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To present a new collocation method for numerically solving partial differential equations (PDEs) in rectangular domains.

The proposed method is based on a Cartesian grid and a 1D integrated‐radial‐basis‐function scheme. The employment of integration to construct the RBF approximations representing the field variables facilitates a fast convergence rate, while the use of a 1D interpolation scheme leads to considerable economy in forming the system matrix and improvement in the condition number of RBF matrices over a 2D interpolation scheme.

The proposed method is verified by considering several test problems governed by second‐ and fourth‐order PDEs; very accurate solutions are achieved using relatively coarse grids.

Only 1D and 2D formulations are presented, but we believe that extension to 3D problems can be carried out straightforwardly. Further, development is needed for the case of non‐rectangular domains.

The contribution of this paper is a new effective collocation formulation based on RBFs for solving PDEs.

The purpose of this paper is to present a new discretisation scheme, based on equation-coupled approach and high-order five-point integrated radial basis function (IRBF) approximations, for solving the first biharmonic equation, and its applications in fluid dynamics.

The first biharmonic equation, which can be defined in a rectangular or non-rectangular domain, is replaced by two Poisson equations. The field variables are approximated on overlapping local regions of only five grid points, where the IRBF approximations are constructed to include nodal values of not only the field variables but also their second-order derivatives and higher-order ones along the grid lines. In computing the Dirichlet boundary condition for an intermediate variable, the integration constants are used to incorporate the boundary values of the first-order derivative into the boundary IRBF approximation.

These proposed IRBF approximations on the stencil and on the boundary enable the boundary values of the derivative to be exactly imposed, and the IRBF solution to be much more accurate and not influenced much by the RBF width. The error is reduced at a rate that is much greater than four. In fluid dynamics applications, the method is able to capture well the structure of steady highly non-linear fluid flows using relatively coarse grids.

The main contribution of this study lies in the development of an effective high-order five-point stencil based on IRBFs for solving the first biharmonic equation in a coupled set of two Poisson equations. A fast rate of convergence (up to 11) is achieved.

This paper is concerned with the application of radial basis function networks (RBFNs) as interpolation functions for all boundary values in the boundary element method (BEM) for the numerical solution of heat transfer problems. The quality of the estimate of boundary integrals is greatly affected by the type of functions used to interpolate the temperature, its normal derivative and the geometry along the boundary from the nodal values. In this paper, instead of conventional Lagrange polynomials, interpolation functions representing these variables are based on the “universal approximator” RBFNs, resulting in much better estimates. The proposed method is verified on problems with different variations of temperature on the boundary from linear level to higher orders. Numerical results obtained show that the BEM with indirect RBFN (IRBFN) interpolation performs much better than the one with linear or quadratic elements in terms of accuracy and convergence rate. For example, for the solution of Laplace's equation in 2D, the BEM can achieve the norm of error of the boundary solution of O(10⁻⁵) by using IRBFN interpolation while quadratic BEM can achieve a norm only of O(10⁻²) with the same boundary points employed. The IRBFN‐BEM also appears to have achieved a higher efficiency. Furthermore, the convergence rates are of O(h^1.38) and O(h^4.78) for the quadratic BEM and the IRBFN‐based BEM, respectively, where h is the nodal spacing.

The purpose of this paper is to develop a new local radial basis function collocation method (LRBFCM) for one‐domain solving of the non‐linear convection‐diffusion equation, as it appears in mixture continuum formulation of the energy transport in solid‐liquid phase change systems.

The method is structured on multiquadrics radial basis functions. The collocation is made locally over a set of overlapping domains of influence and the time stepping is performed in an explicit way. Only small systems of linear equations with the dimension of the number of nodes in the domain of influence have to be solved for each node. The method does not require polygonisation (meshing). The solution is found only on a set of nodes.

The computational effort grows roughly linearly with the number of the nodes. Results are compared with the existing steady analytical solutions for one‐dimensional convective‐diffusive problem with and without phase change. Regular and randomly displaced node arrangements have been employed. The solution is compared with the results of the classical finite volume method. Excellent agreement with analytical solution and reference numerical method has been found.

A realistic two‐dimensional non‐linear industrial test associated with direct‐chill, continuously cast aluminium alloy slab is presented.

A new meshless method is presented which is simple, efficient, accurate, and applicable in industrial convective‐diffusive solid‐liquid phase‐change problems with non‐linear material properties.

This paper aims to introduce a novel algorithm to solve initial/boundary value problems of high-order ordinary differential equations (ODEs) and high-order system of ordinary differential equations (SODEs).

The proposed method is based on Hermite polynomials and extreme learning machine (ELM) algorithm. The Hermite polynomials are chosen as basis function of hidden neurons. The approximate solution and its derivatives are expressed by utilizing Hermite network. The model function is designed to automatically meet the initial or boundary conditions. The network parameters are obtained by solving a system of linear equations using the ELM algorithm.

To demonstrate the effectiveness of the proposed method, a variety of differential equations are selected and their numerical solutions are obtained by utilizing the Hermite extreme learning machine (H-ELM) algorithm. Experiments on the common and random data sets indicate that the H-ELM model achieves much higher accuracy, lower complexity but stronger generalization ability than existed methods. The proposed H-ELM algorithm could be a good tool to solve higher order linear ODEs and higher order linear SODEs.

The H-ELM algorithm is developed for solving higher order linear ODEs and higher order linear SODEs; this method has higher numerical accuracy and stronger superiority compared with other existing methods.

Traditional three-dimensional numerical methods require a long time for transient computational fluid dynamics simulation on oil-filling process of hydrodynamic braking. The purpose of this paper is to investigate reconstruction and prediction methods for the pressure field on blade surfaces to explore an accurate and rapid numerical method to solve transient internal flow in a hydrodynamic retarder.

Dynamic braking performance for the oil-filling process was simulated and validated using experimental results. With the proper orthogonal decomposition (POD) method, the dominant modes of transient pressure distribution on blades were extracted using their spatio-temporal structural features from the knowledge of computed flow data. Pressure field on blades was reconstructed. Based on the approximate model (AM), transient pressure field on blades was predicted in combination with POD. The causes of reconstruction and prediction error were, respectively, analyzed.

Results show that reconstruction with only a few dominant POD modes could represent all flow samples with high accuracy. POD method demonstrates an efficient simplification for accurate prediction of the instantaneous variation of pressure field in a hydrodynamic retarder, especially at the stage of high oil-filling rate.

The paper presents a novel numerical method, which combines POD and AM approaches for rapid and accurate prediction of braking characteristics during the oil-filling period, based on the knowledge of computed flow data.

This paper gives a bibliographical review of the finite element and boundary element parallel processing techniques from the theoretical and application points of view. Topics include: theory – domain decomposition/partitioning, load balancing, parallel solvers/algorithms, parallel mesh generation, adaptive methods, and visualization/graphics; applications – structural mechanics problems, dynamic problems, material/geometrical non‐linear problems, contact problems, fracture mechanics, field problems, coupled problems, sensitivity and optimization, and other problems; hardware and software environments – hardware environments, programming techniques, and software development and presentations. The bibliography at the end of this paper contains 850 references to papers, conference proceedings and theses/dissertations dealing with presented subjects that were published between 1996 and 2002.

The purpose of this paper is to develop an effective numerical approach to assess the nonlinear dynamic responses of a near‐bed submarine pipeline.

A coupled numerical approach is proposed in this paper to assess the nonlinear dynamic responses of this pipeline. The boundary‐element method is first used to get the nonlinear dynamic fluid loading induced by the asymmetric flow. The meshless technique is used to discretize the structure of the pipeline. A numerical example is first presented to verify the effectivity of the present method. Then, the coupled technique is used to simulate the nonlinear dynamic fluid‐structure interaction problem of a near‐bed pipeline. A Newton‐Raphson iteration procedure is used herein to solve the nonlinear system of equations, and the Newmark method is adopted for the time integration.

The presence of seabed results in a large negative lift on a pipeline in a horizontal current. Studies reveal that there exists a critical current velocity, above which the pipeline will become instable, and the critical velocity is significantly affected by the initial gap from the pipeline to the seabed.

The near‐bed submarine pipeline is a widely used structure in marine engineering. This paper originally develops a numerical approach to model this special fluid‐structure interaction problem. It has demonstrated by the examples that the present approach is very effective and has good potential in the practical applications.

The purpose of this paper is to present a new approach of meshless local B-spline based finite difference (FD) method for solving two dimensional transient heat conduction problems.

In the present method, any governing equations are discretized by B-spline approximation which is implemented in the spirit of FD technique using a local B-spline collocation scheme. The key aspect of the method is that any derivative is stated as neighbouring nodal values based on B-spline interpolants. The set of neighbouring nodes are allowed to be randomly distributed thus enhanced flexibility in the numerical simulation can be obtained. The method requires no mesh connectivity at all for either field variable approximation or integration. Time integration is performed by using the Crank-Nicolson implicit time stepping technique.

Several heat conduction problems in complex domains which represent for extended surfaces in industrial applications are examined to demonstrate the effectiveness of the present approach. Comparison of the obtained results with solutions from other numerical method available in literature is given. Excellent agreement with reference numerical method has been found.

The method is presented for 2D problems. Nevertheless, it would be also applicable for 3D problems.

A transient two dimensional heat conduction in complex domains which represent for extended surfaces in industrial applications is presented.

The presented new meshless local method is simple and accurate, while it is also suitable for analysis in domains of arbitrary geometries.

Traditional prediction of braking characteristics of vehicular hydrodynamic retarders is commonly conducted based on braking characteristics model of closed working chamber, namely, closed working chamber model (CWCM). In CWCM, inlet and outlet oil pressures and braking torque are considered to be independent of inlet and outlet flow rates. However, inlet and outlet flow rates can affect internal and external braking characteristics under actual working conditions. This study aims to establish a more accurate braking characteristics model of a hydrodynamic retarder under full oil-charging condition, and then the influence of varying inlet and outlet flow rates on oil pressures and braking torque is investigated in this paper.

A full flow passage of working chamber in a hydrodynamic retarder with inlet and outlets was established, and the reliability of numerical model was analyzed and validated. Pressure rise was introduced to describe the variation of inlet and outlet oil pressures. Then, on the basis of the validation, the CWCM was proposed at different rotor rotational speeds. The inlet and outlet oil pressures and braking torque were numerically computed at different inlet and outlet flow rates with Full Factorial Design experimental method. The results obtained were involved into establishing the braking characteristics model of open working chamber, namely, open working chamber model (OWCM), combined with Radial basis function approximation model. The OWCM with different inlet and outlet flow rates was analyzed and compared with CWCM.

The results show that inlet and outlet flow rates have obvious influence on the variation of inlet and outlet oil pressures in OWCM compared with CWCM. The outlet A pressure rise significantly changes with the inlet and outlet A flow rates, while the pressure rise of outlet B is mainly affected by the outlet B flow rate.

This paper presents an OWCM of hydrodynamic retarders under full oil-charging condition. The model takes into account the impact of oil inflowing and outflowing from the working chamber, which can provide a more accurate prediction of braking characteristics of hydrodynamic retarders.