某些非线性水波问题的同伦分析方法研究
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摘要
非线性波浪理论是目前船舶和海洋工程水动力学的热点研究问题。尽管非线性水波的研究有较长的历史,但是由于其困难性,非线性水波的领域仍然有大量的问题存在。非线性水波领域还有很多的问题亟待解决。随着全球人口的增加,陆地上的资源已经难以满足要求,海洋正在成为人类的资源基地和第二生存空间。海洋资源开发利用首先依赖于海上工程设施,随着海洋开发利用的规模日趋复杂和庞大,港口,海岸以及近海油气开发不断向深水发展,这些海洋平台及船舶有可能承受实际海况下强非线性波浪的作用,这些强非线性波浪的能量很集中,破坏力极大。非线性水波的研究在工程中有着广泛的应用。
求解非线性问题的途径有解析方法、数值方法、定性分析等。在计算机迅速发展的今天,数值计算方法是求解非线性方程的主要方法。数值方法能够求解复杂计算域的非线性问题,而解析方法往往仅适用于具有简单计算区域的问题。但是数值解经常会出现不连续情况,因此计算起来费时费力。另外对于由奇点或多解的情况,数值方法经常会失效。而解析方法则具有以下优势:解析方法能为数值解提供有效验证;解析的表达式可以显示不同物理量的影响和变化趋势,看出变化过程,特别有利于参数的优化;简单的解析表达式能够快速的给出问题的解。摄动方法被广泛地应用于解析求解非线性问题。通过摄动方法,很多有趣的非线性现象和重要属性被揭示出来。摄动方法在科学和工程领域中起着重要的作用。然而,摄动方法也有其自身的局限性。首先,摄动方法必须依赖于某个小的或大的参数。很多非线性问题,尤其是一些强非线性问题,根本不包含这样的小参数或大参数。其次,摄动方法通常只对弱非线性问题有效,对于强非线性问题,经常得不到收敛的解。而近年来发展起来的同伦分析方法完全不依赖小参数的,同时可以调节收敛区间和收敛速度。
本文采用了同伦分析方法求解非线性水波问题,并且对同伦分析方法的应用领域进行了拓展,对同伦分析方法中的初始猜测解给出了相关的讨论。将同伦分析方法拓展到离散的微分方程,建立了新的算法,Difference differential equation-Homotopyanalysis method(DDE-HAM).并且给出了初始猜测解选择的新的方法,是同伦分析方法更具普适性.又将同伦分析方法应用到非线性浅水波和非线性深水波领域,这也是同伦分析方法首次用来解决深水包络孤波和周期波群问题上.
本文所做工作充分表明了同伦分析方法一些显著的特点,如不依赖于非线性方程中的小参数,可以有效地控制和调节级数解的收敛区间和收敛速度。同时突出了同伦分析方法求解非线性问题时的操作方便、适用性广、灵活度高等优点。此外,同伦分析方法所特有的自由性,可以将物理问题与同伦分析方法更紧密的结合,所得到的结果可以作为数值方法和试验结果的有效验证。可以预期同伦分析方法作为复杂非线性问题的有效解析方法将会有更广阔的应用前景。
Nonlinear water wave theory is a hot research topic in the field of naval architecture and ocean engineering hydrodynamics.Although the history of nonlinear water wave study is long,a lot of problems in nonlinear water waves field remain open today,due to their difficulties.
As the population of world increase,the resource of land can not meet the needs of people.Now the ocean is becoming the resource base and the second living space for human being.Exploitation and utilization of sea resources depend on development of ocean engineering.The scale of sea exploitation and utilization is becoming much more complex and bigger,so that exploitation of oil and gas in port,seashore is employed in deep water.The platforms and ships are required to endure strong nonlinear water wave loads in practical cases.The energy of these strong nonlinear water waves is concentrated, of great damage.The study of nonlinear waves has wide applications in ocean engineering.
Among the methods for solving nonlinear problems are analytic method,numerical method and experiment.Nowadays as the computing technology has highly developed, numerical simulation is main technique for solving nonlinear problems.Numerical simulation can solve nonlinear problems with complex computational fields,while analytic method can be merely applied to nonlinear problems with simple computational fields.However,discontinuity often occur in numerical simulation,so it will cost lots of efforts and time for computation.On the other hand,in case of singularity and multiple solutions,numerical simulation will be invalid.Above all,analytic method has the following advantages:Analytic solutions can be used as a test of numerical solutions;analytic solution can reveal the properties of different variables,and it is convenient for parameter optimization.By means of perturbation techniques,a lot of important properties and interesting phenomena of non-linear problems have been revealed.Perturbation techniques are based on the existence of a small/large parameter or variable.Obviously,the existence of perturbation quantities is a cornerstone of perturbation teclmiques.Perturbation method is valid for weak nonlinear problems.For strong nonlinear problems,we can not get the convergent solutions.In recent years,Homotopy analysis method has been proposed,and it can control convergence region and adjust convergence rate.
In this thesis,we apply the homotopy analysis method to solving nonlinear water waves.We proposed a new method for differential-difference equations based on Homotopy analysis method.We called it differential-difference equation-Homotopy analysis method(DDE-HAM).We study the initial guess in homotopy analysis method.And apply it the nonlinear shallow water,that is,one-dimensional,two-dimensional,threedimensional shallow water,we apply it to solve nonlinear Schr(o|¨)dinger equation.Such a kind of explicit,analytic solutions are useful for analyzing periodic wave groups and envelope solitary gravity waves.
The work of this thesis shows the wide applicability of Homotopy analysis method in science and engineeing.Homotopy analysis method has the following advantages, such as it is valid even if a given non-linear problem does not contain any small/large parameters at all,it itself can provide us with a convenient way to control the convergence of approximation series and adjust convergence regions when necessary.The work shows that Homotopy analysis method is more efficient and effective analytic methods in solving nonlinear problems.Especially,some of the results indicates that HAM is a useful tool for those nonlinear problems with analytic solutions as a test of the results by numerical methods or experimental methods.All these examples given in this thesis might be helpful to keep us an open mind for solving nonlinear problems in ocean engineering.
求解非线性问题的途径有解析方法、数值方法、定性分析等。在计算机迅速发展的今天,数值计算方法是求解非线性方程的主要方法。数值方法能够求解复杂计算域的非线性问题,而解析方法往往仅适用于具有简单计算区域的问题。但是数值解经常会出现不连续情况,因此计算起来费时费力。另外对于由奇点或多解的情况,数值方法经常会失效。而解析方法则具有以下优势:解析方法能为数值解提供有效验证;解析的表达式可以显示不同物理量的影响和变化趋势,看出变化过程,特别有利于参数的优化;简单的解析表达式能够快速的给出问题的解。摄动方法被广泛地应用于解析求解非线性问题。通过摄动方法,很多有趣的非线性现象和重要属性被揭示出来。摄动方法在科学和工程领域中起着重要的作用。然而,摄动方法也有其自身的局限性。首先,摄动方法必须依赖于某个小的或大的参数。很多非线性问题,尤其是一些强非线性问题,根本不包含这样的小参数或大参数。其次,摄动方法通常只对弱非线性问题有效,对于强非线性问题,经常得不到收敛的解。而近年来发展起来的同伦分析方法完全不依赖小参数的,同时可以调节收敛区间和收敛速度。
本文采用了同伦分析方法求解非线性水波问题,并且对同伦分析方法的应用领域进行了拓展,对同伦分析方法中的初始猜测解给出了相关的讨论。将同伦分析方法拓展到离散的微分方程,建立了新的算法,Difference differential equation-Homotopyanalysis method(DDE-HAM).并且给出了初始猜测解选择的新的方法,是同伦分析方法更具普适性.又将同伦分析方法应用到非线性浅水波和非线性深水波领域,这也是同伦分析方法首次用来解决深水包络孤波和周期波群问题上.
本文所做工作充分表明了同伦分析方法一些显著的特点,如不依赖于非线性方程中的小参数,可以有效地控制和调节级数解的收敛区间和收敛速度。同时突出了同伦分析方法求解非线性问题时的操作方便、适用性广、灵活度高等优点。此外,同伦分析方法所特有的自由性,可以将物理问题与同伦分析方法更紧密的结合,所得到的结果可以作为数值方法和试验结果的有效验证。可以预期同伦分析方法作为复杂非线性问题的有效解析方法将会有更广阔的应用前景。
Nonlinear water wave theory is a hot research topic in the field of naval architecture and ocean engineering hydrodynamics.Although the history of nonlinear water wave study is long,a lot of problems in nonlinear water waves field remain open today,due to their difficulties.
As the population of world increase,the resource of land can not meet the needs of people.Now the ocean is becoming the resource base and the second living space for human being.Exploitation and utilization of sea resources depend on development of ocean engineering.The scale of sea exploitation and utilization is becoming much more complex and bigger,so that exploitation of oil and gas in port,seashore is employed in deep water.The platforms and ships are required to endure strong nonlinear water wave loads in practical cases.The energy of these strong nonlinear water waves is concentrated, of great damage.The study of nonlinear waves has wide applications in ocean engineering.
Among the methods for solving nonlinear problems are analytic method,numerical method and experiment.Nowadays as the computing technology has highly developed, numerical simulation is main technique for solving nonlinear problems.Numerical simulation can solve nonlinear problems with complex computational fields,while analytic method can be merely applied to nonlinear problems with simple computational fields.However,discontinuity often occur in numerical simulation,so it will cost lots of efforts and time for computation.On the other hand,in case of singularity and multiple solutions,numerical simulation will be invalid.Above all,analytic method has the following advantages:Analytic solutions can be used as a test of numerical solutions;analytic solution can reveal the properties of different variables,and it is convenient for parameter optimization.By means of perturbation techniques,a lot of important properties and interesting phenomena of non-linear problems have been revealed.Perturbation techniques are based on the existence of a small/large parameter or variable.Obviously,the existence of perturbation quantities is a cornerstone of perturbation teclmiques.Perturbation method is valid for weak nonlinear problems.For strong nonlinear problems,we can not get the convergent solutions.In recent years,Homotopy analysis method has been proposed,and it can control convergence region and adjust convergence rate.
In this thesis,we apply the homotopy analysis method to solving nonlinear water waves.We proposed a new method for differential-difference equations based on Homotopy analysis method.We called it differential-difference equation-Homotopy analysis method(DDE-HAM).We study the initial guess in homotopy analysis method.And apply it the nonlinear shallow water,that is,one-dimensional,two-dimensional,threedimensional shallow water,we apply it to solve nonlinear Schr(o|¨)dinger equation.Such a kind of explicit,analytic solutions are useful for analyzing periodic wave groups and envelope solitary gravity waves.
The work of this thesis shows the wide applicability of Homotopy analysis method in science and engineeing.Homotopy analysis method has the following advantages, such as it is valid even if a given non-linear problem does not contain any small/large parameters at all,it itself can provide us with a convenient way to control the convergence of approximation series and adjust convergence regions when necessary.The work shows that Homotopy analysis method is more efficient and effective analytic methods in solving nonlinear problems.Especially,some of the results indicates that HAM is a useful tool for those nonlinear problems with analytic solutions as a test of the results by numerical methods or experimental methods.All these examples given in this thesis might be helpful to keep us an open mind for solving nonlinear problems in ocean engineering.
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