Oscillation limits for weighted residual methods applied to convective diffusion equations

Convection–diffusion equations are difficult to solve when the convection term dominates because most solution methods give solutions which oscillate in space. Previous criteria based on the one-dimensional convection–diffusion equation have shown that finite difference and Galerkin (linear or quadratic basis functions) will not give oscillatory solutions provided the Peclet number times the mesh size (Pe Δx) is below a critical value. These criteria are based on the solution at the nodes, and ensure that the nodal values are monotone. Similar criteria are developed here for other methods: quadratic Galerkin with upwind weighting, cubic Galerkin, orthogonal collocation on finite elements with quadratic, cubic or quartic polynomials using Lagrangian interpolation, cubic or quartic polynominals using Hermite interpolation, and the method of moments. The nodal values do not oscillate for collocation or moments methods with Hermite cubic polynomials regardless of the value of Pe Δx. A new criterion is developed for all methods based on the monotonicity of the solutions throughout the domain. This criterion is more restrictive than one based only on the nodal values. All methods that are second order (Δx²) or better in truncation error give oscillatory solutions (based on the entire domain) unless Pe Δx is below a critical value. This value ranges from 2 for finite difference methods to 4·6 for Hermite, quartic, collocation methods.     

Eigenstrain formulation of boundary integral equations for modeling particle-reinforced composites

A novel computational model is presented using the eigenstrain formulation of the boundary integral equations for modeling the particle-reinforced composites. The model and the solution procedure are both resulted intimately from the concepts of the equivalent inclusion of Eshelby with eigenstrains to be determined in an iterative way for each inhomogeneity embedded in the matrix. The eigenstrains of inhomogeneity are determined with the aid of the Eshelby tensors, which can be readily obtained beforehand through either analytical or numerical means. The solution scale of the inhomogeneity problem with the present model is greatly reduced since the unknowns appear only on the boundary of the solution domain. The overall elastic properties are solved using the newly developed boundary point method for particle-reinforced inhomogeneous materials over a representative volume element with the present model. The effects of a variety of factors related to inhomogeneities on the overall properties of composites as well as on the convergence behaviors of the algorithm are studied numerically including the properties and shapes and orientations and distributions and the total number of particles, showing the validity and the effectiveness of the proposed computational model.     

An integral equation formulation of anti-plane inhomogeneities

In this paper, a boundary integral equation formulation for anti-plane shear inhomogeneous medium is presented to study the interaction between the inhomogeneities and cracks. The proposed boundary integral equation formulation only contains out-of-plane interface displacements and out-of-plane discontinuous displacements over cracks. Numerical implementation is simple since the present formulation has considered the shear equilibrium condition over the interfaces between the matrix and inhomogeneities. Out-of-plane interface displacements and out-of-plane traction integral equations are collocated respectively on the matrix–inhomogeneity interfaces and on one side of the crack surface. Numerical examples are given to show the validity and numerical accuracy of the present method.     

Galerkin formulation and singularity subtraction for spectral solutions of boundary integral equations

A new spectral Galerkin formulation is presented for the solution of boundary integral equations. The formulation is carried out with an exact singularity subtraction procedure based on analytical integrations, which provides a fast and precise way to evaluate the coefficient matrices. The new Galerkin formulation is based on the exact geometry of the problem boundaries and leads to a non-element method that is completely free of mesh generation. The numerical behaviour of the method is very similar to the collocation method; for Dirichlet problems, however, it leads to a symmetric coefficient matrix and therefore requires half the solution time of the collocation method. © 1998 John Wiley & Sons, Ltd.     

Boundary integral equations as applied to an oscillating bubble near a fluid-fluid interface

A new method is presented to describe the behaviour of an oscillating bubble near a fluid-fluid interface. Such a situation can be found for example in underwater explosions (near muddy bottoms) or in bubbles generated near two (biological) fluids separated by a membrane. The Laplace equation is assumed to be valid in both fluids. The fluids can have different density ratios. A relationship between the two velocity potentials just above and below the fluid-fluid interface can be used to update the co-ordinates of the new interface at the next time step. The boundary integral method is then used for both fluids. With the resulting equations the normal velocities on the interface and the bubble are obtained. Depending on initial distances of the bubble from the fluid-fluid interface and density ratios, the bubbles can develop jets towards or away from this interface. Gravity can be important for bubbles with larger dimensions.     

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 formulation of two- and three-dimensional nanoscale inhomogeneities

An integral equation formulation of two- and three-dimensional infinite isotropic medium with nanoscale inhomogeneities is presented in this paper. The Gurtin–Murdoch interface constitutive relation is used to model the continuity conditions along the internal interfaces between the matrix and inhomogeneities. The Poisson’s ratios of both the matrix and inhomogeneities are assumed to be the same. The proposed integral formulation only contains the unknown interface displacements and their derivatives. In order to solve the nanoscale inhomogeneities, the displacement integral equation is used when the source points are acting on the interfaces between the matrix and inhomogeneities. Thus, the resulting system of equations can be formulated so that the interface displacements can be obtained. Furthermore, the stresses at points being in the matrix and nanoscale inhomogeneities can be calculated using the stress integral equation formulation. Numerical results from the present method are in good agreement with those from the conventional sub-domain boundary element method and the analytical method.     

A new iterative integral formulation for semilinear equations based on the generalized quasilinearization theory

This paper presents a new iterative integral approach for solving semilinear equations. The integral formulation is derived based on the generalized quasilinearization theory in which nonlinear equations are replaced by a set of iterative linear equations. An advantage of the new formulation is that its convergence is guaranteed under a given condition and the convergence rate can be quadratic. The effectiveness of the new approach has been demonstrated on several examples of the nonlinear Poisson type. Comparisons with some existing methods and a study of the convergence rate have also been conducted in this work.     

Three-dimensional unsteady transonic flow: An integral equation formulation

The unsteady transonic small perturbation differential equation has been converted into an integro-differential equation by application of the classical Green's function method. After assuming that the motion consists of infinitesimal perturbations around a thin, nearly-planar body a simplified integral equation is obtained for the streamwise velocity component, which is suitable for fast numerical computations.     

An eddy current integral formulation on parallel computer systems

In this paper, we show how an eddy current volume integral formulation can be used to analyse complex 3D conducting structures, achieving a substantial benefit from the use of a parallel computer system. To this purpose, the different steps of the numerical algorithms in view of their parallelization are outlined to enlighten the merits and the limitations of the proposed approach. Numerical examples are developed in a parallel environment to show the effectiveness of the method. Copyright © 2004 John Wiley & Sons, Ltd.     

Effective numerical treatment of boundary integral equations: A formulation for three-dimensional elastostatics

The field equations of three-dimensional elastostatics are transformed to boundary integral equations. The elastic body is divided into subregions, and the surface and interfaces are represented by quadrilateral and triangular elements with quadratic variation of geometry and linear, quadratic or cubic variation of displacement and traction with respect to intrinsic co-ordinates. The integral equation is discretized for each subregion, and a system of banded form obtained. For the integration of kernel-shape function products, Gaussian quadrature formulae are chosen according to upper bounds for error in terms of derivatives of the integrands. Use of the integral formulation is illustrated by the analysis of a prestressed concrete nuclear reactor pressure vessel.     

A numerical method of solving Fredholm integral equations applied to current distribution and inductance in parallel conductors

If the resistance of a conductor is negligible compared with the reactance and if radiation effects are ignored the inductance and current distribution may be obtained by using perfectly-conducting models. In this paper the distribution of current in a system of infinitely-long, perfectly-conducting straight conductors of arbitrary cross-section is shown to satisfy an integral equation of Fredholm type and a general digital procedure for solving this equation, and hence for determining inductance, is given. The relative advantages of stepped, piecewise-linear and piecewise quadratic approximations to the current distribution are studied using arrangements of strip conductors and of isolated rectangular conductors having known current distributions. The advantageous effect of varying the width of the sections used in the computation is also established. It is shown that inductance estimates accurate to within 0·1 per cent can be obtained with a relatively small number of sections and that for a large number of sections the inductance converges on the theoretical value. The paper also examines current distribution and inductance for ‘go-and return’ systems of rectangular conductors, as well as for two paralleled conductors with remote return, and compares the inductances obtained with previous derivations.     

An initial value method for dual integral equations

An initial value method is derived for a set of dual integral equations encountered in solving mixed boundary value problems in mathematical physics with a circular line of separation of boundary conditions. It is shown that the solution itself, not just a transform of the solution, of the dual integral equations satisfies a Fredholm integral equation. The initial value problem is derived from this Fredholm equation.     

An efficient method for solving diffusion equations

An algorithm for solving diffusion equations is described which has an arbitrary order of approximation in space variables, i.e., its accuracy is the higher, the smoother the solution in space variables. This provides an advantage over difference methods, which have a fixed order of approximation in space variables irrespective of the solution smoothness. In practice, this allows one (with smooth initial data) to carry out calculations on coarse space grids by an explicit scheme with applicable time steps. The method is competitive with difference methods in speed and the amount of information stored. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 70, No. 2, pp. 314–317, March–April, 1997.     

An extension of Mercer's theorem to a class of non-symmetric integral equations

Summary In this paper it is shown that certain properties of selfadjoint boundary value problems with positive eigenvalues, like the countability of the discrete eigenvalues and the expand-ability of the problem's Green function, are presevred if the problem is perturbed and becomes non-selfadjoint through an additional term involving a positive, continuously varying parameter. This is so, if the parameter remains small enough and does not transgress the so called stability domain of the problem.

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Eine Erweiterung von Mercers Theorem auf eine Gruppe nichtsymmetrischer Integralgleichungen

Zusammenfassung In dieser Arbeit wird gezeigt, daß gewisse Eigenschaften von selbstadjungierten Randwertproblemen mit positiven Eigenwerten erhalten bleiben, wenn das Problem gestört wird, indem man ein nicht-selbstadjungiertes Glied in der Differentialgleichung hinzufügt, das einen positiven, kontinuierlichen Parameter enthält. Eigenschaften solcher Art sind: Abzählbarkeit der diskreten Eigenwerte und Entwickelbarkeit der Greenschen Funktion. Sie bleiben erhalten, wie behauptet, wenn der Parameter in dem Störungsglied klein genug bleibt und nicht den sogenannten Stabilitätsbereich des Problems verläßt.

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With 3 Figures     

An alternative analytical formulation for the Voigt function applied to resonant effects in nuclear processes

The Voigt function H(a,v) is defined as the convolution of the Gaussian and Lorentzian functions. Recent papers puplished in different areas of physics emphasize the importance of the fast and accurate calculation of the Voigt function for different orders of magnitude of variables a and v. An alternative analytical formulation for the Voigt function is proposed in this paper. This formulation is based on the solution of the non-homogeneous ordinary differential equation, satisfied by the Voigt function, using the Frobenius and parameter variation methods. The functional form of the Voigt function, as proposed, proved simple and precise. Systematic tests are accomplished demonstrating some advantages with other existent methods in the literature and with the numeric method of reference.     

Application of the singularity method for the formulation of plane elastostatical boundary value problems as integral equations

Summary A method for the formulation of elastostatical boundary value problems as integral equations is presented, the basic idea of which consists of superimposing in a suitable fashion singular solutions for the infinite medium. Since mechanical aspects play an important role in the concept of the method, all quantities in the equations can be interpreted physically. The applicability of the method is illustrated by examples of the geometrical and statical boundary value problem ofplane elastostatics for which 32 different formulations as integral equations are established.The second aim of the paper consists of revealing an analogy between the most important notions of the singularity method, viz. between state variables and singularities. The analogy is manifested by certain symmetries of influence functions, and enables the systematical representation of the basic relations and their interpretation within a larger context.

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Anwendung der Singularitätenmethode zur Formulierung von Randwertproblemen der ebenen Elastostatik als Integralgleichungen

Zusammenfassung Es wird eine Methode zur Formulierung von Randwertproblemen der Elastostatik als Integralgleichungen beschrieben, deren Grundgedanke darin besteht, singuläre Lösungen für das unendliche Medium in geeigneter Weise zu überlagern. Da beim Aufstellen der Gleichungen mechanische Gesichtspunkte im Vordergrund stehen, lassen sich alle auftretenden mathematischen Größen physikalisch deuten. Die Anwendbarkeit der Methode wird anhand des geometrischen und des statischen Randwertproblems derebenen Elastostatik erklärt. Es ergeben sich dabei 32 verschiedene Formulierungen der Probleme als Integralgleichungen.Weiterhin wird in dem Aufsatz eine Analogie zwischen den wichtigsten Begriffen der Singularitätenmethode, den Zustandsgrößen und den Singularitäten, aufgedeckt. Die Analogie macht sich durch gewisse Symmetrien der Einflußfunktionen bemerkbar und erlaubt es, die grundlegenden Beziehungen systematisch darzustellen und in einen größeren Zusammenhang einzuordnen.

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Nomenclature

state variables; torso state variables Airy's stress function - u _i displacement vector - _i stress function vector - U _ij distortion tensor - _ij stress tensor - U _i distortion vector - {ei33-3} stress vector - N collective denotation for state variables singularities; torso singularities v wedge dislocation - C _i edge dislocation - R _i single force - G _i Rieder's singularity - F _i Massonnet's singularity - c _i generating vector ofc _ij - r _i generating vector ofr _ij - g _i generating vector ofg _ij - f _i generating vector off _ij - c _ij dipole ofC _i - r _ij dipole ofR _i - g _ij dipole ofG _i - f _ij dipole ofF _i - M collective denotation for singularities - m _i dipole of a singularityM - d moment of a force dipole - b dilation intensity of a force dipole influence functions (NM) collective denotation for influence functions further influence functions: see Chapters 4.1, 4.2 - [NM] collective denotation for proportionality factors of non integral terms corresponding to (NM). further proportionality factors see Chapter 8 geometry x _i , radius vector of the field and source point respectively - ,q _i vector and unit vector respectively in the direction of the connecting line between the field and the source point - distance between the field and the source point - S curve on the infinite plane congruent to the boundary of the elastic body - S ¹,S ² sections ofS - curve equidistant fromS - s, arc length of a field and a source point respectively onS - arc length of a field point on - n_i, normal vector ofS at the field and the source point respectively - curvature ofS further denotations _i , vector of differentiation with respect to field and source point co-ordinates respectively - _ij identity tensor - e _ij permutation tensore ₁₁=e ₂₂=0e ₁₂=–e ₂₁=1 - m Poisson's ratio - G shear modulus - Cauchy principal value - b _i arbitrary constant vector With 10 Figures     

An efficient method to evaluate hypersingular and supersingular integrals in boundary integral equations analysis

An efficient algorithm is employed to evaluated hyper and super singular integral equations encountered in boundary integral equations analysis of engineering problems. The algorithm is based on multiple subtractions and additions to separate singular and regular integral terms in the polar transformation domain, primarily established in Refs. (Guiggiani M, Krishnasamy G, Rudolphi TJ, Rizzo FJ. A general algorithm for the numerical solution of hypersingular boundary integral equations. Trans ASME 1992;59:604–614; Guiggiani M, Casalini P. Direct computation of Cauchy principal value integral in advanced boundary element. Int J Numer Meth Engng 1987;24:1711–1720. Guiggiani M, Gigante A. A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method. J Appl Mech Trans ASME 1990;57:906–915). It can be proved that the regular terms have finite analytical solutions in the range of integration, and the singular terms will be replaced by special periodic kernels in the integral equations. The subtractions involve to multiple derivatives of analytical kernels and the additions require some manipulation to separate the remaining regular terms from singular ones. The regular terms are computed numerically. Three examples on numerical evaluation of singular boundary integrals are presented to show the efficiency and accuracy of the algorithm. In this respect, strongly singular and hypersingular integrals of potential flow problems are considered, followed by a supersingular integral which is extracted from the partial differentiation of a hypersingular integral with respect to the source point.     

An efficient time-truncated boundary element formulation applied to the solution of the two-dimensional scalar wave equation

An efficient time-truncation algorithm applied to the boundary element solution of the two-dimensional scalar wave equation is proposed. Rational interpolation functions are here employed to compute time-domain influence matrices, in appropriate instants of time, improving previous time-truncation strategies presented in the literature. Two numerical examples are considered at the end of the paper, illustrating that the present scheme is fairly suitable for both bounded and unbounded models.