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Based on variable domain variational principle (VP), the variational theory of inverse shape design of heat conduction together with the boundary and the additional boundary conditions enforced on the unknown surface which must be satisfied by the problem is established. Incorporating the boundary and additional conditions into the functional makes the mathematic formulation very concise and compact. Through the established VP, both the temperature field and unknown boundary shape can be determined in a coupled way. The numerical method of the variational principle combined with finite element method is presented in the paper.
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Keywords:
- inverse shape design /
- variational theory /
- finite element method /
- heat conduction
[1] Marquardt D M 1963 Journal of Social Industrial Applied Mathematics 11 431
[2] Huang C H, Chao B H 1997 Int. J. Heat Mass Tran. 40 2045
[3] Sluzalec A, Kleiber M 1996 Int. J. Heat Mass Tran. 39 2609
[4] Li Q, Steven G P, Querin O M, Xie Y M 1999 Int. J. Heat Mass Tran. 42 3361
[5] Sun K K, Bup S J, Woo I L 2007 International Journal of Communications in Heat and Mass Transfer 34 37
[6] Chen H T, Chou J C, Wang H C 2007 Int. J. Heat Mass Tran. 50 45
[7] Yang H T, Hu G J, Xue Q W 2003 Journal of Dalian University of Technology 43 136 (in Chinese) [扬海天、胡国俊、薛齐文2003大连理工大学学报43 136]
[8] Wu Z C 2005 Acta Aerodynamica Sinica 23 114 (in Chinese) [吴兆春2005 空气动力学学报23 114]
[9] Cheng R J, Cheng Y M 2007 Acta Phys. Sin. 56 5569 (in Chinese) [程荣军、程玉民 2007 物理学报56 5569]
[10] Chen l, Cheng Y M 2008 Acta Phys. Sin. 57 6047 (in Chinese) [陈丽、程玉民 2008 物理学报57 6047]
[11] Hung C H, Lo H C 2005 Nume. Heat Tr: A-Appl. 48 1009
[12] You Y X, Miu G P 2002 Acta Phys. Sin.51 270 (in Chinese) [尤云祥、 缪国平2002 物理学报51 270]
[13] Yang K D, Ma Y L 2009 Acta Phys. Sin. 58 1798(in Chinese) [杨坤德、马远良 2009物理学报58 1798]
[14] Sheikh A H, Beck J V 2006 Numer. Heat TR: A-Appl. 50 401
[15] Wu Z C 2009 Commun. Numer. Meth. En. 25 1107
[16] Kong X Q 1998 Application of finite element method in heat transfer
[17] Courant R, Hilbert D 1953 Methods in mathematics physics (New York: Interscience) p260
[18] Qian W C 1980 Calculus of variations and finite element method (Vol.1) (Beijing: Science publishing house ) p116 (in Chinese) [钱伟长 1980 变分法及有限元(上册)( 北京:科学出版社)第116页]
[19] Liu G L 1981 Variational principles for aerodynamics, finite element method and theory of optimization design (lecture note) p60 (in Chinese) [刘高联1981 流体力学变分原理,有限元和优化设计理论(讲义) 第60页]
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[1] Marquardt D M 1963 Journal of Social Industrial Applied Mathematics 11 431
[2] Huang C H, Chao B H 1997 Int. J. Heat Mass Tran. 40 2045
[3] Sluzalec A, Kleiber M 1996 Int. J. Heat Mass Tran. 39 2609
[4] Li Q, Steven G P, Querin O M, Xie Y M 1999 Int. J. Heat Mass Tran. 42 3361
[5] Sun K K, Bup S J, Woo I L 2007 International Journal of Communications in Heat and Mass Transfer 34 37
[6] Chen H T, Chou J C, Wang H C 2007 Int. J. Heat Mass Tran. 50 45
[7] Yang H T, Hu G J, Xue Q W 2003 Journal of Dalian University of Technology 43 136 (in Chinese) [扬海天、胡国俊、薛齐文2003大连理工大学学报43 136]
[8] Wu Z C 2005 Acta Aerodynamica Sinica 23 114 (in Chinese) [吴兆春2005 空气动力学学报23 114]
[9] Cheng R J, Cheng Y M 2007 Acta Phys. Sin. 56 5569 (in Chinese) [程荣军、程玉民 2007 物理学报56 5569]
[10] Chen l, Cheng Y M 2008 Acta Phys. Sin. 57 6047 (in Chinese) [陈丽、程玉民 2008 物理学报57 6047]
[11] Hung C H, Lo H C 2005 Nume. Heat Tr: A-Appl. 48 1009
[12] You Y X, Miu G P 2002 Acta Phys. Sin.51 270 (in Chinese) [尤云祥、 缪国平2002 物理学报51 270]
[13] Yang K D, Ma Y L 2009 Acta Phys. Sin. 58 1798(in Chinese) [杨坤德、马远良 2009物理学报58 1798]
[14] Sheikh A H, Beck J V 2006 Numer. Heat TR: A-Appl. 50 401
[15] Wu Z C 2009 Commun. Numer. Meth. En. 25 1107
[16] Kong X Q 1998 Application of finite element method in heat transfer
[17] Courant R, Hilbert D 1953 Methods in mathematics physics (New York: Interscience) p260
[18] Qian W C 1980 Calculus of variations and finite element method (Vol.1) (Beijing: Science publishing house ) p116 (in Chinese) [钱伟长 1980 变分法及有限元(上册)( 北京:科学出版社)第116页]
[19] Liu G L 1981 Variational principles for aerodynamics, finite element method and theory of optimization design (lecture note) p60 (in Chinese) [刘高联1981 流体力学变分原理,有限元和优化设计理论(讲义) 第60页]
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