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.     

Solution of non‐linear boundary integral equations in complex geometries with auxiliary integral subtraction

The boundary integral equation that results from the application of the reciprocity theorem to non‐linear or non‐homogeneous differential equations generally contains a domain integral. While methods exist for the meshless evaluation of these integrals, mesh‐based domain integration is generally more accurate and can be performed more quickly with the application of fast multipole methods. However, polygonalization of complex multiply‐connected geometries can become a costly task, especially in three‐dimensional analyses. In this paper, a method that allows a mesh‐based integration in complex domains, while retaining a simple mesh structure, is described. Although the technique is intended for the numerical solution of more complex differential equations, such as the Navier–Stokes equations, for simplicity the method is applied to the solution of a Poisson equation, in domains of varying complexity. It is shown that the error introduced by the auxiliary domain subtraction method is comparable to the discretization error, while the complexity of the mesh is significantly reduced. The behaviour of the error in the boundary solution observed with the application of the new method is analogous to the behaviour observed with conventional cell‐based domain integration. Copyright © 2002 John Wiley & Sons, Ltd.     

A new least-squares finite element model for combined Navier–Stokes and Darcy flows in geometrically complicated domains with solid and porous boundaries

In this paper we present a novel method for linking Navier–Stokes and Darcy equations along a porous inner boundary in a flow regime which is governed by both types of these equations. The method is based on a least-squares finite element technique and uses isoparametric C¹ continuous Hermite elements for domain discretization. We show that our technique is superior to previously developed models for the combined Navier–Stokes/Darcy flows. The previous works use weighted residual finite element procedures in conjunction with C⁰ elements which are inherently incapable of linking Navier–Stokes and Darcy equations. The paper includes the application of our model to a geometrically complicated axisymmetric slurry filtration system.     

Source point isolation boundary element method for solving general anisotropic potential and elastic problems with varying material properties

This paper presents a new robust boundary element method, based on a source point isolation technique, for solving general anisotropic potential and elastic problems with varying coefficients. Different types of fundamental solutions can be used to derive the basic integral equations for specific anisotropic problems, although fundamental solutions corresponding to isotropic problems are recommended and adopted in the paper. The use of isotropic fundamental solutions for anisotropic and/or varying material property problems results in domain integrals in the basic integral equations. The radial integration method is employed to transform the domain integrals into boundary integrals, resulting in a pure boundary element analysis algorithm that does not need any internal cells. Numerical examples for 2D and 3D potential and elastic problems are given to demonstrate the correctness and robustness of the proposed method.     

Computational fluid dynamics by boundary–domain integral method

A boundary–domain integral method for the solution of general transport phenomena in incompressible fluid motion given by the Navier–Stokes equation set is presented. Velocity–vorticity formulation of the conservation equations is employed. Different integral representations for conservation field functions based on different fundamental solutions are developed. Special attention is given to the use of subdomain technique and Krylov subspace iterative solvers. The computed solutions of several benchmark problems agree well with available experimental and other computational results. Copyright © 1999 John Wiley & Sons, Ltd.     

Conservative interpolation for the boundary integral solution of the Navier–Stokes equations

The Dual Reciprocity Method (DRM) is a technique to transform the domain integrals that appear in the boundary element method into equivalent boundary integrals. In this approach the non-linear terms are usually approximated by mathematical interpolation applied to the convective terms of the form of the Navier–Stokes equations. In this paper we introduce a conservative interpolation scheme that satisfies the continuity equation and performs better than pure mathematical interpolation. The new scheme together with a subdomain variation of the dual reciprocity method allows better approximation of the non-linear terms in the Navier–Stokes equations for moderate Reynolds number. Received: 21 January 2000     

Radial integration BEM for transient heat conduction problems

In this paper, a new boundary element analysis approach is presented for solving transient heat conduction problems based on the radial integration method. The normalized temperature is introduced to formulate integral equations, which makes the representation very simple and having no temperature gradients involved. The Green's function for the Laplace equation is adopted in deriving basic integral equations for time-dependent problems with varying heat conductivities and, as a result, domain integrals are involved in the derived integral equations. The radial integration method is employed to convert the domain integrals into equivalent boundary integrals. Based on the central finite difference technique, an implicit time marching solution scheme is developed for solving the time-dependent system of equations. Numerical examples are given to demonstrate the correctness of the presented approach.     

A wavenumber domain boundary element method model for the simulation of vibration isolation by periodic pile rows

The wavenumber domain boundary element method (WDBEM) for the interaction between the half-space soil and periodic structures is important for the design of various periodic structures in civil engineering. In this study, a WDBEM model for the half-space soil and periodic pile rows is developed and used in the analysis of the vibration isolation via pile rows. To establish the model, the rigid-body-motion method for the estimation of the Cauchy type singular integrals involved in the WDBEM is established for the first time. In the proposed model, the half-space soil and periodic pile rows are treated as elastic media. Employing the spatial domain boundary integral equations for the half-space soil and pile rows as well as the sequence Fourier transform method, the wavenumber domain boundary integral equations for the soil and pile rows are derived. By using the obtained wavenumber domain boundary integral equations, WDBEM formulations for the half-space soil and periodic pile rows are established. Using the WDBEM formulations as well as the continuity conditions at the pile–soil interfaces, a coupled WDBEM model for the pile–soil system is derived. With the proposed WDBEM model, the influences of the pile length and the shear modulus of the half-space soil on the vibration isolation effect of pile rows are examined. Presented numerical results show that the isolation vibration effect of pile rows is enhanced considerably with increasing length of the piles. Besides, the isolation vibration effect of pile rows is weakened considerably with increasing shear modulus of the half-space soil. Moreover, as expected, multiple pile rows usually produce a better isolation vibration effect than a single pile row.     

Combined single domain and subdomain BEM for 3D laminar viscous flow

A subdomain boundary element method (BEM) using a continuous quadratic interpolation of function and discontinuous linear interpolation of flux is presented for the solution of the vorticity transport equation and the kinematics equation in 3D. By employing compatibility conditions between subdomains an over-determined system of linear equations is obtained, which is solved in a least squares manner. The method, combined with the single domain BEM, is used to solve laminar viscous flows using the velocity vorticity formulation of Navier–Stokes equations. The versatility and accuracy of the method are proven using the 3D lid driven cavity test case.     

Transformation of domain integrals to boundary integrals in BEM analysis of shear deformable plate bending problems

In this paper two techniques, dual reciprocity method (DRM) and direct integral method (DIM), are developed to transform domain integrals to boundary integrals for shear deformable plate bending formulation. The force term is approximated by a set of radial basis functions. To transform domain integrals to boundary integrals using the dual reciprocity method, particular solutions are employed for three radial basis functions. Direct integral method is also introduced in this paper to evaluate domain integrals. Three examples are presented to demonstrate the accuracy of the two methods. The numerical results obtained by using different particular solutions are compared with exact solutions. Received 27 January 1999     

An updated interactive boundary layer method for high Reynolds number flows

The quasi‐simultaneous interactive boundary layer (IBL) method is improved with the iterative correction of an inviscid operator. The updated interactive boundary layer method (UIBL) presented in this work, uses the Hess–Smith panel method (HSPM) as an inviscid operator to update the outer flow calculation and the inviscid velocity in the interaction law (IL). The discretization of the Hilbert integral (HI) from the original method is modified to reduce the error introduced by the calculation of the HI in a restricted domain. The method is tested on a flat plate with a small indentation for two‐dimensional, steady, incompressible and laminar flow. The UIBL method is capable to predict the flow separation and reattachment with good accuracy. The accuracy of the results is competitive with the numerical solution of the Navier–Stokes equations (NSE). Copyright © 2005 John Wiley & Sons, Ltd.     

An efficient and high‐order frequency‐domain approach for transient acoustic–structure interactions in three dimensions

This paper is concerned with numerical solution of the transient acoustic–structure interaction problems in three dimensions. An efficient and higher‐order method is proposed with a combination of the exponential window technique and a fast and accurate boundary integral equation solver in the frequency‐domain. The exponential window applied to the acoustic–structure system yields an artificial damping to the system, which eliminates the wrap‐around errors brought by the discrete Fourier transform. The frequency‐domain boundary integral equation approach relies on accurate evaluations of relevant singular integrals and fast computation of nonsingular integrals via the method of equivalent source representations and the fast Fourier transform. Numerical studies are presented to demonstrate the accuracy and efficiency of the method. Copyright © 2016 John Wiley & Sons, Ltd.     

An indirect Boundary Element Method for solving low Reynolds number Navier–Stokes equations in a three-dimensional cavity

In the present work, we propose an indirect boundary-only integral equation approach for the numerical solution of the Navier–Stokes system of equations in a three-dimensional flow cavity. The formulation is based on an indirect integral representational formula for the permanent Stokes equations, and the use of a particular solution of a nonhomogeneous Stokes system of equations in order to obtain in an iterative way the corresponding complete solution of the problem. Previous boundary-only integral equation approaches to the present problem, using direct boundary elements formulations, result in a series of matrix multiplications that make these approaches computationally costly. Due to the use of an indirect formulation, the present approach is free from those matrix multiplications. © 1998 John Wiley & Sons, Ltd.     

The boundary element method for the solution of Stokes equations in two-dimensional domains

A boundary element method for the solution of Stokes equations governing creeping flow or Stokes flow in the interior of an arbitrary two-dimensional domain is presented. A procedure for introducing pressure data on the boundary of the domain is also included and the integral coefficients of the resulting linear algebraic equations are evaluated analytically. Calculations are performed in a circular domain using a variety of different boundary conditions, including a combination of the fluid velocity and the pressure. Results are presented both on the boundary and inside the solution domain in order to illustrate that the boundary element method developed here provides an efficient technique, in terms of accuracy and convergence, to investigate Stokes flow numerically.     

A dual reciprocity boundary element formulation using the fractional step method for the incompressible Navier–Stokes equations

This paper presents a dual reciprocity boundary element solution method for the unsteady Navier–Stokes equations in two-dimensional incompressible flow, where a fractional step algorithm is utilized for the time advancement. A fully explicit, second-order, Adams–Bashforth scheme is used for the nonlinear convective terms. We performed numerical tests for two examples: the Taylor–Green vortex and the lid-driven square cavity flow for Reynolds numbers up to 400. The results in the former case are compared to the analytical solution, and in the latter to numerical results available in the literature. Overall the agreement is excellent demonstrating the applicability and accuracy of the fractional step, dual reciprocity boundary element solution formulations to the Navier–Stokes equations for incompressible flows.     

The method of fundamental solutions for solving incompressible Navier–Stokes problems

A novel meshless numerical procedure based on the method of fundamental solutions (MFS) is proposed to solve the primitive variables formulation of the Navier–Stokes equations. The MFS is a meshless method since it is free from the mesh generation and numerical integration. We will transform the Navier–Stokes equations into simple advection–diffusion and Poisson differential operators via the operator-splitting scheme or the so-called projection method, instead of directly using the more complicated fundamental solutions (Stokeslets) of the unsteady Stokes equations. The resultant velocity advection–diffusion equations and the pressure Poisson equation are then calculated by using the MFS together with the Eulerian–Lagrangian method (ELM) and the method of particular solutions (MPS). The proposed meshless numerical scheme is a first attempt to apply the MFS for solving the Navier–Stokes equations in the moderate-Reynolds-number flow regimes. The lid-driven cavity flows at the Reynolds numbers up to 3200 for two-dimensional (2D) and 1000 for three-dimensional (3D) are chosen to validate the present algorithm. Through further simulating the flows in the 2D circular cavity with an eccentric rotating cylinder and in the 3D cube with a fixed sphere inside, we are able to demonstrate the advantages and flexibility of the proposed meshless method in the irregular geometry and multi-dimensional flows, even though very coarse node points are used in this study as compared with other mesh-dependent numerical schemes.     

Boundary-only element solutions of 2D and 3D nonlinear and nonhomogeneous elastic problems

This paper presents a robust boundary element method (BEM) that can be used to solve elastic problems with nonlinearly varying material parameters, such as the functionally graded material (FGM) and damage mechanics problems. The main feature of this method is that no internal cells are required to evaluate domain integrals appearing in the conventional integral equations derived for these problems, and very few internal points are needed to improve the computational accuracy. In addition, one of the basic field quantities used in the boundary integral equations is normalized by the material parameter. As a result, no gradients of the field quantities are involved in the integral equations. Another advantage of using the normalized quantities is that no material parameters are included in the boundary integrals, so that a unified equation form can be established for multi-region problems which have different material parameters. This is very efficient for solving composite structural problems.     

BEM simulation of compressible fluid flow in an enclosure induced by thermoacoustic waves

The problem of unsteady compressible fluid flow in an enclosure induced by thermoacoustic waves is studied numerically. Full compressible set of Navier–Stokes equations are considered and numerically solved by boundary-domain integral equations approach coupled with wavelet compression and domain decomposition to achieve numerical efficiency. The thermal energy equation is written in its most general form including the Rayleigh and reversible expansion rate terms. Both, the classical Fourier heat flux model and wave heat conduction model are investigated.The velocity–vorticity formulation of the governing Navier–Stokes equations is employed, while the pressure field is evaluated from the corresponding pressure Poisson equation. Material properties are taken to be for the perfect gas, and assumed to be pressure and temperature dependent.     

Plane stress and plate bending coupling in BEM analysis of shallow shells

In this paper the shear deformable shallow shells are analysed by boundary element method. New boundary integral equations are derived utilizing the Betti's reciprocity principle and coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. Two techniques, direct integral method (DIM) and dual reciprocity method (DRM), are developed to transform domain integrals to boundary integrals. The force term is approximted by a set of radial basis functions. Several examples are presented to demonstrate the accuracy of the two methods. The accuracy of results obtained by using boundary element method are compared with exact solutions and the finite element method. Copyright © 2000 John Wiley & Sons, Ltd.