Indirect Radial Basis Function Research Articles

The numerical investigation of Lagrangian and Eulerian coherent structures for the near wake structure of a hovering Drosophila

An integrated simulation of a Drosophila wing–body combination in hovering flight has been carried out in order to analyze the Lagrangian and Eulerian coherent structures. A parallel unstructured finite volume method based on an arbitrary Lagrangian–Eulerian (ALE) formulation has been initially validated for a flapping rectangular plate and then employed to solve the incompressible unsteady Navier–Stokes equations around a Drosophila wing–body combination. A robust mesh deformation algorithm based on indirect radial basis function method is utilized at each time level while avoiding remeshing. The time variation of the three-dimensional Eulerian coherent structures around a flapping Drosophila in hovering flight is analyzed in the near wake with $$\lambda _{2}$$ -criterion. Meanwhile, the Lagrangian coherent structures are investigated using finite-time Lyapunov exponent fields. In addition, the instantaneous velocity vectors and particle traces are presented along with the aerodynamic parameters including the force, moment and power for a wing–body combination. Furthermore, a wing-only configuration is also investigated in order to show the body effects on aerodynamic loads. The numerical simulations are used to gain insight into the near wake topology as well as their correlations with the aerodynamic force generation. The present fully coupled ALE algorithm is shown to be sufficiently robust to deal with large mesh deformations seen in flapping wings and reveals highly detailed near wake topology which is very useful to study physics in biological flights and can also provide an effective tool for designing bio-inspired MAVs and MFIs.

Two Meshfree Numerical Approaches for Solving High-Order Singular Emden–Fowler Type Equations

In this paper we suggest indirect radial basis function collocation and radial basis function differential quadrature methods for solving high-order singular Emden–Fowler equations. Here, we concentrate on Gaussian (GA, $$\exp (-c^2 r^2)$$ ) as a radial function for approximating the solution of the mentioned equations. In order to overcome the difficulty of the singular point ( $$x=0$$ ), the Head dense points with dense parameter $$\vartheta $$ and shifted Chebyshev points have been handled. The comparison between the numerical and exact results shows the efficiency and accuracy of these methods and also demonstrate these methods have good convergence rate.

Indirect radial basis function approach for bending, free vibration and buckling analyses of functionally graded microbeams

The main aim of this paper is to provide a simple yet accurate solution approach for bending, free vibration and buckling analyses of functionally graded (FG) Euler–Bernoulli and Timoshenko microbeams based on the strain gradient theory (SGT). The non-classical microbeam model incorporates the material length scale parameter which can consequently capture the size effect. The adopted model also can degenerate to the beam model based on the modified couple stress theory (MCST) or the classical beam theory (CBT) by setting two or all material length scale parameters to be zero. The material properties of the beam are assumed to vary continuously with respect to the thickness direction according to the Mori–Tanaka homogenization technique. By using Hamilton’s principle, the governing equations of motion and corresponding boundary condition equations are derived. Then indirect radial basis function approach is applied along axial direction to discretize the highest derivatives appearing in the system of equations. Effects of the number of collocation points and shape parameter on the accuracy of present solutions are investigated and convergence studies are shown that the approach is sensitive to these two parameters. Some of results are compared with the previously published solutions to validate accuracy and reliability of the present approach.

On the Numerical Solution of One-Dimensional Nonlinear Nonhomogeneous Burgers’ Equation

The nonlinear Burgers’ equation is a simple form of Navier-Stocks equation. The nonlinear nature of Burgers’ equation has been exploited as a useful prototype differential equation for modeling many phenomena. This paper proposes two meshfree methods for solving the one-dimensional nonlinear nonhomogeneous Burgers’ equation. These methods are based on the multiquadric (MQ) quasi-interpolation operatorℒ𝒲2and direct and indirect radial basis function networks (RBFNs) schemes. In the present schemes, the Taylors series expansion is used to discretize the temporal derivative and the quasi-interpolation is used to approximate the solution function and its spatial derivatives. In order to show the efficiency of the present methods, several experiments are considered. Our numerical solutions are compared with the analytical solutions as well as the results of other numerical schemes. Furthermore, the stability analysis of the methods is surveyed. It can be easily seen that the proposed methods are efficient, robust, and reliable for solving Burgers’ equation.

An arbitrary Lagrangian–Eulerian formulation for solving moving boundary problems with large displacements and rotations

An Arbitrary Lagrangian–Eulerian (ALE) formulation based on the unstructured finite volume method is proposed for solving moving boundary problems with large displacements and rotations. The numerical method is based on the side-centered arrangement of the primitive variables that does not require any ad-hoc modifications in order to enhance pressure coupling. The continuity equation is satisfied within each element at machine precision and the summation of the continuity equations can be exactly reduced to the domain boundary, which is important for the global mass conservation. A special attention is given to construct an ALE algorithm obeying the discrete geometric conservation law (DGCL). The mesh deformation algorithm is based on the indirect Radial Basis Function (RBF) algorithm at each time level while avoiding remeshing in order to enhance numerical robustness. For the parallel solution of resulting large-scale algebraic equations in a fully coupled form, a matrix factorization is introduced similar to that of the projection method for the whole system and the parallel algebraic multigrid solver BoomerAMG is used for the scaled discrete Laplacian provided by the HYPRE library which we access through the PETSc library. The present numerical algorithm is initially validated for the decaying Taylor–Green vortex flow, the flow past an oscillating circular cylinder in a channel and the flow induced by an oscillating sphere in a cubic cavity. Then the numerical algorithm is applied to the numerical simulation of flow field around a pair of flapping Drosophila wings in hover flight. The time variation of the Eulerian coherent structures in the near wake is shown along with the aerodynamic loads.

A comparison of three explicit local meshless methods using radial basis functions

In this paper, three kinds of explicit local meshless methods are compared: the local method of approximate particular solutions (LMAPS), the local direct radial basis function collocation method (LDRBFCM) which are both first presented in this paper, and the local indirect radial basis function collocation method (LIRBFCM). In all three methods, the time discretization is performed in explicit way, the multiquadric radial basis functions (RBFs) are used to interpolate either initial temperature field and its derivatives or the Laplacian of the initial temperature field. The five-noded sub-domains are used in localization. Numerical results of simple diffusion equation with Dirichlet jump boundary condition are compared on uniform and random node arrangement, the accuracy and stabilities of these three local meshless methods are asserted. One can observe that the improvement of the accuracy with denser nodes and with smaller time steps for all three methods. All methods provide a similar accuracy in uniform node arrangement case. For random node arrangement, the LMAPS and the LDRBFCM perform better than the LIDRBFCM.

Numerical solution of Poisson’s equation using radial basis function networks on the polar coordinate

This paper introduces a variant of direct and indirect radial basis function networks (DRBFNs and IRBFNs) for the numerical solution of Poisson’s equation. We use transformation from Cartesian coordinates to polar ones and use DRBFN and IRBFN methods on the basis of a multiquadric approximation scheme. We have experienced that the result shows better accuracy than previously known ones. Also, our new way of solution does not influence the condition number.

The meshless virtual boundary method and its applications to 2D elasticity problems

A novel numerical method for eliminating the singular integral and boundary effect is processed. In the proposed method, the virtual boundaries corresponding to the numbers of the true boundary arguments are chosen to be as simple as possible. An indirect radial basis function network (IRBFN) constructed by functions resulting from the indeterminate integral is used to construct the approaching virtual source functions distributed along the virtual boundaries. By using the linear superposition method, the governing equations presented in the boundaries integral equations (BIE) can be established while the fundamental solutions to the problems are introduced. The singular value decomposition (SVD) method is used to solve the governing equations since an optimal solution in the least squares sense to the system equations is available. In addition, no elements are required, and the boundary conditions can be imposed easily because of the Kronecker delta function properties of the approaching functions. Three classical 2D elasticity problems have been examined to verify the performance of the method proposed. The results show that this method has faster convergence and higher accuracy than the conventional boundary type numerical methods.

Integrated radial basis functions‐based differential quadrature method and its performance

AbstractIn this paper, indirect radial basis function networks (IRBFN) proposed by Nam and Tranh (Neural Networks 2001; 14(2):185–199; Appl. Math. Modelling 2003; 27:197–220) are incorporated into the differential quadrature (DQ) approximation of derivatives. For simplicity, this new variant of RBF‐DQ approach is named as iRBF‐DQ method. The proposed approach is validated by its application to solve the one‐dimensional Burger's equation, and simulate natural convection in a concentric annulus by solving Navier–Stokes equations. It was found that as compared to the benchmark data, the iRBF‐DQ approach can provide more accurate results than the original RBF‐DQ method. Copyright © 2006 John Wiley & Sons, Ltd.

A meshless IRBFN-based method for transient problems

[Abstract]: The Indirect Radial Basis Function Network (IRBFN) method has been reported to be a highly accurate tool for approximating multivariate functions and solving elliptic partial differential equations (PDEs). The present method is a truly meshless method as defined in [\citet *{Atluri_Shen_02a}]. A recent development of the method for solving transient problems is presented in this paper. Two numerical schemes combining the IRBFN method with different time integration techniques based on either fully or semi-discrete framework are proposed. The two schemes are implemented making use of Hardy's multiquadrics (MQ) and Duchon's thin plate splines (TPS). Some example problems are solved by the present method, and the results compare favorably in terms of accuracy and efficiency with those from other numerical methods such as finite difference (FDM), finite element (FEM), boundary element (BEM) and the Direct Radial Basis Function Network (DRBFN) methods.

An efficient indirect RBFN‐based method for numerical solution of PDEs

AbstractThis article presents an efficient indirect radial basis function network (RBFN) method for numerical solution of partial differential equations (PDEs). Previous findings showed that the RBFN method based on an integration process (IRBFN) is superior to the one based on a differentiation process (DRBFN) in terms of solution accuracy and convergence rate (Mai‐Duy and Tran‐Cong, Neural Networks 14(2) 2001, 185). However, when the problem dimensionalityNis greater than 1, the size of the system of equations obtained in the former is aboutNtimes as big as that in the latter. In this article, prior conversions of the multiple spaces of network weights into the single space of function values are introduced in the IRBFN approach, thereby keeping the system matrix size small and comparable to that associated with the DRBFN approach. Furthermore, the nonlinear systems of equations obtained are solved with the use of trust region methods. The present approach yields very good results using relatively low numbers of data points. For example, in the simulation of driven cavity viscous flows, a high Reynolds number of 3200 is achieved using only 51 × 51 data points. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

Solving high-order partial differential equations with indirect radial basis function networks

This paper reports a new numerical method based on radial basis function networks (RBFNs) for solving high-order partial differential equations (PDEs). The variables and their derivatives in the governing equations are represented by integrated RBFNs. The use of integration in constructing neural networks allows the straightforward implementation of multiple boundary conditions and the accurate approximation of high-order derivatives. The proposed RBFN method is verified successfully through the solution of thin-plate bending and viscous flow problems which are governed by biharmonic equations. For thermally driven cavity flows, the solutions are obtained up to a high Rayleigh number.

Indirect RBFN Method with Scattered Points for Numerical Solution of PDEs

This paper is concerned with the use of the indirect radial basis function network (RBFN) method in solving partial differential equations (PDEs) with scattered points. Indirect RBFNs (Mai-Duy and Tran-Cong, 2001a), which are based on an integration process, are employed to approximate the solution of PDEs via point collocation mechanism in the set of randomly distributed points. The method is tested with the solution of Poisson's equations and the Navier-Stokes equations (Boussinesq material). Good results are obtained using relatively low numbers of data points. For example, the natural convection flow in a square cavity at Rayleigh number of $1.e6$ is simulated successfully using only 1693 random collocation points.

An effective RBFN‐boundary integral approach for the analysis of natural convection flow

AbstractThis paper presents a new radial basis function network‐boundary integral approach for the analysis of natural convection flow. The use of integral equations (IEs) allows the set of simultaneous unknowns to be confined to the boundary only. In this study, all boundary values including geometries are represented by indirect radial basis function networks (IRBFNs), resulting in an effective boundary element method (BEM) especially for the achievement of high Rayleigh numbers with relatively coarse and uniform meshes. Convergence is obtained up to a Rayleigh number of 1.0e7 in the case of a square cavity using a uniform mesh of 31 × 31 and a Rayleigh number of 5.0e4 in the case of a horizontal concentric annulus using a uniform mesh of 11 × 21. Copyright © 2004 John Wiley & Sons, Ltd.

Indirect RBFN method with thin plate splines for numerical solution of differential equations

This paper reports a mesh-free Indirect Radial Basis Function Network method (IRBFN) using Thin Plate Splines (TPSs) for numerical solution of Differential Equations (DEs) in rectangular and curvilinear coordinates. The adjustable parameters required by the method are the number of centres, their positions and possibly the order of the TPS. The first and second order TPSs which are widely applied in numerical schemes for numerical solution of DEs are employed in this study. The advantage of the TPS over the multiquadric basis function is that the former, with a given order, does not contain the adjustable shape parameter (i.e. the RBF's width) and hence TPS-based RBFN methods require less parametric study. The direct TPS-RBFN method is also considered in some cases for the purpose of comparison with the indirect TPS-RBFN method. The TPS-IRBFN method is verified successfully with a series of problems including linear elliptic PDEs, nonlinear elliptic PDEs, parabolic PDEs and Navier-Stokes equations in rectangular and curvilinear coordinates. Numerical results obtained show that the method achieves the norm of the relative error of the solution of O(10 -6) for the case of 1D second order DEs using a density of 51, of O(10 -7) for the case of 2D elliptic PDEs using a density of 20 × 20 and a Reynolds number Re = 200 for the case of Jeffery-Hamel flow with a density of 43 × 12.

Neural networks for BEM analysis of steady viscous flows

AbstractThis paper presents a new neural network‐boundary integral approach for analysis of steady viscous fluid flows. Indirect radial basis function networks (IRBFNs) which perform better than element‐based methods for function interpolation, are introduced into the BEM scheme to represent the variations of velocity and traction along the boundary from the nodal values. In order to assess the effect of IRBFNs, the other features used in the present work remain the same as those used in the standard BEM. For example, Picard‐type scheme is utilized in the iterative procedure to deal with the non‐linear convective terms while the calculation of volume integrals and velocity gradients are based on the linear finite element‐based method. The proposed IRBFN‐BEM is verified on the driven cavity viscous flow problem and can achieve a moderate Reynolds number of 1400 using a relatively coarse uniform mesh. The results obtained such as the velocity profiles along the horizontal and vertical centrelines as well as the properties of the primary vortex are in very good agreement with the benchmark solution. Furthermore, the secondary vortices are also captured by the present method. Thus, it appears that an ability to represent the boundary solution accurately can significantly improve the overall solution accuracy of the BEM. Copyright © 2003 John Wiley & Sons, Ltd.

Approximation of function and its derivatives using radial basis function networks

This paper presents a numerical approach, based on radial basis function networks (RBFNs), for the approximation of a function and its derivatives (scattered data interpolation). The approach proposed here is called the indirect radial basis function network (IRBFN) approximation which is compared with the usual direct approach. In the direct method (DRBFN) the closed form RBFN approximating function is first obtained from a set of training points and the derivative functions are then calculated directly by differentiating such closed form RBFN. In the indirect method (IRBFN) the formulation of the problem starts with the decomposition of the derivative of the function into RBFs. The derivative expression is then integrated to yield an expression for the original function, which is then solved via the general linear least squares principle, given an appropriate set of discrete data points. The IRBFN method allows the filtering of noise arisen from the interpolation of the original function from a discrete set of data points and produces a greatly improved approximation of its derivatives. In both cases the input data consists of a set of unstructured discrete data points (function values), which eliminates the need for a discretisation of the domain into a number of finite elements. The results obtained are compared with those obtained by the feed forward neural network approach where appropriate and the “finite element” methods. In all examples considered, the IRBFN approach yields a superior accuracy. For example, all partial derivatives up to second order of the function of three variables y= x 1 2+ x 1 x 2−2 x 2 2− x 2 x 3+ x 3 2 are approximated with at least an order of magnitude better in the L 2-norm in comparison with the usual DRBFN approach.