On the Cauchy principal value of the surface integral in the boundary integral equation of 3D elasticity

A simple demonstration of the existence of the Cauchy principal value (CPV) of the strongly singular surface integral in the Somigliana Identity at a non-smooth boundary point is presented. First a regularization of the strongly singular integral by analytical integration of the singular term in the radial direction in pre-image planes of smooth surface patches is carried out. Then it is shown that the sum of the angular integrals of the characteristic of the tractions of the Kelvin fundamental solution is zero, a formula for the transformation of angles between the tangent plane of a suface patch and the pre-image plane at smooth mapping of the surface patch being derived 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.     

On the evaluation of hyper-singular integrals arising in the boundary element method for linear elasticity

The boundary element method (BEM) for linear elasticity in its curent usage is based on the boundary integral equation for displacements. The stress field in the interior of the body is computed by differentiating the displacement field at the source point in the BEM formulation, via the strain field. However, at the boundary, this method gives rise to a hypersingular integral relation which becomes numerically intractable. A novel approach is presented here, where hyper-singular kernels for stresses on the boundary are made numerically tractable through the imposition of certain equilibrated displacement modes. Numerical results are also presented for benchmark problems, to illustrate the efficacy of the present approach. Solutions are compared to the commonly used boundary stress algorithm wherein the boundary stresses are computed from known boundary tractions, and derivatives of known displacements tangential to the boundary. An extension of this approach to solve linear elasticity problems using the traction boundary integral equation (TBIE) is also discussed.     

Added mass computations by integral equation methods

This paper presents boundary integral equation procedures for calculating the added mass matrix required to determine hydroelastic vibrational modes of tanks or immersed structures. The liquid is assumed inviscid and incompressible and no surface waves are admitted. Two symmetric variational formulations of the boundary integral equation are derived, including the free surface condition and possible conditions of symmetry. These are approximated by a curved finite element method and numerical results are presented for an axisymmetric tank, immersed cantilever plates and a three-dimensional fuel storage container.     

Analytical representation of linear kernels in collision integral of Boltzmann equation for maxwellian molecules

It is proposed to represent a complicated fivefold collision integral in the Boltzmann equation by a set of relatively simple integral operators. An analytical expression for the kernel of the collision integral can rarely be obtained even in linear cases. One of these cases was studied by Hecke [1] and another case is considered in the present Letter.     

Stability of surface integral equation for left-handed materials

In recent years, left-handed materials (LHMS) have been a hot and debated research topic. Electromagnetic simulation is usually performed for prediction as well as for design. The stability of the surface integral equation method through an insightful and rigorous procedure is analysed. The conclusion is that the PMCHWT formulation at perfectly matched LHM-RHM interface has stability problem, which comes from the plasmon resonance condition. Adding a small amount of loss can greatly improve the condition number of the matrix. Physical and consistent numerical results are shown followed by a brief conclusion     

Boundary integral equation for tangential derivative of flux in Laplace and Helmholtz equations

In this paper, the boundary integral equations (BIEs) for the tangential derivative of flux in Laplace and Helmholtz equations are presented. These integral representations can be used in order to solve several problems in the boundary element method (BEM): cubic solutions including degrees of freedom in flux's tangential derivative value (Hermitian interpolation), nodal sensitivity, analytic gradients in optimization problems, or tangential derivative evaluation in problems that require the computation of such variable (elasticity problems in BEM). The analysis has been developed for 2D formulation. Kernels for tangential derivative of flux lead to high‐order singularities (O(1/r³)). The limit to the boundary analysis has been carried out. Based on this analysis, regularization formulae have been obtained in order to use such BIE in numerical codes. A set of numerical benchmarks have been carried out in order to validate theoretical and practical aspects, by considering known analytic solutions for the test problems. The results show that the tangential BIEs have been properly developed and implemented. Copyright © 2005 John Wiley & Sons, Ltd.     

Boundary integral equation methods for two-dimensional models of crack-field interactions

An introduction to the application of surface integral equation methods to the calculation of eddy current-flaw interactions is presented. Two two-dimensional problems are presented which are solved by the boundary integral equation method. Application of collocation methods reduces the problems to systems of linear algebraic equations. The first problem is that of a closed surface crack in a flat slab with an AC magnetic field parallel to the plane of the crack. The second is that of av-groove crack in the AC field of a pair of parallel wires placed parallel to the vertex of the crack. In both cases, maps of the current densities at the surface are displayed, as well as the impedance changes due to the cracks.     

Construction of kernels G l, 0 l of the nonlinear collision integral in the Boltzmann equation for arbitrary l

It was shown in [1] that kernels L _l (v, v ₁) of linear collision integral and kernels G _{l, 0} ^l (v, v ₁, v ₂) of nonlinear collision integral are related by the Laplace transform. Here, analytical expressions are derived for nonlinear kernels G _{l, 0} ^+l (v, v ₁, v ₂) with arbitrary l for models of hard spheres and pseudo-Maxwellian molecules using the Laplace transform method.     

On an integral equation of viscous flow theory

Summary The integral equation encountered by van de Vooren and Veldman [1] in their study of the Knudsen region near the leading edge of a flat plate is solved by the method of Wiener and Hopf. This exact solution yields the values of certain arbitrary constants which were not determined in [1].     

On an integral equation of radiation-conduction heat transfer

We present a method for solving analytically the linearized integral equation of radiation-conduction heat transfer. Under certain conditions, realized in practice in heat conduction studies of translucent materials, a sufficiently precise solution may be obtained in closed form.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 18, No. 3, pp. 474–480, March, 1970.     

Axisymmetric formulation for boundary integral equation methods in scalar potential problems

Axisymmetric geometries often appear in electromagnetic device studies. The authors present an original formulation for Boundary Integral Equation methods in scalar potential problems. This technique requires only 2D boundary in the r-z plane and evaluation of the equations only on those boundaries.     

On integral equation approaches to the mechanics of fibre-crack interaction

The paper examines the problem of a penny-shaped crack which is formed by the development of a crack in both the fibre and the matrix of a composite consisting of an isolated elastic fibre located in an elastic matrix of infinite extent. The composite region is subjected to a uniform strain field in the direction of the fibre. The paper presents two integral-equation based approaches for the analysis of the problem. The first approach considers the formulation of the complete integral equations governing the associated elasticity problem for a two material region. The second approach considers the boundary integral equation formulation of the problem. Both methods entail the numerical solution of the governing integral equations. The solutions to these integral equations are used to evaluate the stress intensity factor at the boundary of the penny-shaped crack.     

On shape differentiation of discretized electric field integral equation

This work presents shape derivatives of the system matrix representing electric field integral equation discretized with Raviart–Thomas basis functions. The arising integrals are easy to compute with similar methods as the entries of the original system matrix. The results are compared to derivatives computed with automatic differentiation technique and finite differences, and are found to be in an excellent agreement. Furthermore, the derived formulas are employed to analyze shape sensitivity of the input impedance of a planar inverted F-antenna, and the results are compared to those obtained using a finite difference approximation.     

On the convergence of iterative solutions of the integral magnetic field equation

The solution of the integral magnetic field equation by numerical iteration is discussed. Using a simple linear example, it is shown rigorously that relaxation techniques are required to obtain convergence. The range of permissible relaxation parameters is examined and that particular value which yields most rapid convergence is determined. An iterative solution to a simple nonlinear problem is shown to converge rapidly if the relaxation parameter is adjusted appropriately at each step in the iteration. For the general case of a saturable media of complex geometric shape, a relaxation matrix method is proposed in order to achieve rapid convergence.     

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.

     

Doubly periodic volume?surface integral equation formulation for modelling metamaterials

A generalised volume-surface integral equation is extended by way of the periodic Green's function to model arbitrarily complex designs of metamaterials consisting of high-contrast inhomogeneous anisotropic material regions as well as metallic inclusions. The unique aspect of the formulation is the integration of boundary and volume integral equations to increase modelling efficiency and capability. Specifically, the boundary integral approach with equivalent surface currents is adopted over regions consisting of piecewise homogeneous materials as well as metallic perfect electric/magnetic conductor inclusions, whereas the volume integral equation is employed only in inhomogeneous and/or anisotropic material regions. Because the periodic Green's function only needs to be evaluated for the equivalent surface currents enclosing an inhomogeneous and/or anisotropic region, matrix fill time is much less as compared to using a volume formulation. Furthermore, the incorporation of curvilinear finite elements allows for greater geometrical modelling flexibility for arbitrarily shaped high-contrast regions found in typical designs of engineered metamaterials     

Application of the boundary integral equation methods to subsonic flow past bodies and wings

Summary In this paper the boundary integral equation method is applied to the subsonic flow in the presence of three-dimensional bodies or of wings. The integral equation is obtained by the aid of a source distribution on the body surface. The equation discretization is obtained via a collocation method using triangular elements. Numerical tests are performed for the case of the sphere in incompressible fluid (in this case the exact solution is known). Numerical results are also given for the ellipsoid at 0° and 10° incidence for various values of the Mach numberM. The method allows also to consider lifting wings. Numerical determinations are performed for a wing whose section is a NACA64A-008 airfoil.     

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.     

Comparison of two distribution-free methods of integral error characteristic evaluation

Two methods are compared for determining confidence ranges for error characteristics: the bootstrap and l_p ones.Translated from Izmeritel'naya Tekhnika, No. 1, pp. 8–10, January, 1994.