Applied Computational Fluid Dynamics Techniques: An Introduction Based on Finite Element Methods, 2nd Edition

10 EDGE-BASED COMPRESSIBLE FLOW SOLVERS

Consider the typical formation of a RHS using a finite element approximation with shape functions Nⁱ. The resulting integrals to be evaluated are given by

These integrals operate on two sets of data:

• (a) point data, for rⁱ, uⁱ; and
• (b) element data, for volumes, shape functions, etc.

The flow of information is as follows:

1. GATHER point information into the element (e.g. u_i );
2. operate on element data to evaluate the integral in (10.1); and
3. SCATTER-ADD element RHS data to point data to obtain rⁱ.

For many simple flow solvers the effort in step 2 may be minor compared to the cost of indirect addressing operations in steps 1 and 3. A way to reduce the indirect addressing overhead for low-order elements is to change the element-based data structure to an edge-based data structure. This eliminates certain redundancies of information in the element-based data structure. To see this more clearly, consider the formation of the RHS for the Laplacian operator on a typical triangulation. Equation (10.1) may be recast as

This immediately opens three possibilities:

• obtain first the global matrix K^ij and store it in some optimal way (using so-called sparse storage techniques);
• perform a loop over elements, obtaining r_el and adding to r; and
• obtain ...