金属结构航天器陨落过程三维瞬态传热有限元算法研究
|
摘要
构建金属桁架结构航天器陨落再入气动热环境有限元传热模型,是准确预测在轨服役期满大型航天器陨落再入解体过程温度分布的关键。本文采用四节点四面体单元对空间进行离散,依据泛函理论,将传热控制方程离散为代数方程组;利用有限单元法总体合成得到具有对称正定、高度稀疏和非0元素分布的规则性刚度矩阵,发展一维变带宽压缩存贮技术,有效解决大型稀疏矩阵的数据存贮问题;为有效抑制求解过程出现的温度在时间和空间上的振荡问题,发展集中热容矩阵系数处理方法,将热容矩阵的同行或同列元素相加代替对角线元素,使非对角线元素化为0,构造求解三维瞬态温度场的两点向后差分格式、Crank-Nicolson格式和Galerkin格式。通过对正方体瞬态传热计算验证分析,在相同条件下,采用以上三种格式均可获得一致稳定的温度解,并得到与现有ANSYS有限元软件较为吻合的计算结果,验证了所建立三维瞬态传热有限元计算模型的准确可靠性。在此基础上,对铝合金低轨航天器薄壳结构进行了传热计算,给出了类天宫飞行器两舱体陨落飞行107.5 km～90 km不同高度的瞬态温度分布,为这类寿命末期航天器陨落再入解体预报提供理论支撑与可计算模型。
To construct the finite element model for the heat transfer of large-scale complex spacecraft with a metal truss structure during in reentry in aerothermodynamic environment,is the key to accurately predict the temperature distribution of the end-of-life spacecraft during the process of falling reentry and disintegration.In this paper,the four-node tetrahedron element is used to discretize the space,and the heat transfer equation is discretized into a set of algebraic equations.The symmetric positive-definite stiffness matrix with a highly sparse and non-zero element distribution is obtained by the overall synthesis of the finite element method.The one-dimensional variable bandwidth storage technique is developed to effectively resolve the data storage of large sparse matrix.To effectively depress the temperature oscillations appearing in time and space in the solving process,the centralized heat capacity matrix coefficient method is developed.The elements on the same row or column elements of the heat capacity matrix are summed to replace the diagonal elements,so that only the diagonal elements of the new heat capacity matrix are non-zero,and the remaining elements are zero.The two-point backward difference scheme,Crank-Nicolson scheme and Galerkin scheme are constructed to solve the three-dimensional transient temperature field.By computational analysis of transient heat transfer of a rectangular cylinder body,the converged temperature solutions of the above-said three schemes are found to be in good agreement,and the results are consistent with those of the existing finite-element software ANSYS,which confirms the precision and reliability of the present 3 D finite-element model for transient heat transfer.The heat transfer computation is carried on the shell structure of a low-orbit spacecraft made of aluminium alloy,and the transient temperature distribution of the Tiangong's two-capsule structure is simulated and analyzed from the flight altitude of 107.5 km～90 km,which provides the theoretical support and computable model for the forecast of disintegration of the end-of-life spacecraft during falling reentry.
To construct the finite element model for the heat transfer of large-scale complex spacecraft with a metal truss structure during in reentry in aerothermodynamic environment,is the key to accurately predict the temperature distribution of the end-of-life spacecraft during the process of falling reentry and disintegration.In this paper,the four-node tetrahedron element is used to discretize the space,and the heat transfer equation is discretized into a set of algebraic equations.The symmetric positive-definite stiffness matrix with a highly sparse and non-zero element distribution is obtained by the overall synthesis of the finite element method.The one-dimensional variable bandwidth storage technique is developed to effectively resolve the data storage of large sparse matrix.To effectively depress the temperature oscillations appearing in time and space in the solving process,the centralized heat capacity matrix coefficient method is developed.The elements on the same row or column elements of the heat capacity matrix are summed to replace the diagonal elements,so that only the diagonal elements of the new heat capacity matrix are non-zero,and the remaining elements are zero.The two-point backward difference scheme,Crank-Nicolson scheme and Galerkin scheme are constructed to solve the three-dimensional transient temperature field.By computational analysis of transient heat transfer of a rectangular cylinder body,the converged temperature solutions of the above-said three schemes are found to be in good agreement,and the results are consistent with those of the existing finite-element software ANSYS,which confirms the precision and reliability of the present 3 D finite-element model for transient heat transfer.The heat transfer computation is carried on the shell structure of a low-orbit spacecraft made of aluminium alloy,and the transient temperature distribution of the Tiangong's two-capsule structure is simulated and analyzed from the flight altitude of 107.5 km～90 km,which provides the theoretical support and computable model for the forecast of disintegration of the end-of-life spacecraft during falling reentry.
引文
[1] Hoyt R P,Forward R L.Performance of the Terminator Tether for autonomous deorbit of LEO space -craft [A].35t h AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit[C].1999.
[2] 丹枫.和平号的成功坠毁及其影响[J].国际太空,2001(5):6-8.(DAN Feng.The successful crash of Mir and its impact[J].International Space,2001(5):6-8.(in Chinese))
[3] Reyhanoglu M,Alvarado J.Estimation of debris dispersion due to a space vehicle breakup during reentry[J].Acta Astronautica,2013,86:211-218.
[4] Balakrishnan D,Kurian J.Material thermal degradation under reentry aerodynamic heating[J].Journal of Spacecraft and Rockets,2014,51(4):1319-1328.
[5] 李志辉,梁杰,唐志共,等.“航天飞行器跨流域空气动力学与飞行控制关键基础问题研究”项目2014年度进展报告[R].国家科技报告,2014.(LI Zhi-hui,LIANG Jie,TANG Zhi-gong,et al.Progress in 2014 for “Study on Key Basic Problems of Aerodynamics and Flight Control Covering Flow Regimes for Aerospace Craft”[R].National Science and Technology Report,2014.(in Chinese))
[6] Anonymous.DAS user’s guide,version 2.0 [R].NASA,JSC-64047,2007.
[7] Bouslog S A,Ross B P,Madden C B.Space debris reentry risk analysis[A].32n dAerospace Sciences Meeting & Exhibit[C].1994.
[8] Fritsche B,Klinkrad H,Kashkovsky A,et al.Spacecraft disintegration during uncontrolled atmospheric re -entry[J].Acta Astronautica,2000,47(2-9):513-522.
[9] Lips T,Fritsche B.A comparison of commonly used re -entry analysis tools[J].Acta Astronautica,2005,57(2-8):312-323.
[10] Tamma K K,Thornton E A.Re -entry thermal/structural finite -element modeling/analysis of shuttle wing configurations[J].Journal of Spacecraft and Roc-kets,1987,24(2):101-108.
[11] 孔祥谦.有限单元法在传热学中的应用(第三版)[M].北京:科学出版社,1998.(KONG Xiang-qian.Application of Finite Element Method in Heat Transfer (Third Edition) [M].Beijing:Science Press,1998.(in Chinese))
[12] Gould D C.Thermal analysis of a high-speed aircraft wing using p-version finite elements[A].39t hAlAA Aerospace Sciences Meeting & Exhibit[C].2001.
[13] Blosser M.L.Investigation of fundamental modeling and thermal performance issues for a metallic thermal protection system design[A].40t h AIAA Aerospace Sciences Meeting & Exhibit[C].2002.
[14] 姜贵庆.非烧蚀热防护与非烧蚀机理[A].中国空气动力学会近代空气动力学研讨会[C].2005.(JIANG Gui-qing.Non-ablative thermal protection and non-ablative mechanism[A].Symposium on Modern Aerodynamics of China Aerodynamics Society [C].2005.(in Chinese))
[15] 马玉娥.可重复使用运载器热防护系统热/力耦合数值计算研究[D].西北工业大学,2005.(MA Yu-e.Study of Thermo-Mechanical Coupled Computation for Thermal Protection System of Reusable Launch Vehicle[D].Northwestern Polytechnical University,2005.(in Chinese))
[16] 耿湘人,桂业伟,贺立新,等.红外窗口不同冷却方式下的结构传热和热应力特性计算研究[J].空气动力学学报,2008,26(3):329-333.(GENG Xiang-ren,GUI Ye-wei,HE Li-xin,et al.Numerical study on heat transfer and thermal stress for infra-window with externally cooled and internally colled techniques[J].Acta Aerodynamica Sinica,2008,26(3):329-333.(in Chinese))
[17] 刘刚,李长生,刘相华.有限元法求解瞬态温度场时的振荡问题[J].钢铁研究学报,2008,20(7):19-22.(LIU Gang,LI Chang-sheng,LIU Xiang-hua.Oscillation problem during solving for transient temperature field using FEM [J].Journal of Iron and Steel Research,2008,20(7):19-22.(in Chinese))
[18] Ng W H,Friedmann P P,Waas A M.Thermomechanical behavior of a damaged thermal protection system:Finite -element simulations[J].Journal of Aerospace Engineering,2012,25(1):90-102.
[19] 张涛,孙冰.航天器再入全过程轴对称烧蚀热防护数值仿真研究[J].宇航学报,2011,32(5):1195-1204.(ZHANG Tao,SUN Bing.Numerical simulation research on axis-symmetrical ablative thermal protection for spacecraft in whole reentry[J].Journal of Astronautics,2011,32(5):1195-1204.(in Chinese))
[20] Blosser M L,Poteet C C,Chen R R,et al.Development of advanced metallic thermal protection system prototype hardware[J].Journal of Spacecraft and Rockets,2004,41(2):183-194.
[21] 闫长海,孟松鹤,陈贵清,等.金属热防护系统隔热材料的发展与现状[J].导弹与航天运载技术,2006(4):48-52.(YAN Chang-hai,MENG Song-he,CHEN Gui-qing,et al.Development of insulations for meta-llic thermal protection system and its current status[J].Missile and Space Vehcile,2006(4):48-52.(in Chinese))
[22] Darjabeigi K.Design of High Temperature Multi-layer Insulation for Reusable Launch Vehicles[D].University of Virginia,2000.
[23] 张洪武,付振东.考虑辐射效应的多孔材料传热有限元算法[J].航空动力学报,2005,20(1):98-103.(ZHANG Hong-wu,FU Zhen-dong.Finite element method for heat conduction analysis in porous media with consideration of radiative effects[J].Journal of Aerospace Power,2005,20(1):98-103.(in Chinese))
[24] 熊启林,田晓耕,沈亚鹏,等.瞬态热冲击下层合材料板界面的热弹性行为[J].力学学报,2011,43(3):630-634.(XIONG Qi-lin,TIAN Xiao-geng,SHEN Ya-peng,et al.Thermoelastic behavior of interface of composite plate under thermal shock[J].Chinese Journal of Theoretical and Applied Mechanics,2011,43(3):630-634.(in Chinese))
[25] 国义军.炭化材料烧蚀防热的理论分析与工程应用[J].空气动力学学报,1994,12(1):94-100.(GUO Yi-jun.An analysis of a charring ablative thermal protection system with its engineering application[J].Acta Aerodynamica Sinica,1994,12(1):94-100.(in Chinese))
[26] 唐小伟,张顺玉,党雷宁,等.非常规再入/进入问题探讨[J].航天返回与遥感,2015,36(6):11-21.(TANG Xiao -wei,ZHANG Shun-yu,DANG Lei-ning,et al.Discussion on unconventional reentry/entry[J].Spacecraft Recovery & Remotesensing,2015,36(6):11-21.(in Chinese))
[27] Li Z H,Ma Q,Cui J Z.Finite element algorithm for dynamic thermoelasticity coupling problems and application to transient response of structure with strong aerothermodynamic environment[J].Communications in Computational Physics,2016,20(3):773-810.
[28] Ma Q,Cui J Z,Li Z H,et al.Second-order asymptotic algorithm for heat conduction problems of periodic composite materials in curvilinear coordinates[J].Jounal of Computational & Applied Mathematics,2016,306:87-115.
[29] 申跃奎,赵德顺,王秦.考虑流固耦合作用的充气膜结构风压分布研究[J].计算力学学报,2017,34(5):665-671.(SHEN Yue -kui,ZHAO De -shun,WANG Qin.On wind pressure coefficient distribution of air-supported structures considering fluid-structure coupling[J].Chinese Journal of Computational Mecha -nics,2017,34(5):665-671.(in Chinese))
[30] Collier CS.Thermoelastic formulation of stiffened,unsymmetric composite panels for finite element analysis of high speed aircraft [A].AIAA/ASME/ASCE/AHS/ACS 35t h Structures,Dynamics & Materials Confe -rence[C].1994.
[31] Thornton E A,Dechaumphai P,Wieting A R,et al.Integrated transient thermal-structural finite element analysis[A].Proceedings of the AIAA/ASME/ASCE/AHS 22n d Structures,Structural Dynamics and Material Conference [C].1981.
[32] 张顺玉,唐小伟.某大型航天器陨落过程初步研究[A].第三届进入、减速、着陆(EDL)技术全国学术会议[C].长沙,2015.(ZHANG Shun-yu,TANG Xiao -wei.Preliminary study on the falling process of a large spacecraft[A].3r d National Academic Confe -rence on Entry,Deceleration and Landing(EDL)Technology[C].Changsha,2015.(in Chinese))
[33] 梁杰,李志辉,杜波强,等.大型航天器再入陨落时太阳翼气动力/热模拟分析[J].宇航学报,2015,36(12):1348-1355.(LIANG Jie,LI Zhi-hui,DU Bo -qiang,et al.Modeling and analysis of solar array aero -thermodynamics during large-scale spacecraft reentry[J].Journal of Astronautics,2015,36(12):1348-1355.(in Chinese))
[34] 李中华,李志辉,李海燕,等.过渡流区N-S/DSMC耦合计算研究[J].空气动力学学报,2013,31(3):282-287.(LI Zhong-hua,LI Zhi-hui,LI Hai-yan,et al.Research on CFD/DSMC hybrid numerical method in rarefied flows[J].Acta Aerodynamica Sinica,2013,31(3):282-287.(in Chinese))
[35] 李志辉,蒋新宇,吴俊林,等.转动非平衡玻尔兹曼模型方程统一算法与全流域绕流计算应用[J].力学学报,2014,46(3):336-351.(LI Zhi-hui,JIANG Xin-yu,WU Jun-lin,et al.Gas-kinetic unified algorithm for Boltzmann model equation in rotational nonequilibrium and its application to the whole range flow regimes [J].Chinese Journal of Theoretical and Applied Mechanics,2014,46(3):336-351.(in Chinese))
[36] Li Z H,Ma Q,Cui J Z.Second-order two -scale finite element algorithm for dynamic thermo -mechanical coupling problem in symmetric structure[J].Journal of Computational Physics,2016,314:712-748.
[37] 丁丽娟,程杞元.数值计算方法(第二版)[M].北京:北京理工大学出版社,2005.(DING Li-juan,CHENG Qi-yuan.Numerical Methods(2n d Edition)[M].Beijing:Beijing University of Technology Press,2005.(in Chinese))
[38] Ma Q,Cui J Z,Li Z H.Second-order two-scale asymptotic analysis for axisymmetric and spherical symmetric structure with periodic configurations[J].International Journal of Solid and Structures,2016,78-79:77-100.
[2] 丹枫.和平号的成功坠毁及其影响[J].国际太空,2001(5):6-8.(DAN Feng.The successful crash of Mir and its impact[J].International Space,2001(5):6-8.(in Chinese))
[3] Reyhanoglu M,Alvarado J.Estimation of debris dispersion due to a space vehicle breakup during reentry[J].Acta Astronautica,2013,86:211-218.
[4] Balakrishnan D,Kurian J.Material thermal degradation under reentry aerodynamic heating[J].Journal of Spacecraft and Rockets,2014,51(4):1319-1328.
[5] 李志辉,梁杰,唐志共,等.“航天飞行器跨流域空气动力学与飞行控制关键基础问题研究”项目2014年度进展报告[R].国家科技报告,2014.(LI Zhi-hui,LIANG Jie,TANG Zhi-gong,et al.Progress in 2014 for “Study on Key Basic Problems of Aerodynamics and Flight Control Covering Flow Regimes for Aerospace Craft”[R].National Science and Technology Report,2014.(in Chinese))
[6] Anonymous.DAS user’s guide,version 2.0 [R].NASA,JSC-64047,2007.
[7] Bouslog S A,Ross B P,Madden C B.Space debris reentry risk analysis[A].32n dAerospace Sciences Meeting & Exhibit[C].1994.
[8] Fritsche B,Klinkrad H,Kashkovsky A,et al.Spacecraft disintegration during uncontrolled atmospheric re -entry[J].Acta Astronautica,2000,47(2-9):513-522.
[9] Lips T,Fritsche B.A comparison of commonly used re -entry analysis tools[J].Acta Astronautica,2005,57(2-8):312-323.
[10] Tamma K K,Thornton E A.Re -entry thermal/structural finite -element modeling/analysis of shuttle wing configurations[J].Journal of Spacecraft and Roc-kets,1987,24(2):101-108.
[11] 孔祥谦.有限单元法在传热学中的应用(第三版)[M].北京:科学出版社,1998.(KONG Xiang-qian.Application of Finite Element Method in Heat Transfer (Third Edition) [M].Beijing:Science Press,1998.(in Chinese))
[12] Gould D C.Thermal analysis of a high-speed aircraft wing using p-version finite elements[A].39t hAlAA Aerospace Sciences Meeting & Exhibit[C].2001.
[13] Blosser M.L.Investigation of fundamental modeling and thermal performance issues for a metallic thermal protection system design[A].40t h AIAA Aerospace Sciences Meeting & Exhibit[C].2002.
[14] 姜贵庆.非烧蚀热防护与非烧蚀机理[A].中国空气动力学会近代空气动力学研讨会[C].2005.(JIANG Gui-qing.Non-ablative thermal protection and non-ablative mechanism[A].Symposium on Modern Aerodynamics of China Aerodynamics Society [C].2005.(in Chinese))
[15] 马玉娥.可重复使用运载器热防护系统热/力耦合数值计算研究[D].西北工业大学,2005.(MA Yu-e.Study of Thermo-Mechanical Coupled Computation for Thermal Protection System of Reusable Launch Vehicle[D].Northwestern Polytechnical University,2005.(in Chinese))
[16] 耿湘人,桂业伟,贺立新,等.红外窗口不同冷却方式下的结构传热和热应力特性计算研究[J].空气动力学学报,2008,26(3):329-333.(GENG Xiang-ren,GUI Ye-wei,HE Li-xin,et al.Numerical study on heat transfer and thermal stress for infra-window with externally cooled and internally colled techniques[J].Acta Aerodynamica Sinica,2008,26(3):329-333.(in Chinese))
[17] 刘刚,李长生,刘相华.有限元法求解瞬态温度场时的振荡问题[J].钢铁研究学报,2008,20(7):19-22.(LIU Gang,LI Chang-sheng,LIU Xiang-hua.Oscillation problem during solving for transient temperature field using FEM [J].Journal of Iron and Steel Research,2008,20(7):19-22.(in Chinese))
[18] Ng W H,Friedmann P P,Waas A M.Thermomechanical behavior of a damaged thermal protection system:Finite -element simulations[J].Journal of Aerospace Engineering,2012,25(1):90-102.
[19] 张涛,孙冰.航天器再入全过程轴对称烧蚀热防护数值仿真研究[J].宇航学报,2011,32(5):1195-1204.(ZHANG Tao,SUN Bing.Numerical simulation research on axis-symmetrical ablative thermal protection for spacecraft in whole reentry[J].Journal of Astronautics,2011,32(5):1195-1204.(in Chinese))
[20] Blosser M L,Poteet C C,Chen R R,et al.Development of advanced metallic thermal protection system prototype hardware[J].Journal of Spacecraft and Rockets,2004,41(2):183-194.
[21] 闫长海,孟松鹤,陈贵清,等.金属热防护系统隔热材料的发展与现状[J].导弹与航天运载技术,2006(4):48-52.(YAN Chang-hai,MENG Song-he,CHEN Gui-qing,et al.Development of insulations for meta-llic thermal protection system and its current status[J].Missile and Space Vehcile,2006(4):48-52.(in Chinese))
[22] Darjabeigi K.Design of High Temperature Multi-layer Insulation for Reusable Launch Vehicles[D].University of Virginia,2000.
[23] 张洪武,付振东.考虑辐射效应的多孔材料传热有限元算法[J].航空动力学报,2005,20(1):98-103.(ZHANG Hong-wu,FU Zhen-dong.Finite element method for heat conduction analysis in porous media with consideration of radiative effects[J].Journal of Aerospace Power,2005,20(1):98-103.(in Chinese))
[24] 熊启林,田晓耕,沈亚鹏,等.瞬态热冲击下层合材料板界面的热弹性行为[J].力学学报,2011,43(3):630-634.(XIONG Qi-lin,TIAN Xiao-geng,SHEN Ya-peng,et al.Thermoelastic behavior of interface of composite plate under thermal shock[J].Chinese Journal of Theoretical and Applied Mechanics,2011,43(3):630-634.(in Chinese))
[25] 国义军.炭化材料烧蚀防热的理论分析与工程应用[J].空气动力学学报,1994,12(1):94-100.(GUO Yi-jun.An analysis of a charring ablative thermal protection system with its engineering application[J].Acta Aerodynamica Sinica,1994,12(1):94-100.(in Chinese))
[26] 唐小伟,张顺玉,党雷宁,等.非常规再入/进入问题探讨[J].航天返回与遥感,2015,36(6):11-21.(TANG Xiao -wei,ZHANG Shun-yu,DANG Lei-ning,et al.Discussion on unconventional reentry/entry[J].Spacecraft Recovery & Remotesensing,2015,36(6):11-21.(in Chinese))
[27] Li Z H,Ma Q,Cui J Z.Finite element algorithm for dynamic thermoelasticity coupling problems and application to transient response of structure with strong aerothermodynamic environment[J].Communications in Computational Physics,2016,20(3):773-810.
[28] Ma Q,Cui J Z,Li Z H,et al.Second-order asymptotic algorithm for heat conduction problems of periodic composite materials in curvilinear coordinates[J].Jounal of Computational & Applied Mathematics,2016,306:87-115.
[29] 申跃奎,赵德顺,王秦.考虑流固耦合作用的充气膜结构风压分布研究[J].计算力学学报,2017,34(5):665-671.(SHEN Yue -kui,ZHAO De -shun,WANG Qin.On wind pressure coefficient distribution of air-supported structures considering fluid-structure coupling[J].Chinese Journal of Computational Mecha -nics,2017,34(5):665-671.(in Chinese))
[30] Collier CS.Thermoelastic formulation of stiffened,unsymmetric composite panels for finite element analysis of high speed aircraft [A].AIAA/ASME/ASCE/AHS/ACS 35t h Structures,Dynamics & Materials Confe -rence[C].1994.
[31] Thornton E A,Dechaumphai P,Wieting A R,et al.Integrated transient thermal-structural finite element analysis[A].Proceedings of the AIAA/ASME/ASCE/AHS 22n d Structures,Structural Dynamics and Material Conference [C].1981.
[32] 张顺玉,唐小伟.某大型航天器陨落过程初步研究[A].第三届进入、减速、着陆(EDL)技术全国学术会议[C].长沙,2015.(ZHANG Shun-yu,TANG Xiao -wei.Preliminary study on the falling process of a large spacecraft[A].3r d National Academic Confe -rence on Entry,Deceleration and Landing(EDL)Technology[C].Changsha,2015.(in Chinese))
[33] 梁杰,李志辉,杜波强,等.大型航天器再入陨落时太阳翼气动力/热模拟分析[J].宇航学报,2015,36(12):1348-1355.(LIANG Jie,LI Zhi-hui,DU Bo -qiang,et al.Modeling and analysis of solar array aero -thermodynamics during large-scale spacecraft reentry[J].Journal of Astronautics,2015,36(12):1348-1355.(in Chinese))
[34] 李中华,李志辉,李海燕,等.过渡流区N-S/DSMC耦合计算研究[J].空气动力学学报,2013,31(3):282-287.(LI Zhong-hua,LI Zhi-hui,LI Hai-yan,et al.Research on CFD/DSMC hybrid numerical method in rarefied flows[J].Acta Aerodynamica Sinica,2013,31(3):282-287.(in Chinese))
[35] 李志辉,蒋新宇,吴俊林,等.转动非平衡玻尔兹曼模型方程统一算法与全流域绕流计算应用[J].力学学报,2014,46(3):336-351.(LI Zhi-hui,JIANG Xin-yu,WU Jun-lin,et al.Gas-kinetic unified algorithm for Boltzmann model equation in rotational nonequilibrium and its application to the whole range flow regimes [J].Chinese Journal of Theoretical and Applied Mechanics,2014,46(3):336-351.(in Chinese))
[36] Li Z H,Ma Q,Cui J Z.Second-order two -scale finite element algorithm for dynamic thermo -mechanical coupling problem in symmetric structure[J].Journal of Computational Physics,2016,314:712-748.
[37] 丁丽娟,程杞元.数值计算方法(第二版)[M].北京:北京理工大学出版社,2005.(DING Li-juan,CHENG Qi-yuan.Numerical Methods(2n d Edition)[M].Beijing:Beijing University of Technology Press,2005.(in Chinese))
[38] Ma Q,Cui J Z,Li Z H.Second-order two-scale asymptotic analysis for axisymmetric and spherical symmetric structure with periodic configurations[J].International Journal of Solid and Structures,2016,78-79:77-100.