A structural shape optimization system using the two-dimensional boundary contour method

Summary The research recently conducted has demonstrated that the Boundary Contour Method (BCM) is very competitive with the Boundary Element Method (BEM) in linear elasticity Design Sensitivity Analysis (DSA). Design Sensitivity Coefficients (DSCs), required by numerical optimization methods, can be efficiently and accurately obtained by two different approaches using the two-dimensional (2-D) BCM as presented in Refs. [1] and [2]. These approaches originate from the Boundary Integral Equation (BIE). As discussed in [2], the DSCs given by both BIE-based DSA approaches are identical, and thus the users can choose either of them in their applications. In order to show the advantages of this class of DSA in structural shape optimization, an efficient system is developed in which the BCM as well as a BIE-based DSA approach are coupled with a mathematical programming algorithm to solve optimal shape design problems. Numerical examples are presented. Received 20 July 1998; accepted for publication 7 December 1998     

弹性力学问题中一个新的边界积分方程——自然边界积分方程

位移导数边界积分方程一直存在着超奇异积分计算的障碍,该文提出以符号算子δye和εye作用于位移导数边界积分方程,施用一系列变换将边界位移、面力和位移导数转成为新的边界张量,从而得到一个新的边界积分方程－－自然边界积分方程,自然边界积分方奇异性为强奇性,文中给出了相应的Cauchy主值积分算式,自然边界积分方程与位移边界积分方程联合可直接获取边界应力,几个算例表明了自然边界积分方程的正确性。     

断裂力学问题的杂交边界点方法

提出了一种求解断裂力学的新的边界类型无网格方法-杂交边界点法.以修正变分原理和移动最小二乘近似为基础,同时具有边界元法和无网格法的优良特性,求解时仅仅需要边界上离散点的信息.该文将杂交边界点方法应用到弹性断裂问题中,将移动最小二乘方法中的基函数扩充,能更好的模拟裂纹尖端应力场的奇异性,推导了求解断裂力学的杂交边界点法方程,与传统的元网格方法相比,文中方法具有后处理简单,计算精度高的优点.数值算例表明了该方法的稳定性和有效性.     

MESHLESS ANALYSIS FOR THREE-DIMENSIONAL ELASTICITY WITH SINGULAR HYBRID BOUNDARY NODE METHOD

The singular hybrid boundary node method (SHBNM) is proposed for solving three-dimensional problems in linear elasticity. The SHBNM represents a coupling between the hybrid displacement variational formulations and moving least squares (MLS) approximation. The main idea is to reduce the dimensionality of the former and keep the meshless advantage of the later. The rigid movement method was employed to solve the hyper-singular integrations. The 'boundary layer effect', which is the main drawback of the original Hybrid BNM, was overcome by an adaptive integration scheme. The source points of the fundamental solution were arranged directly on the boundary. Thus the uncertain scale factor taken in the regular hybrid boundary node method (RHBNM) can be avoided. Numerical examples for some 3D elastic problems were given to show the characteristics. The computation results obtained by the present method are in excellent agreement with the analytical solution. The parameters that influence the performance of this method were studied through the numerical examples.     

导数场边界积分方程分析近边界应力分布

研究二维弹性力学问题边界积分方程,通过分部积分变换消除了常规导数边界积分方程中的超奇异积分,获得仅含强奇异积分的应力自然边界积分方程.对于近边界应力的计算,进一步运用正则化算法解析计算其中的几乎强奇异积分.较常规边界元法相比,应力自然边界积分方程可以求解离边界更加接近的内点应力值.算例证明了文中方法的可应用性和有效性.     

关于时域边界元法的若干研究

本文从三维弹性动力学方程的基本奇异解着手，导出了适于计算机计算的求解三弹性动力学问题的边界积分方程（ＢＩＥ），并在理论上提出了ＩＢＥ前缘系数矩阵（５）具有（１）准对角特性，（２）其各元素不随时间而变化。据此，本文给出了用时域边界元法求解弹性动力学问题的新方法。最后，数值算例验证了本文方法的正确性。     

弹性理论几类导数边界积分方程之间的变换关系

导数场边界积分方程通常难以应用，因为存在着超奇异主值积分的计算障碍。弹性理论中有几类不同的位移导数边界积分方程，本文采用算子δij和∈ij(排列张量)作用于这些导数边界积分方程，做一系列变换，原有的超奇异积分被正则化为强奇异积分获解。从而建立了这些位移导数边界积分方程之间的转换关系，它们均可以归结为自然边界积分方程。自然边界积分方程仅存在容易计算的Cauchy主值积分。自然边界积分方程分析可直接获得边界应力和位移导数。     

DUAL RECIPROCITY HYBRID BOUNDARY NODE METHOD FOR THREE-DIMENSIONAL ELASTICITY WITH BODY FORCE

Combining Dual Reciprocity Method （DRM） with Hybrid Boundary Node Method （HBNM）, the Dual Reciprocity Hybrid Boundary Node Method （DRHBNM） is developed for three-dimensional linear elasticity problems with body force. This method can be used to solve the elasticity problems with body force without domain integral, which is inevitable by HBNM. To demonstrate the versatility and the fast convergence of this method, some numerical examples of 3-D elasticity problems with body forces are examined. The computational results show that the present method is effective and can be widely applied in solving practical engineering problems.     

A NOVEL BOUNDARY INTEGRAL EQUATION METHOD FOR LINEAR ELASTICITY--NATURAL BOUNDARY INTEGRAL EQUATION

The boundary integral equation (BIE) of displacement derivatives is put at a disadvantage for the difficulty involved in the evaluation of the hypersingular integrals. In this paper, the operators δij and εij are used to act on the derivative BIE. The boundary displacements, tractions and displacement derivatives are transformed into a set of new boundary tensors as boundary variables. A new BIE formulation termed natural boundary integral equation (NBIE) is obtained. The NBIE is applied to solving two-dimensional elasticity problems. In the NBIE only the strongly singular integrals are contained. The Cauchy principal value integrals occurring in the NBIE are evaluated. A combination of the NBIE and displacement BIE can be used to directly calculate the boundary stresses. The numerical results of several examples demonstrate the accuracy of the NBIE.     

A local search algorithm for natural neighbours in the natural element method

A local basis algorithm for searching natural neighbours in Natural Element Method (NEM) is presented for solving the elasticity problems in this paper. Comparison with the global sweep algorithm used in natural element method or Natural Neighbour Method (NNM) for searching natural neighbours, the proposed algorithm is more expedient and convenient in the constructions and computation of natural neighbour interpolations. In the proposed NEM based on local search, the Laplace (non-sibson) interpolations are constructed with respect to the natural neighbour nodes of the given point which have been locally defined. The shape functions from the Laplace approximations have the delta function property and the Laplace interpolants are strictly linear between adjacent nodes, which facilitate imposition of essential boundary conditions and treatment of material discontinuity with ease as it is in the conventional finite element method. The Laplace interpolants derived from the local algorithm and the global algorithm in NEM are identical because of the uniqueness of the Voronoi diagram. Numerical results and convergence studies also show that the present NEM based on local search algorithm possesses the same accuracy and rate of convergence as they are in previous NEM.     

弹性力学平面问题的等价边界积分方程的边界轮廓法

基于边界积分方程中被积函数散度为零的特性,提出了弹性力学平面问题的等价边界积分方程的边界轮廓法,该方法无需进行数值积分,只需要计算单元两结点势函数值之差。实例计算说明,基于传统的边界积分方程的边界轮廓法所得到的面力结果是错误,而本文建立的边界轮廓法则可给出精确的结果。     

C1问题无单元法导函数递推公式

针对无单元方法MLS插值函数，采用正交基函数的导函数进行了详细推演，给出适用于薄板弯曲(C^1)问题的无单元方法导函数递推计算公式．     

弹性力学的一种边界无单元法

首先对移动最小二乘副近法进行了研究，针对其容易形成病态方程的缺点，提出了以带权的正交函数作为基函数的方法－改进的移动最小二乘副近法，改进的移动最小二乘逼近法比原方法计算量小，精度高，且不会形成病态方程组，然后，将弹性力学的边界积分方程方法与改进的移动最小二乘逼近法结合，提出了弹性力学的一种边界无单元法，这种边界无单元法法是边界积分方程的无网格方法，与原有的边界积分方程的无网格方法相比，该方法直接采用节点变量的真实解为基本未知量，是边界积分方程无网格方法的直接解法，更容易引入界条件，且具有更高的精度，最后给出了弹性力学的边界无单元法的数值算例，并与原有的边界积分方程的无网格方法进行了较为详细的比较和讨论。     

The minimal error method for the Cauchy problem in linear elasticity. Numerical implementation for two-dimensional homogeneous isotropic linear elasticity

In this paper, yet another iterative procedure, namely the minimal error method (MEM), for solving stably the Cauchy problem in linear elasticity is introduced and investigated. Furthermore, this method is compared with another two iterative algorithms, i.e. the conjugate gradient (CGM) and Landweber–Fridman methods (LFM), previously proposed by Marin et al. [Marin, L., Háo, D.N., Lesnic, D., 2002b. Conjugate gradient-boundary element method for the Cauchy problem in elasticity. Quarterly Journal of Mechanics and Applied Mathematics 55, 227–247] and Marin and Lesnic [Marin, L., Lesnic, D., 2005. Boundary element-Landweber method for the Cauchy problem in linear elasticity. IMA Journal of Applied Mathematics 18, 817–825], respectively, in the case of two-dimensional homogeneous isotropic linear elasticity. The inverse problem analysed in this paper is regularized by providing an efficient stopping criterion that ceases the iterative process in order to retrieve stable numerical solutions. The numerical implementation of the aforementioned iterative algorithms is realized by employing the boundary element method (BEM) for two-dimensional homogeneous isotropic linear elastic materials.     

A regular boundary element method for fluid flow

The Boundary Element Method is now well established as a valid numerical technique for the solution of field problems, equal to the Finite Element Method in generality and surpassing it in computational efficiency in some cases.¹ In this paper is presented a 'Regular Boundary Element Method' as applied to inviscid laminar fluid flow problems. It involves the formation of a system of regular integral equations obtained by moving the singularity outside the domain of the given problem. It is also shown that non-conforming elements may be used whereby freedoms are not defined at the geometric nodes under the boundary element discretization. A linear element is developed here; higher order variants could easily be defined. Satisfactory numerical results have been obtained using the proposed regular method with both conventional (continuous across the boundary) and non-conforming boundary elements for two-dimensional inviscid laminar fluid flow problems having regular and singular solutions.     

MESHLESS METHOD OF DUAL RECIPROCITY HYBRID RADIAL BOUNDARY NODE METHOD FOR ELASTICITY

Combining the radial point interpolation method （RPIM）, the dual reciprocity method （DRM） and the hybrid boundary node method （HBNM）, a dual reciprocity hybrid radial boundary node method （DHRBNM） is proposed for linear elasticity. Compared to DHBNM, RPIM is exploited to replace the moving least square （MLS） in DHRBNM, and it gets rid of the deficiency of MLS approximation, in which shape functions lack the delta function property, the boundary condition can not be applied easily and directly and it＇s computational expense is high. Besides, different approximate functions are discussed in DRM to get the interpolation property, in which the accuracy and efficiency for different basis functions are compared. Then RPIM is also applied in DRM to replace the conical function interpolation, which can greatly improve the accuracy of the present method. To demonstrate the effectiveness of the present method, DHBNM is applied for comparison, and some numerical examples of 2-D elasticity problems show that the present method is much more effective than DHBNM.     

THE MESHLESS VIRTUAL BOUNDARY METHOD AND ITS APPLICATIONS TO 2D ELASTICITY PROBLEMS

A novel numerical method for eliminating the singular integral and boundary effect is processed. In the proposed method, the virtual boundaries corresponding to the numbers of the true boundary arguments are chosen to be as simple as possible. An indirect radial basis function network (IRBFN) constructed by functions resulting from the indeterminate integral is used to construct the approaching virtual source functions distributed along the virtual boundaries. By using the linear superposition method, the governing equations presented in the boundaries integral equations (BIE) can be established while the fundamental solutions to the problems are introduced. The singular value decomposition (SVD) method is used to solve the governing equations since an optimal solution in the least squares sense to the system equations is available. In addition, no elements are required, and the boundary conditions can be imposed easily because of the Kronecker delta function properties of the approaching functions. Three classical 2D elasticity problems have been examined to verify the performance of the method proposed. The results show that this method has faster convergence and higher accuracy than the conventional boundary type numerical methods.     

A novel non-primitive Boundary Integral Equation Method for three-dimensional and axisymmetric Stokes flows

A new Boundary Integral Equation (BIE) formulation for Stokes flow is presented for three-dimensional and axisymmetrical problems using non-primitive variables, assuming velocity field is prescribed on the boundary. The formulation involves the vector potential, instead of the classical stream function, and all three components of the vorticity are implied. Furthermore, following the Helmholtz decomposition, a scalar potential is added to represent the solenoidal velocity field. Firstly, the BIEs for three-dimensional flows are formulated for the vector potential and the vorticity by employing the fundamental solutions in free space of vector Laplace and biharmonic equations. The equations for axisymmetric flows are then derived from the three-dimensional formulation in a second step. The outcome is a domain integral free BIE formulation for both three-dimensional and axisymmetric Stokes flows with prescribed velocity boundary condition. Numerical results are included to validate and show the efficiency of the proposed axisymmetric formulation.     

Variational principles in elasticity with nonlinear stress-strain relation

Since 1979, a series of papers have been published concerning the variational principles and generalized variational principles in elasticity such as [1] (1979), [6] (1980), [2,3] (1983) and [4,5] (1984). All these papers deal with the elastic body with linear stress-strain relations. In 1985, a book was published on generalized variational principles dealing with some nonlinear elastic body, but never going into detailed discussion. This paper discusses particularly variational principles and generalized variational principles for elastic body with nonlinear stress-strain relations. In these discussions, we find many interesting problems worth while to pay some attention. At the same time, these discussions are also instructive for linear elastic problems. When the strain is small, the high order terms may be neglected, the results of this paper may be simplified to the well-known principles in ordinary elasticity problems.     

The use of displacement potentials in second order elasticity

This paper is concerned with the use of a representation in terms of displacement potentials in second order elasticity for equilibrium problems of homogeneous and isotropic materials. After justifying the adoption of an existing representation for linear elasticity for the purpose at hand, appropriate representations for solutions of second order elasticity problems in terms of displacement potentials (for both compressible and incompressible materials) are discussed. The use of the representations in obtaining complete solutions for equilibrium boundary-value problems is then illustrated by application to two examples of plane strain problems of compressible materials.