3-D magnetostatic field calculation for a magnetic circuit with no(a narrow) air gap

Noting that 3-D magnetostatic field calculations for gapless magnetic circuits are strongly affected by the discretization, the author analyzes this effect for an iron torus using a boundary integral equation based on the surface magnetization current method. The main cause of computational error is the imperfect cancellation of the permeability-free terms in the boundary integral equation due to the improper size of the analytical integration region containing a singular point. A method of reducing the computational error is presented and verified to be valid     

Analytical regularization of hypersingular integral for Helmholtz equation in boundary element method

This paper presents a gradient field representation using an analytical regularization of a hypersingular boundary integral equation for a two-dimensional time harmonic wave equation called the Helmholtz equation. The regularization is based on cancelation of the hypersingularity by considering properties of hypersingular elements that are adjacent to a singular node. Advantages to this regularization include applicability to evaluate corner nodes, no limitation for element size, and reduced computational cost compared to other methods. To demonstrate capability and accuracy, regularization is estimated for a problem about plane wave propagation. As a result, it is found that even at a corner node the most significant error in the proposed method is due to truncation error of non-singular elements in discretization, and error from hypersingular elements is negligibly small.     

On the spectrum of the electric field integral equation and the convergence of the moment method

Existing convergence estimates for numerical scattering methods based on boundary integral equations are asymptotic in the limit of vanishing discretization length, and break down as the electrical size of the problem grows. In order to analyse the efficiency and accuracy of numerical methods for the large scattering problems of interest in computational electromagnetics, we study the spectrum of the electric field integral equation (EFIE) for an infinite, conducting strip for both the TM (weakly singular kernel) and TE polarizations (hypersingular kernel). Due to the self‐coupling of surface wave modes, the condition number of the discretized integral equation increases as the square root of the electrical size of the strip for both polarizations. From the spectrum of the EFIE, the solution error introduced by discretization of the integral equation can also be estimated. Away from the edge singularities of the solution, the error is second order in the discretization length for low‐order bases with exact integration of matrix elements, and is first order if an approximate quadrature rule is employed. Comparison with numerical results demonstrates the validity of these condition number and solution error estimates. The spectral theory offers insights into the behaviour of numerical methods commonly observed in computational electromagnetics. Copyright © 2001 John Wiley & Sons, Ltd.     

Parallel computation of unsteady compressible flows with the EDICT

Recently, the Enhanced-Discretization Interface-Capturing Technique (EDICT) was introduced for simulation of unsteady flow problems with interfaces such as two-fluid and free-surface flows. The EDICT yields increased accuracy in representing the interface. Here we extend the EDICT to simulation of unsteady viscous compressible flows with boundary/shear layers and shock/expansion waves. The purpose is to increase the accuracy in selected regions of the computational domain. An error indicator is used to identify these regions that need enhanced discretization. Stabilized finite-element formulations are employed to solve the Navier-Stokes equations in their conservation law form. The finite element functions corresponding to enhanced discretization are designed to have two components, with each component coming from a different level of mesh refinement over the same computational domain. The primary component comes from a base mesh. A subset of the elements in this base mesh are identified for enhanced discretization by utilizing the error indicator. A secondary, more refined, mesh is constructed by patching together the second-level meshes generated over this subset of elements, and the second component of the functions comes from this mesh. The subset of elements in the base mesh that form the secondary mesh may change from one time level to other depending on the distribution of the error in the computations. Using a parallel implementation of this EDICT-based method, we apply it to test problems with shocks and boundary layers, and demonstrate that this method can be used very effectively to increase the accuracy of the base finite element formulation.     

Some benchmark problems and basic ideas on the accuracy of boundary element analysis

Taken the linear elasticity problems as examples, some benchmark problems are presented to investigate the influence of calculation error and discretization error on the accuracy of boundary element analysis. For the conventional boundary element analysis based on singular kernel function of fundamental solution and using Gaussian elimination method, the main calculation error comes from the integration of kernel and shape function product on each element. Based on some benchmark problems of “simple problem” without discretization error, it can be observed that sometimes a large number of integration points in Gaussian quadrature should be adopted. To guarantee the integration accuracy reliably, an improved adaptive Gaussian quadrature approach is presented and verified. The main error of boundary element analysis is the discretization error, provided the calculation error has been reduced effectively. Based on some benchmark problems, it can be observed that for the bending problems both the constant and linear element are not efficient, the quadratic element with a reasonable refined mesh is required to guarantee the accuracy of boundary element analysis. An error indicator to guide the mesh refinement in the convergence test towards the converged accurate results based on the distribution of boundary effective stress is presented and verified.     

A dynamic algorithm for integration in the boundary element method

The discretization of the boundary in boundary element method generates integrals over elements that can be evaluated using numerical quadrature that approximate the integrands or semi-analytical schemes that approximate the integration path. In semi-analytical integration schemes, the integration path is usually created using straight-line segments. Corners formed by the straight-line segments do not affect the accuracy in the interior significantly, but as the field point approaches these corners large errors may be introduced in the integration. In this paper, the boundary is described by a cubic spline on which an integration path of straight-line segments is dynamically created when the field point approaches the boundary. The algorithm described improves the accuracy in semi-analytical integration schemes by orders of magnitude at insignificant increase in the total solution time by the boundary element method. Results from two indirect BEM and a direct BEM formulation in which the unknowns are approximated by linear and quadratic Lagrange polynomial and a cubic Hermite polynomial demonstrate the versatility of the described algorithm. © 1998 John Wiley & Sons, Ltd.     

Output error estimation strategies for discontinuous Galerkin discretizations of unsteady convection‐dominated flows

We study practical strategies for estimating numerical errors in scalar outputs calculated from unsteady simulations of convection‐dominated flows, including those governed by the compressible Navier–Stokes equations. The discretization is a discontinuous Galerkin finite element method in space and time on static spatial meshes. Time‐integral quantities are considered for scalar outputs and these are shown to superconverge with temporal refinement. Output error estimates are calculated using the adjoint‐weighted residual method, where the unsteady adjoint solution is obtained using a discrete approach with an iterative solver. We investigate the accuracy versus computational cost trade‐off for various approximations of the fine‐space adjoint and find that exact adjoint solutions are accurate but expensive. To reduce the cost, we propose a local temporal reconstruction that takes advantage of superconvergence properties at Radau points, and a spatial reconstruction based on nearest‐neighbor elements. This inexact adjoint yields output error estimates at a computational cost of less than 2.5 times that of the forward problem for the cases tested. The calculated error estimates account for numerical error arising from both the spatial and temporal discretizations, and we present a method for identifying the percentage contributions of each discretization to the output error. Copyright © 2011 John Wiley & Sons, Ltd.     

An integral formulation for steady-state elastoplastic contact over a coated half-plane

 A boundary-domain integral equation for a coated half-space (elastically isotropic homogeneous substratum, possibly anisotropic coating layer) is developed. The half-space fundamental solution is used, so that the discretization is limited to the potential contact zone (boundary elements), the potentially plastic part of the substratum and the coating layer (domain integration cells). Steady-state elastoplastic analysis is implemented within this framework, for plane-strain conditions, for solving rolling and/or sliding contact problems, where at the moment the contact load comes from either a purely elastic contact analysis or is of Hertz type. The constitutive integration is of implicit type. In order to improve accuracy and computational efficiency, infinite elements are used. Comparison of numerical results with other sources, when available, is satisfactory. The present formulation is also used to compute the contact pressure for an isotropic (or anisotropic) coating on an isotropic homogeneous half-space indented by an elastic punch. Received 29 May 2001     

Biharmonic analysis of rectilinear plates by the boundary element method

An improved boundary element formulation (BEM) for two-dimensional non-homogeneous biharmonic analysis of rectilinear plates is presented. A boundary element formulation is developed from a coupled set of Poisson-type boundary integral equations derived from the governing non-homogeneous biharmonic equation. Emphasis is given to the development of exact expressions for the piecewise rectilinear boundary integration of the fundamental solution and its derivatives over several types of isoparametric elements. Incorporation of the explicit form of the integrations into the boundary element formulation improves the computational accuracy of the solution by substantially eliminating the error introduced by numerical quadrature, particularly those errors encountered near singularities. In addition, the single iterative nature of the exact calculations reduces the time necessary to compile the boundary system matrices and also provides a more rapid evaluation of internal point values than do formulations using regular numerical quadrature techniques. The evaluation of the domain integrations associated with biharmonic forms of the non-homogeneous terms of the governing equation are transformed to an equivalent set of boundary integrals. Transformations of this type are introduced to avoid the difficulties of domain integration. The resulting set of boundary integrals describing the domain contribution is generally evaluated numerically; however, some exact expressions for several commonly encountered non-homogeneous terms are used. Several numerical solutions of the deflection of rectilinear plates using the boundary element method (BEM) are presented and compared to existing numerical or exact solutions.     

Infinite boundary elements for electromagnetics

Consider a two‐dimensional plane wave transverse magnetic mode scattering from a perfectly electric conducting ground plane. Let the ground plane be of infinite extent and comprise two regions, a near field and a far field. In the far field, let the ground plane be flat and let us choose the co‐ordinates (x, y) such that it lies on the axis y=0. Over the interior region, let the profile of the ground plane change such that it can lie partially above and also partially below the axis y=0. Finally, let us assume that the source of the excitation lies above the ground plane. To model this general class of problems, a method of moments electric field integral equation formulation is proposed which uses infinite boundary elements to model the far field and boundary elements to model the near field. In the far field, the field variable is approximated by the highest order terms in the far‐field asymptotic expansion. The integrals over the infinite boundary elements are infinite in extent and contain oscillatory terms and hence require special integration rules. The formulation is tested for the specific problem of a semi‐circular cylindrical protrusion of radius a lying above an infinite flat ground plane, such that ka=1 where k is the wave number. This problem is chosen because it has an analytic solution in the form of a Bessel function expansion; hence, the accuracy of the formulation can be tested. In particular, the radar cross section results for various angles of incidence of the plane wave source are calculated and compared with analytic results. Copyright © 2007 John Wiley & Sons, Ltd.     

An element implementation of the boundary face method for 3D potential problems

This work presents a new implementation of the boundary face method (BFM) with shape functions from surface elements on the geometry directly like the boundary element method (BEM). The conventional BEM uses the standard elements for boundary integration and approximation of the geometry, and thus introduces errors in geometry. In this paper, the BFM is implemented directly based on the boundary representation data structure (B-rep) that is used in most CAD packages for geometry modeling. Each bounding surface of geometry model is represented as parametric form by the geometric map between the parametric space and the physical space. Both boundary integration and variable approximation are performed in the parametric space. The integrand quantities are calculated directly from the faces rather than from elements, and thus no geometric error will be introduced. The approximation scheme in the parametric space based on the surface element is discussed. In order to deal with thin and slender structures, an adaptive integration scheme has been developed. An adaptive method for generating surface elements has also been developed. We have developed an interface between BFM and UG-NX(R). Numerical examples involving complicated geometries have demonstrated that the integration of BFM and UG-NX(R) is successful. Some examples have also revealed that the BFM possesses higher accuracy and is less sensitive to the coarseness of the mesh than the BEM.     

High‐order finite volume schemes for the advection–diffusion equation

A high‐order finite volume method based on piecewise interpolant polynomials is proposed to discretize spatially the one‐dimensional and two‐dimensional advection–diffusion equation. Evolution equations for the mean values of each control volume are integrated in time by a classical fourth‐order Runge–Kutta. Since our work focuses on the behaviour of the spatial discretization, the time step is chosen small enough to neglect the time integration error. Two‐dimensional interpolants are built by means of one‐dimensional interpolants. It is shown that when the degree of the one‐dimensional interpolant q is odd, the proper selection of a fixed stencil gives rise to centred schemes of order q+1. In order not to lose precision due to the change of stencil near boundaries, the degree of the interpolants close to boundaries is raised to q+1. Four test cases with small values of diffusion are integrated with high‐order methods. It is shown that the spatial discretization of the advection–diffusion equation with periodic boundary conditions leads to normal discretization matrices, and asymptotic stability must be assured to bound the spatial discretization error. Once the asymptotic stability is assured by means of the spectra of the discretization matrix, the spatial error is of the order of the truncation error. However, it is shown that the discretization of the advection–diffusion equation with arbitrary boundary conditions gives rise to non‐normal matrices. If asymptotic stability is assured, the spatial order of steady solutions is of the order of the truncation error. But, for transient processes, the order of the spatial error is determined by both the truncation error and the norm of the exponential matrix of the spatial discretization. The use of the pseudospectra of the discretization matrix is proposed as a valuable tool to analyse the transient error of the numerical solution. Copyright © 2001 John Wiley & Sons, Ltd.     

Effective stress-based finite element error estimation for composite bodies

This paper presents a discretization error estimator for displacement-based finite element analysis applicable to multi-material bodies such as composites. The proposed method applies a specific stress continuity requirement across the intermaterial boundary consistent with physical principles. This approach estimates the discretization error by comparing the discontinuous finite element effective stress function with a smoothed (C⁰ continuous) effective stress function for non-intermaterial boundary elements with a smoothed pseudo-effective stress function for elements which lie on the intermaterial boundary. Examples are presented which illustrate the effectiveness of the multi-material error estimator. The pointwise pseudo-effective stress and the L₂ norm of the estimated stress error are seen to converge with mesh refinement, while Zienkiewicz and Zhu's error estimator failed to converge for elements on the intermaterial boundary due to the physically admissible stress discontinuities that exist on the intermaterial boundary.     

Boundary element discretization of plane elasticity and plate bending problems

The paper deals with the discretization of the integral equations arising in the boundary formulation of plane elasticity and plate bending problems. Particular attention is paid to the efficiency of the interpolation used in approximating the boundary quantities and to the precision and computational convenience in evaluating the boundary integrals. The proposed discretization model is based on the use of a quadratic B-spline approximation to represent the boundary variables and on the results from the analytical integration to compute the boundary coefficients. The advantages are those of accuracy and the saving of computer time. Some numerical results allow an analysis of the performance of the model.     

Calculation of electric field and force on conductor particles with a surface film

This paper applies the method of images, an analytical method, to calculate electric field on conductor particles with a surface film. The method utilizes the multipole re-expansion and appropriate fundamental solutions. Electric field is repetitively calculated so as to satisfy all the boundary conditions. The main advantage over numerical field-calculation methods is that high accuracy can be realized as neither approximation nor discretization of the particle surface or the film thickness is involved. The calculation results for arrangements of a particle chain under a uniform field show the field intensification due to the film thickness and electrical properties. We have also carried out field calculation by the boundary element method (BEM), and compared the results with the analytical ones. The results by the BEM exhibit higher error with decreasing film thickness. Force and yield stress have been calculated from the electric field and compared with experimental results. The comparison shows a good agreement for the ac field, but significant difference for the dc one.     

Acoustic problems analysis of 3D solid with small holes by fast multipole boundary face method

In this paper, an adaptive fast multipole boundary face method is introduced to implement acoustic problems analysis of 3D solids with open-end small tubular shaped holes. The fast multipole boundary face method is referred as FMBFM. These holes are modeled by proposed tube elements. The hole is open-end and intersected with the outer surface of the body. The discretization of the surface with circular inclusions is achieved by applying several special triangular elements or quadrilateral elements. In the FMBFM, the boundary integration and field variables approximation are both performed in the parametric space of each boundary face exactly the same as the B-rep data structure in standard solid modeling packages. Numerical examples for acoustic radiation in this paper demonstrated the accuracy, efficiency and validity of this method.     

A fast higher-order integral equation method for solution of the full potential equation around airfoils

A method based on an integral equation formulation is described for solution of the full potential equation in terms of the velocity field. In addition to the conventional distribution of singularities over the boundaries of field, a field source distribution is added in the flow region in order to represent the non-linear compressibility effect. The unknown source distribution in the field is calculated from the full potential equation by iteratively updating the normal velocity boundary conditions. In order to treat more complex configurations, local transformations provided by higher-order elements are used. Computation time required for integration of the domain is improved by using a domain decomposition. Results of calculations demonstrate substantial improvement in computation time and are in good agreement with independent results.     

Dynamic response of shear-deformable axisymmetric orthotropic circular plate structures solved by the DQEM and EDQ based time integration schemes

The dynamic response of shear-deformable axisymmetric orthotropic circular plate structures is solved by using the DQEM to the spacial discretization and EDQ to the temporal discretization. In the DQEM discretization, DQ is used to define the discrete element model. Discrete dynamic equilibrium equations defined at interior nodes in all elements, transition conditions defined on the inter-element boundary of two adjacent elements and boundary conditions at the structural boundary form a dynamic equation system at a specified time stage. The dynamic equilibrium equation system is solved by the direct time integration schemes of time-element by time-element method and stages by stages method which are developed by using EDQ and DQ. Numerical results obtained by the developed numerical algorithms are presented. They demonstrate the developed numerical solution procedure.     

Stochastic second-order BEM perturbation formulation

This paper presents the stochastic second order moment perturbation approach to the classical deterministic Boundary Element Method (BEM) formulation. Numerous applications of such a formulation in different problems of stochastic mechanics, especially in the field of computational modeling of structural defects in homogeneous and composite materials occurring randomly in solids and engineering structures, were the main reasons to introduce the proposed model. The stochastic boundary element method (SBEM) formulation of the general linear elasticity boundary value has been provided together with an appropriate discretization. The equations describing the expected values and the covariances of stress and strain tensors for points lying on the boundary and inside the region are considered. This set of equations constitutes a formal mathematical statement of the problem and is suitable for computational implementation.     

A three-dimensional boundary element formulation for the elastoplastic analysis of cracked bodies

In this paper a general boundary element formulation for the three-dimensional elastoplastic analysis of cracked bodies is presented. The non-linear formulation is based on the Dual Boundary Element Method. The continuity requirements of the field variables are fulfilled by a discretization strategy that incorporates continuous, semi-discontinuous and discontinuous boundary elements as well as continuous and semi-discontinuous domain cells. Suitable integration procedures are used for the accurate integration of the Cauchy surface and volume integrals. The explicit version of the initial strain formulation is used to satisfy the non-linearity. Several examples are presented to demonstrate the application of the proposed method. © 1998 John Wiley & Sons, Ltd.