时变磁场下导电结构的电—磁—热—力行为分析
|
摘要
本文针对电-磁-力-热多场耦合问题,对在时变磁场作用下的导电板的振动和应力等特性,建立了相应的理论分析和有限元分析模型,并进行了定量的研究。
首先从导电介质所满足的电磁学、热学、弹性力学的基本定律和方程出发,建立了导电体弹性介质在电磁场与变温场作用下的力学基本方程,包括采用T法建立的与位移、磁场耦合的涡电流初边值问题,由涡电流热效应引起的温度场初边值问题,以及热磁弹性力学的基本方程。在此基础上,针对圆板这一具体结构,给出了其在磁场、温度场作用下的理论分析模型。并利用该理论分析模型,采用磁弹性的线性化理论和贝塞尔级数展开方法,对圆板的磁弹性动力、电磁力、应力、温度变化等问题进行了分析与研究。
然后,进一步分析了导电悬臂梁的在脉冲磁场的磁弹性耦合力学行为；通过对导电结构的磁场、温度场及变形场三个子系统分别建立相应的有限元分析模型,采用迭代算法处理场与场间非线性耦合效应,建立了导电结构的电-磁-力-热多场分析的有限元模型,并对外加磁场中导电矩形板的磁弹性振动实验进行了数值模拟,给出了磁弹性动力响应特征、涡电流密度分布、温度分布和最大挠度变化曲线。这些研究结果为工程实际中电磁工艺的改进以及导电结构的安全设计和可靠性评估提供依据和参考。
This dissertation presents some quantitative researches on electromagneto-thermo-mechanical behaviors of conductive plates subject to time-dependent magnetic fields based on theoretical model and finite element analysis model.
Firstly, based on electromagnetic, thermal and mechanical basic laws and equations of conductive media, the theoretical governing equations for the coupling electro-magnto-thermo-mechanics are employed for the analyses. After degenerating, an analytical solution is given for a conductive circular plate in a time-dependent magnetic field by using a T-method of the eddy current. The solution is expressed in an analytical series form and it is suitable for a circular plate under any arbitrary axisymmetric time-change magnetic fields. Some interesting results including the eddy current, the increment of temperature, and the electromagneto-thermal stresses are investigated respectively in this dissertation. Some characteristics such as the distribution of temperatures, the in-plane electromagnetic stresses and the dynamic response of the conductive plate are first studied and displayed. Secondly, with the methods of iterative arithmetic to solve the nonlinearities among various fields, a nonlinear finite element analysis model is developed for the coupled mechanics-magneto—thermal interaction of conductive plates. Some characteristics on the eddy current, the temperature of plate, the in-plane electromagnetic thermal stresses and the dynamic responses of the conductive plate which is in a time-dependent magnetic field have also been numerically investigated.
This study not only reveals some characteristics on the eddy current, the temperature of plate, the electromagnetic thermal stresses and the dynamic responses of the conductive plate, but presents a new possible way to simplify the analysis of the electromagnetic coupling problem for the conductive structures.
首先从导电介质所满足的电磁学、热学、弹性力学的基本定律和方程出发,建立了导电体弹性介质在电磁场与变温场作用下的力学基本方程,包括采用T法建立的与位移、磁场耦合的涡电流初边值问题,由涡电流热效应引起的温度场初边值问题,以及热磁弹性力学的基本方程。在此基础上,针对圆板这一具体结构,给出了其在磁场、温度场作用下的理论分析模型。并利用该理论分析模型,采用磁弹性的线性化理论和贝塞尔级数展开方法,对圆板的磁弹性动力、电磁力、应力、温度变化等问题进行了分析与研究。
然后,进一步分析了导电悬臂梁的在脉冲磁场的磁弹性耦合力学行为；通过对导电结构的磁场、温度场及变形场三个子系统分别建立相应的有限元分析模型,采用迭代算法处理场与场间非线性耦合效应,建立了导电结构的电-磁-力-热多场分析的有限元模型,并对外加磁场中导电矩形板的磁弹性振动实验进行了数值模拟,给出了磁弹性动力响应特征、涡电流密度分布、温度分布和最大挠度变化曲线。这些研究结果为工程实际中电磁工艺的改进以及导电结构的安全设计和可靠性评估提供依据和参考。
This dissertation presents some quantitative researches on electromagneto-thermo-mechanical behaviors of conductive plates subject to time-dependent magnetic fields based on theoretical model and finite element analysis model.
Firstly, based on electromagnetic, thermal and mechanical basic laws and equations of conductive media, the theoretical governing equations for the coupling electro-magnto-thermo-mechanics are employed for the analyses. After degenerating, an analytical solution is given for a conductive circular plate in a time-dependent magnetic field by using a T-method of the eddy current. The solution is expressed in an analytical series form and it is suitable for a circular plate under any arbitrary axisymmetric time-change magnetic fields. Some interesting results including the eddy current, the increment of temperature, and the electromagneto-thermal stresses are investigated respectively in this dissertation. Some characteristics such as the distribution of temperatures, the in-plane electromagnetic stresses and the dynamic response of the conductive plate are first studied and displayed. Secondly, with the methods of iterative arithmetic to solve the nonlinearities among various fields, a nonlinear finite element analysis model is developed for the coupled mechanics-magneto—thermal interaction of conductive plates. Some characteristics on the eddy current, the temperature of plate, the in-plane electromagnetic thermal stresses and the dynamic responses of the conductive plate which is in a time-dependent magnetic field have also been numerically investigated.
This study not only reveals some characteristics on the eddy current, the temperature of plate, the electromagnetic thermal stresses and the dynamic responses of the conductive plate, but presents a new possible way to simplify the analysis of the electromagnetic coupling problem for the conductive structures.
引文
[1]中国力学学会.力学学科发展现状与趋势.北京：中国科学技术出版社,2007：3-34.
[2]Moon, F. C. and Pao, Y. H., "Magnetoelastic buckling of a thin plate", J. Appl. Mech.,1968, 35(1):53～58.
[3]Moon, F. C. and Pao, Y. H., "Vibration and dynamic instability of a beam-plate in a transverse magnetic field", J. Appl. Mech.,1969,36(1):1～9.
[4]Pao Y. H. and Yeh C. S., "A linear theory for soft ferromagnetic elastic solids", Int. J. Eng. Sci.,1973,11(4):415～436.
[5]T. Kabashima, Y. Ueda, M. Ohto, "A study of the cantilever beam in the varying magnetic field", IEEE Transactions on Magnetics,1990,26(2):563～566.
[6]Van de ven, A. A. F., "Magnetoelastic buckling of a beam of elliptic cross section", Acta Mechanica,1984,51:119～138.
[7]Van Lieshout, P. H., Rongen, P. M. and Van de Ven, A. A. F., "A variational principle for magneto-elastic buckling", J. Eng. Math.,1987,21:227～252.
[8]Zhou, Y. H. and Zheng, X. J., "A general expression of magnetic force for soft ferromagnetic plates in complex magnetic fields", Int. J. Eng. Sci.,1997,35(15):1405～ 1417.
[9]周又和,郑晓静,“铁磁体磁弹性相互作用的广义变分原理与理论模型”,中国科学(A辑),1999,29(1)：61～68.
[10]王省哲.软铁磁薄板与薄壳结构的力-磁-热耦合问题研究.兰州大学博士论文,2000.
[11]Wang X. Z. and Zheng X. J., "Nonlinear analysis of magnetoelastic bending for soft ferromagnetic shells", Proc. ECM'99 International workshop on Experimental and Computational Mechanics in Engineering and Materials Behavior, Urumqi, September 1999,352～357.
[12]王省哲,郑晓静,“非圆Tokamak载流线圈的磁弹性分析”,兰州大学学报(自然科学版),1999,35(1)：34～43.
[13]周又和,郑晓静,王省哲,“悬臂铁磁板磁弹性耦合作用的力学分析”,固体力学学报,1998,19(3)：239-244.
[14]Zheng X. J., Zhou Y. H., Wang X. Z. and Lee J. S., "Bending and buckling of ferroelastic plates", ASCE Journal of Engineering Mechanics,1999,125(2):180～185.
[15]Zheng X. J., Wang X. Z. and Zhou Y. H., "Magnetoelastic analysis of non-circular superconducting partial torus", International Journal Solid and Structures,2000,37:563～ 576.
[16]Zheng, X.J. and Liu, X.N., "Analysis on dynamic characteristics for ferromagnetic conducting plates in a transverse uniform magnetic field", Acta Mechanica Solid Sinica, 2000,13(4):361～368.
[17]Zheng X. J., Zhang J. P., Zhou Y. H., "Dynamic stability of a cantilever conductive plate in
transverse impulsive magnetic field", International Journal of Solids and Structures,2005, 42:2417-2430.
[18]Zhu L.L., Zhang J.P., Zheng X.J., Multi-field coupling behavior of simply-supported conductive plate under the condition of a transverse strong impulsive magnetic field. Acta Mechanica Solida Sinica,2006,19:203-211.
[19]Gou X.F., Zheng X.J., Zhou Y.H." Drift of levitated/suspended body in high-tc superconducting levitation systems under vibration", IEEE Transactions on applied superconductivity,2007,17,(3):3795-3808.
[20]周浩淼,铁磁材料非线性磁弹性耦合理论及其在超磁致伸缩智能材料中的应用,兰州大学博士学位论文2007.
[21]Zheng, X. J., Sun, L., Jin, K'. "A dynamic hysteresis constitutive relation for giant magnetostrictive materials", Mechanics of Advanced Materials and Structures,16:7,516— 521.
[22]Zhou, Y. H. and Miya, K., "Magnetoelastic bending and snapping of large helical coils in a fusion reactor", In:Proc. Spring Annual of Japanese Association of Nuclear Engineering, Tokyo, March 1995.
[23]Zhou, Y. H., Zheng, X. J. and Miya K., "Magnetoelastic bending and buckling of three-coil superconducting partial torus", Fusion Eng. and Design,1996,30(3):275～289.
[24]Zheng X. J., Zhou Y. H. and Lee J. S., "Instability of superconducting partial torus with two pin supports", ASCE Journal of Engineering Mechanics,1999,125(2):174～179.
[25]周又和,郑晓静,“托卡马克聚变堆中环向磁场超导线圈的磁弹性弯曲与稳定性”,核聚变与等离子体物理,1997,7(2)：17～23.
[26]王省哲,郑晓静,“非圆Tokamak载流线圈的磁弹性分析”,兰州大学学报(自然科学版),1999,35(1)：34-43.
[27]X. J. Zheng, Y. H. Zhou, X. P. Yang, and K. Miya, "Analysis of dynamic stability of magnetoelastic coils in a three-coil partial torus", Fusion Eng. and Design,2001,54:31～ 39.
[28]A. Kameari, S. Niikura, "Control of plasma vertical position in tokamak reactors", Nucl. Eng. Des./Fusion,1985,2:365～373.
[29]T.Q. Huan, M.J. Knott, L.R. Turner, and R.B. Wehrle, "Experimental modeling of eddy currents and deflection for Tokamak limiters", Fusion Techology,1986,10:1047～1052.
[30]L. R. Turner and T.Q. Hua, "Experimental study of coupling between eddy currents and deflections in cantilevered beams as models of TOKAMAK limiters", Electromagneto-mechanical Interactions in Deformable Solids and Structures,1987:81～86.
[31]T. Morisue, "Analysis of a coupled problem:The FELIX cantilevered beam", IEEE Trans. On Magn.,1990,26:540～543.
[32]T. Kabashima, Y. Ueda, M. Ohto, "A study of the cantilever beam in the varying magnetic field", IEEE Transactions on Magnetics,1990,26(2):563～566.
[33]T. Takagi, J.Tani, S. Matsuda and S. Kawamura, "Analysis and experiment of dynamic
behavior of a thin conducting plate with a coupling effect", IEEE Trans. Magn.,1992,28: 1259～1262.
[34]Tomoyoshi Horie and Tomoya Niho, "Electromagnetic and structural coupled analysis with the effect of large deflection", IEEE Transactions on Magnetics,1997,33(2):1658～1661.
[35]Michael G. Pantelyat, "Coupled electromagnetic, thermal and elastic-plastic simulation of multi-impulse inductive heating", Int. J. of Applied Electromagnetics and Mechanics,1998, 9:11～24.
[36]Jerzy Zgraja and Michael G. Pantelyat, "Induction heating of large steel disks:coupled electromagnetic, thermal and mechanical simulation", Int. J. of Applied Electromagnetics and Mechanics,1999,10:303～313.
[37]田振国,白象忠,马世麟.“载流圆形薄板的磁弹性应力与变形分析”.力学季刊,2002,23(1)：109-114.
[38]边宇虹,田振国,白象忠,“两边固支条形薄板的非线性磁弹性效应分析”.工程力学,2006,23(8)：6-11.
[39]田振国,白象忠.“载流板壳电磁、热、机械场耦合的弹性效应分析”.应用基础与工程科学学报.2009,17(2)：318-324.
[40]Higuchi M., Kawamura R. Tanigawa,Y. "Magneto-thermo-elastic stresses induced by a transient magnetic field in a conducting solid circular cylinder", Int. J. Solids Struct.44, 5316-5335(2007).
[41]高原文,郭占东,徐榜.”导电矩形板的磁弹性塑性动力失稳特征研究”.振动工程学报,2008,21(3)：228-234.
[42]Yuanwen Gao . Bang Xu . Hoon Huh. "Electromagneto-thermo-mechanical behaviors of conductive circular plate subject to time-dependent magnetic fields", Acta Mech 2010,99-116.
[43]周又和,郑晓静.电磁固体结构力学.北京：科学出版社,1999.
[44]H. Hashizume, T. kurusu and S. Toda, "Numerical analysis of current distribution in type-II superconductors based on T-method", International Journal of Applied Electromagnetics in Materials,1992,3:205～213.
[45]汪映海.电动力学.兰州：兰州大学出版社,1995.
[46]王洪纲.热弹性力学概论.北京：清华大学出版社,1989.
[47]匡震邦.非线性连续介质力学基础.西安：西安交大出版社,1989.
[48]谢德馨.三维涡流场的有限元分析.北京：机械工业出版社,2007
[49]Thomson, W. T., Theory of vibration with applications, Prentice-Hall, Inc., Englewood Cliffs, New Jersey,1972
[50]李庆扬,王能超,易大义.数值分析(第三版).华中科技大学出版社,1986.
[51]O. C. Zeinkiewicz. The finite element method(third edition), McGraw-Hill,1977.
[52]X. Xu, R. F. Cai. "A new plat shell element of 16 nodes 40 degrees of freedom by relative displacement method", Communications in Numerical Methods in Engineering,1993, Vol.9,:15-20,
[53]徐兴,凌道盛,实体退化单元系列,固体力学学报(计算力学专辑),2001,Vol.22：1-12.
[54]凌道盛,张金江,项贻强,徐兴.“虚拟层合单元法及其在桥梁工程中的应用”,土木工程学报,1998,Vol.31：22-29.
[55]丁皓江,谢贻权,何福保,徐兴.弹性和塑性力学中的有限单元法(第2版).北京：机械工业出版社.1989.
[56]王勖成,邵敏.有限单元法基本原理及数值方法(第2版),北京：清华大学出版社,1997.
[57]Bathe, K. J., Finite Element Procedures in Engineering Analysis, Prentice-Hall, Inc., Englewood Cliffs, New Jersey,1982.
[58]Incropera, F.P. and D.P. DeWitt. Fundamentals of Heat and Mass Transfer, Second. Edition, John Wiley and Sons, New York,1985.
[59]Miya K, Akin J, Hanai S. "Finite element analysis of an eddy current induced in thin structures of a magnetic fusion reactor", International Journal for Numerical Methods in Engineering,1981,17:1613-1629.
[2]Moon, F. C. and Pao, Y. H., "Magnetoelastic buckling of a thin plate", J. Appl. Mech.,1968, 35(1):53～58.
[3]Moon, F. C. and Pao, Y. H., "Vibration and dynamic instability of a beam-plate in a transverse magnetic field", J. Appl. Mech.,1969,36(1):1～9.
[4]Pao Y. H. and Yeh C. S., "A linear theory for soft ferromagnetic elastic solids", Int. J. Eng. Sci.,1973,11(4):415～436.
[5]T. Kabashima, Y. Ueda, M. Ohto, "A study of the cantilever beam in the varying magnetic field", IEEE Transactions on Magnetics,1990,26(2):563～566.
[6]Van de ven, A. A. F., "Magnetoelastic buckling of a beam of elliptic cross section", Acta Mechanica,1984,51:119～138.
[7]Van Lieshout, P. H., Rongen, P. M. and Van de Ven, A. A. F., "A variational principle for magneto-elastic buckling", J. Eng. Math.,1987,21:227～252.
[8]Zhou, Y. H. and Zheng, X. J., "A general expression of magnetic force for soft ferromagnetic plates in complex magnetic fields", Int. J. Eng. Sci.,1997,35(15):1405～ 1417.
[9]周又和,郑晓静,“铁磁体磁弹性相互作用的广义变分原理与理论模型”,中国科学(A辑),1999,29(1)：61～68.
[10]王省哲.软铁磁薄板与薄壳结构的力-磁-热耦合问题研究.兰州大学博士论文,2000.
[11]Wang X. Z. and Zheng X. J., "Nonlinear analysis of magnetoelastic bending for soft ferromagnetic shells", Proc. ECM'99 International workshop on Experimental and Computational Mechanics in Engineering and Materials Behavior, Urumqi, September 1999,352～357.
[12]王省哲,郑晓静,“非圆Tokamak载流线圈的磁弹性分析”,兰州大学学报(自然科学版),1999,35(1)：34～43.
[13]周又和,郑晓静,王省哲,“悬臂铁磁板磁弹性耦合作用的力学分析”,固体力学学报,1998,19(3)：239-244.
[14]Zheng X. J., Zhou Y. H., Wang X. Z. and Lee J. S., "Bending and buckling of ferroelastic plates", ASCE Journal of Engineering Mechanics,1999,125(2):180～185.
[15]Zheng X. J., Wang X. Z. and Zhou Y. H., "Magnetoelastic analysis of non-circular superconducting partial torus", International Journal Solid and Structures,2000,37:563～ 576.
[16]Zheng, X.J. and Liu, X.N., "Analysis on dynamic characteristics for ferromagnetic conducting plates in a transverse uniform magnetic field", Acta Mechanica Solid Sinica, 2000,13(4):361～368.
[17]Zheng X. J., Zhang J. P., Zhou Y. H., "Dynamic stability of a cantilever conductive plate in
transverse impulsive magnetic field", International Journal of Solids and Structures,2005, 42:2417-2430.
[18]Zhu L.L., Zhang J.P., Zheng X.J., Multi-field coupling behavior of simply-supported conductive plate under the condition of a transverse strong impulsive magnetic field. Acta Mechanica Solida Sinica,2006,19:203-211.
[19]Gou X.F., Zheng X.J., Zhou Y.H." Drift of levitated/suspended body in high-tc superconducting levitation systems under vibration", IEEE Transactions on applied superconductivity,2007,17,(3):3795-3808.
[20]周浩淼,铁磁材料非线性磁弹性耦合理论及其在超磁致伸缩智能材料中的应用,兰州大学博士学位论文2007.
[21]Zheng, X. J., Sun, L., Jin, K'. "A dynamic hysteresis constitutive relation for giant magnetostrictive materials", Mechanics of Advanced Materials and Structures,16:7,516— 521.
[22]Zhou, Y. H. and Miya, K., "Magnetoelastic bending and snapping of large helical coils in a fusion reactor", In:Proc. Spring Annual of Japanese Association of Nuclear Engineering, Tokyo, March 1995.
[23]Zhou, Y. H., Zheng, X. J. and Miya K., "Magnetoelastic bending and buckling of three-coil superconducting partial torus", Fusion Eng. and Design,1996,30(3):275～289.
[24]Zheng X. J., Zhou Y. H. and Lee J. S., "Instability of superconducting partial torus with two pin supports", ASCE Journal of Engineering Mechanics,1999,125(2):174～179.
[25]周又和,郑晓静,“托卡马克聚变堆中环向磁场超导线圈的磁弹性弯曲与稳定性”,核聚变与等离子体物理,1997,7(2)：17～23.
[26]王省哲,郑晓静,“非圆Tokamak载流线圈的磁弹性分析”,兰州大学学报(自然科学版),1999,35(1)：34-43.
[27]X. J. Zheng, Y. H. Zhou, X. P. Yang, and K. Miya, "Analysis of dynamic stability of magnetoelastic coils in a three-coil partial torus", Fusion Eng. and Design,2001,54:31～ 39.
[28]A. Kameari, S. Niikura, "Control of plasma vertical position in tokamak reactors", Nucl. Eng. Des./Fusion,1985,2:365～373.
[29]T.Q. Huan, M.J. Knott, L.R. Turner, and R.B. Wehrle, "Experimental modeling of eddy currents and deflection for Tokamak limiters", Fusion Techology,1986,10:1047～1052.
[30]L. R. Turner and T.Q. Hua, "Experimental study of coupling between eddy currents and deflections in cantilevered beams as models of TOKAMAK limiters", Electromagneto-mechanical Interactions in Deformable Solids and Structures,1987:81～86.
[31]T. Morisue, "Analysis of a coupled problem:The FELIX cantilevered beam", IEEE Trans. On Magn.,1990,26:540～543.
[32]T. Kabashima, Y. Ueda, M. Ohto, "A study of the cantilever beam in the varying magnetic field", IEEE Transactions on Magnetics,1990,26(2):563～566.
[33]T. Takagi, J.Tani, S. Matsuda and S. Kawamura, "Analysis and experiment of dynamic
behavior of a thin conducting plate with a coupling effect", IEEE Trans. Magn.,1992,28: 1259～1262.
[34]Tomoyoshi Horie and Tomoya Niho, "Electromagnetic and structural coupled analysis with the effect of large deflection", IEEE Transactions on Magnetics,1997,33(2):1658～1661.
[35]Michael G. Pantelyat, "Coupled electromagnetic, thermal and elastic-plastic simulation of multi-impulse inductive heating", Int. J. of Applied Electromagnetics and Mechanics,1998, 9:11～24.
[36]Jerzy Zgraja and Michael G. Pantelyat, "Induction heating of large steel disks:coupled electromagnetic, thermal and mechanical simulation", Int. J. of Applied Electromagnetics and Mechanics,1999,10:303～313.
[37]田振国,白象忠,马世麟.“载流圆形薄板的磁弹性应力与变形分析”.力学季刊,2002,23(1)：109-114.
[38]边宇虹,田振国,白象忠,“两边固支条形薄板的非线性磁弹性效应分析”.工程力学,2006,23(8)：6-11.
[39]田振国,白象忠.“载流板壳电磁、热、机械场耦合的弹性效应分析”.应用基础与工程科学学报.2009,17(2)：318-324.
[40]Higuchi M., Kawamura R. Tanigawa,Y. "Magneto-thermo-elastic stresses induced by a transient magnetic field in a conducting solid circular cylinder", Int. J. Solids Struct.44, 5316-5335(2007).
[41]高原文,郭占东,徐榜.”导电矩形板的磁弹性塑性动力失稳特征研究”.振动工程学报,2008,21(3)：228-234.
[42]Yuanwen Gao . Bang Xu . Hoon Huh. "Electromagneto-thermo-mechanical behaviors of conductive circular plate subject to time-dependent magnetic fields", Acta Mech 2010,99-116.
[43]周又和,郑晓静.电磁固体结构力学.北京：科学出版社,1999.
[44]H. Hashizume, T. kurusu and S. Toda, "Numerical analysis of current distribution in type-II superconductors based on T-method", International Journal of Applied Electromagnetics in Materials,1992,3:205～213.
[45]汪映海.电动力学.兰州：兰州大学出版社,1995.
[46]王洪纲.热弹性力学概论.北京：清华大学出版社,1989.
[47]匡震邦.非线性连续介质力学基础.西安：西安交大出版社,1989.
[48]谢德馨.三维涡流场的有限元分析.北京：机械工业出版社,2007
[49]Thomson, W. T., Theory of vibration with applications, Prentice-Hall, Inc., Englewood Cliffs, New Jersey,1972
[50]李庆扬,王能超,易大义.数值分析(第三版).华中科技大学出版社,1986.
[51]O. C. Zeinkiewicz. The finite element method(third edition), McGraw-Hill,1977.
[52]X. Xu, R. F. Cai. "A new plat shell element of 16 nodes 40 degrees of freedom by relative displacement method", Communications in Numerical Methods in Engineering,1993, Vol.9,:15-20,
[53]徐兴,凌道盛,实体退化单元系列,固体力学学报(计算力学专辑),2001,Vol.22：1-12.
[54]凌道盛,张金江,项贻强,徐兴.“虚拟层合单元法及其在桥梁工程中的应用”,土木工程学报,1998,Vol.31：22-29.
[55]丁皓江,谢贻权,何福保,徐兴.弹性和塑性力学中的有限单元法(第2版).北京：机械工业出版社.1989.
[56]王勖成,邵敏.有限单元法基本原理及数值方法(第2版),北京：清华大学出版社,1997.
[57]Bathe, K. J., Finite Element Procedures in Engineering Analysis, Prentice-Hall, Inc., Englewood Cliffs, New Jersey,1982.
[58]Incropera, F.P. and D.P. DeWitt. Fundamentals of Heat and Mass Transfer, Second. Edition, John Wiley and Sons, New York,1985.
[59]Miya K, Akin J, Hanai S. "Finite element analysis of an eddy current induced in thin structures of a magnetic fusion reactor", International Journal for Numerical Methods in Engineering,1981,17:1613-1629.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |