Radial Basis Function Approximation Research Articles

Boundary elements and the analog equation method for the solution of elastic problems in 3-D non-homogeneous bodies

The application of the Boundary Element Method (BEM) to elastostatic problems involving 3D non-homogeneous materials like Functionally Graded Materials (FGMs) is presented in this paper. The Analog Equation Method (AEM) is used to transform the original problem into a new problem with unknown fictitious source but known Fundamental Solution. By means of this transformation a system of uncoupled Domain/Boundary Integral Equations (D/BIEs) is first obtained, combining standard Boundary Element discretization and Radial Basis Functions (RBFs) approximation for the fictitious source. The application of the original differential operator to the displacement BIE provides the extra equations to compute the unknown fictitious source. The boundary character of the method is maintained since the integrals involved in the equations are limited only to the boundary since the RBFs are selected in such a way that the corresponding analog equation could be solved analytically. Numerical examples for three-dimensional problems in continuously non-homogeneous, isotropic and linear elastic FGMs are presented and discussed.

Application of method of fundamental solutions in solving potential flow problems for ship motion prediction

A novel panel-free approach based on the method of fundamental solutions (MFS) is proposed to solve the potential flow for predicting ship motion responses in the frequency domain according to strip theory. Compared with the conventional boundary element method (BEM), MFS is a desingularized, panel-free and integration-free approach. As a result, it is mathematically simple and easy for programming. The velocity potential is described by radial basis function (RBF) approximations and any degree of continuity of the velocity potential gradient can be obtained. Desingularization is achieved through collating singularities on a pseudo boundary outside the real fluid domain. Practical implementation and numerical characteristics of the MFS for solving the potential flow problem concerning ship hydrodynamics are elaborated through the computation of a 2D rectangular section. Then, the current method is further integrated with frequency domain strip theory to predict the heave and pitch responses of a containership and a very large crude carrier (VLCC) in regular head waves. The results of both ships agree well with the 3D frequency domain panel method and experimental data. Thus, the correctness and usefulness of the proposed approach are proved. We hope that this paper will serve as a motivation for other researchers to apply the MFS to various challenging problems in the field of ship hydrodynamics.

AN IMPLEMENTATION OF THE METHOD OF FUNDAMENTAL SOLUTIONS FOR THE DYNAMIC RESPONSE OF VON KARMAN NONLINEAR PLATE MODEL

The nonlinear plate dynamics is a subject of investigations across many disciplines. Various models have been proposed. In this paper, one of the models derived by von Karman is under consideration. In this case, the initial-boundary value problem of the dynamics of a plate large deflection is described by two nonlinear, coupled partially differential equations of fourth order with two initial conditions and four boundary conditions given at each boundary point. There are various means of simulating the dynamic response of thin nonlinear plate, to various degrees of accuracy. The proposal of this paper is to implement one of the meshfree methods i.e., the method of fundamental solutions (MFS). The problem is solved in discretized time domain. This discretization is done in a conception of the finite difference method. The nonlinearity of the equations, obtained at each time step, is solved by application of Picard iterations. For each iteration step, a boundary value problem is to be solved. Moreover, the equations at each iteration step are inhomogeneous ones. So, the radial basis function approximation is applied and a particular solution of boundary value problems is obtained. The final solution is calculated by implementation of the MFS. The numerical experiment has confirmed that the proposed numerical procedure gives the solutions with demanded accuracy and is a good tool to solve the considered problem.

A new immersed boundary method for compressible Navier–Stokes equations

This paper presents an immersed boundary method for compressible Navier–Stokes equations in irregular domains, based on a local radial basis function approximation. This approach allows one to define a reconstruction of the radial basis functions on each irregular interface cell to treat both the Dirichlet and Neumann boundary conditions accurately on the immersed interfaces. Several numerical examples, including problems with available analytical solutions and the well-documented flow past an airfoil, are presented to test the proposed method. The numerical results demonstrate that the proposed method provides accurate solutions for viscous compressible flows.

Nonintrusive reduced‐order modeling of parametrized time‐dependent partial differential equations

AbstractWe propose a nonintrusive reduced‐order modeling method based on the notion of space‐time‐parameter proper orthogonal decomposition (POD) for approximating the solution of nonlinear parametrized time‐dependent partial differential equations. A two‐level POD method is introduced for constructing spatial and temporal basis functions with special properties such that the reduced‐order model satisfies the boundary and initial conditions by construction. A radial basis function approximation method is used to estimate the undetermined coefficients in the reduced‐order model without resorting to Galerkin projection. This nonintrusive approach enables the application of our approach to general problems with complicated nonlinearity terms. Numerical studies are presented for the parametrized Burgers' equation and a parametrized convection‐reaction‐diffusion problem. We demonstrate that our approach leads to reduced‐order models that accurately capture the behavior of the field variables as a function of the spatial coordinates, the parameter vector and time. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

An adaptive multiquadric radial basis function method for expensive black-box mixed-integer nonlinear constrained optimization

Many real-world optimization problems comprise objective functions that are based on the output of one or more simulation models. As these underlying processes can be time and computation intensive, the objective function is deemed expensive to evaluate. While methods to alleviate this cost in the optimization procedure have been explored previously, less attention has been given to the treatment of expensive constraints. This article presents a methodology for treating expensive simulation-based nonlinear constraints alongside an expensive simulation-based objective function using adaptive radial basis function techniques. Specifically, a multiquadric radial basis function approximation scheme is developed, together with a robust training method, to model not only the costly objective function, but also each expensive simulation-based constraint defined in the problem. The article presents the methodology developed for expensive nonlinear constrained optimization problems comprising both continuous and integer variables. Results from various test cases, both analytical and simulation-based, are presented.

Stable Computation of Differentiation Matrices and Scattered Node Stencils Based on Gaussian Radial Basis Functions

Radial basis function (RBF) approximation has the potential to provide spectrally accurate function approximations for data given at scattered node locations. For smooth solutions, the best accuracy for a given number of node points is typically achieved when the basis functions are scaled to be nearly flat. This also results in nearly linearly dependent basis functions and severe ill-conditioning of the interpolation matrices. Fornberg, Larsson, and Flyer recently generalized the RBF-QR method to provide a numerically stable approach to interpolation with flat and nearly flat Gaussian RBFs for arbitrary node sets in up to three dimensions. In this work, we consider how to extend this method to the task of computing differentiation matrices and stencil weights in order to solve partial differential equations. The expressions for first and second order derivative operators as well as hyperviscosity operators are established, numerical issues such as how to deal with non-unisolvency are resolved, and the accuracy and computational efficiency of the method are tested numerically. The results indicate that using the RBF-QR approach for solving PDE problems can be very competitive compared with using the ill-conditioned direct solution approach or using variable precision arithmetic to overcome the conditioning issue.

High-order Alternating Direction Implicit Method Based on Compact Integrated-RBF Approximations for Unsteady/Steady Convection-Diffusion Equations

In this paper, the alternating direction implicit (ADI) method reported in [You (2006)] for the convection-diffusion equation is implemented in the context of compact integrated radial basis function (CIRBF) approximations. The CIRBF approximations are constructed over 3-point stencils, where extra information is incorporated via two forms: only nodal second-order derivative values (Scheme 1), and both nodal first- and second-order derivative values (Scheme 2). The resultant algebraic systems are sparse, especially for Scheme 2 (tridiagonal matrices). Several steady and non-steady problems are considered to verify the present schemes and to compare their accuracy with some other ADI schemes. Numerical results show that highly accurate results are obtained with the proposed methods.

Hybrid shape parameter strategy for the RBF approximation of vibrating systems

Shape parameters play an important role in radial basis function (RBF) approximations. Therefore, the choice of them is an active field in numerical analysis research. In this paper, first we review the available strategies in the literature for selecting shape parameters. Then, we introduce an alternative approach called hybrid strategy for scaling the RBFs. This strategy is constructed based on the advantages of the older ones and discards their disadvantages. The efficiency of the new strategy is demonstrated by comparing the effects of different strategies on approximating the eigenvalues of ordinary and partial differential equations with different boundary conditions.

Potential approximation in realistic Laplacian computation

ObjectiveThis paper aims to improve the shortcomings of the extant methodologies for realistic Laplacian (RL) computation, and correct the erroneous claims published in the past. MethodsWe implemented several variants of RL computation methods, using various potential approximation techniques and different regularization approaches. The individual variants of the RL computation were tested using simulations based on a realistic head model computed with the boundary element method (BEM). The results which disagreed with previously published works were further analyzed, and the reasons for the disagreement were identified. ResultsWe identified the best regularization techniques for the surface potential approximation, and we showed that once these techniques are used there is often little difference between various potential approximations, which is in contrast with previous claims that promoted the radial basis function (RBF) approximation. Further, our analysis shows that the RBF approximation suffers from Runge phenomenon, which cannot be mitigated simultaneously for both deep and shallow sources; therefore, its good performance is guarantied only if a priori knowledge about the source depth is available. ConclusionsThe previously published methodology for RL computation was not optimal. Improvements are possible if the newly suggested approach is used. SignificanceThe methodology presented in our paper allows more efficient utilization of the RL, providing a useful tool for processing of high density EEG recordings. Presented techniques allow to achieve high EEG spatial resolution, and avoid unnecessary spatial blurring caused by the problems in the previously published RL methodology.

Formula omitted] Bernstein inequalities and inverse theorems for RBF approximation on [formula omitted

Bernstein inequalities and inverse theorems are a recent development in the theory of radial basis function (RBF) approximation. The purpose of this paper is to extend what is known by deriving Lp Bernstein inequalities for RBF networks on Rd. These inequalities involve bounding a Bessel-potential norm of an RBF network by its corresponding Lp norm in terms of the separation radius associated with the network. The Bernstein inequalities will then be used to prove the corresponding inverse theorem.

A simplified approach for imposing the boundary conditions in the local boundary integral equation method

A simplified approach for imposing the boundary conditions in the local boundary integral equation (LBIE) method is presented. The proposed approach employs an integral equation derived using the fundamental solution and the Green's second identity when the collocation node is at the boundary of the solution domain (global boundary). The subdomains for the nodes placed at the global boundary preserve their circular shapes; avoiding in this way any integration over the global boundary. Consequently, the difficulties related to evaluation of singular integrals and determination of intersection points between the global and local circular boundaries are avoided. So far, attempts to avoid these issues have focused on using schemes based on meshless approximations. The downside of such schemes is that the weak formulation is abandoned. In this study the interpolation of field variables over the boundaries of the subdomains is carried out using the radial basis function approximation. Numerical examples show that the proposed approach despite its simplicity, achieves comparable accuracy to the classical treatment of the boundary conditions in the LBIE.

A branch and bound algorithm for the global optimization of Hessian Lipschitz continuous functions

We present a branch and bound algorithm for the global optimization of a twice differentiable nonconvex objective function with a Lipschitz continuous Hessian over a compact, convex set. The algorithm is based on applying cubic regularisation techniques to the objective function within an overlapping branch and bound algorithm for convex constrained global optimization. Unlike other branch and bound algorithms, lower bounds are obtained via nonconvex underestimators of the function. For a numerical example, we apply the proposed branch and bound algorithm to radial basis function approximations.

A new hybrid finite element approach for three-dimensional elastic problems

A new fundamental solution based finite element method (HFS-FEM) is presented for analyzing three-dimensional (3D) elastic problems with body forces in this paper. It begins with deriving formulations of 3D HFS-FEM for elastic problems without body force and then the body force term is handled by means of the method of particular solution and radial basis function approximation. In our analysis, the homogeneous solution is obtained using the proposed HFS-FEM and the particular solution associated with the body force is approximated by radial basis functions. Several standard tests and numerical examples are considered to assess the capability and performance of the proposed method and elements. It is found that, comparing with conventional FEM (ABAQUS), the proposed method can achieve higher accuracy and efficiency when same element meshes are used. It is also found that the elements associated with this method are not very sensitive to mesh distortion and can be employed for problems involving nearly incompressible materials. This new method seems to be promising to deal with problems involving generalized body force, complex geometry, stress concentration and multi-materials.

A dual reciprocity boundary face method for 3D non-homogeneous elasticity problems

The boundary face method is coupled with the dual reciprocity method (DRM) to solve non-homogeneous elasticity problems. We will analyze thin structures based on 3D solid elastic theory rather than the shell theory as in the finite element method (FEM). To circumvent the ill-conditioning problem that occurs in the radial basis function (RBF) approximation in thin structures, a special variation scheme for determining the RBF parameters is proposed. In addition, a new exponential RBF is used which has significantly improved the stability of the RBF, and its particular solution to the elasticity problem is derived for the first time. Comparisons of our method with the traditional DRM, the boundary element method (BEM) and the FEM have been made. Numerical examples have demonstrated that our method outperforms the BEM and FEM with respect to stability, accuracy and efficiency, especially when the structure in question has features of small size, such as thin shells.

Three-dimensional solution for acoustic and transport problems using the radial basis integral equation method

The radial basis integral equations method (RBIEM) has been applied for solution of three-dimensional (3D) acoustic and transport problems. The acoustic problem is often described using the Helmholtz equation, while the transport problems are usually described using the Laplace equation (diffusion only), the Poisson equation (diffusion with sources/sinks) and the convection–diffusion equation. The accuracy of the numerical scheme employing the first and second order Duchon splines augmented by first and second order polynomials, respectively, was examined. The effect of the number of interpolation points used in the radial basis function approximation on the condition number of the system was investigated. Numerical results obtained for the convection–diffusion equation were compared with the solutions obtained using the multi-domain dual reciprocity boundary element method (DRM-MD). The RBIEM formulation was found to be more accurate than the DRM-MD formulation. The implementation does not involve discretization over the boundaries of the subdomains used in the RBIEM formulation when evaluating the integrals.

Local moving least square-one-dimensional integrated radial basis function networks technique for incompressible viscous flows

This paper presents a local moving least square-one-dimensional integrated radial basis function networks method for solving incompressible viscous flow problems using stream function-vorticity formulation. In this method, the partition of unity method is employed as a framework to incorporate the moving least square and one-dimensional integrated radial basis function networks techniques. The major advantages of the proposed method include the following: (i) a banded sparse system matrix which helps reduce the computational cost; (ii) the Kronecker-i property of the constructed shape function which helps impose the essential boundary condition in an exact manner; and (iii) high accuracy and fast convergence rate owing to the use of integration instead of conventional differentiation to construct the local radial basis function approximations. Several examples including two-dimensional (2D) Poisson problems, lid-driven cavity flow and flow past a circular cylinder are considered, and the present results are compared with the exact solutions and numerical results from other methods in the literature to demonstrate the attractiveness of the proposed method.

A Fast Treecode for Multiquadric Interpolation with Varying Shape Parameters

A treecode algorithm is presented for the fast evaluation of multiquadric radial basis function (RBF) approximations. The method is a dual approach to one presented by Krasny and Wang, which applies far-field expansions to clusters of RBF centers (source points). The new approach clusters evaluation points instead and is therefore easily able to cope with basis functions that have different multiquadric shape parameters. The new treecode is able to evaluate an approximation on N centers at M points in $O((N+M) \log M)$ time in the ideal case when evaluation points are uniformly distributed. When coupled with a two-level restricted additive Schwarz preconditioner for GMRES iterations, the treecode is well suited for use within an adaptive RBF iteration, previously described by Driscoll and Heryudono, as is demonstrated by experiments on test functions.

Meshless point collocation for the numerical solution of Navier–Stokes flow equations inside an evaporating sessile droplet

The Navier–Stokes flow inside an evaporating sessile droplet is studied in the present paper, using sophisticated meshfree numerical methods for the computation of the flow field. This problem relates to numerous modern technological applications, and has attracted several analytical and numerical investigations that expanded our knowledge on the internal microflow during droplet evaporation. Two meshless point collocation methods are applied here to this problem and used for flow computations and for comparison with analytical and more traditional numerical solutions. Particular emphasis is placed on the implementation of the velocity-correction method within the meshless procedure, ensuring the continuity equation with increased precision. The Moving Least Squares (MLS) and the Radial Basis Function (RBF) approximations are employed for the construction of the shape functions, in conjunction with the general framework of the Point Collocation Method (MPC). An augmented linear system for imposing the coupled boundary conditions that apply at the liquid–gas interface, especially the zero shear-stress boundary condition at the interface, is presented. Computations are obtained for regular, Type-I embedded nodal distributions, stressing the positivity conditions that make the matrix of the system stable and convergent. Low Reynolds number (Stokes regime), and elevated Reynolds number (Navier–Stokes regime) conditions have been studied and the solutions are compared to those of analytical and traditional CFD methods. The meshless implementation has shown a relative ease of application, compared to traditional mesh-based methods, and high convergence rate and accuracy.

Transient meshless boundary element method for prediction of chloride diffusion in concrete with time dependent nonlinear coefficients

Chloride-induced corrosion of steel reinforcements has been identified as one of the main causes of deterioration of concrete structures. A feasible numerical method is required to predict chloride penetration in concrete structures. A transient meshless boundary element method is proposed to predict chloride diffusion in concrete with time dependent nonlinear coefficient. Taking Green's function as the weighted function, the weighted residue method is adopted to transform the diffusion equation into equivalent integral equations. By the coupling of radial integral method and radial basis function approximation, the domain integrals in equivalent control equations are transformed into boundary integrals. Following the general procedure of boundary element meshing and traditional finite difference method, a set of nonlinear algebraic equations are constructed and are eventually solved with the modified Newtonian iterative method. Several numerical examples are provided to demonstrate the effectiveness and efficiency of the developed model. A comparison of the simulated chloride concentration with the corresponding reported experimental data in a real marine structure indicates the high accuracy and advantage of the time dependent coefficient and nonlinear model over the conventional constant coefficient model.