Preconditioned iterative solution of the 2D Helmholtz equation

Using a finite element method to solve the Helmholtz equation leads to a sparse system of equations which in three dimensions is too large to solve directly. It is also non-Hermitian and highly indefinite and consequently difficult to solve iteratively. The approach taken in this paper is to precondition this linear system with a new preconditioner and then solve it iteratively using a Krylov subspace method. Numerical analysis shows the preconditioner to be effective on a simple 1D test problem, and results are presented showing considerable convergence acceleration for a number of different Krylov methods for more complex problems in 2D, as well as for the more general problem of harmonic disturbances to a non-stagnant steady flow

Preconditioning harmonic unsteady potential flow calculations

This paper considers finite element discretisations of the Helmholtz equation and its generalisation arising from harmonic acoustics perturbations to a non-uniform steady potential flow. A novel elliptic, positive definite preconditioner, with a multigrid implementation, is used to accelerate the iterative convergence of Krylov subspace solvers. Both theory and numerical results show that for a model 1D Helmholtz test problem the preconditioner clusters the discrete system's eigenvalues and lowers its condition number to a level independent of grid resolution. For the 2D Helmholtz equation, grid independent convergence is achieved using a QMR Krylov solver, significantly outperforming the popular SSOR preconditioner. Impressive results are also presented on more complex domains, including an axisymmetric aircraft engine inlet with non-stagnant mean flow and modal boundary conditions

Computer program for nonlinear static stress analysis of shuttle thermal protection system: User's manual

User documentation is presented for a computer program which considers the nonlinear properties of the strain isolator pad (SIP) in the static stress analysis of the shuttle thermal protection system. This program is generalized to handle an arbitrary SIP footprint including cutouts for instrumentation and filler bar. Multiple SIP surfaces are defined to model tiles in unique locations such as leading edges, intersections, and penetrations. The nonlinearity of the SIP is characterized by experimental stress displacement data for both normal and shear behavior. Stresses in the SIP are calculated using a Newton iteration procedure to determine the six rigid body displacements of the tile which develop reaction forces in the SIP to equilibrate the externally applied loads. This user documentation gives an overview of the analysis capabilities, a detailed description of required input data and an example to illustrate use of the program

An automated data management/analysis system for space shuttle orbiter tiles

An engineering data management system was combined with a nonlinear stress analysis program to provide a capability for analyzing a large number of tiles on the space shuttle orbiter. Tile geometry data and all data necessary of define the tile loads environment accessed automatically as needed for the analysis of a particular tile or a set of tiles. User documentation provided includes: (1) description of computer programs and data files contained in the system; (2) definitions of all engineering data stored in the data base; (3) characteristics of the tile anaytical model; (4) instructions for preparation of user input; and (5) a sample problem to illustrate use of the system. Description of data, computer programs, and analytical models of the tile are sufficiently detailed to guide extension of the system to include additional zones of tiles and/or additional types of analyse

Use of an engineering data management system in the analysis of space shuttle orbiter tiles

The use of an engineering data management system to facilitate the extensive stress analyses of the space shuttle orbiter thermal protection system is demonstrated. The methods used to gather, organize, and store the data; to query data interactively; to generate graphic displays of the data; and to access, transform, and prepare the data for input to a stress analysis program are described. Information related to many separate tiles can be accessed individually from the data base which has a natural organization from an engineering viewpoint. The flexible user features of the system facilitate changes in data content and organization which occur during the development and refinement of the tile analysis procedure. Additionally, the query language supports retrieval of data to satisfy a variety of user-specified conditions

Design considerations for attaining 200-knot test velocities at the aircraft landing loads and traction facility

Design studies are presented which consider the important parameters in providing 200 knot test velocities at the landing loads and traction facility. Two major components of this facility, the hydraulic jet catapult and the test carriage structure, are considered. Suitable factors are determined to correlate analytical data for characteristics of the hydraulic jet catapult with data measured from the existing catapult system. The resulting equations are used to calculate test velocities for a range of jet nozzle diameters and carriage masses with both the current 122 m and an increased 183 m catapult stroke. Using the catapult characteristics, a target design point is selected and a carriage structure is sized to meet the target point strength requirements

Eigenmode Analysis of Boundary Conditions for the One-Dimensional Preconditioned Euler Equations

The effect of local preconditioning on boundary conditions is analyzed for the subsonic, one-dimensional Euler equations. Decay rates for the eigenmodes of the initial boundary value problem are determined for different boundary conditions and different preconditioners whose intent is to accelerate low Mach number computations. Riemann invariant boundary conditions based on the unpreconditioned Euler equations are shown to be reflective when used with preconditioning, and asymptotically, at low Mach numbers, initial disturbances do not decay. Other boundary conditions are shown to be perfectly non-reflective in conjunction with preconditioning. Two-dimensional numerical results confirm the trends predicted by the one-dimensional analysis

Mechanism of Reconnection on Kinetic Scales Based on Magnetospheric Multiscale Mission Observations

We examine the role that ions and electrons play in reconnection using observations from the Magnetospheric Multiscale (MMS) mission on kinetic ion and electron scales, which are much shorter than magnetohydrodynamic scales. This study reports observations with unprecedented high resolution that MMS provides for magnetic eld (7.8 ms) and plasma (30 ms for electrons and 150 ms for ions). We analyze and compare approaches to the magnetopause in 2016 November, to the electron diffusion region in the magnetotail in 2017 July followed by a current sheet crossing in 2018 July. Besides magnetic eld reversals, changes in the direction of the ow velocity, and ion and electron heating, MMS observed large uctuations in the electron ow speeds in the magnetotail. As expected from numerical simulations, we have veried that when the eld lines and plasma become decoupled a large reconnecting electric eld related to the Hall current (110 mV/m) is responsible for fast reconnection in the ion diffusion region. Although inertial accelerating forces remain moderate (12 mV/m), the electric elds resulting from the divergence of the full electron pressure tensor provide the main contribution to the generalized Ohms law at the neutral sheet (as large as 200 mV/m). In our view, this illustrates that when ions decouple electron physics dominates. The results obtained on kinetic scales may be useful for better understanding the physical mechanisms governing reconnection processes in various magnetized laboratory and space plasmas

Determination of Drag From Three-Dimensional Viscous and Inviscid Flowfield Computations

A momentum balance approach is used to extract the drag from flowfield computations for wings and wing/bodies in subsonic/transonic flight. The drag is decomposed into vorticity, entropy, and enthalpy components which can be related to the established engineering concepts of induced drag, wave and profile drag, and engine power and efficiency. This decomposition of the drag is useful in formulating techniques for accurately evaluating drag using computational fluid dynamics calculations or experimental data. A formulation for reducing the size of the region of the crossflow plane required for calculating the drag is developed using cut-off parameters for viscosity and entropy. This improves the accuracy of the calculations and decreases the computation time required to obtain the drag results. The improved method is applied to a variety of wings, including the M6, W4, and Ml65 wings, Lockheed Wing A, a NACA 0016 wing, and an Elliptic wing. The accuracy of the resulting drag calculations is related to various computational aspects, including grid type (structured or unstructured), grid density, flow regime (subsonic or transonic), boundary conditions, and the level of the governing equations (Euler or Navier-Stokes). The results show that drag prediction to within engineering accuracy is possible using computational fluid dynamics, and that numerical drag optimization of complex aircraft configurations is possibl