Numerical Modeling of Casting Solidification Using Generalized Finite Difference Method

2010 ◽  
Vol 638-642 ◽  
pp. 2676-2681 ◽  
Author(s):  
Bohdan Mochnacki ◽  
Ewa Majchrzak
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Energy Equation ◽  
Solidification Process ◽  
Temperature Gradients ◽  
Thermal Processes ◽  
Heating Rates ◽  
Collocation Points ◽  
The One

The system casting-mould is considered. The thermal processes proceeding in a casting sub-domain are described using the one domain approach. The model of solidification process is supplemented by the energy equation concerning the mould sub-domain, the continuity conditions given on the contact surface between casting and mould, boundary conditions on the outer surface of the system and the initial ones. To solve the problem the generalized variant of finite difference method (GFDM) is used. Temporary and local values of temperature can be found at the optional set of collocation points from the domain considered. This essential advantage of GFDM allows to locate and thicken nodes at the regions for which the temperature gradients and cooling (heating) rates are considerable. In the final part of the paper, the example of numerical simulation is shown.

scholarly journals MÉTODO DE DIFERENCIAS FINITAS PARA UN PROBLEMA DE VALOR DE FRONTERA UNIDIMENSIONAL

2019 ◽  
pp. 5-8
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Value Problem ◽  
Difference Method ◽  
Sparse Matrices ◽  
Boundary Value ◽  
Bending Moment ◽  
One Dimensional ◽  
Dimensional Boundary ◽  
The One

MÉTODO DE DIFERENCIAS FINITAS PARA UN PROBLEMA DE VALOR DE FRONTERA UNIDIMENSIONAL THE FINITE- DIFERENCE METHOD FOR A ONE-DIMENSIONAL BOUNDARY-VALUE PROBLEM Luis Jaime Collantes Santisteban, Samuel Collantes Santisteban DOI: https://doi.org/10.33017/RevECIPeru2006.0011/ RESUMEN En este trabajo se considera el problema de valor de frontera unidimensional dado en (1). Se aproxima la solución del problema mediante el método de diferencias finitas suponiendo que la función c(x) es no negativa sobre 0,1, lo que permite establecer la convergencia del método de aproximación. El uso del método de diferencias finitas, a la vez, involucra la solución de sistemas de ecuaciones lineales con matrices muy ralas, cuyos ceros están posicionados de una manera remarcable. Dichas matrices son de tipo tridiagonal. Para la solución de dichos sistemas se ha utilizado el método de Thomas. Palabras clave: problema de valor de frontera unidimensional, diferencias finitas, matriz tridiagonal, método de Thomas, momento flexionante. ABSTRACT In this work the one-dimensional boundary-value problem given in (1) is considered. The solution of the problem by means of finite-difference method comes near supposing that the function c(x) is nonnegative on 0,1, which allows to establish the convergence of the considered method of approximation. The use of the finite-difference method, in turn, involves the solution of linear systems with very sparse‟ matrices, whose zeros are arranged in quite remarkable fashion. These matrices are of tridiagonal type. For the solution of these systems the Thomas‟ method has been used. Keywords: one-dimensional boundary-value problem, finite-difference, tridiagonal matrix, Thomas‟ method, bending moment.

A Finite-Difference Method for Calculating Compressible Laminar and Turbulent Boundary Layers

1970 ◽  
Vol 92 (3) ◽  
pp. 523-535 ◽  
Author(s):  
T. Cebeci ◽  
A. M. O. Smith
Keyword(s):  
Boundary Layer ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Turbulent Boundary Layer ◽  
Turbulent Flows ◽  
Difference Method ◽  
Eddy Viscosity ◽  
Energy Equation ◽  
Composite Layer ◽  
Computation Time

This paper presents a finite-difference method for solving laminar and turbulent-boundary-layer equations for incompressible and compressible flows about two-dimensional and axisymmetric bodies and contains a thorough evaluation of its accuracy and computation-time characteristics. The Reynolds shear-stress term is eliminated by an eddy-viscosity concept, and the time mean of the product of fluctuating velocity and temperature appearing in the energy equation is eliminated by an eddy-conductivity concept. The turbulent boundary layer is regarded as a composite layer consisting of inner and outer regions, for which separate expressions for eddy viscosity are used. The eddy-conductivity term is lumped into a “turbulent” Prandtl number that is currently assumed to be constant. The method has been programed on the IBM 360/65, and its accuracy has been investigated for a large number of flows by comparing the computed solutions with the solutions obtained by analytical methods, as well as with solutions obtained by other numerical methods. On the basis of these comparisons, it can be said that the present method is quite accurate and satisfactory for most laminar and turbulent flows. The computation time is also quite small. In general, a typical flow, either laminar or turbulent, consists of about twenty x-stations. The computation time per station is about one second for an incompressible laminar flow and about two to three seconds for an incompressible turbulent flow on the IBM 360/65. Solution of energy equation in either laminar or turbulent flows increases the computation time about one second per station.

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