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## Reduced order model analysis method via proper orthogonal decomposition for variable coefficient of transient heat conduction based on boundary element method

Hu Jin-Xiu, Gao Xiao-Wei
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• #### 摘要

提出了一种基于边界元法求解变系数瞬态热传导问题的特征正交分解(POD)降阶方法, 重组并推导出变系数瞬态热传导问题适合降阶的边界元离散积分方程, 建立了变系数瞬态热传导问题边界元格式的POD降阶模型, 并用常数边界条件下建立的瞬态热传导问题的POD降阶模态, 对光滑时变边界条件瞬态热传导问题进行降阶分析. 首先, 对一个变系数瞬态热传导问题, 建立其边界域积分方程, 并将域积分转换成边界积分; 其次, 离散并重组积分方程, 获得可用于降阶分析的矩阵形式的时间微分方程组; 最后, 用POD模态矩阵对该时间微分方程组进行降阶处理, 建立降阶模型并对其求解. 数值算例验证了本文方法的正确性和有效性. 研究表明: 1) 常数边界条件下建立的低阶POD模态矩阵, 能够用来准确预测复杂光滑时变边界条件下的温度场结果; 2)低阶模型的建立, 解决了边界元法中采用时间差分推进技术求解大型时间微分方程组时求解速度慢、算法稳定性差的问题.

#### Abstract

Boundary element method (BEM) is widely used in engineering analysis, especially in solving the transient heat conduction problem because of the advantage that only boundary of the problem needs to be discretized into elements. The general procedure of solving the variable-coefficient transient heat conduction problem by using the BEM is as follows. First, the governing differential equations are transformed into the boundary-domain integral equations by adopting the basic solution of the linear and homogeneous heat conduction problemGreen function. Second, domain integrals in the integral equation are converted into boundary integrals by the radial integral method or the dual reciprocity method. Finally, the time difference propulsion technology is used to solve the discrete time differential equations. A large number of practical examples verify the correctness and validity of the BEM in solving the variable coefficient of transient heat conduction problem. However, two deficiencies are encountered when the system of time differential equations is solved with the time difference method, i.e., one is the stability of the algorithm, which is closely related to the time step size, and the other is time-consuming when the freedom degree of the problem is large and all specified time steps are considered, because a system of linear equations needs to be solved in each time step. Therefore, in this paper we present a reduced order model analysis method of solving the variable-coefficient transient heat conduction problem based on BEM by using the model reduction method of proper orthogonal decomposition (POD). For variable-coefficient transient heat conduction problems, the discrete integral equations which are suitable for order reduction operation are deduced by using the BEM, the reduced order model is established by using the model reduction method of POD, and a lowdimensional approximate description of the transient heat conduction problem under time-varying boundary condition is obtained by projection of the initial discrete integral equations on some few dominant POD modes obtained from the problem under constant boundary conditions. First, for a variable coefficient transient heat conduction problem, boundary-domain integral equations are established and the domain integrals are transformed into boundary integrals by using the radial integration method. Second, the time differential equations with discrete format which is suitable for order reduction operation are obtained by reorganizing the integral equations. Third, the POD modes are developed by calculating the eigenvectors of an autocorrelation matrix composed of snapshots which are clustered by the given results obtained from experiments, BEM or other numerical methods for transient heat transfer problem with constant boundary conditions. Finally, the reduced order model is established and solved by projecting the time differential equations on reduced POD modes. Examples show that the method developed in this paper is correct and effective. It is shown that 1) the low order POD modes determined under constant boundary conditions can be used to accurately analyze the temperature field of transient heat conduction problems with the same geometric domain but a variety of smooth and time-varying boundary conditions; 2) the establishment of low order model solves the problem of heavy workload encountered in BEM where a set of large linear equations will be formed and solved in each time step when using the time difference method to solve the large time differential equations.

#### 作者及机构信息

###### 通信作者: 高效伟, xwgao@dlut.edu.cn
• 基金项目: 国家自然科学基金(批准号: 11172055)资助的课题.

#### Authors and contacts

###### Corresponding author: Gao Xiao-Wei, xwgao@dlut.edu.cn
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11172055).

#### 参考文献

 [1] Brebbia C A, Dominguez J 1992 Boundary Elements: an Introductory Course (London: McGraw-Hill Book Co.) pp52-57 [2] Gao X W, Davies T G 2002 Boundary Element Programming in Mechanics (Cambridge: Cambridge University Press) pp25-33 [3] Gao X W, Peng H F, Yang K, Wang J 2015 Advanced Boundary Element MethodTheory and Application (Beijing: Science Press) pp110-129 (in Chinese) [高效伟, 彭海峰, 杨凯, 王静 2015 高等边界元法理论与程序 (北京: 科学出版社) 第110129页] [4] Ciskowski R D, Brebbia C A 1991 Boundary Element Methods in Acoustics (Southampton: Elsevier) [5] Gao X W, Hu J X 2012 Acta Mech. Sin. 44 361 (in Chinese) [高效伟, 胡金秀 2012 力学学报 44 361] [6] Zhang A M, Yao X L 2008 Chin. Phys. B 17 927 [7] Li S D, Huang Q B, Li T Y 2012 Acta Phys. Sin. 61 64301 (in Chinese) [李善德, 黄其柏, 李天匀 2012 物理学报 61 64301] [8] Xu J, Xie W H, Deng Y, Wang K, Luo Z Y, Gong H 2013 Acta Phys. Sin. 62 104204 (in Chinese) [许军, 谢文浩, 邓勇, 王侃, 罗召洋, 龚辉 2013 物理学报 62 104204] [9] Li S, He H L 2013 Chin. Phys. B 22 24701 [10] Yang K, Gao X W 2010 Eng. Anal. Bound. Elem. 34 557 [11] Gao X W, Wang J 2009 Eng. Anal. Bound. Elem. 33 539 [12] Gao X W, Peng H F, Liu J 2013 Int. J. Heat Mass Transf. 63 183 [13] Peng H F, Bai Y G, Yang K, Gao X W 2013 Eng. Anal. Bound. Elem. 37 1545 [14] Sutradhar A, Paulino G H 2004 Comput. Meth. Appl. Mech. Eng. 193 4511 [15] Erhart K, Divo E, Kassab A J 2006 Eng. Anal. Bound. Elem. 30 553 [16] Mohammadi M. Hematiyan M R, Marin L 2010 Eng. Anal. Bound. Elem. 34 655 [17] Yu B, Yao W A, Gao X W Gao Q 2014 Numer. Heat Transf. Part B: Fundam. 65 155 [18] Gao X W 2002 Eng. Anal. Bound. Elem. 26 905 [19] Gao X W 2005 J. Comput. Appl. Math. 175 265 [20] Hu J X, Peng H F, Gao X W 2014 Math. Probl. Eng. 2014 284106 [21] Hu J X, Zheng B J, Gao X W 2013 Bound. Elem. Mesh. Reduc. Meth. XXXVI 56 153 [22] Nardini D, Brebbia C A 1982 Boundary Element Methods in Engineering (Berlin: Springer) pp312-326 [23] Jiang Y L 2010 Model Reduction Method (Beijing: Science Press) pp1-4 (in Chinese) [蒋耀林 2010 模型降阶方法 (北京: 科学出版社) 第14页] [24] Chatterjee A 2000 Curr. Sci. 78 808 [25] Liang Y C, Lee H P, Lim S P, Lin W Z, Lee K H Wu C G 2002 J. Sound Vib. 252 527 [26] Fic A, Bialecki R A, Kassab A J 2005 Numer. Heat Transf. Part B: Fundam 48 103 [27] Nie X Y, Yang G W 2015 Acta Aeronaut. Astronaut. Sin. 36 1103 (in Chinese) [聂雪媛, 杨国伟 2015 航空学报 36 1103] [28] Hu J X, Zheng B J, Gao X W 2015 Sci: China Ser. G 45 014602 (in Chinese) [胡金秀, 郑保敬, 高效伟 2015 中国科学G辑 45 014602] [29] Dai B D, Cheng Y M 2007 Acta Phys. Sin. 56 597 (in Chinese) [戴保东, 程玉民 2007 物理学报 56 597]

#### 施引文献

•  [1] Brebbia C A, Dominguez J 1992 Boundary Elements: an Introductory Course (London: McGraw-Hill Book Co.) pp52-57 [2] Gao X W, Davies T G 2002 Boundary Element Programming in Mechanics (Cambridge: Cambridge University Press) pp25-33 [3] Gao X W, Peng H F, Yang K, Wang J 2015 Advanced Boundary Element MethodTheory and Application (Beijing: Science Press) pp110-129 (in Chinese) [高效伟, 彭海峰, 杨凯, 王静 2015 高等边界元法理论与程序 (北京: 科学出版社) 第110129页] [4] Ciskowski R D, Brebbia C A 1991 Boundary Element Methods in Acoustics (Southampton: Elsevier) [5] Gao X W, Hu J X 2012 Acta Mech. Sin. 44 361 (in Chinese) [高效伟, 胡金秀 2012 力学学报 44 361] [6] Zhang A M, Yao X L 2008 Chin. Phys. B 17 927 [7] Li S D, Huang Q B, Li T Y 2012 Acta Phys. Sin. 61 64301 (in Chinese) [李善德, 黄其柏, 李天匀 2012 物理学报 61 64301] [8] Xu J, Xie W H, Deng Y, Wang K, Luo Z Y, Gong H 2013 Acta Phys. Sin. 62 104204 (in Chinese) [许军, 谢文浩, 邓勇, 王侃, 罗召洋, 龚辉 2013 物理学报 62 104204] [9] Li S, He H L 2013 Chin. Phys. B 22 24701 [10] Yang K, Gao X W 2010 Eng. Anal. Bound. Elem. 34 557 [11] Gao X W, Wang J 2009 Eng. Anal. Bound. Elem. 33 539 [12] Gao X W, Peng H F, Liu J 2013 Int. J. Heat Mass Transf. 63 183 [13] Peng H F, Bai Y G, Yang K, Gao X W 2013 Eng. Anal. Bound. Elem. 37 1545 [14] Sutradhar A, Paulino G H 2004 Comput. Meth. Appl. Mech. Eng. 193 4511 [15] Erhart K, Divo E, Kassab A J 2006 Eng. Anal. Bound. Elem. 30 553 [16] Mohammadi M. Hematiyan M R, Marin L 2010 Eng. Anal. Bound. Elem. 34 655 [17] Yu B, Yao W A, Gao X W Gao Q 2014 Numer. Heat Transf. Part B: Fundam. 65 155 [18] Gao X W 2002 Eng. Anal. Bound. Elem. 26 905 [19] Gao X W 2005 J. Comput. Appl. Math. 175 265 [20] Hu J X, Peng H F, Gao X W 2014 Math. Probl. Eng. 2014 284106 [21] Hu J X, Zheng B J, Gao X W 2013 Bound. Elem. Mesh. Reduc. Meth. XXXVI 56 153 [22] Nardini D, Brebbia C A 1982 Boundary Element Methods in Engineering (Berlin: Springer) pp312-326 [23] Jiang Y L 2010 Model Reduction Method (Beijing: Science Press) pp1-4 (in Chinese) [蒋耀林 2010 模型降阶方法 (北京: 科学出版社) 第14页] [24] Chatterjee A 2000 Curr. Sci. 78 808 [25] Liang Y C, Lee H P, Lim S P, Lin W Z, Lee K H Wu C G 2002 J. Sound Vib. 252 527 [26] Fic A, Bialecki R A, Kassab A J 2005 Numer. Heat Transf. Part B: Fundam 48 103 [27] Nie X Y, Yang G W 2015 Acta Aeronaut. Astronaut. Sin. 36 1103 (in Chinese) [聂雪媛, 杨国伟 2015 航空学报 36 1103] [28] Hu J X, Zheng B J, Gao X W 2015 Sci: China Ser. G 45 014602 (in Chinese) [胡金秀, 郑保敬, 高效伟 2015 中国科学G辑 45 014602] [29] Dai B D, Cheng Y M 2007 Acta Phys. Sin. 56 597 (in Chinese) [戴保东, 程玉民 2007 物理学报 56 597]
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##### 出版历程
• 收稿日期:  2015-06-15
• 修回日期:  2015-08-24
• 刊出日期:  2016-01-05

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