Non‐singular boundary integral formulations for plane interior potential problems

In this article, a non‐singular formulation of the boundary integral equation is developed to solve smooth and non‐smooth interior potential problems in two dimensions. The subtracting and adding‐back technique is used to regularize the singularity of Green's function and to simplify the calculation of the normal derivative of Green's function. After that, a global numerical integration is directly applied at the boundary, and those integration points are also taken as collocation points to simplify the algorithm of computation. The result indicates that this simple method gives the convergence speed of order N ^?3 in the smooth boundary cases for both Dirichlet and mix‐type problems. For the non‐smooth cases, the convergence speed drops at O(N ^?1/2) for the Dirichlet problems. Copyright © 2001 John Wiley & Sons, Ltd.     

A complex‐variable boundary element approach to fully developed flow and heat transfer in ducts of general cross‐section

The complex‐variable boundary element method (CVBEM) is used to analyse forced convection in cooling passages with general, convex cross‐sections. Quadratic spline interpolation is employed for the modelling of coupled velocity and temperature fields. The method is well‐suited for ducts with curved surfaces and high geometric aspect ratios. The method is illustrated for ducts with rounded rectangular, elliptical and rounded diamond cross‐sections. General correlations are presented for the fully developed Nusselt number and Moody friction factor, and the Hagenbach entrance region factor. Copyright © 1999 John Wiley & Sons, Ltd.     

A Straightforward Direct Traction Boundary Integral Method for Two-Dimensional Crack Problems Simulation of Linear Elastic Materials

This paper presents a direct traction boundary integral equation method (DTBIEM) for two-dimensional crack problems of materials. The traction boundary integral equation was collocated on both the external boundary and either side of the crack surfaces. The displacements and tractions were used as unknowns on the external boundary, while the relative crack opening displacement (RCOD) was chosen as unknowns on either side of crack surfaces to keep the single-domain merit. Only one side of the crack surfaces was concerned and needed to be discretized, thus the proposed method resulted in a smaller system of algebraic equations compared with the dual boundary element method (DBEM). A new set of crack-tip shape functions was constructed to represent the strain field singularity exactly, and the SIFs were evaluated by the extrapolation of the RCOD. Numerical examples for both straight and curved cracks are given to validate the accuracy and efficiency of the presented method.     

3D NURBS‐enhanced finite element method (NEFEM)

This paper presents the extension of the recently proposed NURBS‐enhanced finite element method (NEFEM) to 3D domains. NEFEM is able to exactly represent the geometry of the computational domain by means of its CAD boundary representation with non‐uniform rational B‐splines (NURBS) surfaces. Specific strategies for interpolation and numerical integration are presented for those elements affected by the NURBS boundary representation. For elements not intersecting the boundary, a standard finite element rationale is used, preserving the efficiency of the classical FEM. In 3D NEFEM special attention must be paid to geometric issues that are easily treated in the 2D implementation. Several numerical examples show the performance and benefits of NEFEM compared with standard isoparametric or cartesian finite elements. NEFEM is a powerful strategy to efficiently treat curved boundaries and it avoids excessive mesh refinement to capture small geometric features. Copyright © 2011 John Wiley & Sons, Ltd.     

Regularized meshless method for multiply-connected-domain Laplace problems

In this paper, the regularized meshless method (RMM) is developed to solve two-dimensional Laplace problem with multiply-connected domain. The solution is represented by using the double-layer potential. The source points can be located on the physical boundary by using the proposed technique to regularize the singularity and hypersingularity of the kernel functions. The troublesome singularity in the traditional methods is avoided and the diagonal terms of influence matrices are easily determined. The accuracy and stability of the RMM are verified in numerical experiments of the Dirichlet, Neumann, and mixed-type problems under a domain having multiple holes. The method is found to perform pretty well in comparison with the boundary element method.     

A Cartesian‐grid collocation technique with integrated radial basis functions for mixed boundary value problems

In this paper, high‐order systems are reformulated as first‐order systems, which are then numerically solved by a collocation method. The collocation method is based on Cartesian discretization with 1D‐integrated radial basis function networks (1D‐IRBFN) (Numer. Meth. Partial Differential Equations 2007; 23 :1192–1210). The present method is enhanced by a new boundary interpolation technique based on 1D‐IRBFN, which is introduced to obtain variable approximation at irregular points in irregular domains. The proposed method is well suited to problems with mixed boundary conditions on both regular and irregular domains. The main results obtained are (a) the boundary conditions for the reformulated problem are of Dirichlet type only; (b) the integrated RBFN approximation avoids the well‐known reduction of convergence rate associated with differential formulations; (c) the primary variable (e.g. displacement, temperature) and the dual variable (e.g. stress, temperature gradient) have similar convergence order; (d) the volumetric locking effects associated with incompressible materials in solid mechanics are alleviated. Numerical experiments show that the proposed method achieves very good accuracy and high convergence rates. Copyright © 2009 John Wiley & Sons, Ltd.     

Global parameterization of a topological surface defined as a collection of trimmed bi‐parametric patches: Application to automatic mesh construction

We describe a new geometric algorithm to map surfaces into a plane convex area. The mapping transformation is bijective; it redefines the whole surface as a unique bi‐parametric patch. Thus this mapping provides a global parametrization of the surface. The surfaces are issued from industrial CAD software; they are usually described by a large number of patches and there are many shortcomings. Indeed, the decomposition into patches depends on the algorithm of the geometric modelling system used for design and usually has no meaning for any technological application. Moreover, in many cases, the surface definition is not compatible, i.e. patches are not well connected, some patches are self‐intersecting or intersect each other. Many applications are hard to address because of these defects. In this paper we show how patch‐independent meshing techniques may be easily automated using a unique metric in a plane parametric space. Thus we provide an automatic procedure to build valid meshes over free‐form surfaces issued from industrial CAD software (Computer Aided Design: this terminology should refer to a large amount of software. For the scope of this paper we only refer to geometric modelling systems. Indeed geometric modelling systems remain the kernel of many CAD software). Copyright © 2002 John Wiley & Sons, Ltd.     

Weighted radial basis collocation method for boundary value problems

This work introduces the weighted radial basis collocation method for boundary value problems. We first show that the employment of least‐squares functional with quadrature rules constitutes an approximation of the direct collocation method. Standard radial basis collocation method, however, yields a larger solution error near boundaries. The residuals in the least‐squares functional associated with domain and boundary can be better balanced if the boundary collocation equations are properly weighted. The error analysis shows unbalanced errors between domain, Neumann boundary, and Dirichlet boundary least‐squares terms. A weighted least‐squares functional and the corresponding weighted radial basis collocation method are then proposed for correction of unbalanced errors. It is shown that the proposed method with properly selected weights significantly enhances the numerical solution accuracy and convergence rates. Copyright © 2006 John Wiley & Sons, Ltd.     

3D散乱数据点分段二次逼近的曲面拟合

为进一步提高曲面重构的保形性及高效性,提出了一种自动构建光顺三角曲面的方法．该法首先通过构建三角形元覆盖边界域来构建一张曲面近似粗网,然后从点集中不断添加新点直至达到指定的容差,在每个插入数据点处构造C1连续的分片二次逼近面片,最终整体的C¹曲面由各三角形上的曲面片拼合而成．最后给出了该方法在真实点集上的运用结果并与其他方法所构造的逼近曲面形状进行了比较,结果表明,该方法对密集3D散乱数据点建模有效,生成的曲面质量高,误差小．该方法也适用于数据精简．     

A spectral collocation method with triangular boundary elements

A spectral collocation method is proposed for solving integral equations arising from boundary integral formulations over surfaces discretized into flat or curved triangular elements. In the numerical approximation, a function of interest defined over a triangular element is approximated with an arbitrary-degree complete polynomial of two local triangle barycentric coordinates. Collocation points are then deployed at the nodes of a triangular Lobatto grid constructed on the basis of the zeros of the Lobatto polynomials, so that the number of collocation points over each element is equal to the number of terms in a complete polynomial expansion. The node interpolation functions are computed from the Proriol polynomial base using the generalized Vandermonde matrix approach. The spectral element method is applied to solve integral equations of the second kind arising from the double-layer representation of a harmonic function in the interior or exterior of a sphere. The numerical results confirm a rapid convergence with respect to the order of the polynomial expansion.     

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.     

The collocation solution of Poisson problems based on approximate Fekete points

The aim of this work is to show how the collocation method may be used for the approximate solution of Poisson problems on planar domains with a smooth boundary in a stable and efficient way. The most important aspect of this work consists in the use of approximate Fekete points recently developed by Sommariva and Vianello. Numerical experiments concerning the collocation solution of Poisson problems defined on the unit disc and an eccentric annulus with the homogeneous Dirichlet boundary conditions are presented. Two sets of trial functions, consisting of algebraic polynomials, satisfying and not satisfying the prescribed boundary conditions are considered. As the presented results show, these easily computable collocation points, giving well-conditioned collocation matrices, open new horizons for the collocation solution of elliptic partial differential equations considered on planar and higher-dimensional domains.     

Bezier三角组合曲面的局域设计

 在三角组合曲面的设计中，需要根据曲面间的拼接条件，协调求解曲面的内部控制顶点．在分析Bezier三角组合曲面设计方法的基础上，提出了Bezier三角曲面GC 连续局域设计方法． 内部分割点采用九参数三次曲面设计方法估算，由内部分割点将三角曲面分割为3个子曲面片，通过构造曲面边界的跨界过渡切矢，推导得出了3个分割子曲面的内部控制顶点的代数表达式．应用该方法，可以简化三角组合曲面的设计过程，提高三角组合曲面的设计计算速度．     

Analysis of singular stress fields at multi‐material corners under thermal loading

The scaled boundary finite‐element method is extended to the modelling of thermal stresses. The particular solution for the non‐homogeneous term caused by thermal loading is expressed as integrals in the radial direction, which are evaluated analytically for temperature changes varying as power functions of the radial coordinate. When applied to model a multi‐material corner, only the boundary of the problem domain is discretized. The boundary conditions on the straight material interfaces and the side‐faces forming the corner are satisfied analytically without discretization. The stress field is expressed semi‐analytically as a series solution. The stress distribution along the radial direction, including both the real and complex power singularity and the power‐logarithmic singularity, is represented analytically. The stress intensity factors are determined directly from their definitions in stresses. No knowledge on asymptotic expansions is required. Numerical examples are calculated to evaluate the accuracy of the scaled boundary finite‐element method. Copyright © 2005 John Wiley & Sons, Ltd.     

Near‐crack contour behaviour and extraction of log‐singular stress terms of the self‐regular traction boundary integral equation

This work contains an analytical study of the asymptotic near‐crack contour behaviour of stresses obtained from the self‐regular traction‐boundary integral equation (BIE), both in two and in three dimensions, and for various crack displacement modes. The flat crack case is chosen for detailed analysis of the singular stress for points approaching the crack contour. By imposing a condition of bounded stresses on the crack surface, the work shows that the boundary stresses on the crack are in fact zero for an unloaded crack, and the interior stresses reproduce the known inverse square root behaviour when the distance from the interior point to the crack contour approaches zero. The correct order of the stress singularity is obtained after the integrals for the self‐regular traction‐BIE formulation are evaluated analytically for the assumed displacement discontinuity model. Based on the analytic results, a new near‐crack contour self‐regular traction‐BIE is proposed for collocation points near the crack contour. In this new formulation, the asymptotic log‐singular stresses are identified and extracted from the BIE. Log‐singular stress terms are revealed for the free integrals written as contour integrals and for the self‐regularized integral with the integration region divided into sub‐regions. These terms are shown to cancel each other exactly when combined and can therefore be eliminated from the final BIE formulation. This work separates mathematical and physical singularities in a unique manner. Mathematical singularities are identified, and the singular information is all contained in the region near the crack contour. Copyright © 2003 John Wiley & Sons, Ltd.     

Optimum interpolation functions for boundary element method

In the Boundary Element Method (BEM) the density functions are approximated by interpolation functions which are chosen to satisfy appropriate continuity requirements. The error of approximation inside an element depends upon the location of the collocation points that are used in constructing the interpolation functions. The location of collocation points also affects the nodal values of the density function and, hence, the total error in the analysis if boundary conditions are satisfied in a collocation sense. In this paper, we minimize the error inside the element using the L₁ norm to obtain the optimum location of collocation points. Results show that irrespective of the continuity requirement at the element end, the location of collocation points computed by the algorithm presented in this paper results in an error that is less than the error corresponding to uniformly spaced collocation points. Results for optimum location of collocation points and the average error are presented for Lagrange polynomials up to order fifteen and for Hermite polynomials that ensure continuity up to the seventh order of derivative at the element end. The information of the optimum location of interpolation points for Lagrange and Hermite polynomials should be useful to other researchers in BEM who could incorporate it into their current programs without making significant changes that would be needed for incorporating the algorithm. The algorithm presented is independent of the BEM application in two-dimensions, provided that the density functions are approximated by polynomials and is applicable to direct and indirect formulations. Two numerical examples show the application of the algorithm to an elastostatic problem in which one boundary is represented by integrals of the Direct BEM while the other boundary by the Indirect BEM and a fracture mechanics problem by Direct method in which the crack is represented by displacement discontinuity density function.     

A hybrid finite difference and moving least square method for elasticity problems

In this paper, a novel hybrid finite difference and moving least square (MLS) technique is presented for the two-dimensional elasticity problems. A new approach for an indirect evaluation of second order and higher order derivatives of the MLS shape functions at field points is developed. As derivatives are obtained from a local approximation, the proposed method is computationally economical and efficient. The classical central finite difference formulas are used at domain collocation points with finite difference grids for regular boundaries and boundary conditions are represented using a moving least square approximation. For irregular shape problems, a point collocation method (PCM) is applied at points that are close to irregular boundaries. Neither the connectivity of mesh in the domain/boundary or integrations with fundamental/particular solutions is required in this approach. The application of the hybrid method to two-dimensional elastostatic and elastodynamic problems is presented and comparisons are made with the boundary element method and analytical solutions.     

A point collocation method based on reproducing kernel approximations

A reproducing kernel particle method with built‐in multiresolution features in a very attractive meshfree method for numerical solution of partial differential equations. The design and implementation of a Galerkin‐based reproducing kernel particle method, however, faces several challenges such as the issue of nodal volumes and accurate and efficient implementation of boundary conditions. In this paper we present a point collocation method based on reproducing kernel approximations. We show that, in a point collocation approach, the assignment of nodal volumes and implementation of boundary conditions are not critical issues and points can be sprinkled randomly making the point collocation method a true meshless approach. The point collocation method based on reproducing kernel approximations, however, requires the calculation of higher‐order derivatives that would typically not be required in a Galerkin method, A correction function and reproducing conditions that enable consistency of the point collocation method are derived. The point collocation method is shown to be accurate for several one and two‐dimensional problems and the convergence rate of the point collocation method is addressed. Copyright © 2000 John Wiley & Sons, Ltd.     

The method of fundamental solutions applied to 3D elasticity problems using a continuous collocation scheme

This paper presents a new formulation of the Method of Fundamental Solutions (MFS) for three-dimensional elasticity problems. The idea of the presented formulation is to integrate both sides of the conventional MFS superposition equation over the body surface. This leads to the proposed “continuous collocation”. Therefore, the surface of the analysed body has to be discretised into boundary elements. In order to make use of the conventional MFS advantage of being a meshless method, a mixed formulation, using the proposed continuous collocation formulation and the conventional meshless formulation, is employed. The present formulation proved its validity and accuracy when analysing problems with non-constant boundary conditions and when computing the tangential stress component on the boundary. It is also shown that the proposed formulation is suitable and accurate for the analysis of thin and slender bodies.     

Boundary element analysis of curved cracked panels with adhesively bonded patches

A new boundary element formulation for analysis of curved cracked panels with adhesively bonded patches is presented in this paper. The effect of the adhesive layer is modelled by distributed body forces (i.e. two in‐plane forces, two moments and one out‐of‐plane force). A coupled boundary integral formulation of a shear deformable plate and two‐dimensional plane stress elasticity is used to determine bending and membrane forces along the adhesive layer taking into consideration the compatibility conditions in the patch area. Two numerical examples are presented to demonstrate the efficiency of the proposed method. It is shown that the out‐of‐plane bending behaviour and panel curvature have significant influence on the magnitude of the stress intensity factors. Copyright © 2003 John Wiley & Sons, Ltd.