BOUNDARY ELEMENT METHOD ANALYSIS OF BOUNDARY VALUE PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS

Abstract

This paper explores non-axisymmetric boundary value problems for the Laplace equation.Neumann's, Dirichlet's and mixed boundary conditions are involved, supposing their periodic behaviour.Boundary value problems arise as auxiliary issues in many practical applications.Among them there are problems related to numerical simulation of vibrations of fluid-filled elastic shells of revolution, coupled vibrations of elastic circular plates resting on a sloshing liquid, crack propagation in elastic mediums, and more.The common feature in these problems is the necessity to obtain the numerical solution of the Laplace equation under different boundary conditions.As these problems are auxiliary, it is necessary to obtain their numerical solutions with high accuracy.The most effective method to solve these problems is the boundary elements method (BEM).Here a new variant of BEM is proposed for the axisymmetric calculation domain with given periodic functions for boundary conditions.The shape of the calculation domain allows us to reduce surface integral equations to one-dimensional ones.In doing so, we must evaluate elliptic-like inner integrals with high accuracy, to elaborate the method of calculation of the outer integrals with logarithmic, Cauchy or Hadamard finite part singularities.An efficient method for evaluating elliptic-like integrals was developed using a special series for integrands, and the quadrature equations were obtained for highprecision calculation of outer integrals.The method developed can be used to determine free vibration modes and frequencies for elastic fluid-filled shells of revolution.