A scaled boundary finite element method applied to electrostatic problems

The scaled boundary finite element method (SBFEM) is a novel semi-analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. In this paper, the SBFEM is firstly extended to solve electrostatic problems. Two new SBFE coordination systems are introduced. Based on Laplace equation of electrostatic field, the derivations (based on a new variational principle formulation) and solutions of SBFEM equations for both bounded domain and unbounded domain problems are expressed in details, the solution for the inclusion of prescribed potential along the side-faces of bounded domain is also presented in details, then the total charges on the side-faces can be semi-analytically solved, and a particular solution for the potential field in unbounded domain satisfying the constant external field is solved. The accuracy and efficiency of the method are illustrated by numerical examples with complicated field domains, potential singularities, inhomogeneous media and open boundaries. In comparison with analytic solution method and other numerical methods, the results show that the present method has strong ability to resolve singularity problems analytically by choosing the scaling centre at the singular point, has the inherent advantage of solving the open boundary problems without truncation boundary condition, has efficient application to the problems with inhomogeneous media by placing the scaling centre in the bi-material interfaces, and produces more accurate solution than conventional numerical methods with far less number of degrees of freedom. The method in electromagnetic field calculation can have broad application prospects.     

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     

Transmission line interpretation of the finite element mesh for a wave propagation problem

A one-dimensional exterior electromagnetic scattering problem is formulated using a differential equation approach followed by a finite element discretization. By interpreting the resulting linear algebraic equations as node voltage equations for a transmission line, a boundary element is obtained which satisfies the requirement of no wave reflection at the edge of the finite element region. Numerical results which show the elimination of non-physical standing waves from the scattered field are presented and discussed.     

Computational mechanics based on the theory of boundary eigensolutions

The theory of boundary eigensolutions for boundary value problems is applied to the development of computational mechanics formulations. The boundary element and finite element methods that result are consistent with the mathematical theory of boundary value problems. Although the approach is quite general, this paper focuses on potential problems. For these problems, both methods employ potential and boundary flux as primary variables. Convergence characteristics of the new flux‐oriented finite element method are also developed. By utilizing suitable boundary weight functions, the formulations are written exclusively in terms of bounded quantities, even for non‐smooth problems involving notches, cracks and mixed boundary conditions. The results of numerical experiments indicate that the algorithms perform in concert with the underlying theory and thus provide an attractive alternative to existing approaches. Beyond this, the approach developed here provides a new perspective from which to view computational mechanics, and can be used to obtain a better understanding of boundary element and finite element methods. Comparisons with closed‐form boundary eigensolutions are also presented in order to provide a means for assessing the numerical methods. Copyright © 2001 John Wiley & Sons, Ltd.     

Interval boundary element method in the presence of uncertain boundary conditions,integration errors,and truncation errors

In engineering, most governing partial differential equations for physical systems are solved using finite element or finite difference methods. Applications of interval methods have been explored in finite element analysis to model systems with parametric uncertainties and to account for the impact of truncation error on the solutions. An alternative to the finite element method is the boundary element method. The boundary element method uses singular functions to reduce the dimension of the domain by transforming the domain variables to boundary variables. In this work, interval methods are developed to enhance the boundary element method for considering causes of imprecision such as uncertain boundary conditions, truncation error, and integration error. Examples are presented to illustrate the effectiveness and potential of an interval approach in the boundary element method.     

Temperature control of TiAl melt during induction skull melting

Abstract

The properties of intermetallic compounds are sensitive to alloy composition and interstitial element content determined by the melting process. Induction skull melting is one of the best methods for melting reactive alloys. For induction heating under vacuum, melt temperature control is problematic. A numerical model for simulating temperature field during induction skull melting has been developed using the direct finite difference method. Factors such as water cooling boundaries, electromagnetic stirring meniscus, and power distribution in the charge were analysed. The chargecrucible interface was taken as a radiation boundary before melting and as a combined radiation and conduction boundary after melting. The free surface was taken as a radiation boundary. The relationship between the height of electromagnetic stirring meniscus and charge weight and melting power was reduced based on the conservation of mass. Based on the conservation of energy, the distribution density of induction current was ascertained. With the program, the relationship between melting power, charge weight, and melt temperature was established. During induction skull melting of gamma-TiAl based alloys, the melt temperature was measured carefully. The theoretical and experimental results were found to be in agreement.     

A finite element method for analysing skin effect in conductors with unknown values of surface boundary conditions

This paper presents a method of analysing the distribution of a two-dimensional electromagnetic field in a conducting medium of known cross-section carrying sinusoidal current and of unknown numerical values of boundary conditions on conductors surfaces. The method combines the finite element and separation of variables methods. An example is given of application of the method to calculation of impedance and electrodynamic forces of two thick, full and round conductors carrying oppositely directed sinusoidal currents. On the basis of numerical computation, graphs of resistance, reactance and electrodynamic forces are plotted.     

Nonlinear Eddy Current Analysis by BEM Utilizing Adaptive Equation Technique

Eddy currents induced in steel are analyzed by the boundary element method. As the periodic electromagnetic quantities in the steel are distorted due to the nonlinear magnetization property, they are expressed by a Fourier series, whose fundamental and harmonic waves are determined by solving the corresponding integral equations with the surface magnetic fields given as the boundary values. In this paper, we introduce a new combined integration scheme that enables efficient, volume-mesh-less treatment of the practical problems typically characterized by the geometrical singularities like edges and corners. Using this procedure, the internal and surface electromagnetic fields are obtained numerically one after another by applying an adaptive equation technique to obtain the boundary values.      

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.     

中大口径电磁流量计无液校准技术的应用研究

基于电磁流量计测量原理,在建立了包含相关物理量的微分方程及边界条件基础上,提出了一种电磁流量计无液校准技术,设计了一套无液校准设备。采用逐点测量管壁磁感应强度及测量传感器结构尺寸的方法,获得确定的边界条件,用数值方法求解微分方程,获得传感器的灵敏度系数,并对2台口径200 mm电磁流量计进行了无液和实流的对比试验,两者所获得的灵敏度系数相对误差不超过0.5%。     

Propagation of transient SH-waves in a dipping structure by finite- and boundary element methods

Summary The boundary and the finite element formulations for the equations of elasticity are presented and applied to the problem of propagation of transient SH-waves in dipping layers overlying a half-space. When the finite element formulation is used, appropriate boundary conditions are imposed on the additional boundary dividing the half-space into a finite and an infinite region. These conditions ensure the transmission of waves across this boundary. When the boundary element method is applied, it is necessary to satisfy the radiation conditions. Theoretical seismograms for the displacement on the surface of the half-space are presented. They show that, for a specific case, the agreement between the two methods is satisfactory. The results can be compared with those found by the exact method of generalized rays in order to check the validity of the finite and the boundary element methods for the specific problem studied in this paper.     

Flaw identification using the boundary element method

In this paper a new boundary element formulation is presented for the identification of the location and size of internal flaws in two-dimensional structures. An introduction to inverse analysis is given, with special reference to methods of flaw identification, along with a brief review of the optimization methods employed. Both the standard boundary element and the dual boundary element method are presented, with the dual boundary element method proposed as the basis for the new formulation. The flaw identification method is presented, along with the computation of the boundary displacement and traction derivatives and the specialized analytical integration used for cracked boundaries. Examples are given to demonstrate the accuracy of the sensitivity values and the performance of flaw location.     

Three-dimensional boundary element analysis of electromagnetic wave scattering by penetrable bodies

A frequency domain boundary element methodology of solving three dimensional electromagnetic wave scattering problems by dielectric particles is reported. The method utilizes a computationally attractive surface integral equation containing only weakly and strongly singular integrals in the contrast to most formulations involving not only strongly singular but hypersingular integrals as well. The main advantage of this integral equation is the fact that its strongly singular part is similar to the one appearing in the corresponding integral equation of dynamic elasticity. Thus, well known advanced integration techniques used successfully in elastic scattering problems can be directly applied to the present analysis. Both continuous and discontinuous quadratic elements are employed in order to accurately treat dielectric scatterers with smooth and piecewise smooth boundaries. Numerical examples dealing with three dimensional electromagnetic wave scattering problems demonstrate the accuracy and efficiency of the proposed boundary element formulation.     

Three-Dimensional Analytical Modeling of Axial-Flux Permanent Magnet Drivers

In this paper, the axial-flux permanent magnet driver is modeled and analyzed in a simple and novel way under three-dimensional cylindrical coordinates. The inherent three-dimensional characteristics of the device are comprehensively considered, and the governing equations are solved by simplifying the boundary conditions. The axial magnetization of the sector-shaped permanent magnets is accurately described in an algebraic form by the parameters, which makes the physical meaning more explicit than the purely mathematical expression in general series forms. The parameters of the Bessel function are determined simply and the magnetic field distribution of permanent magnets and the air-gap is solved. Furthermore, the field solutions are completely analytical, which provides convenience and satisfactory accuracy for modeling a series of electromagnetic performance parameters, such as the axial electromagnetic force density, axial electromagnetic force, and electromagnetic torque. The correctness and accuracy of the analytical models are fully verified by three-dimensional finite element simulations and a 15 kW prototype and the results of calculations, simulations, and experiments under three methods are highly consistent. The influence of several design parameters on magnetic field distribution and performance is studied and discussed. The results indicate that the modeling method proposed in this paper can calculate the magnetic field distribution and performance accurately and rapidly, which affords an important reference for the design and optimization of axial-flux permanent magnet drivers.     

Boundary element analysis of foundation plates in buildings

In this paper, a boundary element method for analyzing building foundation plates is presented. The shear deformation is considered using the Reissner plate theory. The Winkler model is used to model the behavior of the soil. Only the boundary of the plate needs to be discretized. Two practical applications are solved and the results are compared to alternative solutions of different methods, such as, finite difference, finite element and 3D boundary element methods.     

A comparison of localized finite element formulations for two-dimensional wave diffraction and radiation problems

Two of the most promising localized finite element methods are compared: the boundary series element method, in which a series of eigenfunctions is used to represent the far field solution; and the boundary integral element method, in which an integral equation is satisfied at the boundary between localized finite element and outer regions. The methods are applied to water of arbitrary depth. The theory of the two methods is summarized, and typical numerical results are discussed. Consideration is given to the well-known hydrodynamical reciprocal relations, and to the phenomenon of ‘irregular’ frequencies. The relative merits of the two methods are established.     

A mixed hexahedral finite element to model the scattering of a deformed microstrip antenna

Microstrip antennas have a major interest in aeronautical applications due to their low profile. This paper deals with the impact of mechanical strain on the scattering properties of these antennas. Considering a weak coupling between electromagnetism and mechanical behavior, the same 3D hexahedral finite element discretization is used to solve both problems. A node-based approximation is used for mechanical displacement, while for the determination of the electromagnetic fields, a vector finite approximation is implemented to ensure a better consideration of electromagnetic boundary conditions. The weak electromagnetic formulation inducing integrals on open infinite domains, a Boundary Integral Method is used.     

Boundary element modeling of radio base station antennas

The paper presents an accurate and efficient boundary element procedure for the analysis of radiation from base station antennas. The base station antenna system is represented by the vertical antenna array in front of perfectly conducting (PEC) ground plane reflector.The formulation of the problem is based on a set of coupled Pocklington integro-differential equations for vertical antenna array. This set of coupled equations has been numerically treated by the indirect Galerkin–Bubnov boundary element method (GB-IBEM). The numerical results for the currents induced along the wires obtained via GB-IBEM are compared to the results computed via numerical electromagnetic code (NEC).Knowing the current distribution along the antenna array the corresponding radiated field is calculated. Some illustrative numerical results are presented in the paper.     

A comparative study of the boundary and finite element methods for the Helmholtz equation in two dimensions

The performance of the boundary and finite element methods for the Helmholtz equation in two dimensions is investigated. To facilitate the comparison, the system of linear equations arising from the finite element formulation is reduced to a smaller system involving the boundary values of the unknown function and its normal derivative alone. The difference between the boundary and finite element solutions is then expressed in terms of a difference matrix operating on the boundary data. Numerical investigations show that the boundary element method is generally more accurate than the finite element method when the size of the finite elements is comparable to that of the boundary elements, especially for the Dirichlet problem where the boundary values of the solution are specified. Exceptions occur in the neighborhood of isolated points of the Helmholtz constant where eigenfunctions of the boundary integral equation arise and the boundary element method fails to produce a unique solution.     

Virtual boundary element-integral collocation method for the plane magnetoelectroelastic solids

This paper presents a virtual boundary element—integral collocation method (VBEM) for the plane magnetoelectroelastic solids, which is based on the basic idea of the virtual boundary element method for elasticity and the fundamental solutions of the plane magnetoelectroelastic solids. Besides sharing all the advantages of the conventional boundary element method (BEM) over domain discretization methods, it avoids the computation of singular integral on the boundary by introducing the virtual boundary. In the end, several numerical examples are performed to demonstrate the performance of this method, and the results show that they agree well with the exact solutions. The method is one of the efficient numerical methods used to analyze megnatoelectroelastic solids.