Hypersingular Integral Equation Method Research Articles

Regularized hyper-singular boundary integral equation methods for three-dimensional poroelastic problems

This work proposes an accurate hyper-singular boundary integral equation method for dynamic poroelastic problems with Neumann boundary condition in three dimensions and both the direct and indirect methods are adopted to construct combined boundary integral equations. The strongly-singular and hyper-singular integral operators are reformulated into compositions of weakly-singular integral operators and tangential-derivative operators, which allow us to prove the jump relations associated with the poroelastic layer potentials and boundary integral operators in a simple manner. Relying on both the investigated spectral properties of the strongly-singular operators, which indicate that the corresponding eigenvalues accumulate at three points whose values are only dependent on two Lamé constants, and the spectral properties of the Calderón relations of the poroelasticity, we propose low-GMRES-iteration regularized integral equations. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed methodology by means of a Chebyshev-based rectangular-polar solver.

A new integral equation method for calculating interacting stress-intensity factors of multiple holed-cracked anisotropic rock under both far-field and arbitrary surface stresses

Calculation of interacting stress-intensity factors (SIFs) of multiple holed-cracked anisotropic problem is of very importance not only for improving hydraulic fracturing technology in shale-gas exploitation, but also for optimizing crack-arrest holes in rock engineering. Current literatures are mainly focused on interacting SIFs of multiple holed-cracked anisotropic body under far-field stresses regardless of surface stresses on holes and cracks. In this paper, a new integral equation method, based on our derived exact fundamental stress solutions (with simple formulae and high-efficiency calculation process), is proposed to calculate interacting SIFs for multiple holed-cracked anisotropic rock under both far-field and arbitrary surface stresses, where the circular-holes and cracks are of any number, sizes, orientations, relative locations. The new method is proved to be valid by comparing our results of interacting SIFs with those obtained by Green function's method, hyper-singular integral equation method and finite element method. Several computational examples are given to investigate effect of loadings, holed-cracked geometries and orthotropic material properties of shale on interacting SIFs. Research results show that the far-field stress ratio (σx∞/σy∞) greatly affects the neutral crack orientation angle α0of KIA, KIIA and KIB (with null-effect of the circular-hole on the crack) but scarcely affects α0of KIIB (where A, B are inner, outer crack-tips). The surface compressive stresses acting alone on the hole (n) and alone on the crack (p) have little effect on α0. The circular-hole radius (r) is almost irrelative to α0, but relative to the amplifying or shielding degree of interacting SIFs. Young's modulus ratio (E2/E1) and bedding plane orientation angle (φ) of shale decide α0 of KIB but are nearly independent of α0 of KIA and KIIA. In addition, decreasing σx∞/σy∞ or increasing n, p can facilitate crack propagation, and decreasing E2/E1 and φ are helpful for crack arrest. The new method not only avoids the trouble in solving the singular integral equation (without any singularity), but also has high accuracy (exact fundamental stress solutions) and wider application (under both far-field uniform stresses and arbitrarily distributed surface stresses) than the common methods. It is proved to be effective for predicting the multi-crack initiation of rock under hydro-mechanical coupling stress in underground rock engineering for safety assessment and cracking-arrest design, and for optimizing the orientation angle of hydraulic-crack to form better fracturing network in shale-gas exploitation for improvement of shale-gas exploitation efficiency.

An Accurate HyperSingular Boundary Integral Equation Method for Dynamic Poroelasticity in Two Dimensions

This paper is concerned with the boundary integral equation method for solving the exterior Neumann boundary value problem of dynamic poroelasticity in two dimensions. The main contribution of this work consists of two aspescts: the proposal of a novel regularized boundary integral equation, and the presentation of new regularized formulations of the strongly singular and hyper-singular boundary integral operators. First, turning to the spectral properties of the double-layer operator and the corresponding Calderón relation of the poroelasticity, we propose the novel low-GMRES-iteration integral equation whose eigenvalues are bounded away from zero and infinity. Second, with the help of the classical idea of using Günter derivatives and integration-by-parts, we reformulate the strongly singular and hyper-singular integral operators into combinations of the weakly singular operators and the tangential derivatives. The accuracy and efficiency of the proposed methodology are demonstrated through several numerical examples.

О применении метода гиперсингулярных интегральных уравнений к решению задач для упругой плоскости с коллинеарной системой трещин

In the present paper, using the method of hypersingular integral equations, based on the formulas of the inversion of the corresponding singular integral equations, the exact quadrature solution of the classical problems of the mechanics of an elastic plane with a collinear system of cracks is constructed. The elastic plane is in a state of antiplane deformation or plane deformation; in case of antiplane deformation, crack edges are symmetrically loaded by tangential forces, while in case of plane deformation, they are again loaded symmetrically but by normal forces. Mixed boundary-value problems for an elastic half-plane equivalent to these problems are formulated. Under plane deformation, the mixed boundary-value problem for an elastic half-plane is discussed as well when the plane boundary is reinforced by two similar and symmetrically located semi-infinite stringers between which a system of an arbitrarily final number of stringers is situated. It is considered that the stringers are absolutely rigid for expansion and compression and absolutely flexible for bending. A particular case of two similar symmetrically located cracks is considered more in detail. In this case, the exact solution to the problem is also constructed by the method of Chebyshev orthogonal polynomials.

Analysis of 3D planar crack problems in one-dimensional hexagonal piezoelectric quasicrystals with thermal effect. part I: Theoretical formulations

Theoretical and numerical investigations on three-dimensional (3D) planar crack problems in one-dimensional (1D) hexagonal piezoelectric quasicrystals (QCs) with thermal effect are performed systematically. Part I of this work derives a series of theoretical formulations that are then used to study the 3D planar crack problems in the QCs. The simple layer potential functions with the extended displacement discontinuities (EDDs) as the unknown variables and the general solution based on quasi-harmonic functions for the QCs under consideration are used to deduce the boundary equations that govern 3D planar crack problems. The hypersingular integral equation method is used to analyze the asymptotic singularities of the coupled thermal-electrical-phonon-phason fields near the crack edge. Expressions are then presented for the extended stress intensity factors (ESIFs) of a mixed model crack in terms of EDDs for arbitrarily-shaped cracks in the QCs, and the basic relationships between the energy release rate and the ESIFs are established. Closed-form solutions for some typical cracks, including an elliptical crack that is subjected to coupled electrical-phonon-phason loadings and a penny-shaped crack that is subjected to antisymmetric thermal loading, are determined via Fabrikant's analysis method. Additionally, both the physical quantities on the crack plane and the corresponding variables in the coupled thermal-electrical-phonon-phason field in the full space are given. The theoretical formulations derived in this paper provide a fundamental basis for development of the numerical approach proposed in Part II of our work, and can also serve as benchmarks for numerical solutions.

Plane Analysis for an Inclusion in 1D Hexagonal Quasicrystal Using the Hypersingular Integral Equation Method

A model of a thin elastic inclusion embedded in an infinite 1D hexagonal quasicrystal is discussed. The atomic arrangements of the matrix and the inclusion are both periodic along the $$x_{1}$$ -direction and quasiperiodic along the $$x_{2}$$ -direction in the $$ox_{1}x_{2}$$ -coordinate system. Using the hypersingular integral equation method, the inclusion problem is reduced to solving a set of hypersingular integral equations. Based on the exact analytical solution of the singular phonon and phason stresses near the inclusion front, a numerical method of the hypersingular integral equation is proposed using the finite-part integral method. Finally, the numerical solutions for the phonon and phason stress intensity factors of some examples are given.

Scattering of water waves by an inclined plate in a two-layer fluid

The problem of water wave scattering by an inclined thin plate submerged in the lower layer of a two-layer fluid is examined here using linear theory. In a two-layer fluid, corresponding to a particular frequency wave propagates with two different wavenumbers. Thus we determine the reflection and the transmission coefficients and the hydrodynamic force for both the wavenumbers. This leads to two separate problems. These problems are solved using a hypersingular integral equation method. Several numerical results for the physical quantities varying the inclination, depth and length of the plate are presented. Making suitable adjustment of the parameters published results for a vertical plate submerged in a single layer fluid are recovered.

The method of hypersingular integral equations in the problem of electromagnetic wave diffraction by a dielectric body with a partial perfectly conducting coating

AbstractA problem of scattering of a monochromatic electromagnetic wave by a homogeneous dielectric body is considered. A part of the boundary of the body is a perfectly conducting thin surface. The problem is reduced to a system of the boundary integral equations containing integrals with strong singularity, the integrals are understood in terms of Hadamard final part. A numerical solution scheme is constructed on the base of approximate solution of these equations using the methods of piecewise-constant approximations and collocation. The constructed numerical scheme is tested on a model example.

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.

Calculation of the characteristics of reflector antennas by the method of hypersingular integral equations

A method for calculation of the characteristics of reflector antennas (RAs) is proposed. The method is based on the mathematical apparatus of 2D hypersingular integral equations. Radiation patterns of a RA with a reflector formed by a parabolic cylinder as well as distributions of the current density on the surfaces of the reflector and the feed calculated by this method are presented.

Calculation of the input impedance of a microstrip dipole by the method of hypersingular integral equations

A self-consistent method for calculation of the input impedance of a microstrip dipole (MSD) is proposed. A hypersingular integral equation (HSIE) for an unknown function of the distribution of the current density on the MSD surface is derived. A numerical method for calculation of the HSIE is described. Plots of the input impedance as a function of the dipole length and the substrate thickness are presented.

Numerical solution of 3D problems of electromagnetic wave diffraction on a system of ideally conducting surfaces by the method of hypersingular integral equations

We construct a numerical method for solving problems of electromagnetic wave diffraction on a system of solid and thin objects based on the reduction of the problem to a boundary integral equation treated in the sense of the Hadamard finite value. For the construction of such an equation, we construct a numerical scheme on the basis of the method of piecewise continuous approximations and collocations. Unlike earlier known schemes, by using the below-suggested scheme, we have found approximate analytic expressions for the coefficients of the arising system of linear equations by isolating the leading part of the kernel of the integral operator. We present examples of solution of a number of model problems of the diffraction of electromagnetic waves by the suggested method.

Saturated dislocations transient propagation–evolution in olivine structure under ultra high-coupled thermal-force fields

Based on the first principle and flow driven pore-network crack theory, the crystal size saturated dislocations transient (10−4–10−5s) propagation–evolution in olivine structure under ultra high-coupled temperature (200–500°C) and pressure (0.4–1GPa) are studied on the parallel CPU&GPU platform. First, the molecular-micro-scale transient fracture model is established by using hybrid hypersingular integral equation and Lattice Boltzmann method, the hydrogen ion and oxonium ion transport–dehydration (HI–OI–TD) in olivine [(FeMg)SiO4] crystal are explored. The bond-strength–length as function of thermal-force–time fields, the limited thermal-force value for HI–OI–TD through the crystal, and the ion state water adsorption in the crystal are calculated, respectively. Then, based on the above results, the crystal size saturated dislocations/defects propagation–evolution is studied. The relationship between the stress distribution and micro-strain under different velocity–time conditions, the saturated dislocations/defects propagation–evolution as function of coupled thermal-force–time fields are obtained. All these findings can helpful understand the mechanism of the dehydration fracturing shale gas, the coal-gas outbursts, and the coseismic triggering issues.

Analysis of 3D fluid driven crack propagation problem in co-seismic slip under P- and S-waves by hybrid hypersingular integral method

This work reports a new and accurate way of theoretical and numerical description of the extended 3D fluid (electromagnetic and flow) driven crack progression in co-seismic slip under P- and S-waves. First, based on the viscous fluid flow reciprocal work theorem, the hybrid hypersingular integral equation (HIE) method proposed by the author was defined by combined with the coupled extended wave time-domain HIE and the extended diffused interface phase field method. The general extended 3D fluid flow velocity wave solutions are obtained by the extended wave time-domains Green’s function method. The 3D extended dynamic fluid driven crack modeling under fully coupled electromagnetothermoelastic P- and S-wave and flow field was established. Then, the problem is reduced to solving a set of extended hybrid HIEs coupled with nonlinear boundary domain integral equations, in which the unknown functions are the general extended flow velocity discontinuity waves. The behavior of the general extended singular stress indices around the crack front terminating is analyzed by hybrid time-domain main-part analysis. The general extended singular pore stress waves (SPSWs) and the extended dynamic stress intensity factors (DSIFs) on the fluid driven crack surface are obtained from closed-form solutions. In addition, a numerical method for the problem is proposed, in which the extended velocity discontinuity waves are approximated by the product of time-domain density functions and polynomials. The extended DSIFs and general extended SPSWs are calculated, and the results are presented toward demonstrating the applicability of the proposed method.

Wave time-domain hypersingular integral applied to co-seismic slip of P- and S-waves from crack propagation

This work presents a wave time-domain hypersingular integral equation (WTD-HIE) method for modeling the 3D crack propagation problem of co-seismic slip under fully coupled electromagnetothermoelastic P- and S-wave fields through theoretical analysis and numerical simulations. First, the general extended displacement wave solutions are obtained by analyzing wave time-domains using the Green’s function method. Then, based on the wave time-domain boundary element method, the problem is reduced to solving a set of WTD-HIEs coupled with nonlinear boundary integral equations, in which the unknown functions are the general extended displacement discontinuity waves. The behavior of the general extended singular stress indices around the crack front terminating at the slip surface is analyzed by time-domain main-part analysis. The general extended singular stress waves and the extended dynamic stress intensity factors are obtained from closed-form solutions. In addition, a numerical method for the problem is proposed, in which the extended displacement discontinuity waves are approximated by the product of time-domain density functions and polynomials. Finally, the radiation distribution of near-field extended dynamic stress intensity factors for P- and S-waves at the crack surface are calculated, and the results are presented toward demonstrating the applicability of the proposed method.

Mixed‐mode stress intensity factors of a three‐dimensional crack in a bonded bimaterial

PurposeThis paper aims to calculate the mixed‐mode stress intensity factors (SIFs) of a 3D crack meeting the interface in a bimaterial under shear loading by a hypersingular integral equation (HIE) method, And further to assess the accuracy of numerical solutions for the mixed mode SIFs along the crack front.Design/methodology/approachA 3D crack modeling is reduced to solving a set of HIEs. Based on the analytical solutions of the singular stress field around the crack front, a numerical method for the HIEs is proposed by a finite‐part integral method, where the displacement discontinuities of the crack surface are approximated by the product of basic density functions and polynomials. Using FORTRAN program, numerical solutions of the mixed‐mode SIFs of some examples are presented.FindingsThe numerical method is proved to be an effective construction technique. The numerical results show that this numerical technique is successful, and the solution precision is satisfied.Research limitations/implicationsThis work takes further steps to improve the fundamental systems of HIE for its application in the fields of arbitrary shape crack problems. Propose several techniques for numerical implementation, which could increase the efficiency and accuracy of computation.Practical implicationsWhenever there is a structure containing the 3D crack, the analysis method described in this paper can be utilized to find the critical configurations under which the structure may be most vulnerable. In such cases, the strength predictions would be safer if the crack could be taken into account.Originality/valueThis paper is the first to apply HIE method to analyzing the mixed‐mode crack meeting the interface in 3D dissimilar materials. Numerical solutions of the mixed‐mode SIFs can give the satisfied solution precision.

Multiple 3D flaws in coupled electro-magneto-thermo-elastic multiphase composites by extended hypersingular integral equation method

This work presents extended hypersingular integral equation (E-HIE) method to analyze the multiple 3D mixed-mode flaws problem in fully coupled electro-magneto-thermo-elastic multiphase composites under extended electro-magneto-thermo-elastic coupled loads through intricate theoretical analysis and numerical simulations. First, the problem is reduced to solving a set of E-HIEs. Analytical solutions for the extended singular stresses, the extended stress intensity factors (E-SIFs), the extended energy release rate and the extended strain energy density factors (E-SEDFs) near the flaws front are obtained. Then, the numerical method for the E-HIEs for two 3D flaws subjected to extended coupled loads is proposed. Finally, numerical solutions of E-SIFs and E-SEDFs of some examples are given, and the effect of flaws orientation, interaction and shielding is discussed.

The harmonic problem in the plane with cuts under boundary conditions of the third kind

The Dirichlet and Neumann problems for harmonic functions in the plane with cuts of any shape were studied in [1] and [2], respectively; the Dirichlet problem was studied by the method of singular integral equations of the first kind, while the Neumann problem was studied by the method of hypersingular integral equations of the first kind. The problem with skew derivatives, whose special case is the Neumann problem, was studied for harmonic functions in the plane with cuts of any shape in [3, 4], where it was reduced to a uniquely solvable Fredholm integral equation of the second kind. Another generalization of the Neumann problem was considered in [5]. The mixed problem for harmonic functions with the Dirichlet condition on a part of the cuts and the condition with a skew derivative, which generalizes the Neumann condition, on the remaining cuts was studied in [6] by using a reduction to a uniquely solvable Fredholm integral-algebraic equation of the second kind in a Banach space. A special case of this problem [6] is the Dirichlet problem in the plane with cuts of any shape. The Dirichlet and Neumann problems in the plane with cuts for the Helmholtz equation were studied in [7, 8], where they were reduced to uniquely solvable Fredholm equations of the second kind. In this paper, we study the boundary value problem for harmonic functions in the plane with cuts under third-kind boundary conditions on both sides of the cuts. We prove the existence and uniqueness of a solution to this problem. We also obtain an integral representation for the solution in the form of potentials. The density in the potentials are determined from a uniquely solvable system of integral equations. The singularities of the gradient of the solution at the endpoints of the cuts are also studied.

3D modeling of crack growth in electro-magneto-thermo-elastic coupled viscoplastic multiphase composites

This work presents a time-domain hypersingular integral equation (TD-HIE) method for modeling 3D crack growth in electro-magneto-thermo-elastic coupled viscoplastic multiphase composites (EMTE-CVP-MCs) under extended incremental loads rate through intricate theoretical analysis and numerical simulations. Using Green’s functions, the extended general incremental displacement rate solutions are obtained by time-domain boundary element method. Three-dimensional arbitrary crack growth problem in EMTE-CVP-MCs is reduced to solving a set of TD-HIEs coupled with boundary integral equations, in which the unknown functions are the extended incremental displacement discontinuities gradient. Then, the behavior of the extended incremental displacement discontinuities gradient around the crack front terminating at the interface is analyzed by the time-domain main-part analysis method of TD-HIE. Also, analytical solutions of the extended singular incremental stresses gradient and extended incremental integral near the crack fronts in EMTE-CVP-MCs are provided. In addition, a numerical method of the TD-HIE for a 3D crack subjected to extended incremental loads rate is put forward with the extended incremental displacement discontinuities gradient approximated by the product of time-domain basic density functions and polynomials. Finally, examples are presented to demonstrate the application of the proposed method.

Hypersingular integral equation method for a three-dimensional crack in anisotropic electro-magneto-elastic bimaterials

Using Green’s functions, the extended general displacement solutions of a three-dimensional crack problem in anisotropic electro-magneto-elastic (EME) bimaterials under extended loads are analyzed by the boundary element method. Then, the crack problem is reduced to solving a set of hypersingular integral equations (HIE) coupled with boundary integral equations. The singularity of the extended displacement discontinuities around the crack front terminating at the interface is analyzed by the main-part analysis method of HIE, and the exact analytical solutions of the extended singular stresses and extended stress intensity factors (SIFs) near the crack front in anisotropic EME bimaterials are given. Also, the numerical method of the HIE for a rectangular crack subjected to extended loads is put forward with the extended crack opening dislocation approximated by the product of basic density functions and polynomials. At last, numerical solutions of the extended SIFs of some examples are obtained.