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.

A new numerical approach is proposed to address problems of free liquid vibrations in axisymmetric compound rigid shells using a singular boundary method. The liquid is supposed to be perfect and incompressible, and its flow is irrotational so the liquid velocity can be presented as a potential gradient. The approximation of a small fluid surface elevation is used, and the free surface function is presented as a sum of the fluid-filled shell height and the small elevation function. A series expansion of the potential function about stationary states is used. To find the stationary states, an eigenvalue problem is formulated. Eigenvalues and eigenvectors are obtained using the singular boundary method with the origin intensity factor as the singular integral over the singular boundary element. The numerical results for free vibration analysis of cylindrical shells, obtained by singular boundary method and direct boundary element methods, are compared. Different compound rigid shells are considered in numerical simulations of free liquid vibrations.

The paper is devoted to developing the dual reciprocity boundary element method for axisymmetric problems based on usage of axisymmetric polyharmonic splines. The axisymmetric polyharmonic spline is introduced as a linear combination of polyharmonic radial basic functions and polynomial terms. The analytical expressions for proposed axisymmetric polyharmonic radial basic functions are obtained for splines with arbitrary degrees. These expressions include special elliptic integrals that are analyzed and calculated for the first time. The relationships between radial basic functions with positive and negative degree numbers are obtained that allows us to receive the recurrence formulae for specific elliptic integrals with nearest indexes. It reveals the possibility of calculating radial basic functions with arbitrary orders by using the combination of only the first two members in recursive sequence. Implementation of the Gauss well‐known arithmetic‐geometric mean technique provides calculation of the specific elliptic integrals and axisymmetric polyharmonic splines with any given accuracy. Numerical examples for solving two axisymmetric problems in potential theory using the proposed dual reciprocity boundary element method demonstrate high calculation accuracy with low computational costs.

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Singular integral equations arisen in axisymmetric problems of elastostatics are under consideration in this paper. These equations are received after applying the integral transformation and Gauss–Ostrogradsky’s theorem to the Green tensor for equilibrium equations of the infinite isotropic medium. 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 one-dimensional singular integral equations requiring the evaluation of linear integrals only. The thorough analysis of both displacement and traction kernels is accomplished, and similarity in behavior of both kernels is established. The kernels are expressed in terms of complete elliptic integrals of first and second kinds. The second kind elliptic integrals are nonsingular, and standard Gaussian quadratures are applied for their numerical evaluation. Analysis of external integrals proved the existence of logarithmic and Cauchy’s singularities. The numerical treatment of these integrals takes into account the presence of this integrable singularity. The numerical examples are provided to testify accuracy and efficiency of the proposed method including integrals with logarithmic singularity, Catalan’s constant, the Gaussian surface integral. The comparison between analytical and numerical data has proved high precision and availability of the proposed method.