The Geometry of Niggli Reduction I: The Boundary Polytopes of the Niggli Cone

Correct identification of the Bravais lattice of a crystal is an important step in structure solution. Niggli reduction is a commonly used technique. We investigate the boundary polytopes of the Niggli-reduced cone in the six-dimensional space G6 by algebraic analysis and organized random probing of regions near 1- through 8-fold boundary polytope intersections. We limit consideration of boundary polytopes to those avoiding the mathematically interesting but crystallographically impossible cases of 0 length cell edges. Combinations of boundary polytopes without a valid intersection in the closure of the Niggli cone or with an intersection that would force a cell edge to 0 or without neighboring probe points are eliminated. 216 boundary polytopes are found: 15 5-D boundary polytopes of the full G6 Niggli cone, 53 4-D boundary polytopes resulting from intersections of pairs of the 15 5-D boundary polytopes, 79 3-D boundary polytopes resulting from 2-fold, 3-fold and 4-fold intersections of the 15 5-D boundary polytopes, 55 2-D boundary polytopes resulting from 2-fold, 3-fold, 4-fold and higher intersections of the 15 5-D boundary polytopes, 14 1-D boundary polytopes resulting from 3-fold and higher intersections of the 15 5-D boundary polytopes. All primitive lattice types can be represented as combinations of the 15 5-D boundary polytopes. All non-primitive lattice types can be represented as combinations of the 15 5-D boundary polytopes and of the 7 special-position subspaces of the 5-D boundary polytopes. This study provides a new, simpler and arguably more intuitive basis set for the classification of lattice characters and helps to illuminate some of the complexities in Bravais lattice identification. The classification is intended to help in organizing database searches and in understanding which lattice symmetries are "close" to a given experimentally determined cell

The Geometry of Niggli Reduction II: BGAOL -- Embedding Niggli Reduction

Niggli reduction can be viewed as a series of operations in a six-dimensional space derived from the metric tensor. An implicit embedding of the space of Niggli-reduced cells in a higher dimensional space to facilitate calculation of distances between cells is described. This distance metric is used to create a program, BGAOL, for Bravais lattice determination. Results from BGAOL are compared to the results from other metric-based Bravais lattice determination algorithms

Experience with advanced instrumentation in a hot section cascade

The Lewis Research Center gas turbine Hot Section Test Facility was developed to provide a real engine environment with known boundary conditions for the aerothermal performance evaluation and verification of computer design codes. This verification process requires experimental measurements in a hostile environment. The research instruments used in this facility are presented, and their characteristics and how they perform in this environment are discussed. The research instrumentation consisted of conventional pressure and temperature sensors, as well as thin-film thermocouples and heat flux gages. The hot gas temperature was measured by an aspirated temperature probe and by a dual-element, fast-response temperature probe. The data acquisition mode was both steady state and time dependent. These experiments were conducted over a wide range of gas Reynolds numbers, exit gas Mach numbers, and heat flux levels. This facility was capable of testing at temperatures up to 1600 K, and at pressures up to 18 atm. These corresponded to an airfoil exit Reynolds number range of 0.5 x 10(6) to 2.5 x 10(6) based on the airfoil chord of 5.55 cm. The results characterize the performance capability and the durability of the instrumentation. The challenge of making measurements in hostile environments is also discussed. The instruments exhibited more than adequate durability to achieve the measurement profile. About 70 percent of the thin-film thermocouples and the dual-element temperature probe survived several hundred thermal cycles and more than 35 hr at gas temperatures up to 1600 K. Within the experimental uncertainty, the steady-state and transient heat flux measurements were comparable and consistent over the range of Reynolds numbers tested

Converting three-space matrices to equivalent six-space matrices for Delone scalars in S6.

The transformations from the primitive cells of the centered Bravais lattices to the corresponding centered cells have conventionally been listed as three-by-three matrices that transform three-space lattice vectors. Using those three-by-three matrices when working in the six-dimensional space of lattices represented as Selling scalars as used in Delone (Delaunay) reduction, one could transform to the three-space representation, apply the three-by-three matrices and then back-transform to the six-space representation, but it is much simpler to have the equivalent six-by-six matrices and apply them directly. The general form of the transformation from the three-space matrix to the corresponding matrix operating on Selling scalars (expressed in space S6) is derived, and the particular S6matrices for the centered Delone types are listed. (Note: in his later publications, Boris Delaunay used the Russian version of his surname, Delone.)

Heat transfer in aerospace propulsion

Presented is an overview of heat transfer related research in support of aerospace propulsion, particularly as seen from the perspective of the NASA Lewis Research Center. Aerospace propulsion is defined to cover the full spectrum from conventional aircraft power plants through the Aerospace Plane to space propulsion. The conventional subsonic/supersonic aircraft arena, whether commercial or military, relies on the turbine engine. A key characteristic of turbine engines is that they involve fundamentally unsteady flows which must be properly treated. Space propulsion is characterized by very demanding performance requirements which frequently push systems to their limits and demand tailored designs. The hypersonic flight propulsion systems are subject to severe heat loads and the engine and airframe are truly one entity. The impact of the special demands of each of these aerospace propulsion systems on heat transfer is explored

Dynamic programming algorithm for the vehicle routing problem with time windows and EC social legislation

In practice, apart from the problem of vehicle routing, schedulers also face the problem of nding feasible driver schedules complying with complex restrictions on drivers' driving and working hours. To address this complex interdependent problem of vehicle routing and break scheduling, we propose a dynamic programming approach for the vehicle routing problem with time windows including the EC social legislation on drivers' driving and working hours. Our algorithm includes all optional rules in these legislations, which are generally ignored in the literature. To include the legislation in the dynamic programming algorithm we propose a break scheduling method that does not increase the time-complexity of the algorithm. This is a remarkable eect that generally does not hold for local search methods, which have proved to be very successful in solving less restricted vehicle routing problems. Computational results show that our method finds solutions to benchmark instances with 18% less vehicles and 5% less travel distance than state of the art approaches. Furthermore, they show that including all optional rules of the legislation leads to an additional reduction of 4% in the number of vehicles and of 1.5%\ud regarding the travel distance. Therefore, the optional rules should be exploited in practice

The Ultrarelativistic Kerr-Geometry and its Energy-Momentum Tensor

The ultrarelativistic limit of the Schwarzschild and the Kerr-geometry together with their respective energy-momentum tensors is derived. The approach is based on tensor-distributions making use of the underlying Kerr-Schild structure, which remains stable under the ultrarelativistic boost.Comment: 16 pages, (AMS-LaTeX), TUW-94-0