246
8. Complex Geometries
In this method, the solution domain is subdivided into triangular elements. The elements are used to describe the variation of the variables. The
computational nodes are located a t their vertices. Any variable q5 is assumed
to vary linearly within the element, i.e. its shape function is:
q 5 = a x + b y + c .
(8.46)
The coefficients a , b and c are determined by fitting the function to the nodal
values at the vertices. They are thus functions of coordinates and variable
values a t the nodes.
The control volumes are formed around each node by joining the centroids
of the elements and midpoints on element edges, as shown in Fig. 8.13. The
conservation equations in integral form are applied to these CVs as described
above for the finite volume method. The surface and volume integrals are
calculated element-wise: for the CV shown in Fig. 8.13, the CV surface consists of 10 sub-faces, and its volume consists of five sub-volumes (from five
elements which contribute to the CV). Since the variation of variables over
an element is prescribed in form of an analytical function, the integrals can
easily be calculated.
The algebraic equation for a CV involves the node P and its immediate
neighbors (N1 to N5 in Fig. 8.13). Even though the grid consists of triangles
only, the number of neighbors varies in general from one CV to another, leading to irregular matrix structure. This restricts the range of solvers which can
be used; conjugate gradient and Gauss-Seidel solvers are usually employed.
This approach was followed - although only in 2D and using second order approximations - by Baliga and Patankar (1983), Schneider and Raw
(1987), Masson et al. (1994), Baliga (1997), and others. Its extension to 3D
is straightforward, but more complicated.
Actually, one does not have to prescribe the shape functions on the elements. Instead, they can be just used to define polyhedral CVs made up of
more than one element. For example, tetrahedral grids are easy to generate,
but often have bad properties near walls; tetrahedra can be merged to create a polyhedral CV around each tetrahedron vertex as described above. If
one adopts the methods of numerical integration, interpolation, and differentiation described earlier, one can lump the sub-faces common to two CVs
to form one cell face, thus reducing the number of faces and the computing
effort in evaluating surface integrals.
This is sometimes called dual-mesh approach. An example is shown in
Fig. 8.13, where the resulting CV can be compared to that obtained from the
method described above. The number of faces of a cell is now equal to the
number of neighbor CVs. If the underlying triangular mesh is highly nonuniform, the lines connecting node P and its neighbors will not pass through
the cell-face center. However, this can be handled using approach described
in Fig. 8.9 and in Eq. (8.36).
8. Complex Geometries
In this method, the solution domain is subdivided into triangular elements. The elements are used to describe the variation of the variables. The
computational nodes are located a t their vertices. Any variable q5 is assumed
to vary linearly within the element, i.e. its shape function is:
q 5 = a x + b y + c .
(8.46)
The coefficients a , b and c are determined by fitting the function to the nodal
values at the vertices. They are thus functions of coordinates and variable
values a t the nodes.
The control volumes are formed around each node by joining the centroids
of the elements and midpoints on element edges, as shown in Fig. 8.13. The
conservation equations in integral form are applied to these CVs as described
above for the finite volume method. The surface and volume integrals are
calculated element-wise: for the CV shown in Fig. 8.13, the CV surface consists of 10 sub-faces, and its volume consists of five sub-volumes (from five
elements which contribute to the CV). Since the variation of variables over
an element is prescribed in form of an analytical function, the integrals can
easily be calculated.
The algebraic equation for a CV involves the node P and its immediate
neighbors (N1 to N5 in Fig. 8.13). Even though the grid consists of triangles
only, the number of neighbors varies in general from one CV to another, leading to irregular matrix structure. This restricts the range of solvers which can
be used; conjugate gradient and Gauss-Seidel solvers are usually employed.
This approach was followed - although only in 2D and using second order approximations - by Baliga and Patankar (1983), Schneider and Raw
(1987), Masson et al. (1994), Baliga (1997), and others. Its extension to 3D
is straightforward, but more complicated.
Actually, one does not have to prescribe the shape functions on the elements. Instead, they can be just used to define polyhedral CVs made up of
more than one element. For example, tetrahedral grids are easy to generate,
but often have bad properties near walls; tetrahedra can be merged to create a polyhedral CV around each tetrahedron vertex as described above. If
one adopts the methods of numerical integration, interpolation, and differentiation described earlier, one can lump the sub-faces common to two CVs
to form one cell face, thus reducing the number of faces and the computing
effort in evaluating surface integrals.
This is sometimes called dual-mesh approach. An example is shown in
Fig. 8.13, where the resulting CV can be compared to that obtained from the
method described above. The number of faces of a cell is now equal to the
number of neighbor CVs. If the underlying triangular mesh is highly nonuniform, the lines connecting node P and its neighbors will not pass through
the cell-face center. However, this can be handled using approach described
in Fig. 8.9 and in Eq. (8.36).