小波随机有限元方法研究
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摘要
本研究将小波分析方法和随机有限元方法相结合发展了一种用于分析细沟侵蚀模型随机特性的小波随机有限元方法；在时域积分中采用了“精细积分方法”。该工作主要由以下五部分构成：
第一、研究了拟Shannon小波的性质，构造了求解偏微分方程的拟Shannon小波配置法，同时将外推法引入小波配置法，进一步改善了该方法的计算效率和计算精度。在此基础上，根据区间插值小波的概念，构造了拟Shannon区间小波配置法，数值算例表明该方法不但可以消除边界效应，而且可大幅度提高计算精度。
第二、在钟万勰院士提出的“精细积分法”的基础上提出了求解非线性结构动力方程的自适应精细积分法。该方法将外推法引入求解结构动力方程的精细时程积分法中，从而使该方法在求解非线性动力方程中可以自适应选取时间步长；需要指出的是，由于考虑了矩阵指数精细算法和外推法算法在时间离散方法上的一致性，在外推过程中，计算工作量基本没有增加；因此，两种方法的结合有效提高了算法的效率和精度。
第三、将小波配点法和精细积分法相结合提出了求解非线性偏微分方程的自适应插值小波有限元精细积分法。首先基于插值小波变换提出了多层插值小波配置法，该方法相对于原有的多层小波配置法，插值算子构造过程计算量大幅度减少，从而使该方法进一步实用；多层插值小波配置法和自适应精细积分法的结合使求解问题在时间域和空间域实现了全自适应离散；由于自适应小波配置法可以大幅度压缩配点规模，所以即使不使用子域精细积分，计算效率也不会下降很多，从而可避免寻找子域的困难以及可能出现的稳定性问题；另外，在求解偏微分方程时，精细积分方法相对于差分法类方法有更好的稳定性；因此，自适应插值小波有限元精细积分法是一种很有前景的方法。
第四、作为以上方法的应用，同时为了校核方法的稳定性以及计算精度和效率，构造了求解FPK方程的拟Shannon小波配置法，并给出了具体求解平稳FPK方程的具体步骤。分析了Shannon小波配置法和拟Shannon小波配置法在求解FPK方程中的优缺点。同时将该方法成功应用于具有较大梯度解的流体力学中的Burgers方程的求解中，求解结果显示了该方法在求解非线性偏微分方程方面具有重要前景。
第五、将随机有限元方法和小波精细积分法相结合提出了对细沟侵蚀模型的统计特性进行分析的小波随机有限元方法。在该方法中，采用了一种拟摄动法对模型中的非线性方程进行线性化，然后采用小波配置法对模型方程进行空间离散，最后用随机有限元方法求得模型中水流速度，水深和泥沙浓度的统计特性，计算结果和Monte Carlo法模拟结果吻合得很好。
The aim of this study is to develop a Wavelet Stochastic Finite Element Method to be applied in solving partial difference equations. This work includes 5 sections as follows:
A. The properties of Quasi shannon wavelet was studied in this paper, and a wavelet collocation method for partial differential equations was conducted. The extrapolation was used in this method for improving efficiency and accuracy, and the Quasi shannon interval wavelet collocation method was constructed based on the concept of interpolation wavelet transform. This method can handle the problems of complex boundary conditions and improve computation accuracy greatly.
B. Based on the Precise Integration Method for structural dynamic systems proposed by Zhong Wanxie, an adaptive precise integration method for nonlinear time-invariant structural dynamic systems was proposed in this paper. This method was conducted by combining extrapolation method with precise time-integration method. In this method the nonlinear equations can be discreted in time domain adaptively. It should be pointed out that the discrete method in time domain is identical between the precise computation of the matrix exponential and the extrapolation, so that the combination of them would not increase the computation time. Hence this method can improve accuracy and efficiency of the calculation.
C. Based on these work upwards, an adaptively wavelet precise time-invariant integration method was proposed in this paper. In this method, an adaptive multilevel interpolation wavelet collocation method for partial difference equations (PDEs) was conducted, in which the time complexity is less than Oleg V's method, and then the adaptive precise integration method was combined with, so that in this method the adaptively discretes both in time domain and physical domain were realized. The sub-domain precise integration method is not necessary in this method due to the collocation points can be reduced greatly in this method. So the difficulties of looking for sub-domain are avoided, and the stability of the method is ensured..
D. As numerical examples, the Quasi-shannon wavelet collocation for FPK equations was conducted based on the theory of this method. The specific procedures for solving stationary FPK equation, the analysis of the problems raised in this method and the remedies of the problems were presented in this paper. Otherwise, this method was applied into solving Burgers Equation, and the numerical results show that it is a prospective numerical method for nonlinear partial differential equation.
E. Lastly, the wavelet stochastic finite element method for a stochastic model of soil erosion in a rill was proposed. In this method, a new perturbation technique called linearization-correction method was used to linearize the nonlinear equations in the model, and then the wavelet precise integration method was used to calculate the sensitivity of the response. At last the stochastic perturbation method is used to analyse the variance and expectation of sediment concentration, rate and depth of flow. The calculated result was high agreement with that result of Monte Carlo method.
第一、研究了拟Shannon小波的性质，构造了求解偏微分方程的拟Shannon小波配置法，同时将外推法引入小波配置法，进一步改善了该方法的计算效率和计算精度。在此基础上，根据区间插值小波的概念，构造了拟Shannon区间小波配置法，数值算例表明该方法不但可以消除边界效应，而且可大幅度提高计算精度。
第二、在钟万勰院士提出的“精细积分法”的基础上提出了求解非线性结构动力方程的自适应精细积分法。该方法将外推法引入求解结构动力方程的精细时程积分法中，从而使该方法在求解非线性动力方程中可以自适应选取时间步长；需要指出的是，由于考虑了矩阵指数精细算法和外推法算法在时间离散方法上的一致性，在外推过程中，计算工作量基本没有增加；因此，两种方法的结合有效提高了算法的效率和精度。
第三、将小波配点法和精细积分法相结合提出了求解非线性偏微分方程的自适应插值小波有限元精细积分法。首先基于插值小波变换提出了多层插值小波配置法，该方法相对于原有的多层小波配置法，插值算子构造过程计算量大幅度减少，从而使该方法进一步实用；多层插值小波配置法和自适应精细积分法的结合使求解问题在时间域和空间域实现了全自适应离散；由于自适应小波配置法可以大幅度压缩配点规模，所以即使不使用子域精细积分，计算效率也不会下降很多，从而可避免寻找子域的困难以及可能出现的稳定性问题；另外，在求解偏微分方程时，精细积分方法相对于差分法类方法有更好的稳定性；因此，自适应插值小波有限元精细积分法是一种很有前景的方法。
第四、作为以上方法的应用，同时为了校核方法的稳定性以及计算精度和效率，构造了求解FPK方程的拟Shannon小波配置法，并给出了具体求解平稳FPK方程的具体步骤。分析了Shannon小波配置法和拟Shannon小波配置法在求解FPK方程中的优缺点。同时将该方法成功应用于具有较大梯度解的流体力学中的Burgers方程的求解中，求解结果显示了该方法在求解非线性偏微分方程方面具有重要前景。
第五、将随机有限元方法和小波精细积分法相结合提出了对细沟侵蚀模型的统计特性进行分析的小波随机有限元方法。在该方法中，采用了一种拟摄动法对模型中的非线性方程进行线性化，然后采用小波配置法对模型方程进行空间离散，最后用随机有限元方法求得模型中水流速度，水深和泥沙浓度的统计特性，计算结果和Monte Carlo法模拟结果吻合得很好。
The aim of this study is to develop a Wavelet Stochastic Finite Element Method to be applied in solving partial difference equations. This work includes 5 sections as follows:
A. The properties of Quasi shannon wavelet was studied in this paper, and a wavelet collocation method for partial differential equations was conducted. The extrapolation was used in this method for improving efficiency and accuracy, and the Quasi shannon interval wavelet collocation method was constructed based on the concept of interpolation wavelet transform. This method can handle the problems of complex boundary conditions and improve computation accuracy greatly.
B. Based on the Precise Integration Method for structural dynamic systems proposed by Zhong Wanxie, an adaptive precise integration method for nonlinear time-invariant structural dynamic systems was proposed in this paper. This method was conducted by combining extrapolation method with precise time-integration method. In this method the nonlinear equations can be discreted in time domain adaptively. It should be pointed out that the discrete method in time domain is identical between the precise computation of the matrix exponential and the extrapolation, so that the combination of them would not increase the computation time. Hence this method can improve accuracy and efficiency of the calculation.
C. Based on these work upwards, an adaptively wavelet precise time-invariant integration method was proposed in this paper. In this method, an adaptive multilevel interpolation wavelet collocation method for partial difference equations (PDEs) was conducted, in which the time complexity is less than Oleg V's method, and then the adaptive precise integration method was combined with, so that in this method the adaptively discretes both in time domain and physical domain were realized. The sub-domain precise integration method is not necessary in this method due to the collocation points can be reduced greatly in this method. So the difficulties of looking for sub-domain are avoided, and the stability of the method is ensured..
D. As numerical examples, the Quasi-shannon wavelet collocation for FPK equations was conducted based on the theory of this method. The specific procedures for solving stationary FPK equation, the analysis of the problems raised in this method and the remedies of the problems were presented in this paper. Otherwise, this method was applied into solving Burgers Equation, and the numerical results show that it is a prospective numerical method for nonlinear partial differential equation.
E. Lastly, the wavelet stochastic finite element method for a stochastic model of soil erosion in a rill was proposed. In this method, a new perturbation technique called linearization-correction method was used to linearize the nonlinear equations in the model, and then the wavelet precise integration method was used to calculate the sensitivity of the response. At last the stochastic perturbation method is used to analyse the variance and expectation of sediment concentration, rate and depth of flow. The calculated result was high agreement with that result of Monte Carlo method.
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第六章
[1]刘福生，马正飞，王晟．化学工程中分形问题的正交配置法．计算物理，2001，18(3)：235～240
[2]压力依赖于地层渗透率的分形油藏的数值研究．西安石油学院学报．2001，16(2)：21～24
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