A meshless hybrid boundary-node method for Helmholtz problems

By coupling the moving least squares (MLS) approximation with a modified functional, the hybrid boundary node-method (hybrid BNM) is a boundary-only, truly meshless method. Like boundary element method (BEM), an initial restriction of the present method is that non-homogeneous terms accounting for effects such as distributed loads are included in the formulation by means of domain integrals, and thus make the technique lose the attraction of its ‘boundary-only’ character.This paper presents a new boundary-type meshless method dual reciprocity-hybrid boundary node method (DR-HBNM), which is combined the hybrid BNM with the dual reciprocity method (DRM) for solving Helmholtz problems. In this method, the solution of Helmholtz problem is divided into two parts, i.e. the complementary solution and the particular solution. The complementary solution is solved by means of hybrid BNM and the particular one is obtained by DRM. The modified variational formulation is applied to form the discrete equations of hybrid BNM. The MLS is employed to approximate the boundary variables, while the domain variables are interpolated by fundamental solutions. The domain integration is interpolated by radial basis function (RBF). The proposed method in the paper retains the characteristics of the meshless method and BEM, which only requires discrete nodes constructed on the boundary of a domain, several nodes in the domain are needed just for the RBF interpolation. The parameters that influence the performance of this method are studied through numerical examples and known analytical fields. Numerical results for the solution of Helmholtz equation show that high convergence rates and high accuracy are achievable.     

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 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.     

The boundary node method for three‐dimensional problems in potential theory

The boundary node method (BNM) is developed in this paper for solving potential problems in three dimensions. The BNM represents a coupling between boundary integral equations (BIE) and moving least‐squares (MLS) interpolants. The main idea here is to retain the dimensionality advantage of the former and the meshless attribute of the later. This results in decoupling of the ‘mesh’ and the interpolation procedure for the field variables. A general BNM computer code for 3‐D potential problems has been developed. Several parameters involved in the BNM need to be chosen carefully for a successful implementation of the method. An in‐depth and systematic study has been carried out in this paper in order to better understand the effects of various parameters on the performance of the method. Numerical results for spheres and cubes, subjected to different types of boundary conditions, are extremely encouraging. Copyright © 2000 John Wiley & Sons, Ltd.     

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.     

Multi-domain hybrid boundary node method for evaluating top-down crack in Asphalt pavements

Top-down crack is a type of crack that rivals the severity and prevalence of reflective crack. It significantly reduces pavement’s quality service life. The initiation of these cracks was explained by high-contact stresses induced under radial truck tires; however, the mechanism for top-down crack propagation has not been explained. A combination of multi-domain hybrid boundary node method (Hybrid BNM) and fracture mechanics was selected for physical representation and analysis of a pavement with a top-down crack. The hybrid BNM is a boundary-only, 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 ‘energy’. In order to simulate the singularity of the stress on the crack tip, enriched basis functions are used. The factors, which influence the stress intensity factor (SIF) and the expansion path, e.g. horizontal load, thickness of asphalt concrete (AC) layer and base, AC layer and base modulus, are studied through numerical results. It can be concluded that the Hybrid BNM, which has high convergence rates and high accuracy, is able to solve top-down crack problems.     

A new meshless regular local boundary integral equation (MRLBIE) approach

A new meshless method based on a regular local integral equation and the moving least‐squares approximation is developed. The present method is a truly meshless one as it does not need a ‘finite element or boundary element mesh’, either for purposes of interpolation of the solution variables, or for the integration of the ‘energy’. All integrals can be easily evaluated over regularly shaped domains (in general, spheres in three‐dimensional problems) and their boundaries. No derivatives of the shape functions are needed in constructing the system stiffness matrix for the internal nodes, as well as for those boundary nodes with no essential‐boundary‐condition‐prescribed sections on their local boundaries. Numerical examples presented in the paper show that high rates of convergence with mesh refinement are achievable, and the computational results for the unknown variable and its derivatives are very accurate. No special post‐processing procedure is required to compute the derivatives of the unknown variable, as the original result, from the moving least‐squares approximation, is smooth enough. Copyright © 1999 John Wiley & Sons, Ltd.     

Finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions

A new finite element formulation aimed at the solution of problems involving strain localization is presented. The proposed formulation incorporates displacement interpolated embedded localization lines. Results are shown to converge to an ‘exact solution’ when the mesh is refined and also to be quite insensitive to mesh distortions.     

The meshless hypersingular boundary node method for three-dimensional potential theory and linear elasticity problems

The Boundary Node Method (BNM) represents a coupling between Boundary Integral Equations (BIEs) and Moving Least Squares (MLS) approximants. The main idea here is to retain the dimensionality advantage of the former and the meshless attribute of the latter. The result is a ‘meshfree’ method that decouples the mesh and the interpolation procedures. The BNM has been applied to solve 2-D and 3-D problems in potential theory and linear elasticity. The Hypersingular Boundary Element Method (HBEM) has diverse important applications in areas such as fracture mechanics, wave scattering, error analysis and adaptivity, and to obtain a symmetric Galerkin boundary element formulation. The present work presents a coupling of Hypersingular Boundary Integral Equations (HBIEs) with MLS approximants, to produce a new meshfree method — the Hypersingular Boundary Node Method (HBNM). Numerical results from this new method, for selected 3-D problems in potential theory and in linear elasticity, are presented and discussed in this paper.     

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.     

An immersed finite element method with integral equation correction

We propose a robust immersed finite element method in which an integral equation formulation is used to enforce essential boundary conditions. The solution of a boundary value problem is expressed as the superposition of a finite element solution and an integral equation solution. For computing the finite element solution, the physical domain is embedded into a slightly larger Cartesian (box‐shaped) domain and is discretized using a block‐structured mesh. The defect in the essential boundary conditions, which occurs along the physical domain boundaries, is subsequently corrected with an integral equation method. In order to facilitate the mapping between the finite element and integral equation solutions, the physical domain boundary is represented with a signed distance function on the block‐structured mesh. As a result, only a boundary mesh of the physical domain is necessary and no domain mesh needs to be generated, except for the non‐boundary‐conforming block‐structured mesh. The overall approach is first presented for the Poisson equation and then generalized to incompressible viscous flow equations. As an example of fluid–structure coupling, the settling of a heavy rigid particle in a closed tank is considered. Copyright © 2010 John Wiley & Sons, Ltd.     

The boundary node method for three‐dimensional linear elasticity

The Boundary Node Method (BNM) is developed in this paper for solving three‐dimensional problems in linear elasticity. The BNM represents a coupling between Boundary Integral Equations (BIE) and Moving Least‐Squares (MLS) interpolants. The main idea is to retain the dimensionality advantage of the former and the meshless attribute of the later. This results in decoupling of the ‘mesh’ and the interpolation procedure.For problems in linear elasticity, free rigid‐body modes in traction prescribed problems are typically eliminated by suitably restraining the body. However, an alternative approach developed recently for the Boundary Element Method (BEM) is extended in this work to the BNM. This approach is based on ideas from linear algebra to complete the rank of the singular stiffness matrix. Also, the BNM has been extended in the present work to solve problems with material discontinuities and a new procedure has been developed for obtaining displacements and stresses accurately at internal points close to the boundary of a body. Copyright © 1999 John Wiley & Sons, Ltd.     

Integration‐free Coons macroelements for the solution of 2D Poisson problems

Large isoparametric macroelements with closed‐form cardinal global shape functions under the label ‘Coons‐patch macroelements’ (CPM) have been previously proposed and used in conjunction with the finite element method and the boundary element method. This paper continues the research on the performance of CPM in conjunction with the collocation method. In contrast to the previous CPM that was based on a Galerkin/Ritz formulation, no domain integration is now required, a fact that justifies the name ‘integration‐free Coons macroelements’. Therefore, in addition to avoiding mesh generation, and saving human effort, the proposed technique has the additional advantage of further reducing the computer effort. The theory is supported by five test cases concerning Poisson and Laplace problems within 2D smooth quadrilateral domains. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Updated Lagrangian/Arbitrary Lagrangian–Eulerian framework for interaction between a compressible neo‐Hookean structure and an incompressible fluid

We propose a numerical method for a fluid–structure interaction problem. The material of the structure is homogeneous, isotropic, and it can be described by the compressible neo‐Hookean constitutive equation, while the fluid is governed by the Navier–Stokes equations. Our study does not use turbulence model. Updated Lagrangian method is used for the structure and fluid equations are written in Arbitrary Lagrangian–Eulerian coordinates. One global moving mesh is employed for the fluid–structure domain, where the fluid–structure interface is an ‘interior boundary’ of the global mesh. At each time step, we solve a monolithic system of unknown velocity and pressure defined on the global mesh. The continuity of velocity at the interface is automatically satisfied, while the continuity of stress does not appear explicitly in the monolithic fluid–structure system. This method is very fast because at each time step, we solve only one linear system. This linear system was obtained by the linearization of the structure around the previous position in the updated Lagrangian formulation and by the employment of a linear convection term for the fluid. Numerical results are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

The boundary recovery and sliver elimination algorithms of three‐dimensional constrained Delaunay triangulation

A boundary recovery and sliver elimination algorithm of the three‐dimensional constrained Delaunay triangulation is proposed for finite element mesh generation. The boundary recovery algorithm includes two main procedures: geometrical recovery procedure and topological recovery procedure. Combining the advantages of the edges/faces swappings algorithm and edges/faces splittings algorithm presented respectively by George and Weatherill, the geometrical recovery procedure can recover the missing boundaries and guarantee the geometry conformity by introducing fewer Steiner points. The topological recovery procedure includes two phases: ‘dressing wound’ and smoothing, which will overcome topology inconsistency between 3D domain boundary triangles and the volume mesh. In order to solve the problem of sliver elements in the three‐dimensional Delaunay triangulation, a method named sliver decomposition is proposed. By extending the algorithm proposed by Canvendish, the presented method deals with sliver elements by using local decomposition or mergence operation. In this way, sliver elements could be eliminated thoroughly and the mesh quality could be improved in great deal. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical effects in the simulation of Ginzburg–Landau models for superconductivity

Inadequate spatial mesh resolution for simulation of the time‐dependent Ginzburg–Landau equations is shown to give rise to spurious solutions. Phenomenological studies to examine the effect of the physical parameters and boundary conditions in 2D and 3D are presented to illustrate the solution structure and to highlight non‐linear effects related to evolving vortex patterns. We illustrate and explore this issue further by considering a related simplified model in which the number of vortices is equal to the ‘winding number’ that is associated with the applied boundary conditions. Using this model we demonstrate that the solution structure is non‐unique for several values of winding number. Copyright © 2004 John Wiley & Sons, Ltd.     

Application of new error estimators based on gradient recovery and external domain approaches to 2D elastostatics problems

Two new error estimators for the BEM in 2D potential problems were recently presented by the authors. This work extends these two error estimators for 2D elastostatics problems. The first approach involves a local error estimator based on a gradient recovery procedure in which the error function is based on differences between smoothed and non-smoothed rates of change of boundary variables in the local tangential direction. The second approach is associated with the external problem formulation and gives both local and global measures of the error, depending on a choice of the external evaluation point. These approaches are post-processing procedures. Both estimators show consistency with mesh refinement and give similar qualitative results. The error estimator using the gradient recovery approach presents a more general characteristic as its formulation does not rely on an ‘optimal’ choice of an external parameter, such as in the case of the external domain error estimator. Also, the external domain error estimator can be used only for domains in which an exterior region exists. For example, the external domain error estimator cannot be used for an infinite domain with a crack, because a point in the exterior region (inside the crack) will not be at a finite distance to the crack surface.     

Transient analysis of three-dimensional wave propagation using the boundary element method

In the past the time domain solution of the wave equation has been limited to simplified problems. This was due to the limitations of analytical methods and the capacity of computers to manipulate and store ‘large’ blocks of spatial information. With the advent of ‘super computers’ the ability to solve such problems has significantly increased. This paper outlines a method for transient analysis of wave propagation in arbitrary domains using a boundary element method. The technique presented will allow the definition of a domain, the input of impedance conditions on the domain's surface, the specification of inputs on the surface, and the specification of initial conditions within the domain. It will produce a complete solution of the wave equation inside the domain. The techniques are demonstrated using a program with a boundary element formulation of Kirchhoff's equation. The elements used are triangular and compatible.