Hybrid adaptive method for finding the roots of a nonsmooth function in the problem of determining the natural frequencies of fluid vibrations in reservoirs

Abstract

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.