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:
- GATHER point information into the element (e.g. ui );
- operate on element data to evaluate the integral in (10.1); and
- 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 Kij and store it in some optimal way (using so-called sparse storage techniques);
- perform a loop over elements, obtaining rel and adding to r; and
- obtain ...
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