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It is established that the effectiveness of the developed method exceeds the efficiency of both methods - hybrids, when they have applied separately. The developed method is applied to the solution of the characteristic equation in the problem of determining the natural frequencies of oscillations of a liquid in a rigid tank having the form of a shell of revolution. The fluid in the tank is assumed to be perfect and incompressible, and its motion caused by the action of external loads is eddy. Under these assumptions, there exists a velocity potential to describe the fluid motion. The formulation of the problem is given and the method of its reducing to the solution of a nonlinear equation is given. This equation is a characteristic one for the corresponding problem of eigenvalues. The methods of integral singular equations and the boundary element method for their numerical solution are applied. The problem of fluid oscillation in a rigid cylindrical tank is considered. 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引用次数: 0

摘要

本文提出了一种求单变量非光滑函数根的混合自适应方法。提出了非光滑函数的自适应根搜索算法。它假设自适应缩减搜索步骤,并改变搜索方向。发现所提出的方法使我们能够在存在拐点的情况下检测到词根。例如,这对于牛顿法来说是不可能的。该算法的求根精度不依赖于函数的类型、初始近似的选择;在任何情况下,该方法都会以给定的精度找到根。采用二分法、“3/5”法和本文提出的算法对根计算结果进行了比较。结果表明,该方法在单独应用时的有效性超过了两种混合方法的有效性。将该方法应用于确定旋转壳形刚体中液体振荡固有频率问题特征方程的求解。假定罐内流体为完全不可压缩流体,其在外部载荷作用下的运动为涡流。在这些假设下,存在一个速度势来描述流体运动。给出了该问题的表述,并给出了将其化为非线性方程解的方法。这个方程是对应特征值问题的特征方程。应用积分奇异方程的方法和边界元法对其进行数值求解。研究了刚性圆柱形储罐内流体的振荡问题。比较了不同方法对不同节点直径数下流体振荡频率的数值模拟结果。值得注意的是,如果特征方程的根使用近似方法进行局部化,则可以使用所提出的方法进行其细化。
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Hybrid adaptive method for finding the roots of a nonsmooth function in the problem of determining the natural frequencies of fluid vibrations in reservoirs
The article proposes a hybrid adaptive method for finding the roots of a non-smooth function of a single variable. The algorithm of adaptive root search method for non-smooth functions is presented. It assumes both adaptive reduction of a search step, and changing the search direction. It is found that the proposed approach allows us to detect the root even in the presence of a point of inflection. That is, for example, is impossible for the Newton method. The accuracy of finding the root using the proposed algorithm does not depend on the type of functions, the choice of the initial approximation; the method in any case will find the root with the given accuracy. Comparison of the results of the root calculations is performed using the dichotomy method, the "3/5" method and the proposed algorithm. It is established that the effectiveness of the developed method exceeds the efficiency of both methods - hybrids, when they have applied separately. The developed method is applied to the solution of the characteristic equation in the problem of determining the natural frequencies of oscillations of a liquid in a rigid tank having the form of a shell of revolution. The fluid in the tank is assumed to be perfect and incompressible, and its motion caused by the action of external loads is eddy. Under these assumptions, there exists a velocity potential to describe the fluid motion. The formulation of the problem is given and the method of its reducing to the solution of a nonlinear equation is given. This equation is a characteristic one for the corresponding problem of eigenvalues. The methods of integral singular equations and the boundary element method for their numerical solution are applied. The problem of fluid oscillation in a rigid cylindrical tank is considered. The results of numerical simulation of the fluid oscillation frequencies obtained by different methods for different number of nodal diameters are compared. It is noted that if the root of the characteristic equation is localized using approximate methods, then its refinement can be carried out using the proposed approach.
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