复杂热流承载结构拓扑优化设计研究

doi:10.3901/JME.2024.13.092

摘要/Abstract

摘要： 随着多领域高端装备的快速升级与发展，迫切需要面向复杂服役工况的高性能热流承载结构优化设计。然而，缺乏热流耦合物理场的高效求解手段以及考虑复杂边界条件的热流承载结构设计能力不足仍是制约装备更新换代速度和质量的两大关键问题。因此，提出了一种新的无网格粒子方法——等几何粒子流体动力学方法(NURBS-based particle hydrodynamics，NBPH)。通过配置真伪粒子，基于非均匀有理B样条(Non-uniform rational B-splines，NURBS)基函数的插值构建粒子通讯，缓解了偏微分方程数值求解质量与分析域离散化程度之间的强耦合关系，提升了算法的计算效率。其次，将所提出的NBPH方法与固体各向同性材料惩罚(Solid isotropic material with penalization，SIMP)方法结合，搭建了一种新的无网格法拓扑优化框架——NBPH-topology Optimization，用于复杂热流场景的优化设计。根据体素分布构造了粒子流动阻力场，实现了流场与结构场的关联，通过求解连续伴随灵敏度，指导结构拓扑的连续演化。为了验证NBPH-TO框架的有效性和鲁棒性，研究了液冷和风冷两大典型散热场景，完成了优化设计与实验验证，结果表明该研究为复杂热流承载结构拓扑优化设计提供了可行的解决方案和有效的数值工具。

关键词: NBPH, SIMP, 无网格法, 拓扑优化, 热流耦合

Abstract: With the rapid upgrading and development of high-end equipment, there is an urgent need for high-performance heat-flow coupling structure optimization design for complex service conditions. However, the lack of efficient means of solving heat-flow coupling field and the insufficient capability of designing heat-flow coupling structures considering complex boundary conditions are still the two key issues that restrict the speed and quality of equipment upgrading. Therefore, a new meshless particle method is proposed, non-uniform rational B-splines (NURBS) based particle hydrodynamics (NBPH), which achieves the alleviation of the strong coupling between the quality of the numerical solution of partial differential equations and the degree of discretization of the analytical domain through the configuration of real and pseudo particles and the interpolation based on the NURBS basis functions, which improves the computational efficiency of the algorithm. Secondly, the proposed NBPH method is combined with the SIMP method to build a new topology optimization framework based on meshless method, NBPH-Topology Optimization (TO), for optimization and design of complex heat flow scenarios. The particle flow resistance field is constructed based on the voxel distribution to realize the correlation between the flow field and the structural field, and the continuous evolution of the structural topology is guided by solving the continuous Adjoint Sensitivity. To verify the effectiveness and robustness of the NBPH-TO framework, two typical heat dissipation scenarios, liquid-cooled and air-cooled, are investigated, and the optimization design and experimental validation are completed. The results show that the study provides a feasible solution and an effective numerical tool for topology optimization design of the complex heat-flow coupling structures.

Key words: NBPH, SIMP, meshless method, topology optimization, heat flow coupling

中图分类号:

TG156

李宝童, 刘策, 洪军, 刘庆芳, 史萌, 李开泰. 复杂热流承载结构拓扑优化设计研究[J]. 机械工程学报, 2024, 60(13): 92-121.

LI Baotong, LIU Ce, HONG Jun, LIU Qingfang, SHI Meng, LI Kaitai. Topology Optimization Design of Complex Heat-flow Coupling Structures[J]. Journal of Mechanical Engineering, 2024, 60(13): 92-121.

参考文献

[1] Asmussen J, Alexandersen J, Sigmund O, et al. A “poor man's" approach to topology optimization of natural convection problems[J]. Structural and Multidisciplinary Optimization, 2019, 59(4)：1105-1124.
[2] Fawaz A, Hua Y, Le-Corre S, et al. Topology optimization of heat exchangers：A review[J]. Energy, 2022：124053.
[3] Yan Suna, Wang Fengwen, Hong Jun, et al. Topology optimization of microchannel heat sinks using a two-layer model[J]. International Journal of Heat and Mass Transfer, 2019, 143：118462.
[4] 裴元帅, 王定标, 王晓亮, 等. 基于拓扑优化的风冷热沉研究[J]. 机械工程学报, 2020, 56(16)：91-97. PEI Yuanshuai, WANG Dingbiao, WANG Xiaoliang, et al. Research on air-cooled heat sink based on topology optimization[J]. Journal of Mechanical Engineering, 2020, 56(16)：91-97.
[5] 赵曦, 刘义畅, 方喆, 等. 模具冷却通道截面拓扑优化设计[J]. 计算力学学报, 2019, 36(5)：597-602. ZHAO Xi, LIU Yichang, FANG Zhe, et al. Optimized design of cross-section topology of mold cooling channel[J]. Chinese Journal of Computational Mechanics, 2019, 36(5)：597-602.
[6] Hughes T J R, Cottrell J A, Bazilevs Y. Isogeometric analysis：CAD, finite elements, NURBS, exact geometry and mesh refinement[J]. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39-41)：4135-4195.
[7] LEON L B. A numerical approach to the testing of the fission hypothesis[J]. Astronomical Journal, 1977, 82：1013-1024.
[8] Gingold R A, Monaghan J J. Smoothed particle hydrodynamics：Theory and application to non-spherical stars[J]. Monthly Notices of the Royal Astronomical Society, 1977, 181(3)：375-389.
[9] Hu Xiangyu, Adams N A. An incompressible multi- phase SPH method[J]. Journal of Computational Physics, 2007, 227(1)：264-278.
[10] Härdi S, Schreiner M, Janoske U. Enhancing smoothed particle hydrodynamics for shallow water equations on small scales by using the finite particle method[J]. Computer Methods in Applied Mechanics and Engineering, 2019, 344：360-375.
[11] Lian Yanjian, Bui H H, Nguyen G D, et al. A general SPH framework for transient seepage flows through unsaturated porous media considering anisotropic diffusion[J]. Computer Methods in Applied Mechanics and Engineering, 2021, 387：114169.
[12] Duan Guangtao, Sakai M. An enhanced semi-implicit particle method for simulating the flow of droplets with free surfaces[J]. Computer Methods in Applied Mechanics and Engineering, 2022, 389：114338.
[13] Hu Dean, Long Ting, Xiao Yihua, et al. Fluid- structure interaction analysis by coupled FE-SPH model based on a novel searching algorithm[J]. Computer Methods in Applied Mechanics and Engineering, 2014, 276：266-286.
[14] Rakhsha M, Pazouki A, Serban R, et al. Using a half-implicit integration scheme for the SPH-based solution of fluid-solid interaction problems[J]. Computer Methods in Applied Mechanics and Engineering, 2019, 345：100-122.
[15] Hu Wei, Guo Guannan, Hu Xiaozhe, et al. A consistent spatially adaptive smoothed particle hydrodynamics method for fluid-structure interactions[J]. Computer Methods in Applied Mechanics and Engineering, 2019, 347：402-424.
[16] Fuchs S L, Meier C, Wall W A, et al. A novel smoothed particle hydrodynamics and finite element coupling scheme for fluid-structure interaction：The sliding boundary particle approach[J]. Computer Methods in Applied Mechanics and Engineering, 2021, 383：113922.
[17] Zhang Chi, Rezavand M, Hu Xiangyu. A multi- resolution SPH method for fluid-structure interactions[J]. Journal of Computational Physics, 2021, 429：110028.
[18] Hosain M L, Domínguez J M, Fdhila R B, et al. Smoothed particle hydrodynamics modeling of industrial processes involving heat transfer[J]. Applied Energy, 2019, 252：113441.
[19] Wang Zidi, Duan Guangtao, Matsunaga T, et al. Consistent Robin boundary enforcement of particle method for heat transfer problem with arbitrary geometry[J]. International Journal of Heat and Mass Transfer, 2020, 157：119919.
[20] Li Jiao, Wang Guangchun, Zhan Jianghu, et al. Meshless SPH analysis for transient heat conduction in the functionally graded structures[J]. Composites Communications, 2021, 24：100664.
[21] Hopp-Hirschler M, Shadloo M S, Nieken U. A smoothed particle hydrodynamics approach for thermo- capillary flows[J]. Computers & Fluids, 2018, 176：1-19.
[22] ZHANG Zhilang, Walayat K, et al. A finite particle method with particle shifting technique for modeling particulate flows with thermal convection[J]. International Journal of Heat and Mass Transfer, 2019, 128：1245- 1262.
[23] Garoosi F, Shakibaeinia A. Numerical simulation of entropy generation due to natural convection heat transfer using Kernel Derivative-Free (KDF) Incompressible Smoothed Particle Hydrodynamics (ISPH) model[J]. International Journal of Heat and Mass Transfer, 2020, 150：119377.
[24] Yang Pengying, Huang Can, Zhang Zhilang, et al. Simulating natural convection with high rayleigh numbers using the smoothed particle hydrodynamics method[J]. International Journal of Heat and Mass Transfer, 2021, 166：120758.
[25] Koneshwaran S, Thambiratnam D P, Gallage C. Blast response of segmented bored tunnel using coupled SPH-FE method[C]//Structures. Elsevier, 2015, 2：58-71.
[26] Fan Houfu, Li Shaofan. A Peridynamics-SPH modeling and simulation of blast fragmentation of soil under buried explosive loads[J]. Computer Methods in Applied Mechanics and Engineering, 2017, 318：349-381.
[27] Belytschko T, et al. Element-free Galerkin methods[J]. International Journal for Numerical Methods in Engineering, 1994, 37(2)：229-256.
[28] Liu W K, Jun S, Zhang Yifei. Reproducing kernel particle methods[J]. International journal for numerical methods in fluids, 1995, 20(8-9)：1081-1106.
[29] Khan W, Ullah B. Structural optimization based on meshless element free Galerkin and level set methods[J]. Computer Methods in Applied Mechanics and Engineering, 2019, 344：144-163.
[30] Neofytou A, et al. Level set topology optimization with nodally integrated reproducing kernel particle method[J]. Computer Methods in Applied Mechanics and Engineering, 2021, 385：114016.
[31] Zhou Liming, Yang Hao, Ma Long, et al. On the static analysis of inhomogeneous magneto-electro-elastic plates in thermal environment via element-free Galerkin method[J]. Engineering Analysis with Boundary Elements, 2022, 134：539-552.
[32] Ding Rui, Shen Quan, Yao Yuan. The element-free Galerkin method for the dynamic Signorini contact problems with friction in elastic materials[J]. Applied Mathematics and Computation, 2022, 415：126696.
[33] Koshizuka S, Oka Y. Moving-particle semi-implicit method for fragmentation of incompressible fluid[J]. Nuclear Science and Engineering, 1996, 123(3)：421-434.
[34] Atluri S N, Zhu Tulong. A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics[J]. Computational Mechanics, 1998, 22(2)：117-127.
[35] Rabczuk T, Areias P. A meshfree thin shell for arbitrary evolving cracks based on an extrinsic basis[J]. Computer Modeling in Engineering & Sciences, 2006, 16(2)：115-130.
[36] Chen Jiunshyan, Wu Youcai. Stability in Lagrangian and semi-Lagrangian reproducing kernel discretizations using nodal integration in nonlinear solid mechanics[C]// Advances in Meshfree Techniques. Springer Netherlands, 2007：55-76.
[37] Aluru N R. A point collocation method based on reproducing kernel approximations[J]. International Journal for Numerical Methods in Engineering, 2000, 47(6)：1083-1121.
[38] Musavi S H, Ashrafizaadeh M. Meshless lattice Boltzmann method for the simulation of fluid flows[J]. Physical Review E, 2015, 91(2)：023310.
[39] Chen Jiunshyan, Hillman M, Chi Shengwei. Meshfree methods：Progress made after 20 years[J]. Journal of Engineering Mechanics, 2017, 143(4)：04017001.
[40] 强洪夫, 张国星, 王广, 等. SPH方法在宽速域岩石侵彻问题中的应用[J]. 高压物理学报, 2019, 33(5)：174-182. QIANG Hongfu, ZHANG Guoxing, WANG Guang, et al. Application of SPH method to rock intrusion problems in wide velocity domain[J]. Journal of High Pressure Physics, 2019, 33(5)：174-182.
[41] 强洪夫, 孙新亚, 王广, 等. 钢箱内部爆炸破坏的SPH数值模拟[J]. 爆炸与冲击, 2019, 39(5)：24-32. QIANG Hongfu, SUN Xinya, WANG Guang, et al. SPH numerical simulation of explosion damage inside a steel box[J]. Explosion and Shock, 2019, 39(5)：24-32.
[42] 李勇霖, 陈荣华, 田文喜, 等. 对流传热问题的粒子-网格混合方法数值模拟[J]. 原子能科学技术, 2019, 53(5)：811-818. LI Yonglin, CHEN Ronghua, TIAN Wenxi, et al. Numerical simulation of convective heat transfer problem by particle-mesh hybrid method[J]. Atomic Energy Science and Technology, 2019, 53(5)：811-818.
[43] Monaghan J J. An introduction to SPH[J]. Computer Physics Communications, 1988, 48(1)：89-96.
[44] Jin Hongbin, Xin Ding. On criterions for smoothed particle hydrodynamics kernels in stable field[J]. Journal of Computational Physics, 2005, 202(2)：699-709.
[45] Chen Jiunshyan, Wang Huiping. New boundary condition treatments in meshfree computation of contact problems[J]. Computer Methods in Applied Mechanics and Engineering, 2000, 187(3-4)：441-468.
[46] Zhu Tulong, Atluri S N. A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method[J]. Computational Mechanics, 1998, 21(3)：211-222.
[47] Lu Yunyun, Belytschko T, Gu Lei. A new implementation of the element free Galerkin method[J]. Computer Methods in Applied Mechanics and Engineering, 1994, 113(3-4)：397-414.
[48] Kaljević I, Saigal S. An improved element free Galerkin formulation[J]. International Journal for Numerical Methods in Engineering, 1997, 40(16)：2953-2974.
[49] Chen Jiunshyan, Han Weimin, You Yang, et al. A reproducing kernel method with nodal interpolation property[J]. International Journal for Numerical Methods in Engineering, 2003, 56(7)：935-960.
[50] Nitsche J. Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, in：Abhandlungen Aus Dem Math[R]. Semin. Der Univ. Hambg., 1971：9-15. NITSCHE J. For variational principles for solving the Dirichlet problem, see：Mathematical processing when using subdomains without any constraints[R]. Semin. German University. Hamburg, 1971:9-15.
[51] Monaghan J J. Simulating free surface flows with SPH[J]. Journal of Computational Physics, 1994, 110(2)：399-406.
[52] Colagrossi A, Landrini M. Numerical simulation of interfacial flows by smoothed particle hydrodynamics[J]. Journal of Computational Physics, 2003, 191(2)：448-475.
[53] Marrone S, Antuono M, Colagrossi A, et al. δ-SPH model for simulating violent impact flows[J]. Computer Methods in Applied Mechanics and Engineering, 2011, 200(13-16)：1526-1542.
[54] Nguyen V P, Rabczuk T, Bordas S, et al. Meshless methods：a review and computer implementation aspects[J]. Mathematics and Computers in Simulation, 2008, 79(3)：763-813.
[55] Chen Jiunshyan, Wu Chengtang, Yoon S, et al. A stabilized conforming nodal integration for Galerkin mesh-free methods[J]. International Journal for Numerical Methods in Engineering, 2001, 50(2)：435-466.
[56] Guan Paichen, Chi Shengwei, Chen Jiunshyan, et al. Semi-Lagrangian reproducing kernel particle method for fragment-impact problems[J]. International Journal of Impact Engineering, 2011, 38(12)：1033-1047.
[57] Chen Jiunshyan, Hillman M, Rüter M. An arbitrary order variationally consistent integration for Galerkin meshfree methods[J]. International Journal for Numerical Methods in Engineering, 2013, 95(5)：387-418.
[58] Attaway S W, Heinstein M W, Swegle J W. Coupling of smooth particle hydrodynamics with the finite element method[J]. Nuclear Engineering and Design, 1994, 150(2-3)：199-205.
[59] Vuyst T, Vignjevic R, Campbell J C. Coupling between meshless and finite element methods[J]. International Journal of Impact Engineering, 2005, 31(8)：1054-1064.
[60] Johnson G R, Beissel S R, Stryk R A. An improved generalized particle algorithm that includes boundaries and interfaces[J]. International Journal for Numerical Methods in Engineering, 2002, 53(4)：875-904.
[61] Tsuji P, Puso M, Spangler C W, et al. Embedded smoothed particle hydrodynamics[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 366：113003.
[62] Puso M A, Sanders J, Settgast R, et al. An embedded mesh method in a multiple material ALE[J]. Computer Methods in Applied Mechanics and Engineering, 2012, 245：273-289.
[63] Fuchs S L, Meier C, Wall W A, et al. A novel smoothed particle hydrodynamics and finite element coupling scheme for fluid-structure interaction：The sliding boundary particle approach[J]. Computer Methods in Applied Mechanics and Engineering, 2021, 383：113922.
[64] Liu Ce, Li Baotong, Liu Qingfang, et al. A novel implicit meshless particle method：NURBS-based particle hydrodynamics (NBPH)[J]. Computer Methods in Applied Mechanics and Engineering, 2023, 406：115895.
[65] 高亮, 李培根, 黄培, 等. 数字化设计类工业软件发展策略研究[J]. 中国工程科学, 2023, 25(2)：254-262. GAO Liang, LI Peigen, HUANG Pei, et al. Research on the development strategy of industrial software for digital design[J]. China Engineering Science, 2023, 25(2)：254-262.
[66] 张卫红, 周涵, 李韶英, 等. 航天高性能薄壁构件的材料-结构一体化设计综述[J]. 航空学报, 2023, 44(9)：30-46. ZHANG Weihong, ZHOU Han, LI Shaoying, et al. A review of material-structure integration design for high- performance thin-walled components in aerospace[J]. Journal of Aeronautics, 2023, 44(9)：30-46.
[67] Bendsøe M P, Kikuchi N. Generating optimal topologies in structural design using a homogenization method[J]. Computer Methods in Applied Mechanics and Engineering, 1988, 71(2)：197-224.
[68] Bendsøe M P. Optimal shape design as a material distribution problem[J]. Structural Optimization, 1989, 1：193-202.
[69] Stolpe M, Svanberg K. An alternative interpolation scheme for minimum compliance topology optimization[J]. Structural and Multidisciplinary Optimization, 2001, 22(2)：116-124.
[70] Wang M Y, Wang Xiaoming, Guo Dongming. A level set method for structural topology optimization[J]. Computer Methods in Applied Mechanics and Engineering, 2003, 192(1-2)：227-246.
[71] GUO Xu, Zhang Weisheng, Zhong Wenliang. Doing topology optimization explicitly and geometrically—a new moving morphable components based framework[J]. Journal of Applied Mechanics, 2014, 81(8)：081009.
[72] Borrvall T, Petersson J. Topology optimization of fluids in Stokes flow[J]. International Journal for Numerical Methods in Fluids, 2003, 41(1)：77-107.
[73] Gersborg-Hansen A, Sigmund O, Haber R B. Topology optimization of channel flow problems[J]. Structural and Multidisciplinary Optimization, 2005, 30：181-192.
[74] Olesen L H, Okkels F, Bruus H. A high-level programming-language implementation of topology optimization applied to steady-state Navier-Stokes flow[J]. International Journal for Numerical Methods in Engineering, 2006, 65(7)：975-1001.
[75] Dede E M. Multiphysics topology optimization of heat transfer and fluid flow systems[C]//Proceedings of the COMSOL Users Conference. 2009, 715.
[76] Yoon G H. Topological design of heat dissipating structure with forced convective heat transfer[J]. Journal of Mechanical Science and Technology, 2010, 24(6)：1225-1233.
[77] Dede E M, Liu Y. Experimental and numerical investigation of a multi-pass branching microchannel heat sink[J]. Applied Thermal Engineering, 2013, 55(1-2)：51-60.
[78] Yu Minghao, Ruan Shilun, Wang Xinyu, et al. Topology optimization of thermal-fluid problem using the MMC-based approach[J]. Structural and Multidisciplinary Optimization, 2019, 60：151-165.
[79] Zeng Shi, Lee P S. Topology optimization of liquid- cooled microchannel heat sinks：An experimental and numerical study[J]. International Journal of Heat and Mass Transfer, 2019, 142：118401.
[80] Li Baotong, Xie Chenhan, Yin Xinxin, et al. Multidisciplinary optimization of liquid cooled heat sinks with compound jet/channel structures arranged in a multipass configuration[J]. Applied Thermal Engineering, 2021, 195：117159.
[81] Høghøj L C, Nørhave D R, Alexandersen J, et al. Topology optimization of two fluid heat exchangers[J]. International Journal of Heat and Mass Transfer, 2020, 163：120543.
[82] Zeng Shi, Kanargi B, Lee P S. Experimental and numerical investigation of a mini channel forced air heat sink designed by topology optimization[J]. International Journal of Heat and Mass Transfer, 2018, 121：663-679.
[83] Alexandersen J, Aage N, Andreasen C S, et al. Topology optimisation for natural convection problems[J]. International Journal for Numerical Methods in Fluids, 2014, 76(10)：699-721.
[84] Alexandersen J, Sigmund O, Aage N. Large scale three-dimensional topology optimisation of heat sinks cooled by natural convection[J]. International Journal of Heat and Mass Transfer, 2016, 100：876-891.
[85] He Qizhi, Kang Zhan, Wang Yiqiang. A topology optimization method for geometrically nonlinear structures with meshless analysis and independent density field interpolation[J]. Computational Mechanics, 2014, 54：629-644.
[86] Gong Shuguang, Wei Yongbao, Xie Guilan, et al. Study on topology optimization method of particle moving based on element-free Galerkin method[J]. International Journal for Computational Methods in Engineering Science and Mechanics, 2018, 19(5)：305-313.
[87] Zhang Jianping, Wang Shusen, Zhou Guoqiang, et al. Topology optimization of thermal structure for isotropic and anisotropic materials using the element-free Galerkin method[J]. Engineering Optimization, 2020, 52(7)：1097-1118.
[88] Lin Jun, Guan Yanjin, Zhao Guoqun, et al. Topology optimization of plane structures using smoothed particle hydrodynamics method[J]. International Journal for Numerical Methods in Engineering, 2017, 110(8)：726-744.
[89] Li Jiao, Guan Yanjin, Wang Guangchun, et al. A meshless method for topology optimization of structures under multiple load cases[C]//Structures. Elsevier, 2020, 25：173-179.
[90] Gong Shuguang, Du Jiaxi, Liu Xiang, et al. Study on topology optimization under multiple loading conditions and stress constraints based on EFG method[J]. International Journal for Computational Methods in Engineering Science and Mechanics, 2010, 11(6)：328-336.
[91] Luo Zhen, Zhang Nong, Wang Yu, et al. Topology optimization of structures using meshless density variable approximants[J]. International Journal for Numerical Methods in Engineering, 2013, 93(4)：443-464.
[92] Khan W, Ullah B. Structural optimization based on meshless element free Galerkin and level set methods[J]. Computer Methods in Applied Mechanics and Engineering, 2019, 344：144-163.
[93] Neofytou A, Picelli R, Huang Tsunghui, et al. Level set topology optimization for design-dependent pressure loads using the reproducing kernel particle method[J]. Structural and Multidisciplinary Optimization, 2020, 61：1805-1820.
[94] Shobeiri V. Topology optimization using bi-directional evolutionary structural optimization based on the element-free Galerkin method[J]. Engineering Optimization, 2016, 48(3)：380-396.
[95] Gonçalves D C, Lopes J D F, Campilho R, et al. Topology optimization using a natural neighbour meshless method combined with a bi-directional evolutionary algorithm[J]. Mathematics and Computers in Simulation, 2022, 194：308-328.
[96] De Boor C. On calculating with B-splines[J]. Journal of Approximation Theory, 1972, 6(1)：50-62.
[97] Xu Rui, Stansby P, Laurence D. Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach[J]. Journal of Computational Physics, 2009, 228(18)：6703-6725.
[98] Akinci N, Akinci G, Teschner M. Versatile surface tension and adhesion for SPH fluids[J]. ACM Transactions on Graphics (TOG), 2013, 32(6)：1-8.
[99] Nasar A M A, Fourtakas G, Lind S J, et al. High-order velocity and pressure wall boundary conditions in Eulerian incompressible SPH[J]. Journal of Computational Physics, 2021, 434：109793.
[100] Nasar A M A, Fourtakas G, Lind S J, et al. High-order consistent SPH with the pressure projection method in 2-D and 3-D[J]. Journal of Computational Physics, 2021, 444：110563.
[101] Deng Yongbo, Wu Yihui, Liu Zhenyu. Topology optimization theory for laminar flow[M]. Singapore：Springer Singapore, 2018.
[102] 胡长明.以智能制造为主攻方向, 深入推进电子装备企业转型升级[J]. 智能制造, 2022(6)：47-48, 29. Hu Changming. Intelligent manufacturing as the main direction of attack, in-depth promotion of the transformation and upgrading of electronic equipment enterprises [J]. Intelligent Manufacturing, 2022(6)：47-48, 29.
[103] 胡长明, 魏涛, 钱吉裕, 等. 射频微系统冷却技术综述[J]. 现代雷达, 2020, 42(3)：1-11. HU Changming, WEI Tao, QIAN Jiyu, et al. A review of RF microsystem cooling technology[J]. Modern Radar, 2020, 42(3)：1-11.
[104] Asghari T A. PCB thermal via optimization using design of experiments[C]//Thermal and Thermomechanical Proceedings 10th Intersociety Conference on Phenomena in Electronics Systems, 2006. ITHERM 2006. IEEE, 2006：224-228.