SENSITIVITY ANALYSIS FOR BOUNDARY ELEMENT ERROR ESTIMATION AND MESH REFINEMENT

The subject of this paper is the sensitivity analysis of approximate boundary element solutions with respect to the positions of the collocation points. The direct differentiation approach is considered here and the analysis is performed analytically. Since only the collocation points are perturbed, the shape of the body and the corresponding discretization remain unaltered. This aspect makes the present work quite different in spirit with respect to earlier analyses on shape sensitivities. Sensitivities of approximate BEM solutions with respect to the positions of collocation points are shown to be related to the residual of hypersingular integral equations. Numerical results confirm that the present approach can be seen as the analytical counterpart of an adaptive scheme for mesh refinement presented by the same author in some recent papers. Some other advantages of the present approach over the former one are also outlined.     

AN INDIRECT BOUNDARY ELEMENT METHOD FOR PLATE BENDING ANALYSIS

An indirect Boundary Element Method is employed for the static analysis of homogeneous isotropic and linear elastic Kirchhoff plates of an arbitrary geometry. The objectives of this paper consists of a construction and a study of the resulting boundary integral equations as well as a development of stable powerful algorithms for their numerical approximation. These equations involve integrals with high-order kernel singularities. The treatment of singular and hypersingular integrals and a construction of solutions in the neighborhood of the irregular points on the boundary are discussed. Numerical examples illustrate the procedure and demonstrate its advantages. © 1997 by John Wiley & Sons, Ltd.     

SOLVING TRANSIENT DYNAMIC CRACK PROBLEMS BY THE HYPERSINGULAR BOUNDARY ELEMENT METHOD

Abstract— The subject of hypersingular boundary integral equations is a rapidly developing topic due to the advantages which this kind of formulation offers compared to the standard boundary integral method. The hypersingular formulation is particularly well suited for fracture mechanics problems, where there are important gradients of the stress field and singularities. This formulation for time domain antiplane problems has been recently addressed by the authors and in the present paper, the formulation for time domain plane problems is presented and applied for the first time. A mixed Boundary Element approach based on the standard integral equation and the hypersingular integral equation is developed. The mixed formulation allows for a very simple discretization of the problem, where no subregion is needed. Conforming quadratic elements are used for the crack and the external boundaries. The hypersingular integral equation is used for collocation points within the crack elements, while the standard integral representation is used for the external boundaries. Several examples with different crack geometries are studied to illustrate the possibilities of the method. The Stress Intensity Factor (S.I.F.) is very accurately computed from the crack tip opening displacements along the crack tip element. The results show that the proposed approach for S.I.F. evaluation is simple and produces accurate solutions.     

CONTINUITY AND COLLOCATION EFFECTS IN THE BOUNDARY ELEMENT METHOD

The presence of singularities in the integral operators of the boundary element methods requires that the density functions must satisfy certain continuity requirements if the displacements and stresses are to be bounded. Quite often the continuity conditions, particularly on the derivatives of the density functions, are relaxed at the element ends for the sake of simplicity in approximating the unknown density functions. In this paper, a numerical study on the effects of satisfying or violating the continuity requirements and the effect of the boundary condition collocating point on three different BEM formulations is presented. Two are indirect formulations using force singularities and displacement discontinuity singularities, and the third is Rizzo's direct formulation. The two integral operators in the direct BEM appear individually in the two different formulations of the indirect BEM. This makes it possible to study the numerical error and other problems in each integral operator and the interaction of the two integral operators in the direct BEM. The impact of the study on numerical modelling for the three BEM formulations is presented in the paper. © 1997 by John Wiley & Sons, Ltd.     

三维热弹性力学问题的混合边界元法

本本文给出了三维无限大域内点热源作用下的位移、应力场基本解。采用基于虚拟热源法的间接边界元法和直接边界元法的混合边界元法求解三维有限域热弹性力学问题,有效地避免了热弹性力学问题中域内积分的处理。数值计算表明混合边界元法求热弹性力学问题具有简单方便、精度较高的优点。     

三维弹塑性接触边界元法对摩擦的处理

本文建立了三维弹塑性接触问题的边界元法。采用两种摩擦模型计算了轧制过程中弹塑性体和弹性体的接触变形过程，并对两种摩擦模型的计算结果进行了比较。计算结果表明两种摩擦模型均能很好地反映实际情况     

R-ADAPTIVE BOUNDARY ELEMENT METHOD FOR UNSTEADY FREE-SURFACE FLOW ANALYSIS

An adaptive meshing based on the r-method is developed for two-dimensional unsteady non-linear flows with a free surface. Coupling of a boundary element equation and a weighted residual formulation of the pressure equation on the free surface is employed in solving the wave problems. A mesh optimization scheme is constructed for these two matrix equations. The final mesh distribution on the free surface is determined from the weighted average of these two adaptive meshes. Through numerical analyses of a non-linear sloshing problem and solitary waves in a tank, the influence of time interval of the remeshing and the weight factor for two adaptive meshes on the accuracy and efficiency of the proposed approach is investigated.     

EVALUATION OF THE STRESS TENSOR IN 3-D ELASTOPLASTICITY BY DIRECT SOLVING OF HYPERSINGULAR INTEGRALS

A 3-D hypersingular Boundary Integral Equation (BIE) of elastoplasticity is derived. Using this formulation the displacement rate gradients and the complete stress tensor on the boundary can be evaluated directly as opposed to the classical approach, where the shape functions derivatives are to be calculated. The regularization of strongly singular and hypersingular boundary integrals, as well as strongly singular domain integrals for a source point positioned on the boundary is carried out in a general manner. Arbitrary types of elements and arbitrary positions of the source point with respect to continuity requirements can be used. Numerical 3-D elastoplastic examples (notch and crack problems) illustrate the advantages of the proposed method.     

A SPACE–TIME FINITE ELEMENT METHOD FOR THE EXTERIOR STRUCTURAL ACOUSTICS PROBLEM: TIME-DEPENDENT RADIATION BOUNDARY CONDITIONS IN TWO SPACE DIMENSIONS

A time-discontinuous Galerkin space–time finite element method is formulated for the exterior structural acoustics problem in two space dimensions. The problem is posed over a bounded computational domain with local time-dependent radiation (absorbing) boundary conditions applied to the fluid truncation boundary. Absorbing boundary conditions are incorporated as ‘natural’ boundary conditions in the space–time variational equation, i.e. they are enforced weakly in both space and time. Following Bayliss and Turkel, time-dependent radiation boundary conditions for the two-dimensional wave equation are developed from an asymptotic approximation to the exact solution in the frequency domain expressed in negative powers of a non-dimensional wavenumber. In this paper, we undertake a brief development of the time-dependent radiation boundary conditions, establishing their relationship to the exact impedance (Dirichlet-to-Neumann map) for the acoustic fluid, and characterize their accuracy when implemented in our space–time finite element formulation for transient structural acoustics. Stability estimates are reported together with an analysis of the positive form of the matrix problem emanating from the space–time variational equations for the coupled fluid-structure system. Several numerical simulations of transient radiation and scattering in two space dimensions are presented to demonstrate the effectiveness of the space–time method.     

地下隧道工程地震动分析的有限元－人工透射边界方法

采用有限元－人工透射边界方法计算了含有地下隧道工程的地基在ＳＨ波和瑞利波作用下的地震动反应，讨论了各自不同的地震动特性。并通过与边界积分方法比较，验证了有限元—人工透射边界方法对解决此类问题的有效性。     

弹性力学问题的虚边界元－最小二乘法及误差评估

本文从［１］提出的虚边界原理出发,采用最小二乘法建立满足弹性力学问题边界条件的边界积分方程,再用线性虚边界元将其离散化。然后详细地研究了这些离散化的边界积分方程的解折特性。文中引用了误差分析的拉依达（ｐａИＴａ）准则,用来衡量解的误差水平,取得了理想的效果。编制了微机程序,程序中采用高斯积分格式,并考虑了虚,实边界对称条件的具体处理。本文方法不仅可以成功地处理边界条件连续的情况,而且对边界条件不连续的情况也能得出满意的结果。数值算例表明,程序可靠,虚边界变动时算法稳定,具有较高的处理精度。     

A MODEL STUDY OF THE QUALITY OF A POSTERIORI ERROR ESTIMATORS FOR FINITE ELEMENT SOLUTIONS OF LINEAR ELLIPTIC PROBLEMS,WITH PARTICULAR REFERENCE TO THE BEHAVIOR NEAR THE BOUNDARY

In References 1–3 we presented a computer-based theory for analysing the asymptotic accuracy (quality of robustness) of error estimators for mesh-patches in the interior of the domain. In this paper we review the approach employed in References 1–3 and extend it to analyse the asymptotic quality of error estimators for mesh-patches at or near a domain boundary. We analyse two error estimators which were found in References 1–3 to be robust in the interior of the mesh (the element residual with p-order equilibrated fluxes and (p+1)) degree bubble solution or (p+1) degree polynomial solution (ERpB or ERpPp+1; see References 1–3) and the Zienkiewicz–Zhu Superconvergent Patch Recovery (ZZ-SPR; see References 4–7) and we show that the robustness of these estimators for elements adjacent to the boundary can be significantly inferior to their robustness for interior elements. This deterioration is due to the difference in the definition of the estimators for the elements in the interior of the mesh and the elements adjacent to the boundary. In order to demonstrate how our approach can be employed to determine the most robust version of an estimator we analysed the versions of the ZZ estimator proposed in References 9–12. We found that the original ZZ-SPR proposed in References 4–7 is the most robust one, among the various versions tested, and some of the proposed ‘enhancements’ can lead to a significant deterioration of the asymptotic robustness of the estimator. From the analyses given in References 1–3 and in this paper, we found that the original ZZ estimator (given in References 4–7) is the most robust among all estimators analysed in References 1–3 and in this study. © 1997 John Wiley & Sons, Ltd.     

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.     

ELASTOHYDRODYNAMIC LUBRICATION ANALYSIS OF LAYERED LINE CONTACT BY THE BOUNDARY ELEMENT METHOD

This paper presents a numerical routine to compute the contact characteristics of elastomer layered cylinders lubricated by isoviscous liquids. The indentation of the elastic layer is calculated from boundary integral equations which are solved by linear and quadratic boundary element methods for a finite plane model and a circular representation of the junction. The hydrodynamic equation is also transformed into a boundary integral equation and solved by Simpson's rule. Some factors which possibly affect numerical accuracy are examined. Examples for finite plane and circular layer are analysed with reference to parameters for printing press roller contact, in which results are obtained for the indentation, film thickness and liquid pressure, as well as internal stresses through the simultaneous solution of the elasticity and hydrodynamic equations. The results show that high precision is easily achieved and the method is efficient for such layered problems.     

IMPLEMENTATION OF THE BOUNDARY ELEMENT METHOD IN THE DYNAMICS OF FLEXIBLE BODIES

This paper presents the implementation of the Boundary Element Method in the dynamics of flexible multibody systems. Kane's equations are used to formulate the governing boundary initial value problem for an arbitrary three-dimensional elastic body subjected to large overall base motion. Using continuum mechanics principles, direct boundary element incremental formulations are derived. The Galerkin approach was employed to generate the weighted residual statement which serves as a transitory point between continuum mechanics and boundary integral equations. By adapting the updated Langrangian formulation for large displacements analysis and using the Maxwell–Betti reciprocal theorem, integral representations for geometric stiffening were also derived. The non-linear terms were found to be functions of the time-variant stresses associated with the inertial forces at the reference configuration. The domain integrals arising from body forces (such as gravitational loads, inertia loads and thermal loads, etc.) are presented as DRM integrals (Dual-Reciprocity Method). Using the substructuring technique the elastic body is divided into several regions leading to a system of equations whose matrices are sparse (block-banded). The linearized equations of motion were discretized along the boundary of the body, and an algorithm for the integration involving the Houbolt method was used to establish an algebraic system of pseudo-static equilibrium equations. A Newton–Raphson-type iteration scheme was used to solve these discretized balance equations. To take advantage of the sparsity of the matrices, special routines were used to decompose and solve the resulting linear system of equations. An illustrative example is presented to demonstrate the validity of the method as well as how the effects of geometric stiffening effects are captured. The example consists of spin-up manoeuvre of a tapered beam attached to a moving base. The beam was modelled as two-dimensional plane strain problem divided into a number of substructures. Numerical simulation results show how the phenomenon of dynamic stiffening is captured by the present approach.     

ITERATIVE METHODS IN SOLVING NAVIER–STOKES EQUATIONS BY THE BOUNDARY ELEMENT METHOD

The solution of Navier–Stokes equations of time-dependent incompressible viscous fluid flow in planar geometry by the Boundary Domain Integral Method (BDIM) is discussed. The introduction of a subdomain technique to fluid flow problems is considered and improved in order to maintain the stability of BDIM. To avoid problems with flow kinematics computation in the sudomain mesh, a segmentation technique is proposed which combines the original BDIM with its subdomain variant and preserves its numerical stability. In order to reduce the computational cost of BDIM, which greatly depends on the solution of systems of linear equations, iterative methods are used. Conjugate gradient methods, conjugate gradients squared and an improved version of the biconjugate gradient method BiCGSTAB, together with the generalized minimal residual method, are used as iterative solvers. Different types of preconditioning, from simple Jacobi to incomplete LU factorization, are carried out and the performance of chosen iterative methods and preconditioners are reported. Test examples include backward facing step flow and flow through tubular heat exchangers. Test computation results show that BDIM is an accurate approximation technique which, together with the subdomain technique and powerful iterative solvers, can exhibit some significant savings in storage and CPU time requirements.     

APPLICATION OF THE ORDINARY LEAST-SQUARES APPROACH FOR SOLUTION OF COMPLEX VARIABLE BOUNDARY ELEMENT PROBLEMS

Cauchy's theorem is used to generate a Complex Variable Boundary Element Method (CVBEM) formulation for steady, two-dimensional potential problems. CVBEM uses the complex potential, w=ϕ+iψ, to combine the potential function, ϕ, with the stream function, ψ. The CVBEM formulation, using Cauchy's theorem, is shown to be mathematically equivalent to Real Variable BEM which employs Green's second identity and the respective fundamental solution. CVBEM yields an overdetermined system of equations that are commonly solved using implicit and explicit methods that reduce the overdetermined matrix to a square matrix by selectively excluding equations. Alternatively, Ordinary Least Squares (OLS) can be used to minimize the Euclidean norm square of the residual vector that arises due to the approximation of boundary potentials and geometries. OLS uses all equations to form a square matrix that is symmetric, positive definite and diagonally dominant. OLS is more accurate than existing methods and can estimate the approximation error at boundary nodes. The approximation error can be used to determine the adequacy of boundary discretization schemes. CVBEM/OLS provides greater flexibility for boundary conditions by allowing simultaneous specification of both fluid potentials and stream functions, or their derivatives, along boundary elements. © 1997 by John Wiley & Sons, Ltd.     

薄壁杆件翘曲剪应力的边界元精确积分解法

用非连续边界元对薄壁杆件的约束扭转进行了分析,推导出了求解边界点二次翘曲函数值的边界积分方程,给出了边界积分方程数值求解时积分计算的精确表达式。数值算例表明:利用边界积分方程方法分析薄壁杆件的约束扭转问题时效率和精度高,同时采用精确积分可以有效的处理"边界层效应"问题。