C. F. Loeffle¹, L. Zamprogno², W. J. Mansur³, A. Bulcão⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.113, No.3, pp. 367-387, 2017, DOI:10.3970/cmes.2017.113.387

Abstract This study evaluates the effectiveness of a new technique that transforms domain integrals into boundary integrals that is applicable to the boundary element method. Simulations were conducted in which two-dimensional surfaces were approximated by interpolation using radial basis functions with full and compact supports. Examples involving Poisson’s equation are presented using the boundary element method and the proposed technique with compact radial basis functions. The advantages and the disadvantages are examined through simulations. The effects of internal poles, the boundary mesh refinement and the value for the support of the radial basis functions on performance are assessed. More >

U. Hanoglu¹, B. Šarler^1,2,3

CMES-Computer Modeling in Engineering & Sciences, Vol.109-110, No.5, pp. 447-479, 2015, DOI:10.3970/cmes.2015.109.447

Abstract The aim of this paper is to demonstrate the use of the novel Local Radial Basis Function Collocation Method (LRBFCM) [Šarler and Vertnik (2006)] in an industrial coupled thermo-mechanical problem of hot shape rolling of steel. The physical concept of such a large deformation problem is based on a two dimensional traveling slice model [Glowacki (2005)], which assumes deformation and heat flow only in the perpendicular direction to rolling. The solution is performed based on strong formulation. Elliptic Node Generation (ENG) is applied to reposition the nodes over a slice when necessary in order to sustain stability throughout the simulation.… More >

K. N. Grivas¹, M. G. Vavva¹, E. J. Sellountos², D. I. Fotiadis³, D. Polyzos^1,4

CMES-Computer Modeling in Engineering & Sciences, Vol.105, No.2, pp. 87-122, 2015, DOI:10.3970/cmes.2015.105.087

Abstract A simple Local Boundary Integral Equation (LBIE) method for solving the Fisher nonlinear transient diffusion equation in two dimensions (2D) is reported. The method utilizes, for its meshless implementation, randomly distributed nodal points in the interior domain and nodal points corresponding to a Boundary Element Method (BEM) mesh, at the global boundary. The interpolation of the interior and boundary potentials is accomplished using a Local Radial Basis Functions (LRBF) scheme. At the nodes of global boundary the potentials and their fluxes are treated as independent variables. On the local boundaries, potential fluxes are avoided by using the Laplacian companion solution.… More >

S.Yu. Reutskiy¹

CMES-Computer Modeling in Engineering & Sciences, Vol.99, No.4, pp. 327-349, 2014, DOI:10.3970/cmes.2014.099.327

Abstract The paper presents a new meshless numerical method for solving 2D and 3D boundary value problems (BVPs) with elliptic PDEs of general form. The coefficients of the PDEs including the main operator part are spatially dependent functions. The key idea of the method is the use of the basis functions which satisfy the homogeneous boundary conditions of the problem. This allows us to seek an approximate solution in the form which satisfies the boundary conditions of the initial problem with any choice of the free parameters. As a result we separate approximation of the boundary conditions and approximation of the… More >

M. Dehghan¹, M. Abbaszadeh², A. Mohebbi³

CMES-Computer Modeling in Engineering & Sciences, Vol.100, No.5, pp. 399-444, 2014, DOI:10.3970/cmes.2014.100.399

Abstract In this paper three numerical techniques are proposed for solving the system of N-coupled nonlinear Schrödinger (CNLS) equations. Firstly, we obtain a time discrete scheme by approximating the first-order time derivative via the forward finite difference formula, then for obtaining a full discretization scheme, we use the Kansa’s approach to approximate the spatial derivatives via radial basis functions (RBFs) collocation methodology. We introduce the moving least squares (MLS) approximation and radial point interpolation method (RPIM) with their shape functions, separately. It should be noted that the shape functions of RPIM unlike the shape functions of the MLS approximation have kronecker… More >

N. Pham-Sy¹, C.-D. Tran¹, N. Mai-Duy¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.100, No.5, pp. 363-397, 2014, DOI:10.3970/cmes.2014.100.363

Abstract In this paper, a high performance computing method based on the Integrated Radial Basis Function (IRBF), Control Volume (CV) and Domain Decomposition technique for solving Partial Differential Equations is presented. The goal is to develop an efficient parallel algorithm based on the Compact Local IRBF method using the CV approach, especially for problems with non-rectangular domain. The results showed that the goal is achieved as the computational efficiency is quite significant. For the case of square lid driven cavity problem with Renoylds number 100, super-linear speed-up is also achieved. The parallel algorithm is implemented in the Matlab environment using Parallel… More >

T.A. Elgohary¹, L. Dong², J.L. Junkins³, S.N. Atluri⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.100, No.1, pp. 59-84, 2014, DOI:10.3970/cmes.2014.100.059

Abstract In this study, we consider ill-posed time-domain inverse problems for dynamical systems with various boundary conditions and unknown controllers. Dynamical systems characterized by a system of second-order nonlinear ordinary differential equations (ODEs) are recast into a system of nonlinear first order ODEs in mixed variables. Radial Basis Functions (RBFs) are assumed as trial functions for the mixed variables in the time domain. A simple collocation method is developed in the time-domain, with Legendre-Gauss-Lobatto nodes as RBF source points as well as collocation points. The duffing optimal control problem with various prescribed initial and final conditions, as well as the orbital… More >

L. Godinho¹, D. Dias-da-Costa²

CMES-Computer Modeling in Engineering & Sciences, Vol.96, No.5, pp. 293-316, 2013, DOI:10.3970/cmes.2013.096.293

Abstract Transient heat conduction problems can be dealt with using different numerical approaches. In some recent papers, a strategy to tackle these problems using a frequency domain formulation has been presented and successfully applied associated to methods such as the BEM. Here a formulation of the meshless local Petrov-Galerkin (MLPG) is developed and presented to allow the analysis of such problems. The proposed formulation makes use of the RBF-based version of the MLPG and employs the Heaviside step function as the test function, leading to the so-called MLPG(5). In addition, the method is associated with a visibility criterion to allow the… More >

P.H. Wen¹, X.J. Huang¹, M.H. Aliabadi²

CMES-Computer Modeling in Engineering & Sciences, Vol.96, No.3, pp. 199-225, 2013, DOI:10.3970/cmes.2013.096.199

Abstract In this paper, a Local Integral Equation Method (LIEM) is presented for solving two-dimensional nonlocal elasticity problems . The approach is based on the Eringen’s model with LIEM and the interpolation using the radial basis functions to obtain the numerical solutions for 2D problems. A weak form for the set of governing equations with a unit test function is transformed into the local integral equations. The meshless approximation technique with radial basis functions is employed for the implementation of displacements. A set of the local domain integrals is obtained in analytical form for the local elasticity and by using a… More >

T.-T. Hoang-Trieu¹, N. Mai-Duy¹, C.-D. Tran¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.91, No.6, pp. 485-516, 2013, DOI:10.3970/cmes.2013.091.485

Abstract In this paper, compact local integrated radial basis function (IRBF) stencils reported in [Mai-Duy and Tran-Cong (2011) Journal of Computational Physics 230(12), 4772-4794] are introduced into the finite-volume / subregion - collocation formulation for the discretisation of second-order differential problems defined on rectangular and non-rectangular domains. The problem domain is simply represented by a Cartesian grid, over which overlapping compact local IRBF stencils are utilised to approximate the field variable and its derivatives. The governing differential equation is integrated over non-overlapping control volumes associated with grid nodes, and the divergence theorem is then applied to convert volume integrals into surface/line… More >