稳态热传导线源识别反问题的数值通解方法

上海交通大学学报（自然版） ›› 2014, Vol. 48 ›› Issue (02): 265-270.

稳态热传导线源识别反问题的数值通解方法

蒋兰芳a，刘红b，柴国钟b，杨友东a，竺炯林b

(浙江工业大学 a. 之江学院，杭州 310024； b. 机械工程学院，杭州 310032)

收稿日期:2013-05-31
基金资助:
国家自然科学基金(51275471), 浙江省自然科学基金（LQ13A020003）资助项目

Method of Numerical General Solution for Inverse Line Heat Source Problem in Steady State

JIANG Lanfanga,LIU Hongb，CHAI Guozhongb,YANG Youdonga，ZHU Jionglinb

(a. Zhejiang College, Zhejiang University of Technology, Hangzhou 310024, China;b. College of Mechanical Engineering, Zhejiang University of Technology, Hangzhou 310032, China)

Received:2013-05-31

摘要/Abstract

摘要：

提出了一种稳态热传导源项识别反问题的求解方法.构造了基于有限元数值解，满足热传导控制方程和边界条件、离散形式、以热源强度为变量的数值通解显式表达，从而将热传导源项识别反问题转化为多元极值问题，能够方便地反演出热源强度.推导了基本计算公式，讨论了热传导源项识别中热源强度的数学建模问题，进行了算法的误差分析.并以模具加热板的热源设计为算例，验证了算法的正确性与实用性.这对进一步研究热源空间分布识别和多维源项识别反问题具有重要意义.

关键词: 热传导反问题, 源项识别, 热源强度, 数值通解

Abstract:

A method for identification of source terms for inverse heat conduction problems in steady state was presented. Based on finite element numerical solution, an explicit formula for numerical general solution was constructed, which satisfied the heat conduction governing equations and boundary condition, taking the intensity of heat source as the variable with discrete form. The method converted the heat conduction inverse problem to multivariables limit problem, which made the identification of intensity of heat source easy. The basic computational formula was derived, the mathematics modeling problem of identification of intensity of heat source for inverse heat conduction problems was discussed and error analysis was conducted. Taking the heat source design of mould heating plate as an example, the correctness and practicability of the algorithm was validated, which is of importance for further investigation of identification of space of heat source and multidimensional source terms for inverse heat conduction problems.

Key words: inverse heat conduction problem, identification of source terms, intensity of heat source, numerical general solution

中图分类号:

蒋兰芳a，刘红b，柴国钟b，杨友东a，竺炯林b. 稳态热传导线源识别反问题的数值通解方法[J]. 上海交通大学学报（自然版）, 2014, 48(02): 265-270.

JIANG Lanfanga,LIU Hongb，CHAI Guozhongb,YANG Youdonga，ZHU Jionglinb. Method of Numerical General Solution for Inverse Line Heat Source Problem in Steady State[J]. Journal of Shanghai Jiaotong University, 2014, 48(02): 265-270.

参考文献

［1］张小明, 王泽文, 王燕.一类热传导方程多点源反演的唯一性和稳定性［J］.华东理工大学学报, 2009, 32(2):189194.

ZHANG Xiaoming, WANG Zewen, WANG Yan. Uniqueness and stability in determining the point sources for a heat conduction equation［J］. Journal of East China Institute of Technology (Natural Science), 2009, 32(2):189194.

［2］肖庭延, 张俊丽.一类热传导问题源项识别的快速稳定算法［J］.计算物理, 2008, 25(3):335343.

XIAO Tingyan, ZHANG Junli. Stable and fast algorithm for identification of source in heat conduction problems［J］. Computational Physics, 2008, 25(3):335343.

［3］CHENG Wei, ZHAO Lingling, FU Chuli. Source term identification for an axisymmetric inverse heat conduction problem［J］. Computers and Mathematics with Applications, 2010, 59 (1):142148.

［4］Hasanov A. Identification of spacewise and time dependent source terms in 1D heat conduction equation from temperature measurement at a final time［J］. International Journal of Heat and Mass Transfer, 2012, 55(78):20692080.

［5］LIN Shihming. A sequential algorithm and error sensitivity analysis for the inverse heat conduction problems with multiple heat sources［J］. Applied Mathematical Modeling, 2011, 35(6): 26072617.

［6］Kolodziej J A, Mierzwiczak M, Cialkowski M. Application of the method of fundamental solutions and radial basis functions for inverse heat source problem in case of steadystate［J］. International Communications in Heat and Mass Transfer, 2010, 37 (2):121124.

［7］Mierzwiczak M, Kolodziej J A. Application of the method of fundamental solutions and radial basis functions for inverse transient heat source problem［J］. Computer Physics Communications, 2010, 181(12):20352043.

［8］Su J, Silva Neto A J. Twodimensional inverse heat conduction problem of source strength estimation in cylindrical rods［J］. Applied Mathematical Modeling, 2001, 25 (10):861872.

［9］薛齐文, 魏伟.非线性热传导反问题参数辨识［J］.工程力学, 2010, 27(8): 59.

XUE Qiwen, WEI Wei. Parameters identification of nonlinear inverse heat conduction problem［J］. Engineering Mechanics, 2010, 27(8):59.

［10］薛齐文, 杨海天.二阶非定常多宗量热传导反问题的正则解［J］.力学学报, 2007, 39(6):774780.

XUE Qiwen, YANG Haitian. A regularized solution of secondorder transient inverse heat conduction problems with multivariables［J］. Chinese Journal of Theoretical and Applied Mechanics, 2007, 39(6):774780.

［11］杨海天, 薛齐文.两级敏度分析求解非线性稳态多宗量热传导反问题［J］.工热热物理学报, 2003, 24(3):463465.

YANG Haitian, XUE Qiwen. Solving nonlinear inverse heat conduction problems in steady state via 2level sensitivity analysis［J］. Journal of Engineering Thermophysics, 2003, 24(3): 463465.

［12］Silva P M, Orlande H B, Colaco H B. Identification and design of source term in a tworegion heat conduction problem［J］. Inverse Problems in Science and Engineering, 2007, 15 (7):661677.