Numerical Inversion Of Laplace Transformation Research Articles

Efficient computational hybrid method for the solution of 2D multi-term fractional order advection-diffusion equation

Abstract Multi-term time-fractional advection diffusion equations are vital for simulating a wide range of physical phenomena, including fluid dynamics and environmental transport processes. However, due to their natural complexity, these equations pose challenges for conventional numerical approaches. In this article, we develop a high order accurate method to solve the multi-term time-fractional advection diffusion equations. We combine the Laplace transform (LT) to integrate the considered equations in time, with Chebyshev spectral method (CSM) for spatial terms The proposed method produces highly accurate solutions with remarkably low computational cost as compared to finite difference method. The propose numerical scheme first employs the LT which reduces the considered problem into a finite set of elliptic equations which may be solved in parallel. Then, the CSM is employed for the disctrezation of spatial operators, which makes it possibly to accurately represent the solution chebyshev grid. Finally, numerical inversion of LT is used to convert the obtain solution from the Laplace domain into the real domain. This work utilizes the modified Talbot’s method and Stehfest’s method for numerical inversion of the LT. To measure the performance, efficiency, and accuracy of the suggested approach, numerical approximations of three models are acquired and verified against the exact solution. The outcomes presented in tables and figures demonstrate that the modified Talbot’s method performed better as compared to Stehfest’s method.

Transient wave-induced dynamic responses of a circular composite-lined tunnel with imperfect interface

Dynamic perturbation is a significant factor to induce dynamic stress concentrations and vibration velocities in underground engineering. The investigation presented in this paper focuses on the transient responses around a deep-buried composite-lined tunnel with imperfect interface between intermediate liner and rock mass. Based on the integral transform and numerical inversion of Laplace transformation, the time-history variations, temporal and spatial evolutions for stress and velocity around the tunnel are explored theoretically. The results indicate that the incident and shadow sidewalls are characterized by tensile stress concentration around the inner boundary, while the roof and floor are compressive stress concentration areas. The radial and hoop elastic coefficients for the interface have an important influence on the dynamic responses around the tunnel and small elastic coefficients lead to more complicated dynamic responses. The results can provide theoretical basis for the support of underground lined tunnels.

Numerical Inversion of Laplace Transform via Wavelet Operational Matrix and Its Applications to Fractional Differential Equations

Numerical Inversion of Laplace Transform via Wavelet Operational Matrix and Its Applications to Fractional Differential Equations

Fast and Stable Time-Domain Simulation Based on Modified Numerical Inversion of the Laplace Transform

This article presents a new algorithm for time-domain circuit simulation based on numerical inversion of Laplace transform (NILT). The proposed method, labeled NILT n, shows that for the same computational cost of the conventional NILT, referred to as NILT0, the approximation error is reduced by a significant factor, permitting time-marching with much larger time steps and, consequently, saving significant computational cost per time step. Numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed approach.

Time-Domain Analysis of Retarded Partial Element Equivalent Circuit Models Using Numerical Inversion of Laplace Transform

Full-wave time-domain computational electromagnetic (CEM) solvers, which are integral equation (IE)-based, may suffer from what is called “late-time instability” problems. This unstable behavior occurs for CEM solvers for very fast rise-time input signals. A multitude of techniques has been devised by researchers over the years to solve the problem. In this article, we pursue an approach for the stable solution of full-wave partial element equivalent circuit (PEEC) models for fast-rising input waveforms. In particular, step and impulse response will be considered that are the most challenging from a stability point of view. For the solver part, a conventional full-wave PEEC code is used that requires one to use retarded partial elements. Unfortunately, a PEEC, as well as impedance Z-(method of moment) solvers using suitable numerical time-stepping methods have stability problems, especially for fast rising impulse or step inputs. An important step forward is achieved in this work by providing a larger class of stable solutions well above the stability achieved for time-stepping methods in the last 50 years. The time-domain stability is achieved by replacing the stepping integration methods with a numerical inversion of Laplace transform (NILT) technique. The NILT transform starts out by applying it to a frequency-domain PEEC solution. The surprising result is that the NILT-based method has a variable time-dependent bandwidth that is advantageous for the full-wave IE solution stability. In this article, we give several examples that show that a PEEC-NILT solution provides accurate and stable results for impulse, step- and piece-wise linear input waveforms.

Modification of Numerical Inversion of Laplace Transform in Solving Problems of Poroviscoelasticity via BEM

The present paper is dedicated to numerical solving of three dimensional boundary-value problems in poroviscoelastic formulation. The poroviscoelastic formulation is treated as a combination of Biot’s theory of poroelasticity and elastic-viscoelastic correspondence principle. Kelvin-Voigt model and Standard linear solid model are employed in order to describe viscoelastic media properties. Boundary element method and boundary integral equation method are applied to obtain Laplace domain solution of boundary-value problem. Modified Durbin’s algorithm of numerical inversion of Laplace transform is used to perform solutions in time domain. Research is also dedicated to development of numerical modelling technique based on Boundary Element Method (BEM) in Laplace domain for solution of three dimensional transient poroviscoelastic problems. A problem of the three-dimensional poroviscoelastic prismatic solid clamped at one end, and subjected to uniaxial and uniform impact loading at another is considered. Viscosity parameter influence on dynamic responses of displacements and tractions is studied.

Self-Validated Time-Domain Analysis of Linear Systems With Bounded Uncertain Parameters

This brief presents a novel approach to predict the bounds of the time-domain response of a linear system subject to multiple bounded uncertain input parameters. The method leverages the framework of Taylor models in conjunction with the numerical inversion of Laplace transform (NILT). Different formulations of the NILT are reviewed, and their advantages and limitations are discussed. An implementation relying on an inverse fast Fourier transform turns out to be the most efficient and accurate alternative. The feasibility of the technique is validated based on several diverse application examples, namely a control loop, a lossy transmission-line network, and an active low-pass filter.

Transient thermo-electro-elastic contact analysis of a sliding punch acting on a functionally graded piezoelectric strip under non-Fourier heat conduction

Within the framework of the non-Fourier heat conduction law, this paper presents a transient, frictional thermal analysis with heat generated by a sliding punch acting on the surface of a functionally graded piezoelectric strip (FGPS). A continuous variation in the thickness direction is adopted for the material properties of the piezoelectric strip. By using the Laplace and Fourier integral transform techniques and the superposition principle, general solutions of the thermal field, and homogeneous solutions for both real and complex eigenvalues and particular solutions for the electro-elastic field are given. Results in the time-domain are obtained by applying the numerical Laplace-transform inversion method, and the convergence behavior is analyzed. Numerical results demonstrate that the thermal relaxation effect makes the results of the non-Fourier heat conduction model reach their peak values later than those of the Fourier model. The effects of the punch velocity, the friction coefficient, and various gradient parameters on the contact stress and the surface heat flow are revealed in details.

Operational ability of road transport structures in complex hydro geological conditions

Working capacity of traffic facilities in the northern European part of Russia depends on difficult natural climatic conditions. It means deep frost penetration and high ground-water level. Both two factors considerably impact on the amount of frost boil of soils in the roadbed. Because of the analysis of a water-and-thermal regime of soil it is possible to detect it’s characteristic, to adjust initial and boundary conditions in the performance of theoretical tasks based on laws of thermodynamics for description of migration process of heat and damp in the road bed. A solution was found with a numerical inversion of Laplace transformation. Realization of that task makes possible to forecast dampness to the end of winter period and to determine the amount of frost boil of soils. To avoid subgrade deterioration in this period the temporary axle load limitations on public roads are imposed.

Variability Analysis via Parameterized Model Order Reduction and Numerical Inversion of Laplace Transform

A fast algorithm is presented for statistical analysis of large circuits with multiple stochastic parameters. The proposed method combines the merits of the parameterized model order-based techniques and numerical inversion of Laplace transform (NILT). The response of the reduced model at any given time point is expressed as a linear combination of the frequency-domain response at a relatively small number of predetermined complex frequency points. This eliminates the necessity for explicit representation of the dynamic model in the form of a set of differential equations. As a result, the moment vectors associated with frequency are excluded while forming the moments’ subspace, leading to much smaller reduced models. In addition, evaluation of the time-domain response of the reduced-order models using NILT is more efficient and highly parallelizable compared to time-stepping numerical integration techniques. Numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.

Analytical study of freezing behavior of a cavity in thermo-poro-elastic medium

The goal of this paper is to present an analytical solution to have a first insight of the impact of ice formation on the surrounding porous rock on underground cavities like reservoir, pipes, tunnels or wellbores. Among the other analytical solutions found in the literature on this topic, the originality of this work resides in the rigorous theoretical framework of poromechanics, which considers the coupling between liquid water and ice crystal under thermodynamic equilibrium. Liquid water transport, thermal conduction, and elastic properties of the phases are also considered. Two analytical solutions are presented, based on a linearization of the system of governing equations. The first one deals with a spherical cavity within an infinite porous medium leading to an exact analytical solution. It allows validating the Stehfest’s algorithm on the numerical inversion of Laplace Transform, used in the second analytical solution, which considers a cylindrical excavation. The validity of this solution is assessed by comparing its results to that issued from a numerical resolution of the nonlinear system of equations. The analytical solution is then ultimately used to identify the influence of key parameters like the thermal/hydraulic conductivities, the amount of ice formed and the thermal dilatation coefficients on the mechanical response of a cylindrical cavity submitted to an internal frost.

Алгоритм численного обращения преобразования Лапласа в классе обобщенных функций, образующих алгебру со сверткой

Алгоритм численного обращения преобразования Лапласа в классе обобщенных функций, образующих алгебру со сверткой

Algorithm 944

We present Talbot Suite, a C parallel software collection for the numerical inversion of Laplace Transforms, based on Talbot's method . It is designed to fit both single and multiple Laplace inversion problems, which arise in several application and research fields. In our software, we achieve high accuracy and efficiency, making full use of modern architectures and introducing two different levels of parallelism: coarse and fine grained parallelism. They offer a reasonable tradeoff between accuracy, the main aspect for a few inversions, and efficiency, the main aspect for multiple inversions. To take into account modern high-performance computing architectures, Talbot Suite provides different software versions: an OpenMP-based version for shared memory machines and a MPI-based version for distributed memory machines. Moreover, oriented to hybrid architectures, a combined MPI/OpenMP-based implementation is provided too. We describe our parallel algorithms and the software organization. We also report some performance results. Our software includes sample programs to call the Talbot Suite functions from C and from MATLAB.

Numerical Inversion of Laplace Transform for Time Resolved Thermal Characterization Experiment

The aim of this technical brief is to test numerical inverse Laplace transform methods with application in the framework of the thermal characterization experiment. The objective is to find the most reliable technique in the case of a time resolved experiment based on a thermal disturbance in the form of a periodic function or a distribution. The reliability of methods based on the Fourier series methods is demonstrated.

Analysis of Transient Electromagnetic Scattering from Two-Dimensional Open-Ended Structures by Numerical Inversion of Laplace Transform

A novel computational method is proposed to investigate electromagnetic scattering problems. It is error controllable and reliable simulation in time domain can be performed. We apply the proposed method to analysis of transient scattering from open-ended structures and discuss scattering mechanisms.

New approach of electrochemical systems dynamics in the time domain under small-signal conditions: II. Modelling the responses of electrochemical systems by numerical inversion of Laplace transforms

In the first part of this work, we showed that numerical inversion of Laplace transform (NILT) methods could be readily applied to compute the responses of electrochemical systems to small input signals from any reasonable model of the system immittance that is the impedance for current-controlled techniques and the admittance for potential-controlled techniques. The aim of this second article is to provide some applications examples in different fields of electrochemical kinetics. The response of a polymer exchange membrane fuel cell to a small current pulse is computed. The responses of corrosion systems to small potential step/ramp signals are also dealt with. The response of electrochemical systems to a sinusoidal perturbation of electrode potential is investigated. Finally, both Faradaic and non-Faradaic components of the total current are computed by NILT methods under potential- and current-controlled conditions.

Elastodynamic Crack Analysis by Boundary Element Method Using Numerical Inversion of Laplace Transform.

In the present study, an alternative strategy for elasto-dynamic crack problems using boundary element method is discussed. The Laplace transformation is applied to time variation in the formulation. Since the numerical solutions are obtained in the Laplace-transformed domain, the numerical-inverse Laplace transformation is adopted to obtain time histories of the solutions. In the present method, the distributions of the displacement and traction in the vicinity of the crack tip are expressed as the superposition of singular terms and regular ones. The singular terms are the analytical solutions including stress intensity factors of Modes I and II, while the regular terms are approximated by the discrete polynomial functions in general BEM formulations. Additionally, the ligament side is discretized by singular-type crack elements in order to increase the accuracy of the traction near the crack tip. It is shown that the dynamic stress intensity factors (DSIFs) of Modes I and II are determined directly without supplementary procedures such as extrapolation or J-integral scheme. The validity of the present method is confirmed through the analysis of mixed mode dynamic problems of crack.

Study on one-dimensional consolidation of saturated soil with semi-pervious boundaries and under cyclic loading

The variation of effective stress ratio of stratfied soil with semi-pervious boundaries and under cyclic loading was analyzed on the basis of Terzagh's one-dimensional consolidation assumptions. A solution by Laplace Transform was obtained for the case when the soil was under time-varied loading. With numerical inversion of Laplace Transform, some useful results were obtained for several kinds of commonly encountered loadings. The results can be meaningful in engineering practice.

Derivation of continuous molecular weight distributions from the generating function

A method described in the statistical literature for the numerical inversion of Laplace Transforms of real functions has been adapted for the derivation of molecular weight distributions from calculations of the generating function for a polymeric system. It has been shown that a third-order refinement is sufficiently accurate for molecular weight distributors broader than 4.0. This allows the calculations to be carried out with a precision of 16 decimal digits which is commonly used in Fortran. Where higher precision is available, the treatment is applicable to narrower MWDs.

Numerical implementation of the integral-transform solution to Lamb's point-load problem

The present work describes a procedure for the numerical evaluation of the classical integral-transform solution of the transient elastodynamic point-load (axisymmetric) Lamb's problem. This solution involves integrals of rapidly oscillatory functions over semi-infinite intervals and inversion of one-sided (time) Laplace transforms. These features introduce difficulties for a numerical treatment and constitute a challenging problem in trying to obtain results for quantities (e.g. displacements) in the interior of the half-space. To deal with the oscillatory integrands, which in addition may take very large values (pseudo-pole behavior) at certain points, we follow the concept of Longman's method but using as accelerator in the summation procedure a modified Epsilon algorithm instead of the standard Euler's transformation. Also, an adaptive procedure using the Gauss 32-point rule is introduced to integrate in the vicinity of the pseudo-pole. The numerical Laplace-transform inversion is based on the robust Fourier-series technique of Dubner/Abate-Crump-Durbin. Extensive results are given for sub-surface displacements, whereas the limit-case results for the surface displacements compare very favorably with previous exact results.