FMM-accelerated hybrid boundary node method for multi-domain problems

This work presents a fast implementation of the multi-domain hybrid boundary node method (HdBNM) for numerical solution of the Laplace's equation. The preconditioned GMRES is employed to solve the overall system of equations. At each iteration step of the GMRES, the matrix–vector multiplication is split into smaller scale ones at the sub-domain level, and thus accelerated by the fast multipole method independently within individual sub-domains. The computed matrix–vector products at the sub-domain level are then assembled into an overall vector using the equilibrium and continuity conditions at the interfaces. Our method is tested by benchmark examples for three-dimensional potential problems, and high accuracy and efficiency are observed.     

Regular hybrid boundary node method for biharmonic problems

The regular hybrid boundary node method (RHBNM) is a new technique for the numerical solutions of the boundary value problems. By coupling the moving least squares (MLS) approximation with a modified functional, the RHBNM retains the meshless attribute and the reduced dimensionality advantage. Besides, since the source points of the fundamental solutions are located outside the domain, ‘boundary layer effect’ is also avoided. However, an initial restriction of the present method is that it is only suitable for the problems which the governing differential equation is in second order.Now, a new variational formulation for the RHBNM is presented further to solve the biharmonic problems, in which the governing differential equation is in fourth order. The modified variational functional is applied to form the discrete equations of the RHBNM. The MLS is employed to approximate the boundary variables, while the domain variables are interpolated by a linear combination of fundamental solutions of both the biharmonic equation and Laplace’s equation. Numerical examples for some biharmonic problems show that the high accuracy with a small node number is achievable. Furthermore, the computation parameters have been studied. They can be chosen in a wide range and have little influence on the results. It is shown that the present method is effective and can be widely applied in practical engineering.     

Multiple reciprocity hybrid boundary node method for potential problems

Meshless methods have some obvious advantages such as they do not require meshes in the domain and on the boundary, only some nodes are needed in the computation. Furthermore, for the boundary-type meshless methods, the nodes are even not needed in the domain and only distributed on the boundary. Practice shows that boundary-type meshless methods are effective for homogeneous problems. But for inhomogeneous problems, the application of these boundary-type meshless methods has some difficulties and need to be studied further.The hybrid boundary node method (HBNM) is a boundary-only meshless method, which is based on the moving least squares (MLS) approximation and the hybrid displacement variational principle. No cell is required either for the interpolation of solution variables or for numerical integration. It has a drawback of ‘boundary layer effect’, so a new regular hybrid boundary node method (RHBNM) has been proposed to avoid this pitfall, in which the source points of the fundamental solutions are located outside the domain. These two methods, however, can only be used for solving homogeneous problems. Combining the dual reciprocity method (DRM) and the HBNM, the dual reciprocity hybrid boundary node method (DRHBNM) has been proposed for the inhomogeneous terms. The DRHBNM requires a substantial number of internal points to interpolate the particular solution by the radial basis function, where approximation based only on boundary nodes may not guarantee sufficient accuracy.Now a further improvement to the RHBNM, i.e., a combination of the RHBNM and the multiple reciprocity method (MRM), is presented and called the multiple reciprocity hybrid boundary node method (MRHBNM). The solution comprises two parts, i.e., the complementary and particular solutions. The complementary solution is solved by the RHBNM. The particular solution is solved by the MRM, i.e., a sum of high-order homogeneous solutions, which can be approximated by the same-order fundamental solutions. Compared with the DRHBNM, the MRHBNM does not require internal points to obtain the particular solution for inhomogeneous problems. Therefore, the present method is a real boundary-only meshless method, and can be used to deal with inhomogeneous problems conveniently. The validity and efficiency of the present method are demonstrated by a series of numerical examples of inhomogeneous potential problems.     

An improved procedure for solving transient heat conduction problems using the boundary element method

In employing the boundary element method to solve transient heat conduction problems, domain integrals need to be calculated. These integrals can be calculated either by directly discretizing the domain, or indirectly by utilizing a time-marching scheme which requires the time integrations to be evaluated from the initial time. Although the second approach overcomes the need for domain discretization, it has the disadvantage of requiring large storage and CPU time for increasing number of time steps. This paper is concerned with a procedure which approximates the domain integrals without the need for domain discretization. The time-marching scheme is employed so that the domain integrals can be calculated at a particular time with a known weighting and temperature distribution. These integrals can then be utilized to approximate similar domain integrals with a different weighting. It is shown that the method proposed dramatically reduces the storage requirements and CPU time, even for a small number of time divisions.     

Dual reciprocity hybrid boundary node method for acoustic eigenvalue problems

The hybrid boundary node method (HBNM) is a truly meshless method, and elements are not required for either interpolation or integration. The method, however, can only be used for solving homogeneous problems. For the inhomogeneous problem, the domain integration is inevitable. This paper applied the dual reciprocity hybrid boundary node method (DRHBNM), which is composed by the HBNM and the dual reciprocity method (DRM) for solving acoustic eigenvalue problems. In this method, the solution is composed of two parts, i.e. the complementary solution and the particular solution. The complementary solution is solved by HBNM and the particular one is obtained by DRM. The modified variational formulation is applied to form the discrete equations of HBNM. The moving least squares (MLS) is employed to approximate the boundary variables, while the domain variables are interpolated by the fundamental solutions. The domain integration is interpolated by radial basis function (RBF). The Q–R algorithm and Householder algorithm are applied for solving the eigenvalues of the transformed matrix. The parameters that influence the performance of DRHBNM are studied through numerical examples. Numerical results show that high convergence rates and high accuracy are achievable.     

An improved dislocation approach in solving three-dimensional boundary value problems

In this paper a dislocation approach is developed for solving certain three-dimensional boundary-value probléms in an elastic medium. The application of this type of approach is demonstrated by presenting the solutions for certain axisymmetric boundary-value problems for the half-space to which this approach has not been previously applied, as well as to three-dimensional crack problems of arbitrary planar shape. Proofs for these analytic applications are given.     

An improved boundary integral equation method for Helmholtz problems

The eigenvalue problem for the Laplace operator is numerical investigated using the boundary integral equation (BIE) formulation. Three methods of discretization are given and illustrated with numerical examples.     

A hybrid boundary node method

A new variational formulation for boundary node method (BNM) using a hybrid displacement functional is presented here. The formulation is expressed in terms of domain and boundary variables, and the domain variables are interpolated by classical fundamental solution; while the boundary variables are interpolated by moving least squares (MLS). The main idea is to retain the dimensionality advantages of the BNM, and get a truly meshless method, which does not require a ‘boundary element mesh’, either for the purpose of interpolation of the solution variables, or for the integration of the ‘energy’. All integrals can be easily evaluated over regular shaped domains (in general, semi‐sphere in the 3‐D problem) and their boundaries. Numerical examples presented in this paper for the solution of Laplace's equation in 2‐D show that high rates of convergence with mesh refinement are achievable, and the computational results for unknown variables are most accurate. No further integrations are required to compute the unknown variables inside the domain as in the conventional BEM and BNM. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

Transformation of elliptic partial differential equations for solving two-dimensional boundary value problems in fluid flow

It is possible to transform elliptic partial differential equations to exchange the dependent with one of the independent variables. The Laplace equation for a stream function ‘Ψ’ over the X and Y co-ordinate system, for example, can be transformed into a relationship expressing the Y position of streamlines in terms of strem function Ψ and X. Although the resulting new partial differential equation is much more complex it is much more convenient to use in a computer. An irregular Y boundary becomes, with the new relationship, merely the boundary values assigned to the outer streamlines and the computer always need only deal with a rectangular array. The resulting answer is in the form of the position of streamlines which is the information directly required for plotting flow maps.     

A Galerkin boundary node method for biharmonic problems

A Galerkin boundary node method (GBNM) is developed in this paper for solving biharmonic problems. The GBNM combines an equivalent variational form of boundary integral formulations for governing equations with the moving least-squares approximations for construction of the trial and test functions. In this approach, only a nodal data structure on the boundary of a domain is required. In addition, boundary conditions can be implemented accurately and the system matrices are symmetric. The convergence of this method and numerical examples are given to show the efficiency.     

A boundary element method for steady incompressible thermoviscous flow

A boundary element formulation is presented for moderate Reynolds number, steady, incompressible, thermoviscous flows. The governing integral equations are written exclusively in terms of velocities and temperatures, thus eliminating the need for the computation of any gradients. Furthermore, with the introduction of reference velocities and temperatures, volume modelling can often be confined to only a small portion of the problem domain, typically near obstacles or walls. The numerical implementation includes higher order elements, adaptive integration and multiregion capability. Both the integral formulation and implementation are discussed in detail. Several examples illustrate the high level of accuracy that is obtainable with the current method.     

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.     

The hybrid boundary node method accelerated by fast multipole expansion technique for 3D potential problems

This paper presents a fast formulation of the hybrid boundary node method (Hybrid BNM) for solving problems governed by Laplace's equation in 3D. The preconditioned GMRES is employed for solving the resulting system of equations. At each iteration step of the GMRES, the matrix–vector multiplication is accelerated by the fast multipole method. Green's kernel function is expanded in terms of spherical harmonic series. An oct‐tree data structure is used to hierarchically subdivide the computational domain into well‐separated cells and to invoke the multipole expansion approximation. Formulations for the local and multipole expansions, and also conversion of multipole to local expansion are given. And a binary tree data structure is applied to accelerate the moving least square approximation on surfaces. All the formulations are implemented in a computer code written in C++. Numerical examples demonstrate the accuracy and efficiency of the proposed approach. Copyright © 2005 John Wiley & Sons, Ltd.     

A fast multipole hybrid boundary node method for composite materials

This article presents a multi-domain fast multipole hybrid boundary node method for composite materials in 3D elasticity. The hybrid boundary node method (hybrid BNM) is a meshless method which only requires nodes constructed on the surface of a domain. The method is applied to 3D simulation of composite materials by a multi-domain solver and accelerated by the fast multipole method (FMM) in this paper. The preconditioned GMRES is employed to solve the final system equation and precondition techniques are discussed. The matrix–vector multiplication in each iteration is divided into smaller scale ones at the sub-domain level and then accelerated by FMM within individual sub-domains. The computed matrix–vector products at the sub-domain level are then combined according to the continuity conditions on the interfaces. The algorithm is implemented on a computer code written in C + +. Numerical results show that the technique is accurate and efficient.     

The multi-domain hybrid boundary node method for 3D elasticity

In this paper, a multi-domain technique for 3D elasticity problems is derived from the hybrid boundary node method (Hybrid BNM). The Hybrid BNM is based on the modified variational principle and the Moving Least Squares (MLS) approximation. It does not require a boundary element mesh, neither for the purpose of interpolation of the solution variables nor for the integration of energy. This method can reduce the human-labor costs of meshing, especially for complex construction. This paper presents a further development of the Hybrid BNM for multi-domain analysis in 3D elasticity. Using the equilibrium and continuity conditions on the interfaces, the final algebraic equation is obtained by assembling the algebraic equation for each single sub-domain. The proposed multi-domain technique is capable to deal with interface and multi-medium problems and results in a block sparsity of the coefficient matrix. Numerical examples demonstrate the accuracy of the proposed multi-domain technique.     

Meshless analysis of potential problems in three dimensions with the hybrid boundary node method

Combining a modified functional with the moving least‐squares (MLS) approximation, the hybrid boundary node method (Hybrid BNM) is a truly meshless, boundary‐only method. The method may have advantages from the meshless local boundary integral equation (MLBIE) method and also the boundary node method (BNM). In fact, the Hybrid BNN requires only the discrete nodes located on the surface of the domain. The Hybrid BNM has been applied to solve 2D potential problems. In this paper, the Hybrid BNM is extended to solve potential problems in three dimensions. Formulations of the Hybrid BNM for 3D potential problems and the MLS approximation on a generic surface are developed. A general computer code of the Hybrid BNM is implemented in C++. The main drawback of the ‘boundary layer effect’ in the Hybrid BNM in the 2D case is circumvented by an adaptive face integration scheme. The parameters that influence the performance of this method are studied through three different geometries and known analytical fields. Numerical results for the solution of the 3D Laplace's equation show that high convergence rates with mesh refinement and high accuracy are achievable. Copyright © 2004 John Wiley & Sons, Ltd.     

An improved steady inverse method for turbomachinery aerodynamic design

An improved inverse method for turbomachinery design is developed in this paper. A strict ‘no-slip’ boundary condition is imposed during the flow time marching process, and a virtual movement velocity is computed based on characteristics boundary conditions according to the difference of specified and calculated pressure loading, then the camber line of the blade is moved accordingly. A new ‘inverse’ time step calculation method is proposed to make the value of the ‘inverse’ time step case-independent, and a coupled non-uniform rational B-spline (NURBS) smoothing technique is used to improve the robustness of the solver, also guarantee the manufacturability of the obtained blade. Two-dimensional test cases, including compressor and turbine cascades, are adopted to validate the method. Then, the method is extended to three dimensions, and a high-loaded rotor of a fan stage is redesigned by the method in a stage environment. The final results indicate a fairly large performance gain, which demonstrates the effectiveness of the improved method.     

An improved boundary element formulation for wave propagation problems

In this paper the dual reciprocity formulation for scalar wave equations and elastodynamic problems developed by Nardini & Brebbia is extended to the problem of waves propagating in an infinite domain by applying the Sommerfeld's radiation condition on a suitable artificial boundary. The free surface condition of first order can also be taken into consideration. To validate the present scheme, some examples have been worked out and compared with analytical solutions.     

Development of a meshless hybrid boundary node method for Stokes flows

The meshless hybrid boundary node method (HBNM) is a promising method for solving boundary value problems, and is further developed and numerically implemented for incompressible 2D and 3D Stokes flows in this paper. In this approach, a new modified variational formulation using a hybrid functional is presented. The formulation is expressed in terms of domain and boundary variables. The moving least-squares (MLS) method is employed to approximate the boundary variables whereas the domain variables are interpolated by the fundamental solutions of Stokes equation, i.e. Stokeslets. The present method only requires scatter nodes on the surface, and is a truly boundary type meshless method as it does not require the ‘boundary element mesh’, either for the purpose of interpolation of the variables or the integration of ‘energy’. Moreover, since the primitive variables, i.e., velocity vector and pressure, are employed in this approach, the problem of finding the velocity is separated from that of finding pressure. Numerical examples are given to illustrate the implementation and performance of the present method. It is shown that the high convergence rates and accuracy can be achieved with a small number of nodes.