Analytical Solution (analytical + solution)

Distribution by Scientific Domains

Kinds of Analytical Solution

  • approximate analytical solution
  • available analytical solution
  • closed-form analytical solution
  • exact analytical solution

  • Selected Abstracts


    Garey A. Fox
    ABSTRACT: Considerable advancements have been made in the development of analytical solutions for predicting the effects of pumping wells on adjacent streams and rivers. However, these solutions have not been sufficiently evaluated against field data. The objective of this research is to evaluate the predictive performance of recently proposed analytical solutions for unsteady stream depletion using field data collected during a stream/aquifer analysis test at the Tamarack State Wildlife Area in eastern Colorado. Two primary stream/aquifer interactions exist at the Tamarack site: (1) between the South Platte River and the alluvial aquifer and (2) between a backwater stream and the alluvial aquifer. A pumping test is performed next to the backwater stream channel. Drawdown measured in observation wells is matched to predictions by recently proposed analytical solutions to derive estimates of aquifer and streambed parameters. These estimates are compared to documented aquifer properties and field measured streambed conductivity. The analytical solutions are capable of estimating reasonable values of both aquifer and streambed parameters with one solution capable of simultaneously estimating delayed aquifer yield and stream flow recharge. However, for long term water management, it is reasonable to use simplified analytical solutions not concerned with early-time delayed yield effects. For this site, changes in the water level in the stream during the test and a varying water level profile at the beginning of the pumping test influence the application of the analytical solutions. [source]

    An Analytical Solution for Ground Water Transit Time through Unconfined Aquifers

    GROUND WATER, Issue 4 2005
    R. Chesnaux
    An exact, closed-form analytical solution is developed for calculating ground water transit times within Dupuit-type flow systems. The solution applies to steady-state, saturated flow through an unconfined, horizontal aquifer recharged by surface infiltration and discharging to a downgradient fixed-head boundary. The upgradient boundary can represent, using the same equation, a no-flow boundary or a fixed head. The approach is unique for calculating travel times because it makes no a priori assumptions regarding the limit of the water table rise with respect to the minimum saturated aquifer thickness. The computed travel times are verified against a numerical model, and examples are provided, which show that the predicted travel times can be on the order of nine times longer relative to existing analytical solutions. [source]

    Chain Length Distributions of Polyolefins Made with Coordination Catalysts at Very Short Polymerization Times , Analytical Solution and Monte Carlo Simulation

    Joćo B. P. Soares
    Abstract We developed an analytical solution to describe how the CLD of polymers made with coordination polymerization catalysts vary as a function of time for very short polymerization times before the CLD becomes completely developed. We compared the analytical solution with a dynamic Monte Carlo model for validation, obtaining excellent agreement. Our analytical solution can be used to determine when the steady-state hypothesis, commonly used in polymerization models, becomes valid as a function of polymer chain length. We also extended our model to describe polymerization with multiple-site-type catalysts. Depending on the polymerization kinetic parameters of the different site types on the catalyst, the fully developed CLD is reached through very different intermediate CLDs. This modeling approach, although rather simplified, can be used to interpret results from short polymerization time experiments such as the ones done in stopped-flow reactors. [source]

    Analytical approach for the toroidal relaxation of viscoelastic earth

    Hansheng Wang
    SUMMARY This paper is concerned with post-seismic toroidal deformation in a spherically symmetric, non-rotating, linear-viscoelastic, isotropic Maxwell earth model. Analytical expressions for characteristic relaxation times and relaxation strengths are found for viscoelastic toroidal deformation, associated with surface tangential stress, when there are two to five layers between the core,mantle boundary and Earth's surface. The multilayered models can include lithosphere, asthenosphere, upper and lower mantles and even low-viscosity ductile layer in the lithosphere. The analytical approach is self-consistent in that the Heaviside isostatic solution agrees with fluid limit. The analytical solution can be used for high-precision simulation of the toroidal relaxation in five-layer earths and the results can also be considered as a benchmark for numerical methods. Analytical solution gives only stable decaying modes,unstable mode, conjugate complex mode and modes of relevant poles with orders larger than 1, are all excluded, and the total number of modes is found to be just the number of viscoelastic layers between the core,mantle boundary and Earth's surface,however, any elastic layer between two viscoelastic layers is also counted. This confirms previous finding where numerical method (i.e. propagator matrix method) is used. We have studied the relaxation times of a lot of models and found the propagator matrix method to agree very well with those from analytical results. In addition, the asthenosphere and lithospheric ductile layer are found to have large effects on the amplitude of post-seismic deformation. This also confirms the findings of previous works. [source]

    Analytical solution for the electric potential in arbitrary anisotropic layered media applying the set of Hankel transforms of integer order

    E. Pervago
    ABSTRACT The analytical solution and algorithm for simulating the electric potential in an arbitrarily anisotropic multilayered medium produced by a point DC source is here proposed. The solution is presented as a combination of Hankel transforms of integer order and Fourier transforms based on the analytical recurrent equations obtained for the potential spectrum. For the conversion of the potential spectrum into the space domain, we have applied the algorithm of the Fast Fourier Transform for logarithmically spaced points. A comparison of the modelling results with the power-series solution for two-layered anisotropic structures demonstrated the high accuracy and computing-time efficiency of the method proposed. The results of the apparent-resistivity calculation for both traditional pole-pole and tensor arrays above three-layered sequence with an azimuthally anisotropic second layer are presented. The numerical simulations show that both arrays have the same sensitivity to the anisotropy parameters. This sensitivity depends significantly on the resistivity ratio between anisotropic and adjacent layers and increases for the models with a conductive second layer. [source]

    Analytical solution to a bias in the TOPMODEL framework balance

    G.-M. Saulnier
    Abstract The increasing need for distributed hydrological modelling leads to an intense use of spatially distributed predictions of physically based models, such as TOPMODEL as addressed here. The ability of these models to reproduce the internal behaviour of catchments physically is increasingly tested through field experiments (geochemical investigation, distributed measurements network, etc.). This paper will show that, in the case of TOPMODEL, an implicit approximation remains in the classic derivation of the equations that consists in neglecting the surface of saturated areas with respect to the total surface of the catchment. This simplifying, though unnecessary, approximation leads to a systematic underestimation of the catchment water storage deficit and to divergence in the water budget accounting. This may also significantly change the predicted ratio between subsurface and surface water fluxes in the total discharge. An analytical solution is suggested that leads to water balance accounting which is better defined, and more consistent in comparison with field water storage recording. It is expected that this work will ensure more accurate TOPMODEL predictions, consistent with the assumptions of the model. This will then improve the interpretation of comparisons between results of simulation and field experiments. Copyright © 2004 John Wiley & Sons, Ltd. [source]

    Analytical solution of the harmonic waves diffraction by a cylindrical lined cavity in poroelastic saturated medium

    Y. S. Karinski
    Abstract This paper presents a model for the analysis of plane waves diffraction at a cavity in an infinite homogeneous poroelastic saturated medium, lined by a lining composed of four equal segments. An elastic boundary layer is placed between the cavity lining and the infinite porous medium. The boundary layer is simulated by ,elastic boundary conditions' in which the bulk matrix stress is proportional to the relative displacement between the lining and the surrounding medium matrix boundary. In addition, fluid impermeability through the intermediate layer is assumed. For the frequencies, that differ from the pseudoresonanse frequencies, the problem was reduced to the problem of an ideal elastic medium. A closed-form analytical solution of the problem was obtained using Fourier,Bessel series, the convergence of which was proven. It was shown that the number of series terms required to obtain a desired level of accuracy can be determined in advance. The influence of the medium porosity on the medium dynamic stress concentration was studied. Copyright © 2006 John Wiley & Sons, Ltd. [source]

    Erratum to Analytical solution to the transient phase of steady state free precession sequences.

    Magn Reson Med 2009;62:14
    No abstract is available for this article. [source]

    Modeling the effect of high dead-space syringes on the human immunodeficiency virus (HIV) epidemic among injecting drug users

    ADDICTION, Issue 8 2010
    Georgiy V. Bobashev
    ABSTRACT Aims To illustrate the impact of different proportions of injecting drug users (IDUs) sharing high dead-space syringes (HDSS) or low dead-space syringes (LDSS) on the probability of human immunodeficiency virus (HIV) transmission; and thus the impact on injection-related HIV prevalence and incidence. Design A stochastic mathematical model was used to evaluate the impact of HDSS use in high- and low-risk IDU populations. Model parameters were obtained from peer-reviewed publications. Analytical solutions of a simplified deterministic model were obtained to explain the effect of HDSS on HIV endemic states. Findings Simulation analysis shows that the HIV epidemic could be sustained even when a small percentage of sharing (10%) involved HDSS. The effect is much stronger in high-risk compared with low-risk populations. Steady state HIV prevalence increases with the proportion of HDSS, and for high- and low-risk populations reaches around 80% and 20%, respectively. For low-risk populations, the use of LDSS could result in the virtual elimination of HIV. These results are dependent upon an evidence-supported assumption of a significant difference in HIV transmission risk associated with HDSS versus LDSS. Conclusions Our models suggest that injection-related HIV epidemics may not occur when most (e.g. 95% or more) IDUs use LDSS. While these results are based on indirect risk measures and a number of simplifying assumptions, the effect of blood retained in high dead-space syringes on HIV prevalence seems to be very strong, even using relatively conservative assumptions. The findings have potential implications for needle exchange programs and the types of syringes produced and distributed world-wide. [source]

    Analytical Studies on the Impact of Land Reclamation on Ground Water Flow

    GROUND WATER, Issue 6 2001
    Jiu J. Jiao
    Land reclamation has been a common practice to produce valuable land in coastal areas. The impact of land reclamation on coastal environment and marine ecology is well recognized and widely studied. It has not been recognized yet that reclamation may change the regional ground water regime, which may in turn modify the coastal environment, flooding pattern, and stability of slopes and foundations. This paper represents the first attempt to examine quantitatively the effect of reclamation on ground water levels. Analytical solutions are developed to study the ground water change in response to reclamation based on two hypothetical models. In the first model, the ground water flow regime changes only in the hillside around the reclamation areas. In the second model, the ground water regime changes in the entire hill. Both models assume that the ground water flow is in a steady state and satisfies the Dupuit assumptions. Hypothetical examples are used to demonstrate how the ground water level, ground water divide and ground water submarine discharge will change with the scale and hydraulic conductivity of the reclamation materials. The results show that the change of ground water regime depends mainly on the length of the reclaimed area and the values of hydraulic conductivity of the reclaimed materials. It is also seen that the reclamation may impact not only the ground water regime near the coast areas around the reclamation site, but also that in the coast areas opposite the reclamation area. A reclamation site near Tseung Kwan O in the New Territories in Hong Kong, China, is used as a case study to discuss the possible modification of the ground water system caused by reclamation. [source]

    Wave propagation in an inhomogeneous cross-anisotropic medium

    Cheng-Der Wang
    Abstract Analytical solutions for wave velocities and wave vectors are yielded for a continuously inhomogeneous cross-anisotropic medium, in which Young's moduli (E, E,) and shear modulus (G,) varied exponentially as depth increased. However, for the rest moduli in cross-anisotropic materials, , and ,, remained constant regardless of depth. We assume that cross-anisotropy planes are parallel to the horizontal surface. The generalized Hooke's law, strain,displacement relationships, and equilibrium equations are integrated to constitute governing equations. In these equations, displacement components are fundamental variables and, hence, the solutions of three quasi-wave velocities, VP, VSV, and VSH, and the wave vectors, , and , can be generated for the inhomogeneous cross-anisotropic media. The proposed solutions and those obtained by Daley and Hron, and Levin correlate well with each other when the inhomogeneity parameter, k, is 0. Additionally, parametric study results indicate that the magnitudes and directions of wave velocity are markedly affected by (1) the inhomogeneous parameter, k; (2) the type and degree of geomaterial anisotropy (E/E,, G,/E,, and ,/,,); and (3) the phase angle, ,. Consequently, one must consider the influence of inhomogeneous characteristic when investigating the behaviors of wave propagation in a cross-anisotropic medium. Copyright © 2009 John Wiley & Sons, Ltd. [source]

    Analytical solutions for dynamic pressures of coupling fluid,porous medium,solid due to SV wave incidence

    Jin-Ting Wang
    Abstract This paper presents the results of theoretical investigation on the dynamic coupling of an ideal fluid-porous medium-elastic half-space system subjected to SV waves to study the effect of sediment on the seismic response of dams for reservoirs that are deposited with a significant amount of sediment after a long period of operation. The effects of the porous medium and the incident wave angle on dynamic pressures in the overlying ideal fluid are analyzed, and the reflection and transmission coefficients of the wave at the material interfaces are derived using an analytical solution in terms of displacement potentials. The numerical test of modeling shows that the dynamic pressures significantly depend on the properties of porous medium. The fully saturated porous medium reduces the response peaks slightly, while the partially saturated porous medium causes a considerable increase in the resonance peaks. Copyright © 2009 John Wiley & Sons, Ltd. [source]

    Bending of fluid-saturated linear poroelastic beams with compressible constituents

    Zhi-Hua Wang
    Abstract Analytical solutions are presented for fluid-saturated linear poroelastic beams under pure bending. The stress-free boundary condition at the lateral surfaces is satisfied in the St Venant's sense and the Beltrami,Michell compatibility conditions are resolved rigorously, rendering the flexure of the beams analytically tractable. Two sets of formulations are derived based on the coupled and uncoupled diffusion equations respectively. The analytical solutions are compared with three-dimensional finite element simulations. Both sets of analytical formulations are capable of capturing exactly both the initial (undrained) and the steady-state (fully drained) deflection of the beams. However, the analytical solutions are found to be deficient during the transient phase. The cause for the deficiency of the transient analytical solutions is discussed. The accuracy of the analytical solutions improves as Poisson's ratio and the compressibility of the constituents of the porous beam increase, where the St Venant's edge effect at the lateral surfaces is mitigated. Copyright © 2008 John Wiley & Sons, Ltd. [source]

    Localized failure of fibre-reinforced elastic,plastic materials subjected to plane strain loading

    Dunja Peri
    Abstract We consider discontinuous bifurcations as the indicator of a localized failure for a class of composites that are characterized by elastic fibres reinforcing an elastic,plastic matrix. A macroscopic tangent stiffness tensor for the fibre-reinforced composite is developed by consistently homogenizing the contribution of fibres in a spherical representative volume element. Analytical solutions are derived for the critical hardening modulus and corresponding bifurcation directions for the case of plane strain loading. Properties of the solutions are further illustrated on the example of the non-associated Drucker,Prager model at onset of yielding. Results show that presence of fibres decreases the critical hardening modulus, thus inhibiting the onset of strain localization. The rate of decrease in the critical hardening modulus is the highest for pure shear, followed by uniaxial tension, uniaxial compression, biaxial tension and biaxial compression. The main fibre parameters that control the onset of strain localization are their volumetric content and their stiffness modulus whereby very stiff fibres can produce the most significant decrease in the critical hardening modulus, especially for the state of biaxial tension. The critical hardening modulus for the non-associated Drucker,Prager model exhibits a full range of localization modes including compaction bands, dilation bands, and transition in the form of shear bands regardless of the presence of fibres. Presence of fibres affects bifurcation directions, except in the case when Poisson's ratio of the matrix is equal to 0.25. The results demonstrate stabilizing effects of fibres by which they provide the control against the onset of strain localization. Copyright © 2006 John Wiley & Sons, Ltd. [source]

    Analytical solutions for a three-invariant Cam clay model subjected to drained loading histories

    Dunja Peri
    Abstract Analytical solutions are derived for a three-invariant Cam clay model subjected to proportional and circular drained loading histories. The solutions are presented for a specific volume, and volumetric and generalized shear strains. In the case of a proportional loading only straight effective stress paths are considered while in the case of a circular loading the maximum possible change in Lode's angle is ,/3 due to plastic isotropy. Additionally, a concept of deviatoric stiffness is devised and an analytical expression for the generalized hardening modulus is derived. Qualitative and quantitative analyses are carried out in the form of direct comparisons between analytical solutions for drained and undrained loading histories thus offering an improved understanding of the three-invariant model. Copyright © 2006 John Wiley & Sons, Ltd. [source]

    Analytical solutions of linear finite- and small-strain one-dimensional consolidation

    P. H. Morris
    Abstract Analytical solutions are presented for linear finite-strain one-dimensional consolidation of initially unconsolidated soil layers with surcharge loading for both one- and two-way drainage. These solutions complement earlier solutions for initially unconsolidated soil layers without surcharge and initially normally consolidated soil layers with surcharge. Small-strain solutions for the consolidation of initially unconsolidated soil layers with surcharge loading are also presented, and the relationship between the earlier solutions for initially unconsolidated soil without surcharge and the corresponding small-strain solutions, which was not addressed in the earlier work, is clarified. The new solutions for initially unconsolidated soil with surcharge loading can be applied to the analysis of low stress consolidation tests and to the partial validation of numerical solutions of non-linear finite-strain consolidation. They also clarify a formerly perplexing aspect of finite-strain solution charts first noted in numerical solutions. Copyright © 2004 John Wiley & Sons, Ltd. [source]

    Explicit solutions for the instantaneous undrained contraction of hollow cylinders and spheres in porous elastoplastic medium

    A. Giraud
    Abstract In this article we present closed-form solutions for the undrained variations in stress, pore pressure, deformation and displacement inside hollow cylinders and hollow spheres subjected to uniform mechanical pressure instantaneously applied to their external and internal boundary surfaces. The material is assumed to be a saturated porous medium obeying a Mohr,Coulomb model failure criterion, exhibiting dilatant plastic deformation according to a non-associated flow rule which accounts for isotropically strain hardening or softening. The instantaneous response of a porous medium submitted to an instantaneous loading is undrained, i.e. without any fluid mass exchange. The short-term equilibrium problem to be solved is now formally identical to a problem of elastoplasticity where the constitutive equations involve the undrained elastic moduli and particular equivalent plastic parameters. The response of the model is presented (i) for extension and compression undrained triaxial tests, and (ii) for unloading problems of hollow cylinders and spheres through the use of appropriately developed closed-form solutions. Numerical results are presented for a plastic clay stone with strain hardening and an argilite with strain softening. The effects of plastic dilation, of the strain softening law and also of geometry of the cavity on the behaviour of the porous medium have been underlined. Analytical solutions provide valuable benchmarks enabling various numerical methods in undrained conditions with a finite boundary to be verified. Copyright © 2002 John Wiley & Sons, Ltd. [source]

    Numerical solution for consolidation and desiccation of soft soils

    Daniel T. C. Yao
    Abstract The consolidation and desiccation behaviour of soft soils can be described by two time-dependent non-linear partial differential equations using the finite strain theory. Analytical solutions do not exist for these governing equations. In this paper, we develop efficient numerical methods and software for finding the numerical solutions. We introduce a semi-implicit time integration scheme, and show numerically that our method converges. In addition, the numerical solution matches well with the experimental result. A boundary refinement method is also developed to improve the convergence and stability for the case of Neumann type boundary conditions. Interface governing equations are derived to maintain the continuity of consolidation and desiccation processes. This is useful because the soil column can undergo desiccation on top and consolidation on the bottom simultaneously. The numerical algorithms has been implemented into a computer program and the results have been verified with centrifuge test results conducted in our laboratory. Copyright © 2001 John Wiley & Sons, Ltd. [source]

    Wolbachia -induced unidirectional cytoplasmic incompatibility and the stability of infection polymorphism in parapatric host populations

    M. FLOR
    Abstract Wolbachia are intracellular, maternally inherited bacteria that are widespread among arthropods and commonly induce a reproductive incompatibility between infected male and uninfected female hosts known as unidirectional cytoplasmic incompatibility (CI). If infected and uninfected populations occur parapatrically, CI acts as a post-zygotic isolation barrier. We investigate the stability of such infection polymorphisms in a mathematical model with two populations linked by migration. We determine critical migration rates below which infected and uninfected populations can coexist. Analytical solutions of the critical migration rate are presented for mainland-island models. These serve as lower estimations for a more general model with two-way migration. The critical migration rate is positive if either Wolbachia causes a fecundity reduction in infected female hosts or its transmission is incomplete, and is highest for intermediate levels of CI. We discuss our results with respect to local adaptations of the Wolbachia host, speciation, and pest control. [source]

    Path-dependent currency options with mean reversion

    Hoi Ying Wong
    This paper develops a path-dependent currency option pricing framework in which the exchange rate follows a mean-reverting lognormal process. Analytical solutions are derived for barrier options with a constant barrier, lookback options, and turbo warrants. As the analytical solutions are obtained using a Laplace transform, this study numerically shows that the solution implemented with a numerical Laplace inversion is efficient and accurate. The pricing behavior of path-dependent options with mean reversion is contrasted with the Black-Scholes model. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:275,293, 2008 [source]

    Effects of glucose and nitrogen source concentration on batch fermentation kinetics of Lactococcus lactis under hemin-stimulated respirative condition

    Azher Razvi
    Abstract Analytical solutions to the ordinary differential equations governing the kinetics of cell growth, substrate utilization, and product formation of batch fermentation processes were derived and used to study the kinetics of the hemin-stimulated respiratory cultivation of Lactococcus lactis at varied initial glucose concentrations and nitrogen source concentrations. Studies revealed that initial glucose concentration varying in the range of 60 to 90 g/L had no significant substrate inhibitive effect. Furthermore, elevating the concentration of complex nitrogen sources while maintaining glucose concentration at 60% led to a high final biomass concentration of 6.6 g/L, substantially higher than that obtained with the basic medium, which was 4.1 g/L. [source]

    Pounding of structures modelled as non-linear impacts of two oscillators

    K. T. Chau
    Abstract A new formulation is proposed to model pounding between two adjacent structures, with natural periods T1 and T2 and damping ratios ,1 and ,2 under harmonic earthquake excitation, as non-linear Hertzian impact between two single-degree-of-freedom oscillators. For the case of rigid impacts, a special case of our analytical solution has been given by Davis (,Pounding of buildings modelled by an impact oscillator' Earthquake Engineering and Structural Dynamics, 1992; 21:253,274) for an oscillator pounding on a stationary barrier. Our analytical predictions for rigid impacts agree qualitatively with our numerical simulations for non-rigid impacts. When the difference in natural periods between the two oscillators increases, the impact velocity also increases drastically. The impact velocity spectrum is, however, relatively insensitive to the standoff distance. The maximum relative impact velocity of the coupled system can occur at an excitation period Tn* which is either between those of the two oscillators or less than both of them, depending on the ratios T1/T2 and ,1/,2. Although the pounding force between two oscillators has been primarily modelled by the Hertz contact law, parametric studies show that the maximum relative impact velocity is not very sensitive to changes in the contact parameters. Copyright © 2001 John Wiley & Sons, Ltd. [source]

    A risk-based approach for bidding strategy in an electricity pay-as-bid auction

    Javad Sadeh
    Abstract With the reform of electric power industry and the development of electrical energy markets in many countries, it is of significance to develop bidding strategies for generation companies (GenCos). In this environment, one of the most challenging and important tasks for a GenCo is developing effective strategies to optimize hourly offer curve. In this paper, focusing on Iran's electricity market structure, we model the bidding problem from the viewpoint of a GenCo in a pay-as-bid (PAB) auction. Our goal is to present a tool for determining the optimal bidding strategy of a price-taker producer in an electricity PAB auction taking into account the relevant risks. Due to uncertainties in power market, the market-clearing price (MCP) of each hour is assumed to be known as a probability density function (pdf). The optimal solution of bidding problem is obtained analytically based on the classical optimization theory. Also, the analytical solution for a multi-step bid protocol is generalized and the properties of the generalized solution are discussed. A model is developed to consider concept of risk using two different methods. The two proposed methods are then compared and the results interpreted using numerical examples. In addition, the effect of variation of MCP's pdf parameters on supplier's profit is studied. Copyright © 2007 John Wiley & Sons, Ltd. [source]

    Fast analytical short-circuit current calculation of rectifier-fed auxiliary subsystems

    M. Kunz
    The course of time of a three-phase rectifier system which its alternating valve participation can be interpreted as a continuous sequence of alternating switching states. To allow a more convenient calculation, the substitutional circuit with the converter is transformed into state-space coordinates. Hereby each operational mode of the rectifier can be represented by two linear independent space-phasor component networks. In the state-space, an analytical solution for this boundary value system can be carried out. After a retransformation back into the time domain, its time functions can be derived. In contrast to other calculation methods, no assumptions or simplifications have to be made like ideal smooth DC currents. Furthermore, all states of operation of the rectifier bridge can be easily calculated, which cover DC side idle-running to DC short-circuit. [source]

    From unambiguous quantum state discrimination to quantum state filtering

    J.A. Bergou
    Unambiguous discrimination among nonorthogonal but linearly independent quantum states is possible with a certain probability of success. Here, we consider a new variant of that problem. Instead of discriminating among all of the N different states, we now ask for less. We want to unambiguously assign the state to one of two complementary subsets of the set of N given non-orthogonal quantum states, each occurring with given a priori probabilities. We refer to the special case when one subset contains only one state and the other contains the remaining N -1 states as unambiguous quantum state filtering. We present an optimal analytical solution for the special case of N=3, and discuss the optimal strategy to unambiguously distinguish |,1, from the set {|,2,,|,3,}. For unambiguous filtering the subsets need not be linearly independent. We briefly discuss how to construct generalized interferometers (multiports) which provide a fully linear optical implementation of the optimal strategy. [source]

    A finite element analysis of tidal deformation of the entire earth with a discontinuous outer layer

    H. L. Xing
    SUMMARY Tidal deformation of the Earth is normally calculated using the analytical solution with some simplified assumptions, such as the Earth is a perfect sphere of continuous media. This paper proposes an alternative way, in which the Earth crust is discontinuous along its boundaries, to calculate the tidal deformation using a finite element method. An in-house finite element code is firstly introduced in brief and then extended here to calculate the tidal deformation. The tidal deformation of the Earth due to the Moon was calculated for an geophysical earth model with the discontinuous outer layer and compared with the continuous case. The preliminary results indicate that the discontinuity could have different effects on the tidal deformation in the local zone around the fault, but almost no effects on both the locations far from the fault and the global deformation amplitude of the Earth. The localized deformation amplitude seems to depend much on the relative orientation between the fault strike direction and the loading direction (i.e. the location of the Moon) and the physical property of the fault. [source]

    A Bayesian approach to estimating tectonic stress from seismological data

    Richard Arnold
    SUMMARY Earthquakes are conspicuous manifestations of tectonic stress, but the non-linear relationships between the stresses acting on a fault plane, its frictional slip, and the ensuing seismic radiation are such that a single earthquake by itself provides little information about the ambient state of stress. Moreover, observational uncertainties and inherent ambiguities in the nodal planes of earthquake focal mechanisms preclude straightforward inferences about stress being drawn on the basis of individual focal mechanism observations. However, by assuming that each earthquake in a small volume of the crust represents a single, uniform state of stress, the combined constraints imposed on that stress by a suite of focal mechanism observations can be estimated. Here, we outline a probabilistic (Bayesian) technique for estimating tectonic stress directions from primary seismological observations. The Bayesian formulation combines a geologically motivated prior model of the state of stress with an observation model that implements the physical relationship between the stresses acting on a fault and the resultant seismological observation. We show our Bayesian formulation to be equivalent to a well-known analytical solution for a single, errorless focal mechanism observation. The new approach has the distinct advantage, however, of including (1) multiple earthquakes, (2) fault plane ambiguities, (3) observational errors and (4) any prior knowledge of the stress field. Our approach, while computationally demanding in some cases, is intended to yield reliable tectonic stress estimates that can be confidently compared with other tectonic parameters, such as seismic anisotropy and geodetic strain rate observations, and used to investigate spatial and temporal variations in stress associated with major faults and coseismic stress perturbations. [source]

    Estimation of Degradation Rates by Satisfying Mass Balance at the Inlet

    GROUND WATER, Issue 4 2010
    Vedat Batu
    Using a steady-state mass conservative solute transport analytical solution that is based on the third-type (or flux-type or Cauchy) source condition, a method is developed to estimate the degradation parameters of solutes in groundwater. Then, the inadequacy of the methods based on the first-type source-based analytical solute transport solution is presented both theoretically and through an example. It is shown that the third-type source analytical solution exactly satisfies the mass balance constraint at the inlet location. It is also shown that the first-type source (or constant source concentration or Dirichlet) solution fails to satisfy the mass balance constraint at the inlet location and the degree of the failure depends on the value of the degradation as well as the flow and solute transport parameters. The error in the first-type source solution is determined with dimensionless parameters by comparing its results with the third-type source solution. Methods for estimating the degradation parameter values that are based on the first-type steady-state solute transport solution may significantly overestimate the degradation parameter values depending on the values of flow and solute transport parameters. It is recommended that the third-type source solution be used in estimating degradation parameters using measured concentrations instead of the first-type source solution. [source]

    A Stable and Efficient Numerical Algorithm for Unconfined Aquifer Analysis

    GROUND WATER, Issue 4 2009
    Elizabeth Keating
    The nonlinearity of equations governing flow in unconfined aquifers poses challenges for numerical models, particularly in field-scale applications. Existing methods are often unstable, do not converge, or require extremely fine grids and small time steps. Standard modeling procedures such as automated model calibration and Monte Carlo uncertainty analysis typically require thousands of model runs. Stable and efficient model performance is essential to these analyses. We propose a new method that offers improvements in stability and efficiency and is relatively tolerant of coarse grids. It applies a strategy similar to that in the MODFLOW code to the solution of Richard's equation with a grid-dependent pressure/saturation relationship. The method imposes a contrast between horizontal and vertical permeability in gridblocks containing the water table, does not require "dry" cells to convert to inactive cells, and allows recharge to flow through relatively dry cells to the water table. We establish the accuracy of the method by comparison to an analytical solution for radial flow to a well in an unconfined aquifer with delayed yield. Using a suite of test problems, we demonstrate the efficiencies gained in speed and accuracy over two-phase simulations, and improved stability when compared to MODFLOW. The advantages for applications to transient unconfined aquifer analysis are clearly demonstrated by our examples. We also demonstrate applicability to mixed vadose zone/saturated zone applications, including transport, and find that the method shows great promise for these types of problem as well. [source]

    Analytical Methods for Transient Flow to a Well in a Confined-Unconfined Aquifer

    GROUND WATER, Issue 4 2008
    Li-Tang Hu
    Concurrent existence of confined and unconfined zones of an aquifer can arise owing to ground water withdrawal by pumping. Using Girinskii's potential function, Chen (1974, 1983) developed an approximate analytical solution to analyze transient ground water flow to a pumping well in an aquifer that changes from an initially confined system to a system with both unconfined and confined regimes. This article presents the details of the Chen model and then compares it with the analytical model developed by Moench and Prickett (1972) for the same problem. Hypothetical pumping test examples in which the aquifer undergoes conversion from confined to water table conditions are solved by the two analytical models and also a numerical model based on MODFLOW. Comparison of the results suggests that the solutions of the Chen model give better results than the Moench and Prickett model except when the radial distance is very large or aquifer thickness is large compared with drawdown. [source]