Finite difference method

About: Finite difference method is a research topic. Over the lifetime, 21603 publications have been published within this topic receiving 468852 citations. The topic is also known as: Finite-difference methods & FDM.

The physical sense of simulation models of liquid chromatography: propagation through a grid or solution of the mass balance equation

Abstract: Two distinctly different approaches have been used to simulate the movement of bands through a chromatographic column One example of the first approach is the Craig distribution model, which replaces the continuous column with a specific number of discrete equilibration processes Thus it introduces the concept of (theoretical) plates into chromatography, but is not able to explain satisfactorily their significance The second approach is based on the mass balance equation which can be integrated numerically over time and space to give the elution profile In this paper we discuss the physical meaning of the numerical integration process followed by the finite difference methods

A novel method for simulating laser-solid interactions in semiconductors and layered structures

Abstract: We have developed a new implicit finite difference method to simulate the interaction of intense nanosecond laser beams with semiconductors and metal-coated ceramic structures. This method is based upon a higher order implicit finite difference scheme with a smaller truncation error and is not restricted by any stability criterion, thereby allowing faster convergence to the exact solution. The temperature-dependent optical and thermal properties of the irradiated material as well as the temporal variation in the laser intensity have been taken into account. Finite difference equations have been set up for accurate determination of the temperature gradients at the liquid-solid interface, which control the melt-in and resolidification velocities. A new formulation is introduced to accomodate the effect of pulsed laser irradiation on layered composite structures (e.g. metal-coated ceramics) by incorporating the boundary conditions at the composite interface. Using this method, the thermal histories of laser-irradiated materials were predicted. The effects of variation in the pulse energy density, pulse duration and substrate temperature on the maximum melt depths, solidification velocities and surface temperatures were computed. The calculations on the depth of melting were found to be in good agreement with experimental results where complete annealing of the ion implantation damage was used as a measure of the melt depth. The surface temperatures and melt lifetimes in metal-coated ceramics were determined in order to understand the laser mixing process. Simple energy balance considerations were applied to calculate some of the effects of laser irradiation on materials. From these energy considerations, the maximum melt depths as a function of energy density, pulse duration and substrate temperature were obtained and compared with the exact solutions. The maximum surface temperatures, solidification velocities and melt lifetimes were also determined by this analytical method and compared with the detailed calculations. A good agreement between the analytical relations and the detailed numerical calculations provides an excellent guide to researchers in this field.

Fluid Flow and Mixed Convection Transport From a Moving Plate in Rolling and Extrusion Processes

Abstract: The heat transfer arising due to the movement of a continuous heated plate in processes such as hot rolling and hot extrusion has been studied. Of particular interest were the resulting temperature distribution in the solid and the proper imposition of the boundary conditions at the location where the material emerges from a furnace or die. These considerations are important in the simulation and design of practical systems. A numerical study of the thermal transport process has been carried out, assuming a two-dimensional steady circumstance. The boundary layer equations, as well as full governing equations including buoyancey effects, are solved employing finite difference techniques. The effect of various physical parameters, which determine the temperature and flow fields, is studied in detail. The significance of these results in actual manufacturing processes is discussed.

A Finite Difference Method for Unsteady Flow in Variably Saturated Porous Media: Application to a Single Pumping Well

Abstract: Finite difference equations were derived by using the divergence theorem to convert the nonlinear partial differential equation (which approximately describes liquid flow in a variably saturated, elastic porous medium) to an integral equation, and then to integrate around individual mesh volume elements. Original nonlinearity of the differential equation was preserved by keeping saturations and relative conductivities current with hydraulic heads during the iterative matrix solution method. The problem of axisymmetric flow to a water well that penetrates one or more elastic rock units, the upper one of which is unconfmed, provides a convenient vehicle for analysis of the procedural and theoretical study of unconfined and semiconfined flow. Of the three methods tried to solve the matrix equation that resulted from the finite difference equations (which included a form of the direct alternating direction implicit method, the iterative alternating direction implicit method, and the line successive overrelaxation method), the line successive overrelaxation method was the fastest and was selected for use in a general computer program. A comparison with analytical solutions that use Boulton's convolution integral as a velocity boundary condition at the water table for a single aquifer and an aquifer-aquitard system demonstrates close correspondence of the numerical and analytical solutions, even for a case where the water table is lowered appreciably.