Governing Equations of Groundwater Flow and Aquifer Modelling Using Finite Difference Method

2008 ◽  
pp. 201-218
Author(s):  
Shazrah Owais ◽  
S. Atal ◽  
P. D. Sreedevi
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Groundwater Flow ◽  
Difference Method ◽  
Governing Equations

scholarly journals Vibration Analysis of Axially Functionally Graded Non-Prismatic Euler-Bernoulli Beams Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Vibration Analysis ◽  
Mass Density ◽  
Functionally Graded ◽  
Governing Equations ◽  
Mass System ◽  
The Finite Difference Method

This paper presents an approach to the vibration analysis of axially functionally graded (AFG) non-prismatic Euler-Bernoulli beams using the finite difference method (FDM). The characteristics (cross-sectional area, moment of inertia, elastic moduli, and mass density) of AFG beams vary along the longitudinal axis. The FDM is an approximate method for solving problems described with differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. In addition, the boundary conditions and not the governing equations are applied at the beam’s ends. In this paper, differential equations were formulated with finite differences, and additional points were introduced at the beam’s ends and at positions of discontinuity (supports, hinges, springs, concentrated mass, spring-mass system, etc.). The introduction of additional points allowed us to apply the governing equations at the beam’s ends and to satisfy the boundary and continuity conditions. Moreover, grid points with variable spacing were also considered, the grid being uniform within beam segments. Vibration analysis of AFG non-prismatic Euler-Bernoulli beams was conducted with this model, and natural frequencies were determined. Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of AFG non-prismatic Euler-Bernoulli beams, considering the damping. The results obtained in this paper showed good agreement with those of other studies, and the accuracy was always increased through a grid refinement.

Thermo-mechanical vibration, buckling, and bending of orthotropic graphene sheets based on nonlocal two-variable refined plate theory using finite difference method considering surface energy effects

2017 ◽  
Vol 231 (3) ◽  
pp. 111-130 ◽  
Author(s):  
Morteza Karimi ◽  
Ali Reza Shahidi
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Surface Energy ◽  
Difference Method ◽  
Plate Theory ◽  
Small Scale ◽  
Graphene Sheets ◽  
Governing Equations ◽  
Refined Plate Theory ◽  
Temperature Changes

In this article, the influence of temperature change on the vibration, buckling, and bending of orthotropic graphene sheets embedded in elastic media including surface energy and small-scale effects is investigated. To take into account the small-scale and surface energy effects, the nonlocal constitutive relations of Eringen and surface elasticity theory of Gurtin and Murdoch are used, respectively. Using Hamilton’s principle, the governing equations for bulk and surface of orthotropic nanoplate are derived using two-variable refined plate theory. Finite difference method is used to solve governing equations. The obtained results are verified with Navier’s method and validated results reported in the literature. The results demonstrated that for both isotropic and orthotropic material properties, by increasing the temperature changes, the degree of surface effects on the buckling and vibration of nanoplates could enhance at higher temperatures, while it would diminish at lower temperatures. In addition, the effects of surface and temperature changes on the buckling and vibration for isotropic material property are more noticeable than those of orthotropic. On the contrary, these results are totally reverse for bending problem.

A Study of Photo-Thermoelastic Wave in Semiconductor Materials With Spherical Holes Using Analytical-numerical Methods

2021 ◽  
Author(s):  
Faris S. Alzahrani ◽  
Ibrahim Abbas
Keyword(s):  
Finite Difference ◽  
Analytical Solution ◽  
Finite Difference Method ◽  
Difference Method ◽  
Numerical Solutions ◽  
Governing Equations ◽  
Thermoelastic Wave ◽  
Spherical Cavities ◽  
Implicit Finite Difference ◽  
Implicit Finite Difference Method

Abstract Analytical and numerical solutions are two basic tools in the study of photothermal interaction problems in semiconductor medium. In this paper, we compare the analytical solutions with the numerical solutions for thermal interaction in semiconductor mediums containing spherical cavities. The governing equations are given in the domain of Laplace transforms and the eigenvalues approaches are used to obtained the analytical solution. The numerical solutions are obtained by applying the implicit finite difference method (IFDM). A comparison between the numerical solutions and analytical solution are presented. It is found that the implicit finite difference method (IFDM) is applicable, simple and efficient for such problems.

scholarly journals Finite Difference Method of Modelling Groundwater Flow

2011 ◽  
Vol 03 (03) ◽  
pp. 192-198 ◽  
Author(s):  
Magnus.U. Igboekwe ◽  
N. J. Achi
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Groundwater Flow ◽  
Difference Method
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