Hokyung Shim¹, Vinh Thuy Tran Ho^1,,Semyung Wang^1,2, Daniel A. Tortorelli³

CMES-Computer Modeling in Engineering & Sciences, Vol.37, No.2, pp. 175-202, 2008, DOI:10.3970/cmes.2008.037.175

Abstract This paper presents a topological shape optimization technique for electromagnetic problems using a level set method and radial basis functions. The proposed technique is a level set (LS) based optimization dealing with geometrical shape derivatives and topological design. The shape derivative is computed by an adjoint variable method to avoid numerous sensitivity evaluations. A level set model embedded into the scalar function of higher dimensions is propagated to represent the design boundary of a domain. The level set function interpolated into a fixed initial domain is evolved by using the Hamilton-Jacobi equation. The moving free boundaries (dynamic interfaces) represented in… More >

Giorgio Pini¹, Annamaria Mazzia¹, Flavio Sartoretto²,

CMES-Computer Modeling in Engineering & Sciences, Vol.36, No.1, pp. 43-64, 2008, DOI:10.3970/cmes.2008.036.043

Abstract Meshless methods have been explored in many 2D problems and they have been shown to be as accurate as Finite Element Methods (FEM). Compared to the extensive literature on 2D applications, papers on solving 3D problems by meshless methods are surprisingly few. Indeed, a main drawback of these methods is the requirement for accurate cubature rules. This paper focuses on the so called Meshless Local Petrov Galerkin (MLPG) methods. We show that accurate solutions of 3D potential problems can be attained, provided suitable cubature rules are identified, sparse data structures are efficiently stored, and strategies are devised in order to… More >

S. Chantasiriwan¹

CMES-Computer Modeling in Engineering & Sciences, Vol.15, No.3, pp. 137-146, 2006, DOI:10.3970/cmes.2006.015.137

Abstract The multiquadric collocation method is the collocation method based on radial basis function known as multiquadrics. It has been successfully used to solve several linear and nonlinear problems. Although fluid flow problems are among problems previously solved by this method, there is still an outstanding issue regarding the influence of the free parameter of multiquadrics (or the shape parameter) on the performance of the method. This paper provides additional results of using the multiquadric collocation method to solve the lid-driven cavity flow problem. The method is used to solve the problem in the stream function-vorticity formulation and the velocity-vorticity formulation.… More >

S.Y. Wang^1,2, M.Y. Wang³

CMES-Computer Modeling in Engineering & Sciences, Vol.13, No.2, pp. 119-148, 2006, DOI:10.3970/cmes.2006.013.119

Abstract In this paper, an implicit free boundary parametrization method is presented as an effective approach for simultaneous shape and topology optimization of structures. The moving free boundary of a structure is embedded as a zero level set of a higher dimensional implicit level set function. The radial basis functions (RBFs) are introduced to parametrize the implicit function with a high level of accuracy and smoothness. The motion of the free boundary is thus governed by a mathematically more convenient ordinary differential equation (ODE). Eigenvalue stability can be guaranteed due to the use of inverse multiquadric RBF splines. To perform both… More >

C.-D. Tran¹, N. Mai-Duy^1,1, K. Le-Cao¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.85, No.6, pp. 499-520, 2012, DOI:10.3970/cmes.2012.085.499

Abstract A numerical computation of continuum-microscopic model for visco-elastic flows based on the Integrated Radial Basis Function (IRBF) Control Volume and the Stochastic Simulation Techniques (SST) is reported in this paper. The macroscopic flow equations are closed by a stochastic equation for the extra stress at the microscopic level. The former are discretised by a 1D-IRBF-CV method while the latter is integrated with Euler explicit or Predictor-Corrector schemes. Modelling is very efficient as it is based on Cartesian grid, while the integrated RBF approach enhances both the stability of the procedure and the accuracy of the solution. The proposed method is… More >

N. Thai-Quang¹, K. Le-Cao¹, N. Mai-Duy¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.84, No.6, pp. 528-558, 2012, DOI:10.3970/cmes.2012.084.528

Abstract This study is concerned with the development of integrated radial-basis-function (IRBF) method for the simulation of two-dimensional steady-state incompressible viscous flows governed by the pressure-velocity formulation on Cartesian grids. Instead of using low-order polynomial interpolants, a high-order compact local IRBF scheme is employed to represent the convection and diffusion terms. Furthermore, an effective boundary treatment for the pressure variable, where Neumann boundary conditions are transformed into Dirichlet ones, is proposed. This transformation is based on global 1D-IRBF approximators using values of the pressure at interior nodes along a grid line and first-order derivative values of the pressure at the two… More >

Ayşe Gül Kaplan¹, Yılmaz Dereli

CMES-Computer Modeling in Engineering & Sciences, Vol.84, No.5, pp. 423-438, 2012, DOI:10.3970/cmes.2012.084.423

Abstract In this study, the nonlinear symmetric regularized long wave equation was solved numerically by using radial basis functions collocation method. The single solitary wave solution, the interaction of two positive solitary waves and the clash of two solitary waves were studied. Numerical results and simulations of the wave motions were presented. Validity and accuracy of the method was tested by compared with results in the literature. More >

Ali Shokri¹, Mehdi Dehghan¹

CMES-Computer Modeling in Engineering & Sciences, Vol.84, No.4, pp. 333-358, 2012, DOI:10.3970/cmes.2012.084.333

Abstract The Ginzburg-Landau equation has been used as a mathematical model for various pattern formation systems in mechanics, physics and chemistry. In this paper, we study the complex Ginzburg-Landau equation in two spatial dimensions with periodical boundary conditions. The method numerically approximates the solution by collocation method based on radial basis functions (RBFs). To improve the numerical results we use a predictor-corrector scheme. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the accuracy and efficiency of the presented method. More >

T.-T. Hoang-Trieu¹, N. Mai-Duy¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.84, No.2, pp. 171-204, 2012, DOI:10.3970/cmes.2012.084.171

Abstract In this paper, new compact local stencils based on integrated radial basis functions (IRBFs) for solving fourth-order ordinary differential equations (ODEs) and partial differential equations (PDEs) are presented. Five types of compact stencils - 3-node and 5-node for 1D problems and 5×5-node, 13-node and 3×3 -node for 2D problems - are implemented. In the case of 3-node stencil and 3×3-node stencil, nodal values of the first derivative(s) of the field variable are treated as additional unknowns (i.e. 2 unknowns per node for 3-node stencil and 3 unknowns per node for 3×3-node stencil). The integration constants arising from the construction of… More >

P. L. Bishay¹, S.N. Atluri¹

CMES-Computer Modeling in Engineering & Sciences, Vol.84, No.1, pp. 41-98, 2012, DOI:10.3970/cmes.2012.084.041

Abstract Higher-order two-dimensional as well as low and higher-order three-dimensional new Hybrid/Mixed (H/M) finite elements based on independently assumed displacement, and judiciously chosen strain fields, denoted by HMFEM-2, are developed here for applications in macro-mechanics. The idea of these new H/M finite elements is based on collocating the components of the independent strain field, with those derived from the independently assumed displacement fields at judiciously and cleverly chosen collocation points inside the element. This is unlike the other techniques used in older H/M finite elements where a two-field variational principle was used in order to enforce both equilibrium and compatibility conditions… More >