Frontal algorithms for equation-based chemical process flowsheeting on vector and parallel computers

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 flowsheeting 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 simply subdividing the frontal matrix. Flowsheeting results show that the frontal approach performs very well on the CRAY X-MP/416 architecture. However, several pivot steps have to be overlapped on the CRAY-2 computer to ensure that both vector floating-point pipes are kept busy. Results concerning the use of multiple pivot steps indicate that two new assembly-language kernels designed for the CRAY-2 computer prove to be very efficient for performing rank-two and rank-four updates. In fact, the multitasked implementations of these kernels can achieve computational rates well over 1 Gigaflop when used as the nucleus of a frontal solver for EB flowsheeting. Overall, the frontal codes perform very well in comparison to the general sparse code, LU1SOL, and shows considerable promise for the solution of EB flowsheeting matrices on supercomputers.The solution of large, sparse linear equation systems is also a critical phase of the simultaneous solution of interlinked distillation columns. If the modeling equations for a distillation system are grouped by plate, the linear systems take on an almost-block-tridiagonal structure. We apply here the frontal approach to solve these almost-banded linear systems on vector computers. Results for nine interlinked systems show that the frontal approach is up to an order of magnitude faster than the general sparse code, LU1SOL, on the CRAY X-MP computer and is between one and two orders of magnitude faster than LU1SOL on the CDC Cyber 175.U of I OnlyETDs are only available to UIUC Users without author permissio