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为研究红外无损探测稳态多热源反演逆问题, 建立不同形状的均质与非均质稳态热传导模型, 其中内热源个数、位置、强度、面积均为未知项. 基于数值算法中有限元算法对模型进行离散分析, 化简有限元矩阵方程, 最终转化为对Ax = b高度欠定方程的求解. 首次利用分段多项式谱截断奇异值分解法处理内热源逆问题, 并对算法进行改进, 有效改善了该算法在处理多热源反演时存在的严重的热源叠加效应. 根据反演出的内热源信息, 利用有限元算法计算重构出整个模型内所有节点的温度分布. 运用数值仿真Comsol软件和具体实物实验对算法进行有效性评估, 并验证算法在不同热传导模型中的表现. 结果表明, 算法能够准确反演出多热源各参量信息, 在非均质材料模型中仍能准确地反演出热源项, 并有效重构出模型内温度场. 该算法可应用于材料无损检测及人体红外医学成像等领域.
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关键词:
- 无损探测 /
- 多内热源反演 /
- 分段多项式谱截断奇异值分解法 /
- 有限元
This study deals with the case of multiple internal heat source inversion problem of steady-state based on nondestructive infrared detection. We construct homogeneous and heterogeneous steady heat conduction models of different shapes. Neither the number of heat sources, nor their locations, nor their sizes nor their intensities are known. We use the finite element method (FEM) based on numerical algorithm to analyze the two-dimensional model discretely. The internal heat conduction process of model is analyzed. The resultant temperature field can be decomposed into the temperature field caused by the ambient temperature and those given by internal heat sources. We simplify the finite element matrix equation according to decomposition process above. Finally, the problem boils down to solving the highly underdetermined matrix equation of Ax = b. The unknown x item corresponds to internal thermal heat source field. The piecewise polynomial spectral truncated singular value decomposition (PPTSVD) is applied for the first time to the inverse heat source problem. Its regular operator matrix is changed from the original more order differential operator matrix to regional node weighted matrix. After replacement this solution improves the effect of heat source field tending to the boundary. Results of the solution confirms a real heat source field when there are less heat sources or different heat sources are far from each other. But there also exits a serious superimposed effect between neighboring heat sources. We improve the algorithm to study this problem through using the iterative elimination process which complies with the idea of spreading heat source field and then gathering. The iteration tolerance and number of times belonging to one single PPTSVD solving process are reduced. Through iterating the multiple PPTSVD solving process and reconstructing matrixes A and b in each iteration, we obtain the scatter heat source field distribution surrounding real field. Finally, this scattered distribution solution is gathered again. According to the heat source parameter calculated by algorithm, the temperature field of whole model can be reconstructed using FEM. Comsol numerical simulation and real physical experiments are performed to verify the validity and accuracy of the algorithm in different heat conduction models. The results demonstrate that the algorithm can access each parameter of multiple heat sources. Even in the heterogeneous model, it can still obtain accurate results and reconstruct the temperature field of the two-dimensional model. The algorithm can be applied to non-destructive material detection and human infrared medical imaging fields.-
Keywords:
- nondestructive detection /
- multiple internal heat sources inversion /
- piecewise polynomial truncated singular value decomposition methods /
- finite element method
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[2] Li H Q, Lei J, Liu Q B 2012 Int. J. Heat Mass Trans. 55 4442
[3] Kolodziej J A, Mierzwiczak M, Cialkowski M 2010 Int. Commun. Heat. Mass. 37 121
[4] Yan L, Fu C L, Yang F L 2008 Eng. Anal. Bound. Elem. 32 216
[5] Hazanee A, Ismailov M I, Lesnic D, Kerimov N B 2013 Appl. Numer. Math. 69 13
[6] Geng F, Lin Y 2009 Comput. Math. Appl. 58 2098
[7] Shi C, Wang C, Wei T 2014 Appl. Math. Comput. 244 577
[8] Xiong X, Yan Y, Wang J 2011 J. Phys.:Conf. Ser. 290 012017
[9] Li F L, Zhang H Q 2011 Chin. Phys. B 20 100201
[10] Nili Ahmadabadi M, Arab M, Maalek Ghaini F M 2009 Eng. Anal. Bound. Elem. 33 1231
[11] Yang F, Fu C L 2014 J. Comput. Appl. Math. 255 555
[12] Yang F, Fu C L 2010 Comput. Math. Appl. 60 1228
[13] Yang L, Dehghan M, Yu J N, Luo G W 2011 Math. Comput. Simulat. 81 1656
[14] Wu Z, Liu H H, Lebanowski L, Liu Z Q, Hor P H 2007 Phys. Med. Biol. 52 5379
[15] Jiang Z H, Huang S X, Du H D, Liu B 2010 Acta Phys. Sin. 59 8968 (in Chinese) [姜祝辉, 黄思训, 杜华栋, 刘博 2010 59 8968]
[16] Huang Q X, Liu D, Wang F, Yan J H, Chi Y, Cen K F 2007 Acta Phys. Sin. 56 6742 (in Chinese) [黄群星, 刘冬, 王飞, 严建华, 池涌, 岑可法 2007 56 6742]
[17] Dykes L, Reichel L 2014 J. Comput. Appl. Math. 255 15
[18] Alifanov 1994 Inverse Heat Transfer Problems (New York: Springer-Verlag) p150
[19] Liu G R, Lee J H, Patera A T, Yang Z L, Lam K Y 2005 J. Comput. Methods Appl. Mech. Eng. 194 3090
[20] Lu T, Liu B, Jiang P X, Zhang Y W, Li H 2010 Appl. Therm. Eng. 30 1574
[21] Kim D, Lee H, Won Y, Kim D G, Lee Y, Won H 2003 J. Bull. Korean Chem. Soc. 24 967
[22] Hansen P C, Mosegaard K 1996 J. Num. Lin. Alg. Appl. 3 513
[23] Lwaki S, Ueno S 1998 J. Appl. Phys. 83 6441
[24] Xiao H F 2009 Ph. D. Dissertation (Changsha: Central South University) (in Chinese) [肖宏峰 2009 博士学位论文(长沙: 中南大学)]
[25] Zhou M H 2009 Ph. D. Dissertation (in Chinese) (Nanjing: Nanjing University of Science and Technology) [周敏华 2009 博士学位论文(南京: 南京理工大学)]
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[1] Beck J V, Blackwell B, Clair C R 1985 Inverse Heat Conduction: Ill-posed Problems (New York: Wiley) p119
[2] Li H Q, Lei J, Liu Q B 2012 Int. J. Heat Mass Trans. 55 4442
[3] Kolodziej J A, Mierzwiczak M, Cialkowski M 2010 Int. Commun. Heat. Mass. 37 121
[4] Yan L, Fu C L, Yang F L 2008 Eng. Anal. Bound. Elem. 32 216
[5] Hazanee A, Ismailov M I, Lesnic D, Kerimov N B 2013 Appl. Numer. Math. 69 13
[6] Geng F, Lin Y 2009 Comput. Math. Appl. 58 2098
[7] Shi C, Wang C, Wei T 2014 Appl. Math. Comput. 244 577
[8] Xiong X, Yan Y, Wang J 2011 J. Phys.:Conf. Ser. 290 012017
[9] Li F L, Zhang H Q 2011 Chin. Phys. B 20 100201
[10] Nili Ahmadabadi M, Arab M, Maalek Ghaini F M 2009 Eng. Anal. Bound. Elem. 33 1231
[11] Yang F, Fu C L 2014 J. Comput. Appl. Math. 255 555
[12] Yang F, Fu C L 2010 Comput. Math. Appl. 60 1228
[13] Yang L, Dehghan M, Yu J N, Luo G W 2011 Math. Comput. Simulat. 81 1656
[14] Wu Z, Liu H H, Lebanowski L, Liu Z Q, Hor P H 2007 Phys. Med. Biol. 52 5379
[15] Jiang Z H, Huang S X, Du H D, Liu B 2010 Acta Phys. Sin. 59 8968 (in Chinese) [姜祝辉, 黄思训, 杜华栋, 刘博 2010 59 8968]
[16] Huang Q X, Liu D, Wang F, Yan J H, Chi Y, Cen K F 2007 Acta Phys. Sin. 56 6742 (in Chinese) [黄群星, 刘冬, 王飞, 严建华, 池涌, 岑可法 2007 56 6742]
[17] Dykes L, Reichel L 2014 J. Comput. Appl. Math. 255 15
[18] Alifanov 1994 Inverse Heat Transfer Problems (New York: Springer-Verlag) p150
[19] Liu G R, Lee J H, Patera A T, Yang Z L, Lam K Y 2005 J. Comput. Methods Appl. Mech. Eng. 194 3090
[20] Lu T, Liu B, Jiang P X, Zhang Y W, Li H 2010 Appl. Therm. Eng. 30 1574
[21] Kim D, Lee H, Won Y, Kim D G, Lee Y, Won H 2003 J. Bull. Korean Chem. Soc. 24 967
[22] Hansen P C, Mosegaard K 1996 J. Num. Lin. Alg. Appl. 3 513
[23] Lwaki S, Ueno S 1998 J. Appl. Phys. 83 6441
[24] Xiao H F 2009 Ph. D. Dissertation (Changsha: Central South University) (in Chinese) [肖宏峰 2009 博士学位论文(长沙: 中南大学)]
[25] Zhou M H 2009 Ph. D. Dissertation (in Chinese) (Nanjing: Nanjing University of Science and Technology) [周敏华 2009 博士学位论文(南京: 南京理工大学)]
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