Hypersingular Boundary Integral Equations Research Articles

Plane strain problem of an elastic matrix containing multiple Gurtin–Murdoch material surfaces along straight segments

This paper presents the study of the plane strain problem of an infinite isotropic elastic medium subjected to far-field load and containing multiple Gurtin–Murdoch material surfaces located along straight segments. Each material segment represents a membrane of vanishing thickness characterized by its own elastic stiffness and residual surface tension. The governing equations, the jump conditions, and the surface tip conditions are reviewed. The displacements in the matrix are sought as the sum of complex variable single-layer elastic potentials whose densities are equal to the jumps in complex tractions across the segments. The densities are found by solving the system of coupled hypersingular boundary integral equations. The approximations by a series of Chebyshev’s polynomials of the second kind are used with the square root weight functions chosen to satisfy the tip conditions automatically. Numerical examples are presented to illustrate the influence of dimensionless parameters and to study the effects of interactions.

Fracture Analysis of Planar Cracks in 3D Thermal Piezoelectric Semiconductors

The study of piezoelectric semiconductor (PSC) materials and devices is experiencing rapid growth, giving rise to a burgeoning research domain known as piezotronics. Understanding the fracture behavior of PSC materials is essential for designing robust devices and ensuring their long-term functionality. A planar crack of arbitrary shape in the isotropic plane of a three-dimensional (3D) transversely isotropic thermal PSC body subjected to combined thermal-electrical-mechanical loadings is studied via the extended displacement discontinuity (EDD) boundary integral equation method. Under the framework of linearized basic equations and one-way thermal coupling, the hypersingular boundary integral equations with the volume integrals containing the temperature and carrier concentration are obtained and expressed by the EDDs which include displacement, electric potential, carrier concentration and temperature discontinuities across the crack surfaces and are the basic variables in analyzing crack problems. Based on the finite part of hypersingular integrals, the EDDs near the crack border are proved to have the classical behavior of O(r). Furthermore, the asymptotic solutions of extended stresses including the stress, electric displacement, current density and heat flux at the crack front are presented and display classical singular behavior of O(1/r). The extended stress intensity factors are defined and given in terms of EDDs. A 3D numerical model of a penny-shaped crack in the thermal PSC cylinder is established and used to verify the presented formulae.

On the problem of a Gurtin–Murdoch cylindrical material surface embedded in an infinite matrix

The problem of an infinite isotropic elastic matrix subjected to uniform far-field load and containing a Gurtin–Murdoch material surface of cylindrical shape is considered in plane strain setting. The governing equations and the boundary conditions for the problem, reduced to that of an infinite plane containing a material curve along a circular arc, are reviewed. The displacements inside the matrix are sought in the complex variables form of a single layer elastic potential whose density represents the jump in complex tractions across the curve. Exact complex integral representations for the elastic fields everywhere in the material are provided and the problem is further reduced to the system of real variables hypersingular boundary integral equations in terms of the strain and rotation components associated with the curve. The components are then approximated by the series of trigonometric functions that are multiplied by the square root weight functions to allow for automatic incorporation of the tip conditions. The unknown coefficients in the series are found from the system of linear algebraic equations that is solved using standard collocation method. The numerical examples are presented to illustrate the influence of dimensionless parameters with the main focus on the study of curvature-induced effects.

Two-dimensional problem of an infinite matrix reinforced with a Steigmann–Ogden cylindrical surface of circular arc cross-section

The plane strain problem of an elastic matrix subjected to uniform far-field load and containing a Steigmann–Ogden material surface with circular arc cross-section is considered. The governing equations and the boundary conditions for the problem are reviewed. Exact complex integral representations for the elastic fields everywhere in the material are provided. The problem is further reduced to the system of real variables hypersingular boundary integral equations in terms of the first component of the surface stress tensor (surface stress) and the remaining component of that tensor and its second derivative, along with various problem parameters. The two unknowns are then approximated by the series of trigonometric functions that are multiplied by the square root weight functions to allow for automatic incorporation of the tip conditions. The unknown coefficients in series are found from the system of linear algebraic equations that is solved using standard collocation method. The numerical examples are presented to illustrate the influence of dimensionless parameters. The connection of the problem with that of rigid circular arc is discussed.

On accuracy of translations by kernel independent fast multipole methods

The paper concerns with accurate evaluation of local fields in problems, like those of fracture mechanics, for which extreme, rather than mean values, are of prime significance. We focus on large-scale problems, solved iteratively by a boundary element method (BEM) with employing the kernel independent fast multipole method (KI-FMM) for decreasing the complexity of matrix-to-vector multiplications. We aim to study the accuracy of individual translations far-fields for the major types of kernels (weakly singular, singular and hypersingular) entering the boundary integral equations (BIE) of harmonic and elasticity problems in 2D and 3D. It is revealed, analytically and numerically, that (i) the accuracy of individual translations of fields, generated by sources involving the normal at a source point, is notably less than that by sources independent on the normal; (ii) the accuracy is notably less for a hypersingular BIE, then for a singular BIE; and (iii) under the same complexity, the accuracy of translations, performed by using smooth (circular in 2D, spherical in 3D) equivalent surfaces (ES), is notably better than that when using non-smooth (square in 2D, cubic in 3D) ES. These conclusions yield recommendations for minimizing the complexity without loss of the accuracy of translated fields.

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.

Radiation of water waves by a heaving submerged disc in a three-layer fluid

A thin rigid disc is submerged in the infinitely deep lower layer of a three-layer fluid. Using linear water wave theory, the three-dimensional problem of wave radiation due to heaving motion of the disc is considered. The problem is first reduced to solving a hypersingular boundary integral equation. This integral equation is then reduced to a system of one-dimensional Fredholm integral equations of second kind. The solution of each integral equation is associated with a radial component of the difference of potential function across the plate. The solution of these integral equations has been used to compute added mass and damping coefficient numerically. Numerical results for a heaving disc submerged in a single layer, a two-layer, and a three-layer fluid have been depicted. Also, the effects of different parameters like depth of upper layer, middle layer, distance of the disc from the interface, and the density ratios of the fluid layers on these physical quantities have been demonstrated.

On the role of singular and hypersingular BIEs for the BVPs containing a degenerate boundary

In this paper, we find that using either the singular boundary integral equation (BIE) alone or the hypersingular BIE alone may solve the symmetric and anti-symmetric problems containing a degenerate boundary subject to the remote anti-plane shear, respectively. Although the rank of the influence matrix is deficient due to the degenerate boundary, it is still solvable. The mechanism was analytically demonstrated by using the degenerate kernel and numerically performed by using the singular value decomposition (SVD) technique. Besides, solvability was also discussed by using the Fredholm alternative theorem in both the boundary integral equation and boundary element method (BEM). We can analytically prove and numerically explain that the boundary density solved by using the singular BIE alone is different from the exact solution but the field solution is acceptable. The solvability of the linear algebraic system was discussed from the mapping table of the vector space of influence matrix where the domain is composed by the right singular vectors and the vector space of the range is composed of left singular vectors. The corresponding null space and the left null space due to the zero singular value indicates the homogeneous solution and the range deficiency, respectively. Several approaches, the singular or hypersingular BIE alone, adding symmetric or anti-symmetric boundary constraints, the self-regularized approach provided us alternative ways to solve the degenerate-boundary problem without using the dual BEM.

Applications of degenerate kernels to potential flow across circular, elliptical cylinders and a thin airfoil

In this paper, we employ the boundary integral equation (BIE) to analytically derive the solution of potential flow across a circular cylinder, an elliptical cylinder and a thin airfoil. It is found that the symmetric or antisymmetric potential flow problem can be solved by using the UT or the LM equation alone. Both analytical and numerical approaches were considered. The key tool is using the degenerate kernel instead of the closed-form fundamental solution. A closed-form fundamental solution is expressed in terms of degenerate kernel by using polar and elliptical coordinates for a circular cylinder and an elliptical cylinder, respectively. The role of the dual BEM is also examined by showing the singular vectors of influence matrix. Besides, either the singular or the hypersingular BIE are respectively employed alone to solve the problem of symmetric and anti-symmetric flow field. Analytical derivation as well as the BEM is demonstrated. The orientation of the ellipse, as well as the angle of attack for the thin airfoil, is considered. The result shows that the degenerate kernel can be an alternative tool for solving some BVPs, although the complex variable is always employed. Once the degenerate kernel for the closed-form fundamental solution is available, the BIE is nothing more than the linear algebra and an analytical study is possible. The extension to 3-D object is promising once the degenerate kernel is available.

On attached masses of panels oscillating in incompressible medium

The paper discusses small oscillations of a panel in an incompressible medium. Air can be considered an incompressible medium during modal tests of solar array panels for spacecraft deployed on the ground in a lab environment. A panel is represented as a two-sided boundary surface. Conditions are determined for applicability of the potential motion of the medium. Calculation of the attached mass is reduced to the solution of the Neumann boundary value problem. To solve the boundary value problem, the method of boundary elements is used in the piecewise constant approximation variant, which provides a solution of the hypersingular boundary integral equation. Numerical solutions are obtained for the three fundamental modes of rectangular panels. The obtained numerical values are refined using non-linear Shanks transformation. Dependence of attached mass on panel elongation and the amount of the gap between its fragments is studied. For any in-plane oscillation mode of a panel fragment, the attached mass is determined using the principle of linear superposition. An estimate is given of the effect of the distance from the panel to the wall on the attached mass value. Key words: oscillations, incompressible medium, air, attached mass, rectangular panels, boundary elements method.

ON ATTACHED MASSES OF PANELS OSCILLATING IN INCOMPRESSIBLE MEDIUM

The paper discusses small oscillations of a panel in an incompressible medium. Air can be considered an incompressible medium during modal tests of solar array panels for spacecraft deployed on the ground in a lab environment. A panel is represented as a two-sided boundary surface. Conditions are determined for applicability of the potential motion of the medium. Calculation of the attached mass is reduced to the solution of the Neumann boundary value problem. To solve the boundary value problem, the method of boundary elements is used in the piecewise constant approximation variant, which provides a solution of the hypersingular boundary integral equation. Numerical solutions are obtained for the three fundamental modes of rectangular panels. The obtained numerical values are refined using non-linear Shanks transformation. Dependence of attached mass on panel elongation and the amount of the gap between its fragments is studied. For any in-plane oscillation mode of a panel fragment, the attached mass is determined using the principle of linear superposition. An estimate is given of the effect of the distance from the panel to the wall on the attached mass value. Key words: oscillations, incompressible medium, air, attached mass, rectangular panels, boundary elements method.

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.

Integral Equation Analysis of Terahertz Backscattering From Circular Dielectric Rod With Partial Graphene Cover

We study how the backward scattering cross-section (BSCS) of a microsize dielectric rod illuminated by an H-polarized plane wave can be efficiently manipulated using a partial graphene cover and associated plasmon resonances. Our treatment is based on the hyper-singular boundary integral equation, discretized with a Nystrom-type algorithm, which has a mathematically grounded convergence. We compute the BSCS and the extinction cross-section (ECS) in the sub-THz and THz ranges with controlled accuracy and demonstrate that BSCS can be both enhanced and reduced due to the graphene strip.

Iterative dual BEM solution for water wave scattering by breakwaters having perforated thin plates

This study develops an iterative dual BEM (Boundary Element Method) model for water wave scattering by breakwaters having perforated thin plates, which avoids setting artificial sub-domains near the thin plates and thus is very convenient to write general computer codes. In order to directly consider the effect of wave height on the wave energy dissipation by perforated plates, a quadratic (nonlinear) pressure drop condition is imposed on the plates. Then, iterative calculations are conducted to solve the nonlinear boundary condition in the dual BEM model. Also, an analytical method to calculate the integrals of kernel function, which is generated from the hypersingular boundary integral equation in the dual BEM model, is introduced to enhance the calculation efficiency and accuracy. Numerical results show that the iterative calculations in the newly developed dual BEM solution converge rapidly. As examples, wave scattering by several typical breakwaters involving perforated thin plates are solved using the iterative dual BEM model. The corresponding numerical results of the reflection and transmission coefficients are in excellent agreement with the predictions by the analytical solutions. The numerical results also agree reasonably well with experimental data in literature. The present iterative dual BEM model is valuable for analyzing the hydrodynamic performance of breakwaters with perforated thin plates in preliminary engineering design.

Subtraction singularity technique applied to the regularization of singular and hypersingular integrals in high-order curved boundary elements in plane anisotropic elasticity

The numerical solutions of boundary integral equations by the Boundary Element Method (BEM) have been applied in several areas of computational engineering and science such as elasticity and fracture mechanics. The BEM formulations often require the evaluation of complex singular and hypersingular integrals. Therefore, BEM requires special integration schemes for singular elements. The Subtraction Singularity Technique (SST) is a general procedure for evaluating such integrals, which allows for the integral over high-order curved boundary elements. The SST regularises these kernels through the Taylor expansion of the integral kernels around the source point. This study presents the expressions from Taylor expansions, which are required for the regularization of singular and hypersingular boundary integral equations of plane linear anisotropic elasticity. These expressions have been implemented into an academic BEM code, which enables the integration over high-order straight and curved boundary elements. The proposed scheme leads to excellent performance. The results obtained by the proposed scheme are in excellent agreement with reference responses available in the literature.

A dual interpolation Galerkin boundary face method for potential problems

A dual interpolation Galerkin boundary face method (DiGBFM) is applied in this paper by combining the newly developed dual interpolation method with the Galerkin boundary face method (BFM). The dual interpolation method unifies the conforming and nonconforming elements in the BFM implementation. It classifies the nodes of a conventional conforming element into virtual nodes and source nodes. Potentials and fluxes are interpolated using the continuous elements in the same way as conforming BFM, while boundary integral equations (BIEs) are collocated at source nodes, in the same way as nonconforming BFM. In order to arrive at a square linear system, we provide additional constraint equations, which are established by the moving least-squares (MLS) approximation, to condense the degrees of freedom relating to virtual nodes. Compared with the traditional symmetric Galerkin boundary element method (BEM), the symmetry feature of the DiGBFM equations is obtained simply through matrix manipulations, because of the use of the symmetric BEM, and no hypersingular BIE is needed in the DiGBFM. The proposed method has been implemented successfully for solving 2-D steady-state potential problems. Several numerical examples are presented in this paper to show the convergence and accuracy of this new method.

Hypersingular boundary integral equations for plane orthotropic elasticity in terms of first-order stress functions

This paper is intended to present an implementation of the hypersingular boundary integral equations in terms of first-order stress functions for stress computations in plane orthotropic elasticity. In general, the traditional computational technique of the boundary element method used for computing the stress distribution on the boundary and close to it is not as accurate as it should be. In contrast, the accuracy of stress computations on the boundary is greatly increased by applying the hypersingular integral equations. Contrary to the method in which the solution is based on an approximation of displacement field, here the first-order stress functions and the rigid body rotation are the fundamental variables. An advantage of this approach is that the stress components can be obtained directly from the stress functions, there is, therefore, no need for Hooke's law, which should be used when they are computed from displacements. In addition, the computational work can be reduced when the stress distribution is computed at an arbitrary point on the boundary. The numerical examples presented prove the efficiency of this technique.

Isogeometric boundary element method for acoustic scattering by a submarine

Isogeometric analysis with the boundary element method (IGABEM) has recently gained interest. In this paper, the approximability of IGABEM on 3D acoustic scattering problems will be investigated and a new improved BeTSSi submarine will be presented as a benchmark example. Both Galerkin and collocation are considered in combination with several boundary integral equations (BIE). In addition to the conventional BIE, regularized versions of this BIE will be considered. Moreover, the hyper-singular BIE and the Burton–Miller formulation are also considered. A new adaptive integration routine is presented, and the numerical examples show the importance of the integration procedure in the boundary element method. The numerical examples also include comparison between standard BEM and IGABEM, which again verifies the higher accuracy obtained from the increased inter-element continuity of the spline basis functions. One of the main objectives in this paper is benchmarking acoustic scattering problems, and the method of manufactured solution will be used frequently in this regard.

Evaluation of the hypersingular boundary integral equation for acoustic wave-propagation simulations

We have developed a hypersingular boundary integral equation (BIE) approach to simulate wave scattering by inclusions such as karstic cavities and gypsum salts. The solution of conventional BIEs for exterior boundary value problems is of nonuniqueness and numerical instability at characteristic wavenumbers (corresponding to fictitious resonant frequencies). With the hypersingular boundary integral operator, we can handle this problem, but with the difficult-to-calculate hypersingular kernels. Our method to evaluate the singular part in the operator is more efficient than other singularity-elimination techniques. Numerical examples validate the method by comparing with the analytical solution.

Boundary integral equations for anisotropic elasticity of solids containing rigid thread-like inclusions

The paper presents a novel approach in derivation and solution of boundary integral equations of anisotropic elasticity of solids containing thin rigid wires (thread-like inclusions). It proposes to model rigid thread-like inclusions as spatial curves, which can rotate as a rigid one and possess certain rigid displacement. Somigliana identity is written for solids containing rigid thread-like inclusions, which are modeled by spatial curves. Based on this identity the hypersingular boundary integral equations are derived, and it is shown that they also include the non-singular terms. The comparison is made between the models and integral equations for 3D rigid thread-like inclusions and 2D rigid line inclusions. A problem for a single rectilinear rigid thread-like inclusion in an infinite elastic medium is considered. Its solution strategy is proposed based on the boundary element approach. Distribution of forces along the inclusion modeling line and the displacement field near its tip are shown.