Axisymmetric viscoplastic deformation by the boundary element method

This paper presents a boundary element formulation and numerical implementation of the problem of small axisymmetric deformation of viscoplastic bodies. While the extension from planar to axisymmetric problems can be carried out fairly simply for the finite element method (FEM), this is far from true for the boundary element method (BEM). The primary reason for this fact is that the axisymmetric kernels in the integral equations of the BEM contain elliptic functions which cannot be integrated analytically even over boundary elements and internal cells of simple shape. Thus, special methods have to be developed for the efficient and accurate numerical integration of these singular and sensitive kernels over discrete elements. The accurate determination of stress rates by differentiation of the displacement rates presents another formidable challenge.A successful numerical implementation of the boundary element method with elementwise (called the Mixed approach) or pointwise (called the pure BEM or BEM approach) determination of stress rates has been carried out. A computer program has been developed for the solution of general axisymmetric viscoplasticity problems. Comparisons of numerical results from the BEM and FEM, for several illustrative problems, are presented and discussed in the paper. It is possible to get direct solutions for the simpler class of problems for cylinders of uniform cross-section, and these solutions are also compared with the BEM and FEM results for such cases.     

Time‐domain BEM solution of convection–diffusion‐type MHD equations

The two‐dimensional convection–diffusion‐type equations are solved by using the boundary element method (BEM) based on the time‐dependent fundamental solution. The emphasis is given on the solution of magnetohydrodynamic (MHD) duct flow problems with arbitrary wall conductivity. The boundary and time integrals in the BEM formulation are computed numerically assuming constant variations of the unknowns on both the boundary elements and the time intervals. Then, the solution is advanced to the steady‐state iteratively. Thus, it is possible to use quite large time increments and stability problems are not encountered. The time‐domain BEM solution procedure is tested on some convection–diffusion problems and the MHD duct flow problem with insulated walls to establish the validity of the approach. The numerical results for these sample problems compare very well to analytical results. Then, the BEM formulation of the MHD duct flow problem with arbitrary wall conductivity is obtained for the first time in such a way that the equations are solved together with the coupled boundary conditions. The use of time‐dependent fundamental solution enables us to obtain numerical solutions for this problem for the Hartmann number values up to 300 and for several values of conductivity parameter. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical solution of axisymmetric cavity flows using the boundary element method

This paper presents a formulation of the boundary element method (BEM) for solution of axisymmetric cavity flow problems. The governing equation is written in terms of Stokes' stream function, requiring a new fundamental solution to be found. The iterative procedure for adjusting the free-surface position is similar to that used for planar cavity flows. Numerical results are compared with finite difference and finite element solutions, showing the robustness of the BEM model.     

A boundary element method for calculating elastic waves scattered by axisymmetric inclusion

A Boundary Element Method (BEM) is described to compute the scattering of elastic waves by an axisymmetric inclusion in an infinite elastic medium. The boundary loads applied to the inclusion is expanded in terms of Fourier series in an infinite space. The boundary integral equation is solved in the general direction of the axisymmetric inclusion by BEM. The problem of the 3-D scattering of elastic waves is reduced to a 1-Done. According to the geometric features of the axisymmetric in clusion the ring shell elements are adopted in this method. A comparison is made with other BEM methods. The numerical results show this method can reduce the amount of calculation and enhance the speed of convergence. Supported by Foundation of Ph. D Program of State Education Commission of China     

Boundary element method for thermoelastic analysis of three-dimensional transversely isotropic solids

Thermal effects are well known to manifest themselves as additional volume integral terms in the direct formulation of the boundary integral equation (BIE) for linear elastic solids when using the boundary element method (BEM). This domain integral has been successfully transformed in an exact manner to surface ones only in isotropy and in 2D anisotropy, thereby restoring the BEM as a truly boundary solution technique. The difficulties with extending it to 3D general anisotropic solids lie in the mathematical complexity of the Green’s function and its derivatives for such materials. These quantities are required items in the BEM formulation. In this paper, the exact, analytical transformation of the volume integral associated with thermal effects to surface ones is achieved for a transversely isotropic material using a similar approach which the authors have previously employed for the same task in BEM for 2D general anisotropy. A numerical scheme, however, needs to be employed to evaluate some of the new terms introduced in the surface integrals that arise from this process here. The mathematical soundness of the formulation is demonstrated by a few examples; the numerical results obtained are checked by alternative means, including those obtained from the commercial FEM code, ANSYS.     

Analysis of nonlinear large deformation problems by boundary element method

In this paper, the author deduces an approximate solution of nonlinear influence function in rate form for two-dimensional elastic problems on current configuration by the method of comoving coordinate system.Here BEM formulation of large deformation based on Chen’s theory[1] is given. The computational processes of nonlinear boundary integral equation is discussed. The author also compiles a nonlinear computing program NBEM. Numerical examples show that the results presented here is available to the solution of engineering problems.     

A variable stiffness dual boundary element method for mixed-mode elastoplastic crack problems

A dual boundary element formulation is presented for elastoplastic crack problems using a variable stiffness approach. In this approach the Von Mises yield criterion with strain hardening is used and the unknown non-linear terms, as the initial strains, are now defined in function of the scalar flow factors. Dual BEM variable stiffness formulation, based on the utilisation of the traction equation on one of the crack surfaces and the displacement equation on the other, is presented for the solution of general elastoplastic fracture mechanics problems. The validity of the present formulation has been assessed by comparing with the well known iterative dual BEM elastoplastic approach.     

一类新变量的Reissner板弯曲边界元法

文［４］导出了二维弹性力学平面问题的一类新型边界积分方程，本文将该理论和方法推广到三变量的Ｒｅｉｓｓｎｅｒ板弯曲中，给出边界场变量含广义位移和新型广义力的边界积分方程。从而边界弯矩应力张量可直接由离散边界积分方程求出。     

液体自由晃动的数值分析

本文对常重条件下自旋容器内液体的对称和任意阶反对称晃动建立了边界元算法，引入了两种调和函数以确定边界元系数矩阵的主对角元，并且提出了一种对计算机程序的正确性和数值解近似程度的考核方法。     

A boundary element formulation for wear modeling on 3D contact and rolling-contact problems

The present work shows a new numerical treatment for wear simulation on 3D contact and rolling-contact problems. This formulation is based on the boundary element method (BEM) for computing the elastic influence coefficients and on projection functions over the augmented Lagrangian for contact restrictions fulfillment. The constitutive equations of the potential contact zone are Signorini’s contact conditions, Coulomb’s law of friction and Holm–Archard’s law of wear. The proposed methodology is applied to predict wear on different contact and rolling-contact problems. Results are validated with numerical solutions and semi-analytical models presented in the literature. The BEM considers only the degrees of freedom involved on these kind of problems (those on the solids surfaces), reducing the number of unknowns and obtaining a very good approximation on contact tractions using a low number of elements. Together with the formulation, an acceleration strategy is presented allowing to reduce the times of resolution.     

VIBRO-ACOUSTIC BEHAVIOR OF SUBMERGED CYLINDRICAL SHELLS: ANALYTICAL FORMULATION AND NUMERICAL MODEL

We propose both an analytical formulation and a numerical model to study the hydroelastic or vibroacoustic behaviour of cylindrical thin shells immersed in an unbounded, inviscid and heavy fluid. The analytical solution allows us to calculate the dynamic response and the pressure radiated in the far field by a baffled cylinder. This formulation uses the truncated modal basis of the dry structure to expand the displacements of the submerged shell. The analytical model is used as a reference in order to validate a numerical model which couples the finite element method (FEM) to the boundary element method (BEM). As opposed to the analytical formulation which is dedicated to baffled circular cylinders only, the numerical model allows us to treat any axisymmetric shell, such as cylindrical and spherical shells, or more complex shells of revolution. The structure is idealized by two-node ring finite elements and the boundary equation is solved using the method of singularities.     

An efficient iteration algorithm of mixed boundary/finite element method and application to free torsional vibration analysis of bodies of revolution

In this paper,the general formulation of a new proposed iteration algorithm of mixedBEM/FEM for eigenvalue problems of elastodynamics is described.Approximatefundamental solutions of elastodynamics are adopted in the normal mixed BEM/FEMequations.The accuracy of solutions is progressively improved by the iteration procedure.Not only could the awkwar dness of non-algebraic eigenvalue equations be avoided but alsothe accuracy of numerical solutions is almost independent of the interior meshing.All thesegive many advantages in numerical calculation.The algorithm is applied to free torsionalvibration analysis of bodies of revolution.A few cases are studied.All of the numericalresults are very good.     

Multiple reciprocity boundary element analysis of two-dimensional anisotropic thermoelasticity involving an internal arbitrary non-uniform volume heat source

In the direct boundary element method (BEM) formulation of anisotropic thermoelasticity, thermal loads manifest themselves as additional volume integral terms in the boundary integral equation (BIE). Conventionally, this requires internal cell discretisation throughout the whole domain. In this paper, the multiple reciprocity method in BEM analysis is employed to treat the general 2D thermoelasticity problem when the thermal loading is due to an internal non-uniform volume heat source. By successively performing the “volume-to-surface” integral transformation, the general formulation of the associated BIE for the problem is derived. The successful implementation of such a scheme is illustrated by three numerical examples.     

A boundary element application for mixed modeloading idealized sawtooth fracture surface

In this paper, a 2-D elastic-plastic BEM formulation predicting the reduced mode IIand the enhanced mode I stress intensity factors are presented. The dilatant boundary conditions (DBC) are assumed to be idealized uniform sawtooth crack surfaces and an effective Coulombsliding law. Three types of crack face boundary conditions, i.e. (1) BEM sawtooth model-elasticcenter crack tip; (2) BEM sawtooth model-plastic center crack tip; and (3) BEM sawtoothmodel-edge crack with asperity wear are presented. The model is developed to attempt todescribe experimentally observed non-monotonic, non-linear dependence of shear crack behavioron applied shear stress, superimposed tensile stress, and crack length. The asperity sliding isgoverned by Coulombs law of friction applied on the inclined asperity surface which hascoefficient of friction μ. The traction and displacement Greens functions which derive fromNaviers equations are obtained as well as the governing boundary integral equations for an infiniteelastic medium. Accuracy test is performed by comparison stress intensity factors of the BEMmodel with analytical solutions of the elastic center crack tip. The numerical results show thepotential application of the BEM model to investigate the effect of mixed mode loading problemswith various boundary conditions and physical interactions.     

Secondary torsional moment deformation effect in inelastic nonuniform torsion of bars of doubly symmetric cross section by BEM

In this paper a boundary element method is developed for the inelastic nonuniform torsional problem of simply or multiply connected prismatic bars of arbitrarily shaped doubly symmetric cross section, taking into account the secondary torsional moment deformation effect. The bar is subjected to arbitrarily distributed or concentrated torsional loading along its length, while its edges are subjected to the most general torsional boundary conditions. A displacement based formulation is developed and inelastic redistribution is modeled through a distributed plasticity model exploiting three dimensional material constitutive laws and numerical integration over the cross sections. An incremental–iterative solution strategy is adopted to resolve the elastic and plastic part of stress resultants along with an efficient iterative process to integrate the inelastic rate equations. The one dimensional primary angle of twist per unit length, a two dimensional secondary warping function and a scalar torsional shear correction factor are employed to account for the secondary torsional moment deformation effect. The latter is computed employing an energy approach under elastic conditions. Three boundary value problems with respect to (i) the primary warping function, (ii) the secondary warping one and (iii) the total angle of twist coupled with its primary part per unit length are formulated and numerically solved employing the boundary element method. Domain discretization is required only for the third problem, while shear locking is avoided through the developed numerical technique. Numerical results are worked out to illustrate the method, demonstrate its efficiency and wherever possible its accuracy.     

The Domain-Boundary Element Method (DBEM) for hyperelastic and elastoplastic finite deformation: axisymmetric and 2D/3D problems

Summary  This paper presents the solution of geometrically nonlinear problems in solid mechanics by the Domain-Boundary Element Method. Because of the Total-Lagrange approach, the arising domain and boundary integrals are evaluated in the undeformed configuration. Therefore, the system matrices remain unchanged during the solution procedure, and their time-consuming computation needs to be performed only once. While the integral equations for axisymmetric finite deformation problems will be derived in detail, the basic ideas of the formulation in two and three dimensions can be found in [1]. The present formulation includes torsional problems with finite deformations, where additional terms arise due to the curvilinear coordinate system. A Newton–Raphson scheme is used to solve the nonlinear set of equations. This involves the solution of a large system of linear equations, which has been a very time-consuming task in former implementations, [1, 2]. In this work, an iterative solver, i.e. the generalized minimum residual method, is used within the Newton–Raphson algorithm, which leads to a significant reduction of the computation time. Finally, numerical examples will be given for axisymmetric and two/three-dimensional problems. Received 29 August 2000; accepted for publication 10 October 2000     

Fluid flow in an arbitrary cascade on an axisymmetric stream surface in a variable thickness layer

A solution is given for the problem of flow past a cascade on an axisymmetric stream surface in a layer of variable thickness, which is a component part of the approximate solution of the three-dimensional problem for a three-dimensional cascade. Generalized analytic functions are used to obtain the integral equation for the potential function, which is solved via iteration method by reduction to a system of linear algebraic equations. An algorithm and a program for the Minsk-2 computer are formulated. The precision of the algorithm is evaluated and results are presented of the calculation of an example cascade.In the formulation of [1, 3] the problem of flow past a three-dimensional turbomachine cascade is reduced approximately to the joint solution of two-dimensional problems of the averaged axisymmetric flow and the flow on an axisymmetric stream surface in an elementary layer of variable thickness.In the following we solve the second problem for an arbitrary cascade with finite thickness rotating with constant angular velocity in ideal fluid flow: the solution is carried out on a Minsk-2 computer.Many studies have been devoted to this problem. A method for solving the direct problem for a cascade of flat plates in a hyperbolic layer was presented in [2]. Methods were developed in [1, 3] for constructing the flow for the case of a channel with variable thickness; these methods are approximately applicable for dense cascades but yield considerable error for small-load turbomachine cascades. The solution developed in [4], somewhat reminiscent of that of [2], is applicable for thin, slightly curved profiles in a layer with monotonically varying thickness. A solution has been given for a circular cascade for layers varying logarithmically [5] and linearly [6]. Approximate methods for slightly curved profiles in a monotonically varying layer with account for layer variability only in the discharge component were examined in [7–9]. A solution is given in [10] for an arbitrary layer by means of the relaxation method, which yields a roughly approximate flow pattern. The general solution of the problem by means of potential theory and the method of singularities presented in [11] is in error because of neglect of the crossflow through the skeletal line. The computer solution of [12] contains an unassessed error for the calculations in an arbitrary layer. The finite difference method is used in [13] to solve the differential equation of flow, which is illustrated by numerical examples for monotonie layers of axial turbomachines. The numerical solution of [13] is very complex.The solution presented below is found in the general formulation with respect to the geometric parameters of the cascade and the axisymmetric surface and also in terms of the layer thickness variation law.The numerical solution requires about 15 minutes of machine time on the Minsk-2 computer.     

Hydrostatics in weak force fields. Equilibrium shapes of a rotating weightless liquid

We examine the problem of axisymmetric equilibrium forms of the surface of a rotating liquid which has surface tension in the absence of an external force field. We do not assume a priori closure of the equilibrium surface or that it intersects the axis of rotation [1, 2]. We study the properties of the two-parameter family of solutions of the equilibrium equation. Typical forms of the integral curves are constructed from the results of numerical computation on a computer. The methods of boundary layer theory are used to find the approximate expression for the form of the equilibrium surface for large angular rates of rotation.The author wishes to thank L. V. Babenko for writing the program for the numerical calculation and also A. D. Myshkis and A. D. Tyuptsov for discussions of the results and for their valuable advice.     

Energy domain integral applied to solve center and double-edge crack problems in three-dimensions

A three-dimensional Boundary Element Method (BEM) implementation of the energy domain integral for the numerical computation of the crack energy release rate is presented in this paper. The domain expression of the energy release rate is naturally compatible with the BEM, since stresses, strains and derivatives of displacements at internal points can be evaluated using the appropriate boundary integral equations. The pointwise crack energy release rate is evaluated along the three-dimensional crack front over a cylindrical domains that surround a segment of the crack front. The accuracy of the implementation is demonstrated by solving several problems, which include geometries containing straight as well as curved crack fronts.     

Numerical solution of the unsteady Euler equations for airfoils using approximate boundary conditions

This paper presents an efficient numerical method for solving the unsteady Euler equations on stationary rectilinear grids. Boundary conditions on the surface of an airfoil are implemented by using their first-order expansions on the mean chord line. The method is not restricted to flows with small disturbances since there are no restrictions on the mean angle of attack of the airfoil. The mathematical formulation and the numerical implementation of the wall boundary conditions in a fully implicit time-accurate finite-volume Euler scheme are described. Unsteady transonic flows about an oscillating NACA 0012 airfoil are calculated. Computational results compare well with Euler solutions by the full boundary conditions on a body-fitted curvilinear grid and published experimental data. This study establishes the feasibility for computing unsteady fluid-structure interaction problems, where the use of a stationary rectilinear grid offers substantial advantages in saving computer time and program design since it does not require the generation and implementation of time-dependent body-fitted grids.