Chein-Shan Liu¹, Yung-Wei Chen², Jiang-Ren Chang²

The International Conference on Computational & Experimental Engineering and Sciences, Vol.3, No.2, pp. 93-100, 2007, DOI:10.3970/icces.2007.003.093

Abstract A new collocation method is developed here to solve the elliptic boundary value problems with singularities. Specifically, we consider the Motz problem as a test of the performance of the new method, which is found accurate and effective. More >

B. Šarler¹

CMES-Computer Modeling in Engineering & Sciences, Vol.7, No.2, pp. 185-194, 2005, DOI:10.3970/cmes.2005.007.185

Abstract This paper explores the application of the mesh-free radial basis function collocation method for solution of heat transfer and fluid flow problems. The solution procedure is represented for a Poisson reformulated general transport equation in terms of a-symmetric, symmetric and modified (double consideration of the boundary nodes) collocation approaches. In continuation, specifics of a primitive variable solution procedure for the coupled mass, momentum, and energy transport representing the natural convection in an incompressible Newtonian Bussinesq fluid are elaborated. A comparison of different collocation strategies is performed based on the two dimensional De Vahl Davis steady natural convection benchmark with Prandtl… More >

Hongwei Guo³, Xiaoying Zhuang^3,4,5, Timon Rabczuk^1,2,*

CMC-Computers, Materials & Continua, Vol.59, No.2, pp. 433-456, 2019, DOI:10.32604/cmc.2019.06660

Abstract In this paper, a deep collocation method (DCM) for thin plate bending problems is proposed. This method takes advantage of computational graphs and backpropagation algorithms involved in deep learning. Besides, the proposed DCM is based on a feedforward deep neural network (DNN) and differs from most previous applications of deep learning for mechanical problems. First, batches of randomly distributed collocation points are initially generated inside the domain and along the boundaries. A loss function is built with the aim that the governing partial differential equations (PDEs) of Kirchhoff plate bending problems, and the boundary/initial conditions are minimised at those collocation… More >

Yan Gu¹, Wen Chen^1,2, Xiao-Qiao He³

CMC-Computers, Materials & Continua, Vol.29, No.2, pp. 129-154, 2012, DOI:10.3970/cmc.2012.029.129

Abstract This paper applies an improved singular boundary method (SBM) in conjunction with domain decomposition technique to stress analysis of layered elastic materials. For problems under consideration, the interface continuity conditions are approximated in the same manner as the boundary conditions. The multi-layered coating system is decomposed into multiple subdomains in terms of each layer, in which the solution is approximated separately by the SBM representation. The singular boundary method is a recent meshless boundary collocation method, in which the origin intensity factor plays a key role for its accuracy and efficiency. This study also introduces new strong-form regularization formulas to… 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 >

W. M. Abd- Elhameed^1,2, Y. H. Youssri²

CMES-Computer Modeling in Engineering & Sciences, Vol.105, No.5, pp. 375-398, 2015, DOI:10.3970/cmes.2015.105.375

Abstract In this paper, a new spectral algorithm for solving linear and nonlinear fractional-order initial value problems is established. The key idea for obtaining the suggested spectral numerical solutions for these equations is actually based on utilizing the ultraspherical wavelets along with applying the collocation method to reduce the fractional differential equation with its initial conditions into a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. The convergence and error analysis of the suggested ultraspherical wavelets expansion are carefully discussed. For the sake of testing the proposed algorithm, some numerical examples are considered. The numerical results indicate… More >

W. Chen^1,2, C. J. Liu¹, Y. Gu^2,3

CMES-Computer Modeling in Engineering & Sciences, Vol.105, No.4, pp. 251-270, 2015, DOI:10.3970/cmes.2015.105.251

Abstract The singular boundary method (SBM) is a recently-developed meshless boundary collocation method. This method overcomes the well-known fictitious boundary issue associated with the method of fundamental solutions (MFS) while remaining the merits of the later of being truly meshless, integral-free, and easy-to-program. Similar to the MFS, this method, however, produces dense and unsymmetrical coefficient matrix, which although much smaller in size compared with domain discretization methods, requires O(N²) operations in the iterative solution of the resulting algebraic system of equations. To remedy this bottleneck problem for its application to large-scale problems, this paper makes the first attempt to develop a… More >

Tao Zhang^1,2, Yiqian He³, Leiting Dong⁴, Shu Li¹, Abdullah Alotaibi⁵, Satya N. Atluri^2,5

CMES-Computer Modeling in Engineering & Sciences, Vol.97, No.6, pp. 509-533, 2014, DOI:10.3970/cmes.2014.097.509

Abstract In this article, the Meshless Local Petrov-Galerkin (MLPG) Mixed Collocation Method is developed to solve the Cauchy inverse problems of Steady- State Heat Transfer In the MLPG mixed collocation method, the mixed scheme is applied to independently interpolate temperature as well as heat flux using the same meshless basis functions The balance and compatibility equations are satisfied at each node in a strong sense using the collocation method. The boundary conditions are also enforced using the collocation method, allowing temperature and heat flux to be over-specified at the same portion of the boundary. For the inverse problems where noise is… More >

S. Saha Ray¹, A. K. Gupta²

CMES-Computer Modeling in Engineering & Sciences, Vol.103, No.5, pp. 315-341, 2014, DOI:10.3970/cmes.2014.103.315

Abstract In this paper, an efficient numerical schemes based on the Haar wavelet method are applied for finding numerical solution of nonlinear third-order modified Korteweg-de Vries (mKdV) equation as well as modified Burgers' equations. The numerical results are then compared with the exact solutions. The accuracy of the obtained solutions is quite high even if the number of calculation points is small. More >

V. C. Loukopoulos¹, G. C. Bourantas²

CMES-Computer Modeling in Engineering & Sciences, Vol.103, No.1, pp. 49-70, 2014, DOI:10.3970/cmes.2014.103.049

Abstract A numerical solution of the linear and nonlinear time-dependent Schrödinger equation is obtained, using the strong form MLPG Collocation method. Schrödinger equation is replaced by a system of coupled partial differential equations in terms of particle density and velocity potential, by separating the real and imaginary parts of a general solution, called a quantum hydrodynamic (QHD) equation, which is formally analogous to the equations of irrotational motion in a classical fluid. The approximation of the field variables is obtained with the Moving Least Squares (MLS) approximation and the implicit Crank-Nicolson scheme is used for time discretization. For the two-dimensional nonlinear… More >