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Solution of Two-dimensional Linear and Nonlinear Unsteady Schrödinger Equation using “Quantum Hydrodynamics” Formulation with a MLPG Collocation Method ; Bourantas, Georgios in Computer Modeling in Engineering & Sciences (2014), 103(1), 49-70 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 ... [more ▼] 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 equa tions in terms of particle density and velocity potential, by separating the real and imaginary parts of a general solution, called a quantum hydrodynamic (QHD) equa tion, 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 Schrödinger equation, the lagging of coefficients method has been utilized to eliminate the non-linearity of the corresponding examined problem. A Type-I nodal distribution is used in order to provide convergence for the discrete Laplacian operator used at the governing equation. Numerical results are validated, comparing them with analyti cal and numerical solutions. [less ▲] Detailed reference viewed: 84 (0 UL)A node-based smoothed extended finite element method (NS-XFEM) for fracture analysis ; ; et al in Computer Modeling in Engineering & Sciences (2011), 73(4), 331-355 This paper aims to incorporate the node-based smoothed finite element method (NS-FEM) into the extended finite element method (XFEM) to form a novel numerical method (NS-XFEM) for analyzing fracture ... [more ▼] This paper aims to incorporate the node-based smoothed finite element method (NS-FEM) into the extended finite element method (XFEM) to form a novel numerical method (NS-XFEM) for analyzing fracture problems of 2D elasticity. NS-FEM uses the strain smoothing technique over the smoothing domains associated with nodes to compute the system stiffness matrix, which leads to the line integrations using directly the shape function values along the boundaries of the smoothing domains. As a result, we avoid integration of the stress singularity at the crack tip. It is not necessary to divide elements cut by cracks when we replace interior integration by boundary integration, simplifying integration of the discontinuous approximation. The key advantage of the NS-XFEM is that it provides more accurate solutions compared to the XFEM-T3 element. We will show for two numerical examples that the NS-XFEM significantly improves the results in the energy norm and the stress intensity factors. For the examples studied, we obtain super-convergent results. [less ▲] Detailed reference viewed: 19 (1 UL)Numerical Solution of Non-Isothermal Fluid Flows Using Local Radial Basis Functions (LRBF) Interpolation and a Velocity-Correction Method Bourantas, Georgios ; ; et al in Computer Modeling in Engineering & Sciences (2010), 64(2), 187-212 Meshfree point collocation method (MPCM) is developed, solving the velocity-vorticity formulation of Navier-Stokes equations, for two-dimensional, steady state incompressible viscous flow problems in the ... [more ▼] Meshfree point collocation method (MPCM) is developed, solving the velocity-vorticity formulation of Navier-Stokes equations, for two-dimensional, steady state incompressible viscous flow problems in the presence of heat transfer. Particular emphasis is placed on the application of the velocity-correction method, ensuring the continuity equation. The Gaussian Radial Basis Functions (GRBF) interpolation is employed to construct the shape functions in conjunction with the framework of the point collocation method. The cases of forced, natural and mixed convection in a 2D rectangular enclosure are examined. The accuracy and the sta- bility of the proposed scheme are demonstrated through three representative, well known and established benchmark problems. Results are presented for high values of the characteristics non-dimensional numbers of the flow, that is, the Reynolds, the Rayleigh and the Richardson number [less ▲] Detailed reference viewed: 62 (0 UL)Meshfree Point Collocation Schemes for 2D Steady State Incompressible Navier-Stokes Equations in Velocity-Vorticity Formulation for High Values of Reynolds Number Bourantas, Georgios ; ; et al in Computer Modeling in Engineering & Sciences (2010), 59(1), 31-63 A meshfree point collocation method has been developed for the velocity- vorticity formulation of two-dimensional, steady state incompressible Navier-Stokes equations. Particular emphasis was placed on ... [more ▼] A meshfree point collocation method has been developed for the velocity- vorticity formulation of two-dimensional, steady state incompressible Navier-Stokes equations. Particular emphasis was placed on the application of the velocity-correc- tion method, ensuring the continuity equation. The Moving Least Squares (MLS) approximation is employed for the construction of the shape functions, in conjunc- tion with the general framework of the point collocation method. Computations are obtained for regular and irregular nodal distributions, stressing the positivity con- ditions that make the matrix of the system stable and convergent. The accuracy and the stability of the proposed scheme are demonstrated through two representative, well-known, and established benchmark problems. The numerical scheme was also applied to a case with irregular geometry for marginally high Reynolds numbers [less ▲] Detailed reference viewed: 114 (7 UL)Analysis of thermoelastic waves in a two-dimensional functionally graded materials domain by the Meshless Local Petrov-Galerkin (MLPG) method ; ; Bordas, Stéphane et al in Computer Modeling in Engineering & Sciences (2010), 65(1), 27-74 This contribution focuses on the simulation of two-dimensional elastic wave propagation in functionally graded solids and structures. Gradient volume fractions of the constituent materials are assumed to ... [more ▼] This contribution focuses on the simulation of two-dimensional elastic wave propagation in functionally graded solids and structures. Gradient volume fractions of the constituent materials are assumed to obey the power law function of position in only one direction and the effective mechanical properties of the material are determined by the Mori–Tanaka scheme. The investigations are carried out by extending a meshless method known as the Meshless Local Petrov-Galerkin (MLPG) method which is a truly meshless approach to thermo-elastic wave propagation. Simulations are carried out for rectangular domains under transient thermal loading. To investigate the effect of material composition on the dynamic response of functionally graded materials, a metal/ceramic (Aluminum (Al) and Alumina (Al2O3) are considered as ceramic and metal constituents) composite is considered for which the transient thermal field, dynamic displacement and stress fields are reported for different material distributions. [less ▲] Detailed reference viewed: 20 (3 UL)Analysis of thermoelastic waves in a two-dimensional functionally graded materials domain by the Meshless Local Petrov-Galerkin (MLPG) method ; ; Bordas, Stéphane et al in Computer Modeling in Engineering & Sciences (2010), 65(1), 27-74 This contribution focuses on the simulation of two-dimensional elastic wave propagation in functionally graded solids and structures. Gradient volume fractions of the constituent materials are assumed to ... [more ▼] This contribution focuses on the simulation of two-dimensional elastic wave propagation in functionally graded solids and structures. Gradient volume fractions of the constituent materials are assumed to obey the power law function of position in only one direction and the effective mechanical properties of the material are determined by the Mori-Tanaka scheme. The investigations are carried out by extending a meshless method known as the Meshless Local Petrov-Galerkin (MLPG) method which is a truly meshless approach to thermo-elastic wave propagation. Simulations are carried out for rectangular domains under transient thermal loading. To investigate the effect of material composition on the dynamic response of functionally graded materials, a metal/ceramic (Aluminum (Al) and Alumina (Al 2O 3) are considered as ceramic and metal constituents) composite is considered for which the transient thermal field, dynamic displacement and stress fields are reported for different material distributions. Copyright © 2010 Tech Science Press. [less ▲] Detailed reference viewed: 133 (5 UL)Adaptive support domain implementation on the Moving Least Squares approximation for Mfree methods applied on elliptic and parabolic PDE problems using strong-form description Bourantas, Georgios ; ; in Computer Modeling in Engineering & Sciences (2009), 43 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 ... [more ▼] 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 behavior (exact solution on nodes). The present approach is one of the few successful attempts of using MLS approximation [Atluri, Liu, and Han (2006), Han, Liu, Rajendran and Atluri (2006), Atluri and Liu (2006)] instead of RBF approximation for the meshless PC method using strong-form description. To increase the accuracy of the MLS interpolation method and its robustness in problems with natural BCs, a suitable support domain should be chosen in order to ensure an optimized area of coverage for interpolation. To this end, the basis functions are constructed using two different approaches, pertinent to the dimension of the support domain. On one hand, a compact form for the support domain is retained by keeping its radius constant. On the other hand, one can control the number of neighboring nodes as the support domain of each point. The results show that some inaccuracies are present near the boundaries using the first approach, due to the limited number of nodes belonging to the support domain, which results in failed matrix inversion. Instead, the second approach offers capability for fully matrix inversion under many (if not all) circumstances, resulting in basis functions of increased accuracy and robustness. This PC method, applied along with an intelligent adaptive refinement, is demonstrated for elliptic and for parabolic PDEs, related to many flow and mass transfer problems. [less ▲] Detailed reference viewed: 116 (4 UL) |
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