Boundary Galerkin's method for three-dimensional finite element electromagnetic field computation

Electromagnetic field problems are often formulated as boundary value problems in unbounded regions. For this reason, the application of conventional numerical methods, such as the finite element method, is difficult. The paper describes a new technique to circumvent this difficulty. The technique is based on the reduction of the field equations in unbounded space to equivalent boundary Galerkin's criterion. Such criterion can be combined with the volume Galerkin's criterion for regions occupied by conductors. A new quasi-finite-element discretization based on the coupled boundary/volume Galerkin's criterion is presented.     

Boundary elements for three-dimensional elastic crack analysis

In this paper 8-node traction singular boundary elements are employed to represent displacement and traction variations in the vicinity of the crack front in three-dimensional geometries. The numerical procedure suggested for evaluating the singular integrals extending over these special elements is described. The efficiency and accuracy of the special elements and integration procedure are demonstrated by the results obtained in a simple test problem whose analytical solution is known. The interaction of two circular coplanar cracks embedded in an infinite medium under uniform tension loading is also analysed. Finally, the stress intensity factor variation computed for a semi-circular inner surface crack in a pressurized cylinder is presented.     

Boundary element analysis of three-dimensional crack problems in two joined transversely isotropic solids

The present paper presents a boundary element analysis of penny-shaped crack problems in two joined transversely isotropic solids. The boundary element analysis is carried out by incorporating the fundamental singular solution for a concentrated point load in a transversely isotropic bi-material solid of infinite space into the conventional displacement boundary integral equations. The conventional multi-region method is used to analyze the crack problems. The traction-singular elements are employed to capture the singularity around the crack front. The values of the stress intensity factors are obtained by using crack opening displacements. The numerical scheme results are verified with the closed-form solutions available in the literature for a penny-shaped crack parallel to the plane of the isotropy of a homogeneous and transversely isotropic solid of infinite extent. The new problem of a penny-shaped crack perpendicular to the interface of a transversely isotropic bi-material solid is then examined in detail. The crack surfaces are subject to the three normal tractions and the uniform shear traction. The associated stress intensity factor values are obtained and analyzed. The present results can be used for the prediction of the stability of composite structures and the hydraulic fracturing in deep rock strata and reservoir engineering.     

Finite element analysis of 3-D eddy currents

The authors review formulations of three-dimensional (3-D) eddy current problems in terms of various magnetic and electric potentials. The differential equations and boundary conditions are formulated to include the necessary gauging conditions and thus to ensure the uniqueness of the potentials. Different sets of potentials can be used in distinct subregions, thus facilitating an economic treatment of various types of problems. A novel technique for interfacing conducting regions with an electric vector and a magnetic scalar potential to eddy-current-free regions with a magnetic vector potential is described. Finite-element solutions to several large eddy-current problems are presented     

A singular element for three-dimensional fracture mechanics analysis

In this article, a singular boundary element for three-dimensional fracture mechanics analysis is presented. It is a nine-node quadratic element with plane geometry. These nodes are located at one quarter of the distance between two opposite sides of the element. Shape functions with a 11/√r singularity at the crack front are used to represent the tractions. The Stress Intensity Factors are computed as system unknowns appearing (except for a constant) as traction nodal values. Special attention is paid to the development of a simple and accurate integration approach for this singular element. The accuracy of the results obtained with the proposed element is demonstrated by solving several crack problems including edge and embedded cracks with different geometries. The element can be easily implemented and incorporated into existing quadratic boundary element codes. In a companion paper the element is formulated and used for fracture mechanics problems in transversely isotropic materials. Extension to other fields for which boundary element formulations exist, is quite simple.     

Boundary element crack closure calculation of three-dimensional stress intensity factors

A general method for boundary element-crack closure integral calculation of three-dimensional stress intensity factors is presented. An equation for the strain energy release rate in terms of products of nodal values of tractions and displacements is obtained. Embedded and surface cracks of modes I, II, and III are analyzed using the proposed method. The multidomain boundary element technique is introduced so that the crack surface geometry is correctly modeled and the unsymmetrical boundary conditions for mode's II and III crack analysis are handled conveniently. Conventional quadrilateral elements are sufficient for this method and the selection of the size of the crack front elements is independent of the crack mode and geometry. For all of the examples demonstrated in this paper, 54 boundary elements are used, and the most suitable ratio of the width of the crack front elements to the crack depth is 1/10 and the calculation error is kept within ±1.5 percent. Compared to existing analytical and finite element solutions the boundary element-crack closure integral method is very efficient and accurate and it can be easily applied to general three-dimensional crack problems.     

Boundary element analysis of dielectric waveguides

We apply the boundary element method to the analysis of optical waveguides. After summarizing constant and linear element algorithms for both two- and three-dimensional simulations, we introduce a new recursive series procedure for constructing the diagonal matrix elements. We then demonstrate that our method can be employed to minimize the reflectivity of optical waveguide antireflection coatings with both straight and angled facets.     

Boundary element analysis of Navier-Stokes equations

As arrangements, the fundamental solutions of anisotropic convective diffusion equations of transient incompressible viscous fluid flow and boundary elements analysis of the diffusion equation are presented. Secondly, by considering that convective diffusion equations and Navier-Stokes equations are mathematical formulations of mass and momentum conservation law respectively, and that consequently, both physical contents and equation styles are analogous, boundary integral formulations for Navier-Stokes equations are proposed on the basis of formulation of diffusion equations.     

A hybrid boundary element method for three-dimensional fracture analysis

At first, a hybrid boundary element method used for three-dimensional linear elastic fracture analysis is established by introducing the relative displacement fundamental function into the first and the second kind of boundary integral equations. Then the numerical approaches are presented in detail. Finally, several numerical examples are given out to check the proposed method. The numerical results show that the hybrid boundary element method has a very high accuracy for analysis of a three-dimensional stress intensity factor.     

Boundary element analysis of the electromagnet casting mould

A coupled boundary element (BE)-impedance boundary condition (IBC) formulation is used to develop an electromagnetic mold design model, taking into consideration the problem of predicting the shape of the free boundary. The computational model of the electromagnetic mold consists of a right cylindrical conductor, representing the molten metal, coaxial with an induction coil, generally having a single turn, and a field shaping short circuited turn. The IBC is applied to the high-resistivity molten metal region, and the full BE formulation is used for the field shaping turn. An iterative procedure is developed to predict the shape of the molten metal free surface by satisfying the balance between the electromagnetic and gravitational pressures. Results showing the effects of the single turn confinement coil and shielding turn placement on the electromagnetic pressure distributions and on the shape of the free surface are presented     

A three-dimensional least-squares finite element technique for deformation analysis

The development of a three-dimensional least-squares finite element technique suitable for deformation analysis was presented. By adopting a spatial viewpoint, a consistent rate formulation that treats deformation as a process was established. The technique utilized the least-squares variational principle that minimizes the squares of errors encountered in any attempt to meet the field equations exactly. Both velocity and Cauchy stress rate fields were discretized by the same linear interpolation function. The discretization always yields a sparse, symmetric, and positive-definite coefficient matrix. A conjugate gradient iterative solver with incomplete-Choleski preconditioner was used to solve the resulting linear system of equations. Issues such as finite element formulation, mesh design, code efficiency, and time integration were addressed. A set of linear elastic problems was used for patch-test; both homogeneous and non-homogeneous deformations were considered. Additionally, two finite elastic deformation problems were analysed to gauge the overall performance of the technique. The results demonstrated the computational feasibility of a three-dimensional least-squares finite element technique for deformation analysis. © 1997 John Wiley & Sons, Ltd.     

Finite element dispersion analysis for the three-dimensional second-order scalar wave equation

The dispersive properties of finite element semidiscretizations of the three-dimensional second-order scalar wave equation are examined for both plane and spherical waves. This analysis throws light on the performance and limitations of the finite element approximation over the entire spectrum of wavenumbers and provides guidance for optimal mesh discretization as well as mass representation. The 8-node trilinear element, 20-node serendipity element, 27-node triquadratic element and the linear and quadratic spherically symmetric elements are considered.     

Boundary element analysis of inclusions with corners

Boundary element method (BEM) has proven to have very good resolution of large stress gradients such as those that may arise at material interface and reentrant corners. There is, however, a paucity of literature in usage of BEM when the inclusion has a corner. The stress singularity at the corner creates numerical difficulties that need to be addressed. This paper describes: application of BEM to inclusion with and without corners; the numerical modeling difficulties; a methodology for calculation of eigenvalues and stress intensity factors without elaborate analytical expressions; and the future research that is needed for the growth of the boundary element methodology for application to inclusion problems. Numerical results for a rectangular inclusion with sharp and fillet corners that in the limit becomes a circular inclusion demonstrate the potential of the proposed methodology in the analysis of inclusion problems.     

Boundary element analysis of thermoelastic contact problems

A boundary element method (BEM) is applied to thermoelastic contact problems where thermal resistance at the contact interface is not negligible. The displacement, traction, temperature and temperature gradient in the contact zone are unknown quantities to be determined numerically. Due to the existence of thermal resistance, temperature and stress fields are mutually coupled. To solve the problem, two kinds of methods are presented. In the first method, the solution is obtained by minimizing a suitably defined objective function. In the second method, discretized equations of each of the bodies in contact are computed alternately until all prescribed boundary conditions are satisfied. The applicability of these methods to practical problems is examined through several numerical examples.     

Boundary element analysis of anisotropic rock masses

A boundary element formulation is developed for anisotropic elastic rock masses. The boundary element treatment in which the fundamental solutions of Lekhnitskii have been incorporated, and the numerical evaluation of integrals with singularities are discussed. Good agreement found between the numerical and analytical solutions for several example problems demonstrates the capability, accuracy and efficiency of the present formulation. The problem of a deep circular tunnel excavated in a variety of jointed rock masses has also been analysed using the present formulation. The effect of the jointing on the behaviour of the rock mass around the tunnel is evaluated.     

Shape functions and integration formulas for three-dimensional finite element analysis

Shape functions and numerical integration formulas for three-dimensional finite element analysis as found in most finite element reference books are incomplete. For example, shape functions and integration formulas for a pyramid with a quadrilateral base are missing. It is also difficult to find symmetric higher-order integration formulas for triangular and tetrahedral elements. In general, these shape functions and integration formulas cannot be satisfactorily derived as degenerate cases of shape functions and integration formulas for hexahedral (brick) elements. In this paper we present C°-continuous quadrilateral pyramid elements and integration formulas for two- and three-dimensional elements.     

A mixed hybrid finite element for three-dimensional elastic crack analysis

A new three-dimensional crack tip element is proposed, which is based on a mixed hybrid stress/displacement model. A truncated series expansion of eigenfunctions for the straight semi-infinite crack is deduced and assumed for the internal stress and displacement fields in the element. The basic approach of constructing these hybrid elements is outlined. Their good capability, efficiency and accuracy for analyzing three-dimensional elastic crack problems are demonstrated by first numerical examples.

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Résumé On propose un nouveau type d'élément tridimensionnel pour l'extrémité d'une fissure, basé sur un modèle mixte contraintes hybrides/déplacements. On en tire un développement en séries tronquées des eigenfonctions relatives à une fissure droite semi-infinie, et on suppose qu'elle est représentative des champs de contraintes internes et de déplacements dans l'élément. L'approche de base utilisée pour construire ces éléments hybrides est soulignée. On démontre par de premiers exemples numériques qu'ils ont la capacité, l'efficacité et la précision nécessaires à l'analyse des problèmes élastiques et tridimensionnels de fissuration.

     

Boundary element formulation for the coupled stretching-bending analysis of thin laminated plates

The boundary integral equations for the coupled stretching-bending analysis of thin laminated plates involve an integral which will be singular when the field point approaches the source point. To avoid the singular problem occurring in the numerical programming, the boundary integral equations are modified in which the integrals of singular part are integrated analytically. The analytical solutions for the free term coefficients and singular integrals are obtained in explicit closed-form. By dividing the boundary into elements and using suitable interpolation polynomials for basic functions, the set of equations necessary for boundary element programming are written explicitly for regular nodes and corner nodes. The equations for the determination of displacements and stresses at internal points are also presented in this paper.     

Element by element a posteriori error estimation of the finite element analysis for three-dimensional elastic problems

An a posteriori error estimation method for finite element solutions for three-dimensional elastic problems is presented based on the theory developed by the authors for two-dimensional problems.¹ The error is estimated for the finite element solutions obtained using three-dimensional 8-node elements with a linear interpolation function in an arbitrary hexahedron. The method is successfully applied to three-dimensional elastic problems. In order to decrease computing time and memory use, the error is estimated element by element. The major difficulty in the element-wise error estimation technique is satisfying the self-equilibrium condition of applied forces, especially in three-dimensional problems. These forces are mainly due to traction discontinuity on the element boundaries. The difficulty is circumvented by employing an element-wise optimal procedure. It is also shown that a very accurate stress solution can be obtained by adding estimated error to the original finite element solutions.