3D time-domain simulation of electromagnetic diffusion phenomena: A finite-element electric-field approach

Geophysics ◽  
2010 ◽  
Vol 75 (4) ◽  
pp. F115-F126 ◽  
Author(s):  
Evan Schankee Um ◽  
Jerry M. Harris ◽  
David L. Alumbaugh
Keyword(s):  
Finite Element ◽  
Differential Equations ◽  
Electric Field ◽  
Time Domain ◽  
Dirichlet Boundary Conditions ◽  
Earth Model ◽  
Time Step ◽  
Transient Electromagnetic ◽  
Backward Euler ◽  
Diffusion Phenomena

We present a finite-element time-domain (FETD) approach for the simulation of 3D electromagnetic (EM) diffusion phenomena. The finite-element algorithm efficiently simulates transient electric fields and the time derivatives of magnetic fields in general anisotropic earth media excited by multiple arbitrarily configured electric dipoles with various signal waveforms. To compute transient electromagnetic fields, the electric field diffusion equation is transformed into a system of differential equations via Galerkin’s method with homogeneous Dirichlet boundary conditions. To ensure numerical stability and an efficient time step, the system of the differential equations is discretized in time using an implicit backward Euler scheme. The resultant FETD matrix-vector equation is solved using a sparse direct solver along with a fill-in reduced ordering technique. When advancing the solution in time, the FETD algorithm adjusts the time step by examining whether or not the current step size can be doubled without unacceptably affecting the accuracy of the solution. To simulate a step-off source waveform, the 3D FETD algorithm also incorporates a 3D finite-element direct current (FEDC) algorithm that solves Poisson’s equation using a secondary potential method for a general anisotropic earth model. Examples of controlled-source FETD simulations are compared with analytic and/or 3D finite-difference time-domain solutions and are used to confirm the accuracy and efficiency of the 3D FETD algorithm.

Finite-Element Modeling of Time-Domain Controlled-Source Electromagnetic Survey with Weighted Laguerre Polynomials

Geophysics ◽  
2021 ◽  
pp. 1-43
Author(s):  
Qingtao Sun ◽  
Runren Zhang ◽  
Yunyun Hu
Keyword(s):  
Finite Element ◽  
Time Domain ◽  
Laguerre Polynomials ◽  
Time Integration ◽  
Sampling Interval ◽  
Controlled Source ◽  
Time Integration Method ◽  
Transient Electromagnetic ◽  
Backward Euler ◽  
Land Survey

To facilitate the modeling of time-domain controlled-source electromagnetic survey, we propose an efficient finite-element method with weighted Laguerre polynomials, which shows a much lower computational complexity than conventional time integration methods. The proposed method allows sampling the field at arbitrary time steps and also its accuracy is determined by the number of polynomials, instead of the time sampling interval. Analysis is given regarding the optimization of the polynomial number to be used and the criterion of selecting the time scale factor. Two numerical examples in marine and land survey environments are included to demonstrate the superiority of the proposed method over the existing backward Euler time integration method. The proposed method is expected to facilitate the modeling of transient electromagnetic surveys in the geophysical regime.

scholarly journals Numerical Modelling of TEM Fields by the Edge-Based Finite Element Method Using Tetrahedral Element for Central Rectangular Loops

2021 ◽  
Vol 34 (04) ◽  
pp. 1180-1199
Author(s):  
Hossein Shahnazari- Aval ◽  
Mirsattar Meshinchi-Asl ◽  
Mahmoud Mehramuz
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Analytical Solution ◽  
Electric Field ◽  
Time Domain ◽  
Tetrahedral Mesh ◽  
Computational Time ◽  
Transient Electromagnetic ◽  
Edge Based ◽  
Element Method

In this study, we have implemented an edge-based finite element method for the numerical modeling of the transient electromagnetic method. We took the Helmholtz equation of the electric field as the governing equation for the edge-based finite element analysis. The modeling domain was discretized using linear tetrahedral mesh supported by Whitney-type vector basis functions. We inferred the equations by applying the Galerkin method. The system of equation was solved using a corrected version of the Bi Conjugate Gradient Stabilized Method (BiCGStab) algorithm to reduce the computational time. We obtained numerical solution for electric field in the Laplace domain; then the field was transformed into the time domain using the Gaver-Stehfest algorithm. Following this, the impulse response of the magnetic field was obtained through the Faraday law of electromagnetic induction as it is considerably more stable and computationally more efficient than inversion using the Fourier Transform. 3D geoelectric models were used to investigate the convergence of the edge-based finite element method with the analytic solution. The results are in good agreement with the analytical solution value for two resistivity contrasts in the 3D geoelectric brick model. We also compared the results of tetrahedral elements with the brick element in the 3D horizontal sheet and 3D conductive brick model. The results indicated that these two elements show very similar errors, but tetrahedral reflects fewer relative errors. For the low resistivity geoelectric model, numerical checks against the analytical solution, integral-equation method, and finite-difference time-domain solutions showed that the solutions would provide accurate results.

FINITE ELEMENT METHODS FOR NONLINEAR ACOUSTICS IN FLUIDS

2007 ◽  
Vol 15 (03) ◽  
pp. 353-375 ◽  
Author(s):  
TIMOTHY WALSH ◽  
MONICA TORRES
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Acoustics ◽  
Time Step ◽  
Partial Differential ◽  
Time Step Size ◽  
Finite Element Discretizations ◽  
Acoustic Fluid ◽  
Element Discretizations

In this paper, weak formulations and finite element discretizations of the governing partial differential equations of three-dimensional nonlinear acoustics in absorbing fluids are presented. The fluid equations are considered in an Eulerian framework, rather than a displacement framework, since in the latter case the corresponding finite element formulations suffer from spurious modes and numerical instabilities. When taken with the governing partial differential equations of a solid body and the continuity conditions, a coupled formulation is derived. The change in solid/fluid interface conditions when going from a linear acoustic fluid to a nonlinear acoustic fluid is demonstrated. Finite element discretizations of the coupled problem are then derived, and verification examples are presented that demonstrate the correctness of the implementations. We demonstrate that the time step size necessary to resolve the wave decreases as steepening occurs. Finally, simulation results are presented on a resonating acoustic cavity, and a coupled elastic/acoustic system consisting of a fluid-filled spherical tank.

Transient electromagnetic fields of a buried horizontal magnetic dipole

Geophysics ◽  
2016 ◽  
Vol 81 (6) ◽  
pp. E481-E491 ◽  
Author(s):  
Andrei Swidinsky ◽  
Misac Nabighian
Keyword(s):  
Electric Field ◽  
Magnetic Dipole ◽  
Time Domain ◽  
Major Axis ◽  
Secondary Response ◽  
Burial Depth ◽  
Transient Electromagnetic ◽  
The Earth ◽  
Smoke Ring ◽  
Electromagnetic Surveys

Electromagnetic surveys using a vertical transmitter loop are common in land, marine, and airborne geophysical exploration. Most of these horizontal magnetic dipole (HMD) systems operate in the frequency domain, measuring the time derivative of the induced magnetic fields, and therefore a majority of studies have focused on this subset of field measurements. We examine the time-domain electromagnetic response of a HMD including the electric fields and corresponding smoke rings produced in a conductive half-space. Cases of a dipole at the surface and buried within the earth are considered. Results indicate that when the current in the transmitter is rapidly switched off, a single smoke ring is produced within the plane of the vertical transmitter loop, which is then distorted by the air-earth interface. In this situation, the circular smoke ring, which would normally diffuse symmetrically away from the source in a whole space, is approximately transformed into an ellipse, with a vertical major axis at an early time and a horizontal major axis at a late time. As measured from the location of the transmitter, the depth of investigation and lateral footprint of such a system increases with burial depth. It is also observed that the electric field measured in the direction of the magnetic dipole only contains a secondary response related to the charge accumulation on any horizontal conductivity boundaries because the primary field is always absent. This field component can be expressed analytically in terms of a static and time-varying field, the latter term adding spatial complexity to the total horizontal electric field at the earth surface at early times. Applications of this theoretical study include the design of time-domain induction-logging tools, crossborehole electromagnetic surveys, underground mine expansion work, mine rescue procedures, and novel marine electromagnetic experiments.

scholarly journals Improvement of a nonnegative preserved efficient solver for atmospheric chemical kinetic equations

2018 ◽  
Vol 7 (3) ◽  
pp. 6657
Author(s):  
Atika RADID ◽  
Karim RHOFIR
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Atmospheric Chemistry ◽  
Iteration Process ◽  
Iterative Solution ◽  
Chemical Kinetic ◽  
Time Step ◽  
Backward Euler ◽  
Wide Range ◽  
Numerical Solver

Generally, chemical reactions from atmospheric chemistry models are described by a strongly coupled, stiff and nonlinear system of ordinary differential equations, which requires a good numerical solver. Several articles published about the solvers of chemical equations, during the numerical simulation, indicate that one renders the concentration null when it becomes negative. In order to preserve the positivity of the exact solutions, recent works have proposed a new solver called Modified-Backward-Euler (MBE). To improve this solver, we propose in this paper an iterative numerical scheme witch is better fitted to stiff problems. This new approach, called Iterative-Modified-Backward-Euler (IMBE), is based on iterative solution of the P-L structure of the implicit nonlinear ordinary differential equations on each time step. The efficiency of the iteration process is increased by using the Gauss and Successive-Over-Relaxation (SOR). In the case of fast/slow chemical kinetic reactions, we proposed an other variant called Iterative-Quasi-Steady-State-Approximation (IQSSA). The numerical exploration of stiff test problem shows clearly that this formalism is applicable to a wide range of chemical kinetics problems and give a good approximation compared to the recent solver. The numerical procedures give reasonable accurate solutions when compared to exact solution.Generally, chemical reactions from atmospheric chemistry models are described by a strongly coupled, stiff and nonlinear system of ordinary differential equations, which requires a good numerical solver. Several articles published about the solvers of chemical equations, during the numerical simulation, indicate that one renders the concentration null when it becomes negative. In order to preserve the positivity of the exact solutions, recent works have proposed a new solver called Modified-Backward-Euler (MBE). To improve this solver, we propose in this paper an iterative numerical scheme witch is better fitted to stiff problems. This new approach, called Iterative-Modified-Backward-Euler (IMBE), is based on iterative solution of the P-L structure of the implicit nonlinear ordinary differential equations on each time step. The efficiency of the iteration process is increased by using the Gauss and Successive-Over-Relaxation (SOR). In the case of fast/slow chemical kinetic reactions, we proposed an other variant called Iterative-Quasi-Steady-State-Approximation (IQSSA). The numerical exploration of stiff test problem shows clearly that this formalism is applicable to a wide range of chemical kinetics problems and give a good approximation compared to the recent solver. The numerical procedures give reasonable accurate solutions when compared to exact solution.

Electromagnetic modeling of 3-D sources over 2-D inhomogeneities in the time domain

Geophysics ◽  
1992 ◽  
Vol 57 (8) ◽  
pp. 994-1003 ◽  
Author(s):  
Michael Leppin
Keyword(s):  
Differential Equations ◽  
Diffusion Equation ◽  
Time Domain ◽  
Magnetic Induction ◽  
Host Rock ◽  
Vertical Component ◽  
Wavenumber Domain ◽  
Transient Electromagnetic ◽  
Wide Range ◽  
The Time Domain

A numerical method is presented by which the transient electromagnetic response of a two‐dimensional (2-D) conductor, embedded in a conductive host rock and excited by a rectangular current loop, can be modeled. This 2.5-D modeling problem has been formulated in the time domain in terms of a vector diffusion equation for the scattered magnetic induction, which is Fourier transformed into the spatial wavenumber domain in the strike direction of the conductor. To confine the region of solution of the diffusion equation to the conductive earth, boundary values for the components of the magnetic induction on the ground surface have been calculated by means of an integral transform of the vertical component of the magnetic induction at the air‐earth interface. The system of parabolic differential equations for the three magnetic components has been integrated for 9 to 15 discrete spatial wavenumbers ranging from [Formula: see text] to [Formula: see text] using an implicit homogeneous finite‐difference scheme. The discretization of the differential equations on a grid representing a cross‐section of the conductive earth results in a large, sparse system of linear equations, which is solved by the successive overrelaxation method. The three‐dimensional (3-D) response has been computed by an inverse Fourier transformation of the cubic spline interpolated scattered magnetic induction in the wavenumber domain using a digital filtering technique. To test the algorithm, responses have been computed for a two‐layered half‐space and a vertical prism embedded in a conductive host rock. These examples were then compared with results obtained analytically or numerically using frequency‐domain finite‐element and time‐domain integral equation methods. The new numerical procedure gives satisfactory results for a wide range of 2-D conductivity distributions with conductivity ratios exceeding 1:100, provided the grid is sufficiently refined at the corners of the conductivity anomalies.

A finite‐element solution for the transient electromagnetic response of an arbitrary two‐dimensional resistivity distribution

Geophysics ◽  
1986 ◽  
Vol 51 (7) ◽  
pp. 1450-1461 ◽  
Author(s):  
Y. Goldman ◽  
C. Hubans ◽  
S. Nicoletis ◽  
S. Spitz
Keyword(s):  
Finite Element ◽  
Sparse Matrix ◽  
Equation System ◽  
Electromagnetic Response ◽  
Implicit Time ◽  
Two Dimensional ◽  
Time Step ◽  
Transient Electromagnetic ◽  
Exact Boundary ◽  
The Time Domain

We present a numerical method for solving Maxwell’s equations in the case of an arbitrary two‐dimensional resistivity distribution excited by an infinite current line. The electric field is computed directly in the time domain. The computations are carried out in the lower half‐space only because exact boundary conditions are used on the free surface. The algorithm follows the finite‐element approach, which leads (after space discretization) to an equation system with a sparse matrix. Time stepping is done with an implicit time scheme. At each time step, the solution of the equation system is provided by the fast system ICCG(0). The resulting algorithm produces good results even when large resistivity contrasts are involved. We present a test of the algorithm’s performance in the case of a homogeneous earth. With a reasonable grid, the relative error with respect to the analytical solution does not exceed 1 percent, even 2 s after the source is turned off.

Coupled Analysis of Deepwater Floating System Including VIV in Time Domain

2007 ◽  
Author(s):  
Jun-Bumn Rho ◽  
Alexander A. Korobkin ◽  
Jong-Jun Jung ◽  
Hyun-Soo Shin ◽  
Woo-Seob Lee
Keyword(s):  
Finite Element ◽  
Time Domain ◽  
Torsional Stiffness ◽  
Time Step ◽  
Vortex Induced Vibration ◽  
Coupled Equations ◽  
Floating System ◽  
Vessel Motion ◽  
Floating Systems ◽  
The Way

Deepwater floating systems consist of a vessel, risers, and mooring lines. To accurately simulate the floating systems in current, wind, and waves considering (1) bending and torsional stiffness of riser, (2) elongation of the mooring/riser elements, (3) complex end conditions, (4) internal flow effects, and (5) vortex induced vibration, it is necessary to evaluate the vessel motions and mooring/riser behaviors simultaneously in time domain. However, because the size of the system matrix increases significantly as the number of mooring/riser increases, it is quite time-consuming to solve all equations including both mooring/riser and vessel dynamics simultaneously. The present study was performed in order to develop a program for this problem. The 6DOF vessel dynamics is described by the Cummins equation. And the mooring and riser are modeled with the help of finite-element beam. The Newmark method is used as the time marching scheme of the FEM equations for each mooring/riser and the vessel. The coupled equations of the mooring/riser segments and vessel are solved alternatively at each time step. Mooring/riser and the vessel motion affect to each other in the way that the components of the forces at the segment ends are determined as functions of displacements and slopes of them. This procedure makes it possible to consider the coupling effects between vessel and mooring/riser efficiently. Also no iterations are required to match the vessel motion with the riser dynamics. This new approach allows us to use parallel computations and to deal with as many mooring/riser at the same time as necessary. The hydrodynamic forces induced by current are calculated by using the Morison’s formula. The VIV (Vortex Induced Vibration) effects are included in the way that the frequency and the shape of the riser vibration due to VIV are pre-calculated by iterations in the frequency domain. Then the finite element mooring/riser model is modified to consider the hydrodynamic loads including VIV and integrated in the final equations of the floating system in time domain.

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