The non-oscillatory shape functions [1] is written as follows:
(1)
Where 
is Element Degree Limiter, Field Variable Limiter (EDL, FVL), 
is p-order shape functions (in this paper degree of p is 3) and 
, 
and 
are reference functions I used equation (40) of [1] as reference function that is
(2)
Fig. 1. 1D advective equation 
Fig. 2. 1D convection-diffusion equation 
Fig. 3. Grid for flow over a NACA 0012 airfoil
Fig. 4. Pressure distribution on surface of NACA 0012 airfoil, steady-state
Fig. 5. Grid for flow around a cylinder.
Fig. 6. Flow around a cylinder, steady-state
Where 
and 
are liner shape functions, by putting equation (2) in (1) and also 
we have
(3)
Equation (3) is non-oscillatory shape functions that will be used in this paper for CFD problems.
Examples
In this section, I am going to use the equation (3) as shape function for approximating a few equation and compare its result with non-constant 
. As first example, I approximated 1D advective equation by Point Collocation weight function, see figure (1). In second example, I approximated 1D convection-diffusion equation in 
by Galerkin weight function, see figure (2). In third example, I approximated 2D Euler equations for pressure distribution on surface of NACA 0012 airfoil [1] by Galerkin weight function and infinite elements for non-solid boundaries, see figure (4), and flow around a cylinder [1] by Subdomain Collocation weight function see figure (6).
Conclusions
As can be seen from the results, with any number of elements solutions are non-oscillatory, and when we use fine mesh (usually in CFD is used) solutions are very close together. So, using of constant EDL, FVL is useful.
Библиографическая ссылка
Mohammad Reza Akhavan Khaleghi A VERY INEXPENSIVE SCHEME ON RFEM TO USE IN CFD AND OTHER PROBLEMS // European Journal of Natural History. – 2015. – № 4. – С. 3-5;URL: https://world-science.ru/ru/article/view?id=33456 (дата обращения: 25.11.2024).