A Thermo-Mechanical Coupled Topology Optimization Method for Structures under Nonuniform Steady-state Thermal Boundary Conditions

RONG Xuanpei; LUO Weifeng; SHA Jiahui; LIAO Hewei; ZHANG Hailiang

doi:10.7643/issn.1672-9242.2026.07.003

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Equipment Environmental Engineering ›› 2026, Vol. 23 ›› Issue (7) : 22-34. DOI: 10.7643/issn.1672-9242.2026.07.003

Special Topic——Research on Application of Complex Dynamic Environment Analysis and Fine Adaptability Verification Technique for Precise Mother-Daughter UAV System

A Thermo-Mechanical Coupled Topology Optimization Method for Structures under Nonuniform Steady-state Thermal Boundary Conditions

RONG Xuanpei, LUO Weifeng, SHA Jiahui, LIAO Hewei, ZHANG Hailiang^*

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Abstract

The work aims to develop a thermo-mechanical coupled topology optimization method which considers both physical rationality and computational efficiency to address the parasitic effect caused by design-dependent thermal loads in low-density regions, resulting in blurred topological boundaries and insufficient optimization stability. In the proposed framework, structural compliance was minimized under a prescribed material volume constraint. A thermo-mechanical coupled topology optimization model under both mechanical loads and thermal loads was established. The influence of the interpolation relationships among the elastic modulus, thermal expansion coefficient and thermal conductivity on the optimization results was investigated. A multi-parameter SIMP-RAMP hybrid penalization model was proposed. In addition, a GPU-based degree-of-freedom-level matrix-free matrix-vector multiplication strategy was introduced to improve the computational efficiency of finite element analysis and sensitivity calculation. The numerical examples showed that the proposed interpolation model could weaken the contribution of non-physical thermal loads in low-density areas, reduce gray elements, improve the clarity of optimized topologies. The matrix-free parallel computing strategy could avoid explicit assembly of the global stiffness matrix, reduce memory consumption and improve the efficiency iterative solutions. The proposed method can eliminate shortcomings of traditional interpolation models under the action of design-dependent thermal loads, and enhance the stability, physical rationality and computational efficiency of thermo-mechanical coupling topology optimization, providing an effective approach for lightweight structural design in complex thermal environments.

Key words

thermo-mechanical coupling (TMC) / topology optimization (TO) / design-dependent thermal load (DDTL) / parasitic effect (PE) / matrix-free computation (MFC)

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RONG Xuanpei, LUO Weifeng, SHA Jiahui, LIAO Hewei, ZHANG Hailiang. A Thermo-Mechanical Coupled Topology Optimization Method for Structures under Nonuniform Steady-state Thermal Boundary Conditions[J]. Equipment Environmental Engineering. 2026, 23(7): 22-34 https://doi.org/10.7643/issn.1672-9242.2026.07.003

References

[1] ZHU J H, ZHOU H, WANG C, et al.A Review of Topology Optimization for Additive Manufacturing: Status and Challenges[J]. Chinese Journal of Aeronautics, 2021, 34(1): 91-110.
[2] RODRIGUES H, FERNANDES P.A Material Based Model for Topology Optimization of Thermoelastic Structures[J]. International Journal for Numerical Methods in Engineering, 1995, 38(12): 1951-1965.
[3] PEDERSEN P, PEDERSEN N L.Strength Optimized Designs of Thermoelastic Structures[J]. Structural and Multidisciplinary Optimization, 2010, 42(5): 681-691.
[4] GAO T, ZHANG W H.Topology Optimization Involving Thermo-Elastic Stress Loads[J]. Structural and Multidisciplinary Optimization, 2010, 42(5): 725-738.
[5] ZHANG W H, YANG J G, XU Y J, et al.Topology Optimization of Thermoelastic Structures: Mean Compliance Minimization or Elastic Strain Energy Minimization[J]. Structural and Multidisciplinary Optimization, 2014, 49(3): 417-429.
[6] DEATON J D, GRANDHI R V.Stress-Based Design of Thermal Structures via Topology Optimization[J]. Structural and Multidisciplinary Optimization, 2016, 53(2): 253-270.
[7] WU C, FANG J G, LI Q.Multi-Material Topology Optimization for Thermal Buckling Criteria[J]. Computer Methods in Applied Mechanics and Engineering, 2019, 346: 1136-1155.
[8] KRYSKO A V, AWREJCEWICZ J, PAVLOV S P, et al.Topological Optimization of Thermoelastic Composites with Maximized Stiffness and Heat Transfer[J]. Composites Part B: Engineering, 2019, 158: 319-327.
[9] ZHENG J, DING S N, JIANG C, et al.Concurrent Topology Optimization for Thermoelastic Structures with Random and Interval Hybrid Uncertainties[J]. International Journal for Numerical Methods in Engineering, 2022, 123(4): 1078-1097.
[10] JUNKER P, BALZANI D.A New Variational Approach for the Thermodynamic Topology Optimization of Hyperelastic Structures[J]. Computational Mechanics, 2021, 67(2): 455-480.
[11] CHUNG H, AMIR O, KIM H A.Level-Set Topology Optimization Considering Nonlinear Thermoelasticity[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 361: 112735.
[12] Multiscale Topology Optimization of Thermoelastic Structures Using the Level Set Method[C]//AIAA Scitech 2020 Forum. Virginia: AIAA, 2020.
[13] ZHU J H, LI Y, WANG F W, et al.Shape Preserving Design of Thermo-Elastic Structures Considering Geometrical Nonlinearity[J]. Structural and Multidisciplinary Optimization, 2020, 61(5): 1787-1804.
[14] SHI G H, GUAN C Q, QUAN D L, et al.An Aerospace Bracket Designed by Thermo-Elastic Topology Optimization and Manufactured by Additive Manufacturing[J]. Chinese Journal of Aeronautics, 2020, 33(4): 1252-1259.
[15] ALLAIRE G, BIHR M, BOGOSEL B.Support Optimization in Additive Manufacturing for Geometric and Thermo-Mechanical Constraints[J]. Structural and Multidisciplinary Optimization, 2020, 61(6): 2377-2399.
[16] PEDERSEN P, PEDERSEN N L.Interpolation/Penalization Applied for Strength Design of 3D Thermoelastic Structures[J]. Structural and Multidisciplinary Optimization, 2012, 45(6): 773-786.
[17] DEATON J D, GRANDHI R V.Stiffening of Restrained Thermal Structures via Topology Optimization[J]. Structural and Multidisciplinary Optimization, 2013, 48(4): 731-745.
[18] BORRVALL T, PETERSSON J.Large-Scale Topology Optimization in 3D Using Parallel Computing[J]. Computer Methods in Applied Mechanics and Engineering, 2001, 190(46/47): 6201-6229.
[19] PIKLE N K, SATHE S R, VYAVAHARE A Y.Low Occupancy High Performance Elemental Products in Assembly Free FEM on GPU[J]. Engineering with Computers, 2022, 38(3): 2189-2204.
[20] OYARZUN G, BORRELL R, GOROBETS A, et al.MPI-CUDA Sparse Matrix-Vector Multiplication for the Conjugate Gradient Method with an Approximate Inverse Preconditioner[J]. Computers & Fluids, 2014, 92: 244-252.
[21] LOPES P C F, PEREIRA A M B, CLUA E W G, et al. A GPU Implementation of the PCG Method for Large-Scale Image-Based Finite Element Analysis in Heterogeneous Periodic Media[J]. Computer Methods in Applied Mechanics and Engineering, 2022, 399: 115276.
[22] ISUPOV K.Multiple-Precision Sparse Matrix-Vector Multiplication on GPUs[J]. Journal of Computational Science, 2022, 61: 101609.
[23] FEI Y, RONG G D, WANG B, et al.Parallel L-BFGS-B Algorithm on GPU[J]. Computers & Graphics, 2014, 40: 1-9.
[24] SANFUI S, SHARMA D.GPU-Based Mesh Reduction Strategy Utilizing Active Nodes for Structural Topology Optimization[J]. Structures, 2023, 55: 570-586.
[25] WADBRO E, BERGGREN M.Megapixel Topology Optimization on a Graphics Processing Unit[J]. SIAM Review, 2009, 51(4): 707-721.
[26] GAO J Q, ZHOU Y S, HE G X, et al.A Multi-GPU Parallel Optimization Model for the Preconditioned Conjugate Gradient Algorithm[J]. Parallel Computing, 2017, 63: 1-16.
[27] SCHMIDT S, SCHULZ V.A 2589 Line Topology Optimization Code Written for the Graphics Card[J]. Computing and Visualization in Science, 2011, 14(6): 249-256.
[28] RATNAKAR S K, SANFUI S, SHARMA D.Graphics Processing Unit-Based Element-by-Element Strategies for Accelerating Topology Optimization of Three-Dimensional Continuum Structures Using Unstructured All- Hexahedral Mesh[J]. Journal of Computing and Information Science in Engineering, 2022, 22(2): 021013.
[29] MARTÍNEZ-FRUTOS J, MARTÍNEZ-CASTEJÓN P J, HERRERO-PÉREZ D. Efficient Topology Optimization Using GPU Computing with Multilevel Granularity[J]. Advances in Engineering Software, 2017, 106: 47-62.
[30] LIN H J, LIU H, WEI P.A Parallel Parameterized Level Set Topology Optimization Framework for Large-Scale Structures with Unstructured Meshes[J]. Computer Methods in Applied Mechanics and Engineering, 2022, 397: 115112.
[31] ZHU J H, GUO W J, ZHANG W H, et al.Integrated Layout and Topology Optimization Design of Multi-Frame and Multi-Component Fuselage Structure Systems[J]. Structural and Multidisciplinary Optimization, 2017, 56(1): 21-45.
[32] WANG F W, LAZAROV B S, SIGMUND O.Correction: On Projection Methods, Convergence and Robust Formulations in Topology Optimization[J]. Structural and Multidisciplinary Optimization, 2022, 65(10): 278.
[33] PEDERSEN N L.Maximization of Eigenvalues Using Topology Optimization[J]. Structural and Multidisciplinary Optimization, 2000, 20(1): 2-11.
[34] ZHU X F, ZHAO C, WANG X, et al.Temperature-Constrained Topology Optimization of Thermo-Mechanical Coupled Problems[J]. Engineering Optimization, 2019, 51(10): 1687-1709.
[35] ZHANG S S, LI H M, HUANG Y C.An Improved Multi-Objective Topology Optimization Model Based on SIMP Method for Continuum Structures Including Self-Weight[J]. Structural and Multidisciplinary Optimization, 2021, 63(1): 211-230.
[36] ZHU J H, ZHANG W H.Integrated Layout Design of Supports and Structures[J]. Computer Methods in Applied Mechanics and Engineering, 2010, 199(9/10/11/12): 557-569.
[37] GIVOLI D.A Tutorial on the Adjoint Method for Inverse Problems[J]. Computer Methods in Applied Mechanics and Engineering, 2021, 380: 113810.
[38] GUO X, ZHANG W S, ZHONG W L.Doing Topology Optimization Explicitly and Geometrically—A New Moving Morphable Components Based Framework[J]. Journal of Applied Mechanics, 2014, 81(8): 081009.
[39] ZHANG W S, SONG J F, ZHOU J H, et al.Topology Optimization with Multiple Materials via Moving Morphable Component (MMC) Method[J]. International Journal for Numerical Methods in Engineering, 2018, 113(11): 1653-1675.