THESE de DOCTORAT - cerfacs

3.1 Discretizing the Phillips’ equation 43
wardly carried out for tetrahedra. The notation used here is shown in Fig. 3.1. The index e k
stands for a cell element k that is composed by three nodes N i±l . Each face of this triangle is
characterized by its normal vector n. The gradient operator for a quantity ˆp associated to a cell
element e k is discretized following Green’s theorem. It reads
∇ ˆp
⏐ = 1 ∮
ˆp ek ndS (3.2)
ek
V ek ∂e k
where ˆp ek and V ek stands respectively for the pressure and the volume associated to the cell e k .
Let us remind that in the cell-vertex method all quantities are stored in the vertices of the cell.
As a result, ˆp ek must be expressed as an interpolation function g of the neighboring nodes N o .
Hence ˆp ek = g( ˆp o ) where o = ..., i − l i−2 , i − l i−1 , i, i + l i+1 , i + l i+2 , ... is a vector that contains
the neighbor nodes for a given node i and l is a vector that contains the spatial distribution of
matrix elements between the node i and its neighbors 1 . Equation (3.2) is then expressed as:
∇ ˆp
⏐ = 1 ∮
g( ˆp o )ndS = h ek ( ˆp o ) (3.3)
ek
V ek ∂e k
The triangular cell of Fig. 3.1 makes part of a whole set of triangular elements as shown in
Fig. 3.2. Following Green’s theorem, the divergence of ¯c 2 ∇ ˆp for the node N i is discretized as
( ∇ · ¯c 2 ∇ ˆp ) ⏐ ⏐⏐Ni = 1 ∮
¯c 2 N
V i
∇ ˆp
Ni
⏐ · ndS (3.4)
∂N i ek+α
( ∇ · ¯c 2 ∇ ˆp ) ⏐ ⏐ ⏐⏐Ni = 1
V Ni
∮
∂N i
¯c 2 N i
h ek+α ( ˆp o ) · ndS (3.5)
where α is a vector that contains the spatial distribution between the cell e k and its the neighbors.
Note that a volume V Ni is associated to the node i and that this volume is enclosed by surfaces
∂V Ni 2 . This volume V Ni is shown in the shaded region of Fig. 3.2 and ∂N i is represented by the
thick black line enclosing it. Let us consider now what happens when this node N i is placed in
a boundary of the computational domain. Such a case is represented by Fig. 3.3.
The analysis is similar to that of the Eq. (3.5). Nevertheless, the volume of the node N i has been
truncated. The surface enclosing V Ni is still represented by ∂V Ni as before but in the present
case ∂V Ni is splitted in S 1,i and S 2,i , S 2,i being the surface corresponding to the node N i adjacent
1 Let us recall that for unstructured meshes a cell can be composed by nodes that are not numerate consecutivelly
(e.g.: if a node i = 5 has adjacent nodes numbered 9 and 13, then l is 4 and 8 respectively)
2 We talk about a ‘volume’ V Ni being enclosed by ‘surfaces’ ∂V Ni . Of course for the present two dimensional case
the volume and the surface become a surface and an edge respectively.