Minimizing the error near discontinuities in boundary element method

This paper presents two types of discontinuity modeling elements (DME) that minimize the error near a discontinuity. The DME are elements that are discontinuous at one end, can satisfy continuity requirement up to seventh order at the other end, and may have polynomials of order up to fifteen. The error of approximation in the density function is measured by the L₁ norm, which is minimized with respect to the location of collocation points. Results show that the error for optimum location of collocation points in all cases is smaller than those for uniform location of collocation points and the differences in accuracy grows significantly with the order of polynomials. Two tables report the optimum location of collocation points for the DME for use by other researchers. The DME are used in conjunction with the hr-mesh refinement scheme to study modeling of stress near a stress discontinuity. Results of the study show that the recommendations for modeling density functions near a discontinuity are diametrically opposite to those recommendations for modeling of a smooth density function.     

Electric field calculation for HV insulators on the head of transmission tower by coupling CSM with BEM

An approach coupling indirect boundary element method with charge simulation method to calculate the electric fields around the head of transmission tower and its composite insulators is presented. On the composite insulators, the indirect boundary element method is used, while on the transmission tower and the transmission lines, the charge simulation method is used. Therefore, both the heterogeneous medium such as the composite insulators and the filament metal structure such as the transmission tower and the transmission lines are considered all together. The electric field distribution around a 330-kV transmission tower and its composite insulators was analyzed. The method gives the possibility to model very complex geometries.     

Regularized algorithms for the calculation of values on and near boundaries in 2D elastic BEM

It is verified that, under certain continuity conditions, the boundary integral equations (BIEs) of both displacement and displacement derivative can be expressed in a variety of regularized forms with at most weak singularities for any points within the domain or on its boundary. A series of algorithms embedded in the regularized formulations are presented to calculate the field variables within the domain or on its boundary. Detailed numerical results are given to check and compare the validities of the proposed algorithms and some practical effective algorithms are discovered. Due to the character of the at most weak singularities in the formulations, the algorithms require no special numerical treatments, but only the general Gauss quadrature to implement. To the end, the continuity requirements for the field variables and the validities of the algorithms are discussed.     

Identification of boundary displacements in plane elasticity by BEM

The purpose of this study is to present a possible application of BEM for numerical identification of the boundary conditions for Navier equations in plane elasticity with internal measurements, based on insufficient and noisy information for unique identification. The inverse problem is re-formulated as a minimization problem by the direct variational method. The minimization problem is then recast using the gradient method into successive primary and adjoint boundary value problems in the corresponding plane elasticity problem. For numerical solution of the elasticity problems, the conventional direct boundary element method is employed. From the simple numerical examples considered, it is concluded that our identification scheme is stable and the approximate solutions are convergent to the minimum.     

Forward electric field calculation using BEM for time-varying magnetic field gradients and motion in strong static fields

A boundary element method for evaluating the electric fields induced in conducting bodies exposed to magnetic fields varying at low frequency has been developed and applied to sources of magnetic field variation that are of relevance in magnetic resonance imaging. An integral formulation based on constant boundary elements which can be used to study the effects of both temporally varying magnetic field gradients and rigid body movement in a static magnetic field is presented. The validity of this approach has been demonstrated for simple geometries with known analytical solutions and it has also been applied to the evaluation of the induced fields in more realistic models of the human head.     

Internal field error reduction in boundary element analysis for Helmholtz equation

A systematic analysis has been made to clarify the cause of very large errors in the boundary element solutions at internal points near the boundary. Following the results of the analysis, two new ideas have been developed to reduce the errors when solving the Helmholtz equation. The first idea, which is an extended application of the equipotential condition to Helmholtz type problem, is easily applied to any existing program. The second idea which gives us more accurate solutions is based on the analytic integration of boundary integrals by expanding the fundamental solutions into a sum of elementary functions. Test calculations demonstrate that the two present techniques are efficient to reduce the errors near the boundary.     

On the a-priori error analysis and convergence of meshless BEM

This paper deals with the a-priori error analysis and convergence of meshless boundary element methods. The paper investigates the convergence of the particular solution method (PSM) in conjunction with the boundary element method (BEM) as well as the method of fundamental solutions (MFS) and the radial basis functions (RBF)—type techniques. A-priori error estimates for meshless BEMs are also provided, and several illustrating numerical experiments are derived.     

BEM calculation of the complex thermal impedance of microelectronic devices

This paper presents a numerical method for modelling the dynamic thermal behaviour of microelectronic structures in the frequency domain. A boundary element method (BEM) based on a Green's function solution is proposed for solving the 3D heat equation in phasor notation. The method is capable of calculating the AC temperature and heat flux distributions and complex thermal impedance for packages composed of an arbitrary number of bar-shaped components. Various types of boundary conditions, including thermal contact resistance and convective cooling, can be taken into account. A simple benchmark case is investigated and a good convergence towards the analytical solution is obtained. Simulation results for a thin plate under convective cooling are compared with a theoretical model and an excellent agreement is observed. In a second example a more complicated three-layer structure is investigated. The BEM is used to analyse the thermal behaviour if delamination of the package occurs, and a physical explanation for the results is given.     

An error evaluation scheme based on rotation of magnetic field in adaptive finite element analysis

The finite-element analysis is widely used in design stage of electromagnetic apparatuses. The analysis accuracy depends on the characteristics of the finite-element mesh, e.g., number of nodes, number of elements and shape of elements. Recently, the adaptive finite-element analysis is one of the most promising numerical analysis techniques. In process of the adaptive finite-element method, the error evaluation is one of the important schemes. In this paper, a new error evaluation scheme, which is suitable for electromagnetic problems, is proposed. The proposed error evaluation method is then applied to two-dimensional and three-dimensional magnetostatic field problems for its verification.     

On the calculation of boundary stresses in boundary elements

The discontinuity of boundary stresses evaluated using discretized boundary elements is discussed, and stress error bound is derived. A new procedure for calculating interelement stress is proposed to overcome stress discontinuity at the interelement boundary. By employing two middle nodes of the two adjacent elements and the interelement node, a new quadratic element is formed, which leads to a more accurate interelement stress. Two examples are used to study error distributions of displacements and stresses. The comparison study indicates that the new procedure provides a better solution for interelement stresses than the conventional method.     

Ozone and water-vapor measurements by Raman lidar in the planetary boundary layer: error sources and field measurements

A new lidar instrument has been developed to measure tropospheric ozone and water vapor at low altitude. The lidar uses Raman scattering of an UV beam from atmospheric nitrogen, oxygen, and water vapor to retrieve ozone and water-vapor vertical profiles. By numerical simulation we investigate the sensitivity of the method to both atmospheric and device perturbations. The aerosol optical effect in the planetary boundary layer, ozone interference in water-vapor retrieval, statistical error, optical cross talk between Raman-shifted channels, and optical cross talk between an elastically backscattered signal in Raman-shifted signals and an afterpulse effect are studied in detail. In support of the main conclusions of this model study, time series of ozone and water vapor obtained at the Swiss Federal Institute of Technology in Lausanne and during a field campaign in Crete are presented. They are compared with point monitor and balloon sounding measurements for daytime and nighttime conditions.     

BEM treatment of two-dimensional anisotropic field problems by direct domain mapping

Using the method of characteristics, a steady-state anisotropic field problem can be reduced to one governed by Laplace's equation in a mapped plane. No rotation of axes is involved in the process, and the coordinate transformation equations are given in this paper. The advantage of this approach is that the anisotropic problem can be easily solved with any of the readily available boundary element method (BEM) programs for ‘isotropic’ potential theory, albeit on a distorted domain. Two numerical examples are presented to illustrate the scheme.     

An error calculation method for finite element analysis in large displacements

This article describes an error measurement and mesh optimization method for finite elements in non-linear geometry problems. The error calculation is adapted from a method developed by Ladeveze, based on constructing a local statically admissible stress field. The particular difficulty in non-linear geometry lies in choosing a configuration on which the fields is defined. We propose here the lagrangian or reference configuration. The error is then defined as the value in elastic energy of the difference between the two stresses. The optimization used is the type h version. © 1997 John Wiley & Sons, Ltd.     

Efficient calculation of internal results in 2D elasticity BEM

The solution of boundary integral equations in their discretized form requires an accurate treatment of regular as well as singular integrals. The regular integrals are usually solved numerically using Gauss quadrature. Since these integrations make up the major part of the numerical work the choice of the appropriate Gauss order is essential to an accurate and efficient boundary element analysis. Thus, a considerable number of publications is dealing with the subject of choosing a Gauss order suitable to gain efficiency without loosing accuracy. The guidelines determining the choice of the appropriate Gauss order is usually called an integration criterion. This paper presents a study on this topic with emphasis on the accuracy of internal results in 2D elasticity. First the necessity for a new integration criterion is shown. Then a new criterion is derived. This new criterion and various existing criteria from the literature are applied to a standard benchmark problem. The superior performance of the novel criterion is demonstrated.     

On the validity of conforming BEM algorithms for hypersingular boundary integral equations

The widely held notion that the use of standard conforming isoparametric boundary elements may not be used in the solution of hypersingular integral equations is investigated. It is demonstrated that for points on the boundary where the underlying field is C ^1,α continuous, a class of rigorous nonsingular conforming BEM algorithms may be applied. The justification for this class of algorithms is interpreted in terms of some recent criticism. It is shown that the numerical integration in these conforming BEM algorithms using relaxed regularization represents a finite approximation to the standard two-sided Hadamard finite part interpretation of hypersingular integrals. It is also shown that the integration schemes in this class of algorithms are not based upon one-sided finite part interpretations. As a result, the attendant ambiguities associated with the incorrect use of the one-sided interpretations in boundary integral equations pose no problem for this class of algorithms. Additionally, the distinction is made between the analytic discontinuities in the field which place limitations on the applicability of the conforming BEM and the discontinuities resulting from the use of piece-wise C ^1,α interpolations.     

Remedy for round-off error accumulation observed in a neutron diffusion calculation using the multiple reciprocity boundary element method

A round-off error accumulation is observed in a multiple reciprocity computation for a fission neutron source iteration problem when a certain convergence condition is not satisfied. The present paper presents a reformulation of the multiple reciprocity method to remove the numerical problem. The neutron diffusion equation (NDE) is arranged using Wielandt's spectral shift technique in such a way that the convergence condition is always satisfied through source iterations. The boundary integral equation in this case requires the fundamental solution to the standard Helmholtz equation, while the fundamental solution corresponding to the correspoding to the original NDE was one to the modified Helmholtz equation. Except for this, the new multiple reciprocity formulation is identical to the original one for the fission source iteration problems. Some test calculation results indicate that a rapid and stable convergence can be realized using the new method and no round-off error accumulation is observed any more.     

An extended BEM formulation for plates reinforced by rectangular beams

This article presents a BEM formulation developed particularly for analysis of plates reinforced by rectangular beams. This is an extended version of a previous paper that only took into account bending effects. The problem is now re-formulated to consider bending and membrane force effects. The effects of the reinforcements are taken into account by using a simplified scheme that requires application of an initial stress field to locally correct the bending and stretching stiffness of the reinforcement regions. The domain integrals due to the presence of the reinforcements are then transformed to the reinforcement/plate interface. To reduce the number of degrees of freedom related to the presence of the reinforcement, the proposed model was simplified to consider only bending and stretching rigidities in the direction of the beams. The complete model can be recovered by applying all six internal force correctors, corresponding to six degrees of freedom per node. Examples are presented to confirm the accuracy of the formulation and to illustrate the level of simplification introduced by this strong reduction in the number of degrees of freedom.     

Transfer coefficients near the boundary of thermodynamic stability

It has been established according to the data of the molecular-dynamic calculations of the transfer coefficients in the stable and metastable states of the Lennard-Jones fluid that in the variables p, T the families of the lines of constant value of the self-diffusion coefficient D, excess thermal conductivity Δλ and excess shear viscosity Δη have an envelope, which coincides with the spinodal. Thus, (?D/?p) _T → ∞, (?Δλ/? p) _T → ∞, (?Δη/?p) _T → ∞ when approaching the spinodal.