An integral equation method for the solution of 3-dimensional,non-linear,magnetostatic problems

A finite element method for computing the resultant magnetic field arising from a given source field in the presence of a magnetic material of variable permeability is described; in this method finite element approximations to the scalar potential of the resulting field and the magnetic susceptibility, in the region occupied by the magnetic material, are determined from the non-linear integral equation for the scalar potential and the constitutive susceptibility relation, using a collocation scheme. The method is used to compute the shielding effect of a thin rectangular plate of variable permeability on a given source field. The plate is subdivided uniformly into brick elements; the resulting translational invariance of the integrals required in the calculations is exploited to achieve major computational savings. A consequence of the thinness of the plate is that the calculation of the requisite integrals by analytic methods leads to considerable loss of accuracy by differencing; this difficulty is overcome by using a scheme which combines both analytic and quadrature techniques. The resulting system of non-linear algebraic equations is solved by Powell's hybrid method; an efficient scheme for calculating an initial approximation to the Jacobian, which utilizes the structure of the equations, is presented. The results of the calculations are discussed.     

Numerical solution of the variation boundary integral equation for inverse problems

In this paper a procedure to solve the identification inverse problems for two‐dimensional potential fields is presented. The procedure relies on a boundary integral equation (BIE) for the variations of the potential, flux, and geometry. This equation is a linearization of the regular BIE for small changes in the geometry. The aim in the identification inverse problems is to find an unknown part of the boundary of the domain, usually an internal flaw, using experimental measurements as additional information. In this paper this problem is solved without resorting to a minimization of a functional, but by an iterative algorithm which alternately solves the regular BIE and the variation BIE. The variation of the geometry of the flaw is modelled by a virtual strainfield, which allows for greater flexibility in the shape of the assumed flaw. Several numerical examples demonstrate the effectiveness and reliability of the proposed approach. Copyright © 2000 John Wiley & Sons, Ltd.     

An improved boundary integral equation method for Helmholtz problems

The eigenvalue problem for the Laplace operator is numerical investigated using the boundary integral equation (BIE) formulation. Three methods of discretization are given and illustrated with numerical examples.     

An integral equation method for dynamic crack growth problems

Within the assumptions of linear elastic fracture mechanics, dynamic stresses generated by a crack growth event are examined for the case of an infinite body in the state of plane strain subjected to mode I loading.The method of analysis developed in this paper is based on an integral equation in one spatial coordinate and in time. The kernel of this equation, i.e., the influence or Green's function, is the response of an elastic half-space to a concentrated unit impulse acting on its edge. The unknown function is the normal stress distribution in the plane of the crack, while the free term represents the effect of external loading.The solution for the stresses is obtained with the assumption that its spatial distribution contains a square root singularity near the tip of the crack, while its intensity is an unknown function of time. Thus, the orginal integral equation in space and time reduces to Volterra's integral equation of the first kind in time. The equation is singular, with the singularity of the kernel being a combined effect of the singularity of the influence function and the singularity of the dynamic stresses at the tip of the crack. Its solution is obtained numerically with the aid of a combination of quadrature and product integration methods. The case of a semi-infinite crack moving with a prescribed velocity is examined in detail.The method can be readily extended to problems involving mode II and mixed mode crack propagation as well as to problems of dynamic external loadings.     

Efficient integral equation method for the solution of 3-D magnetostatic problems

Nonlinear three-dimensional (3-D) magnetostatic field problems are solved using integral equation methods (IEM). Only the nonlinear material itself has to be discretized. This results in a system of nonlinear equations with a relative small number of unknowns. To keep computational costs low the fully dense system matrix is compressed with the fast multipole method. The accuracy of the applied indirect IEM formulation is improved significantly by the use of a difference field concept and a special treatment of singularities at edges. An improved fixed point solver is used to ensure convergence of the nonlinear problem.     

An alternative integral equation approach for curved and kinked cracks

An integral-equation representation of cracks is presented which differs from the well-known dislocation-layer representation in that the equations are written in terms of the displacement-discontinuity across the crack surfaces rather than derivatives of the displacement-discontinuity. The advantages of such a representation is that, unlike the dislocation layer, the displacement discontinuity is not singular at crack tips and kinks. The method is demonstrated for two-dimensional infinite domains.

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Résumé On propose une représentation de fissures par une équation intégrale qui diffère de la représentation bien connue des couches de dislocation en ce que les équations constitutives sont écrites en termes de discontinuité de déplacement au travers de la section de la fissure, plutôt que dérivées de ces discontinuités de déplacement. L'avantage de cette représentation est que, à la différence de la couche de dislocation, le discontinuité de déplacement n'est pas singulière à l'extrémité ou aux branchements de la fissure. La méthode est soumise à démonstration pour des domaines infinis à deux dimensions.

     

Boundary integral equation solution of moving boundary phase change problems

Boundary integral equation methods are presented for the solution of some two-dimensional phase change problems. Convection may enter through boundary conditions, but cannot be considered within phase boundaries. A general formulation based on space-time Green's functions is developed using the complete heat equation, followed by a simpler formulation using the Laplace equation. The latter is pursued and applied in detail. An elementary, noniterative system is constructed, featuring linear interpolation over elements on a polygonal boundary. Nodal values of the temperature gradient normal to a phase change boundary are produced directly in the numerical solution. The system performs well against basic analytical solutions, using these values in the interphase jump condition, with the simplest formulation of the surface normal at boundary vertices. Because the discretized surface changes automatically to fit the scale of the problem, the method appears to offer many of the advantages of moving mesh finite element methods. However, it only requires the manipulation of a surface mesh and solution for surface variables. In some applications, coarse meshes and very large time steps may be used, relative to those which would be required by fixed grid domain methods. Computations are also compared to original lab data, describing two-dimensional soil freezing with a time-dependent boundary condition. Agreement between simulated and measured histories is good.     

An integral equation formulation of plate bending problems

Summary The mathematical theory of thin elastic plates loaded by transverse forces leads to biharmonic boundary value problems. These may be formulated in terms of singular integral equations, which can be solved numerically to a tolerable accuracy for any shape of boundary by digital computer programs. Particular attention is devoted to clamped and simply-supported rectangular plates. Our results indicate support for the generally accepted treatment of such plates and for the intuitive picture of deflection behaviour at a corner.     

An integral equation approach to eddy-current calculations

An integral-equation approach has been used to solve eddy current problems. The conducting material is represented by a network of current-carrying line elements. Consequently, Maxwell's field equations can be replaced by Kirchhoff's circuit rules. The loop equations for voltages, supplemented by the node equations for the currents, comprise a set of linear equations that can be solved repeatedly to give the time development of the eddy currents. Currents, magnetic fields, and power are calculated at each step. For a two-dimensional geometry, either thin plates or infinite cylinders can be calculated. Rectangular and circular cross sections have been calculated with good agreement to analytical expressions. Thin curved shells have also been calculated.     

The boundary integral equation method for the solution of problems involving elastic slabs

Three boundary integral equations for the solution of an important class of elastic slab problems are considered. Some numerical examples are examined in order to illustrate the application of the integral equations to particular boundary-value problems.     

Uncoupled superposition of finite element and integral equation methods for the solution of crack problems

A direct superposition procedure is used in order to solve the problem of a generally cracked body: The regular field is generated by the finite element method whereas the singular field is by the singular integral equation method. This process, combined in a one-step technique, was applied to three typical plane problems of cracked sheets and the results are compared with those from other numerical methods.

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Résumé On utilise une procédure de superposition directe en vue de résoudre le problème d'un corps fissuré de manière complète. La méthode des éléments finis donne lieu à un champ régulier, tandis qu'un champ singulier résulte de l'application de la méthode des intégrales singulières. Combinée à une technique de ressaut simple, cette procédure a été appliquée à trois problèmes typiques plans de feuillards fissurés, et les résultats en sont comparés avec ceux que fournissent d'autres méthodes numériques.

     

An integral equation method for non-self-adjoint eigenvalue problems and its applications to non-conservative stability problems

The aim of the work reported in this paper is to present the new formulation of the integral equation method for non-self-adjoint problems and to apply the method to stability problems of elastic continua subjected to non-conservative loadings. A general non-self-adjoint eigenvalue problem stated in terms of differential operators is transformed into a set of coupled integral equations. Our derivation of integral equations is based on an inverse formulation of a canonical form for the original problem and the corresponding fundamental solution pair. Three well-known non-conservative stability problems in elasticity are examined by this integral equation method as illustrative examples. The approximate values of the critical parameters of sample problems demonstrate a sufficient accuracy through a comparison of other values.     

The first-kind and the second-kind boundary integral equation systems for solution of frictionless contact problems

An original approach to the numerical solution of displacement boundary integral equation (BIE) and traction hypersingular boundary integral equation (HBIE) by the boundary element method (BEM) for contact problems is given. The main point is to show, how the contact conditions are used to formulate the first-kind and the second-kind BIE systems in the case of frictionless two-body elastic contact. The solution of the first-kind BIE is performed by symmetric Galerkin BEM; the second-kind BIE is solved by an appropriate collocation BEM. The contact problem in itself is solved by the method of subsequent approximations of contact region. Both forms of BIE system are compared in several numerical examples. This comparison is made for different kinds of contact problem. The major emphasis is put on the evaluation of contact pressure. The obtained results are compared with referenced numerical and with the analytical ones.     

The DRM‐MD integral equation method: an efficient approach for the numerical solution of domain dominant problems

This work presents a multi‐domain decomposition integral equation method for the numerical solution of domain dominant problems, for which it is known that the standard Boundary Element Method (BEM) is in disadvantage in comparison with classical domain schemes, such as Finite Difference (FDM) and Finite Element (FEM) methods. As in the recently developed Green Element Method (GEM), in the present approach the original domain is divided into several subdomains. In each of them the corresponding Green's integral representational formula is applied, and on the interfaces of the adjacent subregions the full matching conditions are imposed. In contrast with the GEM, where in each subregion the domain integrals are computed by the use of cell integration, here those integrals are transformed into surface integrals at the contour of each subregion via the Dual Reciprocity Method (DRM), using some of the most efficient radial basis functions known in the literature on mathematical interpolation. In the numerical examples presented in the paper, the contour elements are defined in terms of isoparametric linear elements, for which the analytical integrations of the kernels of the integral representation formula are known. As in the FEM and GEM the obtained global matrix system possesses a banded structure. However in contrast with these two methods (GEM and non‐Hermitian FEM), here one is able to solve the system for the complete internal nodal variables, i.e. the field variables and their derivatives, without any additional interpolation. Finally, some examples showing the accuracy, the efficiency, and the flexibility of the method for the solution of the linear and non‐linear convection–diffusion equation are presented. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical solution of Lippmann-Schwinger integral equation

A method is discussed for obtaining a numerical solution of an equation similar to the equation for the transport of radiant energy for a steady radiation field [1].Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 33, No. 2, pp. 329–331, August, 1977.     

An improved form of the hypersingular boundary integral equation for exterior acoustic problems

An improved form of the hypersingular boundary integral equation (BIE) for acoustic problems is developed in this paper. One popular method for overcoming non-unique problems that occur at characteristic frequencies is the well-known Burton and Miller (1971) method [7], which consists of a linear combination of the Helmholtz equation and its normal derivative equation. The crucial part in implementing this formulation is dealing with the hypersingular integrals. This paper proposes an improved reformulation of the Burton–Miller method and is used to regularize the hypersingular integrals using a new singularity subtraction technique and properties from the associated Laplace equations. It contains only weakly singular integrals and is directly valid for acoustic problems with arbitrary boundary conditions. This work is expected to lead to considerable progress in subsequent developments of the fast multipole boundary element method (FMBEM) for acoustic problems. Numerical examples of both radiation and scattering problems clearly demonstrate that the improved BIE can provide efficient, accurate, and reliable results for 3-D acoustics.     

Solution of periodic group crack problems by using the Fredholm integral equation approach

Summary Periodic group cracks composed of infinitely many groups numbered from j = -∞,...-2,-1,0,1,2,...to j = ∞ placed periodically in an infinite plate are studied. The same loading condition and the same geometry are assumed for cracks in all groups. The Fredholm integral equation is formulated for the cracks of the central group (or the 0-th group) collecting the influences from the infinite neighboring groups. The influences from many neighboring groups on the central group are evaluated exactly, and those from remote groups approximately summed up into one term. The stress intensity factors can be directly evaluated from the solution of the Fredholm integral equation. Numerical examples show that the suggested technique provides very accurate results. Finally, several numerical examples are presented, and the interaction between the groups is addressed.     

Solutions of multiple crack problems of a circular plate or an infinite plate containing a circular hole by using Fredholm integral equation approach

Presented is an elementary solution, which is a particular solution of the circular plate containing one crack. The solution consists of two parts and satisfies the following conditions: (i) the first part corresponds to a pair of normal and tangential concentrated forces acting at a prescribed point on both edges of a single crack; (ii) the second part corresponds to some distributed tractions along both edges of the crack; (iii) the obtained elementary solution, i.e. the sum of the first and second parts, satisfies a traction free condition on the circular boundary. Using this elementary solution and taking some undetermined density of the elementary solution along each crack, a system of Fredholm integral equations of multiple crack problems can always be obtained. The multiple crack problems of an infinite plate containing a circular hole can be solved in a similar way. Several numerical examples are given in this paper.     

A Grid-based integral approach for quasilinear problems

For non-homogeneous and nonlinear problems, a major difficulty in applying the Boundary Element Method is the treatment of the volume integrals that arise. A recent proposed method, the grid-based integration method (GIM), uses a 3-D uniform grid to reduce the complexity of volume discretization, i.e., the discretization of the whole domain is avoided. The same grid is also used to accelerate both surface and volume integration. The efficiency of the GIM has been demonstrated on 3-D Poisson problems. In this paper, we report our work on the extension of this technique to quasilinear problems. Numerical results of a 3-D Helmholtz problem and a quasilinear Laplace problem on a multiply-connected domain with Dirichlet boundary conditions are presented. These results are compared with analytic solutions. The performance of the GIM is measured by plotting the L₂-norm error as a function of the overall CPU time and is compared with the auxiliary domain method in the Helmholtz problem.