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  • Open Access


    Stable PDE Solution Methods for Large Multiquadric Shape Parameters

    Arezoo Emdadi1, Edward J. Kansa2, Nicolas Ali Libre1,3, Mohammad Rahimian1, Mohammad Shekarchi1

    CMES-Computer Modeling in Engineering & Sciences, Vol.25, No.1, pp. 23-42, 2008, DOI:10.3970/cmes.2008.025.023

    Abstract We present a new method based upon the paper of Volokh and Vilney (2000) that produces highly accurate and stable solutions to very ill-conditioned multiquadric (MQ) radial basis function (RBF) asymmetric collocation methods for partial differential equations (PDEs). We demonstrate that the modified Volokh-Vilney algorithm that we name the improved truncated singular value decomposition (IT-SVD) produces highly accurate and stable numerical solutions for large values of a constant MQ shape parameter, c, that exceeds the critical value of c based upon Gaussian elimination. More >

  • Open Access


    Improving Volume Element Methods by Meshless Radial Basis Function Techniques

    P. Orsini1, H. Power1,2, H. Morvan1

    CMES-Computer Modeling in Engineering & Sciences, Vol.23, No.3, pp. 187-208, 2008, DOI:10.3970/cmes.2008.023.187

    Abstract In this work, we present a modified Control Volume (CV) method that uses a Radial Basis Function (RBF) interpolation to improve the prediction of the flux accuracy at the faces of the CV. The method proposed differs from classical CV methods in the way that the flux at the cell surfaces is computed. A local RBF interpolation of the field variable is performed at the centres of the cell being integrated and its neighbours. This interpolation is then used to reconstruct the solution and its gradient in the integration points which support the flux computation. In addition, it is required… More >

  • Open Access


    A Hybrid Multi-Region BEM / LBIE-RBF Velocity-Vorticity Scheme for the Two-Dimensional Navier-Stokes Equations

    E.J. Sellountos1, A. Sequeira1

    CMES-Computer Modeling in Engineering & Sciences, Vol.23, No.2, pp. 127-148, 2008, DOI:10.3970/cmes.2008.023.127

    Abstract In this work a hybrid velocity-vorticity scheme for the solution of the 2D Navier-Stokes equations is presented. The multi-region Local Boundary Integral Equation (LBIE) combined with Radial Basis Functions (RBF) interpolation is used for the solution of the kinematics and the multi-region BEM for the solution of the transport kinetics. The final system of equations is in band form for both methods. The issue of RBF discontinuities is resolved by constructing the RBF matrix locally in every region. The kinematics integral equation is used in three different forms, for coupling the velocity field on the boundary, on interior points and… More >

  • Open Access


    Geometrically Nonlinear Analysis of Reissner-Mindlin Plate by Meshless Computation

    P. H. Wen1, Y. C. Hon2

    CMES-Computer Modeling in Engineering & Sciences, Vol.21, No.3, pp. 177-192, 2007, DOI:10.3970/cmes.2007.021.177

    Abstract In this paper, we perform a geometrically nonlinear analysis of Reissner-Mindlin plate by using a meshless collocation method. The use of the smooth radial basis functions (RBFs) gives an advantage to evaluate higher order derivatives of the solution at no cost on extra-interpolation. The computational cost is low and requires neither the connectivity of mesh in the domain/boundary nor integrations of fundamental/particular solutions. The coupled nonlinear terms in the equilibrium equations for both the plane stress and plate bending problems are treated as body forces. Two load increment schemes are developed to solve the nonlinear differential equations. Numerical verifications are… More >

  • Open Access


    An Unconditionally Time-Stable Level Set Method and Its Application to Shape and Topology Optimization

    S.Y. Wang1,2, K.M. Lim2,3, B.C. Khoo2,3, M.Y. Wang4

    CMES-Computer Modeling in Engineering & Sciences, Vol.21, No.1, pp. 1-40, 2007, DOI:10.3970/cmes.2007.021.001

    Abstract The level set method is a numerical technique for simulating moving interfaces. In this paper, an unconditionally BIBO (Bounded-Input-Bounded-Output) time-stable consistent meshfree level set method is proposed and applied as a more effective approach to simultaneous shape and topology optimization. In the present level set method, the meshfree infinitely smooth inverse multiquadric Radial Basis Functions (RBFs) are employed to discretize the implicit level set function. A high level of smoothness of the level set function and accuracy of the solution to the Hamilton-Jacobi partial differential equation (PDE) can be achieved. The resulting dynamic system of coupled Ordinary Differential Equations (ODEs)… More >

  • Open Access


    Computation of transient viscous flows using indirect radial basis function networks

    N. Mai-Duy1, L. Mai-Cao2, T. Tran-Cong3

    CMES-Computer Modeling in Engineering & Sciences, Vol.18, No.1, pp. 59-78, 2007, DOI:10.3970/cmes.2007.018.059

    Abstract In this paper, an indirect/integrated radial-basis-function network (IRBFN) method is further developed to solve transient partial differential equations (PDEs) governing fluid flow problems. Spatial derivatives are discretized using one- and two-dimensional IRBFN interpolation schemes, whereas temporal derivatives are approximated using a method of lines and a finite-difference technique. In the case of moving interface problems, the IRBFN method is combined with the level set method to capture the evolution of the interface. The accuracy of the method is investigated by considering several benchmark test problems, including the classical lid-driven cavity flow. Very accurate results are achieved using relatively low numbers… More >

  • Open Access


    Meshless Local Petrov-Galerkin (MLPG) Approaches for Solving the Weakly-Singular Traction & Displacement Boundary Integral Equations

    S. N. Atluri1, Z. D. Han1, S. Shen1

    CMES-Computer Modeling in Engineering & Sciences, Vol.4, No.5, pp. 507-518, 2003, DOI:10.3970/cmes.2003.004.507

    Abstract The general Meshless Local Petrov-Galerkin (MLPG) type weak-forms of the displacement & traction boundary integral equations are presented, for solids undergoing small deformations. These MLPG weak forms provide the most general basis for the numerical solution of the non-hyper-singular displacement and traction BIEs [given in Han, and Atluri (2003)], which are simply derived by using the gradients of the displacements of the fundamental solutions [Okada, Rajiyah, and Atluri (1989a,b)]. By employing the various types of test functions, in the MLPG-type weak-forms of the non-hyper-singular dBIE and tBIE over the local sub-boundary surfaces, several types of MLPG/BIEs are formulated, while also… More >

  • Open Access


    Meshless BEM for Three-dimensional Stokes Flows

    C.C. Tsai1, D.L. Young2, A.H.-D. Cheng3

    CMES-Computer Modeling in Engineering & Sciences, Vol.3, No.1, pp. 117-128, 2002, DOI:10.3970/cmes.2002.003.117

    Abstract This paper describes a combination of the dual reciprocity method (DRM) and the method of fundamental solution (MFS) as a meshless BEM (DRM-MFS) to solve three-dimensional Stokes flow problems by the velocity-vorticity formulation, where the DRM is based on the compactly supported, positive definite radial basis functions (CS-PD-RBF). In the velocity-vorticity formulation, both of the diffusion type vorticity equations and the Poisson type velocity equations are solved by DRM-MFS. Here a typical internal cubic cavity flow and an external flow past a sphere are presented. The results are acceptable. Furthermore, this paper provides a preliminary work for applications to the… More >

  • Open Access


    An Efficient Mesh-Free Method for Nonlinear Reaction-Diffusion Equations

    M.A. Golberg1, C.S. Chen2

    CMES-Computer Modeling in Engineering & Sciences, Vol.2, No.1, pp. 87-96, 2001, DOI:10.3970/cmes.2001.002.087

    Abstract The purpose of this paper is to develop a highly efficient mesh-free method for solving nonlinear diffusion-reaction equations in Rd, d=2, 3. Using various time difference schemes, a given time-dependent problem can be reduced to solving a series of inhomogeneous Helmholtz-type equations. The solution of these problems can then be further reduced to evaluating particular solutions and the solution of related homogeneous equations. Recently, radial basis functions have been successfully implemented to evaluate particular solutions for Possion-type equations. A more general approach has been developed in extending this capability to obtain particular solutions for Helmholtz-type equations by using polyharmonic spline… More >

  • Open Access


    A Numerical Study of Strain Localization in Elasto-Thermo-Viscoplastic Materials using Radial Basis Function Networks

    P. Le1, N. Mai-Duy1, T. Tran-Cong1, G. Baker2

    CMC-Computers, Materials & Continua, Vol.5, No.2, pp. 129-150, 2007, DOI:10.3970/cmc.2007.005.129

    Abstract This paper presents a numerical simulation of the formation and evolution of strain localization in elasto-thermo-viscoplastic materials (adiabatic shear band) by the indirect/integral radial basis function network (IRBFN) method. The effects of strain and strain rate hardening, plastic heating, and thermal softening are considered. The IRBFN method is enhanced by a new coordinate mapping which helps capture the stiff spatial structure of the resultant band. The discrete IRBFN system is integrated in time by the implicit fifth-order Runge-Kutta method. The obtained results are compared with those of the Modified Smooth Particle Hydrodynamics (MSPH) method and Chebychev Pseudo-spectral (CPS) method. More >

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