Laplace Inversion Method Research Articles

Consolidation settlement of vertically loaded pile groups in multilayered poroelastic soils

Pile groups are commonly used as the foundations of many structures including those used in transportation infrastructures. Consolidation settlement of a pile foundation is an important design parameter. A theoretical model is developed in this study to estimate the consolidation settlement and axial load transfer of vertically loaded pile groups in multilayered poroelastic soils. The multilayered saturated soil is modeled according to Biot’s poroelasticity theory. In order to determine quasi-static response of pile groups, the interaction problem is first formulated in the Laplace transform domain. Vertical displacement compatibility is enforced at the pile-soil interface to simulate the pile group-soil interaction. Axial deformation of each pile is represented by an exponential series with undetermined coefficients, which are obtained from a variational approach. Vertical displacement influence functions due to a buried uniform vertical load applied to the layered soil are required in the formulation. The application of an exact stiffness matrix method yields the required influence functions. Time-domain solutions are obtained by employing a numerical Laplace inversion method. Numerical results for time-dependent vertical stiffness and consolidation settlement are presented for different pile group configurations, layer profiles, pile elastic moduli and pile lengths.

On the Bimodal Radial Solute Transport in Dual‐Permeability Porous Media

AbstractPorous media usually exhibit distinct transport properties of a dual domain, which is attributed to layering, fracturing, or aggregation. In this study, a mobile‐mobile (MM) model of radial solute transport consisting of two advection‐dispersion equations (ADEs) is presented to describe the bimodal (or double‐peaked) breakthrough curves (BTCs) through two distinct pore domains consisting of interrelated fast‐flow and slow‐flow domains. Transport processes, including advection, dispersion, and first‐order mass transfer between the two domains, were considered in the model. Semi‐analytical solutions were derived using a Laplace transform, matrix diagonalization and numerical Laplace inversion method, and the solutions were verified against the existing analytical solution for the mobile‐immobile (MIM) model under the special case of zero velocity in the slow‐flow domain. In addition, Markov Chain Monte Carlo is applied to estimate the MM model parameters using the experimental data. The results indicate that a larger transfer coefficient between the fast‐flow and the slow‐flow domains resulted in a weaker bimodal phenomenon for concentration distribution curves, which was notably more apparent when there was a greater variation in pore flow velocities between the two domains. Also, the MM model can be used to investigate other types of abnormal transport features such as late arrival and early arrival. Finally, the ADE, MIM, and MM models were used to interpret the experimental data conducted by Fernàndez‐Garcia et al. (2004, http://doi.org.10.1029/2004wr003112), and it was found that the proposed MM model can explain the bimodal feature of BTCs while the ADE and MIM models fail to do so.

A mathematical framework for multiphase poromechanics in multiple porosity media

Unconventional geomaterials often exhibit multi-modal pore size distribution. We have developed a comprehensive framework for porous media exhibiting multiple porosity scales that are saturated with one or two types of fluids using mixture theory. Both the governing equations and constitutive laws have been clearly derived and identified, respectively. The effective stress σ′ emerged from the energy balance equation is adoptable for both elastic and elastoplastic deformations, in which pore fractions and saturations play a central role. The proposed model is general in a sense that it works for both uncoupled simulation and coupled simulation. The field equations for uncoupled flow simulation are solved using the Laplace transform and numerical Laplace inversion methods. By visualizing the dimensionless results, we can gain a quantitative insight of the different stages in the depletion process of a naturally fractured reservoir. For coupled flow and geomechanics simulation, a strip load problem and a two-phase flow in a deformable 3D reservoir problem illustrate the impacts of plasticity, multiple porosity, inter-porosity exchange, and capillary pressure on the system response.

УЧЕТ ТРЕНИЯ НА НАЧАЛЬНОМ ЭТАПЕ ВЕРТИКАЛЬНОГО ВНЕДРЕНИЯ ВЫПУКЛОГО УДАРНИКА В УПРУГУЮ ПОЛУПЛОСКОСТЬ

The initial stage of indentation of a convex rigid punch into isotropic elastic half-plane with friction is considered. A closed mathematical formulation in Cartesian coordinates is presented by equations of motion in the absence of a body force for homogeneous isotropic elastic medium in terms of displacement potentials, Cauchy equations for deformations, Hooke law and translational equation for the punch. Initial conditions are homogeneous. Outside of contact region surface is free from stresses. Inside contact region normal displacements of the surface of a half-space and the surface of a punch are taken to be equal and the relation between tangential and normal stresses in a form of Coulomb friction law is given. Due to the short duration of the supersonic stage and smallness of the contact region radius the tangential stress direction is considered to be point independent. Resolving functional equations are given in a form of convolutions with influence functions. The latter is the solution the original problem for half-plane with a special boundary condition in a form of the Dirac delta function. This solution is obtained in the domain of Laplace transform by time and Fourier transform by spatial coordinate. The original is constructed with the combined Fourier–Laplace inversion method. Solution is given with consideration to the initial stage characteristic property of supersonic velocity of contact region expansion (no smaller than propagation velocity of expansion-compression waves). It is shown that kinematic parameters of the punch, resulting force and discontinuities of the first kind on boundaries of the contact region are independent of friction unlike contact pressure. Calculation examples for different values of friction coefficient are presented.

Analytical Model for Leakage Detection in CO2 Sequestration in Deep Saline Aquifers: Application to ex Situ and in Situ CO2 Sequestration Processes.

One of the prominent methods for carbon dioxide sequestration is disposal into deep saline aquifers. This is mainly because deep saline aquifers provide significant capacity for storage of unwanted fluids underground for a long period. However, saline aquifers may have a leaky cap rock. The sealing capacity of a cap rock must, therefore, be evaluated to ensure the integrity and safety of its storage media; hence robust classifications of the cap rock are required even before starting the storage/disposal operations. Aqueous fluids can be injected into a target storage aquifer, and pressure changes owing to leakage can be monitored in an upper aquifer separated by a cap rock for a short period. The measurement of pressure responses in the monitoring aquifer can be used to identify and characterize any leakage path in the cap rock. This paper provides analytical models in the Laplace domain for both in situ and ex situ CO2 sequestration methods. Using the numerical Laplace inverse method called the “Stehfest” method, the analytical solution in the real domain is calculated. The analytical solutions developed can be used for determining both dimensionless pressure changes in monitoring and storage aquifers due to leakages and dimensionless leakage rates.

An exact analytical model for fluid flow through finite rock matrix block with special saturation function

An exact analytical solution for one-dimensional fluid flow through rock matrix block is presented. The nonlinearity induced from flow functions makes the governing equations describing this mechanism difficult to be analytically solved. In this paper, an analytical solution to the infiltration problems considering non-linear relative permeability functions is presented for finite depth, despite its profound and fundamental importance. Elimination of the nonlinear terms in the equation, as a complex and tedious task, is done by applying several successive mathematical manipulations including: Hopf-Cole transformation to obtain a diffusive type PDE; an exponential type transformation to get a convective-diffusive type PDE with suitable boundary conditions; Laplace transformation; Integral transformation to homogenize the boundary condition in the Laplace domain; and finally Laplace inversion method to find the time domain solution. The obtained solution is used for developing a new matrix-fracture transfer function. The developed analytical equations are used for prediction of drying front in the case of evaporation from deep water table. The model results are in close agreement with the experimental data and numerical simulation of the process. Results of sensitivity analysis of different parameters on recovery rate and ultimate production based on design of experiment method are also discussed.

Transient Dean flow in a channel with suction/injection: A semi-analytical approach

The mathematical model responsible for fully developed laminar transient flow formation between the gaps of two stationary concentric porous tubes due to the imposition of azimuthal pressure gradient (Dean flow) is solved semi-analytically. The tubes walls are porous so that a radial flow can be superimposed. The solution of the momentum and continuity equations are obtained semi-analytically by using the combination of Laplace transform technique and a Laplace inversion method called Riemann sum approximation method. The solutions for skin friction at [Formula: see text] (outer surface of the inner porous tube) and [Formula: see text] (inner surface of the outer porous tube) are presented. The impact of suction/injection parameter and the ratio of the radii of the tubes are examined for the velocity profile and skin friction. Results show that the velocity profile decreases with increase in suction/injection parameter for various values of time, [Formula: see text] and at large value of time, ([Formula: see text]), the velocity and the skin friction attain a steady state.

Hyperpolarized Laplace NMR.

Laplace nuclear magnetic resonance (NMR), dealing with NMR relaxation and diffusion experiments, reveals details of molecular motion and provides chemical resolution complementary to NMR spectra. Laplace NMR has witnessed a great progress in past decades due to the development of methodology and signal processing, and it has lots of extremely interesting applications in various fields, including chemistry, biochemistry, geology, archaeology, and medicine. The aim of this minireview is to give a pedagogically oriented overview of Laplace NMR. It does not provide a full literature review of the field, but, instead, it elucidate the benefits and features of Laplace NMR methods through few selected examples. The minireview describes also recent progress in multidimensional Laplace NMR and Laplace inversion methods. Furthermore, the potential of modern hyperpolarization methods as well as ultrafast approach to increase the sensitivity and time-efficiency of the Laplace NMR experiments is highlighted.

Performance of some numerical Laplace inversion methods on American put option formula

Numerical inversion approaches of Laplace transform is used to obtain a semianalytic solution. Some of the mathematical inversion methods such as Durbin-Crump, Widder, and Papoulis can be used to calculate American put options through the optimal exercise price in the Laplace space. The comparison of methods on some simple functions is aimed to know the accuracy and parameters which used in the calculation of American put options. The result obtained is the performance of each method regarding accuracy and computational speed. The Durbin-Crump method has an average error relative of 2.006e-004 with computational speed of 0.04871 seconds, the Widder method has an average error relative of 0.0048 with computational speed of 3.100181 seconds, and the Papoulis method has an average error relative of 9.8558e-004 with computational speed of 0.020793 seconds.

Probing numerical Laplace inversion methods for two and three-site molecular exchange between interconnected pore structures

Two-dimension (2D) Nuclear Magnetic Resonance relaxometry experiments are a powerful tool extensively used to probe the interaction among different pore structures, mostly in inorganic systems. The analysis of the collected experimental data generally consists of a 2D numerical inversion of time-domain data where T2-T2 maps are generated. Through the years, different algorithms for the numerical inversion have been proposed. In this paper, two different algorithms for numerical inversion are tested and compared under different conditions of exchange dynamics; the method based on Butler–Reeds–Dawson (BRD) algorithm and the fast-iterative shrinkage-thresholding algorithm (FISTA) method. By constructing a theoretical model, the algorithms were tested for a two- and three-site porous media, varying the exchange rates parameters, the pore sizes and the signal to noise ratio. In order to test the methods under realistic experimental conditions, a challenging organic system was chosen. The molecular exchange rates of water confined in hierarchical porous polymeric networks were obtained, for a two- and three-site porous media. Data processed with the BRD method was found to be accurate only under certain conditions of the exchange parameters, while data processed with the FISTA method is precise for all the studied parameters, except when SNR conditions are extreme.

Performance of Gahver-Stehfest Numerical Laplace Inversion Method on Option Pricing Formulas

In this paper we study the performance of Gahver-Stehfest numerical Laplace inversion method. The method is applied to some simple functions which have analytical Laplace inversion and the option pricing formulas which their analytical inversions are not available. The accuracy and efficiency of the methods for each functions are presented.

On the exact maximum likelihood inference of Fisher\u2013Bingham distributions using an adjusted holonomic gradient method

Holonomic function theory has been successfully implemented in a series of recent papers to efficiently calculate the normalizing constant and perform likelihood estimation for the Fisher–Bingham distributions. A key ingredient for establishing the standard holonomic gradient algorithms is the calculation of the Pfaffian equations. So far, these papers either calculate these symbolically or apply certain methods to simplify this process. Here we show the explicit form of the Pfaffian equations using the expressions from Laplace inversion methods. This improves on the implementation of the holonomic algorithms for these problems and enables their adjustments for the degenerate cases. As a result, an exact and more dimensionally efficient ODE is implemented for likelihood inference.

Application of Numerical Laplace Inversion Methods in Chemical Engineering with Maple®

Abstract In this paper, the procedure of choice of the effective numerical inversion method of Laplace transform for solving of the mathematical model of gas flow was shown. Two inversion methods are evaluated in this work: the Gaver-Stehfest method, and the Dubner and Abate method. The numerical tests has been made using test functions. Based on the test results the fast and accurate method was selected and applied for modeling of the gas flow. The Gaver-Stehfest method has proven the most effective. The model was developed using the computer algebra system Maple®. Implementation of the methods were done by authors.

Ghost calibration and the pricing of barrier options and CDS in spectrally one-sided Lévy models: the parabolic Laplace inversion method

Recently, the advantages of conformal deformations of the contours of integration in pricing formulas were demonstrated in the context of wide classes of Lévy models and the Heston model. In the present paper, we construct efficient conformal deformations of the contours of integration in the pricing formulas for barrier options and CDS in the setting of spectrally one-sided Lévy models, taking advantage of Rogers’s trick [J. Appl. Prob., 2000, 37, 1173–1180] that greatly simplifies calculation of the Wiener-Hopf factors. We extend the trick to wide classes of Lévy processes of infinite variation with zero diffusion component. In the resulting formulas (both in the finite variation and the infinite variation cases), we make quasi-parabolic deformations as in Boyarchenko and Levendorskiĭ [Int. J. Theor. Appl. Finance, 2013, 16 (3), 1350011], which greatly increase the rate of convergence of the integrals. We demonstrate that in many cases the proposed method is more accurate than the standard realization of Laplace inversion. We also exhibit examples in which the standard realization is so unstable that it cannot be used for any choice of the error control parameters. This may lead to a ghost calibration: a situation where a parameter set of a model is declared to be a ‘good fit’ to the data only because the errors of calibration and of the numerical method used for pricing (almost) cancel each other out.

Ghost Calibration and Pricing Barrier Options and CDS in Spectrally One-Sided L'evy Models: The Parabolic Laplace Inversion Method

Recently, the advantages of conformal deformations of the contours of integration in pricing formulas were demonstrated in the context of wide classes of Levy models and the Heston model. In the present paper, we construct efficient conformal deformations of the contours of integration in the pricing formulas for barrier options and CDS in the setting of spectrally one-sided Levy models, taking advantage of Rogers’s trick [J. Appl. Prob., 2000, 37, 1173–1180] that greatly simplifies calculation of the Wiener-Hopf factors. We extend the trick to wide classes of Levy processes of infinite variation with zero diffusion component. In the resulting formulas (both in the finite variation and the infinite variation cases), we make quasi-parabolic deformations as in Boyarchenko and Levendorskiĭ [Int. J. Theor. Appl. Finance, 2013, 16 (3), 1350011], which greatly increase the rate of convergence of the integrals. We demonstrate that in many cases the proposed method is more accurate than the standard realization o...

Discrete-Time CPPI Under Transaction Cost and Regime Switching

This paper studies the performance of discrete-time constant proportional portfolio insurance (CPPI) under proportional trading cost and regime switching. Explicit formulas are developed for a variety of measures for the performance of a CPPI portfolio, and a double-sided Laplace inversion method is developed to compute the Omega measure of a CPPI portfolio. The established formulas can be easily implemented for sensitivity analysis on performance of a CPPI portfolio, and a numerical example with a real data set of S&P 500 index is presented to illustrate the eects the regime switching feature of the nancial market and the existence of transaction cost can exert on the performance of a CPPI portfolio.

DUAL PHASE LAG HEAT CONDUCTION IN FUNCTIONALLY GRADED HOLLOW SPHERES

In the present work, the dual phase lag heat conduction in functionally graded hollow spheres is investigated under spherically symmetric and axisymmetric thermal loading. The heat conduction equation is given based on the dual phase lag theory to consider the details of energy transport in the material in comparison with the non-Fourier hyperbolic heat conduction. All the material properties of the sphere are taken to vary continuously along the radial direction following a power-law with arbitrary non-homogeneity indices except the phase lags which are assumed to be constant for simplicity. The specified spherically symmetric and axisymmetric boundary conditions of the sphere lead to a 1D and 2D heat conduction problem, respectively. Employing the Laplace transform to eliminate the time dependency of the problem, analytical solutions are obtained for the temperature and heat flux. The final results in the time domain are obtained by a numerical Laplace inversion method. The speed of thermal wave in the functionally graded sphere based on the dual phase lag is compared with that of the hyperbolic heat conduction. Furthermore, the numerical results are shown to clarify the effects of phase lags and non-homogeneity indices on the thermal response. The current results are verified with those reported in the literature.

Pricing Barrier Options and Credit Default Swaps (CDS) in Spectrally One-Sided Levy Models: The Parabolic Laplace Inversion Method

Recently, the advantages of conformal deformations of the contours of integration in pricing formulas were demonstrated in the context of wide classes of Levy models and the Heston model. In the present paper we construct efficient conformal deformations of the contours of integration in the pricing formulas for barrier options and CDS in the setting of spectrally one-sided Levy models, taking advantage of Rogers's trick (J. Appl. Prob. 2000) that greatly simplifies calculation of the Wiener-Hopf factors. We extend the trick to wide classes of Levy processes of infinite variation with zero diffusion component. In the resulting formulas (both in the finite variation and the infinite variation cases), we make quasi-parabolic deformations as in S. Boyarchenko and Levendorskii (IJTAF 2013), which greatly increase the rate of convergence of the integrals. We demonstrate that the proposed method is more accurate than the standard realization of Laplace inversion in many cases. We also exhibit examples in which the standard realization is so unstable that it cannot be used for any choice of the error control parameters.

An analytical solution for dynamic behavior of thick doubly curved functionally graded smart panels

In this paper, an analytical solution is presented for free vibration and dynamic behavior of doubly curved laminated shell consisting of a functionally graded core layer and surface attached functionally graded piezoelectric layers. Shell through-thickness kinematics is based on higher order shear deformation theory of shells, whereas a quadratic variation is assumed for electric potential. Using Hamilton’s principle and Maxwell’s equation, the governing equations of motion under mechanical loads are derived as seven highly coupled partial differential equations. Implementing Laplace transformation, doing few mathematical operations and using Laplace inverse method, time dependencies of displacement components are expressed in explicit phrases. Besides numerical results for shell natural frequencies, effects of different material properties and shell geometries on the transient response are discussed in details. In particular the influence of shell curvature on the spectra of maximax response has been studied. It is found that, these parameters play major roles on determining the time-histories of the electric potential and displacement components.

On three methods for analytic Laplace inversion in the framework of Brownian motion and their excursions

Working in a framework originating with Brownian motion and its excursions, this paper establishes a two-step Laplace inversion method for determining a function which is known through its transform after a convolution with another function with a known transform. The first step here has as its domain the class of parabolic cylinder functions, and it develops analytic Laplace inversion of their reciprocals. The second step pertains to convolutions on the positive reals with analytic factors where one of them is of exponential-order decay to zero at the origin; it develops two Laplace-inversion-based methods for handling these by asymptotic expansions. The results are shown to have applications to finance, yielding series representations and asymptotic expansions for the valuation and hedging of Parisian barrier options.