39. [39]

    and Dykstra, Richard L

    Boyle, James P. and Dykstra, Richard L. , year =. A Method for Finding Projections onto the Intersection of Convex Sets in Hilbert Spaces , ISBN =. doi:10.1007/978-1-4613-9940-7_3 , booktitle =

    work page doi:10.1007/978-1-4613-9940-7_3

  40. [40]

    Dykstras algorithm with

    Bauschke, Heinz H and Lewis, Adrian S , year =. Dykstras algorithm with. Optimization , publisher =. doi:10.1080/02331930008844513 , number =

    work page doi:10.1080/02331930008844513

  41. [41]

    2025 , issn =

    The latent variable proximal point algorithm for variational problems with inequality constraints , journal =. 2025 , issn =. doi:https://doi.org/10.1016/j.cma.2025.118181 , author =

    work page doi:10.1016/j.cma.2025.118181 2025

  42. [42]

    Tyrrell and Wets, Roger J

    Rockafellar, R. Tyrrell and Wets, Roger J. B. , year =. Variational Analysis , ISBN =. doi:10.1007/978-3-642-02431-3 , publisher =

    work page doi:10.1007/978-3-642-02431-3

  43. [43]

    , title =

    Sukumar, N. , title =. International Journal for Numerical Methods in Engineering , volume =. doi:https://doi.org/10.1002/nme.1193 , year =

    work page doi:10.1002/nme.1193

  44. [44]

    IEEE Transactions on Magnetics , title =

    Cherri\`. IEEE Transactions on Magnetics , title =. 2025 , volume =

    work page 2025

  45. [45]

    Bauschke and Patrick L

    Bauschke, Heinz H. and Combettes, Patrick L. , year =. Convex Analysis and Monotone Operator Theory in Hilbert Spaces , ISBN =. doi:10.1007/978-3-319-48311-5 , journal =

    work page doi:10.1007/978-3-319-48311-5

  46. [46]

    and Jarratt, P

    Dowell, M. and Jarratt, P. , year =. A modified regula falsi method for computing the root of an equation , volume =. BIT , publisher =. doi:10.1007/bf01934364 , number =

    work page doi:10.1007/bf01934364

  47. [47]

    SIAM Journal on Imaging Sciences , volume =

    Beck, Amir and Teboulle, Marc , title =. SIAM Journal on Imaging Sciences , volume =. 2009 , doi =

    work page 2009

  48. [48]

    Byrd, Peihuang Lu, and Jorge Nocedal

    Zhu, Ciyou and Byrd, Richard H. and Lu, Peihuang and Nocedal, Jorge , title =. ACM Trans. Math. Softw. , month = dec, pages =. 1997 , issue_date =. doi:10.1145/279232.279236 , abstract =

    work page doi:10.1145/279232.279236 1997

  49. [49]

    2014 , publisher =

    Variational analysis in Sobolev and BV spaces: applications to PDEs and optimization , author =. 2014 , publisher =

    work page 2014

  50. [50]

    Multi-material topology optimization using Wachspress interpolations for designing a 3-phase electrical machine stator , volume =

    Cherrière, Théodore and Laurent, Luc and Hlioui, Sami and Louf, Fran. Multi-material topology optimization using Wachspress interpolations for designing a 3-phase electrical machine stator , volume =. Structural and Multidisciplinary Optimization , publisher =. doi:10.1007/s00158-022-03460-1 , number =

    work page doi:10.1007/s00158-022-03460-1

  51. [51]

    The Minimum Euclidean-Norm Point in a Convex Polytope: Wolfe's Combinatorial Algorithm is Exponential , journal =

    De Loera, Jes\'. The Minimum Euclidean-Norm Point in a Convex Polytope: Wolfe's Combinatorial Algorithm is Exponential , journal =. 2020 , doi =

    work page 2020

  52. [52]

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

    2021 , title =. doi:10.1016/bs.hna.2020.10.004 , author =

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

  53. [53]

    and Dapogny, C

    Allaire, G. and Dapogny, C. and Delgado, G. and Michailidis, G. , title =. ESAIM: Control, Optimisation and Calculus of Variations , pages =. 2014 , publisher =. doi:10.1051/cocv/2013076 , zbl =

    work page doi:10.1051/cocv/2013076 2014

  54. [54]

    Frédéric and Shapiro, Alexander , year =

    Bonnans, J. Frédéric and Shapiro, Alexander , year =. Perturbation Analysis of Optimization Problems , ISBN =. doi:10.1007/978-1-4612-1394-9 , publisher =

    work page doi:10.1007/978-1-4612-1394-9

  55. [55]

    10.1051/cocv/2022078

    An abstract Lagrangian framework for computing shape derivatives , DOI = "10.1051/cocv/2022078", url = "https://doi.org/10.1051/cocv/2022078", journal =

    work page doi:10.1051/cocv/2022078

  56. [56]

    Analysis and application of a lower envelope method for sharp-interface multiphase problems , url =

    Laurain, Antoine , doi =. Analysis and application of a lower envelope method for sharp-interface multiphase problems , url =. Control and Cybernetics , number =

    work page

  57. [57]

    Gangl , keywords =

    P. Gangl , keywords =. A multi-material topology optimization algorithm based on the topological derivative , journal =. 2020 , issn =. doi:https://doi.org/10.1016/j.cma.2020.113090 , url =

    work page doi:10.1016/j.cma.2020.113090 2020

  58. [58]

    Extended level set method: A multiphase representation with perfect symmetric property, and its application to multi-material topology optimization , journal =

    Masaki Noda and Yuki Noguchi and Takayuki Yamada , keywords =. Extended level set method: A multiphase representation with perfect symmetric property, and its application to multi-material topology optimization , journal =. 2022 , issn =. doi:https://doi.org/10.1016/j.cma.2022.114742 , url =

    work page doi:10.1016/j.cma.2022.114742 2022

  59. [59]

    and Hinterm\"

    Adam, L. and Hinterm\". A. Optimization and Engineering , publisher =. 2018 , month = jul, pages =. doi:10.1007/s11081-018-9394-5 , number =

    work page doi:10.1007/s11081-018-9394-5 2018

  60. [60]

    2025 , eprint =

    Multi-material structural optimization for additive manufacturing based on a phase field approach , author =. 2025 , eprint =

    work page 2025

  61. [61]

    Bendsøe and Ole Sigmund.Topology Optimization

    Bendsøe, Martin P. and Sigmund, Ole , year =. Topology Optimization , ISBN =. doi:10.1007/978-3-662-05086-6 , publisher =

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

  62. [62]

    Monotonicity-preserving interproximation of

    Clemens Pechstein and Bert Jüttler , doi =. Monotonicity-preserving interproximation of. Journal of Computational and Applied Mathematics , keywords =

    work page

  63. [63]

    Optimization of a Multiphysics Problem in Semiconductor Laser Design , volume =

    Adam, Luk\'. Optimization of a Multiphysics Problem in Semiconductor Laser Design , volume =. SIAM Journal on Applied Mathematics , publisher =. 2019 , month = jan, pages =. doi:10.1137/18m1179183 , number =

    work page doi:10.1137/18m1179183 2019

  64. [64]

    1974 , publisher=

    Conjugate duality and optimization , author=. 1974 , publisher=

    work page 1974