基于网格自适应的飞行器防热材料热传导系数辨识
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摘要
为准确辨识高超声速飞行器防热材料热传导系数,构造了针对热传导偏微分方程系统的约束泛函极值问题,基于表征多项式逼近性能的Weierstrass定理,采用Lagrange多项式对热传导系数进行参数化建模,进而将无穷维的约束泛函极值问题转化为有限维的非线性规划问题,再利用基于动态优化方程的优化方法将此非线性规划问题转化为常微分方程初值问题进行求解。为从较有限的测量数据中准确地"学习"热传导系数,建立采用"过拟合"判别准则的网格自适应迭代算法改善辨识精度。研究表明本文采用的辨识策略与优化方法有效,辨识结果可以较准确地反映材料热传导系数的变化规律。
To accurately identify the thermal conductivity of a hypersonic vehicle's heat shield material, the constrained functional minimization problem for the heat conduction partial differential equation system is developed and effectively solved. Based on the Weierstrass theorem, the Lagrange polynomial is used to parameterize the thermal conductivity to obtain the finite-dimensional nonlinear programming(NLP) problem from the original infinite-dimensional problem. With the dynamic optimization equation(DOE) based method, the NLP is transformed to the initial-value problem(IVP) to be solved. In order to accurately learn the thermal conductivity from the limited measurement data, the adaptive mesh refinement method that prevents the over-fitting is proposed to enhance the identification accuracy. The research suggests that the identification approach and the optimization method are effective, and the identified results may accurately simulate the true thermal conductivity.
To accurately identify the thermal conductivity of a hypersonic vehicle's heat shield material, the constrained functional minimization problem for the heat conduction partial differential equation system is developed and effectively solved. Based on the Weierstrass theorem, the Lagrange polynomial is used to parameterize the thermal conductivity to obtain the finite-dimensional nonlinear programming(NLP) problem from the original infinite-dimensional problem. With the dynamic optimization equation(DOE) based method, the NLP is transformed to the initial-value problem(IVP) to be solved. In order to accurately learn the thermal conductivity from the limited measurement data, the adaptive mesh refinement method that prevents the over-fitting is proposed to enhance the identification accuracy. The research suggests that the identification approach and the optimization method are effective, and the identified results may accurately simulate the true thermal conductivity.
引文
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[22] 张涵信,沈孟育.计算流体力学:差分方法的原理和应用[M].北京:国防工业出版社,2002.
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[25] 张志国,余梦伦,耿光有,等.应用伪谱法的运载火箭在线制导方法研究[J].宇航学报,2017,38(3):262-269.[Zhang Zhi-guo,Yu Meng-lun,Geng Guang-you,et al.Research on application of pseudo-spectral method in online guidance method for a launch vehicle[J].Journal of Astronautics,2017,38(3):262-269.]
[26] Patterson M A,Hager W W,Rao A V.A ph mesh refinement method for optimal control.Optimal control application and methods[J].2015,36(4):398-421.
[27] 雷虎民,刘滔,李炯,等.网格缩减的自适应hp伪谱法[J].控制理论与应用,2016,33(8):1061-1067.[Lei Hu-min,Liu Tao,Li Jiong,et al.Adaptive mesh refinement of hp pseudospectral method using mesh size reduction[J].Control Theory & Applications,2016,33(8):1061-1067.]
[28] Vapnik V N.统计学习理论的本质[M].张学工,译.北京:清华大学出版社,2000.
[29] Nocedal J,Wright S J.Numerical optimization [M].New York:Springer Science,2006.
[2] 何开锋,汪清,钱炜祺,等.高超声速飞行器气动力/热参数辨识研究综述[J].实验流体力学,2011,25(5):99-104.[He Kai-feng,Wang Qing,Qian Wei-qi,et al.Review of aerodynamic and aero-thermodynamic parameter estimation research for hypersonic aircraft[J].Journal of Experiments in Fluid Mechanics,2011,25(5):99-104.]
[3] Ozisik M N,Orlande H R B.Inverse heat transfer-fundamentals and applications [M].New York:Taylor & Francis,2000.
[4] 易法军,张巍,孟松鹤,等.C_C复合材料高温热物理性能实验研究[J].宇航学报,2002,23(5):85-88.[Yi Fa-jun,Zhang Wei,Meng Song-he,et al.An experimental study on thermophysical properties of C/C composites at elevated temperature[J].Journal of Astronautics,2002,23(5):85-88.]
[5] 赵晓琳.多孔介质有效导热系数的算法研究[D].大连:大连理工大学,2009.[Zhao Xiao-lin.The research of effective conductivity in the porous media[D].Dalian:Dalian University of Technology,2009.]
[6] 周焕林,徐兴盛,李秀丽,等.反演二维瞬态热传导问题随温度变化的导热系数[J].应用数学和力学,2014,35(12):1341-1351.[Zhou Huan-Lin,Xu Xing-sheng,Li Xiu-li,et al.Identification of temperature-dependent thermal conductivity for 2-D transient heat conduction problems[J].Applied Mathematics and Mechanics,2014,35(12):1341-1351.]
[7] 唐中华,钱国红,钱炜祺.材料热传导系数随温度变化函数的反演方法[J].计算力学学报,2011,28(3):377-382.[Tang Zhong-hua,Qian Guo-hong,Qian Wei-qi.Estimation of temperature-dependent function of thermal conductivity for a material[J].Chinese Journal of Computational Mechanics,2011,28(3):377-382.]
[8] 周宇,钱炜祺,邵元培.考虑约束的材料热传导系数辨识方法[J/OL].空气动力学报,[2017-12-29].http:/www.kqdlxxb.com/CN/abstract/abstract12107.shtml.[Zhou Yu,Qian Wei-qi,Shao Yuan-pei.Constrained estimation method of thermal conductivity of a material[J/OL].Acta Aerodynamica Sinica,[2017-12-29].http:/www.kqdlxxb.com/CN/abstract/abstract12107.shtml.]
[9] Hastie T,Tibshirani R,Friedman J.统计学习基础—数据挖掘、推理与预测[M].范明,柴玉美,昝红英,译.北京:电子工业出版社:2004.
[10] Gill P E,Murray W,Saunders M A.SNOPT:An SQP algorithm for large-scale constrained optimization[J].SIAM Review,2005,47(1):99-131.
[11] Forsgren A,Gill P E,Wright M H.Interior methods for nonlinear optimization[J].SIAM Review,2002,44(4):525-597.
[12] Courant R.Variational methods for the solution of problems of equilibrium and vibrations[J].Bull.Amer.Math.Soc.,1943,49:1-23.
[13] Botsaris C A.Differential gradient methods[J].J.Math.Anal.Appl.,1978,63,177-198.
[14] Alvarez F,Perez J M.A dynamical system associated with newton’s method for parametric approximations of convex mini-mization problems[J].Appl.Math.Optim.,1998,38(2):193-217.
[15] Tanabe K.An algorithm for constrained maximization in nonlinear programming[J].Journal of Operations Research Society of Japan,1974,17:184-201.
[16] Yamashita H.A differential equation approach to nonlinear programming[J].Math.Prog.1980,18:155-168.
[17] Evtushenko Y G,Zhadan V G.Stable barrier projection and barrier Newton methods in nonlinear programming[J].Optim.Methods Softw,1994,3:237-256.
[18] Schropp J.A dynamical systems approach to constrained minimization[J].Numer.Math.,2000,21:537-551.
[19] Zhang S,Li W B,Qian W Q,et al.Transform the non-linear programming problem to the initial-value problem to solve [Z].arXiv preprint,arXiv:1804.09829 [math.OC],2018.
[20] 钱炜祺,蔡金狮.再入航天飞机表面热流密度辨识[J].宇航学报,2000,21(4):1-6.[Qian Wei-qi,Cai Jin-shi.Surface heat flux identification of reentry space shuttle[J].Journal of Astronautics,2000,21(4):1-6.]
[21] 马忠辉,孙秦,王小军,等.热防护系统多层隔热结构传热分析及性能研究[J].宇航学报,2003,24(5):543-546.[Ma Zhong-hui,Sun Qin,Wang Xiao-jun,et al.TPS multi-layer insulation thermal analysis and performance study[J].Journal of Astronautics,2003,24(5):543-546.]
[22] 张涵信,沈孟育.计算流体力学:差分方法的原理和应用[M].北京:国防工业出版社,2002.
[23] Jackson D.A proof of Weierstrass’s theorem[J].The American Mathematical Monthly,1934,41(5):309-312.
[24] Garg D,Patterson M A,Hager W W,et al.A unified framework for the numerical solution of optimal control problems using pseudospectral methods[J].Automatica,2010,46(11):1843-1851.
[25] 张志国,余梦伦,耿光有,等.应用伪谱法的运载火箭在线制导方法研究[J].宇航学报,2017,38(3):262-269.[Zhang Zhi-guo,Yu Meng-lun,Geng Guang-you,et al.Research on application of pseudo-spectral method in online guidance method for a launch vehicle[J].Journal of Astronautics,2017,38(3):262-269.]
[26] Patterson M A,Hager W W,Rao A V.A ph mesh refinement method for optimal control.Optimal control application and methods[J].2015,36(4):398-421.
[27] 雷虎民,刘滔,李炯,等.网格缩减的自适应hp伪谱法[J].控制理论与应用,2016,33(8):1061-1067.[Lei Hu-min,Liu Tao,Li Jiong,et al.Adaptive mesh refinement of hp pseudospectral method using mesh size reduction[J].Control Theory & Applications,2016,33(8):1061-1067.]
[28] Vapnik V N.统计学习理论的本质[M].张学工,译.北京:清华大学出版社,2000.
[29] Nocedal J,Wright S J.Numerical optimization [M].New York:Springer Science,2006.