### Frontal Algorithms for Equation-Based Chemical Process
Flowsheeting on Vector and Parallel Computers

by

Stephen E. Zitney and Mark A. Stadtherr

#### ABSTRACT

We present several frontal algorithms for solving the large, sparse,
linear equation systems arising in equation-based (EB) chemical process
flowsheeting. The frontal approach is generally used as a banded matrix
solver in connection with finite element problems. We adapt it here to
the more general process flowsheeting matrix. The motivation is that the
frontal approach exploits vector computer architectures by treating parts
of the sparse matrix as full submatrices, thereby allowing arithmetic operations
to be performed with full-matrix code (without indirect addressing). On
parallel computers, the inner loops of the elimination phase can be multitasked
by a simple subdivision of the frontal matrix.

The flowsheeting results in this study show that the frontal approach
performs very efficiently on the CRAY X-MP computer. High performance rates
are obtained because the dense SAXPY operation in the inner loop can utilize
the hardware chaining feature and three paths-to-memory of the CRAY X-MP
system to keep both vector floating-point pipes busy at all times. Similar
performance is achieved on the one path-to-memory architecture of the CRAY-2
computer by performing several steps of Gaussian elimination together using
multiple-rank updates of the frontal matrix. Results for assembly-language
implementations of rank-two and rank-four updates indicate that they perform
with great effectiveness on the CRAY-2 system. In fact, the multitasked
versions of these kernels can achieve computation rates well over one gigaflop
when used as the nucleus of a frontal solver for EB flowsheeting. Overall,
the frontal codes developed here significantly outperform the usual general
sparse code, LU1SOL, and show considerable promise for the solution of
large EB flowsheeting matrices on advanced computer architectures.

Comput. Chem. Eng., 17, 319-338 (1993)