William Y. Crutchfield

A wideband fast multipole method for the Helmholtz equation in three dimensions

We describe a wideband version of the Fast Multipole Method for the Helmholtz equation in three dimensions. It unifies previously existing versions of the FMM for high and low frequencies into an algorithm which is accurate and efficient for any frequency, having a CPU time of O(N) if low-frequency computations dominate, or O(NlogN) if high-frequency computations dominate. The performance of the algorithm is illustrated with numerical examples.

Parallelization of structured, hierarchical adaptive mesh refinement algorithms

Abstract.We describe an approach to parallelization of structured adaptive mesh refinement algorithms. This type of adaptive methodology is based on the use of local grids superimposed on a coarse grid to achieve sufficient resolution in the solution. The key elements of the approach to parallelization are a dynamic load-balancing technique to distribute work to processors and a software methodology for managing data distribution and communications. The methodology is based on a message-passing model that exploits the coarse-grained parallelism inherent in the algorithms. The approach is illustrated for an adaptive algorithm for hyperbolic systems of conservation laws in three space dimensions. A numerical example computing the interaction of a shock with a helium bubble is presented. We give timings to illustrate the performance of the method.

An Adaptive Projection Method for Unsteady, Low-Mach Number Combustion

We present an adaptive projection method for modeling unsteady, low-Mach reacting flow in an unconfined region. The equations are based on a model for low-Mach number combustion that consists of evolution equations coupled with a constraint on the divergence of the flow. The algorithm is based on a projection methodology in which we first advance the evolution equations and then solve an elliptic equation to enforce the divergence constraint. The adaptive mesh refinement (AMR) scheme uses a time-varying hierarchy of rectangular grids. The integration scheme is a recursive procedure in which coarse grids are advanced, fine grids are advanced to the same time as the coarse grids, and the coarse and fine grid data are then synchronized. The method is currently implemented for laminar, axisymmetric flames with a reduced kinetics mechanism and a Lewis number of unity. Three methane-air flames, two steady and one flickering, are presented as numerical examples.

Approximate Projection Methods: Part I. Inviscid Analysis

The use of approximate projection methods for modeling low Mach number flows avoids many of the numerical complications associated with exact projection methods, but introduces additional design choices in developing a robust algorithm. In this paper we first explore these design choices in the setting of inviscid incompressible flow using several computational examples. We then develop a framework for analyzing the behavior of the different design variations and use that analysis to explain the features observed in the computations. As part of this work we introduce a new variation of the approximate projection algorithm that combines the advantages of several of the previous versions.

Fast, accurate integral equation methods for the analysis of photonic crystal fibers I: Theory

We present a new integral equation method for calculating the electromagnetic modes of photonic crystal fiber (PCF) waveguides. Our formulation can easily handle PCFs with arbitrary hole geometries and irregular hole distributions, enabling optical component manufacturers to optimize hole designs as well as assess the effect of manufacturing defects. The method produces accurate results for both the real and imaginary parts of the propagation constants, which we validated through extensive convergence analysis and by comparison with previously published results.

An adaptive multifluid interface-capturing method for compressible flow in complex geometries

We present a numerical method for solving the multifluid equations of gas dynamics using an operator-split second-order Godunov method for flow in complex geometries in two and three dimensions. The multifluid system treats the fluid components as thermodynamically distinct entities and correctly models fluids with different compressibilities. This treatment allows a general equation-of-state (EOS) specification and the method is implemented so that the EOS references are minimized. The current method is complementary to volume-of-fluid (VOF) methods in the sense that a VOF representation is used, but no interface reconstruction is performed. The Godunov integrator captures the interface during the solution process. The basic multifluid integrator is coupled to a Cartesian grid algorithm that also uses a VOF representation of the fluid-body interface. This representation of the fluid-body interface allows the algorithm to easily accommodate arbitrarily complex geometries. The resulting single grid multifluid-Cartesian grid integration scheme is coupled to a local adaptive mesh refinement algorithm that dynamically refines selected regions of the computational grid to achieve a desired level of accuracy. The overall method is fully conservative with respect to the total mixture. The method will be used for a simple nozzle problem in two-dimensional axisymmetric coordinates.

An Adaptive Semi-Implicit Scheme for Simulations of Unsteady Viscous Compressible Flows

A numerical scheme for simulation of unsteady, viscous, compressible flows is considered. The scheme employs an explicit discretization of the inviscid terms of the Navier-Stokes equations and an implicit discretization of the viscous terms. The discretization is second order accurate in both space and time. Under appropriate assumptions, the implicit system of equations can be decoupled into two linear systems of reduced rank. These are solved efficiently using a Gauss-Seidel method with multigrid convergence acceleration. When coupled with a solution-adaptive mesh refinement technique, the hybrid explicit-implicit scheme provides an effective methodology for accurate simulations of unsteady viscous flows. The methodology is demonstrated for both body-fitted structured grids and for rectangular (Cartesian) grids.

Sensitivity analysis of photonic crystal fiber.

Photonic crystal fibers are well-known to offer a number of unusual properties, including supercontinuum generation, large mode-areas and controllable dispersion behavior. Their manufacturability would be enhanced by a more detailed understanding of how small perturbations in the fibers geometric structure cause variations in the fibers fundamental modes. In this paper, we demonstrate that such sensitivity analysis is feasible using highly accurate boundary integral techniques.

Numerical simulation of a wave-guide mixing layer on a Cray C-90

The development of a three-dimensional spatially evolving compressible mixing layer is investigated numerically using a parallel implementation of Adaptive Mesh Refinement (AMR) on a Cray C-90. The parallel implementation allowed the flow to be highly resolved while significantly reducing the wall-clock runtime. A sustained computation rate of 5.3 Gigaflops including I/O was obtained for a typical production run on a 16 processor machine. A novel mixing layer configuration is investigated where a pressure mismatch is maintained between the two inlet streams. In addition, the sonic character of the two streams is sufficiently different so that the pressure relief wave is trapped in the high speed stream. The trapped wave forces the mixing layer to form a characteristic cellular pattern. The cellular structure introduces curvature into the mixing layer that excites centrifugal instabilities characterized by large-scale counter-rotating vortical pairs embedded within the mixing layer. These are the dominant feature of the flow. Visualizations of these structures in cross-section show the pumping action which lifts dense fluid up into light gas. This effect has a strong impact on mixing enhancement as monitored by a conserved scalar formulation. Once the large-scale structures axe well established in the flow and undergo intensification from favorable velocity gradients, the time-averaged integrated product shows almost a four-fold increase. A spectral analysis of the flow-field over the cellular structures, as part of a full space-time analysis, shows these structures to be zero-frequency modes that develop from low level essentially broad-banded noise. This characterization of the vortical structures and their appearance is consistent with a recent linear stability analysis, of a mixing layer over a curved wall that predicts the most unstable modes to be zero frequency streamwise vortices.