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arxiv: 2603.26437 · v1 · submitted 2026-03-27 · ⚛️ physics.app-ph

A fractal geometry enhanced topology optimization design for high-performance liquid cooling plates

Zixu Han , Kairan Yang , Peng Zhang This is my paper

Pith reviewed 2026-05-14 22:33 UTC · model grok-4.3

classification ⚛️ physics.app-ph
keywords topology optimizationfractal geometryliquid cooling plateheat transfer areaconvective coolingdensity-based optimizationthermal performancepressure drop
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The pith

Adding fractal dimension to topology optimization produces liquid cooling plates with 46% more heat transfer area and 15-17 K lower temperatures.

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

The paper proposes fractal geometry topology optimization (FGTO), which adds fractal dimension as an extra design variable inside the standard density-based topology optimization framework used for liquid cooling plates. Conventional TO treats the domain as porous media and cannot directly optimize convective heat transfer because it lacks an explicit expression for the solid-fluid interface area. FGTO incorporates the fractal dimension, controlled by a parameter s, so that the objective function can directly penalize or reward the heat transfer area. This change produces more intricate solid topologies that separate solid from fluid more effectively and escape local optima. The resulting designs deliver 15.6 K lower average temperature and 16.9 K lower maximum temperature than conventional TO while trading off against higher pressure drop as s increases.

Core claim

Embedding fractal dimension as an additional design freedom into density-based topology optimization explicitly includes the convective heat transfer area in the objective function. The method therefore generates cooling-plate topologies with 46% larger heat transfer area than standard TO, yielding average and maximum temperature reductions of 15.6 K and 16.9 K respectively. Increasing the input parameter s that sets the fractal dimension further improves thermal performance at the expense of pressure drop, and the added sensitivity difference between solid and liquid phases promotes better solid-liquid separation.

What carries the argument

The fractal geometry topology optimization (FGTO) method, in which fractal dimension (controlled by input parameter s) is introduced as an extra design variable inside density-based TO to make the solid-liquid interface area explicit in the objective function.

If this is right

  • Raising the fractal parameter s improves thermal performance while increasing pressure drop, allowing tunable trade-offs between cooling and pumping power.
  • The intensified difference in objective-function sensitivity between solid and liquid phases helps the optimizer escape local optima and achieve clearer solid-liquid separation.
  • The resulting topologies are more complex than those from conventional TO and deliver measurably lower average and peak temperatures.
  • Direct optimization of heat transfer area becomes possible once fractal dimension is treated as a design variable.

Where Pith is reading between the lines

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

  • The same fractal-dimension variable could be added to topology optimization for other devices where interface area dominates performance, such as heat exchangers or porous reactors.
  • Real prototypes tested under turbulent flow would be needed to check whether the simulated gains survive manufacturing tolerances and actual fluid dynamics.
  • Tuning s could serve as a practical knob for balancing thermal resistance against hydraulic resistance in engineering design workflows.

Load-bearing premise

Varying the fractal parameter s will produce manufacturable structures whose simulated temperature reductions hold up in physical tests without being erased by unmodeled effects such as turbulence or fabrication limits.

What would settle it

Fabricate one conventional TO plate and one FGTO plate at the same s value, then measure their steady-state average and maximum surface temperatures under identical heat input and coolant flow rate; a difference near 15-17 K would support the claim.

Figures

Figures reproduced from arXiv: 2603.26437 by Kairan Yang, Peng Zhang, Zixu Han.

Figure 1
Figure 1. Figure 1: The FGTO model for liquid cooling plate design and performance manipulation. (a) application scenarios for liquid cooling plates, including power electronic components, CPUs/GPUs, and batteries [48- 50], (b) schematic of the 3D numerical calculation model for performance evaluation, (c) schematic of the 2D FGTO model [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of the conventional TO and FGTO methods. (a) conventional TO method, (b) the FGTO method incorporating porous medium model by fractal geometry theory [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the thermal objectives by the FGTO and conventional TO. The FGTO targets maximizing heat dissipation by convective heat transfer through lateral heat transfer area in porous media, while the conventional TO targets maximizing heat dissipation by heat conduction [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the experimental results in [62] and numerical results. (a) variation of average (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows the FGTO results at various s at wt=0.7. By varying s, the fractal dimension within the design domain can be modified according to Eq. (1), influencing the evolution of design variables and consequently yielding different structural topologies of the optimized liquid cooling plates. It can be observed from [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Variation of the dimensionless sensitivity of thermal objective function ߝ௧ with γ at different s [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Variation of the dimensionless objective functions with iteration steps by the FGTO at wt=0.7 and s=3000, 5000, 8000, and the conventional TO at wt=0.7. (a) dimensionless thermal objective function, (b) dimensionless hydraulic objective function [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Iteration and convergence processes of the FGTO and conventional TO. (a) the conventional TO at wt=0.7, (b) the FGTO at s=5000 and wt=0.7 [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The temperature contours of different liquid cooling plates at Tin=308.15 K, uin=0.2 m/s and heat flux of 10 W/cm2 . (a) temperature contours of the bottom surface on the heated plate, (b) temperature contours of the middle plane on the direction of thickness [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The results of the 3D performance evaluation at Tin=308.15 K, uin=0.2 m/s and heat flux of 10 W/cm2 . (a) average temperature and maximum temperature of the bottom surface and the Nusselt number, (c) pressure drop and the PEC [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Variation of the average temperature, maximum temperature, pressure drop and the PEC with inlet velocity. (a) average temperature, (b) maximum temperature, (c) pressure drop, (d) the PEC [PITH_FULL_IMAGE:figures/full_fig_p029_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of the effects of wt on the conventional TO and FGTO. (a) design variable field of the conventional TO results at various wt, (b) design variable field of the FGTO results at various wt at s=3000 [PITH_FULL_IMAGE:figures/full_fig_p031_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The FGTO results at s=8000, wt=0.8 and s=8000, wt=0.9 As discussed earlier, increasing either s or wt alone can enhance heat dissipation capabilities of the FGTO liquid cooling plates at the cost of higher flow resistance. s=8000 wt = 0.8 s=8000 wt = 0.9 wavy shapes  [PITH_FULL_IMAGE:figures/full_fig_p032_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The temperature contours on the bottom surface of the FGTO result at s=8000 and the conventional TO result at various wt, with inlet velocity of 0.2 m/s, inlet temperature of 308.15 K and heat flux of 10 W/cm2 3.4 Performance manipulation by the conventional TO and FGTO As discussed in section 3.3, simply varying wt by the conventional TO struggles to efficiently manipulate thermal and hydraulic performan… view at source ↗
Figure 15
Figure 15. Figure 15: Performance comparison of the optimized liquid cooling plates by the FGTO at s=500, 1000, 2000, 3000, 4000, 5000, 8000 at wt =0.7 and by the conventional TO at varying wt=0.3, 0.4, 0.5, 0.7, 0.8, 0.9, under the operating condition of Tin=308.15 K, uin=0.2 m/s and heat flux of 10 W/cm2 . (a) average temperature and pressure drop, (b) the Nusselt number and the PEC [PITH_FULL_IMAGE:figures/full_fig_p035_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Performance comparison of the liquid cooling plates by the FGTO method and conventional TO method in [24]. (a) the FGTO results and the conventional TO results, (b) average wall temperature, pressure drop and the PEC of the FGTO results and conventional TO results [PITH_FULL_IMAGE:figures/full_fig_p037_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Comparison of the conventional method in [64] and the FGTO method. (a) the TO results obtained by the conventional and FGTO methods, (b) variation of the maximum temperature of the bottom plate with flow rate, (c) variation of the pressure drop with flow rate, (d) variation of the PEC with flow rate 3.6 Feasibility and adaptability of the FGTO method Although the FGTO method can yield the optimized liquid… view at source ↗
Figure 18
Figure 18. Figure 18: The FGTO results for different solid to liquid thermal conductivity ratios K and coolant viscosities μ at s=5000 and wt=0.7, at inlet velocity of 0.01 m/s, inlet temperature of 308.15 K and heat flux of 1 W/cm2 . (a) design variable field at various K, (b) design variable field at various μ [PITH_FULL_IMAGE:figures/full_fig_p040_18.png] view at source ↗
read the original abstract

The density-based bi-objective topology optimization (TO) has been widely adopted in liquid cooling plate design, where the design domain is treated as porous media with porosity as the design variable. However, conventional TO method struggles to directly optimize the convective heat transfer due to its incapabilities of explicitly depicting the heat transfer area in objective function, which limits the optimization of thermal performance. In this study, a fractal geometry topology optimization (FGTO) method is proposed, which incorporates fractal dimension as an additional design freedom into the density-based TO framework. Different from the conventional TO methods, the FGTO explicitly describes the heat transfer area, and achieves a direct optimization of convective heat transfer through the objective function. Compared to the conventional TO, the FGTO achieves a more complex structural topology in the optimized liquid cooling plate with a 46% improvement in heat transfer area. The fractal dimension is manipulated by varying the input parameter s, and increasing s can improve thermal performance of the FGTO results at the cost of larger pressure drop. Superior thermal-hydraulic performance can be achieved by varying s, with the average and maximum temperatures of the FGTO results reduced by 15.6 K and 16.9 K, respectively, compared with those of the conventional TO results. The integration of fractal geometry into the TO intensifies the difference in objective function sensitivity between solid and liquid phases, which is conducive to facilitating solid-liquid separation and contributes to escape from local optimal solutions.

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

3 major / 2 minor

Summary. The manuscript proposes a fractal geometry topology optimization (FGTO) method that augments conventional density-based bi-objective TO for liquid cooling plates by treating fractal dimension as an additional design freedom controlled by input parameter s. This is claimed to enable explicit optimization of convective heat transfer area in the objective function, yielding more complex topologies with a 46% increase in heat transfer area and reductions of 15.6 K (average) and 16.9 K (maximum) in temperature relative to standard TO, at the expense of higher pressure drop; the approach is said to intensify solid-liquid phase sensitivity differences and help escape local optima.

Significance. If the numerical gains prove robust, the FGTO framework could provide a practical extension to topology optimization for thermal management devices, offering a route to higher-performance liquid cooling plates in electronics and power systems by directly targeting heat transfer area rather than relying solely on porosity variables.

major comments (3)
  1. [Abstract and Results] Abstract and Results section: The central claims of 46% heat transfer area improvement and temperature reductions (15.6 K average, 16.9 K maximum) are presented without mesh convergence studies, error bars on the CFD results, or explicit baseline details on the conventional TO implementation (including turbulence model and boundary conditions), which are load-bearing for establishing superiority.
  2. [Methodology] Methodology: The mechanism by which varying s controls fractal dimension and intensifies objective-function sensitivity between solid and liquid phases to facilitate escape from local optima is asserted but not supported by explicit equations, sensitivity derivations, or analysis showing how the modification alters the optimization landscape.
  3. [Validation] Validation: No physical prototypes, experimental flow-loop tests, or measured pressure-drop and temperature data are reported, leaving open whether the simulated gains persist under real turbulence, fabrication limits, or interface effects that are not captured in the idealized model.
minor comments (2)
  1. [Notation] The range and physical interpretation of parameter s should be stated explicitly in the early methodology paragraphs to aid reproducibility.
  2. [Figures] Topology figures would benefit from side-by-side quantitative overlays (e.g., area values) and clearer indication of fractal features versus conventional results.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the constructive and detailed comments, which have helped us identify areas for improvement in the manuscript. We provide point-by-point responses below and have revised the manuscript accordingly where feasible.

read point-by-point responses

  1. Referee: [Abstract and Results] The central claims of 46% heat transfer area improvement and temperature reductions (15.6 K average, 16.9 K maximum) are presented without mesh convergence studies, error bars on the CFD results, or explicit baseline details on the conventional TO implementation (including turbulence model and boundary conditions), which are load-bearing for establishing superiority.

    Authors: We agree that these elements are essential for rigorously establishing the reported improvements. In the revised manuscript, we will add mesh convergence studies confirming that the key metrics (heat transfer area and temperatures) are mesh-independent. Error bars derived from variations in numerical tolerances and solver settings will be included on the CFD results. We will also expand the Methodology section to explicitly document the conventional TO baseline, including the k-ε turbulence model, all boundary conditions, and solver settings used for comparison. These additions will appear in both the Methodology and Results sections. revision: yes

  2. Referee: [Methodology] The mechanism by which varying s controls fractal dimension and intensifies objective-function sensitivity between solid and liquid phases to facilitate escape from local optima is asserted but not supported by explicit equations, sensitivity derivations, or analysis showing how the modification alters the optimization landscape.

    Authors: We acknowledge that the original description was insufficiently detailed. The revised manuscript will include the explicit mathematical formulation of how the input parameter s modulates the fractal dimension within the density-based topology optimization framework. We will derive and present the sensitivity expressions of the objective function with respect to the design variables, demonstrating the intensified phase sensitivity between solid and liquid regions. Additionally, we will add an analysis (with supporting equations and a new figure) illustrating how this modification alters the optimization landscape to aid escape from local optima. revision: yes

  3. Referee: [Validation] No physical prototypes, experimental flow-loop tests, or measured pressure-drop and temperature data are reported, leaving open whether the simulated gains persist under real turbulence, fabrication limits, or interface effects that are not captured in the idealized model.

    Authors: The referee is correct that the study is purely numerical and does not include experimental validation. We will add a new subsection in the Conclusions explicitly discussing the limitations of the idealized CFD model, including assumptions on turbulence modeling and solid-liquid interfaces. We will also state that experimental fabrication and flow-loop testing of the optimized fractal topologies is planned as future work to verify performance under real conditions. However, such experiments require substantial additional resources and fabrication capabilities beyond the scope of this numerical development paper. revision: partial

standing simulated objections not resolved
  • Absence of experimental validation data (physical prototypes and flow-loop measurements), which cannot be generated within the current numerical study.

Circularity Check

0 steps flagged

No circularity: FGTO adds independent fractal parameter s to standard density-based TO without reducing results to fitted inputs or self-definitions

full rationale

The paper extends conventional density-based topology optimization by introducing fractal dimension as an explicit additional design variable controlled by input parameter s. Reported gains (46% heat-transfer-area increase and temperature drops of 15.6 K / 16.9 K) are obtained from comparative numerical simulations of the optimized topologies, not from any algebraic reduction, parameter fitting to the target metrics, or self-citation chain. No equations are shown that equate the objective-function sensitivity or area improvement directly to the definition of s itself. The method therefore remains self-contained against external simulation benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The claim rests on the standard porous-media model of density-based TO plus the new assumption that fractal dimension can be varied via s to directly control and optimize convective area.

free parameters (1)
  • s
    Input parameter varied to control fractal dimension and trade thermal performance against pressure drop.
axioms (1)
  • domain assumption Density-based TO treats the design domain as porous media with porosity as the design variable.
    Explicitly stated as the foundation of both conventional and FGTO methods.
invented entities (1)
  • fractal dimension as additional design freedom no independent evidence
    purpose: To explicitly describe and optimize heat transfer area in the objective function.
    Introduced as the core novelty of FGTO.

pith-pipeline@v0.9.0 · 5562 in / 1313 out tokens · 43477 ms · 2026-05-14T22:33:35.813265+00:00 · methodology

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