System Of Hypersingular Integral Equations Research Articles

Error analysis of Chebyshev polynomial-based numerical method for system of hypersingular integral equations

Error analysis of Chebyshev polynomial-based numerical method for system of hypersingular integral equations

Dynamic Instability of Shallow Shells Interacting with a Three-Dimensional Potential Gas Flow

To study the interaction of a vibrating shallow shell with a three-dimensional subsonic gas flow, we deduce a system of hypersingular integral equations for the aerodynamic derivatives of the pressure drop. This system of equations is convenient for the solution of the problems of aeroelasticity. We solve the system of hypersingular integral equations by using a numerical approach based on the discrete vortex method. To model vibrations of shallow shells, we deduce a system of ordinary differential equations with the help of the assumed-mode method. We also perform the numerical investigation of dynamic instability of the equilibrium state of a shallow shell in the subsonic gas flow.

Radiation of waves by a submerged nearly circular rough plate in ice‐covered ocean

AbstractA rough nearly circular (and/or circular) plate submerged in some ice‐covered ocean is undergoing a heaving motion. This radiation problem is reduced to a hypersingular integral equation over the surface of the rough plate. Perturbation method further simplifies the governing equation to get a system of hypersingular integral equations. The geometry of nearly circular plate has been conformally mapped onto a unit disc and the modified equations have been solved by a spectral method proved to be efficient for the case of a circular disc. Both rough disc and rough nearly circular plate have been considered for the numerical results. The effects of the flexural rigidity of ice cover and submergence depth of the plate on added mass and damping coefficient have been investigated along with the effect of roughness of the plate on these hydrodynamic coefficients.

Method for determining the two-dimensional current density distribution function over the radiating structure based on chiral metamaterials

The article is devoted to the development of a method of electrodynamic analysis and a two-dimensional mathematical model of strip radiating structures based on the apparatus of hypersingular equations in order to ensure the correct calculation of their characteristics when using relatively small computational resources. A system of hypersingular integral equations with respect to the unknown transverse and longitudinal components of the current density distribution functions is obtained. This system of hypersingular equations was solved using the collocation method, where Gaussian nodes (zeros of Legendre polynomials) were used as collocation points. This approach allows for faster convergence compared to uniform partitioning. Numerical results of calculations of current density distribution functions for various parameters of the radiating structure based on chiral metamaterials are obtained. It is shown that in the case of wide emitters, it is necessary to take into account both components of the current density distribution function. The advantage of this method in comparison with universal analogues is the ability to accurately calculate the characteristics of radiating structures based on chiral metamaterials with wide emitters.

Numerical solution for the thermally insulated cracks in bonded dissimilar materials using hypersingular integral equations

The new system of hypersingular integral equations (HSIEs) for the thermally insulated inclined cracks and thermally insulated circular arc cracks subjected to remote shear stress in bonded dissimilar materials was formulated by using the modified complex potentials (MCPs) function method with the continuity conditions of the resultant force, displacement and heat conduction functions. This new system of HSIEs is derived from the elasticity problem and heat conduction problem by using crack opening displacement (COD) function and temperature jump along the crack faces. The appropriate quadrature formulas were used to solve numerically the new system of HSIEs for the unknown COD function and the known traction along the crack as the right hand term. Numerical solutions for the value of nondimensional stress intensity factors (SIFs) at all the cracks tips are illustrated. The comparison of nondimensional SIFs for the cracks with and without thermal is also illustrated.

Stress Intensity Factors for Crack Problems in Bonded Dissimilar Materials

The inclined crack problem in bonded dissimilar materials was considered in this study. The system of hypersingular integral equations (HSIEs) was formulated using the modified complex potentials (MCP) function method, where the continuity conditions of the resultant force and the displacement are applied. In the equations, the crack opening displacement (COD) serves as the unknown function and the traction along the cracks as the right-hand terms. By applying the curved length coordinate method and the appropriate quadrature formulas, the HSIEs are reduced to the system of linear equations. It was found that the nondimensional stress intensity factors (SIF) at the crack tips depend on the ratio of elastic constants, the crack geometries and the distance between the crack and the boundary.

Radiation of waves by a thin cap submerged in ice-covered ocean

SummaryThe present article is concerned with the radiation of flexural gravity waves due to a thin cap submerged in the ice-covered ocean. The problem is reduced to a system of hypersingular integral equations using the boundary perturbation method. The first-order approximation has only been considered. The effects of the rigidity of the ice sheet and depth of submergence on the added mass and damping coefficient have been analysed. Two types of caps (for example, concave upwards and concave downwards) have been considered for the numerical results. The effect of the concavity on added mass and damping coefficient has also been studied. The present study should be helpful to understand the nature of waves generated by a heaving submerged body in an ice-covered ocean.

Mode Stresses for the Interaction between Two Inclined Cracks in Bonded Two Half Planes

The various mode of stresses for the interaction between two inclined cracks in the upper part of bonded two half planes which are normal stress (Mode I), shear stress (Mode II), tearing stress (Mode III) and mixed stress was studied. For this problem, the modified complex potentials (MCPs) method was used to develop the new system of hypersingular integral equations (HSIEs) by applying the conditions for continuity of resultant force and displacement functions with the unknown variable of crack opening displacement (COD) function and the right hand terms are the tractions along the crack. The curve length coordinate method and Gauss quadrature rules were used to solve numerically the obtained HSIEs to compute the stress intensity factors (SIFs) in order to determine the strength of the materials containing cracks. Numerical solutions presented the characteristic of nondimensional SIFs at the cracks tips. It is obtained that the various stresses and the elastic constants ratio are influences to the value of nondimensional SIFs at the crack tips.

Stress intensity factors for bonded two half planes weakened by thermally insulated cracks

The thermally insulated inclined or circular arc cracks problems subjected to axial stress in the upper part of bonded two half planes are considered. The modified complex potential function method with the continuity conditions of the resultant force, displacement, and heat conduction functions is used to develop the new system of hypersingular integral equations (HSIEs) for the problems. The new system of HSIEs is solved numerically for the unknown crack opening displacement function and the known traction along the crack as the right-hand term using the appropriate quadrature formulas. Numerical results for the nondimensional stress intensity factors (SIFs) at all cracks tips are presented. A comparison between the nondimensional SIFs for cracks with and without thermal influence is also given

Numerical solution and its analysis for a system of hypersingular integral equations

A system of hypersingular integral equations occurs quite naturally in several branches of science and engineering during the formulation of many boundary value problems. The analytical solution is known for the system of dominant equations. However, there are many real world problems such as crack problems occur in the field of fracture mechanics which may not be formulated as system of dominant equations. Therefore, we propose a numerical method to find the approximate solution for such generalized form. The convergence of the proposed method is proved in L N 2 space. This convergence helps to derive theoretical error bound for the error between the exact and the approximate solution. An application of the proposed method in finding numerical solution of hypersingular integral equation over the curves is shown. Finally, the derived theoretical error bound is numerically calculated and validated with the help of numerical examples. • A numerical method is proposed to find solution for the system of HSIE. • The error bound is derived and convergence of approximate solutions is proved. • All derived theoretical results are validated through test examples.

One-Mode Propagation of Elastic Waves through a Doubly Periodic Array of Cracks

The article is devoted to the derivation of analytical expressions for the reflection and propagation coefficients, when a plane longitudinal wave falls on a system of a finite number of consecutively located identical flat gratings, each of which consists of a periodic array of rectilinear cracks in an elastic isotropic medium. The problem is solved in a flat statement. In the mode of single-mode frequency range, the problem is reduced to a system of hypersingular integral equations, the solution of which gives the reflection and propagation coefficients, as well as the representation of the wave field inside the medium.

Stress intensity factor for multiple inclined or curved cracks problem in circular positions in plane elasticity

AbstractThe problems of multiple inclined or curved cracks in circular positions is treated by using the hypersingular integral equation method. The cracks center are placed at the edge of a virtual circle with radius R. The first crack is fixed on the x‐axis while the second crack is located on the boundary of a circle with the varying angle, θ. A system of hypersingular integral equations is formulated and solved numerically for the stress intensity factor (SIF). Numerical examples demonstrate the effect of interaction between two cracks in circular positions are given. It is found that, the severity at the second crack tips are significant when the ratio length of the second to the first crack is small and it is placed at a small angle of θ.

2D Low-frequency Penetration of Elastic Waves Through a Double Periodic Array of Cracks

The paper is devoted to derivation of analytic expressions for coefficients of reflection and transmission when a plane wave interacts with a system of three identical twodimensional gratings, each of which consists of a periodic array of rectangular cracks of the elastic isotropic medium in 2-D. In a one-mode range approximation the problem is reduced to the system of hypersingular integral equations, solution of which gives the coefficients of reflection and propagation, and an explicit representation of the wave field inside the structure.

A Numerical Method for Solving a System of Hypersingular Integral Equations of the Second Kind

A numerical method for solving a system of hypersingular integral equations of the second kind is presented. The theorem on the existence and uniqueness of a solution to such a system is proved. The rate of convergence of an approximate solution to the exact solution is estimated.

The optimal design of a flap gate array in front of a straight vertical wall: Resonance of the natural modes and enhancement of the exciting torque

We consider a wave energy converter made of an array of Q neighbouring floating flap gates with finite thickness in front of a straight vertical wall in constant depth. Solutions of the radiation and scattering problems are achieved by application of Green's theorem and Green's function yielding a system of hypersingular integral equations for the velocity potential expanded in terms of Legendre polynomials. We investigate how the distance between the array and the vertical wall affects the performance of the array under the action of monochromatic and random waves. We show that large values of the exciting torque on the gates can be obtained by tuning the wall distance with the resonance of the natural modes of the array; this in turn yields large power extraction for a wide range of frequencies.

Flap gate farm: From Venice lagoon defense to resonating wave energy production. Part 2: Synchronous response to incident waves in open sea

We consider a flap gate farm, i.e. a series of P arrays, each made of Q neighbouring flap gates, in an open sea of constant depth, forced by monochromatic incident waves. The effect of the gate thickness on the dynamics of the system is taken into account. By means of Green's theorem a system of hypersingular integral equations for the velocity potential in the fluid domain is solved in terms of Legendre polynomials. We show that synchronous excitation of the natural frequencies of Sammarco et al. (2013) yields large amplitude response of gate motion. This aspect is fundamental for the optimisation of the gate farm for energy production.

Magnetoelectroelastodynamic interaction of multiple arbitrarily oriented planar cracks

The problem of multiple arbitrarily oriented planar cracks in an infinite magnetoelectroelastic space under dynamic loadings is considered. An explicit solution to the problem is given in the Laplace transform domain in terms of suitable exponential Fourier integral representations. The unknown functions in the Fourier integrals are directly related to the Laplace transform of the jumps in the displacements, electric potential and magnetic potential across opposite crack faces and are to be determined by solving a system of hypersingular integral equations. Once the hypersingular integral equations are solved, the displacements, electric potential, magnetic potential and other quantities of interest such as the crack tip intensity factors may be easily computed in the Laplace transform domain and recovered in the physical space with the help of a suitable algorithm for inverting Laplace transforms.

Dynamic response of planar cracks in an infinitely long piezoelectric strip

The problem of determining the electro-elastic fields around an array of parallel planar cracks in an infinitely long piezoelectric strip is considered. The cracks, acted upon by dynamic loads, are either electrically impermeable or permeable. A semi-analytic method based on the theory of exponential Fourier transformation is presented for solving the problem in the Laplace transform domain. The Laplace transforms of the jumps in the displacements and electric potential across opposite crack faces are determined by solving a system of hypersingular integral equations. Once these displacement and electric potential jumps are obtained, the displacements and electric potential and other physical quantities of interest, such as the crack tip stress and electric displacement intensity factors, can be recovered with the help of a suitable algorithm for inverting Laplace transforms. The stress and electric displacement intensity factors are computed for some specific cases of the problem.

Influence of a circular crack on the electromagnetic field in a conductor

The problem of scattering of low-frequency electromagnetic fields on circular cracks is reduced by the method of small parameter to a system of hypersingular integral equations of the Newton-potential type. For the solution of these equations, we propose to use the method of orthogonal polynomials of two variables. It is shown that the electric dipole whose moment is proportional to the normal component of the sounding field is an electrodynamic analog of the crack.

Dynamic interaction of multiple arbitrarily oriented planar cracks in a piezoelectric space: A semi-analytic solution

The problem of determining the electro-elastic fields around arbitrarily oriented planar cracks in an infinite piezoelectric space is considered. The cracks which are acted upon by a transient load are either electrically impermeable or permeable. A semi-analytic method based on the theory of exponential Fourier transformation is proposed for solving the problem in the Laplace transform domain. The Laplace transforms of the jumps in the displacements and electric potential across opposite crack faces are determined by solving a system of hypersingular integral equations. Once these displacement and electric potential jumps are obtained, the displacements and electric potential and other physical quantities of interest, such as the crack tip stress and electric displacement intensity factors, can be computed with the help of a suitable algorithm for inverting Laplace transforms. The stress and electric displacement intensity factors are computed for some specific cases of the problem.