A novel algorithm for shape parameter selection in radial basis functions collocation method

Many Radial Basis Functions (RBFs) contain a free shape parameter that plays an important role for the application of Meshless method to the analysis of multilayered composite and sandwich plates. In most papers the authors end up choosing this shape parameter by trial and error or some other ad hoc means. In this paper a novel algorithm for shape parameter selection, based on a convergence analysis, is presented. The effectiveness of this algorithm is assessed by static analyses of laminated composite and sandwich plates.     

A point interpolation meshless method based on radial basis functions

A point interpolation meshless method is proposed based on combining radial and polynomial basis functions. Involvement of radial basis functions overcomes possible singularity associated with the meshless methods based on only the polynomial basis. This non‐singularity is useful in constructing well‐performed shape functions. Furthermore, the interpolation function obtained passes through all scattered points in an influence domain and thus shape functions are of delta function property. This makes the implementation of essential boundary conditions much easier than the meshless methods based on the moving least‐squares approximation. In addition, the partial derivatives of shape functions are easily obtained, thus improving computational efficiency. Examples on curve/surface fittings and solid mechanics problems show that the accuracy and convergence rate of the present method is high. Copyright © 2002 John Wiley & Sons, Ltd.     

Weighted radial basis collocation method for boundary value problems

This work introduces the weighted radial basis collocation method for boundary value problems. We first show that the employment of least‐squares functional with quadrature rules constitutes an approximation of the direct collocation method. Standard radial basis collocation method, however, yields a larger solution error near boundaries. The residuals in the least‐squares functional associated with domain and boundary can be better balanced if the boundary collocation equations are properly weighted. The error analysis shows unbalanced errors between domain, Neumann boundary, and Dirichlet boundary least‐squares terms. A weighted least‐squares functional and the corresponding weighted radial basis collocation method are then proposed for correction of unbalanced errors. It is shown that the proposed method with properly selected weights significantly enhances the numerical solution accuracy and convergence rates. Copyright © 2006 John Wiley & Sons, Ltd.     

Application of variational mesh generation approach for selecting centers of radial basis functions collocation method

In this paper, a two-dimensional variational mesh generation method is applied to obtain adaptive centers for radial basis functions (RBFs). At first, a set of uniform centers is distributed in the domain, then mesh generation differential equations are used to move the centers to region with high gradients. An iterative algorithm is introduced to solve steady-state mesh generation differential equations with RBFs. Functions with steep variation in the domains are used to validate the adaptive centers generation method. In addition to the centers adaption process is applied to solve elliptic partial differential equations via RBFs collocation method. Numerical results of Helmholtz differential equation show a clear reduction in the error, when the adaptive centers are used for RBFs.     

A local Heaviside weighted meshless method for two-dimensional solids using radial basis functions

 A meshless method is developed for the stress analysis of two-dimensional solids, based on a local weighted residual method with the Heaviside step function as the weighting function over a local subdomain. Trial functions are constructed using radial basis functions (RBF). The present method is a truly meshless method based only on a number of randomly located nodes. No domain integration is needed, no element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions. Effects of the sizes of local subdomain and interpolation domain on the performance of the present method are investigated. The behaviour of shape parameters of multiquadrics (MQ) has been systematically studied. Example problems in elastostatics are presented and compared with closed-form solutions and show that the proposed method is highly accurate and possesses no numerical difficulties. Received: 10 November 2002 / Accepted: 5 March 2003     

A comparison of approximate response functions in structural reliability analysis

In order to reduce computational costs in structural reliability analysis, it has been suggested to utilize approximate response functions for reliability assessment. One well-established class of methods to deal with suitable approximations is the Response Surface Method. The basic idea in utilizing the response surface method is to replace the true limit state function by an approximation, the so-called response surface, whose function values can be computed more easily. The functions are typically chosen to be first- or second-order polynomials. Higher-order polynomials on the one hand tend to show severe oscillations, and on the other hand they require too many support points. This may be overcome by applying smoothing techniques such as the moving least-squares method. An alternative approach is given by Artificial Neural Networks. In this approach, the input and output parameters are related by means of relatively simple yet flexible functions, such as linear, step, or sigmoid functions which are combined by adjustable weights. The main feature of this approach lies in the possibility of adapting the input–output relations very efficiently. A further possibility lies in the utilization of radial basis functions. This method also allows for a flexible adjustment of the interpolation scheme. In all approaches as presented it is essential to achieve high quality of approximation primarily in the region of the random variable space which contributes most significantly to the probability of failure. The paper presents an overview of these approximation methods and demonstrates their potential by application to several examples of nonlinear structural analysis.     

A mesh-free minimum length method for 2-D problems

A mesh-free minimum length method (MLM) has been proposed for 2-D solids and heat conduction problems. In this method, both polynomials as well as modified radial basis functions (RBFs) are used to construct shape functions for arbitrarily distributed nodes based on minimum length procedure, which possess Kronecker delta property. The shape functions are then used to formulate a mesh-free method based on weak-form formulation. Both Gauss integration (GI) and stabilized nodal integration (NI) are employed to numerically evaluate Galerkin weak form. The numerical examples show that the MLM achieves better accuracy than the 4-node finite elements especially for problems with steep gradients. The method is easy to implement and works well for irregularly distributed nodes. Some numerical implementation issues for MLM are also discussed in detail.     

A hybrid radial boundary node method based on radial basis point interpolation

The hybrid boundary node method (HBNM) retains the meshless attribute of the moving least squares (MLS) approximation and the reduced dimensionality advantages of the boundary element method. However, the HBNM inherits the deficiency of the MLS approximation, in which shape functions lack the delta function property. Thus in the HBNM, boundary conditions are implemented after they are transformed into their approximations on the boundary nodes with the MLS scheme.This paper combines the hybrid displacement variational formulation and the radial basis point interpolation to develop a direct boundary-type meshless method, the hybrid radial boundary node method (HRBNM) for two-dimensional potential problems. The HRBNM is truly meshless, i.e. absolutely no elements are required either for interpolation or for integration. The radial basis point interpolation is used to construct shape functions with delta function property. So unlike the HBNM, the HRBNM is a direct numerical method in which the basic unknown quantity is the real solution of nodal variables, and boundary conditions can be applied directly and easily, which leads to greater computational precision. Some selected numerical tests illustrate the efficiency of the method proposed.     

Subdomain radial basis collocation method for fracture mechanics

The direct approximation of strong form using radial basis functions (RBFs), commonly called the radial basis collocation method (RBCM), has been recognized as an effective means for solving boundary value problems. Nevertheless, the non‐compactness of the RBFs precludes its application to problems with local features, such as fracture problems, among others. This work attempts to apply RBCM to fracture mechanics by introducing a domain decomposition technique with proper interface conditions. The proposed method allows (1) natural representation of discontinuity across the crack surfaces and (2) enrichment of crack‐tip solution in a local subdomain. With the proper domain decomposition and interface conditions, exponential convergence rate can be achieved while keeping the discrete system well‐conditioned. The analytical prediction and numerical results demonstrate that an optimal dimension of the near‐tip subdomain exists. The effectiveness of the proposed method is justified by the mathematical analysis and demonstrated by the numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Solutions of partial differential equations with random Dirichlet boundary conditions by multiquadric collocation method

Problems described by deterministic partial differential equations with random Dirichlet boundary conditions are considered. Formulation of the solution to such a problem by the global collocation method using multiquadrics is presented. The quality of the solution to a stochastic problem depends on both its expected value and its variance. It is proposed that the shape parameter of multiquadrics should be chosen to optimize both the accuracy and the variance of the solution. Test problems described by the Poisson, the Helmholtz, and the diffusion–convection equations with random Dirichlet boundary conditions are solved by the multiquadric collocation method. It is found that there is a trade-off between solution accuracy and solution variance for each problem.     

Reproducing kernel enhanced local radial basis collocation method

Standard radial basis functions (RBFs) offer exponential convergence, however, the method is suffered from the large condition numbers due to their ‘nonlocal’ approximation. The nonlocality of RBFs also limits their applications to small‐scale problems. The reproducing kernel functions, on the other hand, provide polynomial reproducibility in a ‘local’ approximation, and the corresponding discrete systems exhibit relatively small condition numbers. Nonetheless, reproducing kernel functions produce only algebraic convergence. This work intends to combine the advantages of RBFs and reproducing kernel functions to yield a local approximation that is better conditioned than that of the RBFs, while at the same time offers a higher rate of convergence than that of reproducing kernel functions. Further, the locality in the proposed approximation allows its application to large‐scale problems. Error analysis of the proposed method is also provided. Numerical examples are given to demonstrate the improved conditioning and accuracy of the proposed method. Copyright © 2007 John Wiley & Sons, Ltd.     

A numerical method for heat transfer problems using collocation and radial basis functions

Simple, mesh/grid free, numerical schemes for the solution of heat transfer problems are developed and validated. Unlike the mesh or grid-based methods, these schemes use well-distributed quasi-random collocation points and approximate the solution using radial basis functions. The schemes work in a similar fashion as finite differences but with random points instead of a regular grid system. This allows the computation of problems with complex-shaped boundaries in higher dimensions with no extra difficulty. © 1998 John Wiley & Sons, Ltd.     

Boundary collocation method for acoustic eigenanalysis of three-dimensional cavities using radial basis function

 In this paper, a theoretical formulation based on the collocation method is presented for the eigenanalysis of arbitrarily shaped acoustic cavities. This article can be seen as the extension of non-dimensional influence function (NDIF) method proposed by Kang et al. (1999, 2000a) extending from two-dimensional to three-dimensional case. Unlike the conventional collocation techniques in the literature, approximate functions used in this paper are two-point functions of which the argument is only the distance between the two points. Based on this radial basis expansion, the acoustic field can be represented more exactly. The field solution is obtained through the linear superposition of radial basis function, and boundary conditions can be applied at the discrete points. The influence matrix is symmetric regardless of the boundary shape of the cavity, and the calculated eigenvalues rapidly converge to the exact values by using only a few boundary nodes. Moreover, the method results in true and spurious boundary modes, which can be obtained from the right and left unitary vectors of singular value decomposition, respectively. By employing the updating term and document of singular value decomposition (SVD), the true and spurious eigensolutions can be sorted out, respectively. The validity of the proposed method are illustrated through several numerical examples. Received: 29 August 2001 / Accepted: 27 June 2002 Financial support from the National Science Council under Grant No. NSC-90-2211-E-019-006 for National Taiwan Ocean University is gratefully acknowledged.     

A note on the finite element method with singular basis functions

In this note, we make a few comments concerning the paper of Hughes and Akin (Int. J. Numer. Meth. Engng., 15 , 733–751 (1980)). Our primary goal is to demonstrate that the rate of convergence of numerical solutions of the finite element method with singular basis functions depends upon the location of additional collocation points associated with the singular elements. Copyright © 1999 John Wiley & Sons, Ltd.     

Aeroacoustic source term computation based on radial basis functions

In low Mach number aeroacoustics, the known disparity of length scales makes it possible to apply well-suited simulation models using different meshes for flow and acoustics. The workflow of these hybrid methodologies include performing an unsteady flow simulation, computing the acoustic sources, and simulating the acoustic field. Therefore, hybrid methods seek for robust and flexible procedures, providing a conservative mesh to mesh interpolation of the sources while ensuring high computational efficiency. We propose a highly specialized radial basis function interpolation for the challenges during hybrid simulations. First, the computationally efficient local radial basis function interpolation in conjunction with a connectivity-based neighbor search technique is presented. Second, we discuss the computation of spatial derivatives based on radial basis functions. These derivatives are computed in a local-global approach, using a Gaussian kernel on local point stencils. Third, radial basis function interpolation and derivatives are used to compute complex aeroacoustic source terms. These ingredients are necessary to provide flexible source term calculations that robustly connect flow and acoustics. Finally, the capabilities of the presented approach are shown in a numerical experiment with a co-rotating vortex pair.     

An efficient ensemble of radial basis functions method based on quadratic programming

Radial basis function (RBF) surrogate models have been widely applied in engineering design optimization problems to approximate computationally expensive simulations. Ensemble of radial basis functions (ERBF) using the weighted sum of stand-alone RBFs improves the approximation performance. To achieve a good trade-off between the accuracy and efficiency of the modelling process, this article presents a novel efficient ERBF method to determine the weights through solving a quadratic programming subproblem, denoted ERBF-QP. Several numerical benchmark functions are utilized to test the performance of the proposed ERBF-QP method. The results show that ERBF-QP can significantly improve the modelling efficiency compared with several existing ERBF methods. Moreover, ERBF-QP also provides satisfactory performance in terms of approximation accuracy. Finally, the ERBF-QP method is applied to a satellite multidisciplinary design optimization problem to illustrate its practicality and effectiveness for real-world engineering applications.     

Moving least‐squares approximation with discontinuous derivative basis functions for shell structures with slope discontinuities

Moving least‐squares approximation with discontinuous derivative basis functions (MLSA‐DBF) is introduced for analysis of shell structures with slope discontinuities. To deal with shells with arbitrary slope discontinuities, the Cartesian coordinate is introduced in the construction of MLSA on the shell surface. The possible causes of singularity in the moment matrix of MLSA on the shell surface with slope discontinuities are identified, and the Moore–Penrose pseudoinverse is used to obtain the generalized inverse of the singular moment matrix resulting from linear dependency and insufficient influence nodes in the MLSA. Following the proposed formulations for shear deformable shell structures with slope discontinuities in the Cartesian coordinates, several numerical examples are analyzed to demonstrate the performance, validity, accuracy, and convergence properties of the proposed MLSA‐DBF approach. Copyright © 2007 John Wiley & Sons, Ltd.     

A mesh-free method using exponential basis functions for laminates modeled by CLPT, FSDT and TSDT – Part II: Implementation and results

In this part of the paper we give the details of the implementation of the method presented in the first part. Also the solutions of several benchmark plate problems with various geometries are presented to validate the results. It has been observed that the method can perform excellently in a wide range of problems defined for the bending analysis of laminated plates based on various plate theories. For further use, some explicit expressions are given for the exponential basis functions suitable for the solution of symmetric cross-ply laminates.     

Mesh free Galerkin method based on natural neighbors and conformal mapping

In this work, a robust mesh free method has been presented for the analysis of two dimensional problems. An efficient natural neighbor algorithm for construction of polygonal support domains has been used in this method that takes into account the nonuniform nodal discretization in the element free Galerkin formulation. The use of natural neighbors for determining the compact support is shown to overcome some of the shortcomings of the conventional distance metric based methods. For nonuniform nodal discretization there is a need for evaluating weights that have anisotropic compact supports. The smoothness and conformance of the weight function to the support domain obtained from natural neighbor algorithm is achieved through an efficient conformal mapping procedure such as Schwarz-Christoffel mapping. Numerical examples demonstrate that the proposed mesh free method gives good estimates of the stress/strain fields.     

Radial basis functions methods for solving Fokker-Planck equation

In this paper two numerical meshless methods for solving the Fokker-Planck equation are considered. Two methods based on radial basis functions to approximate the solution of Fokker-Planck equation by using collocation method are applied. The first is based on the Kansa's approach and the other one is based on the Hermite interpolation. In addition, to conquer the ill-conditioning of the problem for big number of collocation nodes, two time domain Discretizing schemes are applied. Numerical examples are included to demonstrate the reliability and efficiency of these methods. Also root mean square and N_e errors are obtained to show the convergence of the methods. The errors show that the proposed Hermite collocation approach results obtained by the new time-Discretizing scheme are more accurate than the Kansa's approach.