Hybrid Boundary Node Method Research Articles

ACA-accelerated hybrid boundary node method applied to multi-domain steady heat conduction problems

This paper presents a fast implementation of the hybrid boundary node method (HBNM) to solve the multi-domain steady heat conduction problems using the Adaptive Cross Approximation (ACA). It is well known that the matrices arising from the discretization of boundary integral equation are fully populated and require special compression techniques for the efficient treatment. The ACA using only few of the original entries for the approximation of the whole matrix is a purely algebraic algorithm which is kernel-independent and easy to be applied directly for various problems. Accelerated by the ACA, several multi-domain steady heat conduction problems have been solved by the HBNM. The results demonstrate the potentials of the ACA-HBNM for solving multi-domain steady heat conduction problems.

The Erdogan fundamental solution-based hybrid boundary node method for fracture problems

Based on the Erdogan fundamental solutions for infinite cracked plates, a new hybrid boundary node method for linear fracture problems is proposed in this paper. The origin singular intensity factor for the Erdogan fundamental solutions is developed to overcome the singular when the field points are coincided with the source points, and no virtual source points are needed, a new scheme for calculating the origin singular intensity factor for the Erdogan fundamental solutions is developed. Based on the Erdogan fundamental solutions, the zero traction boundary condition on crack surfaces is naturally and strictly satisfied in this method, and no nodes are arranged on the crack surface in the entire calculation process. Based on the Erdogan fundamental solution of stress intensity factor for the mixed mode crack, the stress intensity factor of the present method can be easily interpolated by the Erdogan fundamental solutions. As a result, no complex scheme for calculating stress intensity factor is needed. Based on those theories and methods, the proposed method is further applied to analyze some linear crack problems, and the computational accuracy, convergence rate and the versatility of the present method are demonstrated in details.

A dual singular hybrid boundary node method based on origin singular intensity factor

In this work, a dual singular hybrid boundary node method(SBNM) is proposed for two-dimensional potential problems. The present SBNM is developed as an improved method of the traditional dual hybrid boundary node method to give a method to deal with the boundary layer effect, instead of the traditional rigid body displacement or auxiliary source points, and improves the numerical stability of the traditional dual hybrid boundary node method. The origin singular intensity factor is employed to deal with the singular and near singular when the source points and the calculating points are very close or coincident with each other, by this method, the source points are located on physical boundary and coincide with the calculating points, and an adaptive interpolating method is developed to calculate the origin singular intensity factor to overcome numerical instability in large-scale simulations caused by singular and near singular, based on origin singular intensity factor, a new scheme of internal variables interpolation is proposed. Some studies are done to given to verify that the origin singular intensity factors exist and are not infinite, and they have the same values for the same types of problems on the same referring point with the same calculating geometry. Numerical examples are shown that the present method is efficient and accurate, and it can be used for some other large practical engineering.

Dual reciprocity hybrid boundary node method for nonlinear problems

In this paper, a boundary type meshless method of dual reciprocity hybrid boundary node method (DHBNM) is proposed to solve complicate Poisson type linear and nonlinear problems. Firstly, the solutions are divided into the complementary solutions related to homogeneous equation and the particular solutions solved by nonhomogeneous terms, for the latter, they are approximated by the radial basis function interpolation based on dual reciprocity method, and the complementary solutions are obtained based on simple Poisson's equation by hybrid boundary node method, by which a simple fundamental solution of the Laplacian operator is employed instead of some other complicated ones; then a function of field functions and their derivatives on any point can be easily obtained, employing the concept of the analog equation of Katsikadelis, the field functions and their derivatives can be expressed as the function of unknown series of coefficients, and a series of nonlinear equivalent equations can be established by collocating the original governing equation at discrete points in the interior and on boundary of the domain. As a result, a new meshless method of dual reciprocity hybrid boundary node method is proposed to solve nonlinear Poisson type problems, because of the usage of those techniques, the boundary type meshless properties can be kept for any type of nonlinear equations. Different types of classical nonlinear problems are presented to validate the effectiveness and the accuracy of the present method.

Continuous–discontinuous hybrid boundary node method for frictional contact problems

This paper presents a continuous–discontinuous hybrid boundary node method for frictional contact problems. In this method, the outer and internal boundaries are divided into several individual segments, for a continuous segment on outer boundary, the radial point interpolation method (RPIM) is employed for shape function construction, for discontinuous segments, the enriched discontinuous basis functions combined with RPIM are developed, in order to reflect the local field property of displacement and stress around crack tip, different basis functions for displacement and traction are developed for shape function construction on discontinuous segments individually. And the near tip asymptotic field functions and Heaviside function are employed for simulating the high gradient of stress field and discontinuous displacement field on contact surfaces. Besides a frictional contact theory and complementation detail for the present method is proposed, and some additional equations are developed for frictional contact iteration. Based on above technique and theory, a continuous–discontinuous hybrid boundary node method is proposed for frictional contact problems. Some numerical examples are shown that the present method is effective and can be widely used for some frictional contact engineering.

A continuous–discontinuous hybrid boundary node method for solving stress intensity factor

A novel boundary type meshless method called continuous–discontinuous hybrid boundary node method is proposed in this paper, in which the enriched discontinuous shape function is developed to solve linear elastic crack problems. Firstly, the whole boundary is divided into several individual segments, and variables on each one of those segments are interpolated, respectively. For continuous segments, radial point interpolation method is employed. In regard to discontinuous segments, the enriched discontinuous basis functions combining with radial point interpolation method are developed for simulating the discontinuity of displacement and stress field on surfaces of crack, and the near tip asymptotic field functions are employed for simulating the high gradient of stress field around crack tip, so that high accuracy and discontinuity property of a crack can be easily described. Stress intensity factors are calculated directly using displacement extrapolation by displacement field near crack tip. Some numerical examples are shown that the present method is effective and can be widely applied in some practical engineering.

A new dual reciprocity hybrid boundary node method based on Shepard and Taylor interpolation method and Chebyshev polynomials

A new dual reciprocity hybrid boundary node method (DHBNM) is proposed in this paper, in which the Shepard and Taylor interpolation method (STIM) and Chebyshev polynomials interpolation are proposed. Firstly, the Shepard interpolation is used to construct zero level shape function, and the high-power shape functions are constructed through the Taylor expansion, and through those two methods, no inversion is needed in the whole process of the shape function construction. Besides, Chebyshev polynomials are used as the basis functions for particular solution interpolation instead of the conical function, radial basis functions, and the analytical solutions of the basic form of particular solutions related to Chebyshev polynomials for elasticity are obtained, by means of this method, no internal node is needed, and interpolation coefficients can be given as explicit functions, so no inversion is needed for particular solution interpolation, which costs a large amount of computational expense for the traditional method. Based on those two methods, a new dual reciprocity hybrid boundary node method is developed, compared to the traditional DHBNM, no inversion is needed for both shape function construction and particular solution interpolation, which greatly improves the computational efficiency, and no internal node is needed for particular solution interpolation. Numerical examples are given to illustrate that the present method is accurate and effective.

A rigid-inclusion model for fiber-reinforced composites by fast multipole hybrid boundary node method

A hybrid boundary node method for mechanical analysis of fiber-reinforced composites based on a rigid-inclusion model is proposed. Numerical simulation for fiber-reinforced composites usually demands extremely large and expensive computer resources. In a fiber-reinforced composite, the fibers have much higher values of stiffness compared with the matrix. In this case, the fibers can be considered as rigid ones and it can significantly reduce the computational scale for the analysis. In fact, the matrix is the only elastic domain which needs to be modeled in the proposed modeling, while the fibers are treated as rigid inclusions with only six rigid-body displacements. The fast multipole method is coupled with the rigid model to further reduce the computational time for large scale computation. Numerical examples demonstrate the efficiency of the proposed approach.

A new hybrid boundary node method based on Taylor expansion and the Shepard interpolation method

SummaryA novel meshless method based on the Shepard and Taylor interpolation method (STIM) and the hybrid boundary node method (HBNM) is proposed. Based on the Shepard interpolation method and Taylor expansion, the STIM is developed to construct the shape function of the HBNM. In the STIM, the Shepard shape function is used as the basic function, which is the zero‐level shape function, and the high‐power basic functions are constructed through Taylor expansion. Four advantages of the STIM are the interpolation property, the arbitrarily high‐order consistency, the absence of inversion for the whole process of shape function construction, and the low computational expense. These properties are desirable in the implementation of meshless methods. By combining the STIM and the HBNM, a much more effective meshless method is proposed to solve the elasticity problems. Compared with the traditional HBNM, the STIM can improve accuracy because of the use of high‐power basic functions and can also improve the computational efficiency because there is no inversion for the shape function construction process. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.

Mechanical Analysis of 3D Composite Materials byHybrid Boundary Node Method

Mechanical Analysis of 3D Composite Materials byHybrid Boundary Node Method

An improved hybrid boundary node method for 2D crack problems

The meshless hybrid boundary node method (HBNM) is a promising method for solving boundary value problems and is further developed and numerically implemented for 2D crack problems in this paper, in which the displacement discontinuity method (DDM) is introduced, and an iterative hybrid technique is employed. In this approach, the original problem is decomposed into two subsidiary problems, and the HBNM is used to model the finite domain of the body without crack, while DDM is utilized to represent the cracks. The results will be added and compared with the boundary conditions of the original problem. Iteration will be performed between the external boundaries and crack faces until all of the boundary conditions are satisfied. Thus, the advantages of the component methods are effectively combined. Numerical examples are given to illustrate the implementation and performance of the present method. It is shown that the high accuracy can be achieved with a small number of nodes, and the present iterative hybrid approach is very suitable for modeling complex multi-cracks and branched cracks problems and is also very easy to be extended to solve the crack propagation.

A new model for the analysis of reinforced concrete members with a coupled HdBNM/FEM

As a truly boundary-type meshless method, the hybrid boundary node method (HdBNM) does not require ‘boundary element mesh’, either for the purpose of interpolation of the solution variables or for the integration of ‘energy’. In this paper, the HdBNM is coupled with the finite element method (FEM) for predicting the mechanical behaviors of reinforced concrete. The steel bars are considered as body forces in the concrete. A bond model is presented to simulate the bond-slip between the concrete and steels using fictitious spring elements. The computational scale and cost for meshing can be further reduced. Numerical examples, in 2D and 3D cases, demonstrate the efficiency of the proposed approach.

Hybrid Boundary Node Method for Analysis of 3D Thin Structure

The hybrid boundary node method (Hybrid BNM) is a boundary type meshless method which based on the modified variational principle and the Moving Least Squares (MLS) approximation. This paper apply the Hybrid BNM to thin structures. No plate and shell theories are used. Theory analysis of the feasibility of the Hybrid BNM for thin structures is proposed. Numerical result shows that the Hybrid BNM is suitable for thin structures and high accuracy can be observed.

Numerical Simulation of Heat Conduction Problems by a New Fast Multipole Hybrid Boundary-Node Method

The fast multipole method (FMM) is an effective technique to reduce the computational cost in solving large-scale problems. In this article, a new fast multipole hybrid boundary-node method (FM-HBNM) is presented to solve three-dimensional heat conduction problems. In the new FM-HBNM, a diagonal form for translation operators is used and the computational cost of the multipole to local (M2L) translation is further reduced. Formulations for the new FM-HBNM are derived. The computational costs for the original and new FM-HBNM are estimated. The numerical results show that a speed-up about 2–3 times can be achieved by the new FM-HBNM.

Thermal analysis of 3D composites by a new fast multipole hybrid boundary node method

This paper applies the hybrid boundary node method (Hybrid BNM) for the thermal analysis of 3D composites. A new formulation is derived for the inclusion-based composites. In the new formulation, the unknowns of the interfaces are assembled only once in the final system equation, which can reduce nearly one half of degrees of freedom (DOFs) compared with the conventional multi-domain solver when there are lots of inclusions. A new version of the fast multipole method (FMM) is also coupled with the new formulation and the technique is applied to thermal analysis of composites with many inclusions. In the new fast multipole hybrid boundary node method (FM-HBNM), a diagonal form for translation operators is used and the method presented can be applied to the computation of more than 1,000,000 DOFs on a personal computer. Numerical examples are presented to analyze the thermal behavior of composites with many inclusions.

A new formulation for thermal analysis of composites by hybrid boundary node method

In this paper, a new formulation based on the hybrid boundary node method (Hybrid BNM) is proposed for thermal analysis of composites. The Hybrid BNM is a boundary type meshless method which based on the modified variational principle and the Moving Least Squares (MLS) approximation. In the new formulation, continuity conditions are used as the conventional multi-domain solver and the unknowns of the interfaces are assembled only once in the final system equation, which can reduce both the computational time and memory required. The new formulation is quiet suitable for the inclusion-based composites, especially for the case when the inclusions are solid and totally embedded in the matrix domain. The carbon nanotubes (CNTs) based composites are also discussed and studied by the new formulation. It shows that the thickness of the CNT has little influence on the thermal properties of the composites. Numerical examples are presented to verify the new formulation and the results have shown the accuracy and efficiency of the new formulation.

Development of a meshless hybrid boundary node method for Stokes flows

Abstract 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.

The regular hybrid boundary node method in the bending analysis of thin plate structures subjected to a concentrated load

This paper proposes a computational procedure based on the regular hybrid boundary node method (RHBNM) for solving the thin plate subjected to a concentrated load. The solution is decomposed into the particular solution arising from the concentrated load and the complementary solution for the homogeneous equation . In the solution procedure, the particular solution is first obtained by the fundamental solution. For the latter, we employ the RHBNM to solve. The RHBNM is a promising boundary type meshless method based on a modified variational principle and the moving least squares (MLS) approximation, and exploits the meshless attributes of the MLS and the reduced dimensionality advantages of the boundary element method (BEM). In this paper, a modified variational functional of the thin plate is developed, in which the independent variables are the generalized displacements and generalized tractions on the boundary and the lateral deflection in the domain. The MLS method is employed to approximate the boundary variables whereas the domain variables are interpolated by a linear combination of fundamental solutions of both the biharmonic equation and Laplace's equation . Some numerical tests illustrate the validity and efficiency of the present method. ► The meshless RHBNM is extended to thin plate subjected to a concentrated load. ► Particular solution arising from concentrated load is obtained by fundamental solution. ► The complementary solution for the homogeneous equation is obtained by the RHBNM. ► A modified variational is developed to derive the system of equations. ► The numerical tests show that the accuracy and convergence rate are very high.

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.

A Hybrid Radial Boundary Node Method for Biharmonic Problems

The hybrid radial boundary node method is applied to solve the biharmonic problems. Based on modified variational principle, the variational formula of the biharmonic problems is established. The radial basis point interpolation is employed to approximate the boundary variables, while the domain variables are interpolated by a combination of the fundamental solution of the laplace equation and the biharmonic equation. Compared to the regular hybrid boundary node method, as the shape function has the delta function property, the boundary conditions of the original problem can be easily implemented, and the fictitious source points are not involved. Numerical examples show that this method is efficient for solving the biharmonic equation.