Analysis of crack closure problem using the dual boundary element method

A numerical procedure of the crack closure problem solved by the dual boundary element method is developed in this paper. The dual boundary element method is used to allow for the solution to a general mixed-mode crack problem with a single regional formulation. The frictional contact problem on the crack surface is formulated with the complementary problem adapting the Coulomb's friction law. Several examples are shown to demonstrate the validity of the present procedure.     

Complementary variational principles for steady heat conduction with mixed boundary conditions

Summary New stationary, maximum and minimum principles associated with the boundary value problem of steady heat conduction with general boundary conditions are derived in a unified manner from the theory of complementary variational principles. One of the results contains the Brand-Lahey [3] stationary principle as a special case.     

Boundary element-linear complementary equation solution to polygonial unilaterally simply supported plates

This article describes a method for the bending analysis of simply supported plates allowing for lifting at corners. The boundary element-linear complementary equation method (BE–LCEM) is described and used to solve the underlying problem. Following numerical discretisation using the boundary integral equation method for this contact problem, an effective linear complementary equation (LCE) is then established with two complementary variables for each contact node. Complementary variables are taken as the normal contact force and boundary deflection of the plate. The resulting LCEs are solved by using mathematical programming. Details of the transformation are described. A number of examples are presented for different shapes of plates to demonstrate the effectiveness of the features as implemented. Results are compared of simply supported plates with corners allowing for and restrained from lifting.     

Boundary element analysis of unilateral supported Reissner plates on elastic foundations

 A boundary element–linear complementary equation method (BE–LCEM) is developed for the bending of thick plates with free edges on unilateral elastic foundations with particular emphasis on the non-contact phenomenon between the plates and the subgrades. The theory of thick plate was used, and three boundary conditions on free edge have been adopted. Following numerical discretization by using the boundary integral equation method for this contact problem, an effective linear complementary equation is then established with two complementary variables for each contact node. Complementary variables are taken as the normal contact force and the relative deflection between the plate and the foundation. The solution of which can be obtained using mathematical programming. A number of examples are presented to demonstrate the effectiveness of the features as implemented. Two types of foundations (Winkler and half-space) are examined and the method is shown to provide good agreement with available analytical solutions obtained by other investigators. Received 16 July 2000     

正交压电复合材料层板各类边界的解析解

压电复合材料层板由压电片与纤维层叠合而成。基于等效单层理论的位移场和电势场,针对正交铺层压电复合材料层板柱面弯曲问题,建立了力电耦合平衡方程,获得了一般边界的解析解。解析解由特解和通解两部分组成,特解对应于简支边界条件,通解由其他各类边界条件确定。平衡方程的变量仅4个,且不随层数变化。如采用相应的位移和电势分布函数,可以得到一阶理论、高阶理论、指数型理论等多种理论的解析解。算例中给出了各种边界条件下位移、应力和电势的解,讨论了各种理论的精度,观察到了固支边界的应力奇异现象。     

印刷全画面检测图像定位辅助标记研究

目的研究基于全画面检测系统图像的定位辅助标记及其选用原则。方法制作四色油墨的渐变印刷样条,用分光密度计测试密度值,然后扫描输出RGB值,研究不同颜色辅助标记在不同密度值下,对CCD图像传感器RGB响应值的影响规律,并推导出图像定位边界特征显著性描述的相关公式。结果黄色、青色、品红色油墨分别与CCD相机的蓝色、红色、绿色响应呈良好的补色关系,黑色油墨与CCD相机的蓝色、红色、绿色响应呈良好的中性关系;补色通道上的标记响应与主体图像的响应反差较大,其定位边界显著性更突出。结论一般情况下,选择与主色调呈补色关系的单色油墨作为定位辅助标记,当主色调较难分辨时,可考虑使用较为通用的黑色标记。     

Universal theorems for total energy of the dynamics of linearly elastic heterogeneous solids

In this paper we consider a sample of a linearly elastic heterogeneous composite in elastodynamic equilibrium and present universal theorems which provide lower bounds for the total elastic strain energy plus the kinetic energy, and the total complementary elastic energy plus the kinetic energy. For a general heterogeneous sample which undergoes harmonic motion at a single frequency, we show that, among all consistent boundary data which produce the same average strain, the uniform-stress boundary data render the total elastic strain energy plus the kinetic energy an absolute minimum. We also show that, among all consistent boundary data which produce the same average momentum in the sample, the uniform velocity boundary data render the total complementary elastic energy plus the kinetic energy an absolute minimum. We do not assume statistical homogeneity or material isotropy in our treatment, although they are not excluded. These universal theorems are the dynamic equivalent of the universal theorems already known for the static case [Nemat-Nasser and Hori, 1993] and [Nemat-Nasser and Hori, 1995]. It is envisaged that the bounds on the total energy presented in this paper will be used to formulate computable bounds on the overall dynamic properties of linearly elastic heterogeneous composites with arbitrary microstructures.     

Stability analysis of plates by a complementary energy method

A complementary energy method for the stability analysis of plates is presented. Buckling loads of rectangular plates with different boundary conditions are obtained, and the convergence of the solution is studied.     

Boundary integral equation method for shape optimization of elastic structures

A general method for shape design sensitivity analysis as applied to plane elasticity problems is developed with a direct boundary integral equation formulation, using the material derivative concept and adjoint variable method. The problem formulation is very general and a complete consideration is given to describing the boundary variation by including the tangential component of the velocity field. The method is then applied to obtain the sensitivity formula for a general stress constraint imposed over a small part of the boundary. The accuracy of the design sensitivity analysis is studied with a fillet and an elastic ring design problem. Among the several numerical implementations tested, the second order boundary elements with a cubic spline representation of the moving boundary have shown the best accuracy. A smooth characteristic function is found to be better than a plateau function for localization of the stress constraint. Optimal shapes for the two problems are presented to show numerical applications.     

A new BEM formulation for the acoustic eigenfrequency analysis

A new boundary element formulation has been developed for two- and three-dimensional acoustic eigenfrequency analyses. The formulation is based on the well known method of constructing a solution of a differential equation in terms of a complementary function and particular integral. An advanced isoparametric implementation with automatic error control in the integration is used. A number of realistic examples of application to automotive acoustic cavities are described.     

Fourier series solution for a rectangular thick plate with free edges on an elastic foundation

A Fourier series solution is presented for a system of first-order partial differential equations which describe the linear elastic behaviour of a thick rectangular plate resting on an elastic foundation and carrying an arbitrary transverse load. The lateral edges of the plate are unstressed. A central step in the method for solving the system of equations is to combine a complementary function with a particular solution of the system in order to satisfy the boundary conditions. The complementary function is the sum of two series. The terms of the first series are products of a Fourier term in one space variable with the solution of an eigenvalue problem in the other space variable. The second series is similar and comes from reversing the roles of the space variables.     

Heat conduction in an inhomogeneous half-space subject to a nonlinear boundary condition. Application of bergman-type series

Bergman-type series solutions involving iterated complementary error integrals are constructed for nonlinear boundary value problems for heat conduction in an inhomogeneous half-space. In particular, a small-time solution is developed when the nonlinear boundary condition is of the Stefan-Boltzmann type.     

Shape design sensitivity analysis via material derivative-adjoint variable technique for 3-D and 2-D curved boundary elements

A general approach to shape design sensitivity analysis of three- and two-dimensional elastic solid objects is developed using the material derivative-adjoint variable technique and boundary element method. The formulation of the problem is general and first-order sensitivities in the form of boundary integrals for the effect of boundary shape variations are derived for an arbitrary performance functional. Second-order quadrilateral surface elements (for 3-D problems) and quadratic boundary elements (for 2-D problems) are employed in the solution of primary and adjoint systems and discretization of the boundary integral expressions for sensitivities. The accuracy of sensitivity information is studied for selected global performance functionals and also for boundary state fields at discrete points. Numerical results are presented to demonstrate the accuracy and efficiency of this approach.     

Thermal convection in a fluid layer sandwiched between two porous layers of different permeabilities

Summary A linear stability analysis is carried out for the convective instability problem in a horizontal fluid layer sandwiched between two porous layers of different permeabilities. The velocity boundary condition is a general one and is via-media between the free and rigid boundary conditions. The thermal condition at the porous-fluid interface is assumed to be neither constant heat flux nor constant temperature, but a condition leading to a third-type of boundary condition. The principle of exchange of stability is valid for the problem and the critical eigenvalue is obtained for the general boundary condition using the single-term Rayleigh-Ritz technique. The results of several works are recovered as limiting cases of the present study.     

Comparison of highly efficient absorbing boundary conditions for the beam propagation method

Numerical experiments using the paraxial finite-difference beam propagation method have been performed with the following boundary conditions: perfectly matched layer, Higdon absorbing boundary conditions, complementary operators method, and extended complementary operators method. We have shown that Higdon operators must be modified for the paraxial wave equation to take into account the spectrum of incident rays on the boundaries of the computational domain. Reflection coefficients, accuracy, numerical dissipation/ gain, memory requirements, and time computation are compared and discussed for these absorbing techniques.     

Assumed stress function finite element method: Two-dimensional elasticity

A complementary energy-based finite element formulation, using assumed stress functions as the approximating functions, is developed for the linear elastic two-dimensional stress analysis. It features the use of blending function interpolants, enabling the convenient representation of traction boundary conditions, which in the past have posed difficulties. A family of rectangular elements is constructed. Numerical results assessing the behaviour of these elements are presented. An advantage of this approach is in the accurate prediction of stress distributions.     

A SINGLE-DOMAIN BOUNDARY ELEMENT METHOD FOR 3-D ELASTOSTATIC CRACK ANALYSIS USING CONTINUOUS ELEMENTS

A boundary element method is presented for single-domain analysis of cracked three-dimensional isotropic elastostatic solids. A numerical treatment for the hypersingular Boundary Integro-Differential Equation (BIDE) for displacement derivatives is described, in which continuous boundary elements may be used. Hadamard principal values of the hypersingular integrals arising in the formulation are evaluated using polar co-ordinates defined on the tangent planes at the source point, and the free term coefficients are calculated directly using a numerical technique. The forms of the Boundary Integral Equation (BIE) and the BIDE are considered for a source point on the coincident surfaces of a crack, and a scheme is given for defining the Traction Boundary Integral Equation TBIE so that it optimally incorporates the traction information deficient in its complementary partner, the BIE. Numerical results for some example mixed-mode crack problems are presented.     

A direct traction BIE approach for three-dimensional crack problems

A boundary element (BE) approach based on the traction boundary integral equation for the general solution of three-dimensional (3D) crack problems is presented. The hypersingular and strongly singular integrals appearing in the formulation are analytically transformed to yield line and surface integrals which are at most weakly singular. Regularization and analytical transformation of the boundary integrals is done prior to any boundary discretization. The integration process does not require of any change of coordinates and the resulting integrals can be numerically evaluated in a simple and efficient way. In order to show the generality, simplicity and robustness of the proposed approach, different flat and curved crack problems in infinite and finite domains are analyzed. A simple BE discretization strategy is adopted. The results obtained using rather course meshes are very accurate. The emphasis of this paper is on the effective application of the proposed BE approach and it is pretended to contribute to the transformation of hypersingular boundary element formulation in something as clear, general and easy to handle as the classical formulation but much better suited for fracture mechanics problems.     

Numerical solutions of hypersingular integral equation for curved cracks in circular regions

In this paper, numerical solutions of a hypersingular integral equation for curved cracks in circular regions are presented. The boundary of the circular regions is assumed to be traction free or fixed. The suggested complex potential is composed of two parts, the principle part and the complementary part. The principle part can model the property of a curved crack in an infinite plate. For the case of the traction free boundary, the complementary part can compensate the traction on the circular boundary caused by the principle part. Physically, the proposed idea is similar to the image method in electrostatics. By using the crack opening displacement (COD) as the unknown function and traction as right hand term in the equation, a hypersingular integral equation for the curved crack problems in the circular regions is obtained. The equation is solved by using the curve length coordinate method. In order to prove that the suggested method can be used to solve more complicated cases of the curved cracks, several numerical examples are given.     

Direct method of solution for general boundary value problem of the Laplace equation

A general boundary value problem for two-dimensional Laplace equation in the domain enclosed by a piecewise smooth curve is considered. The Dirichlet and the Neumann data are prescribed on respective parts of the boundary, while there is the second part of the boundary on which no boundary data are given. There is the third part of the boundary on which the Robin condition is prescribed. This problem of finding unknown values along the whole boundary is ill posed. In this sense we call our problem an inverse boundary value problem. In order for a solution to be identified the inverse problem is reformulated in terms of a variational problem, which is then recast into primary and adjoint boundary value problems of the Laplace equation in its conventional form. A direct method for numerical solution of the inverse boundary value problem using the boundary element method is presented. This method proposes a non-iterative and unified treatment of conventional boundary value problem, the Cauchy problem, and under- or over-determined problems.