摘要
在随机载荷作用下海洋结构物响应概率密度曲线端部区域内的值被称为极端响应值(extreme responses),人们在设计中常常要遇到的一类问题是如何才能精确预报海洋结构物慢漂纵荡运动的极端响应。如果海洋结构物在横摇自由度上的极端响应超过一临界值,海洋结构物就会丧失稳性而倾覆。上述的极端响应和倾覆行为可统称为海洋结构物的极端行为(extreme behavior),本文研究用几种不同的方法来分析海洋结构物的上述极端行为。
为了研究随机激励下海洋结构物的慢漂极端响应和倾覆行为,需要对海洋结构物进行动力分析。在对海洋结构物进行动力分析时,海洋结构物的附加质量、海洋结构物所受的阻尼和复原力、以及海洋结构物所受的单位波幅的波浪激励力(或力矩)都是可用一些成熟的商用水动力软件来计算的,这样就可建立海洋结构物运动的非线性随机微分方程,本文的主要工作就是要研究这些非线性随机微分方程的有效解法。
在对海洋结构物进行动力分析时,人们常常采用数值仿真的方法—即对海洋结构物的运动微分方程进行数值积分,并对求得的各响应历经做统计处理。这在原理上是十分简明的,执行起来也很方便,而且由此得出的响应概率密度曲线中部绝大部分范围内的值也是很精确的。数值仿真的上述优越性使有些研究人员在分析海洋结构物的极端行为时也采用它,本文对这一做法做出了反思。首先,我们在研究中把仿真的效率与仿真预报的慢漂极端响应的精度这两个问题统筹起来考虑,用一个顺应式近海结构物实例说明了仿真预报的海洋结构物慢漂极端响应的精度是不高的。为提高慢漂极端响应预报的精度,人们会想将仿真的次数增加。我们在研究中首次用实例向人们展示:当慢漂极端响应的精度提高到一定程度后,再继续增加仿真的次数就几乎不起作用了,而此时的慢漂极端响应值的精度还并未达到令人满意的程度。为了克服数值仿真的这一局限性,我们因而有必要研究用其它方法来预报海洋结构物在随机载荷作用下的慢漂极端响应。本文也对数值仿真分析海洋结构物(特别是船舶)的稳性的表现做了定量的研究。给定一个激励力频率和一个激励力幅值,取不同的船舶初始横摇角和初始横摇角速度,我们发现此时用数值仿真研究船舶倾覆的瞬态行为(transient behavior)的效率是极为低下的。本文也研究了当运动的初始条件被固定,用数值仿真分析激励力频率和激励力幅值变化时的船舶倾覆行为的表现,发现用数值仿真在这些控制参数变化时研究船舶的倾覆行为时的效率也是低下的。综合以上分析的结果表明:除非运动的初始条件和海洋结构物系统的控制参数被详尽地研究,否则根据数值仿真的结果一般是难以对海洋结构物的稳性下结论的。我们因而有必要研究用其他的方法来理性地分析海洋结构物的稳性。
本文接下来研究了用路径积分法分析受随机激励的海洋结构物的慢漂极端响应时的表现,在用路径积分法求解海洋结构物运动的非线性随机微分方程时,先是由原始运动方程导出控制海洋结构物响应概率密度的Fokker-Plank-Kolmogorov方程,然后再数值求解Fokker-Plank-Kolmogorov方程得出海洋结构物响应的联合概率密度,进而可得出海洋结构物位移响应的边际概率密度(marginal probability densities)。路径积分法是建立在马尔科夫扩散过程理论和数值插值程序的基础之上的,所以我们首先系统地研究了马尔科夫扩散过程理论,也对用于进行路径积分的各种数值插值程序进行了对比研究分析,并重点研究了一种高效的数值插值程序—高斯-勒让德插值程序。从基于高斯-勒让德积分规则的路径积分法被开发出来以后用该方法所处理的问题还只限于弱非线性的情况。本文的作者成功地用基于高斯-勒让德积分规则的路径积分法来处理了一个强非线性随机振荡问题,发现在非线性很强的情况下极端响应的预报精度也是很高的。接下来本文用路径积分法分析了一受随机激励的海洋结构物的慢漂极端响应,为便于比较,我们采用了曾用数值仿真分析过的同一海洋结构物实例,即一个近海顺应式结构物的慢漂纵荡运动。首先,在结构物的系统方程有解析解的特殊情况下,我们比较了路径积分解和数值仿真解的精度,我们比较了数值仿真5100次和路径积分运行25次各自预报得到的该海洋结构物的慢漂位移极端响应的边际概率密度值,发现在预报随机激励下海洋结构物的慢漂极端响应时,用路径积分法预报得到的结果比用数值仿真预报得到的结果精度要高得多。同时发现路径积分法预报结构物慢漂极端响应的效率比数值仿真预报的效率要高。接下来,在结构物的系统方程无解析解的一般情况下,我们用路径积分法获得了此近海顺应式结构物的慢漂极端响应的精确预报。我们看到,采用路径积分法的另一大优点是该海洋结构物的慢漂极端响应的超越概率可被连带获得,而且在计算机程序运行时几乎不需要额外的时间。
在用路径积分法求海洋结构物的慢漂极端响应时,海洋结构物慢漂响应的初始概率密度是由研究者自己选取的,初始概率密度选取的好坏对路径积分法运行的时间是有很大影响的。为找到一条优选初始概率密度的高效的途径来进一步提高路径积分法的效率,本文提出了一种复合路径积分法,其思想是先用少量的数值仿真并结合正态分布的3原则来粗估一个初始概率密度,这样得到的初始概率密度与最终结果就会很接近,然后将得到的初始概率密度带入路径积分法的源程序中运行,就可节省路径积分的次数。我们在结构物的系统方程有解析解的特殊情况下,比较了复合路径积分法和原路径积分法的效率和精度,证明了复合路径积分法的精确性和高效性。接下来,在结构物的系统方程无解析解的一般情况下,我们用复合路径积分法来计算了一系泊漂浮圆柱体的慢漂极端响应。本文的研究显示:在有合适的商用水动力软件配合的前提下,路径积分法可为船舶和海洋工程师们提供一强有力的工具来预报系泊海洋结构物在随机激励下慢漂振荡的极端响应。σ
本文最后研究了用已有的迈尔尼可夫法分析受随机激励的海洋结构物(船舶)的稳性时的表现。迈尔尼科夫法并不直接求解船舶横摇运动的微分方程,取而代之的是,迈尔尼科夫法专注于研究系统的质的方面的行为(或者更精确地说,质的方面不同的行为间的转变)。在研究中我们用人们已验证了的迈尔尼可夫衡准来分析了一艘驳船的动态稳性。为此我们首先计算了本船横摇运动微分方程中有关的各水动力参数,接下来把有关的各水动力参数无因次化,然后我们根据已有的迈尔尼科夫衡准公式计算了驳船为克服临界波浪激励力矩值而需要的非线性阻尼值。我们发现迈尔尼科夫衡准公式将临界横摇激励力矩与船舶阻尼值联系了起来,这使得设计者可以根据临界横摇激励力矩值来调整船舶的阻尼值,例如增大舭龙骨尺寸,或调整船舶的其它主要参数。这一点在船舶初始设计阶段设计需要反复循环时显得尤为重要。我们同时比较了用数值仿真在激励力幅值控制参数变化时研究船舶的倾覆行为时的效率,发现迈尔尼科夫分析比数值仿真既具有效率上的优越性,又具有易下结论的优越性。我们接着用迈尔尼科夫法分析了带初始横倾的船舶的倾覆行为,给定一个横倾量值,我们利用随机迈尔尼科夫均方值分析得出了对应于不同阻尼值的临界波浪激励力矩值。我们同时也比较了在船舶的横摇初始条件变化时用数值仿真研究船舶的倾覆行为的效率。发现在船舶初始设计阶段用迈尔尼科夫法分析带初始横倾的船舶的倾覆行为是高效的。综合我们的研究结果,我们认为在船舶初始设计阶段,在有合适的商用水动力软件配合的前提下,迈尔尼可夫法可被用作为一个分析船舶动态稳性的高效的辅助工具。
在上述研究过程中笔者编制了一些Mathematica程序包。在研究的最后我们给出了在这一领域内进一步研究的方向。
The values in the tail region of the response probability density curve of ocean structures under random loading are called extreme responses. One kind of problem often met in design is how one can accurately predict ocean structures’slow drift surge extreme responses. If the extreme rolling response of an ocean structure exceeds a critical value, the structure will lose its stability and capsize. The above mentioned extreme response and capsizing behavior can be called the extreme behavior of an ocean structure. This dissertation uses several different methods to analyze the extreme behavior of an ocean structure.
To study the slow drift extreme response and the capsizing behavior of ocean structures, dynamic analysis is required on the ocean structures. When performing dynamic analysis on these structures, the added mass, damping, restoring force and wave excitation of unit wave amplitude can all be calculated by some mature commercial software. With these information the nonlinear random differential equation of motion of the ocean structure can be established. The main work of this dissertation is to study the effective methods for solving these nonlinear random differential equations of motion.
When performing dynamic analysis on ocean structures, the method of numerical simulation is often used by the people, i. e. to perform numerical integration on the ocean structure’s differential equation of motion, and then statistically processing the aquired response time histories. The principle for doing this is very simple and clear, and the execution is also very convenient. Meanwhile, the predicted response values in the large scope of the central part of the probability density curve are also very accurate by this means. The above advantages of numerical simulation make some researchers use it also for analyzing the extreme behavior of ocean structures. This dissertation makes counter thinking on this. Firstly, this research considers concurrently the problem of the simulation efficiency and the problem of the precision of the predicted extreme response. An example of a compliant offshore structure is used to demonstrate that the slow drift extreme responses of the ocean structure predicted by numerical simulation are not very accurate. To improve the accuracy of the predicted slow drift extreme responses, some people will consider to increase the simulation numbers. Using an example, this research first demonstrates that:when a specific level of accuracy of the slow drift extreme responses is reached, increasing the number of simulations will no longer work anymore,and the accuracy of the slow drift extreme responses is not to the satisfaction to the people. To overcome the limitation of this kind of brute force simulation, it is therefore necessary for us to study using other methods for predicting the slow drift extreme responses of ocean structures under random excitations. This dissertation also quantitatively studies of the performance of analyzing the stability behavior of ocean structures (ships) by numerical simulation. It is found that the efficiency is very low when thoroughly investigating the ship rolling initial conditions’phase space via numerical simulation, i.e. the efficiency is very low when studying the ship capsizing transient behavior via numerical simulation. This dissertation also studies the performance of numerical simulation for analyzing the ship capsizing behavior when the exciting frequency and amplitude are varied and the motion initial conditions are fixed. It is found that the efficiency is very low when using numerical simulation for studying the ship capsizing behavior in the control parameter space. Summarizing the above analysis results shows that the numerical simulations results are generally inconclusive on the ocean structures’stability behavior unless the motion initial conditions and the ocean structure system’s control parameter space are thoroughly studied. Therefore, it is necessary for us to study using other methods in order to rationally analyze the stability of ocean structures.
Next, this dissertation studies the performance of the path integral solution method for analyzing the slow drift extreme responses of ocean structures under random excitations. When using the path integral solution method to solve the nonlinear random differential equation of motion of an ocean structure, the Fokker-Plank-Kolmogorov equation governing the response probability density of the ocean structure is first derived from the original equation of motion. Then the Fokker-Plank-Kolmogorov equation is numerically solved to obtain the response joint probability densities of the ocean structure. Thereafter, the response displacement marginal probability densities of the ocean structure are obtained.
The path integral solution method is established upon the Markov diffusion process theory and the numerical interpolation scheme. Therefore, we first systematically studies the Markov diffusion process theory. Comparison study and analysis are also conducted on various kinds of numerical interpolation schemes for doing path integration. Emphasis is put on studying a kind of high efficiency numerical interpolation scheme-the Gauss-Legendre interpolation scheme. The problems solved by the path integral solution based on Gauss-Legendre interpolation scheme are limited to weakly nonlinear cases. The author of the dissertation successfully solve a strongly nonlinear random oscillation problem by the path integral solution based on Gauss-Legendre interpolation scheme. It is found that the accuracy of the predicted extreme responses is very high even when the nonlinearity is very strong. Next, the slow drift extreme responses of an ocean structure under random excitations are analyzed via path integral solution method in this dissertation . For comparison, we adopt the same ocean structure example once analyzed via numerical simulation , i. e. the slow drift surge motion of a compliant offshore structure. First, we compare the accuracy of the path integral solutions and the numerical simulation solutions in the special case that the system equation has analytical solutions. We compare the ocean structure’s slow drift displacement extreme response marginal probability density values predicted via 5100 rounds numerical simulations and 25 rounds path integral solutions. It is found that the accuracy of the results of the slow drift extreme responses of the ocean structure predicted by the path integral solution method is much higher than those predicted by the numerical simulation. The efficiency for predicting the slow drift extreme responses of the structure by the path integral solution method is higher than that of the numerical simulation. Next, in the general case that the structure system equation has no analytical solutions, we obtain accurate prediction of the slow drift extreme responses of the compliant offshore structure via the path integral solution method. We find that another advantage of path integral solution is that the ocean structure’s slow drift extreme response exceeding probabilities can be obtained as a by-product. Also almost no extra computer running time is needed for doing this.
When using the path integral solution method for solving the slow drift extreme responses of an ocean structure, the initial probability density of the slow drift response of the ocean structure is decided by the researcher. Whether a good choice is made has great influence on the running time of the path integration. To find an optimal way for selecting the initial probability density in order to further improve the efficiency of the path integral method, this dissertation proposes a compound path integral solution method. The idea of the compound path integral solution is to estimate a rough initial probability density using small number of numerical simulations and the 3 sigma rules of normal distribution. The initial probability density obtained by this means will be very similar as the final result. Then the obtained initial probability density is inserted to the source code of the path integral solution method. The number of numerical path integrations will be reduced via this means. We compare the accuracy of the compound path integral solutions and original path integral solutions in the special case that the system equation has analytical solutions. The accuracy and efficiency of the compound path integral solution method is validated. Next, in the general case that the structure system equation has no analytical solutions, we calculate the slow drift extreme responses of a moored floating cylinder. This research demonstrates that the path integral solution method together with suitable commercial hydrodynamic software can provide a powerful tool for naval architects and ocean engineers for predicting the extreme responses of slow drift oscillations of moored ocean structures under random excitations.
This dissertation finally studies the performance of the existing Melnikov method for analyzing the stability behavior of ocean structures (ships) under random excitations. Instead of directly solving the rolling motion differential equation of a ship, the Melnikov method concentrates on studying the qualitative behavior of the system ( or more precisely, the changes in qualitatively different behaviors.). In the research we utilize the Melnikov method that has been validated by others for analyzing the dynamic stability of a barge. To do so we first calculate the hydrodynamic force coefficients in the ship’s rolling differential equation. Next, we non-dimensionalize the hydrodynamic force coefficients, and calculate the nonlinear damping needed for overcoming the critical wave exciting moment using the existing Melnikov formulas . We find that the critical rolling exciting moment and the ship’s damping are connected via the Melnikov criteria formula. This makes it possible for the designer to adjust the ship’s damping value according to the critical rolling exciting moment value, e.g.. to increase the dimensions of the bilge keel, or to change other main parameters of the ship. This aspect is particularly important in the initial design phase of the design circle. After comparing the efficiency of the numerical simulation for studying the ship’s capsizing behavior in the control parameter space, we find that Melnikov analysis is efficient and conclusive, and has advantages over numerical simulation. Next, we use the Melnikov method to analyze the capsizing behavior of a biased ship. Given a bias value, we utilize the stochastic Melnikov mean square value analysis to obtain the critical wave exciting moment values corresponding to various damping values. We also compare the efficiency of using numerical simulation to thoroughly study the ship’s rolling initial conditions’phase space. It is found that the efficiency of using the Melnikov method for analyzing the biased ship’s capsizing behavior in the ship initial design phase is high. Summarizing the research results, we believe that together with suitable commercial hydrodynamic software the Melnikov method can be used as a highly efficient auxiliary tool for analyzing the dynamic stability of ship.
Some Mathematica software packages have been developed by the author during the research project. In the final stage of this research we give some further research directions in this field.
引文
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