A simple and efficient BEM algorithm for planar cavity flows

This paper presents a boundary element formulation for solution of planar Riabouchinsky cavity flow problems. An iterative procedure for adjusting the free surface position is developed and shown to be stable and convergent. Numerical results are compared with finite difference and finite element solutions, showing the superior accuracy of the BEM models.     

Neural networks for BEM analysis of steady viscous flows

This paper presents a new neural network‐boundary integral approach for analysis of steady viscous fluid flows. Indirect radial basis function networks (IRBFNs) which perform better than element‐based methods for function interpolation, are introduced into the BEM scheme to represent the variations of velocity and traction along the boundary from the nodal values. In order to assess the effect of IRBFNs, the other features used in the present work remain the same as those used in the standard BEM. For example, Picard‐type scheme is utilized in the iterative procedure to deal with the non‐linear convective terms while the calculation of volume integrals and velocity gradients are based on the linear finite element‐based method. The proposed IRBFN‐BEM is verified on the driven cavity viscous flow problem and can achieve a moderate Reynolds number of 1400 using a relatively coarse uniform mesh. The results obtained such as the velocity profiles along the horizontal and vertical centrelines as well as the properties of the primary vortex are in very good agreement with the benchmark solution. Furthermore, the secondary vortices are also captured by the present method. Thus, it appears that an ability to represent the boundary solution accurately can significantly improve the overall solution accuracy of the BEM. Copyright © 2003 John Wiley & Sons, Ltd.     

Boundary element analysis of three-dimensional mixing flow of Newtonian and viscoelastic fluids

The boundary element method (BEM) is implemented for the simulation of three-dimensional transient flows of typical relevance to mixing. Creeping Newtonian and viscoelastic fluids of the Maxwell type are examined. A boundary-only formulation in the time domain is proposed for linear viscoelastic flows. Special emphasis is placed on cavity flows involving simple- and multiple-connected moving domains. The BEM becomes particularly suited in multiple-connected flows, where part of the boundary (stirrer or rotor) is moving, and the remaining outer part (cavity or barrel) is at rest. In this case, conventional methods, such as the finite element method (FEM), generally require remeshing or mesh refinement of the three-dimensional fluid volume as the flow evolves and the domain of computation changes with time. The BEM is shown to be much easier to implement since the kinematics of the elements bounding the fluid is known (imposed). It is found that, for simple cavity flow induced by a rotating vane at constant angular velocity, the tractions at the vane tip and cavity face exhibit non-linear periodic dynamical behavior with time for fluids obeying linear constitutive equations. © 1998 John Wiley & Sons, Ltd.     

Time-dependent hydroelastic analysis of cavitating propulsors

A 3-D potential-based boundary element method (BEM) is coupled with a 3-D finite element method (FEM) for the time-dependent hydroelastic analysis of cavitating propulsors. The BEM is applied to evaluate the moving cavity boundaries and fluctuating pressures, as well as the added mass and hydrodynamic damping matrices. The FEM is applied to analyze the dynamic blade deformations and stresses due to pressure fluctuations and centrifugal forces. The added mass and hydrodynamic damping matrices are superimposed onto the structural mass and damping matrices, respectively, to account for the effect of fluid–structure interaction. The problem is solved in the time-domain using an implicit time integration scheme. An overview of the formulation for both the BEM and FEM is presented, as well as the BEM/FEM coupling algorithm. Numerical and experiment validation studies are shown. The effects of fluid–structure interaction on the propeller performance are discussed.     

Application of the reciprocity theorem to scattering of surface waves by a cavity

Scattering of surface waves by a cylindrical cavity at the surface of a homogenous, isotropic, linearly elastic half-space is analyzed in this paper. In the usual manner, the scattered field is shown to be equivalent to the radiation from a distribution of tractions, obtained from the incident wave on the surface of the cavity. For the approximation used in this paper, these tractions are shifted to tractions applied to the projection of the cavity on the surface of the half-space. The radiation of surface waves from a normal and a tangential line load, recently determined by the use of the reciprocity theorem, is employed to obtain the field scattered by the cavity from the superposition of displacements due to the distributed surface tractions. The vertical displacement at some distance from the cavity is compared with the solution of the scattering problem obtained by the boundary element method (BEM) for various depths and widths of the cavity. Comparisons between the analytical and BEM results are graphically displayed. The limitations of the approximate approach are discussed based on the comparisons with the BEM results.     

FEM‐BEM coupling for the exterior Stokes problem with non‐conforming finite elements and an application to small droplet deformation dynamics

A non‐conforming, discontinuous Galerkin finite element–boundary element coupling procedure is presented for the exterior planar Stokes problem. The novel coupled formulation is developed using that for the conforming case as a guide to the introduction of extra mortar variables used to couple a discontinuous interior finite element solution with a continuous exterior boundary element solution. Convergence results for the new scheme are presented, for a range of different interior penalties, on computational domains discretized with regular structured meshes. To illustrate an application, the excitations required to model two‐phase droplet deformations in an extensional flow, under simple surface tension, with the new scheme are also presented. For a selection of different drop viscocities and exterior flows, with and without a rotational component, the progression to a steady‐state deformation of initially undeformed circular drops is calculated and the results compared with those from both a conforming FEM‐BEM equivalent scheme and from a small perturbation analysis where available. Copyright © 2011 John Wiley & Sons, Ltd.     

Natural convection in porous media—dual reciprocity boundary element method solution of the Darcy model

This paper describes the solution of a steady state natural convection problem in porous media by the dual reciprocity boundary element method (DRBEM). The boundary element method (BEM) for the coupled set of mass, momentum, and energy equations in two dimensions is structured by the fundamental solution of the Laplace equation. The dual reciprocity method is based on augmented scaled thin plate splines. Numerical examples include convergence studies with different mesh size, uniform and non‐uniform mesh arrangement, and constant and linear boundary field discretizations for differentially heated rectangular cavity problems at filtration with Rayleigh numbers of Ra*=25, 50, and 100 and aspect ratios of A=1/2, 1, and 2. The solution is assessed by comparison with reference results of the fine mesh finite volume method (FVM). Copyright © 2000 John Wiley & Sons, Ltd.     

Boundary Integral Procedures for Unsaturated Flow Problems

Abstract. A novel numerical scheme based on the singular integral theory of the boundary element method. (BEM) is presented for the solution of transient unsaturated flow in porous media. The effort in the present paper is directed in facilitating the application of the boundary integral theory to the solution of the highly non-linear equations that govern unsaturated flow. The resulting algorithm known as the Green element method (GEM) presents a robust attractive method in the state-of -the-art application of the boundary element methodology. Three GEM models based on their different methods of handling the non-linear diffusivity, illustrate the suitability and robustness of this approach for solving highly non-linear 1-D and 2-D flows which would have proved cumbersome or too difficult to implement with the classical BEM approach.     

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.     

Solution of the Navier–Stokes equations in velocity–vorticity form using a Eulerian–Lagrangian boundary element method

This paper describes the Eulerian–Lagrangian boundary element model for the solution of incompressible viscous flow problems using velocity–vorticity variables. A Eulerian–Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian–Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection‐dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two‐dimensional Navier–Stokes problems with low–medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd.     

Research on the companion solution for a thin plate in the meshless local boundary integral equation method

The meshless local boundary integral equation method is a currently developed numerical method, which combines the advantageous features of Galerkin finite element method ( GFEM ), boundary element method (BEM) and element free Galerkin method (EFGM), and is a truly meshless method possessing wide prospects in engineeringapplications. The companion solution and all the other formulas required in the meshless local boundary integral equation for a thin plate were presented, in order to make this method apply to solve the thin plate problem.     

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.     

A coupled FEM/BEM approach and its accuracy for solving crack problems in fracture mechanics

The finite element (FEM) and the boundary element methods (BEM) are well known powerful numerical techniques for solving a wide range of problems in applied science and engineering. Each method has its own advantages and disadvantages, so that it is desirable to develop a combined finite element/boundary element method approach, which makes use of their advantages and reduces their disadvantages. Several coupling techniques are proposed in the literature, but until now the incompatibility of the basic variables remains a problem to be solved. To overcome this problem, a special super-element using boundary elements based on the usual finite element technique of total potential energy minimization has been developed in this paper. The application of the most commonly used approaches in finite element method namely quarter-point elements and J-integrals techniques were examined using the proposed coupling FEM–BEM. The accuracy and efficiency of the proposed approach have been assessed for the evaluation of stress intensity factors (SIF). It was found that the FEM–BEM coupling technique gives more accurate values of the stress intensity factors with fewer degrees of freedom.     

A subdomain boundary element method for high‐Reynolds laminar flow using stream function‐vorticity formulation

The paper presents a new formulation of the integral boundary element method (BEM) using subdomain technique. A continuous approximation of the function and the function derivative in the direction normal to the boundary element (further ‘normal flux’) is introduced for solving the general form of a parabolic diffusion‐convective equation. Double nodes for normal flux approximation are used. The gradient continuity is required at the interior subdomain corners where compatibility and equilibrium interface conditions are prescribed. The obtained system matrix with more equations than unknowns is solved using the fast iterative linear least squares based solver. The robustness and stability of the developed formulation is shown on the cases of a backward‐facing step flow and a square‐driven cavity flow up to the Reynolds number value 50 000. Copyright © 2004 John Wiley & Sons, Ltd.     

A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes

This paper proposes a hybrid vertex-centered finite volume/finite element method for solution of the two dimensional (2D) incompressible Navier-Stokes equations on unstructured grids.An incremental pressure fractional step method is adopted to handle the velocity-pressure coupling.The velocity and the pressure are collocated at the node of the vertex-centered control volume which is formed by joining the centroid of cells sharing the common vertex.For the temporal integration of the momentum equations,an implicit second-order scheme is utilized to enhance the computational stability and eliminate the time step limit due to the diffusion term.The momentum equations are discretized by the vertex-centered finite volume method (FVM) and the pressure Poisson equation is solved by the Galerkin finite element method (FEM).The momentum interpolation is used to damp out the spurious pressure wiggles.The test case with analytical solutions demonstrates second-order accuracy of the current hybrid scheme in time and space for both velocity and pressure.The classic test cases,the lid-driven cavity flow,the skew cavity flow and the backward-facing step flow,show that numerical results are in good agreement with the published benchmark solutions.     

Spectral/hp penalty least‐squares finite element formulation for unsteady incompressible flows

In this paper, we present spectral/hp penalty least‐squares finite element formulation for the numerical solution of unsteady incompressible Navier–Stokes equations. Pressure is eliminated from Navier–Stokes equations using penalty method, and finite element model is developed in terms of velocity, vorticity and dilatation. High‐order element expansions are used to construct discrete form. Unlike other penalty finite element formulations, equal‐order Gauss integration is used for both viscous and penalty terms of the coefficient matrix. For time integration, space–time decoupled schemes are implemented. Second‐order accuracy of the time integration scheme is established using the method of manufactured solution. Numerical results are presented for impulsively started lid‐driven cavity flow at Reynolds number of 5000 and transient flow over a backward‐facing step. The effect of penalty parameter on the accuracy is investigated thoroughly in this paper and results are presented for a range of penalty parameter. Present formulation produces very accurate results for even very low penalty parameters (10–50). Copyright © 2008 John Wiley & Sons, Ltd.     

A boundary element analysis of multiply connected three‐dimensional cavity mixing flow of polymer solutions

The emergence of non‐linear dynamics in cavity mixing is examined using the boundary element method (BEM). The method is implemented for the simulation of three‐dimensional transient creeping flow of Newtonian or linear viscoelastic fluids of the Jeffreys type. A boundary only formulation in the time domain is proposed for viscoelastic flow. Special emphasis is placed on cavity flow involving multiply connected moving domains. The BEM becomes particularly suited for this case, when part of the boundary (stirrer or rotor) is moving, and the remaining outer part (cavity) is at rest. In contrast to conventional volume methods, the BEM is shown to be much easier to implement since the kinematics of the elements bounding the fluid is known (imposed). It is found that, for a simple cavity flow induced by a rotating vane at constant angular velocity, the tractions at the vane tip and cavity face exhibit non‐linear periodic dynamical behaviour with time for fluids obeying linear constitutive equations. Copyright © 1999 John Wiley & Sons, Ltd.     

Application of scaled boundary finite element method in static and dynamic fracture problems

The scaled boundary finite element method (SBFEM) is a recently developed numerical method combining advantages of both finite element methods (FEM) and boundary element methods (BEM) and with its own special features as well. One of the most prominent advantages is its capability of calculating stress intensity factors (SIFs) directly from the stress solutions whose singularities at crack tips are analytically represented. This advantage is taken in this study to model static and dynamic fracture problems. For static problems, a remeshing algorithm as simple as used in the BEM is developed while retaining the generality and flexibility of the FEM. Fully-automatic modelling of the mixed-mode crack propagation is then realised by combining the remeshing algorithm with a propagation criterion. For dynamic fracture problems, a newly developed series-increasing solution to the SBFEM governing equations in the frequency domain is applied to calculate dynamic SIFs. Three plane problems are modelled. The numerical results show that the SBFEM can accurately predict static and dynamic SIFs, cracking paths and load-displacement curves, using only a fraction of degrees of freedom generally needed by the traditional finite element methods.The project supported by the National Natural Science Foundation of China (50579081) and the Australian Research Council (DP0452681)The English text was polished by Keren Wang.     

Finite element analysis of creeping flows using marker particles

The penalty function formulation of the finite element method is described for the analysis of transient incompressible creeping flows. Marker particles are utilized to represent moving free surfaces and to visualize the flow patterns. For determining the movement of markers from element to element, the area coordinate system of the linear triangular element is introduced. With the method presented, a punch indentation problem and an injection problem for an L-shaped cavity are solved for Newtonian and power-law fluids.     

基于球面细分法的高效高精度近奇异积分计算

精确高效地计算近奇异积分,对边界元法的成功实施至关重要,也是边界元法在实际工程计算中面临的主要障碍之一。论文提出了一种基于球面细分技术的近奇异积分计算方法,可以精确计算任意基本解类型、任意单元形状和任意源点位置的近奇异积分。该方法首先通过计算源点到单元的最近最远距离,来确定球面细分的初始半径和终止半径;然后通过一系列半径呈指数级增长的球面来分割积分单元,得到一系列三角形和四边形子单元;最后把细分后得到的子单元变成弧形状,即三角形和四边形子单元分别变成扇形和环形子单元。由于球面细分是直接在三维笛卡尔坐标系下进行的,所以它适用于任何类型的单元。此外,由于基本解主要是源点到场点距离的函数,因此在同等精度下,近奇异积分在子单元的环向上所需要的高斯积分点数将大大减少。在径向方向上,由于球半径系列呈指数级变化,各个子块可以做到等精度高斯积分。数值算例表明,与传统近奇异积分计算方法相比,论文提出的方法更加稳定,精度更高。