Laplace Transform for Finite Element Analysis of Electromagnetic Interferences in Underground Metallic Structures

A numerical methodology is proposed for the calculation of transient electromagnetic interference induced by overhead high-voltage power lines in metallic structures buried in soil—pipelines for oil or gas transportation. A series of 2D finite element simulations was employed to sample the harmonic response of a given geometry section. The numerical inverse Laplace transform of the results allowed obtaining the time domain evolution of the induced voltages and currents in the buried conductors, for any given condition of the power line.

Solution for the Multigroup Neutron Space Kinetics Equations by Source Iterative Method

In this work, we used a modified Picard’s method to solve the Multigroup Neutron Space Kinetics Equations (MNSKE) in Cartesian geometry. The method consists in assuming an initial guess for the neutron flux and using it to calculate a fictitious source term in the MNSKE. A new source term is calculated applying its solution, and so on, iteratively, until a stop criterion is satisfied. For the solution of the fast and thermal neutron fluxes equations, the Laplace Transform technique is used in time variable resulting in a first order linear differential matrix equation, which are solved by classical methods in the literature. After each iteration, the scalar neutron flux and the delayed neutron precursors are reconstructed by polynomial interpolation. We obtain the fluxes and precursors through Numerical Inverse Laplace Transform by Stehfest method. We present numerical simulations and comparisons with available results in literature.

Coseismic and post-seismic gravity disturbance induced by seismic sources using a 2.5-D spectral element method

SUMMARY I present a prescription for computing free-air coseismic and post-seismic gravity changes induced by seismic sources in a viscoelastic earth model. I assume a spherical earth geometry and a 2.5-D calculation, that is, 3-D motions that satisfy the equations of quasi-static equilibrium on a 2-D viscoelastic structure. The prescription permits application to regional gravity computations where a 2-D structure adequately represents the structural heterogeneity. I use a hybrid approach where deformation is computed on a discretized domain and the resulting density perturbations are expanded with spherical harmonics to produce the free-air gravity field. Starting with a solution to the equations of quasi-static displacements in the Laplace transform domain for a given dislocation source, I solve Poisson’s equation using Lagrangian interpolation on spectral element nodes to compute the required deformation quantities that contribute to free-air gravity. A numerical inverse Laplace transform then yields time domain results. This methodology is tested with analytic solutions on a spherically stratified viscoelastic structure, then applied to evaluate the effect of a descending slab of relatively high viscosity on post-seismic gravity in a megathrust faulting setting.

Time-Dependent Propagation of Tsunami-Generated Acoustic–Gravity Waves in the Atmosphere

Tsunami-generated linear acoustic–gravity waves in the atmosphere with altitude-dependent vertical stratification and horizontal background winds are studied with the long-term goal of real-time tsunami warning. The initial-value problem is examined using Fourier–Laplace transforms to investigate the time dependence and to compare the cases of anelastic and compressible atmospheres. The approach includes formulating the linear propagation of acoustic–gravity waves in the vertical, solving the vertical displacement of waves and pressure perturbations numerically as a set of coupled ODEs in the Fourier–Laplace domain, and employing den Iseger’s algorithm to carry out a fast and accurate numerical inverse Laplace transform. Results are presented for three cases with different atmospheric and tsunami profiles. Horizontal background winds enhance wave advection in the horizontal but hinder the vertical transmission of internal waves through the whole atmosphere. The effect of compressibility is significant. The rescaled vertical displacement of internal waves at 100-km altitude shows an arrival at the early stage of wave development due to the acoustic branch that is not present in the anelastic case. The long-term displacement also shows an O(1) difference between the compressible and anelastic results for the cases with uniform and realistic stratification. Compressibility hence affects both the speed and amplitude of energy transmitted to the upper atmosphere because of fast acoustic waves.

Thermal Memory Response in Magneto-thermoelastic Medium Having Long Cylindrical Cavity

The present paper deals with the memory response on thermal disturbances emanating from a cylindrical cavity in an unbounded thermoelastic solid. Here we have theoretically demonstrated the memory response of thermal disturbances in the generalized magneto-thermo-elastic materials. Firstly, the characteristics of thermoelastic disturbances originated from the cavity in an unbounded elastic solid under the light of generalized magnetothermoelasticity theory with memory dependent derivatives (MDD). For numerical computation, cylindrical-polar coordinate system with radial symmetry subjected to two different types of heat sources into the cavity are considered. An integral transform method and, while in inverse transformation, an efficient and pragmatic NILT (Numerical Inverse Laplace Transform) is adopted. Finally, parameter studies are performed to evaluate the effect of the kernel function and time delay. For thermal wave the results show appreciable differences with those in the usual magneto-thermoelasticity theory.

The Rational Spectral Method Combined with the Laplace Transform for Solving the Robin Time-Fractional Equation

In this paper, the rational spectral method combined with the Laplace transform is proposed for solving Robin time-fractional partial differential equations. First, a time-fractional partial differential equation is transformed into an ordinary differential equation with frequency domain components by the Laplace transform. Then, the spatial derivatives are discretized by the rational spectral method, the linear equation with the parameter s is solved, and the approximation Ux,s is obtained. The approximate solution at any given time, which is the numerical inverse Laplace transform, is obtained by the modified Talbot algorithm. Numerical experiments are carried out to demonstrate the high accuracy and efficiency of our method.

Semi-analytical solution for consolidation of vertical drain to unsaturated soils considering well resistance and smear effect under time-varying loadings

In this paper, based on equal-strain assumption a semi-analytical solution, considering well re-sistance, smear effect and time-varying loading, is deduced for radial consolidation aided by vertical drain (VD) to unsaturated soils. Firstly, by employing the general integration, Laplace transform, decoupling methods and numerical inverse Laplace transform, the semi-analytical solution in the time domain is ob-tained. Then, its validity is verified by the special cases of the proposed solution under instantaneous loading. Finally, the case analysis show that the dissipation of excess pore pressures is accelerated with the decrease of smear coefficients (αa or αw) or well resistance factors (Ga or Gw). In addition, when the well resistance factor is less than 1, the barrier of VD material to flow can be ignored. Furthermore, a smaller value of the loading frequency of cyclic loading, the bigger the amplitude, and the less fluctuation period in the dissipa-tion rates. Moreover, the current solution can analyse the consolidation characteristics of unsaturated soils with VDs under arbitrary time-varying loadings, including cyclic loadings.