Multi-Domain Boundary Element Method for Axisymmetric Problems in Potential Theory and Linear Isotropic Elasticity

Abstract

The paper presents an approach based on reduced boundary element methods to resolve axisymmetric problems in potential and linear isotropic elasticity theories. The singular integral equations for these problems are received using fundamental solutions. Initially three-dimensional problems expressed in Cartesian coordinates are transformed to cylindrical ones and integrated with respect to the circumference coordinate. So the three-dimensional axisymmetric problems are reduced to systems of singular integral equations requiring the evaluation of linear integrals only. The fundamental solutions and their derivatives are expressed in terms of complete elliptic integrals. The effective algorithm for treatment of the singular integrals is proposed. The multi-domain boundary element method is applied for the numerical simulation. As examples, the following problems are considered: fluid induced vibrations of a compound cylindrical-spherical elastic shell partially filled with an ideal incompressible liquid, and axisymmetric elasticity problems for an isotropic body with rigid or elastic circular cylindrical inclusions.

Singular Boundary Method in a Free Vibration Analysis of Compound Liquid-Filled Shells

“…In the BEM simulation, the one-dimensional axisymmetrical formulation is in use [15], [16], and the SBM is described beforehand. The equal numbers of elements on the free surface (m) and on wetted parts (3m) of the shell are applied both in the BEM and SBM.…”

“…It is ascertained that the diagonal elements of matrixes These expressions are called origin intensity factors in singular boundary methods. In the axisymmetric case, we can calculate surface integrals as curve integrals of Green's function integrated by the angle variable [15]. Green's function and its normal derivative, integrated by the angle variable, are expressed in terms of complete elliptic integrals of the first and second kind [15], [16] using the following formulae [17]…”

BEM analysis of gravitational–capillarity waves on free surfaces of compound shells of revolution

“…Taking the time derivative of eqn (8) and using condition (9), we have (10) Operator p ∆ here is Laplacian in the plane perpendicular to Oz axis. It can be expressed through the standard Laplacian by the following formula:…”

“…Three-dimensional statements of the problem were considered in [8,9]. It should be noted that interest to coupled gravity-capillarity interaction has notably increased in recent decades due to developing new mathematical tools together with advanced numerical methods and experimental techniques [10,11].…”

“…26)Considering the shells of revolution, replace Cartesian coordinates (x, y, z) with cylindrical ones (r, q, z) in eqn (26) with respect to the circumference coordinate q as in[16], taking into account that operators R and Q can be expressed on terms of complete elliptic integrals as follows[10]:( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) are the components of the external unit normal n to the surface s in the r and z directions, respectively, and ( ) ( )…”