A cell-less BEM formulation for axisymmetric elastoplasticity via particular integrals

This study deals with the particular integral formulation for purely axisymmetric elastoplastic analysis. The axisymmetric elastostatic equation is used for the complementary solution. The axisymmetric particular integrals for displacement and strain rates are derived by integrating three-dimensional formulation along the circumferential direction leading to elliptic integrals. The particular integrals for stress and traction rates are obtained by using the stress–strain and traction–stress relations. The Newton–Raphson algorithm for the plastic multiplier is used to solve the system equation. The numerical results for four example problems are given and compared with their analytical solutions or those by other BEM and FEM programs to demonstrate the accuracy of the present formulation. Generally, agreement among all of those results is satisfactory.     

Analytical time integration for BEM axisymmetric acoustic modelling

In this work, a numerical time‐domain approach to model acoustic wave propagation in axisymmetric bodies is developed. The acoustic medium is modelled by the boundary element method (BEM), whose time convolution integrals are evaluated analytically, employing the concept of finite part integrals. All singularities for space integration, present in the expressions generated by time integration, are treated adequately. Some applications are presented to demonstrate the validity of the analytical expressions generated for the BEM, and the results obtained with the present approach are compared with those generated by applying numerical time integration. Copyright © 2007 John Wiley & Sons, Ltd.     

A FEM/BEM for axisymmetric electromagnetic and thermal modelling of induction furnaces

In this paper, we propose a numerical method to solve a mathematical model for axisymmetric induction furnaces used for melting materials like metals or silicon. A finite/boundary element method (FEM/BEM) is introduced to solve the eddy current problem giving the electromagnetic field. In order to solve the non‐linear heat transfer problem involving change of state, we use an enthalpy formulation and propose an iterative algorithm to solve the corresponding finite element approximation. Numerical results for an industrial induction heating system are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

An advanced BEM for electromagnetic wave scattering problems with axisymmetric dielectric particles

An advanced boundary element/fast Fourier transform (FFT) methodology for solving axisymmetric electromagnetic wave scattering problems with general, non-axisymmetric boundary conditions is presented. The incident field as well as the boundary quantities of the problem are expanded in complex Fourier series with respect to the circumferential direction. Each of the expanding coefficients satisfies a surface integral equation which, due to axisymmetry, is reduced to a line integral along the surface generator of the body and an integral over the angle of revolution. The first integral is evaluated by discretizing the meridional line of the body into isoparametric elements and employing Gauss quadrature. The integration over the angle of revolution is performed simultaneously for all the expanding coefficients through the FFT. The singular integrals are computed directly with high accuracy. Representative numerical examples demonstrate the accuracy of the proposed boundary element formulation.     

Calculation of mode III stress intensity factor by BEM for cracked axisymmetric bodies

This paper is concerned with the numerical calculation of Mode III stress intensity factor by BEM for cracked axisymmetric bodies, under torsion. Mode III stress intensity factors K _III are obtained using the asymptotic displacement field in the vicinity of the crack border. The asymptotic field is derived by integration along the boundary of the meridian of the cylinder. For traction free cracks no discretization of the crack surface was found necessary. Numerical results proving the efficiency of the proposed method are presented and compared with results given in the literature and with those obtained by FEM.     

Conjugate gradient methods for three-dimensional BEM systems of equations

A fundamental advantage of the boundary element method (BEM) is that the dimensionality of the problems is reduced by one. However, this advantage has to be weighted against the difficulty in solving the resulting systems of algebraic linear equations whose matrices are dense, non-symmetric and sometimes ill conditioned. For large three-dimensional problems the application of the classical direct methods becomes too expensive.This paper studies the comparative performance of iterative techniques based on conjugate gradient solvers as bi-conjugate gradient (Bi-CG), generalized minimal residual (GMRES), conjugate gradient squared (CGS), quasi-minimal residuals (QMR) and bi-conjugate gradient stabilized (Bi-CGStab) for potential and exterior problems. Preconditioning is also considered and assessed.Two examples, one from electrostatics and other from fluid mechanics, were employed to test these methods, which proved to be effective and competitive as solvers for BEM linear algebraic systems of equations.     

Constructing efficient substructure-based preconditioners for BEM systems of equations

In this work, a generic substructuring algorithm is employed to construct global block-diagonal preconditioners for BEM systems of equations. In this strategy, the allowable fill-in positions are those on-diagonal block matrices corresponding to each BE subregion. As these subsystems are independently assembled, the preconditioner for a particular BE model, after the LU decomposition of all subsystem matrices, is easily formed. So as to highlight the efficiency of the preconditioning proposed, the Bi-CG solver, which presents a quite erratic convergence behavior, is considered. In the particular applications of this paper, 3D representative volume elements (RVEs) of carbon-nanotube (CNT) composites are analyzed. The models contain up to several tens of thousands of degrees of freedom. The efficiency and relevance of the preconditioning technique is also discussed in the context of developing general (parallel) BE codes.     

Analytic solutions for axisymmetric magnetostatic systems involving iron

The analytic solutions are given for three simple axisymmetric systems involving toroidal conductors and iron. The solutions can be used for testing and gauging the accuracy of three-dimensional numerical schemes. In each case it is demonstrated how the fields can be calculated from a spherical harmonic expansion. This provides a unified and efficient method for calculating the fields.     

A self-regularized approach for rank-deficient systems in the BEM of 2D Laplace problems

The Laplace problem subject to the Dirichlet or Neumann boundary condition in the direct and indirect boundary element methods (BEM) sometimes both may result in a singular or ill-conditioned system (some special situations) for the interior problem. In this paper, the direct and indirect BEMs are revisited to examine the uniqueness of the solution by introducing the Fichera’s idea and the self-regularized technique. In order to construct the complete range of the integral operator in the BEM lacking a constant term in the case of a degenerate scale, the Fichera’s method is provided by adding the constraint and a slack variable to circumvent the problem of degenerate scale. We also revisit the Fredholm alternative theorem by using the singular value decomposition (SVD) in the discrete system and explain why the direct BEM and the indirect BEM are not indeed equivalent in the solution space. According to the relation between the SVD structure and Fichera’s technique, a self-regularized method is proposed in the matrix level to deal with non-unique solutions of the Neumann and Dirichlet problems which contain rigid body mode and degenerate scale, respectively, at the same time. The singularity and proportional influence matrices of 3 by 3 are studied by using the property of the symmetric circulant matrix. Finally, several examples are demonstrated to illustrate the validity and the effectiveness of the self-regularized method.     

A hybrid polynomial dimensional decomposition for uncertainty quantification of high-dimensional complex systems

This paper presents a novel hybrid polynomial dimensional decomposition (PDD) method for stochastic computing in high-dimensional complex systems. When a stochastic response does not possess a strongly additive or a strongly multiplicative structure alone, then the existing additive and multiplicative PDD methods may not provide a sufficiently accurate probabilistic solution of such a system. To circumvent this problem, a new hybrid PDD method was developed that is based on a linear combination of an additive and a multiplicative PDD approximation, a broad range of orthonormal polynomial bases for Fourier-polynomial expansions of component functions, and a dimension-reduction or sampling technique for estimating the expansion coefficients. Two numerical problems involving mathematical functions or uncertain dynamic systems were solved to study how and when a hybrid PDD is more accurate and efficient than the additive or the multiplicative PDD. The results show that the univariate hybrid PDD method is slightly more expensive than the univariate additive or multiplicative PDD approximations, but it yields significantly more accurate stochastic solutions than the latter two methods. Therefore, the univariate truncation of the hybrid PDD is ideally suited to solving stochastic problems that may otherwise mandate expensive bivariate or higher-variate additive or multiplicative PDD approximations. Finally, a coupled acoustic-structural analysis of a pickup truck subjected to 46 random variables was performed, demonstrating the ability of the new method to solve large-scale engineering problems.     

A Laplace-transform-based three-dimensional BEM for poroelasticity

This paper presents the formulation and the numerical implementation of a three-dimensional direct boundary element method for the Biot theory of poroelasticity. To avoid the need of time-stepping and volume integration, the solution is performed in the Laplace transform space. Solution in time is obtained via numerical inversion. Several examples, including the settlement of a rectangular footing and a modified Mandel problem, are examined.     

A fast wavelet-multipole method for direct BEM

A fast wavelet-multipole method (WMM) has been developed to achieve further speedup for the boundary element method in solving the direct boundary integral equations. The main idea is to compute the right-hand-side vector by the fast multipole method and to solve the linear system by the wavelet compression method. By using the variable order moments, almost linear complexity can be obtained. The primary advantages of the present WMM lie in that it (1) permits efficient implementation; (2) is universal in handling practical problems with complicated geometries. Numerical examples with around 1 million unknowns, performed on nontrivial geometries, clearly show that the WMM can shorten the total computational time by reducing the time for computing the right-hand-side.     

A total ALE FE formulation for axisymmetric and torsional problems (including viscoplasticity,thermodynamic coupling and dynamics)

This paper—first—extends a recent ‘assumed enhanced deformation gradient’ finite ring(segment) element (Int. J. Numer. Methods Eng. 2001; 50 :899–918.) to Arbitary Lagrange Euler (ALE) computations, by setting up the assumed tensor on the computational configuration, and—second—shows an elegant way of incorporating dynamics into real ALE computations (no splitting into purely—Lagrange steps and then—remeshing steps), by introducing material mesh velocities and accelerations and spatial mesh velocities and accelerations. Copyright © 2006 John Wiley & Sons, Ltd.     

Stress analysis for multilayered coating systems using semi-analytical BEM with geometric non-linearities

For a long time, most of the current numerical methods, including the finite element method, have not been efficient to analyze stress fields of very thin structures, such as the problems of thin coatings and their interfacial/internal mechanics. In this paper, the boundary element method for 2-D elastostatic problems is studied for the analysis of multi-coating systems. The nearly singular integrals, which is the primary obstacle associated with the BEM formulations, are dealt with efficiently by using a semi-analytical algorithm. The proposed semi-analytical integral formulas, compared with current analytical methods in the BEM literature, are suitable for high-order geometry elements when nearly singular integrals need to be calculated. Owing to the employment of the curved surface elements, only a small number of elements need to be divided along the boundary, and high accuracy can be achieved without increasing more computational efforts. For the test problems studied, very promising results are obtained when the thickness of coated layers is in the orders of 10⁻⁶–10⁻⁹, which is sufficient for modeling most coated systems in the micro- or nano-scales.     

A polynomial algorithm for uniqueness of normal forms of linear shallow term rewrite systems

Term rewrite systems are useful in many areas of computer science. Two especially important areas are decision procedures for the word problem of some algebraic systems and rule-based programming. One of the most studied properties of rewrite systems is confluence, and one of the primary benefits of having a confluent rewrite system is that the system also has uniqueness of normal forms. However, uniqueness of normal forms is an interesting property in its own right and well studied. Also, confluence can be too strong a requirement for some applications. In this paper, we study the decidability of uniqueness of normal forms. Uniqueness of normal forms is decidable for ground rewrite systems, but is undecidable in general. This paper shows that the uniqueness of normal forms problem is decidable for the class of linear shallow term rewrite systems, and gives a decision procedure that is polynomial as long as the arities of the function symbols are bounded or the signature is fixed.     

A new BEM formulation for the acoustic eigenfrequency analysis

A new boundary element formulation has been developed for two- and three-dimensional acoustic eigenfrequency analyses. The formulation is based on the well known method of constructing a solution of a differential equation in terms of a complementary function and particular integral. An advanced isoparametric implementation with automatic error control in the integration is used. A number of realistic examples of application to automotive acoustic cavities are described.     

A time-dependent incompressible viscous BEM for moderate Reynolds numbers

The boundary element method is applied to transient viscous incompressible flow. The time-domain formulation allows a boundary-only solution for linear Stokes flow. For higher speed flows in which the non-linear convective effects cannot be ignored, a volume integral must be retained. However, the introduction of reference velocities often limits the non-linear region to the vicinity of obstacles or bounding surfaces. Additionally, the volume terms are rewritten to eliminate the need for the calculation of velocity gradients. A general purpose numerical implementation of this new formulation then produces a very attractive tool for engineering analysis. This implementation includes a Newton-Raphson algorithm, permitting accurate solutions up to the moderate Reynolds number range. Several numerical examples are provided to validate the present approach.     

A simple and efficient element for axisymmetric shells

A two noded, straight element which includes shear deformation effects is presented and shown to be extremely efficient in the analysis of axisymmetric shells. A single point of numerical integration is essential for its success when applied to thin shells where the results compare favourably with those achieved with more complex curved elements.