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

Gnitko, V.; Degtyariov, Kyryl; Karaiev, Artem; Strelnikova, Elena

doi:10.2495/be420171

Cited by 15 publications

(9 citation statements)

References 14 publications

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“…To get the closed system for obtaining the unknown potential, we must include the relationship between Euler's and Lagrange's velocities on the free surface [14]:…”

Section: Problem Statementmentioning

confidence: 99%

“…Solution of eqns (13) and (14) allows us to receive the potential j that describes both gravitational and capillarity effects on the free surface of the liquid in partially fluid-filled shells.…”

Section: Problem Statementmentioning

confidence: 99%

“…3). The requisite number of boundary elements for evaluating unknown functions with given accuracy was estimated by the authors in [14]. It was shown that applying 180 boundary elements along the wetted parts of shells and 100 elements along the free surface radius is sufficient for reaching accuracy e = 10 −3 .…”

Section: Numerical Simulation Of Axisymmrtric Liquid-free Vibrationsmentioning

confidence: 99%

See 2 more Smart Citations

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

Gnitko¹,

Karaiev²,

Myronenko³

et al. 2021

Int. J. CMEM

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The paper presents a problem of gravitational-capillarity wave propagation in the frame of boundary integral equations. The wave propagation is considered in rigid compound shells of revolution. The liquid is supposed to be an ideal and incompressible one, and its flow is irrotational. The boundary value problem is formulated for Laplace's equation to obtain the velocity potential. Non-penetration boundary conditions are used at the shell's wetted surface, as well as kinematic and dynamic boundary conditions are given on the free liquid surface. Effects of surface tension are included in the Bernoulli's equation as additional pressure that is proportional to the free surface mean curvature. It allows us to consider coupled effects of both gravitational and capillarity waves. The problem is reduced to a system of singular integral equations. For their numerical simulation, the boundary element method is in use. The singular integral equations in implementation of a discrete model are transformed to linear algebraic ones, and eigenvalue problems are solved for different capillarity length numbers. Benchmark numerical investigations are presented including different kinds of compound rigid shells.

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“…To get the closed system for obtaining the unknown potential, we must include the relationship between Euler's and Lagrange's velocities on the free surface [14]:…”

Section: Problem Statementmentioning

confidence: 99%

Section: Problem Statementmentioning

confidence: 99%

Section: Numerical Simulation Of Axisymmrtric Liquid-free Vibrationsmentioning

confidence: 99%

See 1 more Smart Citation

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

Gnitko¹,

Karaiev²,

Myronenko³

et al. 2021

Int. J. CMEM

Self Cite

View full text Add to dashboard Cite

show abstract

“…The boundary element method (BEM) [1] has become an efficient and popular alternative to the finite element and finite differences methods, because one of the greatest benefits of using BEM [2–4], or other boundary‐based methods is that we need to make a discretization for body's boundary only, rather than the entire domain. But if we consider the BEM application to inhomogeneous problems or boundary value problems with body forces, time dependent effects or certain class of non‐linearities then it generally leads to governing integral equations that contain additional domain integrals.…”

Section: Introductionmentioning

confidence: 99%

Axisymmetric polyharmonic spline approximation in the dual reciprocity method

Karaiev

Strelnikova

2021

Z Angew Math Mech

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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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“…In modelling mechanical processes, systems of differential equations with periodic boundary value conditions are usually involved. Analytical solutions of these equations have been obtained for some simple cases, but advanced numerical methods are currently needed in mechanical and engineering applications [8], [9]. For successive applications of numerical methods to solve periodic boundary value problems (PBVP), theorems are needed regarding periodic behaviour of the solutions.…”

Section: Introductionmentioning

confidence: 99%

Boundary Element Method Analysis of Boundary Value Problems With Periodic Boundary Conditions

Gnitko

Karaiev

Choudhary

et al. 2021

WIT Transactions on Engineering Sciences

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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.

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

Cited by 15 publications

References 14 publications

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

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

Axisymmetric polyharmonic spline approximation in the dual reciprocity method

Boundary Element Method Analysis of Boundary Value Problems With Periodic Boundary Conditions

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