Basic Integral Equation Research Articles

On the Influence of Interlayer Nonuniformity on Contact Characteristics during the Interaction of a Pipe and a Rigid Cylindrical Insert

Contact problem for viscoelastic aging pipe with a longitudinally nonuniform thin elastic internal coating and a rigid cylindrical insert is considered in the paper. The basic integral equation with integral operators of different types (mixed integral equation) is given. It's analytical solution for contact stresses in insert area is presented. The solution is constructed in such a way that the function describing the inner coating nonuniformity is distinguished by a separate factor. This fact allows one to perform accurate calculations even in cases where the coating properties are described by rapidly changing and even discontinuous functions. Other known analytical methods do not allow one achieving such a results.

Boundary element methods in diffraction of a point-source acoustic wave by a rigid infinite wedge

The paper presents some boundary element methods (BEM) applied to diffraction of the harmonic acoustic wave generated by a point source. The authors first consider a basic boundary integral equation (BIE). Its analytical analysis permits development of some explicit conclusions. Then, some boundary element methods arise as a result of the dicretization of the BIE by various quadrature formulas. The matrices of the arising systems of linear algebraic equations (SLAE) are studied in the aspect of the structure of their spectrum. It is shown that the spectrum of the matrices is located in the right half-plane, hence the matrices are positive-definite. This predetermines a rapid convergence of the iterative algorithms applied, to solve the SLAEs.

НЕСТАЦИОНАРНОЕ КОНТАКТНОЕ ВЗАИМОДЕЙСТВИЕ АБСОЛЮТНО ТВЕРДОГО УДАРНИКА И УПРУГОГО ПОЛУПРОСТРАНСТВА С ЗАГЛУБЛЕННОЙ ПОЛОСТЬЮ

A closed mathematical formulation of plane non-stationary contact problems for rigid dies and an elastic half-space with deepened cavities is constructed. Using the dynamic theorem of reciprocity of works, a system of resolving equations is obtained. It includes the basic boundary integral equation arising from the principle of reciprocity and boundary conditions, as well as the equations of translational and rotational motion of the die. The fundamental and singular solutions for the elastic plane are the cores of the main resolving equation. They determine displacements and stresses in the elastic plane from the applied single instantaneous concentrated force. An original solution algorithm based on the method of boundary integrals with an additional iterative procedure that allows one to take into account the partial separation of the boundary surfaces of the die and the half-space in the contact area has been developed and implemented on a computer. In this case, at each moment of time in the half-space, there is a region that is in a perturbed state and outside of which there are no perturbations. The integral operators of the resolving system of equations are replaced by discrete analogs in the spatial variable and in time. A parametric study of the process of unsteady contact of an absolutely rigid rectangular die with a half-space having a recessed cavity is carried out. Analysis of the calculation results revealed the manifestation of a significant effect of the cavity on the process of non-stationary contact interaction. The influence of the cavity begins to show itself from the moment the reflected waves arrive from its boundary. In this case, the nature of the unsteady stress-strain state and displacements changes significantly. The developed calculation algorithm can be used in engineering practice by design and research organizations in the process of designing and calculating buildings and structures under the influence of natural and man-made vibrations propagating in the soil.

The Boundary Element Method applied to the solution of the diffusion-wave problem

A Boundary Element Method formulation is developed for the solution of the two-dimensional diffusion-wave problem, which is governed by a partial differential equation presenting a time fractional derivative of order α, with 1.0 < α < 2.0. In the proposed formulation, the fractional derivative is transferred to the Laplacian through the Riemann–Liouville integro-differential operator; then, the basic integral equation of the method is obtained through the Weighted Residual Method, with the fundamental solution of the Laplace equation as the weighting function. In the final expression, the presence of additional terms containing the history contribution of the boundary variables constitutes the main difference between the proposed formulation and the standard one. The proposed formulation, however, works well for 1.5 ≤ α < 2.0, producing results with good agreement with the analytical solutions and with the Finite Difference ones.

Study of the ionization efficiency for nuclear recoils in pure crystals

We study the basic integral equation in Lindhard's theory describing the energy given to atomic motion by nuclear recoils in a pure material when the atomic binding energy is taken into account. The numerical solution, which depends only on the slope of the velocity-proportional electronic stopping power and the binding energy, leads to an estimation of the ionization efficiency which is in good agreement with the available experimental measurements for Si and Ge. In this model, the quenching factor for nuclear recoils features a cut-off at an energy equal to twice the assumed binding energy. We argue that the model is a reasonable approximation for Ge even for energies close to the cutoff, while for Si is valid up to recoil energies greater than ~500 eV.

Plane problem of contact interaction for foundation with multilayer nonuniform coating

Present article is devoted to solving the problem of contact interaction between a rigid punch and a viscoelastic aging foundation with a thin coating, which consists of several elastic layers. The nonuniformities of layer depend on longitudial coordinate. They can be described be discontinuous or rapidly changing unctions. The basic mixed integral equation and integral additional conditions for this problem are presented. The analytic solutions for all versions of the problem are presented.

Wear of foundation with a nonuniform coating by rough punch

Plane problem of contact interaction for a viscoelastic foundation with multilayer coating and a rigid punch is considered. It is assumed that thin coating consists of several different elastic layers. The rigidity of the each layer depends on longitudial coordinate and can be described by own rapidly changing function. We obtain basic integral equation and additional conditions for this problem. The analytic solution of the problem is presented.

Methods for the Identification of the Magnetization of the Permanent Magnets on the Basis of Integral Equations. Research and Application Examples

Methods for the Identification of the Magnetization of the Permanent Magnets on the Basis of Integral Equations. Research and Application Examples

Electrodynamic treatment by electric current pulses as effective method of control of stress-strain states and improvement of life of welded structures

The present work was aimed at the development of the new technological process: a postweld electrodynamic treatment of welded joints by pulses of a high-density current. Basing on the numerical solution of Maxwell’s equations for the case of a pulsed electromagnetic field the calculation model of distribution of field components was designed on the basis of integral equations, which allow determining the electromagnetic forces P and current density j in aluminium alloy AMg6 during the process of capacitive energy storage discharge. The calculation data allow accounting for influence of dynamic stresses and electric plasticity effect on change in a stressed state of a metal during the electrodynamic treatment. Comparison of calculated and experimental results showed a good correlation of the obtained data. Using the model, the parameters of the electrodynamic treatment, providing the reduction of residual stresses in welded joints of alloy AMg6, were determined. Generators of the pulse current and electrode devices, realizing the preset parameters of electrodynamic effects at single- and two-channel schemes were designed and manufactured. Results of experimental investigations, carried out on specimens of base metal and welded joints showed that the electrodynamic treatment has a positive effect on structure of the treated metal and allows decreasing greatly the residual stresses in welded joints, increasing their resistance to fatigue and brittle fracture, as well as eliminating the distortion deformations in thin-walled elements of structures. The developed technologies and equipment gave an opportunity to perform the electrodynamic treatment of welded joints of hull structures of ships of aluminium alloy AMg6 and repair welds of an intermediate casing of the aircraft engine of heat-resistant magnesium alloy ML10, thus promoting the increase in their service reliability and life.

Strong anharmonicity in pristine graphene

The thermodynamic coefficients of a free standing infinite graphene monolayer are calculated using the quasi-classical unsymmetrized self-consistent field method (USF). The basic nonlinear integral equations of this theory are solved numerically in the strong anharmonic approximation. The isothermal and adiabatic elastic bulk moduli, the isochoric and isobaric heat capacities, the thermal expansion, thermal pressure coefficient, and the macroscopic Gruneisen parameter are calculated in terms of the derivatives of a specifically chosen interatomic potential function for different values of stress and for temperatures ranging from below room temperature up to the point of loss of thermodynamic stability. The nearest-neighbor distances vary from ≳1.4 Å to ≲1.8 Å for zero stress. Under stress, these distances decrease. At room temperature the molar heat capacities are ∼5.0 Jmol−1K−1. The elasticity moduli vary from 15.0 eV up to zero at the temperature of loss of stability and are increased by stress. The thermal expansion coefficient has a strong dependence on the temperature and is negative for temperatures lower than ∼340 K. For high temperatures it monotonically increases and decreases with stress. The macroscopic Gruneisen parameter has a strong nonlinear dependence with temperature and is estimated in about at ∼340 K its value decreases to ∼1.0 K and for even lower temperature it shows a peak and deep structure similar to what has been earlier reported for fullerene C60.

A fast numerical algorithm for a basic dual integral equation of the flapping wing in a flow of non-viscous incompressible fluid

The present paper proposes a new approach to the classical problem of the harmonic oscillations of a thin wing in a flow of non-viscous incompressible fluid. The problem is reduced to a dual integral equation, permitting application of numerical methods. The numerical experiments are performed by using some advanced fast non-stationary iterative methods, with the help of the two-dimensional Fast Fourier Transform. There is given a brief survey on the iterative methods, to evaluate the most efficient algorithms in application to the considered problem of the flapping wing theory.

Axisymmetric contact between a rigid punch and a coated foundation with rough surfaces

Axisymmetric contact problem for a coated foundation and a rigid punch is considered. It is assumed that the shape of the thin upper layer (coating) and the shape of the punch surface can be described by different rapidly changing functions. We obtain basic integral equation for this problem and additional condition. A projection method is developed for this problem. It allows one to obtain the solution of the equation with high accuracy that cannot be done by known methods.

Radial integration boundary element method for heat conduction problems with convective heat transfer boundary

ABSTRACTA new boundary domain integral equation with convective heat transfer boundary is presented to solve variable coefficient heat conduction problems. Green’s function for the Laplace equation is used to derive the basic integral equation with varying heat conductivities, and as a result, domain integrals are included in the derived integral equations. The existing domain integral is converted into an equivalent boundary integral using the radial integration method by expressing the normalized temperature as a series of radial basis functions. This treatment results in a pure boundary element analysis algorithm and requires no internal cells to evaluate the domain integral. Numerical examples are presented to demonstrate the accuracy and efficiency of the present method.

Numerical analysis of an advective diffusion domain coupled with a diffusive heat source

This paper presents a formulation of the boundary element method (BEM) for the study of heat diffusion and advective effect in isotropic and homogeneous media. The proposed formulation has a time independent fundamental solution obtained from the two-dimensional Laplace equation. Consequently, the formulation is called D-BEM since it has domain integrals in the basic integral equation. The first order time derivative that appears in the integral equations is approximated by a backward finite difference scheme. Homogeneous subregions are considered in the analysis in a specific model simulating a nonuniform flow passing by a circular obstacle under internal heat generation. Results from numerical models are compared with the available analytical solutions. The correlation estimator R2 is employed to validate the numerical model and to demonstrate the accuracy of the proposed formulation.

A triple reciprocity method in Laplace transform boundary element method for three-dimensional transient heat conduction problems

In this paper, a boundary element method employing the Laplace transform is developed to solve the three-dimensional transient heat conduction problems. The fundamental solution for the modified Helmholtz equation is adopted in order to derive the basic integral equations. Due to the effects of initial temperature and heat generation, the domain integrals appearing in the integral equation will degrade the advantages of boundary element method. Therefore, a new triple reciprocity formulation in Laplace domain is proposed to convert the domain integrals into boundary integrals. The higher order fundamental solutions required in the triple reciprocity method can be obtained from the original fundamental solution by multiplying a constant term, which leads to a much simpler formulation in the Laplace domain. Then, the inverse transformation can be used to obtain the solutions in the time domain. Several numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method. In summary, the triple reciprocity formulation is a useful approach to capture the transient heat conduction responses.

Numerical Modeling of Uniform Slab Brightness Coefficients based on the Mirror Reflection Principle and Linear Singular Integral Equations Solutions

Based on the mirror reflection principle and solutions of modified linear singular integral equations, the numerical modeling of the unified exit function of the outside radiation field and photometrical invariants of brightness coefficients for a uniform slab of finite optical thickness has been carried out. The efficacy of applying the angular discretization method for problems of numerical modeling of outer radiation fields in the «atmosphere — underlying surface» system has been proved. This new approach allows generalizing the basic results in the particular case of a semi-infinite uniform slab. In this connection the main mathematical aspects and computational peculiarity of the numerical realization of the angular discretization method have been considered. Due to linearity of the used basic integral equations, the conducted analysis can be generalized to the case of scalar and polarized inner radiation fields taking into account the multiple anisotropic scattering of photons and their reflection from an arbitrary horizontally uniform underlying surface.

Convolution Quadrature based BEM in acoustics for absorbing boundary conditions

AbstractIn many fields of engineering the acoustic behavior has to be determined, e.g. the sound distribution in a room or the sound radiation into the surrounding. Often, the goal is to obtain a sound pressure field such that disturbing noise is reduced to an acceptable level. In room acoustics, sound absorbing materials are often used to obtain this goal. The mathematical description is done with the wave equation and absorbing boundary conditions. The numerical treatment can be done with Boundary Element methods, where the absorbing boundary results in a Robin boundary condition. This boundary condition connects the Neumann trace with the Dirichlet trace of the time derivative.Here, an indirect formulation, which uses the single layer potential, is used as basic boundary integral equation. The convolution quadrature method is applied for time discretisation, which allows a simple formulation of the Robin boundary condition in the Laplace domain. Convergence studies with a refinement in space and time show the expected rates. A realistic example for indoor acoustics, the computation of the sound pressure level in a staircase of the University of Zurich, show the suitability of this approach in determining the indoor acoustics. The absorbing boundary condition shows the expected behavior. (© 2016 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

A boundary integral equation over the thin rotating blade of a wind turbine

In the present paper we study the aerodynamics of a thin rotating Wind Turbine (WT) blade. Under the assumption of high specific speed (i.e. large angular velocity), by using the classical potential theory, the problem is reduced to a dual integral equation over the surface of the blade. Since a certain value of the axial component of the velocity vector, which is not known a priori, is present in the right-hand side of the basic dual integral equation, the method proposed is based on an iteration scheme. This rapidly converges in two-three steps. The method is tested on example of a small WT, by a comparison with the results obtained by the ANSYS-CFX finite-volume software.

Current Distribution of Dipole Antenna forDifferent Lengths Using Different Types ofBasis Functions Applying To Method ofMoment

In this paper, the generation of current distribution of a planar dipole antenna for different length of the antenna using different types of basic functions applying Method Of Moment (MOM) is discussed. Here, we give a brief description on MOM for generating the current distribution of the dipole antenna over the entire length of the antenna so that we became enable to calculate the self impedance of the dipole antenna. The calculation starts from the basic Pocklingtons' Integral Equation (PIE) of Greens' function. The basic functions are chosen as the rooftop function and the pulse function and the further progress has been done.

A mathematical model for the propulsive thrust of the thin elastic wing harmonically oscillating in a flow of non-viscous incompressible fluid

In the present paper there is proposed an analytical approach to study vibration of a rectangular elastic wing in the stationary stream of non-viscous fluid. We first develop a basic two-dimensional integral equation. Then a series expansion along the short coordinate is applied. This reduces the problem to an infinite set of one-dimensional integral equations which is studied asymptotically with respect to the large aspect ratio parameter. An example of optimization of thickness of the wing is demonstrated, to test the efficiency of the proposed method in applications.