Computer simulation algorithms, vectorization

The vectorization of computer simulation algorithms for new generations of supercomputers is described by Heyes and Fincham in Vol. 63 of this series. Algorithm descriptions are avaUable in the Quarterly Review of the... 

Tables S.l and 5.2 list the computational requirements for the new dynamic simulation algorithm, using the most efficient algorithms known for each calculation for different values of N. The computations are tabulated in toms of the matrix and vector quantities which are found in the first three stq>s of the algorithm. The requited scalar opoations (multiplications, additions) are given for an AT-link, serial manipulator with simple revolute and prismatic joints only. The efficient matrix transformations and oth simplifications described in Chapter 3 have been applied in each stq>, and the computations necessary to determine the individual link transformation matrices have also been included.

In the develc ment of the simulation algorithm in the previous section, the objective was to decouple the simple closed-chain mechanism by computing the spatial force vectors exerted by the chains on the reference membo. The spatial tip forces computed in that algorithm are real, measurable forces, associated with the general jdnts which connect the reference membo and each chain tip. Once these forces are known, the chains are effectively decoupled from the refnence member, and the general joint accelerations may be computed for each chain separately. 

Since this approach maps all possible interactions to processors, no communication is required during force calculation. Moreover, the row assignments are completed before the first step of the simulation. The computation of the bounds for each processor require O(P ) time, which is very negligible compared to N (for N S> P). The communication required at the end of each step to update the position and velocity vectors is done by reducing force vectors of length N, and therefore scales as 0 N) per node per time step. Thus the overall complexity of this algorithm is. 

Vector and Parallel Algorithms for the Molecular Dynamics Simulation of Macromolecules on Shared-Memory Computers. 

F. Mueller-Plathe and D. Brown, Comput. Phys. Commun., 64, 7 (1991). Multi-Color Algorithms m Molecular Simulation Vectorization and Parallelization of Internal Forces and Constraints. 

Salmi (25) set up equations needed to simulate the transient response of both the PFR and the CSTR. The balance equations and the generahzed equations for the rates of the elementary steps are compactly expressed in vector and matrix notation. Details of the computational algorithms are discussed, and they are applied to the N2O decomposition (Eqs. 5 and 6). In another paper (26) these equations are used to simulate (for both PFRs and CSTRs) the responses of sysfems following many mechanisms Eley-Rideal, Langmuir-Hinshelwood. a combination of the two. with and without dissociative adsorption, etc. These curves can be added to those of Kobayashi (22), to expand the general view of how various systems respond. 

The Ewald summation is computationally quite expensive to implement. Under conditions of constant a (which will give the same density of reciprocal vectors, k) then it scales as the square of the number of particles in the central simulation cell. If a is allowed to vary then the algorithm can be made to scale as though the consequent value of a might make the... 

With these newly developed 0(N) TBMD algorithms, simulations with more than 1000 atoms can be performed on sequential computers such as the IBM RISC-6000 workstation or vector computers such as the Cray. Qiu et al. have applied the 0 N) method to study the structure and energetics of giant fullerenes. Shown in Fig. 26 is the optimized geometry of a 1620-atom fullerene obtained using an IBM RISC-6000 workstation. Mauri and Galli have applied the 0 N) TBMD to study the structure and dynamic of C q striking a diamond surface [131] 1140 carbon atoms have been used in their simulation. 

J. E. Mertz, D. J. Tobias, C. L. Brooks 111, and U. C. Singh, J. Comput. Chem., 12, 1270 (1991). Vector and Parallel Algorithms for the Molecular Dynamics Simulation of Macromolecules on Shared-Memory Computers. 

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