Singular Boundary Integral Equations Research Articles

Structural Shape Optimization Based on Multi-Patch Weakly Singular IGABEM and Particle Swarm Optimization Algorithm in Two-Dimensional Elastostatics

In this paper, a multi-patch weakly singular isogeometric boundary element method (WSIGABEM) for two-dimensional elastostatics is proposed. Since the method is based on the weakly singular boundary integral equation, quadrature techniques, dedicated to the weakly singular and regular integrals, are applied in the method. A new formula for the generation of collocation points is suggested to take full advantage of the multi-patch technique. The generated collocation points are essentially inside the patches without any correction. If the boundary conditions are assumed to be continuous in every patch, no collocation point lies on the discontinuous boundaries, thus simplifying the implementation. The multi-patch WSIGABEM is verified by simple examples with analytical solutions. The features of the present multi-patch WSIGABEM are investigated by comparison with the traditional IGABEM. Furthermore, the combination of the present multi-patch WSIGABEM and the particle swarm optimization algorithm results in a shape optimization method in two-dimensional elastostatics. By changing some specific control points and their weights, the shape optimizations of the fillet corner, the spanner, and the arch bridge are verified to be effective.

NUMERICAL MODELLING OF ELECTROMAGNETIC WAVE SCATTERING ON GRATINGS WITH ONE-DIMENSIONAL QUASI-FRACTAL PERIOD STRUCTURE

Numerical modelling of the properties of E-polarised and H-polarised waves scattered on periodic screened quasi-fractal gratings is carried out. The location of the band system at each period is determined by the principle of constructing a generalized symmetric Cantor set at a certain step of the algorithm. A mathematical model of the problems based on systems of boundary singular integral equations of the first kind was used in the study. These systems of equations were obtained using the method of parametric representations of singular and hypersingular integral operators. The systems of singular integral equations were solved numerically using the computational schemes of the method of discrete singularities. The solutions of these equations are used to obtain the main characteristics of the electric and magnetic fields. This experiment proved the possibility of using the MDS computational scheme to analyse systems containing 8 – 16 bands at different distances from each other. Graphs of the dependence of harmonic amplitudes on the wavenumber, point plots of absolute values of all non-zero harmonics at resonant wavenumber values, and maps of electric and magnetic field components in the region above the grating were obtained. It is confirmed that the overall field structure in the case of normal incidence is significantly influenced by all harmonics with absolute numbers from 0 to 50. The harmonics had a large number of resonances that were observed at different values of the wavenumber. This led to a complex structure of the isolines of absolute values of the scattered electric and magnetic field amplitudes in the region above the structure, and a significant difference in amplitude values with small changes in coordinates. In the future, it is planned to carry out computer simulations for imperfectly conducting structures and compare the results with the numerical results for the ideal case considered in this paper. The proposed structure may be of interest for the design of multimode broadband antennas.

Dvumernye zadachi fil'tratsii zhidkosti s granichnymi istochnikami v anizotropnom neodnorodnom sloe

We study the first and second boundary value problems and the transmission problem for the complex potential of a two-dimensional filtration flow in an anisotropic and inhomogeneous (variable permeability and thickness) porous layer. The flow sources are arbitrary discrete and can generally be located both on the boundaries and outside the boundaries. The boundaries are modeled by arbitrary smooth (piecewise smooth) closed lines, and the flow sources are singularities (isolated singular points) of the complex potential. The presence of a system of sources on the boundaries leads to a fundamentally new generalization (complication) of the boundary conditions, which are characterized by singular functions with isolated singular points. In the case of an anisotropic homogeneous (constant permeability and thickness) layer and rectilinear boundaries, the solutions of the problems are presented in closed form. In the general case, when an arbitrary smooth closed curve models a boundary with sources located on it, a generalized Cauchy type integral for the complex flow potential is used. This permitted reducing the second boundary value problem and the transmission problem to boundary singular integral equations. The problems studied are mathematical models of two-dimensional filtration processes in layered porous media, which are of interest, for example, for the practice of extracting fluids (oil, water) from natural anisotropically heterogeneous soil layers.

Potential-flow computations in infinite or zero gravity limits via a weakly singular boundary integral equation

Potential flow computations, based on a weakly singular integral equation, are considered in the infinite-gravity limit g=∞ and the zero-gravity limit g=0. The weakly singular integral equation, obtained via a straightforward and rigorous application of Green’s basic identity, involves a distribution of weak dipoles that is continuous across the body surface. The integral equation is solved via a low-order panel method, which is validated for a half sphere and applied to the S60 ship model in translation along the x, y or z-axis. A systematic numerical analysis of the convergence of numerical predictions shows that the number of panels that is required to obtain a 0.1% accuracy is significantly different for the infinite-gravity or zero-gravity flows, and for translation of the S60 hull along the x, y or z-axis. In particular, a very large number of panels is required for translation of the S60 ship model along the horizontal x or y-axis in the zero-gravity limit.

Analysis of near-interface cracks in three-dimensional anisotropic multi-materials by efficient BIEM

This paper presents the full analysis of near-interface cracks in three-dimensional linear elastic multimaterial bodies by a weakly singular boundary integral equation method (BIEM). The proposed technique was implemented in a general framework that allowed the treatment of finite cracked bodies with general configurations, material anisotropy, and mode-mixity. A system of integral equations governing unknown data on the boundary, crack surface, and material interface was established using a pair of weakly singular weak-form displacement and traction integral equations together with continuity along the material interface. A symmetric Galerkin boundary element method along with the finite element technique were implemented to solve the governing integral equations. In addition, the special near-front interpolation was employed to enhance the approximation of the relative crack-face displacement in the neighborhood of the crack front. The solved crack-face data were used directly to post-process the stress intensity factors and T-stresses along the crack front. An extensive numerical study was carried out not only to demonstrate the accuracy, convergence, and capability of the proposed technique, but also to explore the influence of material stiffness and distance to the material interface on fracture data along the crack front.

Convergence analysis of oversampled collocation boundary element methods in 2D

Collocation boundary element methods for integral equations are easier to implement than Galerkin methods because the elements of the discretisation matrix are given by lower-dimensional integrals. For that same reason, the matrix assembly also requires fewer computations. However, collocation methods typically yield slower convergence rates and less robustness, compared to Galerkin methods. We explore the extent to which oversampling can improve both robustness and convergence rates of collocation for singular boundary integral equations. We show that in some cases convergence rates can actually be higher than the corresponding Galerkin method, although this requires oversampling at a faster than linear rate. In most cases of practical interest, oversampling at least lowers the error by a constant factor. This can still be a substantial improvement: we analyse an example where linear oversampling by a constant factor J (leading to a rectangular system of size JN × N) improves the error at a cubic rate in the constant J. Furthermore, the oversampled collocation method is much less affected by a poor choice of collocation points, as we show how oversampling can lead to guaranteed convergence. Numerical experiments are included for the two-dimensional Helmholtz equation.

Plane Boundary Value Problems of Thermoelasticity and Their Generalized Solutions

Nonstationary boundary value problems of uncoupled thermoelasticity are considered. A method of boundary integral equations in the initial space-time has been developed for solving boundary value problems of thermoelasticity by plane deformation by use general functions method (GFM). The generalized solutions of boundary value problems are constructed in the space of generalized vector functions. Fundamental Green tensor, stress tensor and their antiderivatives over time are used for construction their regular integral representations. By use analogue of Gauss formula for fundamental stress the singular boundary integral equations are constructed to determine the unknown boundary functions. The resulting formulas have important engineering application. They make possible to determine the thermal stressed state of the medium by the boundary values of stresses, displacements, temperature and heat flux, without solving singular boundary integral equations. Because for real engineering problems these physical characteristics can be experimentally measured at the boundary. Moreover, the formulas allow to calculate the influence of each of these characteristics of the process on its stressstrain state. The last one is very important in designing structures made of thermoelastic materials.

Колебания слоя с расслоением в рамках градиентной теории упругости

The direct problem of antiplane oscillations of a layer with delamination in the context of the gradient theory of elasticity is considered. The gradient model proposed by Aifantis is taken as a mathematical model. The main attention has been paid to the analysis of mechanical fields at the crack bank and at its tips - stress concentrators. The study is carried out using the method of boundary integral equations (BIE). The BIE for the gradient of displacement field on the crack is constructed. The analysis of the constructed BIE is carried out and the cubic singularity is explicitly revealed. The solution of singular BIE is constructed using approximating Chebyshev polynomials. A study for cracks of small relative length - asymptotic approach is carried out, simple semi-analytical expressions for constructing the crack swap function are obtained, the range of efficiency of the asymptotic approach is obtained. The stress fields in the area of the crack tips are constructed. Numerical results of computational experiments are presented.

Interpolation method for evaluating weakly singular kernels

The solution of initial, boundary, and mixed value problems through the integral equation method yields certain boundary singular integral equations. In many scientific and industrial applications in artificial intelligence, biological systems, scattering, radiation, and image processing, there is a need to evaluate such types of singular integrals, especially the weakly singular kernels. This study presents a new numerical method for evaluating weakly singular kernels based on some advanced matrix-vector barycentric Lagrange interpolation formulas. We developed these formulas to be applied to numerically evaluating singular integrals. At the same time, we created three computational rules to determine the optimal locations for the distribution of the interpolation nodes to be within the integration domain and never be outside for any value of the interpolant degree. These rules are devised so that the equidistant nodes depend on the step-sizes, which are defined as functions of the interpolant degree by some small real number greater than or equal to zero. Thus, we overcame the singularity of the kernels on the whole integration domain and obtained uniform interpolation. Moreover, the presented method gives the kernel's values and the kernel's integral values at the singular points, whereas the numerical or exact values do not exist. The solutions to the illustrated four examples are shown in the given tables and figures. The interpolant solutions which we obtain by low-degree interpolants are faster to converge to the numerical or exact ones (if they exist). This confirms the originality of the presented method and its effectiveness in obtaining high-precision results.

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.

On the regularization of Cauchy-type integral operators via the density interpolation method and applications

This paper presents a regularization technique for the high order efficient numerical evaluation of nearly singular, principal-value, and finite-part Cauchy-type integral operators. By relying on the Cauchy formula, the Cauchy-Goursat theorem, and on-curve Taylor interpolations of the input density, the proposed methodology allows to recast the Cauchy and associated integral operators as smooth contour integrals. As such, they can be accurately evaluated everywhere in the complex plane—including at problematic points near and on the contour—by means of elementary quadrature rules. Applications of the technique to the evaluation of the Laplace layer potentials and related integral operators, as well as to the computation conformal mappings, are examined in detail. The former application, in particular, amounts to a significant improvement over the recently introduced harmonic density interpolation method. Spectrally accurate discretization approaches for smooth and piecewise smooth contours are presented. A variety of numerical examples, including the solution of weakly singular and hypersingular Laplace boundary integral equations, and the evaluation of challenging conformal mappings, demonstrate the effectiveness and accuracy of the density interpolation method in this context.

Analysis of Cracked Body Strengthened by Adhesively Bonded Patches by BEM-FEM Coupling

This paper presents an efficient numerical technique capable of handling the stress analysis of three-dimensional cracked bodies strengthened by adhesively bonded patches. The proposed technique is implemented within the framework of the coupling of the weakly singular boundary integral equation method and the standard finite element procedure. The former is applied to efficiently treat the elastic body containing cracks, whereas the latter is adopted to handle both the adhesive layers and patches. The approximation of the near-front relative crack-face displacement is enhanced by using local interpolation functions that can capture the right asymptotic behavior. This also offers the direct calculation of the stress intensity factors along the crack front. A selected set of results is reported to demonstrate the capability of the proposed technique and the influence of various parameters on the performance of the strengthening.

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.

Investigation of Generalized SIFs of cracks in 3D piezoelectric media under various crack-face conditions

This paper investigates the influence of crack geometry, crack-face and loading conditions, and the permittivity of a medium inside the crack gap on intensity factors of planar and non-planar cracks in linear piezoelectric media. A weakly singular boundary integral equation method together with the near-front approximation is adopted to accurately determine the intensity factors. Obtained results indicate that the non-flat crack surface, the electric field, and the permittivity of a medium inside the crack gap play a crucial role on the behavior of intensity factors. The mode-I stress intensity factors (KI) for two representative non-planar cracks under different crack-face conditions are found significantly different and they possess both upper and lower bounds. In addition, KI for impermeable and semi-permeable non-planar cracks treated depends strongly on the electric field whereas those of impermeable, permeable, and semi-permeable penny-shaped cracks are identical and independent of the electric field. The stress/electric intensity factors predicted by permeable and energetically consistent models are, respectively, independent of and dependent on the electric field for the penny-shaped crack and the two representative non-planar cracks. Also, the permittivity of a medium inside the crack gap strongly affects the intensity factors for all crack configurations considered except for KI of the semi-permeable penny-shaped crack.

Radial basis functions for solving the singular boundary integral equation of the compressible fluid flow around obstacles

Abstract In this paper we find a numerical solution for the Singular Boundary Integral Equation (SBIE), with sources distribution, of the compressible fluid flow around obstacles. The numerical solution is obtained by using linear boundary elements, radial basis functions for the unknown approximation and the Cauchy Principal Value for the treatment of singularities that appear. The proposed method is implemented into a computer code, made in Mathcad programming language, and, for some particular cases, numerical solutions are found. An analytic checking of the computer code is also made, in order to validate the proposed approach.

Boundary element analysis of 3D shell‐like rigid electrically conducting inclusions in anisotropic thermomagnetoelectroelastic solids

AbstractThe paper presents general boundary element approach for analysis of thermomagnetoelectroelastic solids containing shell‐like electrically conducting inclusions with high magnetic permittivity. The latter are modeled with opened surfaces with certain boundary conditions on their faces. Rigid displacement and rotation, along with constant electric and magnetic potentials of inclusions are accounted for in these boundary conditions. Formulated boundary value problem is reduced to a system of singular boundary integral equations, which is solved numerically by the boundary element method. Special attention is paid to the field singularity at the front line of a shell‐like inclusion. Special shape functions are introduced, which account for this square‐root singularity and allow accurate determination of field intensity factors. Approaches for fast and accurate numerical evaluation of anisotropic thermomagnetoelectroelastic kernels are discussed, which are crucial in derivation of fast and precise boundary element approach. Numerical examples are presented.

Analysis for T-stress of cracks in 3D anisotropic elastic media by weakly singular integral equation method

This paper proposes an efficient numerical procedure for computing T-stresses of cracks in three-dimensional, linearly elastic, infinite media. The technique is established in a broad framework allowing a medium made of generally anisotropic materials and cracks of arbitrary shape and under general loading conditions to be treated. A pair of weakly singular, weak-form, displacement and traction boundary integral equations is utilized to formulate the key equations governing the unknown crack-face fields. Besides the basic benefits such as the reduction of spatial dimensions of the solution space and the efficient treatment of unbounded domains and remote boundary data, use of such integral equations in the formulation offers additional positive features including the computational simplicity resulting from the weakly singular nature of all involved integrals and the involvement of a complete set of crack-face displacement fields. A weakly singular symmetric Galerkin boundary element method together with the special near-front approximation is utilized to solve for the unknown relative crack-face displacement whereas the sum of the crack-face displacement is obtained from the displacement boundary integral equation for cracks via the Galerkin technique. The latter step is one of the essential aspects of the present study that provides the direct means for determining the T-stress data in terms of the sum of the crack-face displacement in the neighborhood of the crack-front. An extensive numerical study is conducted for various scenarios and a selected set of results is reported to demonstrate both the accuracy and capability of the proposed technique.

Method of discrete singularities in problems of filler vibrations in fuel tanks under overloads and at low gravity

The paper is about free vibration problems in fuel tanks with different levels of fillers both under overloads and low gravity. The fuel tanks are considered as rigid shells of revolution. The liquid, contained in the tanks, is supposed to be an ideal and incompressible one, and its flow, induced by external excitations, is irrotational. The problem of free axisymmetric harmonic oscillations of the fluid-filled rigid compound shell is considered. The mixed boundary value problem is formulated, and boundary conditions are received on the boundaries of the fluid domain. The non-penetration conditions is formulated on the rigid shell boundaries, and on the free liquid surface there are kinematic and dynamic boundary conditions. The flow fluctuations are described by using the velocity potential that for ideal and incompressible liquids satisfies the Laplace equation. For its solution, the integral representation is in use. But there are two unknown functions, the velocity potential and the function describing the shape and position of the free surface during time. The transformation of the boundary conditions leads to eliminating one of these unknown functions. So, the system of boundary singular integral equations is obtained to determine the velocity potential. The discrete singularity method and the boundary element method are applied for its numerical solution. The problem of determining own modes and frequencies is solved by using the techniques, where the surface tension effects are neglected. Then these modes are considered as basic functions to determine the modes and frequencies of the liquid taking into account the surface tension. Thus, in this work the method is developed which takes into account the surface tension effect on the frequency of fluid fluctuations in the rigid tank under low gravity conditions. The surface of the interaction between liquid media and gas is considered as a thin membrane, whose thickness is neglected. The fluid pressure on this surface is determined by the Laplace-Young equation. The developed method is useful for the investigation of free and forced fluid oscillations in rigid compound shells with arbitrary meridians.

Boundary Element Modeling of Pyroelectric Solids with Shell Inclusions

Abstract The paper presents general boundary element approach for analysis of thermoelectroelastic (pyroelectric) solids containing shell-like electricity conducting permittive inclusions. The latter are modeled with opened surfaces with certain boundary conditions on their faces. Rigid displacement and rotation, along with constant electric potential of inclusions are accounted for in these boundary conditions. Formulated boundary value problem is reduced to a system of singular boundary integral equations, which is solved numerically by the boundary element method. Special attention is paid to the field singularity at the front line of a shell-like inclusion. Special shape functions are introduced, which account for this square-root singularity and allow accurate determination of field intensity factors. Numerical examples are presented.

Low Frequency Sloshing Analysis of Cylindrical Containers with Flat and Conical Baffles

This paper presents an analysis of low-frequency liquid vibrations in rigid partially filled containers with baffles. The liquid is supposed to be an ideal and incompressible one and its flow is irrotational. A compound shell of revolution is considered as the container model. For evaluating the velocity potential the system of singular boundary integral equations has been obtained. The single-domain and multi-domain reduced boundary element methods have been used for its numerical solution. The numerical simulation is performed to validate the proposed method and to estimate the sloshing frequencies and modes of fluid-filled cylindrical shells with baffles in the forms of circular plates and truncated cones. Both axisymmetric and non-axisymmetric modes of liquid vibrations in baffled and un-baffled tanks have been considered. The proposed method makes it possible to determine a suitable place with a proper height for installing baffles in tanks by using the numerical experiment.