Multi-material robust topology optimization for high-performance heat exchangers

doi:10.1007/s10483-026-3425-7

Abstract

Abstract:

To design an efficient and robust integrated heat exchanger configuration, this work develops a multi-material robust topology optimization (RTO) model, and simultaneously considers the distribution of solid, fluid, and metal foam. In the proposed RTO model, the inlet flow velocity and the metal foam porosity are both treated as random variables, while two pseudo-density fields function as design variables. The objective is to maximize both the heat exchange performance mean and thermal robustness, subject to the constraints that the flow dissipation mean $(μ J)$ and the structural average strain energy mean $(μ C)$ are less than their thresholds. To solve the RTO model efficiently, polynomial chaos expansion is adopted to propagate random uncertainties from the input variables to the multi-physical response of the heat exchanger. The two-dimensional (2D) multi-material-based RTO results demonstrate that compared with the deterministic topology optimization (DTO) configuration, when $μ J$ is constrained to 10 and 20 times that of the straight channel configuration, the heat exchange performance mean of the RTO configuration is improved by 13.85% and 18.01%, and the robustness is improved by 29.54% and 29.65%. Notably, the metal foam in the configuration obtained by the RTO is distributed around the flow channels. For the three-dimensional (3D) multi-material-based RTO, with $μ J$ and $μ C$ constrained to 0.8 and 1.2 times the corresponding values of the straight multi-channel configuration, the mean performance and robustness are improved by 33.35%-42.54% and 53.71%-59.88%, respectively. Furthermore, compared with the conventional bi-material RTO method, the 3D heat exchanger configuration obtained by the proposed multi-material RTO method possesses superior heat transfer performance mean and enhanced thermal robustness, while achieving lower flow dissipation and reduced average structural strain energy.

Key words: high-performance heat exchanger, multi-material robust topology optimization (RTO), two pseudo-density fields, thermal-fluid-structural, polynomial chaos expansion

2010 MSC Number:

O342

Zelin WANG, Zhenzhou LU, Yizhou CHEN, Jiangji QU, Xingcheng WANG. Multi-material robust topology optimization for high-performance heat exchangers. Applied Mathematics and Mechanics (English Edition), 2026, 47(9): 2089-2108.

TrendMD

Add to citation manager EndNote|Reference Manager|ProCite|BibTeX|RefWorks

URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3425-7

https://www.amm.shu.edu.cn/EN/Y2026/V47/I9/2089

Figures/Tables 15

Fig. 1

Table 1

Left-hand sides of Eqs. (29)–(31)"

Term	J	$T ¯ f$	C
$− ∂ Q ∂ p$	$u ⋅ ⁢ n$	0	0
$− ∂ P ∂ ∇ ⁢ u T ⋅ ⁢ n − ∂ Q ∂ u$	$(p + ⁢ u 2 2) ⁢ ⁢ n + (u ⋅ ⁢ n) ⁢ u$	0	0
$∂ P ∂ T − ∇ ⋅ ∂ P ∂ ∇ T$	0	γ	0
$− ∂ P ∂ ⁢ D + ∇ ⋅ ∂ P ∂ ∇ ⁢ D$	0	0	$2 ⁢ (⁢ D ⋅ ⁢ n) ⁢ ⁢ n$

Table 1

Fig. 2

Fig. 3

Table 2

Material properties of the solid and coolant water in this work"

Material	$c / (J ⋅ (kg ⋅ K) − 1)$	$k / (W ⋅ (m ⋅ K) − 1)$	$ρ / (kg ⋅ m − 3)$	$η / (N ⋅ s ⋅ m − 2)$	$E / (N ⋅ m − 2)$	ν
Solid	880	237	2 700	-	$7 × 1010$	0.33
Water	4 180	0.61	1 000	0.001	-	-

Table 2

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Table 3

Heat exchange performance comparison of configurations by bimaterial RTO[33] and three-component RTO"

RTO method	$J lim$	$C lim$	$μ P$ / $(W ⋅ m − 3)$	$σ P / (W ⋅ m − 3)$
Bimaterial RTO	1.0J₀	1.5C₀	15 254.4	1 296.1
Multi-material RTO	0.8J₀	1.2C₀	17 304.0	963.2
Bimaterial RTO	2.0J₀	1.5C₀	17 589.4	1 015.3
Multi-material RTO	1.6J₀	1.2C₀	20 025.6	804.4

Table 3

Fig. 12

[1]	NAIR, V., BABY, A., ANOOP, M. B., INDRAJITH, S., MURALI, M., and NAIR, M. B. A comprehensive review of air-cooled heat sinks for thermal management of electronic devices. International Communications in Heat and Mass Transfer, 159, 108055 (2024)
[2]	SHENG, L., WANG, H. B., ZHANG, C. F., and ZHANG, X. J. Experimental study on immersion cooling performance of a lithium-ion battery module for commercial/industrial energy storage at different ambient temperatures. International Journal of Thermal Sciences, 221, 110473 (2026)
[3]	ANITHA, L. and GIREESHA, B. J. Convective flow of Jeffrey nanofluid along an upright microchannel with Hall current and Buongiorno model: an irreversibility analysis. Applied Mathematics and Mechanics (English Edition), 44(9), 1613–1628 (2023) https://doi.org/10.1007/s10483-023-3029-6
[4]	ZHAO, Q. K., TAO, L. B., and XU, H. Analysis of periodic pulsating nanofluid flow and heat transfer through a parallel-plate channel in the presence of magnetic field. Applied Mathematics and Mechanics (English Edition), 44(11), 1957–1972 (2023) https://doi.org/10.1007/s10483-023-3048-7
[5]	ZHANG, X. L., JI, Z., WANG, J. F., and LV, X. Research progress on structural optimization design of microchannel heat sinks applied to electronic devices. Applied Thermal Engineering, 235, 121294 (2023)
[6]	VOCALE, P. and MORINI, G. L. Impact of cross-section geometry on microchannel heat sink performance in the presence of slip and temperature jump boundary conditions. International Journal of Heat and Mass Transfer, 251, 127408 (2025)
[7]	ZHANG, Z., ZHANG, G. M., ZHANG, Y., and TIAN, M. C. Experimental study on the effect of hydraulic diameter on the flow boiling characteristics in microchannels. International Journal of Heat and Mass Transfer, 241, 126736 (2025)
[8]	PAVITHRA, C. G., GIREESHA, B. J., SUSHMA, S., and GOWTHAM, K. J. Analysis of convective-radiative heat transfer in dovetail longitudinal fins with shape-dependent hybrid nanofluids: a study using the Hermite wavelet method. Applied Mathematics and Mechanics (English Edition), 46(2), 357–372 (2025) https://doi.org/10.1007/s10483-025-3218-9
[9]	WANG, X. Z., LEI, T. M., LIU, J. Q., WO, L., MAO, X., and XU, R. J. Heat transfer performance and optimization of a microchannel heat exchanger with insert louvered fins. Applied Thermal Engineering, 279, 127644 (2025)
[10]	WANG, Z. L., LAI, B. Z., WANG, H., XIAO, H. Y., and MING, P. W. Effects of micropore characteristics in the metal skeleton on heat and mass transfer in an open foam structure for thermal management in the hydrogen UAV. International Journal of Thermal Sciences, 179, 107628 (2022)
[11]	IASIELLO, M., BIANCO, N., CHIU, W. K. S., and NASO, V. Thermal conduction in open-cell metal foams: anisotropy and representative volume element. International Journal of Thermal Sciences, 137, 399–409 (2019)
[12]	HSU, C. T. A closure model for transient heat conduction in porous media. Journal of Heat Transfer, 121(3), 733–739 (1999)
[13]	YANG, X. H., KUANG, J. J., LU, T. J., HAN, F. S., and KIM, T. A simplistic analytical unit cell based model for the effective thermal conductivity of high porosity open-cell metal foams. Journal of Physics D: Applied Physics, 46(25), 255302 (2013)
[14]	WANG, F. Z. and LI, X. L. The stagnant thermal conductivity of porous media predicted by the random walk theory. International Journal of Heat and Mass Transfer, 107, 520–533 (2017)
[15]	HAN, H. T., HAN, Y. M., LIN, Y., WANG, C. M., KORVINK, J. G., and DENG, Y. B. Topology optimization of microchannel heat sinks for laminar flows of thermal-fluid. Applied Thermal Engineering, 270, 126153 (2025)
[16]	LI, L. J., XU, J. N., and LI, S. J. Topology optimization of microchannel structures for enhanced heat flow in liquid cooling garments. Physics of Fluids, 36(12), 127140 (2024)
[17]	YU, M. H., RUAN, S. L., GU, J. F., REN, M. K., LI, Z., WANG, X. Y., and SHEN, C. Y. Three-dimensional topology optimization of thermal-fluid-structural problems for cooling system design. Structural and Multidisciplinary Optimization, 62(6), 3347–3366 (2020)
[18]	WU, J. P., LIU, H. L., LI, C. C., LI, C. H., and XIE, G. N. Topological optimization and thermal performance of cold plates for lithium-ion battery with non-uniform heat sources. Applied Thermal Engineering, 254, 123922 (2024)
[19]	OZGUC, S., TEAGUE, T. F. G., PAN, L., and WEIBEL, J. A. Experimental study of topology optimized, additively manufactured microchannel heat sinks designed using a homogenization approach. International Journal of Heat and Mass Transfer, 209, 124108 (2023)
[20]	FRAGNITO, A., ANDREASEN, C. S., IASIELLO, M., MAURO, G. M., and BIANCO, N. A multi-step topology optimization framework of fin structures for accelerated melting in a phase change material (PCM)-filled cavity for thermal management. International Journal of Heat and Mass Transfer, 254, 127694 (2026)
[21]	OZGUC, S., TEAGUE, T. F. G., PAN, L., and WEIBEL, J. A. Experimental study of topology optimized, additively manufactured microchannel heat sinks designed using a homogenization approach. International Journal of Heat and Mass Transfer, 209, 124108 (2023)
[22]	HUANG, X. M., CHEN, Y., HOU, L., MIAO, C. M., and LI, Y. Stress-related multi-material structures topology optimization with gradient interfaces. Composite Structures, 365, 119176 (2025)
[23]	ZHANG, J. P., ZHAO, L., DONG, J. Y., LI, Y. X., LIU, B. X., ZHAO, Y. P., and WU, S. Y. Multi-material topology optimization of transient heat transfer structure based on element-free Galerkin method. International Communications in Heat and Mass Transfer, 167, 109383 (2025)
[24]	SAURABH, S., GUPTA, A., CHOWDHURY, R., and PODUGU, P. Robust topology optimization for transient dynamic response minimization. Computer Methods in Applied Mechanics and Engineering, 426, 117009 (2024)
[25]	ZHENG, J., CHEN, H., and JIANG, C. Robust topology optimization for structures under thermo-mechanical loadings considering hybrid uncertainties. Structural and Multidisciplinary Optimization, 65(1), 29 (2022)
[26]	BIANCO, N., CHERELLA, N., FRAGNITO, A., IASIELLO, M., and MAURO, G. M. Multi-material topology optimization of innovative microchannel heat sinks equipped with metal foams. International Journal of Heat and Mass Transfer, 222, 125201 (2024)
[27]	LEE, D., CHANG, S., and LEE, J. Generalized polynomial chaos expansion by reanalysis using static condensation based on substructuring. Applied Mathematics and Mechanics (English Edition), 45(5), 819–836 (2024) https://doi.org/10.1007/s10483-024-3108-8
[28]	BIANCO, N., FRAGNITO, A., IASIELLO, M., and MAURO, G. Topology optimization of a pseudo 3D microchannel heat sink: influence of the initial guess. Journal of Physics: Conference Series, 2685(1), 012004 (2024)
[29]	BORRVALL, T. and PETERSSON, J. Topology optimization of fluids in Stokes flow. International Journal for Numerical Methods in Fluids, 41(1), 77–107 (2003)
[30]	SHAH, R. K. and LONDON, A. L. Laminar Flow Forced Convection in Ducts, Academic Press, New York (1978)
[31]	RYU, J. H., CHOI, D. H., and KIM, S. J. Numerical optimization of the thermal performance of a microchannel heat sink. International Journal of Heat and Mass Transfer, 45(13), 2823–2827 (2002)
[32]	FARAHANI, S. D., MAMOEI, A. J., and ALIZADEH, A. Thermal performance of microchannel heat sink integrated with porous medium, slip coefficient and phase change material and machine learning approach. Journal of Energy Storage, 74, 109357 (2023)
[33]	WANG, Z. L. and LU, Z. Z. Robust thermal-fluid-structural topology optimization for porous heat exchangers in hydrogen aircrafts. Applied Thermal Engineering, 282, 128871 (2026)