Análise matemática de soluções descontínuas de leis de conservação hiperbólicas e resoluções numéricas para a captura de ondas de choque em escoamentos multifásicos em meios porosos

O processo de recuperação secundária de petróleo é comumente realizado com a injeção de água ou gás no reservatório a fim de manter a pressão necessária para sua extração. Para que o investimento seja viável, os gastos com a extração precisam ser menores do que o retorno financeiro obtido com a produção de petróleo. Objetivando-se estudar possíveis cenários para o processo de exploração, costuma-se utilizar simulações dos processos de extração. As equações que modelam esse processo de recuperação são de caráter hiperbólico e não lineares, as quais podem ser interpretadas como Leis de Conservação, cujas soluções são complexas por suas naturezas descontínuas. Essas descontinuidades ou saltos são conhecidas como ondas de choque. Neste trabalho foi abordada uma análise matemática para os fenômenos oriundos de leis de conservação, para em seguida utilizá-la no entendimento do referido problema. Foram estudadas soluções fracas que, fisicamente, podem ser interpretadas como ondas de choque ou rarefação, então, para que fossem distinguidas as fisicamente admissíveis, foi considerado o princípio de entropia, nas suas diversas formas. As simulações foram realizadas nos âmbitos dos escoamentos bifásicos e trifásicos, em que os fluidos são imiscíveis e os efeitos gravitacionais e difusivos, devido à pressão capilar, foram desprezados. Inicialmente, foi feito um estudo comparativo de resoluções numéricas na captura de ondas de choque em escoamento bifásico água-óleo. Neste estudo destacam-se o método Composto LWLF-k, o esquema NonStandard e a introdução da nova função de renormalização para o esquema NonStandard, onde obteve resultados satisfatórios, principalmente em regiões onde a viscosidade do óleo é muito maior do que a viscosidade da água. No escoamento bidimensional, um novo método é proposto, partindo de uma generalização do esquema NonStandard unidimensional. Por fim, é feita uma adaptação dos métodos LWLF-4 e NonStandard para a simulação em escoamentos trifásicos em domínios unidimensional. O esquema NonStandard foi considerado mais eficiente nos problemas abordados, uma vez que sua versão bidimensional mostrou-se satisfatória na captura de ondas de choque em escoamentos bifásicos em meios porosos.

Finite volume distance field solution applied to medial axis transform

Accurate and efficient computation of the nearest wall distance d (or level set) is important for many areas of computational science/engineering. Differential equation-based distance/ level set algorithms, such as the hyperbolic-natured Eikonal equation, have demonstrated valuable computational efficiency. Here, in the context, as an 'auxiliary' equation to the main flow equations, the Eikonal equation is solved efficiently with two different finite volume approaches (the cell vertex and cell-centered). Application of the distance solution is studied for various geometries. Moreover, a procedure using the differential field to obtain the medial axis transform (MAT) for different geometries is presented. The latter provides a skeleton representation of geometric models that has many useful analysis properties. As an alternative approach to the pure geometric methods (e.g. the Voronoi approach), the current d-MAT procedure bypasses many difficulties that are usually encountered by pure geometric methods, especially in three dimensional space. It is also shown that the d-MAT approach provides the potential to sculpt/control the MAT form for specialized solution purposes. Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc.

Eliminating discharge imbalances in predicting shallow water flows with shocks

The shallow water equations are widely used in modelling environmental flows. Being a hyperbolic system of differential equations, they admit shocks that represent hydraulic jumps and bores. Although the water surface can be solved satisfactorily with the modern shock-capturing schemes, the predicted flow rate often suffers from imbalances where shocks occur, eg the mass conservation is violated by failing to maintain a constant discharge rate at every cross-section in a steady open channel flow. A total-variation-diminishing Lax-Wendroff scheme is developed, and used to demonstrate how to achieve an exact flux balance. The performance of the proposed methods is inspected through some test cases, which include 1- and 2-dimensional, flat and irregular bed scenarios. The proposed methods are shown to preserve the mass exactly, and can be easily extended to other shock-capturing models.

Stiffness of sands through a laboratory test database

Deformations of sandy soils around geotechnical structures generally involve strains in the range small (0·01%) to medium (0·5%). In this strain range the soil exhibits non-linear stress-strain behaviour, which should be incorporated in any deformation analysis. In order to capture the possible variability in the non-linear behaviour of various sands, a database was constructed including the secant shear modulus degradation curves of 454 tests from the literature. By obtaining a unique S-shaped curve of shear modulus degradation, a modified hyperbolic relationship was fitted. The three curve-fitting parameters are: an elastic threshold strain γe, up to which the elastic shear modulus is effectively constant at G0; a reference strain γr, defined as the shear strain at which the secant modulus has reduced to 0·5G0; and a curvature parameter a, which controls the rate of modulus reduction. The two characteristic strains γe and γr were found to vary with sand type (i.e. uniformity coefficient), soil state (i.e. void ratio, relative density) and mean effective stress. The new empirical expression for shear modulus reduction G/G0 is shown to make predictions that are accurate within a factor of 1·13 for one standard deviation of random error, as determined from 3860 data points. The initial elastic shear modulus, G0, should always be measured if possible, but a new empirical relation is shown to provide estimates within a factor of 1·6 for one standard deviation of random error, as determined from 379 tests. The new expressions for non-linear deformation are easy to apply in practice, and should be useful in the analysis of geotechnical structures under static loading.

Development of realistic liquid equations of state for multi-phase CFD modelling

Several equations of state (EOS) have been incorporated into a novel algorithm to solve a system of multi-phase equations in which all phases are assumed to be compressible to varying degrees. The EOSs are used to both supply functional relationships to couple the conservative variables to the primitive variables and to calculate accurately thermodynamic quantities of interest, such as the speed of sound. Each EOS has a defined balance of accuracy, robustness and computational speed; selection of an appropriate EOS is generally problem-dependent. This work employs an AUSM+-up method for accurate discretisation of the convective flux terms with modified low-Mach number dissipation for added robustness of the solver. In this paper we show a newly-developed time-marching formulation for temporal discretisation of the governing equations with incorporated time-dependent source terms, as well as considering the system of eigenvalues that render the governing equations hyperbolic.

Does temporal discounting explain unhealthy behavior? A systematic review and reinforcement learning perspective.

The tendency to make unhealthy choices is hypothesized to be related to an individual's temporal discount rate, the theoretical rate at which they devalue delayed rewards. Furthermore, a particular form of temporal discounting, hyperbolic discounting, has been proposed to explain why unhealthy behavior can occur despite healthy intentions. We examine these two hypotheses in turn. We first systematically review studies which investigate whether discount rates can predict unhealthy behavior. These studies reveal that high discount rates for money (and in some instances food or drug rewards) are associated with several unhealthy behaviors and markers of health status, establishing discounting as a promising predictive measure. We secondly examine whether intention-incongruent unhealthy actions are consistent with hyperbolic discounting. We conclude that intention-incongruent actions are often triggered by environmental cues or changes in motivational state, whose effects are not parameterized by hyperbolic discounting. We propose a framework for understanding these state-based effects in terms of the interplay of two distinct reinforcement learning mechanisms: a "model-based" (or goal-directed) system and a "model-free" (or habitual) system. Under this framework, while discounting of delayed health may contribute to the initiation of unhealthy behavior, with repetition, many unhealthy behaviors become habitual; if health goals then change, habitual behavior can still arise in response to environmental cues. We propose that the burgeoning development of computational models of these processes will permit further identification of health decision-making phenotypes.

Differential equation based length scales to improve DES and RANS simulations

Surprisingly expensive to compute wall distances are still used in a range of key turbulence and peripheral physics models. Potentially economical, accuracy improving differential equation based distance algorithms are considered. These involve elliptic Poisson and hyperbolic natured Eikonal equation approaches. Numerical issues relating to non-orthogonal curvilinear grid solution of the latter are addressed. Eikonal extension to a Hamilton-Jacobi (HJ) equation is discussed. Use of this extension to improve turbulence model accuracy and, along with the Eikonal, enhance Detached Eddy Simulation (DES) techniques is considered. Application of the distance approaches is studied for various geometries. These include a plane channel flow with a wire at the centre, a wing-flap system, a jet with co-flow and a supersonic double-delta configuration. Although less accurate than the Eikonal, Poisson method based flow solutions are extremely close to those using a search procedure. For a moving grid case the Poisson method is found especially efficient. Results show the Eikonal equation can be solved on highly stretched, non-orthogonal, curvilinear grids. A key accuracy aspect is that metrics must be upwinded in the propagating front direction. The HJ equation is found to have qualitative turbulence model improving properties. © 2003 by P. G. Tucker.

Spatial structure of internal tidal waves generated on weak topographies

The generation of internal gravity waves by barotropic tidal flow passing over a two-dimensional topography is investigated. Rather than calculating the conversion of tidal energy, this study focuses on delineating the geometric characteristics of the spatial structure of the resulting internal wave fields (i.e., the configurations of the internal beams and their horizontal projections) which have usually been ignored. it is found that the various possible wave types can be demarcated by three characteristic frequencies: the tidal frequency, wo; the buoyancy frequency, N; and the vertical component of the Coriolis vector or earth's rotation.f. When different possibilities arising from the sequence of these frequencies are considered, there occur 12 kinds of wave structures in the full 3D space in contrast to the 5 kinds identified by the 2D theory. The constant wave phase lines may form as ellipses or hyperbolic lines on the horizontal plane, provided the buoyancy frequency is greater or less than the tidal frequency. The effect that stems from the consideration of the basic flow is also found, which not only serves as the reason for the occurrence of higtter harmonics but also increases the wave strength in the direction of basic flow. (C) 2009 Elsevier B.V. All rights reserved.

锦屏电站地下厂房卸荷岩体参数研究

It is a basic work to ascertain the parameters of rock mass for evaluation about stability of the engineering. Anisotropism、inhomogeneity and discontinuity characters of the rock mass arise from the existing of the structural plane. Subjected to water、weathering effect、off-loading, mechanical characters of the rock mass are greatly different from rock itself, Determining mechanical parameters of the rock mass becomes so difficult because of structure effect、dimension effect、rheological character, ‘Can’t give a proper parameter’ becomes one of big problems for theoretic analysis and numerical simulation. With the increment of project scale, appraising the project rock mass and ascertaining the parameters of rock mass becomes more and more important and strict. Consequently, researching the parameters of rock mass has important theoretical significance and actual meaning. The Jin-ping hydroelectric station is the first highest hyperbolic arch dam in the world under construction, the height of the dam is about 305m, it is the biggest hydroelectric station at lower reaches of Yalong river. The length of underground factory building is 204.52m, the total height of it is 68.83m, the maximum of span clearance is 28.90m. Large-scale excavation in the underground factory of Jin-ping hydroelectric station has brought many kinds of destructive phenomenon, such as relaxation、spilling, providing a precious chance for study of unloading parameter about rock mass. As we all know, Southwest is the most important hydroelectric power base in China, the construction of the hydroelectric station mostly concentrate at high mountain and gorge area, basically and importantly, we must be familiar with the physical and mechanical character of the rock mass to guarantee to exploit safely、efficiently、quickly, in other words, we must understand the strength and deformation character of the rock mass. Based on enough fieldwork of geological investigation, we study the parameter of unloading rock mass on condition that we obtain abundant information, which is not only important for the construction of Jin-ping hydroelectric station, but also for the construction of other big hydroelectric station similar with Jin-ping. This paper adopt geological analysis、test data analysis、experience analysis、theory research and Artificial Neural Networks （ANN） brainpower analysis to evaluate the mechanical parameter, the major production is as follows: (1)Through the excavation of upper 5-layer of the underground powerhouse and the statistical classification of the main joints fractures exposed, We believe that there are three sets of joints, the first group is lay fracture, the second group and the fourth group are steep fracture. These provide a strong foundation for the following calculation of and analysis; (2)According to the in-situ measurement about sound wave velocity、displacement and anchor stress, we analyses the effects of rock unloading effect，the results show a obvious time-related character and localization features of rock deformation. We determine the depth of excavation unloading of underground factory wall based on this. Determining the rock mass parameters according to the measurement about sound wave velocity with characters of low- disturbing、dynamic on the spot, the result can really reflect the original state, this chapter approximately the mechanical parameters about rock mass at each unloading area; (3)Based on Hoek-Brown experienced formula with geological strength index GSI and RMR method to evaluate the mechanical parameters of different degree weathering and unloading rock mass about underground factory, Both of evaluation result are more satisfied; (4)From the perspective of far-field stress, based on the stress field distribution ideas of two-crack at any load conditions proposed by Fazil Erdogan (1962)，using the strain energy density factor criterion (S criterion) proposed by Xue changming(1972)，we establish the corresponding relationship between far-field stress and crack tip stress field, derive the integrated intensity criterion formula under the conditions of pure tensile stress among two line coplanar intermittent jointed rock，and establish the corresponding intensity criterion for the exploratory attempt; (5)With artificial neural network, the paper focuses on the mechanical parameters of rock mass that we concerned about and the whole process of prediction of deformation parameters, discusses the prospect of applying in assessment about the parameters of rock mass，and rely on the catalog information of underground powerhouse of Jinping I Hydropower Station, identifying the rock mechanics parameters intellectually，discusses the sample selection, network design, values of basic parameters and error analysis comprehensively. There is a certain significance for us to set up a set of parameters evaluation system，which is in construction of large-scale hydropower among a group of marble mass.

地震波场延拓+菲涅尔体叠加方法及成像效果

In the increasingly enlarged exploration target, deep target layer(especially for the reservoir of lava) is a potential exploration area. As well known, the reflective energy becomes weak because the seismic signals of reflection in deep layer are absorbed and attenuate by upper layer. Caustics and multi-values traveltime in wavefield are aroused by the complexity of stratum. The ratio of signal to noise is not high and the fold numbers are finite(no more than 30). All the factors above affect the validity of conventional processing methods. So the high S/N section of stack can't always be got with the conventional stack methods even if the prestack depth migration is used. So it is inevitable to develop another kind of stack method instead. In the last a few years, the differential solution of wave equation was hold up by the condition of computation. Kirchhoff integral method rose in the initial stages of the ninetieth decade of last century. But there exist severe problems in it, which is are too difficult to resolve, so new method of stack is required for the oil and gas exploration. It is natural to think about upgrading the traditionally physic base of seismic exploration methods and improving those widely used techniques of stack. On the other hand, great progress is depended on the improvement in the wave differential equation prestack depth migration. The algorithm of wavefield continuation in it is utilized. In combination with the wavefield extrapolation and the Fresnel zone stack, new stack method is carried out It is well known that the seismic wavefield observed on surface comes from Fresnel zone physically, and doesn't comes from the same reflection points only. As to the more complex reflection in deep layer, it is difficult to describe the relationship between the reflective interface and the travel time. Extrapolation is used to eliminate caustic and simplify the expression of travel time. So the image quality is enhanced by Fresnel zone stack in target. Based on wave equation, high-frequency ray solution and its character are given to clarify theoretical foundation of the method. The hyperbolic and parabolic travel time of the reflection in layer media are presented in expression of matrix with paraxial ray theory. Because the reflective wave field mainly comes from the Fresnel Zone, thereby the conception of Fresnel Zone is explained. The matrix expression of Fresnel zone and projected Fresnel zone are given in sequence. With geometrical optics, the relationship between object point in model and image point in image space is built for the complex subsurface. The travel time formula of reflective point in the nonuniform media is deduced. Also the formula of reflective segment of zero-offset and nonzero offset section is provided. For convenient application, the interface model of subsurface and curve surface derived from conventional stacks DMO stack and prestack depth migration are analyzed, and the problem of these methods was pointed out in aspects of using data. Arc was put forward to describe the subsurface, thereby the amount of data to stack enlarged in Fresnel Zone. Based on the formula of hyperbolic travel time, the steps of implementation and the flow of Fresnel Zone stack were provided. The computation of three model data shows that the method of Fresnel Zone stack can enhance the signal energy and the ratio of signal to noise effectively. Practical data in Xui Jia Wei Zhi, a area in Daqing oilfield, was processed with this method. The processing results showed that the ability in increasing S/N ratio and enhancing the continuity of weak events as well as confirming the deep configuration of volcanic reservoir is better than others. In deeper target layer, there exists caustic caused by the complex media overburden and the great variation of velocity. Travel time of reflection can't be exactly described by the formula of travel time. Extrapolation is bring forward to resolve the questions above. With the combination of the phase operator and differential operator, extrapolating operator adaptable to the variation of lateral velocity is provided. With this method, seismic records were extrapolated from surface to any different deptlis below. Wave aberration and caustic caused by the inhomogenous layer overburden were eliminated and multi-value curve was transformed into the curve.of single value. The computation of Marmousi shows that it is feasible. Wave field continuation extends the Fresnel Zone stack's application.

Direct Computation of 3D Shape Invariants and the Focus of Expansion

Structure from motion often refers to the computation of 3D structure from a matched sequence of images. However, a depth map of a surface is difficult to compute and may not be a good representation for storage and recognition. Given matched images, I will first show that the sign of the normal curvature in a given direction at a given point in the image can be computed from a simple difference of slopes of line-segments in one image. Using this result, local surface patches can be classified as convex, concave, parabolic (cylindrical), hyperbolic (saddle point) or planar. At the same time the translational component of the optical flow is obtained, from which the focus of expansion can be computed.

The Alignment of Objects With Smooth Surfaces: Error Analysis of the Curvature Method

The recognition of objects with smooth bounding surfaces from their contour images is considerably more complicated than that of objects with sharp edges, since in the former case the set of object points that generates the silhouette contours changes from one view to another. The "curvature method", developed by Basri and Ullman [1988], provides a method to approximate the appearance of such objects from different viewpoints. In this paper we analyze the curvature method. We apply the method to ellipsoidal objects and compute analytically the error obtained for different rotations of the objects. The error depends on the exact shape of the ellipsoid (namely, the relative lengths of its axes), and it increases a sthe ellipsoid becomes "deep" (elongated in the Z-direction). We show that the errors are usually small, and that, in general, a small number of models is required to predict the appearance of an ellipsoid from all possible views. Finally, we show experimentally that the curvature method applies as well to objects with hyperbolic surface patches.

Birkhoff normal forms for Fourier integral operators II

Iantchenko, A.; Sj?strand, J., (2001) 'Birkhoff normal forms for Fourier integral operators II', American Journal of Mathematics 124(4) pp.817-850 RAE2008

The Synthesis of Arbitrary Stable Dynamics in Non-linear Neural Networks II: Feedback and Universality

We wish to construct a realization theory of stable neural networks and use this theory to model the variety of stable dynamics apparent in natural data. Such a theory should have numerous applications to constructing specific artificial neural networks with desired dynamical behavior. The networks used in this theory should have well understood dynamics yet be as diverse as possible to capture natural diversity. In this article, I describe a parameterized family of higher order, gradient-like neural networks which have known arbitrary equilibria with unstable manifolds of known specified dimension. Moreover, any system with hyperbolic dynamics is conjugate to one of these systems in a neighborhood of the equilibrium points. Prior work on how to synthesize attractors using dynamical systems theory, optimization, or direct parametric. fits to known stable systems, is either non-constructive, lacks generality, or has unspecified attracting equilibria. More specifically, We construct a parameterized family of gradient-like neural networks with a simple feedback rule which will generate equilibrium points with a set of unstable manifolds of specified dimension. Strict Lyapunov functions and nested periodic orbits are obtained for these systems and used as a method of synthesis to generate a large family of systems with the same local dynamics. This work is applied to show how one can interpolate finite sets of data, on nested periodic orbits.

Equity trading volume and volatility: Latent information arrivals and common long-run dependencies

This article examines the behavior of equity trading volume and volatility for the individual firms composing the Standard & Poor's 100 composite index. Using multivariate spectral methods, we find that fractionally integrated processes best describe the long-run temporal dependencies in both series. Consistent with a stylized mixture-of-distributions hypothesis model in which the aggregate "news"-arrival process possesses long-memory characteristics, the long-run hyperbolic decay rates appear to be common across each volume-volatility pair.