An Adaptive Fast Multipole Boundary Element Method for Three-dimensional Potential Problems

An adaptive fast multipole boundary element method (FMBEM) for general three-dimensional (3-D) potential problems is presented in this paper. This adaptive FMBEM uses an adaptive tree structure that can balance the multipole to local translations (M2L) and the direct evaluations of the near-field integrals, and thus can reduce the number of the more costly direct evaluations. Furthermore, the coefficients used in the preconditioner for the iterative solver (GMRES) are stored and used repeatedly in the direct evaluations of the near-field contributions. In this way, the computational efficiency of the adaptive FMBEM is improved significantly. The adaptive FMBEM can be applied to both the original FMBEM formulation and the new FMBEM with diagonal translations. Several numerical examples are presented to demonstrate the efficiency and accuracy of the adaptive FMBEM for studying large-scale 3-D potential problems. The adaptive FMBEM is found to be about 50% faster than the non-adaptive version of the new FMBEM in solving the model (with 558,000 elements) for porous materials studied in this paper. The computational efficiencies and accuracies of the FMBEM as compared with the finite element method (FEM) are also studied using a heat-sink model. It is found that the adaptive FMBEM is especially advantageous in modeling problems with complicated domains for which free meshes with much more finite elements would be needed with the FEM.     

Mechanical Analysis of 3D Composite Materials by Hybrid Boundary Node Method

In this paper, an improved multi-domain model based on the hybrid boundary node method (Hybrid BNM) is proposed for mechanical analysis of 3D composites. The Hybrid BNM is a boundary type meshless method which based on the modified variational principle and the Moving Least Squares (MLS) approximation. The improved multi-domain model can reduce the total degrees of freedom (DOFs) compared with the conventional multi-domain solver. It is very suitable for the inclusion-based composites, especially for the composites when the inclusions are solid and totally embedded in the matrix domain. Numerical examples are presented to verify the improved multi-domain model and the results have shown the accuracy and efficiency of the improved model.     

Fast HdBNM for large-scale thermal analysis of CNT-reinforced composites

Because of their high thermal conductivities, carbon nanotubes (CNT) have promising potential in development of fundamentally new composites. To study the influence of CNTs distribution on the overall properties of a composite, the modeling of a Representative Volume Element (RVE) including a large number of CNTs that are randomly distributed and oriented is necessary. However, analysis of such a RVE using standard numerical methods faces two severe difficulties, namely the discretization of the geometry and a very large computational scale. In this paper, the first difficulty is alleviated by employing the Hybrid Boundary Node Method (HdBNM), which is a form of the boundary type meshless methods. To overcome the second difficulty, the Fast Multipole Method (FMM) is combined with the HdBNM to solve a simplified mathematical model. RVEs containing various numbers of CNTs with different lengths, shapes and alignments have been analyzed, resulting in valuable insights gained into the thermal behavior of the composite material.     

A Meshless Local Petrov-Galerkin (MLPG) Approach for 3-Dimensional Elasto-dynamics

A Meshless Local Petrov-Galerkin (MLPG) method has been developed for solving 3D elasto-dynamic problems. It is derived from the local weak form of the equilibrium equations by using the general MLPG concept. By incorporating the moving least squares (MLS) approximations for trial and test functions, the local weak form is discretized, and is integrated over the local sub-domain for the transient structural analysis. The present numerical technique imposes a correction to the accelerations, to enforce the kinematic boundary conditions in the MLS approximation, while using an explicit time-integration algorithm. Numerical examples for solving the transient response of the elastic structures are included. The results demonstrate the efficiency and accuracy of the present method for solving the elasto-dynamic problems; and its superiority over the Galerkin Finite Element Method.     

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.     

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.     

Nonlinear Dynamic Analysis of Three-Dimensional Elasto-Plastic Solids by the Meshless Local Petrov-Galerkin (MLPG) Method

The meshless local Petrov-Galerkin approach is proposed for the nonlinear dynamic analysis of three-dimensional (3D) elasto-plastic problems. Galerkin weak-form formulation is applied to derive the discrete governing equations. A weak formulation for the set of governing equations is transformed into local integral equations on local sub-domains by using a unit test function and local weak-form formulation in three dimensional continua for the general dynamic problems is derived. Three dimensional Moving Least-Square (MLS) approximation is considered as shape function to approximate the field variable of scattered nodes in the problem domain. Normality hypothesis of plasticity is adopted to define the stress-strain relation in elasto-plastic analysis and the unknown plastic multiplier is obtained by the consistency condition. Von Mises yield criterion in three dimensional space is used as a yield function to determine whether the material has yielded. The Newmark time integration method in an incremental form is used to solve the final system of nonlinear second order Ordinary Differential Equations (ODEs). Several numerical examples are given to demonstrate the accuracy and effectiveness of the present numerical approach.     

Fast single domain–subdomain BEM algorithm for 3D incompressible fluid flow and heat transfer

In this paper acceleration and computer memory reduction of an algorithm for the simulation of laminar viscous flows and heat transfer is presented. The algorithm solves the velocity–vorticity formulation of the incompressible Navier–Stokes equations in 3D. It is based on a combination of a subdomain boundary element method (BEM) and single domain BEM. The CPU time and storage requirements of the single domain BEM are reduced by implementing a fast multipole expansion method. The Laplace fundamental solution, which is used as a special weighting function in BEM, is expanded in terms of spherical harmonics. The computational domain and its boundary are recursively cut up forming a tree of clusters of boundary elements and domain cells. Data sparse representation is used in parts of the matrix, which correspond to boundary‐domain clusters pairs that are admissible for expansion. Significant reduction of the complexity is achieved. The paper presents results of testing of the multipole expansion algorithm by exploring its effect on the accuracy of the solution and its influence on the non‐linear convergence properties of the solver. Two 3D benchmark numerical examples are used: the lid‐driven cavity and the onset of natural convection in a differentially heated enclosure. Copyright © 2008 John Wiley & Sons, Ltd.     

A Differential Privacy Based (<i>k</i>-Ψ)-Anonymity Method for Trajectory Data Publishing

In recent years, mobile Internet technology and location based services have wide application. Application providers and users have accumulated huge amount of trajectory data. While publishing and analyzing user trajectory data have brought great convenience for people, the disclosure risks of user privacy caused by the trajectory data publishing are also becoming more and more prominent. Traditional k-anonymous trajectory data publishing technologies cannot effectively protect user privacy against attackers with strong background knowledge. For privacy preserving trajectory datapublishing, we propose a differential privacy based (k-Ψ)-anonymity method to defend against re-identification and probabilistic inference attack. The proposed method is divided into two phases: in the first phase, a dummy-based (k-Ψ)-anonymous trajectory data publishing algorithm is given, which improves (k-δ)-anonymity by considering changes of threshold δ on different road segments and constructing an adaptive threshold set Ψ that takes into account road network information. In the second phase, Laplace noise regarding distance of anonymous locations under differential privacy is used for trajectory perturbation of the anonymous trajectory dataset outputted by the first phase. Experiments on real road network dataset are performed and the results show that the proposed method improves the trajectory indistinguishability and achieves good data utility in condition of preserving user privacy.     

Boundary element‐free method (BEFM) for two‐dimensional elastodynamic analysis using Laplace transform

In this paper, we present a direct meshless method of boundary integral equation (BIE), known as the boundary element‐free method (BEFM), for two‐dimensional (2D) elastodynamic problems that combines the BIE method for 2D elastodynamics in the Laplace‐transformed domain and the improved moving least‐squares (IMLS) approximation. The formulae for the BEFM for 2D elastodynamic problems are given, and the numerical procedures are also shown. The BEFM is a direct numerical method, in which the basic unknown quantities are the real solutions of the nodal variables, and the boundary conditions can be implemented directly and easily that leads to a greater computational precision. For the purpose of demonstration, some selected numerical examples are solved using the BEFM. Copyright © 2005 John Wiley & Sons, Ltd.     

The Global Nonlinear Galerkin Method for the Solution of von Karman Nonlinear Plate Equations: An Optimal & Faster Iterative Method for the Direct Solution of Nonlinear Algebraic Equations F(x) = 0, using x<sup style='margin-left:-6.5px'>·</sup> = λ[αF + (1 - α)B^TF]

The application of the Galerkin method, using global trial functions which satisfy the boundary conditions, to nonlinear partial differential equations such as those in the von Karman nonlinear plate theory, is well-known. Such an approach using trial function expansions involving multiple basis functions, leads to a highly coupled system of nonlinear algebraic equations (NAEs). The derivation of such a system of NAEs and their direct solutions have hitherto been considered to be formidable tasks. Thus, research in the last 40 years has been focused mainly on the use of local trial functions and the Galerkin method, applied to the piecewise linear system of partial differential equations in the updated or total Lagrangean reference frames. This leads to the so-called tangent-stiffness finite element method. The piecewise linear tangent-stiffness finite element equations are usually solved by an iterative Newton-Raphson method, which involves the inversion of the tangent-stiffness matrix during each iteration. However, the advent of symbolic computation has made it now much easier to directly derive the coupled system of NAEs using the global Galerkin method. Also, methods to directly solve the NAEs, without inverting the tangent-stiffness matrix during each iteration, and which are faster and better than the Newton method are slowly emerging. In a previous paper [Dai, Paik and Atluri (2011a)], we have presented an exponentially convergent scalar homotopy algorithm to directly solve a large set of NAEs arising out of the application of the global Galerkin method to von Karman plate equations. While the results were highly encouraging, the computation time increases with the increase in the number of NAEs-the number of coupled NAEs solved by Dai, Paik and Atluri (2011a) was of the order of 40. In this paper we present a much improved method of solving a larger system of NAEs, much faster. If F(x) = 0 [F_i(x_j) = 0] is the system of NAEs governing the modal amplitudes x_j [j = 1, 2...N], for large N, we recast the NAEs into a system of nonlinear ODEs: x^· = λ[αF + (1 - α)B^TF], where λ and α are scalars, and B_ij = ∂F_i / ∂x_j. We derive a purely iterative algorithm from this, with optimum value for λ and α being determined by keeping x on a newly defined invariant manifold [Liu and Atluri (2011b)]. Several numerical examples of nonlinear von Karman plates, including the post-buckling behavior of plates with initial imperfections are presented to show that the present algorithms for directly solving the NAEs are several orders of magnitude faster than those in Dai, Paik and Atluri (2011a). This makes the resurgence of simple global Galerkin methods, as alternatives to the finite element method, to directly solve nonlinear structural mechanics problems without piecewise linear formulations, entirely feasible.     

Three-dimensional (3D) interconnected networks fabricated via <Emphasis Type="Italic">in-situ</Emphasis> growth of N-doped graphene/carbon nanotubes on Co-containing carbon nanofibers for enhanced oxygen reduction

The strategy of combining highly conductive frameworks with abundant active sites is desirable in the preparation of alternative catalysts to commercial Pt/C for the oxygen reduction reaction (ORR). In this study, N-doped graphene (NG) and carbon nanotubes (CNT) were grown in-situ on Co-containing carbon nanofibers (CNF) to form three-dimensional (3D) interconnected networks. The NG and CNT bound the interlaced CNF together, facilitating electron transfer and providing additional active sites. The 3D interconnected fiber networks exhibited excellent ORR catalytic behavior with an onset potential of 0.924 V (vs. reversible hydrogen electrode) and a higher current density than Pt/C beyond 0.720 V. In addition, the hybrid system exhibited superior stability and methanol tolerance to Pt/C in alkaline media. This method can be extended to the design of other 3D interconnected network architectures for energy storage and conversion applications.

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