A Numerical Algorithm Based on Quadratic Finite Element for Two-Dimensional Nonlinear Time Fractional Thermal Diffusion Model

In this article,a high-order scheme,which is formulated by combining the quadratic finite element method in space with a second-order time discrete scheme,is developed for looking for the numerical solution of a two-dimensional nonlinear time fractional thermal diffusion model.The time Caputo fractional derivative is approximated by using the L2-1formula,the first-order derivative and nonlinear term are discretized by some second-order approximation formulas,and the quadratic finite element is used to approximate the spatial direction.The error accuracy O(h3+
t2)is obtained,which is verified by the numerical results.

Non-Split PML Boundary Condition for Finite Element Time-Domain Modeling of Ground Penetrating Radar

As a highly efficient absorbing boundary condition, Perfectly Matched Layer (PML) has been widely used in Finite Difference Time Domain (FDTD) simulation of Ground Penetrating Radar (GPR) based on the first order electromagnetic wave equation. However, the PML boundary condition is difficult to apply in GPR Finite Element Time Domain (FETD) simulation based on the second order electromagnetic wave equation. This paper developed a non-split perfectly matched layer (NPML) boundary condition for GPR FETD simulation based on the second order electromagnetic wave equation. Taking two-dimensional TM wave equation as an example, the second order frequency domain equation of GPR was derived according to the definition of complex extending coordinate transformation. Then it transformed into time domain by means of auxiliary differential equation method, and its FETD equation is derived based on Galerkin method. On this basis, a GPR FETD forward program based on NPML boundary condition is developed. The merits of NPML boundary condition are certified by compared with wave field snapshots, signal and reflection errors of homogeneous medium model with split and non-split PML boundary conditions. The comparison demonstrated that the NPML algorithm can reduce memory occupation and improve calculation efficiency. Furthermore, numerical simulation of a complex model verifies the good absorption effects of the NPML boundary condition in complex structures.

Proper orthogonal decomposition based seismic source wavefield reconstruction for finite element reverse time migration

The large storage requirement is a critical issue in cross-correlation imaging-condition based reverse time migration(RTM),because it requires the operation of the source and receiver wavefields at the same time.The boundary value method(BVM),based on the finite difference method(FDM),can be used to reconstruct the source wavefield in the reverse time propagation in the same way as the receiver wavefield,which can reduce the storage burden of the RTM data.Considering that the FDM cannot well handle models with discontinuous material properties and rough interfaces,we develop a source wavefield reconstruction strategy based on the finite element method(FEM),using proper orthogonal decomposition(POD)to enhance computational efficiency.In this method,we divide the whole time period into several segments,and construct the POD basis functions to get a reduced order model(ROM)for the source wavefield reconstruction in each segment.We show the corresponding quantitative analysis of the storage requirement of the POD-FEM.Numerical tests on the homogeneous model show the effectiveness of the proposed method,while the layered model and part of the Marmousi model tests indicate that the POD-FEM can keep an excellent balance between computational efficiency and memory usage compared with the full-stored method(FSM)and the BVM,and can be effectively applied in imaging.

Kinematic source model for simulation of near-fault ground motion field using explicit finite element method

This paper briefly reviews the characteristics and major processes of the explicit finite element method in modeling the near-fault ground motion field. The emphasis is on the finite element-related problems in the finite fault source modeling. A modified kinematic source model is presented, in which vibration with some high frequency components is introduced into the traditional slip time function to ensure that the source and ground motion include sufficient high frequency components. The model presented is verified through a simple modeling example. It is shown that the predicted near-fault ground motion field exhibits similar characteristics to those observed in strong motion records, such as the hanging wall effect, vertical effect, fling step effect and velocity pulse effect, etc.

Study on generalized magneto-thermoelastic problems by FEM in time domain

This paper presents an investigation of temperature, displacement, stress, and induced magnetic field in a half space perfectly-conductive plate. Finite element equations regarding generalized magneto-thermoelasticity problems with two relaxation times （i.e., the G-L theory） are derived using the principle of virtual work. For avoiding numerical complication involved in inverse Laplace and Fourier transformation and low precision thereof, the equations are solved directly in time-domain. As a numerical example, the derived equation is used to investigate the generalized magneto-thermoelastic behavior of a semi-infinite plate under magnetic field and subjecting to a thermal shock loading. The results demonstrate that FEM can faithfully predict the deformation of the plate and the induced magnetic field, and most importantly can reveal the sophisticated second sound effect of heat conduction in two-dimensional generalized thermoelastic solids, which is usually difficult to model by routine transformation methods. A peak can be observed in the distribution of stress and induced front and the magnitude of magnetic field at the heat wave the peak decreases with time, which can not be obtained by transformation methods. The new method can also be used to study generalized piezo-thermoelastic problems.

Benchmarking of two three-dimensional numerical models in time/space domain to predict railway-induced ground vibrations

In the last 30 years,the scientific community has developed and proposed different models and numerical approaches for the study of vibrations induced by railway traffic.Most of them are formulated in the frequency/wave number domain and with a 2.5D approach.Three-dimensional numerical models formulated in the time/space domain are less frequently used,mainly due to their high computational cost.Notwithstanding,these models present very attractive characteristics,such as the possibility of considering nonlinear behaviors or the modelling of excess pore pressure and non-homogeneous and non-periodic geometries in the longitudinal direction of the track.In this study,two 3D numerical approaches formulated in the time/space domain are compared and experimentally validated.The first one consists of a finite element approach and the second one of a finite difference approach.The experimental validation in an actual case situated in Carregado(Portugal)shows an acceptable fitting between the numerical results and the actual measurements for both models.However,there are some differences among them.This study therefore includes some recommendations for their use in practical soil dynamics and geotechnical engineering.

A seismic free field input model for FE-SBFE coupling in time domain

A seismic free field input formulation of the coupling procedure of the finite elelnent(FE)and the scaled boundary finite-element(SBFE)is proposed to perform the unbounded soil-structure interaction analysis in time domain. Based on the substructure technique,seismic excitation of the soil-structure system is represented by the free-field motion of an elastic half-space.To reduce the computational effort,the acceleration unit-impulse response function of the unbounded soil is decomposed into two functions:linear and residual.The latter converges to zero and can be truncated as required. With the prescribed tolerance parameter,the balance between accuracy and effMency of the procedure can be controlled. The validity of the model is verified by the scattering analysis of a hemi-spherical canyon subjected to plane harmonic P,SV and SH wave incidence.Numerical results show that the new procedure is very efficient for seismic problems within a nor- real range of frequency.The coupling procedure presented herein can be applied to linear and nonlinear earthquake re- sponse analysis of practical structures which are built on unbounded soil.

High Precision Time Domain Forward Modeling for Crosshole Electromagnetic Tomography

To improve the resolution of crosshole electromagnetic tomography, high precision of forward modeling is necessary. A pseudo-spectral time domain （PSTD） forward modeling was used to simulate electromagnetic wave propagation between two boreholes. The PSTD algorithm is based on the finite difference time domain （FDTD） method and uses the fast Fourier transform （FFT） algorithm for spatial derivatives in Maxwell＇s equations. Besides having the strongpoint of the FDTD method, the calculation precision of the PSTD algorithm is higher than that of the FDTD method under the same calculation condition. The forward modeling using the PSTD method will play an important role in enhancing the resolution of crosshole electromagnetic tomography.

Time Domain Simulation of Transient Responses of Very Large Floating Structures Under Unsteady External Loads

A time domain finite element method （FEM） for the analysis of transient elastic response of a very large floating structure （VLFS） subjected to arbitrary time-dependent external loads is presented. This method is developed directly in time domain and the hydrodynamic problem is formulated based on linear, inviscid and slightly compressible fluid theory and the structural response is analyzed on the thin plate assumption. The time domain finite element procedure herein is validated by comparing numerical results with available experimental data. Finally, the transient elastic response of a pontoon-type VLFS under the landing of an airplane is computed by the proposed time domain FEM. The time histories of the applied force and the position and velocity of an airplane during landing are modeled with data from a Boeing 747-400 jumbo jet.

2.5-D modeling of cross-hole electromagnetic measurement by nite element method

A finite element method is developed for simulating frequency domain electromagnetic responses due to a dipole source in the 2-D conductive structures. Computing costs are considerably minimized by reducing the full three-dimensional problem to a series of two-dimensional problems. This is accomplished by transforming the problem into y-wave number （Ky） domain using Fourier transform and the y-axis is parallel to the structural strike. In the Ky domain, two coupled partial differential equations for magnetic field Hy and electric field Ey are derived. For a specific value of Ky, the coupled equations are solved by the finite element method with isoparametric elements in the x-z plane. Application of the inverse Fourier transform to the Ky, domain provides the electric and magnetic fields in real space. The equations derived can be applied to general complex two-dimensional structures containing either electric or magnetic dipole source in any direction. In the modeling of the electromagnetic measurement, we adopted a pseudo-delta function to distribute the dipole source current and circumvent the problem of singularity at the source point. Moreover, the suggested method used isoparametric finite elements to accommodate the complex subsurface formation. For the large scale linear system derived from the discretization of the Maxwell＇s equations, several iterative solvers were used and compared to select the optimal one. A quantitative test of accuracy was presented which compared the finite element results with analytic solutions for a dipole source in homogeneous space for different ranges and different wave numbers Ky. to validate the addressed the effects of the distribution range τ of the homogeneous medium. code and check its effectiveness. In addition, we pseudo-delta function on the numerical results in

Finite Element Analysis to Two-Dimensional Nonlinear Sloshing Problems

A two-dimensional nonlinear sloshing problem is analyzed by means of the fully nonlinear theory and time domain second order theory of water waves. Liquid sloshing in a rectangular container Subjected to a horizontal excitation is simulated by the finite element method. Comparisons between the two theories are made based on their numerical results. It is found that good agreement is obtained for the case of small amplitude oscillation and obvious differences occur for large amplitude excitation. Even though, the second order solution can still exhibit typical nonlinear features of nonlinear wave and can be used instead of the fully nonlinear theory.

An Experiment to Improve the Model of Lumped Element Circuit

The model of lumped element circuit ignores the finite time of signals to propagate around a circuit. However, using modern oscilloscope, the time of nanoseconds in a circuit can be measured. Then the speed of alternating electricity can be obtained in a RL circuit. A typical RL circuit is formed by a power source, wire, resistance and inductance. The basic formula is: U(t)=I(t)R+LdI(t)/dt. It can be derived from the Ohm’s law and Kirchhoff laws. Based on our experimental results, this paper has discussed the new explanation of this equation in a RL circuit. As a result, the speed of alternating electricity is greater than light in a special RL circuit. The model of lumped element circuit can be improved when considering the finite time of signals.

3-D Modelling of the Confederation Bridge Using Data of Full Scale Tests

Long-span bridges are special structures that require advanced analysis techniques to examine their performance. This paper presents a procedure developed to model the Confederation Bridge using 3-D beam elements. The model was validated using the data collected before the opening of the bridge to the public. The bridge was instrumented to conduct fullscale static and dynamic tests. The static tests were to measure the deflection of the bridge pier while the dynamic tests to measure the free vibrations of the pier due to a sudden release of the static load. Confederation Bridge is one of the longest reinforced concrete bridges in the world. It connects the province of Prince Edward Island and the province of New Brunswick in Canada. Due to its strategic location and vital role as a transportation link between these two provinces, it was designed using higher safety factors than those for typical highway bridges. After validating the present numerical model, a procedure was developed to evaluate the performance of similar bridges subjected to traffic and seismic loads. It is of interest to note that the foundation stiffness and the modulus of elasticity of the concrete have significant effects on the structural responses of the Confederation Bridge.

A varying time-step explicit numerical inte-gration algorithm for solving motion equa-tion

If a traditional explicit numerical integration algorithm is used to solve motion equation in the finite element simulation of wave motion, the time-step used by numerical integration is the smallest time-step restricted by the stability criterion in computational region. However, the excessively small time-step is usually unnecessary for a large portion of computational region. In this paper, a varying time-step explicit numerical integration algorithm is introduced, and its basic idea is to use different time-step restricted by the stability criterion in different computational region. Finally, the feasibility of the algorithm and its effect on calculating precision are verified by numerical test.

Finite element analysis of second order wave radiation by a group of cylinders in the time domain

A finite element based numerical method is employed to analyze the wave radiation by multiple or a group of cylinders in the time domain. The nonlinear free surface and body surface boundary conditions are satisfied based on the perturbation method up to the second order. The first- and second-order velocity potential problems at each time step are solved through a Finite Element Method （FEM）. The matrix equation of the FEM is solved through iteration and the initial solution is obtained from the result at the previous time step. The three-dimensional （3-D） mesh required is generated based on a two-dimensional （2-D） hybrid mesh on a horizontal plane and its extension in the vertical direction. The hybrid mesh is generated by combining an unstructured grid away from cylinders and two structured grids near the cylinder and the artificial boundary. The fluid velocity on the free surface and the cylinder surface are calculated by using a differential method. Results for various configurations including the cases of two cylinders and four cylinders and a group of eighteen cylinders are obtained to show the joint influences of cylinders on the first- and second- order waves and forces, including the effects of spacing ratios and wave frequency on the second order waves and the mean force, in particular.

Experimental and Numerical Investigations on Vibration Characteristics of a Loaded Ship Model

In this paper, the vibration characteristics of the structure in the finite fluid domain are analyzed using a coupled finite element method. The added mass matrix is calculated with finite element method （FEM） by 8-node acoustic fluid elements. The vibration characteristics of the structure in the finite fluid domain are calculated combining structure FEM mass matrix. By writing relevant programs, the numerical analysis on vibration characteristics of a submerged cantilever rectangular plate in finite fluid domain and loaded ship model is performed. A modal identification experiment for the loaded ship model in air and in water is conducted and the experiment results verify the reliability of the numerical analysis. The numerical method can be used for further research on vibration characteristics and acoustic radiation problems of the structure in the finite fluid domain.

A Comparison of Deterministic and Stochastic Susceptible-Infected-Susceptible (SIS) and Susceptible-Infected-Recovered (SIR) Models

In this paper we build and analyze two stochastic epidemic models with death. The model assumes that only susceptible individuals (S) can get infected (I) and may die from this disease or a recovered individual becomes susceptible again (SIS model) or completely immune (SIR Model) for the remainder of the study period. Moreover, it is assumed there are no births, deaths, immigration or emigration during the study period;the community is said to be closed. In these infection disease models, there are two central questions: first it is the disease extinction or not and the second studies the time elapsed for such extinction, this paper will deal with this second question because the first answer corresponds to the basic reproduction number defined in the bibliography. More concretely, we study the mean-extinction of the diseases and the technique used here first builds the backward Kolmogorov differential equation and then solves it numerically using finite element method with FreeFem++. Our contribution and novelty are the following: however the reproduction number effectively concludes the extinction or not of the disease, it does not help to know its extinction times because example with the same reproduction numbers has very different time. Moreover, the SIS model is slower, a result that is not surprising, but this difference seems to increase in the stochastic models with respect to the deterministic ones, it is reasonable to assume some uncertainly.

基于时域有限差分法的电性源感应-极化效应三维数值模拟

时域电性源电磁探测法是一种可以有效快速探测矿产资源的方法。极化效应会导致电性源电磁响应快速衰减,发生符号反转现象。本文采用Cole-Cole模型描述极化效应,利用整数阶有理逼近算法实现任意分数阶Cole-Cole模型有理化;采用Yee氏网格对仿真区域进行剖分,基于时域有限差分(finite-difference time-domain, FDTD)法实现电性源感应-极化效应三维数值模拟。本文对三种典型模型(均匀半空间模型、极化半空间模型和三维极化体模型)的电磁响应进行数值模拟,结果表明:均匀半空间模型的电磁响应与解析解基本吻合,并且相对误差小于10%,证明了三维数值模拟方法的正确性;极化半空间模型与三维极化体模型的电磁响应均在晚期出现了负响应,与极化理论结果相符合。