基于本征正交分解的波浪力随机场降维模拟
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摘要
基于平稳多变量随机过程理论,推导了基于正交随机变量的本征正交分解(POD)公式,在此基础上,通过定义POD公式中正交随机变量集的随机函数形式,建议了平稳多变量随机过程模拟的POD-随机函数方法。该方法与传统基于随机相位角的POD都是基于正交随机变量POD的特例,然而该方法仅需两个基本随机变量即可实现对平稳多变量随机过程的精细表达。根据线性化的Morison方程,推导了波浪力多变量随机过程的功率谱密度函数矩阵,利用POD-随机函数方法,实现了波浪力多变量随机过程模拟的高效降维。最后,结合P-M波浪谱,给出了水平向波浪力多变量随机过程的功率谱密度矩阵,并对水平向波浪力随机场进行数值模拟分析。算例表明,该方法所需的基本随机变量最少,生成的代表性样本数量少且构成一个完备的概率集,在模拟精度方面亦具有显著优势。
Due to the theory of multivariate stationary stochastic process, the proper orthogonal decomposition(POD) formula based on the orthogonal random variables was derived. On the basis of this, by defining the random function form of the orthogonal random variable set in the POD formula, a POD-random function approach was suggested for simulating the multivariate stationary stochastic process. Note that the proposed approach and the conventional approach based on the random phase angles are both the special cases of the POD approach based on the orthogonal random variables. However, the proposed approach can accurately represent the multivariate stationary stochastic process with just two elementary random variables. The power spectrum density matrix involved in the multivariate stationary stochastic process of wave force was derived according to the linearized Morison formula. Then by means of the proposed approach, effective dimension reduction simulation for the multivariate stationary stochastic process of wave force could thus be realized. Finally, combined with the P-M wave spectrum, the power spectrum density matrix involved in the multivariate stochastic process of the horizontal wave force was derived. And the numerical simulation analysis of the horizontal stochastic wave force field was carried out. The example shows that the proposed approach requires the least number of the elementary random variables and a small number of representative samples which constitute a complete set of probabilities. It has significant advantages in terms of simulation accuracy.
Due to the theory of multivariate stationary stochastic process, the proper orthogonal decomposition(POD) formula based on the orthogonal random variables was derived. On the basis of this, by defining the random function form of the orthogonal random variable set in the POD formula, a POD-random function approach was suggested for simulating the multivariate stationary stochastic process. Note that the proposed approach and the conventional approach based on the random phase angles are both the special cases of the POD approach based on the orthogonal random variables. However, the proposed approach can accurately represent the multivariate stationary stochastic process with just two elementary random variables. The power spectrum density matrix involved in the multivariate stationary stochastic process of wave force was derived according to the linearized Morison formula. Then by means of the proposed approach, effective dimension reduction simulation for the multivariate stationary stochastic process of wave force could thus be realized. Finally, combined with the P-M wave spectrum, the power spectrum density matrix involved in the multivariate stochastic process of the horizontal wave force was derived. And the numerical simulation analysis of the horizontal stochastic wave force field was carried out. The example shows that the proposed approach requires the least number of the elementary random variables and a small number of representative samples which constitute a complete set of probabilities. It has significant advantages in terms of simulation accuracy.
引文
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[24] LI Jie,CHEN Jianbing.The number theoretical method in response analysis of nonlinear stochastic structures[J].Computational Mechanics,2007,39(6):693-708.
[2] PIERSON W J.An interpretation of the observable properties of sea waves in terms of the Energy of the Gaussian record[J].Eos Transactions American Geophysical Union,1954,35(5):747-757.
[3] BORGMAN L E.Spectral analysis of ocean wave forces on piling[J].Journal of the Waterways & Harbors Division,1967,93(2):129-156.
[4] DI PAOLA M,PISANO A A.Multivariate stochastic wave generation[J].Applied Ocean Research,1996,18(6):361-365.
[5] REN B,WANG Yongxue.Numerical simulation of random wave slamming on structures in the splash zone[J].Ocean Engineering,2004,31(5):547-560.
[6] DONG Guohai,XU Tiaojian,ZHAO Yunpeng,et al.Numerical simulation of hydrodynamic behavior of gravity cage in irregular waves[J].Aquacultural Engineering,2010,42(2):90-101.
[7] LONGUET-HIGGINS M S,CARTWRIGHT D E,SMITH N D.Observations of the directional spectrum of sea waves using the motions of a floating buoy [J].Ocean Wave Spectra,1963:111-132.
[8] REN Lin,MAO Zhihua,HUANG Haiqing,et al.Satellite-based RAR performance simulation for measuring directional ocean wave spectrum based on SAR inversion spectrum[J].Acta Oceanologica Sinica,2010,29(4):13-20.
[9] 白连平,陈秀真.三维随机海浪的数值模拟[J].海洋工程,2000,18(4):32-35.BAI Lianping,CHEN Xiuzhen.Numerical simulation of three-dimensional random wave[J].Ocean Engineering,2000,18(4):32-35.
[10] 张寒元,于定勇.三维波峰的数值模拟[J].海洋工程,2004,22(1):52-58.ZHANG Hanyuan,YU Dingyong.Numerical simulation of three dimensional wave crest[J].Ocean Engineering,2004,22(1):52-58.
[11] 张思将,杨洁,欧阳艺.基于方向谱的海浪三维数值模拟[J].舰船电子对抗,2013,36(4):54-57.ZHANG Sijiang,YANG Jie,OUYANG Yi.3D Numerical simulation of sea wave based on directional spectrum[J].Shipboard Electronic Countermeasure,2013,36(4):54-57.
[12] 赵振民,孔民秀,杜志江,等.船舶振荡运动仿真的方向谱方法[J].哈尔滨工业大学学报,2010,42 (5):742-745.ZHAO Zhenmin,KONG Minxiu,DU Zhijiang,et al.Simulation method for ship oscillation based on directional spectrum[J].Journal of Harbin Institute of Technology,2010,42(5):742-745.
[13] 刘章军,刘玲.随机海浪过程模拟的随机函数方法[J].振动与冲击,2014,33(20):1-6.LIU Zhangjun,LIU Ling.Simulation of stochastic ocean states by random function methods[J].Journal of Vibration and Shock,2014,33(20):1-6.
[14] LIU Zhangjun,LIU Wei,PENG Yongbo.Random function based spectral representation of stationary and non-stationary stochastic processes[J].Probabilistic Engineering Mechanics,2016,45:115-126.
[15] LI Jie,CHEN Jianbing.Stochastic dynamics of structures[M].Singapore:John Wiley & Sons,2009.
[16] LIU Zhangjun,LIU Zixin,CHEN Denghong.Probability density evolution of a nonlinear concrete gravity dam subjected to non-stationary seismic ground motion[J].Journal of Engineering Mechanics,ASCE,2018,144(1):04017157.
[17] DI PAOLA M,ZINGALES M.Digital simulation of multivariate earthquake ground motions[J].Earthquake Engineering & Structural Dynamics,2000,29:1011-1027.
[18] CHEN Xinzhong,KAREEM A.Proper orthogonal decomposition-based modeling analysis and simulation of dynamic wind load effects on structures[J].Journal of Engineering Mechanics,2005,131(4):325-339.
[19] LIU Zhangjun,LIU Zhenghui,PENG Yongbo.Simulation of multivariate stationary stochastic processes using dimension-reduction representation methods[J].Journal of Sound and Vibration,2018,418:144-162.
[20] 王树青,梁丙辰.海洋工程波浪力学[M].青岛:中国海洋大学出版社,2013.
[21] 港口与航道水文规范:JTS 145—2015 [S].北京:人民交通出版社,2015.
[22] 俞聿修,柳淑学.随机波浪及其工程应用[M].大连:大连理工大学出版社,2011.
[23] 宋金宝,徐德伦,孙孚.计算孤立小桩柱上随机波浪力的一个线性化的Morison公式[J].海洋学报(中文版),1997,19(3):133-141.SONG Jinbao,XU Delun,SUN Fu.A linearized morison formula for calculating the random wave forces on an isolated small pile column[J].Acta Oceanologica Sinica,1997,19(3):133-141.
[24] LI Jie,CHEN Jianbing.The number theoretical method in response analysis of nonlinear stochastic structures[J].Computational Mechanics,2007,39(6):693-708.