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Space-Time Multigrid Methods Suitable for Topology Optimisation of Transient Heat Conduction

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Pith's one-line read One scalar, the geometric mean of min and max diffusivity, reliably picks space- or time-coarsening at each multigrid level; a solver built on it converged for the primal and adjoint systems throughout a 1D topology optimisation.

desk verdict A solid, honest empirical study: the geometric-mean effective anisotropy parameter for semi-coarsening is a genuinely new idea for high-contrast STMG, and the eight-way rediscretisation comparison is useful, but the main heuristic was tuned on the same 1D test family used to validate it, so its external validity is the weak link. read the letter →

arxiv 2505.10168 v1 pith:6NZW6MCU submitted 2025-05-15 cs.CE

Magnus Appel , Joe Alexandersen This is my paper · ORCID
classification cs.CE MSC 65K1065M5565M2265M3280M1080M50
keywords topologyoptimisationspace-timemultigridparallelintimetransientheatconductionhigh-contrasteffectiveanisotropyparametersemi-coarsening
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Topology optimisation of transient heat conduction is expensive because the heat equation must be solved many times while the material layout evolves. This paper proposes space-time multigrid (STMG) methods to make those solves fast and parallelisable in time, and addresses the obstacle that makes STMG hard for design problems: diffusivity contrasts of $10^4$ between conductor and insulator mean the same grid may need time-coarsening in one region and space-coarsening in another. The central proposal is a single indicator, the effective anisotropy parameter $\lambda_{\mathrm{eff}} = \sqrt{\min_e \lambda_e \max_e \lambda_e}$, which at each multigrid level decides whether to coarsen in space or in time. Across six one-dimensional test problems the crossover between the two choices fell in a narrow band of $\lambda_{\mathrm{eff}}$, and the resulting solver converged for both the state and adjoint equations at every cycle of a one-dimensional topology optimisation. Should the indicator transfer to two and three dimensions, the payoff would be a dependable, parallel-in-time solver for design problems that currently take days of computation.

What carries the argument

The load-bearing object is the effective anisotropy parameter, $\lambda_{\mathrm{eff}}$, the geometric mean of the minimum and maximum element-level anisotropy values. It compresses a spatially varying anisotropy field $\lambda_e = D_e \Delta t/\Delta x^2$ that spans orders of magnitude in high-contrast problems into the single number that the coarsening strategy (Algorithm 3) compares against the threshold $\lambda_{\mathrm{crit}} = 0.25$: coarsen in time if $\lambda_{\mathrm{eff}} < \lambda_{\mathrm{crit}}$, otherwise coarsen in space. The second mechanism is the rediscretisation pair: causal, forward-in-time-only prolongation stencils, and coarse-level system matrices reassembled by averaging the thermal resistivity $\rho = 1/k$ rather than the conductivity, the discrete analogue of combining series resistors, which is why it suits one spatial dimension. A third, optional mechanism (Algorithm 4) forces time-coarsening whenever space-coarsening would drop the spatial element count below a threshold $M$, trading coarse-grid resolution of small features against the solver's sensitivity to large anisotropy.

What would settle it

Run the same two-grid STMG method, with the same pointwise Jacobi smoother, on a two-dimensional high-contrast problem containing a thin insulating layer, and measure the spread of $\lambda_{\mathrm{eff}}$ values at which the space-coarsening and time-coarsening convergence curves cross. If that crossover band is wider than the roughly one decade ($2^{-3}$ to $2^{-1}$) observed in one dimension, the geometric-mean indicator does not transfer beyond the tested setting; equivalently, repeating the six test problems with a realistic pulsed heat load instead of the synthetic load of Equation (15) would reveal how much of the reliability is load-dependent.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that, for the all-at-once backward-Euler finite-element discretisation of transient heat conduction with strongly heterogeneous diffusivity, the question of whether the next multigrid level should coarsen in space or in time can be answered level by level by comparing a single scalar against a fixed threshold. The scalar is the effective anisotropy parameter $\lambda_{\mathrm{eff}} = \sqrt{\min_e(\lambda_e) \max_e(\lambda_e)}$, where $\lambda_e = D_e \Delta t/\Delta x^2$ is the per-element anisotropy parameter and $D_e = k_e/c_e$ the thermal diffusivity; equivalently $\lambda_{\mathrm{eff}} = D_{\mathrm{eff}} \Delta t/\Delta x^2$ with $D_{\mathrm{eff}}$ the geometric mean of the minimum and maximum diffusivity. The paper reports that this expression was selected by trial and error over six one-dimensional test problems with material contrasts up to $10^4$ in conductivity and/or heat capacity, and that it located the space-versus-time coarsening crossover in the narrow band $2^{-3} \le \lambda_{\mathrm{eff}} \le 2^{-1}$ across all six problems, beating seven alternative candidate expressions. It further establishes that, among eight rediscretisation combinations, averaging the thermal resistivity $\rho = 1/k$ when assembling coarse levels performed best in one dimension, because a 1D heat path behaves as resistors in series, and that causal prolongation (transferring information only forwards in time) beat bilinear interpolation for problems without small features. Applied to a one-dimensional topology optimisation, the STMG solver converged in at most 80 cycles in every optimisation cycle, for both the primal system and the adjoint system, even though the adjoint problem's time direction is backwards.

Load-bearing premise

The load-bearing premise is that the six one-dimensional test problems, each run with a single synthetic heat load, one smoother (damped pointwise Jacobi, damping factor $1/2$), and uniform Cartesian meshes, are representative enough that the geometric-mean indicator and coarsening rule, selected by trial and error on them, transfer to other loads, smoothers, and dimensions without further tuning.

Editorial extensions

If this is right

  • On the six one-dimensional test problems, the crossover between space- and time-coarsening lies within $2^{-3} \le \lambda_{\mathrm{eff}} \le 2^{-1}$ despite diffusivity contrasts of $10^4$, so a fixed threshold $\lambda_{\mathrm{crit}} = 0.25$ is a workable default.
  • Because $\lambda_{\mathrm{eff}}$ can be defined from the material extremes alone, the coarsening path for an entire optimisation can be frozen: prolongation, restriction, and coarse-level storage are built once and reused in every optimisation cycle.
  • In the 1D optimisation test, the STMG solver converged within at most 80 cycles in every optimisation cycle, for both the primal and the adjoint system, with warm restarts consistently reducing the cycle count.
  • Causal (forward-in-time-only) prolongation works for the adjoint problem even though adjoint time runs backwards, and this holds also when Galerkin projection embeds the prolongation directly in the coarse-level operators.
  • For one-dimensional problems with thin features, averaging thermal resistivities on the coarse levels outperforms averaging conductivities, averaging the design field, and Galerkin projection, and the resistivity method's advantage is largest for very small features ($F \le 1/128$).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's own series-versus-parallel argument implies the ranking of reassembly methods should flip in two or three dimensions: a finned or branched heat sink conducts mostly in parallel, so averaging conductivities or using Galerkin projection should beat averaging resistivities there, a direct, testable extension the authors flag but do not demonstrate.
  • Because the design-independent version of $\lambda_{\mathrm{eff}}$ uses only the two material extremes, the whole coarsening path can be precomputed before an optimisation starts; an extension of this observation is that a general-purpose space-time solver library could choose its semi-coarsening direction without ever reading the design field.
  • The fact that forward-only prolongation works for the backwards-running adjoint problem suggests the causal stencil is matched to the one-sided time coupling of backward Euler, not to the physical direction of time; for symmetric time integrators such as Crank-Nicolson the same choice may need re-examination.
  • Because $\lambda_{\mathrm{eff}}$ depends only on the extremes of the diffusivity field, it is blind to where the contrast sits; layered test problems may not probe geometries where a high-diffusivity island is surrounded by insulator, and a two-dimensional analogue of problem 8 would show whether the observed resolution heuristic $M \approx 1/F$ survives.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes space-time multigrid (STMG) methods for solving the all-at-once linear systems that arise in density-based topology optimization of transient heat conduction. The central methodological contribution is an effective anisotropy parameter, defined as the geometric mean of the minimum and maximum element diffusivity (Eq. 20), which is used by Algorithm 3 to choose between semi-coarsening in space and semi-coarsening in time for high-contrast heterogeneous diffusivity. The paper also compares four coarse-level reassembly schemes (averaging of conductivity, design field, resistivity, and Galerkin projection) with two interpolation stencils, and introduces a resolution-guarded coarsening strategy (Algorithm 4) intended for designs with small features. The proposed methods are applied as solvers for both the primal and adjoint systems in a one-dimensional, transient-thermal-compliance optimization problem, and converge in every optimization cycle.

Significance. If the proposed effective-anisotropy criterion is valid, it provides a simple, inexpensive way to guide semi-coarsening in STMG for heterogeneous diffusion problems, which is directly relevant to parallel-in-time algorithms for topology optimization. The paper's systematic comparison of rediscretisation methods and its demonstration that STMG works for the adjoint problem, including with a forward-in-time causal prolongation, are useful contributions. The authors are transparent about several limitations, explicitly noting the one-dimensional setting and the trial-and-error origin of the effective anisotropy parameter. However, the central heuristic was selected from the same family of test problems used to validate it, and the optimization demonstration replaces the per-design criterion with a global-extrema variant; these gaps materially limit the strength of the general claim and should be addressed before publication.

major comments (3)
  1. [Section 4, Eq. (20), Table 2] The effective anisotropy parameter λeff in Eq. (20) was selected through trial and error among the candidate expressions in Table 2 using the same six test problems (Table 1) that are then used to demonstrate that it "works well". All six problems share the design-field family in Eq. (23) with α = 10, the artificial heat load in Eq. (15), the same damped pointwise Jacobi smoother with ω = 1/2, and uniform Cartesian meshes. The claim that λeff reliably determines the coarsening direction is therefore an in-sample statement, and the reported range of intersection points in Figure 5 is not an independent estimate of predictive performance. Because Algorithm 3 applies λeff recursively at every level, this selection-on-validation effect is load-bearing for the central claim. The authors should either test λeff on problems outside this family (for example, multiple alternating features, greyscale transitions, different smoothers or heat loads), or explicitly scope the claim to the tested configuration; the concluding caveat that the authors do not know whether the definition works for other discretisations and multigrid components should be reflected in the abstract and at the point where the criterion is proposed.
  2. [Section 7, Eqs. (42)-(44)] The topology-optimization demonstration does not actually use Eq. (20) as stated. In Eqs. (42)-(44), the per-design extrema over element diffusivities are replaced by global extrema of DSIMP over the full range χ ∈ [0,1], which are simply the pure-material diffusivities. This freezes the coarsening path for every optimization cycle and makes the criterion independent of the evolving design. Consequently, the convergence results in Figure 13 validate a different, less local coarsening criterion, not the central per-design λeff rule of Algorithm 3. The manuscript should either recompute λeff from the current design each cycle and report whether the frozen path differs from the adaptive one, or explicitly state that the optimization study evaluates only the frozen-extrema variant.
  3. [Sections 5-7] All numerical evidence in the paper is restricted to one spatial dimension, uniform Cartesian space-time meshes, and a single smoother (damped pointwise Jacobi with ω = 1/2). The rediscretisation comparison in Section 5.3 attributes the superiority of the resistivity-averaging method to the fact that one-dimensional thermal resistances combine in series, which implies that the ranking of reassembly methods is dimension-dependent. The abstract and intro frame the methods as "suitable for topology optimisation" of transient heat conduction, which in practice is a two- or three-dimensional problem. The current evidence does not establish transferability to higher dimensions or to other smoothers, so the claims of suitability should be explicitly scoped to the one-dimensional configuration, or supplemented by at least one two-dimensional test case.
minor comments (5)
  1. [Section 4, Table 2] The comparison of candidate expressions for λeff is reported only qualitatively as "Yes" or "No" without giving the actual ranges of intersection points; reporting the quantitative ranges, as done for the chosen expression in Figure 5, would make the selection argument more convincing.
  2. [Section 5.1, Eqs. (9), (25), (26)] The stencil matrices in Eqs. (9), (25), and (26) appear to be typeset with reversed brackets and unclear orientation, making it difficult to identify which stencil direction corresponds to space and which to time; please fix the LaTeX formatting.
  3. [Section 6.1, Figure 11] The statement that the M ≤ 8 and M = 16 curves are identical for the CR-method is not visible from the figure alone; adding markers or separating the curves would make this claim verifiable.
  4. [Section 7.3, Figure 13] The vertical scale of Figure 13 differs between panels (a) and (b), which reach about 20 cycles, and panel (c), which reaches about 80 cycles; a common scale or explicit annotation would ease cross-panel comparison.
  5. [Section 2.2, Eq. (5)] The all-at-once matrix in Eq. (5) is large and presented without row-block annotations; marking the Dirichlet row and the C/Δt + K blocks would improve readability.

Circularity Check

1 steps flagged · score 6.0 of 10

The λeff criterion in Eq. 20 is selected by trial and error on the same six 1D problems used to validate it, so its reliability claim is an in-sample fit rather than an independent prediction.

  1. fitted input called prediction [Section 4, Eq. (20), Table 2, Figure 5; Section 8 Conclusion]
    "Through trial and error it was found that the following “effective anisotropy parameter” works well as an indicator for when to use x- or t-coarsening when considering high-contrast problems: λeff = sqrt(min_e λe max_e λe) (20). ... All the tested expressions are listed in Table 2, along with a short summary of the findings of the tests. ... There is only one expression which works well for both equal and unequal amounts of each material, which is the one at the top of the list. As such, this expression was adopted as being the definition of λeff."

    Eq. (20) was selected by testing the candidate expressions in Table 2 on the six problems of Table 1 and keeping the one whose x- and t-coarsening convergence-factor intersections fell in the narrowest range. Figure 5 then presents the same six two-grid experiments as the demonstration that λeff is ‘a reasonably reliable indicator’. The selection criterion and the validation criterion are the same, so the claim that Eq. (20) works well is an in-sample report of the fit, not an independent prediction. The paper discloses the trial-and-error origin and its limited scope, and later sections give some indirect support, so the circularity is partial rather than total.

full rationale

The main derivation chain is otherwise self-contained. The smoother, coarsening types, prolongation/restriction operators, and rediscretisation methods are standard multigrid components and are not defined in terms of the target convergence results. The comparison of rediscretisation methods (Section 5) and the small-feature coarsening strategy (Section 6) are independent empirical studies, and the Section 7 optimization is a separate robustness demonstration. No load-bearing self-citation occurs: the authors’ own [41] is cited only as a prior Parareal approach, and [12] is the code base. The one genuine circular step is the λeff indicator: Eq. (20) was chosen from the candidates in Table 2 by trial and error on exactly the six problems later used to demonstrate its reliability (Table 1, Figure 5), so the narrow range of intersection points is the selection metric being reported as confirmation. The paper’s own hedges — “the authors do not know if the presented definition of the effective anisotropy parameter will work reliably for other choices of discretisations and multigrid method components,” and the replacement of per-design min/max by global extrema in Eqs. (42)–(44) — further limit the support for the general claim. Because this is a disclosed, empirical selection-on-validation rather than a definitional or self-citation chain, the score is 6 (partial circularity) rather than higher.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The paper's central claims rest on an empirical heuristic (λeff) and several hand-chosen solver parameters, rather than on a derivation from first principles. No new physical entities are introduced, and the background discretisation is standard. The main burden is the representativeness of the test problem class used to select the heuristic.

free parameters (5)
  • Damping factor ω = 0.5
    Hand-chosen in Section 3.1; observed to work reasonably well, but convergence behavior is sensitive to it.
  • Coarsening threshold λcrit = 0.25
    Set in Sections 5.2 and 7.2; approximates the crossover between x- and t-coarsening for the considered method.
  • Smoothing steps ν = 5 (20 in Sections 6-7)
    Default 5 in Section 3.1; raised to 20 in later tests to stabilize the spatial-resolution strategy.
  • Spatial resolution threshold M = 8 to 128, varying with feature size
    User-defined in Algorithm 4; optimal M correlates with feature size F, so it is a problem-dependent free parameter.
  • Artificial heat load parameters = amplitude 10^6 W/m^3, frequency 200 in Eq. (15)
    Chosen to expose many frequencies; the paper notes convergence estimates depend on the heat load, so this is a hand-picked test input.
assumptions (5)
  • standard math Galerkin FE with linear elements and backward Euler gives a valid discretisation of the heat equation
    Used throughout Section 2; standard discretisation, not proved.
  • domain assumption Pointwise damped Jacobi smoothing with ω=1/2 is a stable smoother for the multigrid hierarchy
    Assumed in Section 3.1; observed to work on test problems, but the paper notes the BP-method becomes unstable, so stability is not guaranteed.
  • ad hoc to paper The artificial heat load q in Eq. (15) is representative of the frequency content needed to measure worst-case multigrid convergence
    The paper defines this load to obtain pessimistic convergence factor estimates; the validity of this choice for general problems is not established.
  • ad hoc to paper The effective anisotropy parameter λeff = sqrt(min λ max λ) is a valid scalar summary of a heterogeneous λ field
    Introduced in Section 4 based on trial and error; no derivation, and the authors state it may not work for other smoothers or discretisations.
  • domain assumption For the adjoint problem, a forward-in-time prolongation operator remains effective despite the reversed time direction
    Observed in Section 7.3; no explanation beyond empirical observation, and the paper notes it is not due to source scaling or interpolation direction for bilinear.

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Pith. "Pith review of Space-Time Multigrid Methods Suitable for Topology Optimisation of Transient Heat Conduction." pith.science (2026). https://pith.science/paper/6NZW6MCU

@misc{pith2026250510168,
  author       = {Pith},
  title        = {Pith review of: Space-Time Multigrid Methods Suitable for Topology Optimisation of Transient Heat Conduction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6NZW6MCU}},
  note         = {Machine review of arXiv:2505.10168}
}
read the original abstract

This paper presents Space-Time MultiGrid (STMG) methods which are suitable for performing topology optimisation of transient heat conduction problems. The proposed methods use a pointwise smoother and uniform Cartesian space-time meshes. For problems with high contrast in the diffusivity, it was found that it is beneficial to define a coarsening strategy based on the geometric mean of the minimum and maximum diffusivity. However, other coarsening strategies may be better for other smoothers. Several methods of discretising the coarse levels were tested. Of these, it was best to use a method which averages the thermal resistivities on the finer levels. However, this was likely a consequence of the fact that only one spatial dimension was considered for the test problems. A second coarsening strategy was proposed which ensures spatial resolution on the coarse grids. Mixed results were found for this strategy. The proposed STMG methods were used as a solver for a one-dimensional topology optimisation problem. In this context, the adjoint problem was also solved using the STMG methods. The STMG methods were sufficiently robust for this application, since they converged during every optimisation cycle. It was found that the STMG methods also work for the adjoint problem when the prolongation operator only sends information forwards in time, even although the direction of time for the adjoint problem is backwards.

Figures

Figures reproduced from arXiv: 2505.10168 by the authors.

Figure 1
Figure 1. Illustrations of the three different types of coarsening which are defined in Section 3.1. Algorithm 1 VCycle(l, u (l) , b (l) ) 1: if l = Nl then 2: Solve for u (l) in the equation J (l)u (l) = b (l) 3: return u(l) 4: else 5: Perform ν smoothing steps on J (l)u (l) = b (l) 6: b (l+1) ← R (l)  b (l) − J (l)u (l)  7: z (l+1) ← VCycle(l + 1, 0, b (l+1)) 8: u (l) ← u (l) + P (l) z (l+1) 9: Perform ν smoothing steps o… view at source ↗
Figure 2
Figure 2. Imposed heat load, q(x, t), used for the test problems considered in this paper, as defined in Equation (15). The initial guess for the vector of unknowns is u = 0. The relative residual after n cycles is defined as: rn = kJun − bk kbk (12) where un is u after the n th cycle and k·k is the Euclidian norm. The multigrid method is terminated after the n th cycle if the following condition is true: rn < 10−9 or rn > 10… view at source ↗
Figure 3
Figure 3. Estimated convergence factors of the space-time m [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Design field, χ, defined for problem 3 in Section 4. To demonstrate that λeff works well as an indicator for which coarsening type to choose, a series of test problems with moderate/high contrast in the material parameters are considered. In each problem, the design fi…
Figure 5
Figure 5. Figure 5: Estimated convergence factors for the two-grid sp [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: The design field, χ, which is defined in Equation (35), with L = 0.1 m and F = 0.1. The considered STMG method is applied to the above test problems where Nl (the number of levels) is varied between 2 and 10. For this, the coarsening strategy in Algorithm 3 is used, an…
Figure 7
Figure 7. Figure 7: Results of the tests of different rediscretisation methods applied to problem 7 as defined in Section 5.2, which is a problem where all spatial features are easy to resolve. These rediscretisation methods are defined in Section 5.1. The vertical line at 4 levels is the…
Figure 8
Figure 8. Figure 8: Results of the tests of different rediscretisation methods applied to problem 8 as defined in Section 5.2, which considers two conducting regions separated by a thin insulating region. These rediscretisation methods are defined in Section 5.1. The rediscretisation meth…
Figure 9
Figure 9. Figure 9: Results of the tests of different rediscretisation methods applied to problem 9 as defined in Section 5.2, which considers two insulating regions separated by a thin conducting region. These rediscretisation methods are defined in Section 5.1. The rediscretisation meth…
Figure 10
Figure 10. Figure 10: Example of a coarsening path which can be chosen by [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: Results of the coarsening strategy defined in Algo [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: Evolution of the design field over the optimisatio [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]
Figure 13
Figure 13. Figure 13: Number of multigrid cycles per optimisation cycl [PITH_FULL_IMAGE:figures/full_fig_p026_13.png]

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Reference graph

Works this paper leans on

64 extracted references · 38 canonical work pages · cited by 1 Pith paper

  1. [1]

    M. P . Bendsøe, O. Sigmund, Topology optimization: theor y, methods, and applications, Springer V erlag, 2003

    work page 2003

  2. [2]

    J. D. Deaton, R. V . Grandhi, A survey of structural and mul tidisciplinary continuum topology optimization: post 2000, Structural and Multidisciplinary Optimization 49 (2 014) 1–38

    work page 2000

  3. [4]

    Zhuang, Z

    C. Zhuang, Z. Xiong, A global heat compliance measure bas ed topology optimization for the tran- sient heat conduction problem, Numerical heat transfer. Pa rt B, Fundamentals 65 (2014) 445–471. doi:10.1080/10407790.2013.873309

    arXiv 2014

  4. [5]

    Dbouk, A review about the engineering design of optima l heat transfer sys- tems using topology optimization, Applied Thermal Enginee ring 112 (2017) 841–854

    T. Dbouk, A review about the engineering design of optima l heat transfer sys- tems using topology optimization, Applied Thermal Enginee ring 112 (2017) 841–854. URL: https://www.sciencedirect.com/science/article/pii/S135943111632645X. doi:https://doi.org/10.1016/j.applthermaleng.2016.10.134

    doi:10.1016/j.applthermaleng.2016.10.134 2017

  5. [6]

    S. Wu, Y . Zhang, S. Liu, Transient thermal dissipation e fficiency based method for topology opti- mization of transient heat conduction structures, Interna tional Journal of Heat and Mass Transfer 170 (2021) 121004. URL: https://www.sciencedirect.com/science/article/pii/S0017931021001071. doi:https://doi.org/10.1016/j.ijheatmasstransfer.2021.121004

    work page arXiv 2021

  6. [7]

    S. Wu, Y . Zhang, S. Liu, Topology optimization for minimi zing the maximum temperature of transient heat conduction structure, Structural and Multidisciplinary O ptimization 60 (2019) 69–82

    work page 2019

  7. [8]

    T. Zeng, H. Wang, M. Y ang, J. Alexandersen, Topology opti mization of heat sinks for instantaneous chip cooling using a transient pseudo-3d thermofluid model, Inte rnational Journal of Heat and Mass Transfer 154 (2020). doi:10.1016 /j.ijheatmasstransfer.2020.119681

    arXiv 2020

  8. [9]

    P . Coffin, K. Maute, A level-set method for steady-state and transie nt natural convection problems, Structural and Multidisciplinary Optimization 53 (2016) 1047–1067. d oi:10.1007/s00158-015-1377-y

    doi:10.1007/s00158-015-1377-y 2016

Show all 64 references
  1. [10]

    Elesin, B

    Y . Elesin, B. S. Lazarov, J. S. Jensen, O. Sigmund, Desig n of robust and e fficient photonic switches using topology optimization, Photonics and Nanostructures - Fun damentals and Applications 10 (2012) 153–165. doi:https://doi.org/10.1016/j.photonics.2011.10.003

    2012 doi

  2. [11]

    Mukherjee, D

    S. Mukherjee, D. Lu, B. Raghavan, P . Breitkopf, S. Dutta, M. Xiao, W . Zhang, Accelerating large-scale topology optimization: state-of-the-art and challenges, Archives of Computational Methods in Engineering (2021) 1–23. doi:10.1007/s11831-021-09544-3

    2021 doi

  3. [12]

    O. Amir, N. Aage, B. S. Lazarov, On multigrid-cg for e fficient topology optimization, Structural and Multidis- ciplinary Optimization 49 (2014) 815–829

    2014

  4. [13]

    N. Aage, E. Andreassen, B. S. Lazarov, Topology optimiz ation using petsc: An easy-to-use, fully parallel, open source topology optimization framework, Structural and Mu ltidisciplinary Optimization 51 (2015) 565–572

    2015

  5. [14]

    Herrero-Pérez, P

    D. Herrero-Pérez, P . J. Martínez Castejón, Multi-gpu a cceleration of large-scale density-based topology opti- mization, Advances in Engineering Software 157-158 (2021) 103006. doi:10.1016/j.advengsoft.2021.103006

    2021

  6. [15]

    Peetz, A

    D. Peetz, A. Elbanna, On the use of multigrid preconditi oners for topology optimization, Structural and Multi- disciplinary Optimization 63 (2021) 835–853. doi:10.1007 /s00158-020-02750-w

    2021

  7. [16]

    M. R. Hestenes, E. Stiefel, Methods of conjugate gradie nts for solving linear systems, Journal of Research of the National Bureau of Standards 49 (1952) 409–436. 28

    1952

  8. [17]

    Y . Saad, M. H. Schultz, Gmres: A generalized minimal res idual algorithm for solving nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computi ng 7 (1986) 856–869. doi:10.1137 /0907058

    1986

  9. [18]

    Alexandersen, O

    J. Alexandersen, O. Sigmund, N. Aage, Large scale three -dimensional topology optimisation of heat sinks cooled by natural convection, International Journal of Heat and Mass Transfer 100 (2016) 876–891. doi:10.1016/j.ijheatmasstransfer.2016.05.013

    2016 doi

  10. [19]

    N. Aage, E. Andreassen, B. S. Lazarov, O. Sigmund, Giga- voxel computational morphogenesis for structural design, Nature 550 (2017) 84–86. doi:10.1038 /nature23911

    2017

  11. [20]

    L. C. Høghøj, D. R. Nørhave, J. Alexandersen, O. Sigmund , C. S. Andreasen, Topology optimiza- tion of two fluid heat exchangers, International Journal of H eat and Mass Transfer 163 (2020) 120543. doi:10.1016/j.ijheatmasstransfer.2020.120543

    2020

  12. [21]

    E. A. Trä ff, O. Sigmund, N. Aage, Topology optimization of ultra high re solution shell structures, Thin-Walled Structures 160 (2021) 107349. doi:10.1016 /j.tws.2020.107349

    2021

  13. [22]

    Herrero-Pérez, S

    D. Herrero-Pérez, S. G. Picó-Vicente, A parallel geome tric multigrid method for adaptive topology optimization, Structural and Multidisciplinary Optimization 66 (2023) 2 25. doi:10.1007/s00158-023-03675-w

    2023 doi

  14. [23]

    S. Luo, F. Y ang, Y . Wang, An e fficient isogeometric topology optimization based on the adap - tive damped geometric multigrid method, Advances in Engine ering Software 196 (2024) 103712. doi:10.1016/j.advengsoft.2024.103712

    2024

  15. [24]

    Y . Zhou, C. Y e, Y . Liu, S. Fu, E. T. Chung, A robust solver f or large-scale heat transfer topology optimization,

  16. [25]

    B. S. Lazarov, Topology optimization using multiscale finite element method for high-contrast media, in: I. Lirkov, S. Margenov, J. Wa ´sniewski (Eds.), Large-Scale Scientific Computing, Spring er Berlin Heidelberg, Berlin, Heidelberg, 2014, pp. 339–346

    2014

  17. [26]

    Alexandersen, B

    J. Alexandersen, B. S. Lazarov, Topology optimisation of manufacturable microstruc- tural details without length scale separation using a spect ral coarse basis precondi- tioner, Computer Methods in Applied Mechanics and Engineer ing 290 (2015) 156–182. URL: https://www.scienc...

    2015 doi

  18. [27]

    Alexandersen, B

    J. Alexandersen, B. S. Lazarov, Tailoring Macroscale R esponse of Mechanical and Heat Transfer Systems by Topology Optimization of Microstructural Details, Springer International Publishing, Cham, 2015, pp. 267–288. doi:10.1007/978-3-319-18320-6_15

    2015 doi

  19. [28]

    Zambrano, S

    M. Zambrano, S. Serrano, B. S. Lazarov, J. Galvis, Fast m ultiscale contrast independent preconditioners for linear elastic topology optimization problems, Journa l of Computational and Applied Mathematics 389 (2021) 113366. URL: https://www.sciencedirect.com/science/article/p...

    2021

  20. [29]

    Kristiansen, N

    H. Kristiansen, N. Aage, An open-source framework for l arge-scale transient topology optimization using petsc, Structural and Multidisciplinary Optimization 65 (2022) 2 95. doi:10.1007/s00158-022-03312-y

    2022 doi

  21. [30]

    M. J. Gander, 50 years of time parallel time integration , in: T. Carraro, M. Geiger, S. Körkel, R. Rannacher (Eds.), Multiple Shooting and Time Domain Decomposition Me thods, Springer International Publishing, Cham, 2015, pp. 69–113

    2015

  22. [31]

    B. W . Ong, J. B. Schroder, Applications of time parallel ization, Computing and Visualization in Science 23 (2020) 11. doi:10.1007 /s00791-020-00331-4. 29

    2020

  23. [32]

    Lions, Y

    J.-L. Lions, Y . Maday, G. Turinici, Résolution d’edp pa r un schéma en temps pararéel, Comptes Rendus de l’Académie des Sciences - Series I - Mathem atics 332 (2001) 661–

    2001

  24. [33]

    Friedho ff, R

    S. Friedho ff, R. D. Falgout, T. Kolev, S. MacLachlan, J. B. Schroder, A mul tigrid-in-time algorithm for solving evolution equations in parallel, Technical Report, Lawren ce Livermore National Lab.(LLNL), Livermore, CA (United States), 2012

    2012

  25. [34]

    D. E. Womble, A time-stepping algorithm for parallel co mputers, SIAM Journal on Scientific and Statistical Computing 11 (1990) 824–837. URL: https://doi.org/10.1137/0911049. doi:10.1137/0911049

    1990 doi

  26. [35]

    Emmett, M

    M. Emmett, M. Minion, Toward an e fficient parallel in time method for partial di fferential equations, Commu- nications in Applied Mathematics and Computational Scienc e 7 (2012) 105–132

    2012

  27. [36]

    Hahne, B

    J. Hahne, B. Polenz, I. Kulchytska-Ruchka, S. Friedho ff, S. Ulbrich, S. Schöps, Parallel-in-time optimization of induction motors, Journal of Mathematics in Industry 13 (20 23) 6

  28. [37]

    X. Du, M. Sarkis, C. E. Schaerer, D. B. Szyld, Inexact and truncated parareal-in-time krylov subspace methods for parabolic optimal control problems, Electronic Transa ctions on Numerical Analysis 40 (2013) 36–57

    2013

  29. [38]

    Götschel, M

    S. Götschel, M. L. Minion, Parallel-in-time for parabo lic optimal control problems using pfasst, in: P . E. Bjørstad, S. C. Brenner, L. Halpern, H. H. Kim, R. Kornhuber, T. Rahman, O. B. Widlund (Eds.), Domain Decomposition Methods in Science and Engineering XXIV , Springer I...

    2018

  30. [39]

    N. R. G. S. Günther, J. B. Schroder, A non-intrusive para llel-in-time approach for simulta- neous optimization with unsteady pdes, Optimization Metho ds and Software 34 (2019) 1306–

    2019

  31. [40]

    M. J. B. Theulings, R. Maas, L. Noël, F. van Keulen, M. Lan gelaar, Reducing time and memory require- ments in topology optimization of transient problems, Inte rnational Journal for Numerical Methods in En- gineering 125 (2024) e7461. URL: https://onlinelibrary.wiley.com/doi/...

    2024 doi

  32. [41]

    Appel, J

    M. Appel, J. Alexandersen, One-shot parareal approach for topology optimisation of transient heat flow, 2024. URL: https://arxiv.org/abs/2411.19030. arXiv:2411.19030

    2024 arXiv

  33. [42]

    Hackbusch, Parabolic multi-grid methods, in: R

    W . Hackbusch, Parabolic multi-grid methods, in: R. Glo winski, J.-L. Lions (Eds.), Computing methods in applied sciences and engineering VI, Elsevier science publ ishers B.V ., 1983, pp. 189–197

    1983

  34. [43]

    R. D. Falgout, S. Friedho ff, T. V . Kolev, S. P . MacLachlan, J. B. Schroder, S. V andewalle, Multigrid methods with space–time concurrency, Computing and Visualization in Science 18 (2017) 123–143. doi:10.1007 /s00791- 017-0283-9

    2017

  35. [44]

    Horton, S

    G. Horton, S. V andewalle, A space-time multigrid metho d for parabolic partial di fferential equations, SIAM Journal on Scientific Computing 16 (1995) 848–864

    1995

  36. [45]

    S. R. Franco, F. J. Gaspar, M. A. Villela Pinto, C. Rodrig o, Multigrid method based on a space-time approach with standard coarsening for parabolic problems, Applied Mathematics and Computation 317 (2018) 25–34. URL: https://www.sciencedirect.com/science/article/pii/S00963003...

    2018 doi

  37. [46]

    M. J. Gander, M. Neumüller, Analysis of a new space-time parallel multigrid algorithm for parabolic problems, SIAM Journal on Scientific Computing 38 (2016) A2173–A2208. URL: https://doi.org/10.1137/15M1046605. doi:10.1137/15M1046605. 30

    2016 doi

  38. [47]

    H. Li, M. F. Wheeler, Space-time geometric multigrid me thod for nonlinear advection–di ffusion problems, Applicable Analysis 101 (2022) 4194–4202. URL: https://doi.org/10.1080/00036811.2022.2039387. doi:10.1080/00036811.2022.2039387

    2022

  39. [48]

    Margenberg, P

    N. Margenberg, P . Munch, A space-time multigrid method for space-time finite element discretizations of parabolic and hyperbolic pdes, 2024. URL: https://arxiv.org/abs/2408.04372. arXiv:2408.04372

    2024 arXiv

  40. [49]

    Schwalsberger, A Space-Time Parallel Multigrid Met hod for the Transient Eddy-Current Equation, Master’s thesis, Johannes Kepler University Linz, 2018

    M. Schwalsberger, A Space-Time Parallel Multigrid Met hod for the Transient Eddy-Current Equation, Master’s thesis, Johannes Kepler University Linz, 2018. URL: https://epub.jku.at/download/pdf/3140299.pdf

    2018

  41. [50]

    Neumüller, M

    M. Neumüller, M. Schwalsberger, A parallel space-time multigrid method for the eddy-current equation, 2019. URL: https://arxiv.org/abs/1911.09548. arXiv:1911.09548

    2019 arXiv

  42. [51]

    M. P . Bendsøe, Optimal shape design as a material distri bution problem, Structural optimization 1 (1989) 193–202

    1989

  43. [52]

    G. I. Rozvany, M. Zhou, T. Birker, Generalized shape opt imization without homogenization, Structural opti- mization 4 (1992) 250–252

    1992

  44. [53]

    Moucha, R

    R. Moucha, R. C. Bailey, An accurate and robust multigri d algorithm for 2d for- ward resistivity modelling, Geophysical Prospecting 52 (2 004) 197–212. URL: https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1365-2478.2004.00411.x. doi:https://doi.org/10.1111/j.1365-2478.2004.00411.x

    2004

  45. [54]

    P . O. Frederickson, Totally parallel multilevel algor ithms, Technical Report, Research Institute for Advanced Computer Science, 1988

    1988

  46. [55]

    Wesseling, Theoretical and practical aspects of a mu ltigrid method, SIAM Journal on Scientific and Statistical Computing 3 (1982) 387–407

    P . Wesseling, Theoretical and practical aspects of a mu ltigrid method, SIAM Journal on Scientific and Statistical Computing 3 (1982) 387–407. URL: https://doi.org/10.1137/0903025. doi:10.1137/0903025

    1982 doi

  47. [56]

    Subramaniam, T

    V . Subramaniam, T. Dbouk, J.-L. Harion, Topology optim ization of conductive heat trans- fer devices: An experimental investigation, Applied Therm al Engineering 131 (2018) 390–

    2018

  48. [57]

    Bourdin, Filters in topology optimization, Interna tional Jour- nal for Numerical Methods in Engineering 50 (2001) 2143–215 8

    B. Bourdin, Filters in topology optimization, Interna tional Jour- nal for Numerical Methods in Engineering 50 (2001) 2143–215 8. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.116. doi:https: //doi.org/10.1002/nme.116. arXiv:https://onlinelibrary.wiley.com/doi/pdf/1...

    2001 doi

  49. [58]

    Svanberg, The method of moving asymptotes—a new meth od for structural opti- mization, International Journal for Numerical Methods in E ngineering 24 (1987) 359–

    K. Svanberg, The method of moving asymptotes—a new meth od for structural opti- mization, International Journal for Numerical Methods in E ngineering 24 (1987) 359–

    1987

  50. [59]

    Michaleris, D

    P . Michaleris, D. A. Tortorelli, C. A. Vidal, Tangent op erators and design sensitiv- ity formulations for transient non-linear coupled problem s with applications to elastoplas- ticity, International Journal for Numerical Methods in Eng ineering 37 (1994) 2471–

    1994

  51. [373]

    doi:https://doi.org/10.1002/nme.1620240207

    URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.1620240207. doi:https://doi.org/10.1002/nme.1620240207

    doi

  52. [411]

    doi:https://doi.org/10.1016/j.applthermaleng.2017.12.026

    URL: https://www.sciencedirect.com/science/article/pii/S1359431117361501. doi:https://doi.org/10.1016/j.applthermaleng.2017.12.026

    2017 doi

  53. [668]

    doi:https://doi.org/10.1016/S0764-4442(00)01793-6

    URL: https://www.sciencedirect.com/science/article/pii/S0764444200017936. doi:https://doi.org/10.1016/S0764-4442(00)01793-6

    doi

  54. [1321]

    doi:10.1080/10556788.2018.1504050

    URL: https://doi.org/10.1080/10556788.2018.1504050. doi:10.1080/10556788.2018.1504050. arXiv:https://doi.org/10.1080/10556788.2018.1504050

    2018

  55. [2025]

    arXiv:2410.06850

    URL: https://arxiv.org/abs/2410.06850. arXiv:2410.06850

    arXiv

  56. [2499]

    doi:https://doi.org/10.1002/nme.1620371408

    URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.1620371408. doi:https://doi.org/10.1002/nme.1620371408. 31

    doi

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