Selection and sequencing heuristics to reduce variance in gas turbine engine nozzle assemblies

During the maintenance of gas turbine engines, the selection and placement of nozzle guide vanes about the circumference of a turbine nozzle greatly affects the efficiency and reliable operation of these engines. A number of research studies have addressed these selection and sequencing problems by essentially solving each problem independent of the other. The selection problem considers the maximization of the number of assembled nozzles that are feasible with respect to exit area requirements. Whereas, the sequencing problem considers the assignment of vanes, such that the variation in the throat area between adjacent vanes is minimized. However, the ability to obtain a sequence of vanes with a reasonably low variation in adjacent throat areas can, and often does, depend largely on the vanes that have been selected. Consequently, we propose and develop a selection methodology that not only accounts for feasibility with respect to exit area constraints, but also for the impact on the variation of the resulting sequence of vanes. The proposed methods are empirically compared to those of previous studies. These comparisons show that orders of magnitude reductions in expected engine operating costs are very likely.     

Measure theoretical approach for optimal shape design of a nozzle

In this paper we present a new method for designing a nozzle. In fact the problem is to find the optimal domain for the solution of a linear or nonlinear boundary value PDE, where the boundary condition is defined over an unspecified domain. By an embedding process, the problem is first transformed to a new shape-measure problem, and then this new problem is replaced by another in which we seek to minimize a linear form over a subset of linear equalities. This minimization is global, and the theory allows us to develop a computational method to find the solution by a finite-dimensional linear programming problem.     

Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle

In this paper, we study the stability of supersonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle of varying cross-sections. We formulate the problem as an initial–boundary value problem with the contact discontinuity as a free boundary. To deal with the free boundary value problem, we employ the Lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem. We develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain. Finally, by using the inverse Lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state, we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity.     

Global entropy solutions to the compressible Euler equations in the isentropic nozzle flow for large data: Application of the generalized invariant regions and the modified Godunov scheme

We study the motion of isentropic gas in nozzles. This is a major subject in fluid dynamics. In fact, the nozzle is utilized to increase the thrust of rocket engines. Moreover, the nozzle flow is closely related to astrophysics. These phenomena are governed by the compressible Euler equations, which are one of crucial equations in inhomogeneous conservation laws.In this paper, we consider its unsteady flow and devote to proving the global existence and stability of solutions to the Cauchy problem for the general nozzle. The theorem has been proved in Tsuge (2013). However, this result is limited to small data. Our aim in the present paper is to remove this restriction, that is, we consider large data. Although the subject is important in Mathematics, Physics and engineering, it remained open for a long time. The problem seems to rely on a bounded estimate of approximate solutions, because we have only method to investigate the behavior with respect to the time variable. To solve this, we first introduce a generalized invariant region. Compared with the existing ones, its upper and lower bounds are extended constants to functions of the space variable. However, we cannot apply the new invariant region to the traditional difference method. Therefore, we invent the modified Godunov scheme. The approximate solutions consist of some functions corresponding to the upper and lower bounds of the invariant regions. These methods enable us to investigate the behavior of approximate solutions with respect to the space variable. The ideas are also applicable to other nonlinear problems involving similar difficulties.     

Transonic shock wave in an infinite nozzle asymptotically converging to a cylinder

We construct a single transonic shock wave pattern in an infinite nozzle asymptotically converging to a cylinder, which is close to a uniform transonic shock wave. In other words, suppose there is a uniform transonic shock wave in an infinite cylinder nozzle which can be constructed easily, if we perturbed the supersonic incoming flow and the infinite nozzle a little bit, we can obtain a transonic wave near the uniform one. As a consequence, we can show that the uniform transonic wave is stable with respect to the perturbation of the incoming flow and nozzle wall. Based on the theory of [G.Q. Chen, M. Feldman, Existence and stability of multi-dimensional transonic flows through an infinite nozzle of arbitrary cross-sections, Arch. Ration. Mech. Anal. 184 (2007) 185-242], the crucial parts of this paper are to derive the uniform Schauder estimates of the linear elliptic equation for the infinite nozzle asymptotically converging to a cylinder.     

Inviscid and viscous stationary waves of gas flow through contracting-expanding nozzles

In this work we study steady states of one-dimensional viscous isentropic compressible flows through a contracting-expanding nozzle. Treating the viscosity coefficient as a singular parameter, the steady-state problem can be viewed as a singularly perturbed system. For a contracting-expanding nozzle, a complete classification of steady states is given and the existence of viscous profiles is established via the geometric singular perturbation theory. Particularly interesting is the existence of a maximal sub-to-super transonic wave and its role in the formation of other complicated transonic waves consisting of a sub-to-super portion.     

Transonic nozzle flows and free boundary problems for the full Euler equations

We establish the existence and uniqueness of transonic flows with a transonic shock through a two-dimensional nozzle of slowly varying cross-sections. The transonic flow is governed by the steady, full Euler equations. Given an incoming smooth flow that is close to a constant supersonic state (i.e., smooth Cauchy data) at the entrance and the subsonic condition with nearly horizontal velocity at the exit of the nozzle, we prove that there exists a transonic flow whose downstream smooth subsonic region is separated by a smooth transonic shock from the upstream supersonic flow. This problem is approached by a one-phase free boundary problem in which the transonic shock is formulated as a free boundary. The full Euler equations are decomposed into an elliptic equation and a system of transport equations for the free boundary problem. An iteration scheme is developed and its fixed point is shown to exist, which is a solution of the free boundary problem, by combining some delicate estimates for the elliptic equation and the system of transport equations with the Schauder fixed point argument. The uniqueness of transonic nozzle flows is also established by employing the coordinate transformation of Euler-Lagrange type and detailed estimates of the solutions.     

Clinic scheduling models with overbooking for patients with heterogeneous no-show probabilities

Clinical overbooking is intended to reduce the negative impact of patient no-shows on clinic operations and performance. In this paper, we study the clinical scheduling problem with overbooking for heterogeneous patients, i.e. patients who have different no-show probabilities. We consider the objective of maximizing expected profit, which includes revenue from patients and costs associated with patient waiting times and physician overtime. We show that the objective function with homogeneous patients, i.e. patients with the same no-show probability, is multimodular. We also show that this property does not hold when patients are heterogeneous. We identify properties of an optimal schedule with heterogeneous patients and propose a local search algorithm to find local optimal schedules. Then, we extend our results to sequential scheduling and propose two sequential scheduling procedures. Finally, we perform a set of numerical experiments and provide managerial insights for health care practitioners.     

A meta-heuristic algorithm for heterogeneous fleet vehicle routing problems with two-dimensional loading constraints

The two-dimensional loading heterogeneous fleet vehicle routing problem (2L-HFVRP) is a variant of the classical vehicle routing problem in which customers are served by a heterogeneous fleet of vehicles. These vehicles have different capacities, fixed and variable operating costs, length and width in dimension, and two-dimensional loading constraints. The objective of this problem is to minimize transportation cost of designed routes, according to which vehicles are used, to satisfy the customer demand. In this study, we proposed a simulated annealing with heuristic local search (SA_HLS) to solve the problem and the search was then extended with a collection of packing heuristics to solve the loading constraints in 2L-HFVRP. To speed up the search process, a data structure was used to record the information related to loading feasibility. The effectiveness of SA_HLS was tested on benchmark instances derived from the two-dimensional loading vehicle routing problem (2L-CVRP). In addition, the performance of SA_HLS was also compared with three other 2L-CVRP models and four HFVRP methods found in the literature.     

A linear approximation for the regular reflection of a weak shock at a wedge satisfying sonic condition

The problem of shock reflection by a wedge, which the flow is dominated by the unsteady potential flow equation, is a important problem. In weak regular reflection, the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream. The reflected shock is transonic. Its position is a free boundary for the unsteady potential equation, which is degenerate at the sonic line in self-similar coordinates. Applying the special partial hodograph transformation used in [Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle I, 2-D case, Comm. Pure Appl. Math. 57 (2004) 1-51; Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle II, 3-D case, IMS, preprint (2003)], we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain. When the angle, which between incident shock and wedge, is small, we can see that weak regular reflection as the disturbance of normal reflection as in [Shuxing Chen, Linear approximation of shock reflection at a wedge with large angle, Comm. Partial Differential Equations 21 (78) (1996) 1103-1118]. By linearizing the resulted nonlinear equation and boundary conditions with above viewpoint, we obtain a linear degenerate elliptic equation with mixed boundary conditions and a linear degenerate elliptic equation with oblique boundary conditions in a curved quadrilateral domain. By means of elliptic regularization techniques, delicate a priori estimate and compact arguments, we show that the solution of linearized problem with oblique boundary conditions is smooth in the interior and Lipschitz continuous up to the degenerate boundary.     

Optimal Hölder Continuity for a Class of Degenerate Elliptic Problems with an Application to Subsonic-Sonic Flows

In this paper, we first study a class of elliptic equations with anisotropic boundary degeneracy. Besides establishing the existence, uniqueness and comparison principle, we obtain the optimal Hölder estimates for weak solutions by the estimates in the Campanato space. Based on such Hölder estimates, we then investigate subsonic-sonic flows with singularities at the sonic curves in a symmetric convergent nozzle with straight wall for an approximate model of the potential flow equation. It is proved that the perturbation problem of the symmetric subsonic-sonic flow is solvable and the symmetric subsonic-sonic flow is stable.     

Continuous and discontinuous solutions for optimum thrust nozzles of given length

In its first sections, the paper deals with optimum thrust nozzles of given length and exit radius for flows with swirl. The computation is based on a modification of methods familiar for flows without swirl. Rather extensive numerical results show that the swirl does not impair the specific impulse attainable at a given nozzle length. The analysis suggests that the assumption of isentropic continuous flows, on which this approach is based, may sometimes be too restrictive. A survey of plane nozzles shows, on the other hand, that discontinuities need to be admitted only if, besides the length, a rather large radius of the nozzle is prescribed. Discontinuous solutions have been thoroughly investigated by Shmyglevskiy. At least in principle, we use the same line of thought, but considerable simplifications are possible if one starts with the variational formulation of Rao. In its numerical discussion and also in some analytical details, the present paper goes beyond Shmyglevskiy's results. The problem is conveniently discussed in astate plane, which has the local state of the flow (flow direction and speed or Mach number) as independent variables. By taking into account second variations, one can determine the boundary of the region for which continuous solutions give the (local) maximum. This boundary coincides with the locus of points at which the solution in the physical plane would fold back into itself. Another limitation of the original approach emerges if one asks under which conditions the thrust can be increased by admitting along the control surface values of the entropy that are higher than those of the oncoming flow. The conditions for isentropic and nonisentropic jumps are formulated and evaluated next, and a survey of the discontinuities which satisfy conditions for isentropic and also for selected nonisentropic jumps is given. Up to this point, the analysis is concerned only with the state distribution along the control surface. Jumps of the state in the interior require the occurrence of centered compression waves. Sample computations show that, in most cases, flow fields of this character can be generated by the choice of the nozzle shape. In some cases, no nozzle contours exist which generate the optimizing state distribution along the control surface as determined by the present analysis. It would then be necessary to include from the very beginning conditions for the realizability of the flow field.     

Iterative Denoising

One problem in many fields is knowledge discovery in heterogeneous, high-dimensional data. As an example, in text mining an analyst often wishes to identify meaningful, implicit, and previously unknown information in an unstructured corpus. Lack of metadata and the complexities of document space make this task difficult. We describe Iterative Denoising, a methodology for knowledge discovery in large heterogeneous datasets that allows a user to visualize and to discover potentially meaningful relationships and structures. In addition, we demonstrate the features of this methodology in the analysis of a heterogeneous Science News corpus.     

Continuous Subsonic-Sonic Flows in a Convergent Nozzle

This paper concerns continuous subsonic-sonic potential flows in a two-dimensional convergent nozzle. It is shown that for a given nozzle which is a perturbation of a straight one, a given point on its wall where the curvature is zero, and a given inlet which is a perturbation of an arc centered at the vertex, there exists uniquely a continuous subsonic-sonic flow whose velocity vector is along the normal direction at the inlet and the sonic curve, which satisfies the slip conditions on the nozzle walls and whose sonic curve intersects the upper wall at the given point. Furthermore, the sonic curve of this flow is a free boundary, where the flow is singular in the sense that the speed is only C^1/2 Hölder continuous and the acceleration blows up. The perturbation problem is solved in the potential plane, where the flow is governed by a free boundary problem of a degenerate elliptic equation with two free boundaries and two nonlocal boundary conditions, and the equation is degenerate at one free boundary.     

Tuning distributed control algorithms for optimal functioning

In this paper, we present a model which characterizes distributed computing algorithms. The goals of this model are to offer an abstract representation of asynchronous and heterogeneous distributed systems, to present a mechanism for specifying externally observable behaviours of distributed processes and to provide rules for combining these processes into networks with desired properties (good functioning, fairness...). Once these good properties are found, the determination of the optimal rules are studied.Subsequently, the model is applied to three classical distributed computing problems: namely the dining philosophers problem, the mutual exclusion problem and the deadlock problem, (generalizing results of our previous publications [1], [2]). The property of fairness has a special position that we discuss.     

A complementary tool to enhance the effectiveness of existing methods for heterogeneous fixed fleet vehicle routing problem

The heterogeneous fixed fleet vehicle routing problem (HFFVRP) is a variant of the standard vehicle routing problem (VRP), in which the vertices have to be served using a fixed number of vehicles that could be different in size and fixed or variable costs. In this article, we propose an integer linear programming-based heuristic approach in order to solve the HFFVRP that could be used as a complementary tool to improve the performance of the existing methods of solving this problem. Computational results show the effectiveness of the proposed method.     

Radial profiles of deposition nozzles for high uniformity

An important design problem is concernedwith uniform wetting of plane areas using arrays of spraying nozzles. Good uniformity requires that the radial profile of deposition from the nozzles, as well as the array spacing, be carefully chosen. The purpose of this paper is to present a technique for computing nozzle profiles which result in high uniformity when used in particular array configurations. The method is based on a spline model for the nozzle, with coefficients determined by minimizing the absolute deviation from mean deposition. Results for square, rectangular, and triangular arrays are presented.     

An interactive GRAMPS algorithm for the heterogeneous fixed fleet vehicle routing problem with and without backhauls

In this article, a visual interactive approach based on a new greedy randomised adaptive memory programming search (GRAMPS) algorithm is proposed to solve the heterogeneous fixed fleet vehicle routing problem (HFFVRP) and a new extension of the HFFVRP, which is called heterogeneous fixed fleet vehicle routing problem with backhauls (HFFVRPB). This problem involves two different sets of customers. Backhaul customers are pickup points and linehaul customers are delivery points that are to be serviced from a single depot by a heterogeneous fixed fleet of vehicles, each of which is restricted in the capacity it can carry, with different variable travelling costs.     

A Carleman estimate for the linear magnetoelastic waves system and an inverse source problem in a bounded conductive medium

ABSTRACT

In this paper, we consider an inverse problem for the simultaneous diffusion process of elastic and electromagnetic waves in an isotropic heterogeneous elastic body which is identified with an open bounded domain. From the mathematical point of view, the system under consideration can be viewed as the coupling between the hyperbolic system of elastic waves and a parabolic system for the magnetic field. We study an inverse problem of determining the external source terms by observations data in a neighborhood of the boundary and we prove the Hölder stability. For the proof, we show a Carleman estimate for the displacement and the magnetic field of the magnetoelastic system.     

关于含裂纹的复合材料的一个细观断裂模型

本文研究复合材料的小范围屈服断裂过程.首先,提出了含裂纹复合材料的一个细观断裂模型,其中对裂纹顶端的外部地区采用连续介质各向异性描述,对裂纹顶端的小范围地区则采用多相的描述.多相区中考虑三种材料成分,即纤维、界面和基体,它们都可被看作是非线性的,并计入有限变形的影响.我们应用边界层——非线性有限元方法作出本问题的解.