The convergence of a mixed finite element method for steady convective incompressible viscous flow

Some results on the convergence of the assumed deviatoric stress-pressure-velocity mixed finite element method for steady, convective, incompressible, viscous flow are given. An abstract error estimate is proved, which shows that the same LBB conditions for hybrid finite element method for Stokes flow are also applicable to the present method. An unusual term appears in the estimate, the rate of convergence for this term is examined. To make our idea clear, the same finite element method is applied to single elliptic equations first.This work was supported by the Science Foundation of Academia Sinica, No. (84)-103     

On the superiority of the mixed element free Galerkin method for solving the steady incompressible flow problems

The element free Galerkin (EFG) method is a promising method for solving flow problems, but it meets the difficulty of volumetric locking for solving the incompressible flow problems. In this paper, a mixed EFG method is proposed for solving the steady incompressible flow problems, which avoids the volumetric locking and inherits the meshfree properties. The method employs two sets of nodes, one for the velocity approximation and the other for the pressure approximation. Specially, the ratio between the velocity node number and the pressure node number is taken as the only indicator for the locking behavior of the mixed EFG method. And inf–sup tests are carried out to investigate the relationship between the ratio and locking behavior. By two numerical examples, the accuracy, rate of convergence and efficiency of the mixed EFG method are also carefully studied. The results show that the accuracy, convergence and efficiency of the mixed EFG method are superior to that of the time-related fractional step methods.     

Optimization method for steady conduction in special geometry using a boundary element method

In some steady heat conduction problems in special geometries which consist of a closely spaced surface and circular holes in an infinite domain, thermal system designers may want to optimize the configuration of circular holes in terms of their radii and locations to achieve the goal of uniform temperature distribution over a closely spaced surface. In this paper, an efficient optimization procedure for this kind of problem is proposed utilizing (i) the special boundary element analysis, (ii) the corresponding design sensitivity analysis and (iii) the CONMIN algorithm. Three sample problems were solved to demonstrate the efficiency and the usefulness of the proposed optimization procedure. Some industrial engineering examples of such problems can be found in the injection molding process, the compression molding process, and so on. © 1998 John Wiley & Sons, Ltd.     

A comparison of boundary element method formulations for steady state anisotropic heat conduction problems

In this paper the boundary element method (BEM) is numerically implemented in order to solve steady state anisotropic heat conduction problems. Various types of elements, namely, constant elements, continuous and discontinuous linear elements and continuous and discontinuous quadratic elements are used. The performances of these various BEM formulations are compared for both the direct well-posed Dirichlet problem and the inverse ill-posed Cauchy problem, revealing several features of the BEM. Furthermore, previously undetermined analytical solutions for the integrals associated with linear and quadratic elements are presented.     

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     

A boundary element method without internal cells for solving viscous flow problems

In this paper, a new boundary element method without internal cells is presented for solving viscous flow problems, based on the radial integration method (RIM) which can transform any domain integrals into boundary integrals. Due to the presence of body forces, pressure term and the non-linearity of the convective terms in Navier–Stokes equations, some domain integrals appear in the derived velocity and pressure boundary-domain integral equations. The body forces induced domain integrals are directly transformed into equivalent boundary integrals using RIM. For other domain integrals including unknown quantities (velocity product and pressure), the transformation to the boundary is accomplished by approximating the unknown quantities with the compactly supported fourth-order spline radial basis functions combined with polynomials in global coordinates. Two numerical examples are given to demonstrate the validity and effectiveness of the proposed method.     

A high order finite element for completely incompressible creeping flow

A finite element method for simulating the creeping flow of an incompressible material is presented. The method allows for (1) a quadratic approximation of the velocity field, (2) material incompressibility everywhere within an element and (3) the ability to follow the flow through large changes of the material boundaries. A candle slowly bending under its own weight is simulated for illustrative purposes.     

A three-step finite element method for unsteady incompressible flows

This paper describes a three-step finite element method and its applications to unsteady incompressible fluid flows. The stability analysis of the one-dimensional purely convection equation shows that this method has third-order accuracy and an extended numerical stability domain in comparison with the Lax-Wendroff finite element method. The method is cost effective for incompressible flows, because it permits less frequent updates of the pressure field with good accuracy. In contrast with the Taylor-Galerkin method, the present three-step finite element method does not contain any new higher-order derivatives, and is suitable for solving non-linear multi-dimensional problems and flows with complicated outlet boundary conditions. The three-step finite element method has been used to simulate unsteady incompressible flows, such as the vortex pairing in mixing layer. The properties of the flow fields are displayed by the marker and cell technique. The obtained numerical results are in good agreement with the literature.     

An improved hybrid boundary node method for solving steady fluid flow problems

This paper describes the application of an improved hybrid boundary node method (hybrid BNM) for solving steady fluid flow problems. The hybrid BNM is a boundary type meshless method, which combined the moving least squares (MLS) approximation and the modified variational principle. It only requires nodes constructed on the boundary of the domain, and does not require any ‘mesh’ neither for the interpolation of variables nor for the integration. As the variables inside the domain are interpolated by the fundamental solutions, the accuracy of the hybrid BNM is rather high. However, shape functions for the classical MLS approximation lack the delta function property. Thus in this method, the boundary condition cannot be enforced easily and directly, and its computational cost is high for the inevitable transformation strategy of boundary condition. In the method we proposed, a regularized weight function is adopted, which leads to the MLS shape functions fulfilling the interpolation condition exactly, which enables a direct application of essential boundary conditions without additional numerical effort. The improved hybrid BNM has successfully implemented in solving steady fluid flow problems. The numerical examples show the excellent characteristics of this method, and the computation results obtained by this method are in a well agreement with the analytical solutions, which indicate that the method we introduced in this paper can be implemented to other problems.     

Scaled boundary finite element method for compressible and nearly incompressible elasticity over arbitrary polytopes

In this paper, a purely displacement-based formulation is presented within the framework of the scaled boundary finite element method to model compressible and nearly incompressible materials. A selective reduced integration technique combined with an analytical treatment in the nearly incompressible limit is employed to alleviate volumetric locking. The stiffness matrix is computed by solving the scaled boundary finite element equation. The salient feature of the proposed technique is that it neither requires a stabilization parameter nor adds additional degrees of freedom to handle volumetric locking. The efficiency and the robustness of the proposed approach is demonstrated by solving various numerical examples in two and three dimensions.     

A new iterative technique of the finite element method for the analysis of the steady state incompressible viscous flow based on the transformation of variables

In this paper the convergence acceleration for solving steady-state incompressible flows, by using iterative solvers, is explored. The variable transformation: u = u – , p = –r, where u and are the rotational velocity and the velocity potential, respectively, is applied to the finite element discretized equations so as to get diagonal-dominant equations. The effectivity of the present techniques is demonstrated on the 2D lid-driven flow and the 3D flow in a disk-cylinder system.     

A 3-D indirect boundary element method for bounded creeping flow of drops

The simulation of the flow of emulsions in porous media presents formidable challenges, due to the extremely complex evolving geometry. Methods based on boundary integral equations, suitable for creeping flows, reduce the effort dedicated to geometry representation, but can become computationally expensive. An efficient indirect boundary integral formulation representing deformable drops in a bounded Stokes flow, resulting in a set of Fredholm integral equations of the second kind, is presented. The boundary element method (BEM) based on the formulation employs an accurate numerical integration scheme for the singular kernels involved, an effective and accurate curvature and normal calculation method, and an adaptive remeshing method to simulate interfacial deformation of drops. Two benchmark problems are used to assess the accuracy of the method, and to investigate its behavior for large problems. The method is found to provide accurate results combined with well-posedness, making it suitable for use in accelerated fast multipole method algorithms.     

A hybrid element method for steady linearized free-surface flows

It is first shown that the two-dimensional linearized ship wave problem can be recast as the sum of a radiation and a diffraction problem for simple harmonic waves. Each problem can be solved by a hybrid element method (HEM) where conventional finite elements are used near the body and analytical solutions are used in the remaining infinite regions (super-elements). Variational principles which incorporate the matching conditions between regular and super-elements as natural conditions are derived. Numerical examples are presented. The theoretical aspects for extending the above ideas to a three-dimensional ship wave problems are also described.     

A dual-time method for two-dimensional unsteady incompressible flow calculations

A method for computing unsteady incompressible viscous flows on moving or deforming meshes is described. It uses a well-established time-marching finite-volume flow solver, developed for steady compressible flows past rigid bodies. Time-marching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic. Such methods can be extended to steady incompressible flows using an artificial compressibility scheme. A time-accurate scheme for unsteady incompressible flows is achieved by using an implicit real-time discretization and a dual-time approach, which uses a technique similar to the artificial compressibility scheme. Results are presented for test cases on both fixed and deforming meshes. Experimental, numerical and theoretical data have been included for comparison where available and reasonable agreement has been achieved. © 1998 John Wiley & Sons, Ltd.     

A parallel multipolar indirect boundary element method for the Neumann interior Stokes flow problem

A multipolar expansion technique is applied to the indirect formulation of the boundary element method in order to solve the two‐dimensional internal Stokes flow second kind boundary value problems. The algorithm is based on a multipolar expansion for the far field and numerical evaluation for the near field. Due to the nature of the algorithm, it is necessary to resort to the use of an iterative solver for the resulting algebraic linear system of equations. A parallel implementation is designed to take advantage of the natural domain decomposition of fast multipolar techniques and bring further improvements. Copyright © 2000 John Wiley & Sons, Ltd.     

A time domain boundary element method for poroelasticity

The development of a general boundary element method (BEM) for two- and three-dimensional quasistatic poroelasticity is discussed in detail. The new formulation, for the complete Biot consolidation theory, operates directly in the time domain and requires only boundary discretization. As a result, the dimensionality of the problem is reduced by one and the method becomes quite attractive for geotechnical analyses, particularly those which involve extensive or infinite domains. The presentation includes the definition of the two key ingredients for the BEM, namely, the fundamental solutions and a reciprocal theorem. Then, once the boundary integral equations are derived, the focus shifts to an overview of the general purpose numerical implementation. This implementation includes higher-order conforming elements, self-adaptive integration and multi-region capability. Finally, several detailed examples are presented to illustrate the accuracy and suitability of this boundary element approach for consolidation analysis.     

A boundary element method for transient piezoelectric analysis

In this paper, a boundary element (BE) formulation is developed originally which treats three-dimensional problems of transient piezoelectricity. The approach at hand uses the fundamental solution of the static piezoelectric operator instead of the transient one. This results in a domain integral appearing in the representation formula, which contains the inertia term. This domain integral can be transformed to the boundary using the dual reciprocity method (DRM), which leads to a system of ordinary differential equations in time domain, similar to the systems obtained in standard finite element methods (FEM). The DRM has been chosen because of the difficulties and big computational effort involved in a BE implementation, which makes use of the anisotropic transient piezoelectric fundamental solution. It is an approach that appears to be much too time-consuming for use in a commercial BE code, in which computational costs is an important issue. The method presented in this paper is validated by a numerical example for transient piezoelectricy, which demonstrates excellent agreement with FE computations for the generalized displacements, and an improved accuracy for the flux quantities such as electric field and elastic stresses.     

A semi-analytic boundary element method for parabolic problems

A new semi-analytic solution method is proposed for solving linear parabolic problems using the boundary element method. This method constructs a solution as an eigenfunction expansion using separation of variables. The eigenfunctions are determined using the dual reciprocity boundary element method. This separation of variables-dual reciprocity method (SOV-DRM) allows a solution to be determined without requiring either time-stepping or domain discretisation. The accuracy and computational efficiency of the SOV-DRM is found to improve as time increases. These properties make the SOV-DRM an attractive technique for solving parabolic problems.