Finite difference method

Abstract: We introduce numerical methods for simulating the diffusive motion of rigid bodies of arbitrary shape immersed in a viscous fluid. We parameterize the orientation of the bodies using normalized quaternions, which are numerically robust, space efficient, and easy to accumulate. We construct a system of overdamped Langevin equations in the quaternion representation that accounts for hydrodynamic effects, preserves the unit-norm constraint on the quaternion, and is time reversible with respect to the Gibbs-Boltzmann distribution at equilibrium. We introduce two schemes for temporal integration of the overdamped Langevin equations of motion, one based on the Fixman midpoint method and the other based on a random finite difference approach, both of which ensure that the correct stochastic drift term is captured in a computationally efficient way. We study several examples of rigid colloidal particles diffusing near a no-slip boundary and demonstrate the importance of the choice of tracking point on the measured translational mean square displacement (MSD). We examine the average short-time as well as the long-time quasi-two-dimensional diffusion coefficient of a rigid particle sedimented near a bottom wall due to gravity. For several particle shapes, we find a choice of tracking point that makes the MSD essentially linear with time, allowing us to estimate the long-time diffusion coefficient efficiently using a Monte Carlo method. However, in general, such a special choice of tracking point does not exist, and numerical techniques for simulating long trajectories, such as the ones we introduce here, are necessary to study diffusion on long time scales.

Abstract: A numerical method is presented for analyzing the transonic potential flow past a lifting, swept wing. A finite-difference approximation to the full potential equation is solved in a coordinate system which is nearly conformally mapped from the physical space in planes parallel to the symmetry plane, and reduces the wing surface to a portion of one boundary of the computational grid. A coordinate invariant, rotated difference scheme is used, and the difference equations are solved by relaxation. The method is capable of treating wings of arbitrary planform and dihedral, although approximations in treating the tips and vortex sheet make its accuracy suspect for wings of small aspect ratio. Comparisons of calculated results with experimental data are shown for examples of both conventional and supercritical transport wings. Agreement is quite good for both types, but it was found necessary to account for the displacement effect of the boundary layer for the supercritical wing, presumably because of its greater sensitivity to changes in effective geometry.

Abstract: The present study develops a numerical simulation program to predict the transient behavior of fiber orientations together with a mold filling simulation for short-fiber-reinforced thermoplastics in arbitrary three-dimensional injection mold cavities. The Dinh-Armstrong model including an additional stress due to the existence of fibers is incorporated into the Hele-Shaw equation to result in a new pressure equation governing the filling process. The mold filling simulation is performed by solving the new pressure equation and energy equation via a finite element/finite difference method as well as evolution equations for the second-order orientation tensor via the fourth-order Runge-Kutta method. The fiber orientation tensor is determined at every layer of each element across the thickness of molded parts with appropriate tensor transformations for arbitrary three-dimensional cavity space.