A hypersingular integral equation for current density on the surface of a microstrip vibrator

The problem of finding the current-density distribution on the surface of a microstrip vibrator (MV) in the framework of a thin ideally conducting strip, deposited on a dielectric substrate with one-sided metallization, is reduced to solution of a hypersingular integral equation (HSIE). The dependences of the input impedance on the vibrator length and substrate thickness are presented.     

Solution of a singular integral equation

Summary A direct method is presented for solving a singular integral equation which is a generalisation of one occurring in viscous flow theory and for which other methods of solution have been described by Brown [1] and Boersma [2].     

A direct boundary integral equation formulation for the Oseen flow past a two-dimensional cylinder of arbitrary cross-section

Summary A novel boundary integral formulation is presented for the direct solution of the classical problem of slow flow past a two-dimensional cylinder of arbitrary cross section in an unbounded viscous medium, the equations of motion having first been linearised by the Oseen approximation. It is shown how the governing partial differential equations of motion, together with the no-slip boundary conditions on the cylinder, may be reformulated as a pair of coupled integral equations of the second kind, which may be manipulated further to yield the lift and drag coefficients explicitly, as well as flow characteristics anywhere in the flowfield.The present formulation requires a non-iterative numerical solution procedure which is applicable to low Reynolds number flows. The method is not restricted in its ability to deal with complicated cylinder geometries, as the discretisation of only the cylinder surface is required.Results of the present method are shown to be in good agreement with those of previous analytical and numerical investigations.With 2 Figures     

A new singular integral equation for the classical crack problem in plane and antiplane elasticity

A new singular integral equation (with a kernel with a logarithmic singularity) is proposed for the crack problem inside an elastic medium under plane or antiplane conditions. In this equation the integral is considered in the sense of a finite-part integral of Hadamard because the unknown function presents singularities of order ?3/2 at the crack tips. The Galerkin and the collocation methods are proposed for the numerical solution of this equation and the determination of the values of the stress intensity factors at the crack tips and numerical results are presented. Finally, the advantages of this equation are also considered.     

A weakly singular boundary integral equation in elastodynamics for heterogeneous domains mitigating fictitious eigenfrequencies

A boundary element formulation applied to dynamic soil–structure interaction problems with embedded foundations may give rise to inaccurate results at frequencies that correspond to the eigenfrequencies of the finite domain embedded in an exterior domain of semi-infinite extent. These frequencies are referred to as fictitious eigenfrequencies. This problem is illustrated and mitigated modifying the original approach proposed by Burton and Miller for acoustic problems, which combines the boundary integral equations in terms of the displacement and its normal derivative using a complex coupling parameter . Hypersingular terms in the original boundary integral equation are avoided by replacing the normal derivative by a finite difference approximation over a characteristic distance h, still leading to an exact boundary integral equation. A proof of the uniqueness of this formulation for small h and a smooth boundary is given, together with a parametric study for the case of a rigid massless cylindrical embedded foundation. General conclusions are drawn for the practical choice of the dimensionless coupling parameter and the dimensionless distance      

A method for the numerical solution of singular integral equations with a principal value integral

A method for the numerical solution of singular integral equations with kernels having a singularity of the Cauchy type is presented. The singular behavior of the unknown function is explicitly built into the solution using the index theorem. The integral equation is replaced by integral relations at a discrete set of points. The integrand is then approximated by piecewise linear functions involving the value of the unknown function at a finite set of points. This permits integration in a closed form analytically. Thus the problem is reduced to a system of linear algebraic equations. The results obtained in this way are compared with the more sophisticated procedures based on Gauss-Chebyshev and Lobatto-Chebyshev quadrature formulae. An integral equation arising in a crack problem of the classical theory of elasticity is used for this purpose.     

Accurate integration of singular kernels in boundary integral formulations for Helmholtz equation

Boundary integral formulations for the 2D Helmholtz equation involve kernels in the form of modified Bessel functions. Accurate schemes for evaluating integrals of the kernels and their derivatives are presented. Special attention is paid to integrals involving singular and near singular kernels. Both boundary and domain integrals are considered. It is shown that, with the use of series expansion functions for the modified Bessel functions, the boundary integrals can be evaluated analytically in the neighbourhood of the singularity. For domain integrals, the behaviour of the kernels in the vicinity of the singularity is used to construct accurate numerical quadrature schemes. A transient heat conduction problem is formulated as a Helmholtz equation, solved, and compared against analytic solution to demonstrate the effectiveness of these schemes in relation to traditional methods. References are made to previous work to advocate the utility of the boundary integral method for non-linear and time-transient problems.     

Continuity requirements for density functions in the boundary integral equation method

Two methods of forming regular or hypersingular boundary integral equations starting from an interior integral representations are discussed. One method involves direct treatment of the singularities such as Cauchy principal value and/or finite-part interpretation of the integrals and the other does not. By either approach, theory places the same restrictions on the smoothness of the density function for the integrals to exist, assuming sufficient smoothness of the geometrical boundary itself. Specifically, necessary conditions on the smoothness of the density function for meaningful boundary integral formulas to exist as required for the collocation procedure are established here. Cases for which such conditions may not be sufficient are also mentioned and it is understood that with Galerkin techniques, weaker smoothness requirements may pertain. Finally, the bearing of these issues on the choice of boundary elements, to numerically solve a hypersingular boundary integral equation, is explored and numerical examples in 2D are presented.     

An integral equation method for the diffraction of oblique waves by an infinite cylinder

A hybrid integral equation method is formulated to study the diffraction of oblique waves by an infinite cylinder. The water depth and the geometry of the floating cylinder are assumed to be uniform in the y-direction (one of the horizontal axes). Numerical discretization and integrations are performed in the vertical plane. Analytical solutions are used in far fields such that radiation boundary conditions are satisfied. Numerical results are obtained for the case of wave scattering by a floating rectangular cylinder in a constant water depth. The accuracy and efficiency of present method are compared with those obtained by other numerical techniques.     

Approximate solution of a singular integral equation relating to the subsonic flow past oscillating wings

An approximate method has been developed to solve the singular integral equation occurring in the theory of three-dimensional wings oscillating harmonically in subsonic flow with arbitrary frequencies. If the oscillations are slow enough and the Mach number not too near unity, Garner[1] has shown that the governing differential equation of the complex amplitude of the enthalpy can be approximated by the Laplace equation. However, if the frequency of the oscillating wing is not low, the acceleration potential satisfies the homogeneous form of the Helmholtz equation. The solution of this differential equation with the known oscillatory boundary condition then leads to the complicated singular integral equation obtained by several authors [2, 3]. The presence of the exponential term in the chordwise integrals together with the infinite limit of integration complicates matters. By splitting up the range of integration in a particular manner it is shown in the present paper that the integral equation can be solved for any arbitrary frequency. The method has been applied to the calculation of pitching derivatives with first order frequency effects for sweptback wings in subsonic flow and the results compared with those given by Garner [1] to check the accuracy of the calculations.     

Determination of higher order coefficients and zones of dominance using a singular integral equation approach

A method to determine higher order coefficients from the solution of a singular integral equation is presented. The coefficients are defined by , which gives the radial stress at a distance, r, in front of the crack tip. In this asymptotic series the stress intensity factor, k₀, is the first coefficient, and the T-stress, T₀, is the second coefficient. For the example of an edge crack in a half space, converged values of the first 12 mode I coefficients (k_n and T_n, n = 0, … , 5) have been determined, and for an edge crack in a finite width strip, the first six coefficients are presented. Coefficients for an internal crack in a half space are also presented. Results for an edge crack in a finite width strip are used to quantify the size of the k-dominant zone, the kT-dominant zone and the zones associated with three and four terms, taking into account the entire region around the crack tip.     

Analysis of fractures in 3D piezoelectric media by a weakly singular integral equation method

A weakly singular, symmetric Galerkin boundary element method (SGBEM) is established to compute stress and electric intensity factors for isolated cracks in three-dimensional, generally anisotropic, piezoelectric media. The method is based upon a weak-form integral equation, for the surface traction and the surface electric charge, which is established by means of a systematic regularization procedure; the integral equation is in a symmetric form and is completely regularized in the sense that its integrand contains only weakly singular kernels of (hence allowing continuous interpolations to be employed in the numerical approximation). The weakly singular kernels which appear in the weak-form integral equation are expressed explicitly, for general anisotropy, in terms of a line integral over a unit circle. In the numerical implementation, a special crack-tip element is adopted to discretize the region near the crack front while the remainder of the crack surface is discretized by standard continuous elements. The special crack-tip element allows the relative crack-face displacement and electric potential in the vicinity of the crack front to be captured to high accuracy (even with relatively large elements), and it has the important feature that the mixed-mode intensity factors can be directly and independently extracted from the crack front nodal data. To enhance the computational efficiency of the method, special integration quadratures are adopted to treat both singular and nearly singular integrals, and an interpolation strategy is developed to approximate the weakly singular kernels. As demonstrated by various numerical examples for both planar and non-planar fractures, the method gives rise to highly accurate intensity factors with only a weak dependence on mesh refinement.     

A practical method for singular integral equations of the second kind

A convenient and efficient numerical method is presented for the treatment of Cauchy type singular integral equations of the second kind. The solution is achieved by splitting the Cauchy singular term into two parts, allowing one of the parts to be determined in a closed-form while the other part is evaluated by standard Gauss-Jacobi mechanical quadrature. Since the Cauchy singularity is removed after this manipulation, the quadrature abscissas and weights may be readily available and the placement of the collocation points is flexible in the present method. The method is exact when the unknown function can be expressed as the product of a fundamental function and a polynomial of degree less than the number of the integration points. The proposed algorithm can also be extended to the case where the singularities are complex and is found equally effective. The proposed algorithm is easy to implement and provides a shortcut for programming the numerical solution to the singular integral equation of the second kind.     

Transport ac loss of a superconducting cylinder with field dependent critical current density

The transport ac loss per cycle per unit length of a hard superconducting cylinder is calculated from the critical state model assuming a Kim-type and an exponential field dependent critical current density. Without such dependence, the results are consistent with Norris’ equations for an ellipse bar, in which the critical current density is assumed not to depend on the flux density. Based on Norris’ equations, the expressions of the loss are derived for a finite length cylinder. It is shown that the field dependence decreases and increases the loss at low and high ac currents, respectively, and the effects of the parameter p on the loss are related to the magnetization process. Compared to Norris’ prediction, the results for the Kim and exponential model show the same trend with respect to the external transport current.     

A simplified integral equation method for the calculation of the eigenvalues of Helmholtz equation

An integral equation method is presented for the numerical calculation of the eigenvalues of the scalar Helmholtz equation. By using a particular solution instead of Green's function, the calculations could be simplified due to the elimination of complex numbers.     

Coupling of singular integral equation methods and finite elements in 2-D elasticity

The coupling of the singular integral operators method (S.I.O.M.) and the finite element method (F.E.M.) is proposed for the solution of 2-D elasticity problems. Such a combined numerical method is especially used for engineering problems with unbounded domains or regions of high stress concentration, where singularities are present. Then special solutions can be determined by the S.I.O.M. in areas with infinite domain or when singularities occur and these to be combined with corresponding solutions by finite elements. An application of 2-D elasticity is finally given to the determination of the stress field around a circular hole subjected to internal pressure. For the solution of the above problem is used the coupling method of S.I.O.M. and F.E.M.Kombination der Singulären-Integral-Methode mit der Finite-Elemente-Methode für die Lösung von 2D-Elastizitätsproblemen

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Zusammenfassung In dem vorliegenden Papier wird die Kombination der sog. Singulären-Integral-Operatoren Methode (SIOM) mit der Finite-Elemente-Methode (FEM) zur Lösung von 2D-Elastizitätsberechnungsproblemen vorgeschlagen. Derartige kombinierte numerische Methoden werden besonders zur Lösung von Berechnungsproblemen mit unbegrenzter Ausdehnung oder mit Bereichen hoher Spannungskonzentration eingesetzt, wenn Singularitäten auftreten. Spezielle Lösungen können mit Hilfe der S.I.O.M. in Bereichen sehr geringer Abmessungen oder mit Singularitäten erhalten werden und mit der FEM zum Erhalt der gesamten Lösung kombiniert werden. Die Anwendung der hier beschriebenen Methodik wird anhand der Bestimmung des Spannungsfelds (2D) einer ebenen Platte im Bereich eines runden Lochs dargestellt, welche einer Innendruckbelastung ausgesetzt ist.

     

A note on the Gauss-Jacobi quadrature formulae for singular integral equations of the second kind

A fast and efficient numerical method based on the Gauss-Jacobi quadrature is described that is suitable for solving Fredholm singular integral equations of the second kind that are frequently encountered in fracture and contact mechanics. Here we concentrate on the case when the unknown function is singular at both ends of the interval. Quadrature formulae involve fixed nodal points and provide exact results for polynomials of degree 2n − 1, where n is the number of nodes. Finally, an application of the method to a plane problem involving complete contact is presented.