Chein-Shan Liu¹, Weichung Yeih², Satya N. Atluri³

CMES-Computer Modeling in Engineering & Sciences, Vol.44, No.3, pp. 281-312, 2009, DOI:10.3970/cmes.2009.044.281

Abstract Here we develop a general purpose pre/post conditionerT, to solve an ill-posed system of linear equations,Ax=b. The conditionerTis obtained in the course of the solution of the Laplace equation, through a boundary-collocation Trefftz method, leading to:Ty=x, whereyis the vector of coefficients in the Trefftz expansion, andxis the boundary data at the discrete points on a unit circle. We show that the quality of the conditionerTis greatly enhanced by using multiple characteristic lengths (Multiple Length Scales) in the Trefftz expansion. We further show thatTcan be multiplicatively decomposed into a dilationTDand a rotationTR. For an odd-orderedA, we More >

Sirajul Haq^1,2, Siraj-Ul-Islam³, Marjan Uddin^1,4

CMES-Computer Modeling in Engineering & Sciences, Vol.44, No.2, pp. 115-136, 2009, DOI:10.3970/cmes.2009.044.115

Abstract A mesh free method for the numerical solution of the nonlinear Schrodinger (NLS) and coupled nonlinear Schrodinger (CNLS) equation is implemented. The presented method uses a set of scattered nodes within the problem domain as well as on the boundaries of the domain along with approximating functions known as radial basis functions (RBFs). The set of scattered nodes do not form a mesh, means that no information of relationship between the nodes is needed. Error norms L₂, L_∞ are used to estimate accuracy of the method. Stability analysis of the method is given to demonstrate its More >

R. Vertnik¹, B. Šarler²

CMES-Computer Modeling in Engineering & Sciences, Vol.44, No.1, pp. 65-96, 2009, DOI:10.3970/cmes.2009.044.065

Abstract The application of the mesh-free Local Radial Basis Function Collocation Method (LRBFCM) in solution of incompressible turbulent flow is explored in this paper. The turbulent flow equations are described by the low - Re number k-emodel with Jones and Launder [Jones and Launder (1971)] closure coefficients. The involved velocity, pressure, turbulent kinetic energy and dissipation fields are represented on overlapping 5-noded sub-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBF's. The velocity, turbulent kinetic energy and… More >

G. C. Bourantas¹, E. D. Skouras^2,3,4, G. C. Nikiforidis¹

CMES-Computer Modeling in Engineering & Sciences, Vol.43, No.1, pp. 1-26, 2009, DOI:10.3970/cmes.2009.043.001

Abstract The extent of application of meshfree methods based on point collocation (PC) techniques with adaptive support domain for strong form Partial Differential Equations (PDE) is investigated. The basis functions are constructed using the Moving Least Square (MLS) approximation. The weak-form description of PDEs is used in most MLS methods to circumvent problems related to the increased level of resolution necessary near natural (Neumann) boundary conditions (BCs), dislocations, or regions of steep gradients. Alternatively, one can adopt Radial Basis Function (RBF) approximation on the strong-form of PDEs using meshless PC methods, due to the delta function… More >

Y.-M. Guo¹

CMES-Computer Modeling in Engineering & Sciences, Vol.41, No.3, pp. 177-194, 2009, DOI:10.3970/cmes.2009.041.177

Abstract In this paper, for analyses of the rigid-plastic metal forming problems, a hybrid PCM/FEM is developed. By introducing a boundary layer of finite element in boundary domain of workpiece, unsatisfactory issue of the positivity conditions of boundary points can be avoided, and the complicated boundary conditions can be easily imposed with the boundary layer of finite element. A plane strain upsetting process is analyzed by using the hybrid PCM/FEM. More >

Brent L. Adams, Joshua Kacher

CMC-Computers, Materials & Continua, Vol.14, No.3, pp. 185-196, 2009, DOI:10.3970/cmc.2009.014.185

Abstract Consideration is given to the resolution of dislocation density afforded by EBSD-based scanning electron microscopy. Comparison between the conventional Hough- and the emerging high-resolution cross-correlation-based approaches is made. It is illustrated that considerable care must be exercised in selecting a step size (Burger's circuit size) for experimental measurements. Important variables affecting this selection include the dislocation density and the physical size and density of dislocation dipole and multi-pole components of the structure. It is also illustrated that simulations can be useful to the interpretation of experimental recoveries. More >

Chien-Ching Ma^1,2, Wen-Cha Wu²

CMC-Computers, Materials & Continua, Vol.11, No.2, pp. 79-108, 2009, DOI:10.3970/cmc.2009.011.079

Abstract The two-dimensional problem of a planar transversely isotropic piezoelectric layered half-plane subjected to generalized line forces and edge dislocations in the layer is analyzed by using the Fourier-transform method and the series expansion technique. The full-field solutions for displacements, stresses, electrical displacements and electric fields are expressed in explicit closed forms. The complete solutions consist only of the simplest solutions for an infinite piezoelectric medium with applied loadings. It is shown in this study that the physical meaning of this solution is the image method. The explicit solutions include Green's function for originally applied loadings… More >

Leevan Ling¹

The International Conference on Computational & Experimental Engineering and Sciences, Vol.8, No.4, pp. 133-138, 2008, DOI:10.3970/icces.2008.008.133

Abstract The history of meshless collocation methods featured plenty of nicely calculated practical solutions, but a solid mathematical basis was long missing for the most popular asymmetric technique introduced by E. Kansa. Thus the impact of this work will be to supply a lasting mathematical foundation which will also improve our general understanding of such technique. Our previous research gave a convergent algorithm. More >

Z. Aksamija^1,2, U. Ravaioli³

CMES-Computer Modeling in Engineering & Sciences, Vol.37, No.1, pp. 45-64, 2008, DOI:10.3970/cmes.2008.037.045

Abstract Finding the scalar potential from the Poisson equation is a common, yet challenging problem in semiconductor modeling. One of the central problems in traditional mesh-based methods is the assignment of charge to the regular mesh imposed for the discretisation. In order to avoid this problem, we create a mesh-free algorithm which starts by assigning each mesh point to each particle present in the problem. This algorithm is based on a Fourier series expansion coupled with point matching. An efficient algorithm for repeatedly solving the Poisson problem for moving charge distributions is presented. We demonstate that More >

G.J.A. Loeven^1,2, H. Bijl³

CMES-Computer Modeling in Engineering & Sciences, Vol.36, No.3, pp. 193-212, 2008, DOI:10.3970/cmes.2008.036.193

Abstract In this paper a Two-Step approach is presented for uncertainty quantification for expensive problems with multiple uncertain parameters. Both steps are performed using the Probabilistic Collocation method. The first step consists of a sensitivity analysis to identify the most important parameters of the problem. The sensitivity derivatives are obtained using a first or second order Probabilistic Collocation approximation. For the most important parameters the probability distribution functions are propagated using the Probabilistic Collocation method using higher order approximations. The Two-Step approach is demonstrated for flow around a NACA0012 airfoil with eight uncertain parameters in the More >