Method Of Singular Integral Equations Research Articles

Multi-stage Hydro-fracture Trajectories: Modelling by the SIE Method

This study is aimed at simulation of numerous curvilinear hydro-fracture trajectories in the framework of the mechanics of brittle fracture. The numerical model is based on the method of singular integral equations, SIE. The process of fracture propagation is considered as quasi-static, it is presented by iterations of the stationary step-by-step states, for which the SIE is solved at every step to calculate the stress intensity factors for the current configuration of the fracture system. The trajectory of the fracture is found by applying the criterion of maximum tensile stress at the crack tips and the crack closure is controlled at every step. Two types of fracture growth have been analysed: simultaneous and subsequent for different values of spacing and in-situ stress ratio.

Well-posedness of the dirichlet problem in a cylindrical domain for degenerating multidimensional elliptic equations

For elliptic equations, boundary-value problems on the plane were shown to be well posed in [1] by using methods from the theory of analytic functions of a complex variable. When the number of independent variables is greater than two, difficulties of a fundamental nature arise. The highly attractive and convenient method of singular integral equations can hardly be applied, because the theory of multidimensional singular integral equations is still incomplete [2]. In the present paper, using a method from [3], we establish the unique solvability of the classical solution of the Dirichlet problem in a cylindrical domain for degenerating multidimensional elliptic equations.

Crack behaviour in zinc coating and at the interface zinc-hot galvanised TRIP steel 800

In the present work, two paths of crack propagation in the bimaterial structure of zinc-TRIP steel 800 were investigated. Abaqus numerical simulation and singular integral equation method were used for the estimation of the crack behaviour at the interface. It was found that the energy release rate decreases as soon as the crack approaches the interface and increases when crossing it. The computation of the strain energies of deflection and penetration obtained from elastic modelling showed that the crack is more prone to deflect into the interface rather than to penetrate it. The same behaviour was supported by the volumetric approach results. Finally, analytical and numerical findings were in a good agreement with microscopic examination obtained from SEM observations.

Interfacial fracture analysis of a piezoelectric–polythene composite cylindrical shell patch under axial shear

The present work studies the interfacial fracture in a piezoelectric cylindrical shell patch. The problem is solved by the methods of infinite trigonometric series and Cauchy singular integral equation, and the numerical results of the stress intensity factor (SIF) are obtained. The effects of the interfacial radius and crack’s location on the SIF are explained through the effects of the free surface, interfacial curvature, crack length, and interface end, respectively. An optimal stiffness matching relationship between the piezoelectric layer and dielectric substrate is suggested. The effects of the piezoelectric and dielectric coefficients are explained through the mechanism of piezoelectric stiffening.

Torsion of an elastic space containing an axially symmetric narrow cavity

By the method of singular integral equations, we solve the axially symmetric problem of torsion for an elastic space containing a smooth cavity. We obtain the stress distributions and their maximum values at the edges of cavities of various shapes. We also compute the stress concentration factors on the surfaces of axially symmetric cavities in the entire range of values of the radii of rounding of their tips under the conditions of torsion of the elastic body. The numerical results are obtained for cavities of different configurations.

Study and solution of BEM-singular integral equation method in the case of concentrated loads

The Boundary Element Method has been widely used in order to obtain numerical solutions in many problems in science and engineering. On the other hand the singular integral equation method can treat successfully the singular fields appearing in engineering problems when the right hand side of it is a regular function. In this paper we consider the case where the right hand side of a Boundary Integral Equation, singular integral equation is a distribution. This problem arises in the case of concentrated forces and/or moments. Our main scope is to propose appropriate methods for the numerical solution of singular integral equations when the loading is a concentrated force or a moment. Before that we are obliged to analyze the solvability of such equations. The main key for the proof of the solvability is the proof of existence of Cauchy integrals of distributions. As a by product we find a very interesting result that the solution behaves as 1/r or 1/r2 at the point where the concentrated force or moment, respectively, is applied. Obviously the existence of essential singularities in the solution creates difficulties in the numerical treatment of integral equations. Thus two appropriate numerical methods and an interpolatory quadrature method are proposed for the numerical solution of BIE–SIE. Numerical examples illustrate the importance of the proposed methods.

Fully discretized collocation methods for nonlinear singular Volterra integral equations

We consider a nonlinear weakly singular Volterra integral equation arising from a problem studied by Lighthill (1950) [1]. A series expansion for the solution is obtained and shown to be convergent in a neighbourhood of the origin. Owing to the singularity of the solution at the origin, the global convergence order of product integration and collocation methods is not optimal. However, the optimal orders can be recovered if we use the fully discretized collocation methods based on graded meshes. A theoretical proof is given and we present some numerical results which illustrate the performance of the methods.

On the Convergence Problem of One-Dimensional Hypersingular Integral Equations

We develop the expansion method of singular integral equation (SIE) for hypersingular integral equation (HSIE). Relating the hypersingular integrals to Cauchy principal-value integrals, we interpolate the kernel and the density functions to the truncated Chebyshev series of the second kind. The corresponding convergence results for the functions <svg style="vertical-align:-3.56265pt;width:103.45px;" id="M1" height="20.549999" version="1.1" viewBox="0 0 103.45 20.549999" width="103.45" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,16.05)"><path id="x1D453" d="M619 670q0 -13 -9 -26t-18 -19q-13 -10 -25 2q-36 38 -66 38q-31 0 -54.5 -50t-45.5 -185h120l-20 -31l-107 -12q-23 -138 -57 -293q-27 -122 -55 -184.5t-75 -109.5q-60 -61 -114 -61q-25 0 -47.5 15t-22.5 31q0 17 31 44q11 8 20 -1q10 -11 31 -19t35 -8q26 0 47 19&#xA;q34 34 71 253l54 315h-90l-3 12l31 30h70q28 138 90 204q35 37 75 57.5t70 20.5q26 0 45 -14t19 -28z" /></g><g transform="matrix(.017,-0,0,-.017,15.684,16.05)"><path id="x2208" d="M448 1h-83q-118 0 -201.5 74.5t-83.5 179.5t83.5 179.5t201.5 74.5h83v-50h-84q-87 0 -150.5 -51.5t-73.5 -127.5h308v-50h-308q10 -76 73.5 -127.5t150.5 -51.5h84v-50z" /></g><g transform="matrix(.017,-0,0,-.017,29.385,16.05)"><path id="x1D436" d="M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5&#xA;t325.5 101.5q114 0 204 -38z" /></g> <g transform="matrix(.012,-0,0,-.012,41.375,7.888)"><path id="x2113" d="M363 114l21 -20q-62 -106 -161 -106q-61 0 -94 40t-33 138v56l-54 -40l-14 27l68 57v89q0 112 15.5 166t49.5 91q43 46 96 46q44 0 71 -32.5t27 -88.5q0 -41 -17.5 -81.5t-49.5 -76t-55.5 -57t-55.5 -46.5v-87q0 -138 81 -138q60 0 105 63zM177 421v-101q117 105 117 217&#xA;q0 38 -15 60.5t-41 22.5q-34 0 -47.5 -46.5t-13.5 -152.5z" /></g> <g transform="matrix(.017,-0,0,-.017,46.887,16.05)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,52.769,16.05)"><path id="x5B" d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" /></g><g transform="matrix(.017,-0,0,-.017,58.634,16.05)"><path id="x2212" d="M535 230h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,68.612,16.05)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g><g transform="matrix(.017,-0,0,-.017,76.771,16.05)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,83.469,16.05)"><use xlink:href="#x31"/></g><g transform="matrix(.017,-0,0,-.017,91.628,16.05)"><path id="x5D" d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" /></g><g transform="matrix(.017,-0,0,-.017,97.493,16.05)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g> </svg> and <svg style="vertical-align:-2.3205pt;width:201.77499px;" id="M2" height="19" version="1.1" viewBox="0 0 201.77499 19" width="201.77499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,16.05)"><path id="x1D43E" d="M764 650l-7 -26q-61 -10 -91.5 -22.5t-76.5 -48.5l-235 -184q104 -152 208 -276q31 -37 54.5 -48.5t68.5 -16.5l-7 -28h-156q-138 170 -213 284q-19 29 -33.5 35t-33.5 -7l-29 -173q-15 -75 -5.5 -90.5t74.5 -20.5l-5 -28h-260l7 28q58 5 74 21.5t30 89.5l70 378&#xA;q12 68 2 83t-72 22l6 28h257l-7 -28q-62 -7 -76 -21.5t-27 -83.5l-32 -172q29 11 78 46q112 84 217 179q18 16 23.5 25.5t-1 14.5t-26.5 8l-30 4l5 28h249z" /></g><g transform="matrix(.017,-0,0,-.017,13.254,16.05)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,19.135,16.05)"><path id="x1D461" d="M324 430l-26 -36l-112 -4l-55 -265q-13 -66 7 -66q13 0 44.5 20t50.5 40l17 -24q-38 -40 -85.5 -73.5t-87.5 -33.5q-50 0 -21 138l55 262h-80l-2 8l25 34h66l25 99l78 63l10 -9l-37 -153h128z" /></g><g transform="matrix(.017,-0,0,-.017,24.932,16.05)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,31.629,16.05)"><path id="x1D465" d="M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37&#xA;q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z" /></g><g transform="matrix(.017,-0,0,-.017,41.131,16.05)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,51.739,16.05)"><use xlink:href="#x2208"/></g><g transform="matrix(.017,-0,0,-.017,65.44,16.05)"><use xlink:href="#x1D436"/></g> <g transform="matrix(.012,-0,0,-.012,77.425,7.887)"><use xlink:href="#x2113"/></g> <g transform="matrix(.017,-0,0,-.017,82.938,16.05)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,88.819,16.05)"><use xlink:href="#x5B"/></g><g transform="matrix(.017,-0,0,-.017,94.684,16.05)"><use xlink:href="#x2212"/></g><g transform="matrix(.017,-0,0,-.017,104.662,16.05)"><use xlink:href="#x31"/></g><g transform="matrix(.017,-0,0,-.017,112.821,16.05)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,119.519,16.05)"><use xlink:href="#x31"/></g><g transform="matrix(.017,-0,0,-.017,127.678,16.05)"><use xlink:href="#x5D"/></g><g transform="matrix(.017,-0,0,-.017,137.334,16.05)"><path id="xD7" d="M528 54l-36 -38l-198 201l-198 -201l-36 38l197 200l-197 201l36 38l198 -202l198 202l36 -38l-197 -201z" /></g><g transform="matrix(.017,-0,0,-.017,151.085,16.05)"><use xlink:href="#x5B"/></g><g transform="matrix(.017,-0,0,-.017,156.95,16.05)"><use xlink:href="#x2212"/></g><g transform="matrix(.017,-0,0,-.017,166.928,16.05)"><use xlink:href="#x31"/></g><g transform="matrix(.017,-0,0,-.017,175.088,16.05)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,181.785,16.05)"><use xlink:href="#x31"/></g><g transform="matrix(.017,-0,0,-.017,189.945,16.05)"><use xlink:href="#x5D"/></g><g transform="matrix(.017,-0,0,-.017,195.809,16.05)"><use xlink:href="#x29"/></g> </svg>, <svg style="vertical-align:-0.546pt;width:34.6875px;" id="M3" height="11.9875" version="1.1" viewBox="0 0 34.6875 11.9875" width="34.6875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.25)"><use xlink:href="#x2113"/></g><g transform="matrix(.017,-0,0,-.017,11.758,11.25)"><path id="x2265" d="M531 285l-474 -214v56l416 183l-416 184v56l474 -215v-50zM531 -40h-474v50h474v-50z" /></g><g transform="matrix(.017,-0,0,-.017,26.462,11.25)"><use xlink:href="#x31"/></g> </svg>, are derived in an appropriate <svg style="vertical-align:-3.276pt;width:60.900002px;" id="M4" height="16.15" version="1.1" viewBox="0 0 60.900002 16.15" width="60.900002" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path id="x1D43F" d="M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z" /></g> <g transform="matrix(.012,-0,0,-.012,9.763,16.087)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g> <g transform="matrix(.017,-0,0,-.017,16.1,12.012)"><use xlink:href="#x5B"/></g><g transform="matrix(.017,-0,0,-.017,21.965,12.012)"><use xlink:href="#x2212"/></g><g transform="matrix(.017,-0,0,-.017,31.943,12.012)"><use xlink:href="#x31"/></g><g transform="matrix(.017,-0,0,-.017,40.102,12.012)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,46.8,12.012)"><use xlink:href="#x31"/></g><g transform="matrix(.017,-0,0,-.017,54.959,12.012)"><use xlink:href="#x5D"/></g> </svg> norm to the true solution of the weight function. Numerical examples are also presented to validate the theoretical findings.

Stress Concentration near Holes in the Elastic Plane Subjected to Antiplane Deformation

By the method of singular integral equations, we determine the solutions of antiplane problems of the theory of elasticity for a plane weakened by curvilinear holes with rounded and sharp vertices. The stress intensity factors at sharp vertices are found as a result of the limit transition from rounded vertices by using the relation between the stress intensity factors and stress concentration factors at the sharp and rounded tips of a V-shaped notch. We also obtain numerical results for the holes of various configurations: elliptic, in the form of physical slots, oval, rhombic, and rectangular.

Collocation for Cauchy singular integral equations

The paper is concerned with the stability of collocation methods for Cauchy singular integral equations with piecewise continuous coefficients on an interval, where these methods look for an approximate solution of the form μ(x)pn(x) with a Jacobi weight μ(x) and a polynomial pn(x). Here, the Chebyshev weight μ(x)=1-x1+x and collocation with respect to Chebyshev nodes of first and fourth kind are considered.

Electrodynamic analysis of resonance tags for radio-frequency identification of objects based on the method of singular integral equations

Electrodynamic analysis of resonance radio-frequency identification tags in the form of a system of plane concentric metal rings is presented. The analysis is based on the method of singular integral equations. The patterns of the structure and frequency dependences of the response of tags with various code combinations are obtained and studied. The results provided by the method of singular integral equations and finite-difference time-domain technique are compared. It is shown that the proposed method makes it possible to substantially optimize the analysis of the considered identification tags through minimization of the computation time.

Distribution of stresses over the contour of a rounded V-shaped notch under antiplane deformation

The solution of the antiplane problem of the theory of elasticity for a plane with semiinfinite rounded V-shaped notch is obtained by the method of singular integral equations. On this basis, we deduce the relationships between the stress concentration and stress intensity factors for sharp and rounded notches. The obtained solution is compared with the well-known solution of a similar problem for a hyperbolic notch. It is shown that both the radius of rounding of the notch tip and the shape of its neighborhood strongly affect the distribution of stresses on the boundary contour.

A plastic crack in a smectic liquid crystal – a possible model

A plastic crack model for smectic A liquid crystals is suggested in this article. The solution for the crack problem is based on the dislocation pile-up principle and the singular integral equation method. From the solution we can determine the size of the plastic zone at the crack tip and the crack tip opening (sliding) displacement, which are parameters relevant to the stability/instability of materials. The results may be useful in developing soft-matter mechanics.

Numerical Modelling of Steady-State Flow in 2D Cracked Anisotropic Porous Media by Singular Integral Equations Method

The equations governing plane steady-state flow in heterogeneous porous media containing curved-line intersecting cracks (Pouya and Ghabezloo in Transp Porous Media 84:511–532, 2010) and the potential solution obtained for these equations are considered here. The theoretical results are first completed for the mass balance at crack intersections points. Then, a numerical procedure based on a singular integral equations method is described concretely to derive this solution for cracked materials. Closed-form expressions of elementary integrals for special choice of collocation points lead to a very quick and easy numerical method. It is shown that this method can be applied efficiently to the study of the steady-state flow in cracked materials with anisotropic matrix permeability and a dense distribution of curved-line intersecting cracks. Some applications of this method to the permeability of cracked materials are given.

Inverse scattering in guides

We present statements and a method of solution to the inverse scattering problem of reconstructing permittivity of a dielectric inclusion in a 2D or 3D waveguide from the transmission characteristics. The approach employs a volume singular integral equation (VSIE) method. The unique solvability of VSIE is established. The inverse problem is solved by the method of iterations applied to VSIE; the convergence of the method is proved

Distribution of stresses near V-shaped notches in the complex stressed state

By the method of singular integral equations, we obtain the solution of a two-dimensional problem of the elasticity theory for a plane containing a semiinfinite rounded V-notch under complex loading. On the basis on this solution, we establish the relationships between the stress intensity factors at the tip of the sharp V-notch and the maximum stresses and their gradients at the tip of the corresponding rounded notch. For finite bodies with V-notches, the presented solutions are obtained as asymptotic dependences for small radii of rounding of the tips. The deduced relationships can be used to perform the limit transitions and find the stress intensity factors at the tips of sharp V-notches on the basis of the solutions obtained for the corresponding rounded stress concentrators. The efficiency of the method is illustrated for the problem of determination of the stress intensity factors at the vertices of a rectangular hole in the elastic plane.

含裂纹功能梯度材料热接触的奇异积分方程方法

含裂纹功能梯度材料热接触的奇异积分方程方法

Longitudinal shear of an elastic medium with a thin rectilinear sharp-pointed piezoelectric inclusion of low rigidity

UDC 539.3 We propose a method for the investigation of the stress-strain state near the edges of a sharp-pointed, thin, rectilinear, piezoelectric inclusion of varying thickness and low rigidity located in an elastic isotropic medium. The method is based on the combination of an asymptotic analysis of solutions of the problem and the method of singular integral equations, the numerical realization of which is based on the Kantorovich regularization procedure of divergent integrals and the collocation method.

A quadrature method for Cauchy singular integral equations with index -1

A quadrature method for Cauchy singular integral equations with index -1

Calculation of open resonators with the help of a modified method of extended boundary conditions

The problem on the eigenmodes of a 2D open resonator is solved with the help of the method of extended boundary conditions. A modification of this method that makes it possible to substantially enhance the accuracy of computation is proposed. The results obtained by means of approximate techniques are compared to the solution obtained with the use of the singular integral equation method.