Numerical analysis of the inclusion-crack interactions using an integral equation

 A general-purpose integral formulation is proposed for the analysis of the interaction between inclusions and cracks embedded in an elastic isotropic homogeneous infinite medium subjected to a remote loading. This formulation is tailored for the inclusions of arbitrary shapes with the presence of cracks. The discretization is limited to the inclusions (with continuous quadratic triangular and quadrilateral elements) and the cracks (using discontinuous quadratic elements). For the calculation of the stress intensity factors at the crack tips, special crack tip elements are used to model the variation of the displacements near the crack tips. Maximum circumferential stress criterion is adopted to determine the crack propagating direction. Numerical results of benchmark examples are compared with other available methods. Received: 8 January 2002 / Accepted: 24 September 2002     

The integral equation formulations of an infinite elastic medium containing inclusions, cracks and rigid lines

The integral equation formulations of an infinite homogeneous isotropic medium containing various inclusions, cracks and rigid lines are presented. The present integral equation formulations contain the displacements (no tractions) over the inclusion-matrix interfaces, the discontinuous displacements over crack surfaces and the axial and the shear forces along rigid-line inclusions. Besides, the sub-domain boundary element method is also used in the present research. Numerical results from the present method and the sub-domain boundary element method are compared and discussed.     

Interaction between cracks and rigid-line inclusions by an integral equation approach

 An integral equation approach is presented to investigate the interaction between cracks and rigid-line inclusions embedded in an infinite isotropic elastic matrix subject to remote loading. The relevant fundamental solutions in the integral formulation are presented. Special tip elements are used to simulate the variation of the discontinuous displacements over the crack surfaces, and the axial and shear forces along the rigid-line inclusions. The stress intensity factors at the crack tips and at the ends of the rigid-line inclusions are computed and compared with available solutions. Received: 6 August 2002 / Accepted: 3 February 2003 The work described in this paper was partially supported by a grant from the Research Grant Council of the Hong Kong Special Administration Region, China (Project No.: HKU 7011/01E). The authors would like to thank two reviewers for their constructive comments and suggestions to the paper. The comments of Professor H.P. Hong at the Department of Civil and Environmental Engineering of the University of Western Ontario of Canada are also appreciated.     

Boundary-integral equation analysis of twisted internally cracked axisymmetric bimaterial elastic solids

 Green's function is obtained for the infinite bimaterial elastic solid, containing an internal circular interface crack, loaded by a unit tangential co-axial circular source. An axisymmetric direct boundary integral equation (BIE) is used for the analysis of a finite bimaterial axisymmetric body containing an internal circular interface crack and a finite homogeneous cracked cylinder, both under torsional loading. Using the proposed technique, no discretization of the crack surface is necessary. Numerical results for both examples as obtained by the proposed method are presented and discussed. Received: 29 October 2001 / Accepted: 29 May 2002     

Study on the propagation of inlet flow distortion in axial compressor using an integral method

 An integral method is investigated and developed in the current work. The effects of the parameters of inlet distortions on the trend of downstream flow feature in compressor are simulated. Other than the drag-to-lift ratio of the blade and the inlet incidence angle, it is found that the distorted inlet velocity is another essential parameter to control the distortion in propagation. Based on this study, a novel critical distortion line and corresponding critical distortion factor are proposed to express the effect of the two essential inlet parameters on the propagation of distortion, namely, the inlet incidence angle and the distorted inlet velocity. From the viewpoint of compressor efficiency, the propagation of inlet flow distortion is further described by a compressor critical performance and its critical characteristic. The results present a useful physical insight to an axial flow compressor behavior and asymptotic behavior of the propagation of inlet distortion, and confirm the active role of compressor in determining the velocity distribution when compressor responds to an inlet flow distortion. Received: 20 December 2001 / Accepted: 21 August 2002 The authors would like to thank HQ RSAF for permission to publish this work, their financial support and encouragement. The first author wants to acknowledge Prof. Frank Marble of California Institute of Technology, for bringing the problem to the author's attention and for his helpful discussion.     

A semi-analytical approach to three-dimensional normal contact problems with friction

 We present a semi-analytical approach for three-dimensional elastostatic normal contact problems with friction. The numerical approach to iteration on contact area and stick zone size is supported by an underlying analytical solution relating normal and tangential surface tractions to surface displacements in three coordinate directions. The governing equations are fully coupled. The analytical surface displacement solutions for a basic loading element have been derived elsewhere (Li and Berger 2001), and the total surface displacements are constructed as a superposition of deflections due to overlapping pyramid load segments. This approach requires no interpolation scheme for the field variables, which distinguishes it from other numerical techniques such as the FEM, BEM, and meshless methods. A background grid is defined only on the contact surfaces, and iteration approaches are used to determine a convergent configuration for contact domain and stick zone size. The approach is exercised on several normal contact problems, with and without friction, and the results compare favorably to existing analytical and numerical solutions. Received: 10 July 2002 / Accepted: 3 December 2002 The authors appreciate the support of the UC Department of Mechanical Engineering and the UC Office of the Vice President for Research, who jointly provided funds for this work.     

A Trefftz function approximation in local boundary integral equations

 In the present paper the Trefftz function as a test function is used to derive the local boundary integral equations (LBIE) for linear elasticity. Since Trefftz functions are regular, much less requirements are put on numerical integration than in the conventional boundary integral method. The moving least square (MLS) approximation is applied to the displacement field. Then, the traction vectors on the local boundaries are obtained from the gradients of the approximated displacements by using Hooke's law. Nodal points are randomly spread on the domain of the analysed body. The present method is a truly meshless method, as it does not need a finite element mesh, either for purposes of interpolation of the solution variables, or for the integration of the energy. Two ways are presented to formulate the solution of boundary value problems. In the first one the local boundary integral equations are written in all nodes (interior and boundary nodes). In the second way the LBIE are written only at the interior nodes and at the nodes on the global boundary the prescribed values of displacements and/or tractions are identified with their MLS approximations. Numerical examples for a square patch test and a cantilever beam are presented to illustrate the implementation and performance of the present method. Received 6 November 2000     

The ponding problem on elastic membranes: an analog equation solution

 In this paper, the nonlinear response of elastic membranes with arbitrary shape under partial and full ponding loads has been analyzed. Large deflections are considered, which result from nonlinear kinematic relations. The problem is formulated in terms of the displacements components and the three coupled nonlinear governing equations are solved using the analog equation method (AEM). The membrane may be prestressed either by prescribed boundary displacements or tractions. Using the concept of the analog equation the three coupled nonlinear equations are replaced by three uncoupled Poisson's equations with fictitious sources under the same boundary conditions. Subsequently, the fictitious sources are established using a procedure based on the BEM and the displacement components as well as the stress resultants at any point of the membrane are evaluated from their integral representations. In addition to the geometrical nonlinearity, the ponding problem is itself nonlinear, because the ponding load depends on the deflection surface that it produces. Iterative schemes are developed which converge to the equilibrium state of the membrane under the ponding loads. Several membranes are analyzed which illustrate the method and demonstrate its efficiency and accuracy. The method has all the advantages of the pure BEM, since the discretization and integration is limited only to the boundary. Received 28 July 2001     

A localized differential quadrature (LDQ) method and its application to the 2D wave equation

 Differential Quadrature (DQ) is a numerical technique of high accuracy, but it is sensitive to grid distribution and requires that the number of grid points cannot be too large. These two requirements greatly restrict wider applications of DQ method. Through a simplified stability analysis in this paper, it is concluded that these two limitations are due to stability requirements. This analysis leads us to propose to localize differential quadrature to a small neighbourhood so as to keep the balance of accuracy and stability. The derivatives at a grid point are approximated by a weighted sum of the points in its neighbourhood rather than of all grid points. The method is applied to the one- and two-dimensional wave equations. Numerical examples show the present method produces very accurate results while maintaining good stability. The proposed method enables us to solve more complicated problems and enhance DQ's flexibility significantly. Received: 23 October 2001 / Accepted: 3 July 2002     

Boundary integral equation method for two dissimilar elastic inclusions in an infinite plate

This paper provides a numerical solution for an infinite plate containing two dissimilar elastic inclusions, which is based on complex variable boundary integral equation (CVBIE). The original problem is decomposed into two problems. One is an interior boundary value problem (BVP) for two elastic inclusions, while other is an exterior BVP for the matrix with notches. After performing discretization for the coupled boundary integral equations (BIEs), a system of algebraic equations is formulated. The inverse matrix technique is suggested to solve the relevant algebraic equations, which can avoid using the assembling of some matrices. Several numerical examples are carried out to prove the efficiency of suggested method and the hoop stress along the interface boundary is evaluated.     

Parallelized FVM algorithm for three-dimensional viscoelastic flows

 A parallel implementation for the finite volume method (FVM) for three-dimensional (3D) viscoelastic flows is developed on a distributed computing environment through Parallel Virtual Machine (PVM). The numerical procedure is based on the SIMPLEST algorithm using a staggered FVM discretization in Cartesian coordinates. The final discretized algebraic equations are solved with the TDMA method. The parallelisation of the program is implemented by a domain decomposition strategy, with a master/slave style programming paradigm, and a message passing through PVM. A load balancing strategy is proposed to reduce the communications between processors. The three-dimensional viscoelastic flow in a rectangular duct is computed with this program. The modified Phan-Thien–Tanner (MPTT) constitutive model is employed for the equation system closure. Computing results are validated on the secondary flow problem due to non-zero second normal stress difference N ₂. Three sets of meshes are used, and the effect of domain decomposition strategies on the performance is discussed. It is found that parallel efficiency is strongly dependent on the grid size and the number of processors for a given block number. The convergence rate as well as the total efficiency of domain decomposition depends upon the flow problem and the boundary conditions. The parallel efficiency increases with increasing problem size for given block number. Comparing to two-dimensional flow problems, 3D parallelized algorithm has a lower efficiency owing to largely overlapped block interfaces, but the parallel algorithm is indeed a powerful means for large scale flow simulations. Received: 2 July 2002 / Accepted: 15 November 2002 This research is supported by an A^⋆STAR Grant EMT/00/011.     

Domain element local integral equation method for potential problems in anisotropic and functionally graded materials

An efficient numerical method is proposed for 2-d potential problems in anisotropic media with continuously variable material coefficients. The method is based on the local integral relationships (integral form of balance equation and/or integral equations utilizing fundamental solutions) and consistent approximation of field variable using standard domain-type elements. The accuracy and convergence of the proposed method is tested by several examples and compared with benchmark analytical solutions.     

Modeling of 3D transversely piezoelectric and elastic bimaterials using the boundary element method

 This paper presents a numerical model for three-dimensional transversely isotropic bimaterials based on the boundary element formulation. The point force solutions expressed in a united-form for distinct eigenvalues are studied for transversely isotropic piezoelectricity and pure elasticity. A boundary integral formulation is implemented for the modeling of two-phase materials. In this study, the stress distributions are computed for a near interface flaw. The influences of the shape and location of the flaw on the the stress concentration are examined. The accuracy of the numerical procedures is validated through selected example problems and comparison studies. Received 3 October 2001 / Accepted 9 April 2002     

气动声学Lighthill方程的Kirchhoff积分解分析

Lighthill的声类比(acoustic analogy)是目前气动声定量预测中应用最为广泛的一种方法。使用非齐次波动方程的Kirchhoff积分公式对Ligthhill方程进行求解。Kirchhoff公式中的延迟时间表示不同位置点声源对场点声压叠加时的相位作用,推导时强调延迟时间函数的导数运算。基于Kirchhoff积分公式对于有物体存在于流场中的情况,详细推导了Curle解,并对Curle公式中的各声源项进行了分析。文章有助于气动声学初学者正确地认识声类比理论,加深对Curle公式的理解。     

A study on time schemes for DRBEM analysis of elastic impact wave

 The precise integration and differential quadrature methods are two new unconditionally stable numerical schemes to approximate time derivative with more than the second order accuracy. Recent studies showed that compared with the Houbolt and Newmark methods, they produced more accurate solutions with large time step for the problems where response is primarily dominated by low and intermediate frequency modes. This paper aims to investigate these time schemes in the context of the dual reciprocity BEM (DRBEM) formulation of various shock-excited scalar elastic wave problems, where high modes have important affect on traction response. The Houbolt method was widely recommended in many literatures for such DRBEM dynamic formulations. However, this study found that the damped Newmark algorithm was the most efficient and accurate for impact traction analysis in conjunction with the DRBEM. The precise integration and differential quadrature methods are shown inapplicable for such shock-excited problems due to the absence of numerical damping. On the other hand, we also found that to achieve the same order of accuracy, the differential quadrature method required much less computing effort than the precise integration method due to the use of the Bartels–Stewart algorithm solving the resulting Lyapunov matrix analogization equation. Received 6 November 2000     

An Efficient Algebraic Algorithm for the Geometric Completion to Involution

 We describe an adaption of a differential algebraic completion algorithm for linear systems of partial differential equations that allows us to deduce intrinsic differential geometric information like the number of prolongations and projections needed for the completion. This new hybrid algorithm represents a much more efficient realisation of the classical Cartan–Kuranishi completion than previous purely geometric ones. A classical problem in geometric completion theory is the existence of δ-singular coordinate systems in which the algorithms do not terminate. We develop a new and a very simple criterion for δ-singularity based on a comparison of the Janet and the Pommaret division. This criterion can also be used for the direct construction of δ-regular coordinates. Received: July 28, 2000; revised version: October 16, 2001     

The boundary contour method for two-dimensional Stokes flow and incompressible elastic materials

 A variant of the boundary element method, called the boundary contour method (BCM), offers a further reduction in dimensionality. Consequently, boundary contour analysis of two-dimensional (2-D) problems does not require any numerical integration at all. While the method has enjoyed many successful applications in linear elasticity, the above advantage has not been exploited for Stokes flow problems and incompressible media. In order to extend the BCM to these materials, this paper presents a development of the method based on the equations of Stokes flow and its 2-D kernel tensors. Potential functions are derived for quadratic boundary elements. Numerical solutions for some well-known examples are compared with the analytical ones to validate the development. Received 28 August 2001 / Accepted 15 January 2002     

The least-squares meshfree method for solving linear elastic problems

 A meshfree method based on the first-order least-squares formulation for linear elasticity is presented. In the authors' previous work, the least-squares meshfree method has been shown to be highly robust to integration errors with the numerical examples of Poisson equation. In the present work, conventional formulation and compatibility-imposed formulation for linear elastic problems are studied on the convergence behavior of the solution and the robustness to the inaccurate integration using simply constructed background cells. In the least-squares formulation, both primal and dual variables can be approximated by the same function space. This can lead to higher rate of convergence for dual variables than Galerkin formulation. In general, the incompressible locking can be alleviated using mixed formulations. However, in meshfree framework these approaches involve an additional use of background grids to implement lower approximation space for dual variables. This difficulty is avoided in the present method and numerical examples show the uniform convergence performance in the incompressible limit. Therefore the present method has little burden of the requirement of background cells for the purposes of integration and alleviating the incompressible locking. Received: 16 December 2001 / Accepted: 4 November 2002     

Dynamic analysis of nonlinear membranes by the analog equation method: a boundary-only solution

 A boundary-only solution is presented for dynamic analysis of elastic membranes under large deflections. The solution procedure is based on the analog equation method (AEM). According to this method, the three coupled nonlinear second order hyperbolic partial differential equations in terms of displacements, which govern the response of the membrane, are replaced with three Poisson's quasi-static equations under fictitious time dependent sources. The fictitious sources are established using a BEM-based procedure and the displacements as well as the stress resultants at any point are evaluated from their integrals representations. Numerical examples are presented which illustrate the method. Received 16 December 2000 / Accepted 25 April 2002