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## Analytic Solution (analytic + solution)
## Selected Abstracts## STABILITY CONDITION OF DISTRIBUTED DELAY SYSTEMS BASED ON AN ANALYTIC SOLUTION TO LYAPUNOV FUNCTIONAL EQUATIONS ASIAN JOURNAL OF CONTROL, Issue 1 2006Young Soo SuhABSTRACT An analytic solution to Lyapunov functional equations for distributed delay systems is derived. The analytic solution is computed using a matrix exponential function, while conventional computation has been relied on numerical approximations. Based on the analytic solution, a necessary and sufficient stability condition for distributed delay systems with unknown but bounded constant delay is proposed. [source] ## Analytic solution for the nucleolus of a three-player cooperative game, NAVAL RESEARCH LOGISTICS: AN INTERNATIONAL JOURNAL, Issue 7 2010Mingming LengAbstract The nucleolus solution for cooperative games in characteristic function form is usually computed numerically by solving a sequence of linear programing (LP) problems, or by solving a single, but very large-scale, LP problem. This article proposes an algebraic method to compute the nucleolus solution analytically (i.e., in closed-form) for a three-player cooperative game in characteristic function form. We first consider cooperative games with empty core and derive a formula to compute the nucleolus solution. Next, we examine cooperative games with nonempty core and calculate the nucleolus solution analytically for five possible cases arising from the relationship among the value functions of different coalitions. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010 [source] ## Recombination lines and free-free continua formed in asymptotic ionized winds: Analytic solution for the radiative transfer ASTRONOMISCHE NACHRICHTEN, Issue 7 2009R. IgnaceAbstract In dense hot star winds, the infrared and radio continua are dominated by free-free opacity and recombination emission line spectra. In the case of a spherically symmetric outflow that is isothermal and expanding at constant radial speed, the radiative transfer for the continuum emission from a dense wind is analytic. Even the emission profile shape for a recombination line can be derived. Key to these derivations is that the opacity scales with only the square of the density. These results are well-known. Here an extension of the derivation is developed that also allows for line blends and the inclusion of an additional power-law dependence beyond just the density dependence. The additional power-law is promoted as a representation of a radius dependent clumping factor. It is shown that differences in the line widths and equivalent widths of the emission lines depend on the steepness of the clumping power-law. Assuming relative level populations in LTE in the upper levels of He II, an illustrative application of the model to Spitzer/IRS spectral data of the carbon-rich star WR 90 is given (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Sound propagation in a channel with a discontinuity in the form of a semicircle MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2003Volkmar WeiseAbstract For the description of wave propagation in ducts with variable cross-sections it is necessary to determine reflection and transmission at the points of discontinuity in the cross-sectional contour. Analytic solutions are well known for rectangular and circular duct cross-sections with rotational symmetric arrangements. The aim of this paper is the description of sound propagation in cylindrical ducts with semicircular discontinuities and thus non-rotationally symmetric discontinuities of the cross-sectional contour. In particular, excitation with higher azimuthal and radial modes also will be included. This solution is achieved by using the orthogonality properties of the trigonometric and Bessel functions. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Unquenched flavors in the Klebanov-Strassler theory FORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 7-9 2008Article first published online: 21 JUL 200, S. CremonesiAbstract We present an analytic solution of type IIB supergravity plus D7-branes, describing the addition of any number of flavors to the Klebanov-Strassler background. The dual field theory and its self-similar RG flow, described by a cascade of Seiberg dualities also in the presence of flavors, are discussed. The solution indicates that the dual gauge theory enjoys a duality wall in the UV. We stress the correspondence between Seiberg duality on the field theory side, and large gauge transformations on the RR and NSNS potentials on the gravity side. This contribution is mostly based on [1]. [source] ## Characterization of a Multilayer Aquifer Using Open Well Dilution Tests GROUND WATER, Issue 1 2007L. Jared WestAn approach to characterization of multilayer aquifer systems using open well borehole dilution is described. The approach involves measuring observation well flow velocities while a nearby extraction well is pumped by introducing a saline tracer into observation wells and collecting dilution vs. depth profiles. Inspection of tracer profile evolution allows discrete permeable layers within the aquifer to be identified. Dilution profiles for well sections between permeable layers are then converted into vertical borehole flow velocities and their evolution, using an analytic solution to the advection-dispersion equation applied to borehole flow. The dilution approach is potentially able to measure much smaller flow velocities that would be detectable using flowmeters. Vertical flow velocity data from the observation wells are then matched to those generated using a hydraulic model of the aquifer system, "shorted" by the observation wells, to yield the hydraulic properties of the constituent layers. Observation well flow monitoring of pumping tests represents a cost-effective alternative or preliminary approach to pump testing each layer of a multilayer aquifer system separately using straddle packers or screened wells and requires no prior knowledge of permeable layer depths and thicknesses. The modification described here, of using tracer dilution rather than flowmeter logging to obtain well flow velocities, allows the approach to be extended to greater well separations, thus characterizing a larger volume of the aquifer. An example of the application of this approach to a multilayer Chalk Aquifer in Yorkshire, Northeast England, is presented. [source] ## Analysis of Steady Ground Water Flow Toward Wells in a Confined-Unconfined Aquifer GROUND WATER, Issue 4 2006Chen Chong-XiA confined aquifer may become unconfined near the pumping wells when the water level falls below the confining unit in the case where the pumping rate is great and the excess hydraulic head over the top of the aquifer is small. Girinskii's potential function is applied to analyze the steady ground water flow induced by pumping wells with a constant-head boundary in a mixed confined-unconfined aquifer. The solution of the single-well problem is derived, and the critical radial distance at which the flow changes from confined to unconfined condition is obtained. Using image wells and the superposition method, an analytic solution is presented to study steady ground water flow induced by a group of pumping wells in an aquifer bounded by a river with constant head. A dimensionless function is introduced to determine whether a water table condition exists or not near the pumping wells. An example with three pumping wells is used to demonstrate the patterns of potentiometric surface and development of water table around the wells. [source] ## Traffic flow continuum modeling by hypersingular boundary integral equations INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2010Luis M. RomeroAbstract The quantity of data necessary in order to study traffic in dense urban areas through a traffic network, and the large volume of information that is provided as a result, causes managerial difficulties for the said model. A study of this kind is expensive and complex, with many sources of error connected to each step carried out. A simplification like the continuous medium is a reasonable approximation and, for certain dimensions of the actual problem, may be an alternative to be kept in mind. The hypotheses of the continuous model introduce errors comparable to those associated with geometric inaccuracies in the transport network, with the grouping of hundreds of streets in one same type of link and therefore having the same functional characteristics, with the centralization of all journey departure points and destinations in discrete centroids and with the uncertainty produced by a huge origin/destination matrix that is quickly phased out, etc. In the course of this work, a new model for characterizing traffic in dense network cities as a continuous medium, the diffusion,advection model, is put forward. The model is approached by means of the boundary element method, which has the fundamental characteristic of only requiring the contour of the problem to be discretized, thereby reducing the complexity and need for information into one order versus other more widespread methods, such as finite differences and the finite element method. On the other hand, the boundary elements method tends to give a more complex mathematical formulation. In order to validate the proposed technique, three examples in their fullest form are resolved with a known analytic solution. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## A numerical scheme for strong blast wave driven by explosion INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2006Kaori KatoAbstract After the detonation of a solid high explosive, the material has extremely high pressure keeping the solid density and expands rapidly driving strong shock wave. In order to simulate this blast wave, a stable and accurate numerical scheme is required due to large density and pressure changes in time and space. The compressible fluid equations are solved by a fractional step procedure which consists of the advection phase and non-advection phase. The former employs the Rational function CIP scheme in order to preserve monotone signals, and the latter is solved by interpolated differential operator scheme for achieving the accurate calculation. The procedure is categorized into the fractionally stepped semi-Lagrangian. The accuracy of our scheme is confirmed by checking the one-dimensional plane shock tube problem with 103 times initial density and pressure jump in comparison with the analytic solution. The Sedov,Taylor blast wave problem is also examined in the two-dimensional cylindrical coordinate in order to check the spherical symmetry and the convergence rates. Two- and three-dimensional simulations for the blast waves from the explosion in the underground magazine are carried out. It is found that the numerical results show quantitatively good agreement with the experimental data. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Linearized and non-linear acoustic/viscous splitting techniques for low Mach number flows INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2003Mohammad FarshchiAbstract Computation of the acoustic disturbances generated by unsteady low-speed flow fields including vortices and shear layers is considered. The equations governing the generation and propagation of acoustic fluctuations are derived from a two-step acoustic/viscous splitting technique. An optimized high order dispersion,relation,preserving scheme is used for the solution of the acoustic field. The acoustic field generated by a corotating vortex pair is obtained using the above technique. The computed sound field is compared with the existing analytic solution. Results are in good agreement with the analytic solution except near the centre of the vortices where the acoustic pressure becomes singular. The governing equations for acoustic fluctuations are then linearized and solved for the same model problem. The difference between non-linear and linearized solutions falls below the numerical error of the simulation. However, a considerable saving in CPU time usage is achieved in solving the linearized equations. The results indicate that the linearized acoustic/viscous splitting technique for the simulation of acoustic fluctuations generation and propagation by low Mach number flow fields seems to be very promising for three-dimensional problems involving complex geometries. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## An approximate projection method for incompressible flow INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2002David E. StevensThis paper presents an approximate projection method for incompressible flows. This method is derived from Galerkin orthogonality conditions using equal-order piecewise linear elements for both velocity and pressure, hereafter Q1Q1. By combining an approximate projection for the velocities with a variational discretization of the continuum pressure Poisson equation, one eliminates the need to filter either the velocity or pressure fields as is often needed with equal-order element formulations. This variational approach extends to multiple types of elements; examples and results for triangular and quadrilateral elements are provided. This method is related to the method of Almgren et al. (SIAM J. Sci. Comput. 2000; 22: 1139,1159) and the PISO method of Issa (J. Comput. Phys. 1985; 62: 40,65). These methods use a combination of two elliptic solves, one to reduce the divergence of the velocities and another to approximate the pressure Poisson equation. Both Q1Q1 and the method of Almgren et al. solve the second Poisson equation with a weak error tolerance to achieve more computational efficiency. A Fourier analysis of Q1Q1 shows that a consistent mass matrix has a positive effect on both accuracy and mass conservation. A numerical comparison with the widely used Q1Q0 (piecewise linear velocities, piecewise constant pressures) on a periodic test case with an analytic solution verifies this analysis. Q1Q1 is shown to have comparable accuracy as Q1Q0 and good agreement with experiment for flow over an isolated cubic obstacle and dispersion of a point source in its wake. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## MAP fusion method for superresolution of images with locally varying pixel quality INTERNATIONAL JOURNAL OF IMAGING SYSTEMS AND TECHNOLOGY, Issue 4 2008Kio KimAbstract Superresolution is a procedure that produces a high-resolution image from a set of low-resolution images. Many of superresolution techniques are designed for optical cameras, which produce pixel values of well-defined uncertainty, while there are still various imaging modalities for which the uncertainty of the images is difficult to control. To construct a superresolution image from low-resolution images with varying uncertainty, one needs to keep track of the uncertainty values in addition to the pixel values. In this paper, we develop a probabilistic approach to superresolution to address the problem of varying uncertainty. As direct computation of the analytic solution for the superresolution problem is difficult, we suggest a novel algorithm for computing the approximate solution. As this algorithm is a noniterative method based on Kalman filter-like recursion relations, there is a potential for real-time implementation of the algorithm. To show the efficiency of our method, we apply this algorithm to a video sequence acquired by a forward looking sonar system. © 2008 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 18, 242,250, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). [source] ## Simulation of EM wave propagation in magnetoelectric media using TLM INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS, Issue 6 2001Christos ChristopoulosAbstract In this paper, the simulation of general frequency-dependent magnetoelectric material properties in time-domain TLM is described. The formulation is developed from Maxwell's equations and the constitutive relations using bilinear ,,-transform methods leading to a general Padé system. The approach is applicable to all frequency-dependent linear materials including those displaying anisotropic and bianisotropic behaviour. The method is validated by the example of a chiral slab having an analytic solution. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## H, preview tracking in output feedback setting INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 7 2004Akira KojimaAbstract The performance of preview control is investigated in terms of H, -criterion. In the output feedback setting, an H, preview control problem is discussed and the analytic solution is characterized by introducing matrix Riccati equations. The strength and the limitation of the H, preview performance are illustrated with numerical examples. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Estimation of the hindered settling function R(,) from batch-settling tests AICHE JOURNAL, Issue 4 2005Daniel R. LesterAbstract The hindered settling function R(,) is a material function that quantifies the interphase drag of colloidal suspensions for all solids volume fractions ,. A method is presented to estimate R(,) from batch-settling tests for solids volume fractions between the initial solids volume fraction, ,0, and the solids volume fraction at which the suspension forms a continuously networked structure, ,g, known as the gel point. The method is based on an analytic solution of the associated inverse problem. Techniques are presented to address initialization mechanics observed in such tests as well as experimental noise and discrete data. Analysis of synthetic and experimental data suggests that accurate estimates of R(,) are possible in most cases. These results provide scope for characterization of suspension dewaterability from batch-settling tests alone. © 2005 American Institute of Chemical Engineers AIChE J, 2005 [source] ## TRIVARIATE SUPPORT OF FLAT-VOLATILITY FORWARD LIBOR RATES MATHEMATICAL FINANCE, Issue 2 2010Farshid JamshidianThis paper investigates the multivariate support of forward Libor rates in the one-factor, constant volatilities Libor market model. The comparatively simple bivariate case was solved in Jamshidian (2008) in connection to the recent finding by Davis and Mataix-Pastor (2007) of positive probability of negative Libor rates in the swap market model. The approach here builds on Jamshidian (2008) but becomes really effective only in the trivariate case, and there particularly for a special "flat-volatility" case, leading to an analytic solution. The main idea is a certain recursion in the Libor market model by means of which the calculation of the support is reduced to a calculus of variation problem (with bounds on the slope). [source] ## Application of parameter differentiation for flow of a third grade fluid past an infinite porous plate NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2010M. SajidAbstract This article investigates the analytic solution for the flow of a third grade fluid past an infinite porous plate. The method of parameter differentiation is used to linearized the governing flow equation. The solution of the obtained linear equation is developed by differential transform method in combination with the method of superposition. The obtained results are compared with existing results in the literature and an excellent agreement is found. This shows that the parameter differentiation is a powerful technique for solving nonlinear problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010 [source] ## Optimal auditing in the banking industry OPTIMAL CONTROL APPLICATIONS AND METHODS, Issue 2 2008T. BoschAbstract As a result of the new regulatory prescripts for banks, known as the Basel II Capital Accord, there has been a heightened interest in the auditing process. Our paper considers this issue with a particular emphasis on the auditing of reserves, assets and capital in both a random and non-random framework. The analysis relies on the stochastic dynamic modeling of banking items such as loans, reserves, Treasuries, outstanding debts, bank capital and government subsidies. In this regard, one of the main novelties of our contribution is the establishment of optimal bank reserves and a rate of depository consumption that is of importance during an (random) audit of the reserve requirements. Here the specific choice of a power utility function is made in order to obtain an analytic solution in a Lévy process setting. Furthermore, we provide explicit formulas for the shareholder default and regulator closure rules, for the case of a Poisson-distributed random audit. A property of these rules is that they define the standard for minimum capital adequacy in an implicit way. In addition, we solve an optimal auditing time problem for the Basel II capital adequacy requirement by making use of Lévy process-based models. This result provides information about the optimal timing of an internal audit when the ambient value of the capital adequacy ratio is taken into account and the bank is able to choose the time at which the audit takes place. Finally, we discuss some of the economic issues arising from the analysis of the stochastic dynamic models of banking items and the optimization procedure related to the auditing process. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Threshold values of random K -SAT from the cavity method RANDOM STRUCTURES AND ALGORITHMS, Issue 3 2006Stephan MertensAbstract Using the cavity equations of Mézard, Parisi, and Zecchina Science 297 (2002), 812; Mézard and Zecchina, Phys Rev E 66 (2002), 056126 we derive the various threshold values for the number of clauses per variable of the random K -satisfiability problem, generalizing the previous results to K , 4. We also give an analytic solution of the equations, and some closed expressions for these thresholds, in an expansion around large K. The stability of the solution is also computed. For any K, the satisfiability threshold is found to be in the stable region of the solution, which adds further credit to the conjecture that this computation gives the exact satisfiability threshold.© 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006 [source] ## Limitations of a linear model for the hurricane boundary layer THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 641 2009Stefanie VoglAbstract The linear model for the steady boundary layer of a rapidly rotating axisymmetric vortex is derived from a detailed scale analysis of the full equations of motion. The previously known analytic solution is re-appraised for vortices of hurricane scale and strength. The internal consistency of the linear approximation is investigated for such a vortex by calculating from the solution the magnitude of the nonlinear terms that are neglected in the approximation compared with the terms retained. It is shown that the nonlinear terms are not negligibly small in a large region of the vortex, a feature that is consistent with the scale analysis. We argue that the boundary-layer problem is well-posed only at outer radii where there is subsidence into the layer. At inner radii, where there is ascent, only the radial pressure gradient may be prescribed and not the wind components at the top of the boundary layer, but the linear problem cannot be solved in these circumstances. We examine the radius at which the vertical flow at the top of the boundary layer changes sign for different tangential wind profiles relevant to hurricanes and show that this is several hundred kilometres from the vortex centre. This feature represents a further limitation of the linear model applied to hurricanes. While the present analysis assumes axial symmetry, the same limitations presumably apply to non-axisymmetric extensions to the linear model. Copyright © 2009 Royal Meteorological Society [source] ## Instability of finite-amplitude lower-neutral Eady waves THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 620 2006M. FantiniAbstract The problem of the stability of the two-dimensional neutral Eady wave, studied numerically by Fantini and Davolio (2001) as a possible mechanism leading to strong orographic cyclogenesis, is re-examined. An approximate analytic solution is found, and a simple condition for the appearance of meridionally structured unstable secondary modes is proposed, based on the relative vorticity of the primary wave. The approximate analytic solution compares well with the numerical simulations and extends previous results. Copyright © 2006 Royal Meteorological Society [source] ## Explicit nonlinear predictive control for a magnetic levitation system, ASIAN JOURNAL OF CONTROL, Issue 3 2010A. UlbigAbstract The paper presents a methodology for the construction of an explicit nonlinear control law via approximation of the nonlinear constrained finite-time optimal control (CFTOC). This is achieved through an approximate mapping of a general nonlinear system in a set of linear piecewise affine (PWA) systems. The key advantages of this methodology are two-fold. First, the construction of an analytic solution of the CFTOC problem leads to an efficient explicit implementation. Second, by taking advantage of model predictive control's systematic fashion to handle constraints, an improved performance can be obtained for the closed-loop system. The proposed theory is applied in real-time for a system with fast dynamics: a magnetic levitation benchmark. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society [source] ## STABILITY CONDITION OF DISTRIBUTED DELAY SYSTEMS BASED ON AN ANALYTIC SOLUTION TO LYAPUNOV FUNCTIONAL EQUATIONS ASIAN JOURNAL OF CONTROL, Issue 1 2006Young Soo SuhABSTRACT An analytic solution to Lyapunov functional equations for distributed delay systems is derived. The analytic solution is computed using a matrix exponential function, while conventional computation has been relied on numerical approximations. Based on the analytic solution, a necessary and sufficient stability condition for distributed delay systems with unknown but bounded constant delay is proposed. [source] ## An Adaptive Strategy for the Local Discontinuous Galerkin Method Applied to Porous Media Problems COMPUTER-AIDED CIVIL AND INFRASTRUCTURE ENGINEERING, Issue 4 2008Esov S. VelázquezDG methods may be viewed as high-order extensions of the classical finite volume method and, since no interelement continuity is imposed, they can be defined on very general meshes, including nonconforming meshes, making these methods suitable for h-adaptivity. The technique starts with an initial conformal spatial discretization of the domain and, in each step, the error of the solution is estimated. The mesh is locally modified according to the error estimate by performing two local operations: refinement and agglomeration. This procedure is repeated until the solution reaches a desired accuracy. The performance of this technique is examined through several numerical experiments and results are compared with globally refined meshes in examples with known analytic solutions. [source] ## A General Formula for Valuing Defaultable Securities ECONOMETRICA, Issue 5 2004P. Collin-DufresnePrevious research has shown that under a suitable no-jump condition, the price of a defaultable security is equal to its risk-neutral expected discounted cash flows if a modified discount rate is introduced to account for the possibility of default. Below, we generalize this result by demonstrating that one can always value defaultable claims using expected risk-adjusted discounting provided that the expectation is taken under a slightly modified probability measure. This new probability measure puts zero probability on paths where default occurs prior to the maturity, and is thus only absolutely continuous with respect to the risk-neutral probability measure. After establishing the general result and discussing its relation with the existing literature, we investigate several examples for which the no-jump condition fails. Each example illustrates the power of our general formula by providing simple analytic solutions for the prices of defaultable securities. [source] ## Functionalized-Silk-Based Active Optofluidic Devices ADVANCED FUNCTIONAL MATERIALS, Issue 7 2010Konstantinos TsiorisAbstract Silk protein from the silkworm Bombyx mori has excellent chemical and mechanical stability, biocompatibility, and optical properties. Additionally, when the protein is purified and reformed into materials, the biochemical functions of dopants entrained in the protein matrix are stabilized and retained. This unique combination of properties make silk a useful multifunctional material platform for the development of sensor devices. An approach to increase the functions of silk-based devices through chemical modifications to demonstrate an active optofluidic device to sense pH is presented. Silk protein is chemically modified with 4-aminobenzoic acid to add spectral-color-responsive pH sensitivity. The functionalized silk is combined with the elastomer poly(dimethyl siloxane) in a single microfluidic device. The microfluidic device allows spatial and temporal control of the delivery of analytic solutions to the system to provide the optical response of the optofluidic device. The modified silk is stable and spectrally responsive over a wide pH range from alkaline to acidic. [source] ## A unified continuum representation of post-seismic relaxation mechanisms: semi-analytic models of afterslip, poroelastic rebound and viscoelastic flow GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2010Sylvain BarbotSUMMARY We present a unified continuum mechanics representation of the mechanisms believed to be commonly involved in post-seismic transients such as viscoelasticity, fault creep and poroelasticity. The time-dependent relaxation that follows an earthquake, or any other static stress perturbation, is considered in a framework of a generalized viscoelastoplastic rheology whereby some inelastic strain relaxes a physical quantity in the material. The relaxed quantity is the deviatoric stress in case of viscoelastic relaxation, the shear stress in case of creep on a fault plane and the trace of the stress tensor in case of poroelastic rebound. In this framework, the instantaneous velocity field satisfies the linear inhomogeneous Navier's equation with sources parametrized as equivalent body forces and surface tractions. We evaluate the velocity field using the Fourier-domain Green's function for an elastic half-space with surface buoyancy boundary condition. The accuracy of the proposed method is demonstrated by comparisons with finite-element simulations of viscoelastic relaxation following strike-slip and dip-slip ruptures for linear and power-law rheologies. We also present comparisons with analytic solutions for afterslip driven by coseismic stress changes. Finally, we demonstrate that the proposed method can be used to model time-dependent poroelastic rebound by adopting a viscoelastic rheology with bulk viscosity and work hardening. The proposed method allows one to model post-seismic transients that involve multiple mechanisms (afterslip, poroelastic rebound, ductile flow) with an account for the effects of gravity, non-linear rheologies and arbitrary spatial variations in inelastic properties of rocks (e.g. the effective viscosity, rate-and-state frictional parameters and poroelastic properties). [source] ## Energy flux in viscoelastic anisotropic media GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2006Vlastislav, ervenýSUMMARY We study properties of the energy-flux vector and other related energy quantities of homogeneous and inhomogeneous time-harmonic P and S plane waves, propagating in unbounded viscoelastic anisotropic media, both analytically and numerically. We propose an algorithm for the computation of the energy-flux vector, which can be used for media of unrestricted anisotropy and viscoelasticity, and for arbitrary homogeneous or inhomogeneous plane waves. Basic part of the algorithm is determination of the slowness vector of a homogeneous or inhomogeneous wave, which satisfies certain constraints following from the equation of motion. Approaches for determination of a slowness vector commonly used in viscoelastic isotropic media are usually difficult to use in viscoelastic anisotropic media. Sometimes they may even lead to non-physical solutions. To avoid these problems, we use the so-called mixed specification of the slowness vector, which requires, in a general case, solution of a complex-valued algebraic equation of the sixth degree. For simpler cases, as for SH waves propagating in symmetry planes, the algorithm yields simple analytic solutions. Once the slowness vector is known, determination of energy flux and of other energy quantities is easy. We present numerical examples illustrating the behaviour of the energy-flux vector and other energy quantities, for homogeneous and inhomogeneous plane P, SV and SH waves. [source] ## Solution of the unsaturated soil moisture equation using repeated transforms INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 15 2001S. G. FityusAbstract An alternative method of solution for the linearized ,theta-based' form of the Richards equation of unsaturated flow is developed in two spatial dimensions. The Laplace and Fourier transformations are employed to reduce the Richards equation to an ordinary differential equation in terms of a transformed moisture content and the transform variables, s and ,. Separate analytic solutions to the transformed equation are developed for initial states which are either in equilibrium or dis-equilibrium. The solutions are assembled into a finite layer formulation satisfying continuity of soil suction, thereby facilitating the analysis of horizontally stratified soil profiles. Solution techniques are outlined for various boundary conditions including prescribed constant moisture content, prescribed constant flux and flux as a function of moisture change. Example solutions are compared with linearized finite element solutions. The agreement is found to be good. An adaptation of the method for treating the quasilinearized Richards equation with variable diffusivity is also described. Comparisons of quasilinear solutions with some earlier semi-analytical, finite element and finite difference results are also favourable. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## Discrete element method for modelling solid and particulate materials INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2007Federico A. TavarezAbstract The discrete element method (DEM) is developed in this study as a general and robust technique for unified two-dimensional modelling of the mechanical behaviour of solid and particulate materials, including the transition from solid phase to particulate phase. Inter-element parameters (contact stiffnesses and failure criteria) are theoretically established as functions of element size and commonly accepted material parameters including Young's modulus, Poisson's ratio, ultimate tensile strength, and fracture toughness. A main feature of such an approach is that it promises to provide convergence with refinement of a DEM discretization. Regarding contact failure, an energy criterion based on the material's ultimate tensile strength and fracture toughness is developed to limit the maximum contact forces and inter-element relative displacement. This paper also addresses the issue of numerical stability in DEM computations and provides a theoretical method for the determination of a stable time-step. The method developed herein is validated by modelling several test problems having analytic solutions and results show that indeed convergence is obtained. Moreover, a very good agreement with the theoretical results is obtained in both elastic behaviour and fracture. An example application of the method to high-speed penetration of a concrete beam is also given. Copyright © 2006 John Wiley & Sons, Ltd. [source] |