We develop an efficient multiple time step ͑MTS͒ force splitting scheme for biological applications in the AMBER program in the context of the particle-mesh Ewald ͑PME͒ algorithm. Our method applies a symmetric Trotter factorization of the Liouville operator based on the position-Verlet scheme to Newtonian and Langevin dynamics. Following a brief review of the MTS and PME algorithms, we discuss performance speedup and the force balancing involved to maximize accuracy, maintain long-time stability, and accelerate computational times. Compared to prior MTS efforts in the context of the AMBER program, advances are possible by optimizing PME parameters for MTS applications and by using the position-Verlet, rather than velocity-Verlet, scheme for the inner loop. Moreover, ideas from the Langevin/MTS algorithm LN are applied to Newtonian formulations here. The algorithm's performance is optimized and tested on water, solvated DNA, and solvated protein systems. We find CPU speedup ratios of over 3 for Newtonian formulations when compared to a 1 fs single-step Verlet algorithm using outer time steps of 6 fs in a three-class splitting scheme; accurate conservation of energies is demonstrated over simulations of length several hundred ps. With modest Langevin forces, we obtain stable trajectories for outer time steps up to 12 fs and corresponding speedup ratios approaching 5. We end by suggesting that modified Ewald formulations, using tailored alternatives to the Gaussian screening functions for the Coulombic terms, may allow larger time steps and thus further speedups for both Newtonian and Langevin protocols; such developments are reported separately.

We present a bifurcation study of the incompressible Navier-Stokes equations in a model complex geometry: a spatially periodic array of cylinders in a channel. The dynamics of the flow include a Hopf bifurcation from steady to oscillatory flow at an approximate Reynolds number R of 350 and the appearance of a second frequency at approximately R 890. The multiple frequency dynamics include a substantial increase in spatial and temporal scales with Reynolds number as compared with the simple limit cycle oscillation present close to R = 350. Numerical bifurcation studies of the dynamics are performed using three forms of global eigenfunction expansions. The first basis set is derived through principal factor analysis (Karhunen-Loève expansion) of snapshots from accurate direct spectral element numerical solutions of the Navier-Stokes equations. The second set is obtained from the eigenfunctions of the Stokes operator for this geometry. Finally eigenfunctions are derived from a singular Stokes operator, i.e., the Stokes operator modified to include a variable coefficient which vanishes at the domain boundaries. Truncated systems of (∼ 100) ODEs are obtained through projection of the Navier-Stokes equations onto the basis sets, and a comparative study of the resulting dynamical models is performed.

A measurement program currently underway at Arvin/Calspan Advanced Technology Center has been used in the evaluation of observed engine behavior during dust ingestion. The Pratt and Whitney TF33 turbofan and J57 turbojet were used in the investigation. Solid particle ingestion was found to erode the compressor blades and result in substantial performance deterioration. The engines were found to have increased susceptibility to surge at low power settings. The roles that anti-ice and intercompressor bleed airplay in surge avoidance are discussed. A discussion of the fuel controller behavior in a deteriorated engine and its effect during steady-state engine operation is also presented. Experimental data obtained during testing were compared to a predictive capability developed to describe deteriorated engine response. The effects of tip clearance, blade profile, and secondary flows were taken into account. The results show good agreement with experimentally observed engine behavior.