3D finite-element forward modeling of electromagnetic data using vector and scalar potentials and unstructured grids

Geophysics ◽  
2014 ◽  
Vol 79 (4) ◽  
pp. E149-E165 ◽  
Author(s):  
Seyedmasoud Ansari ◽  
Colin G. Farquharson
Keyword(s):  
Finite Element ◽  
Electric Field ◽  
Half Space ◽  
Normal Component ◽  
Forward Modeling ◽  
Computational Domain ◽  
Disk Model ◽  
Weighted Residuals ◽  
Physical Scale ◽  
Scalar Potentials

We present a finite-element solution to the 3D electromagnetic forward-modeling problem in the frequency domain. The method is based on decomposing the electric field into vector and scalar potentials in the Helmholtz equation and in the equation of conservation of charge. Edge element and nodal element basis functions were used, respectively, for the vector and scalar potentials. This decomposition was performed with the intention of satisfying the continuity of the tangential component of the electric field and the normal component of the current density across the interelement boundaries, therefore finding an efficient solution to the problem. The computational domain was subdivided into unstructured tetrahedral elements. The system of equations was discretized using the Galerkin variant of the weighted residuals method, with the approximated vector and scalar potentials as the unknowns of a sparse linear system. A generalized minimum residual solver with an incomplete LU preconditioner was used to iteratively solve the system. The solution method was validated using five examples. In the first and second examples, the fields generated by small dipoles on the surface of a homogeneous half-space were compared against their corresponding analytic solutions. The third example provided a comparison with the results from an integral equation method for a long grounded wire source on a model with a conductive block buried in a less conductive half-space. The fourth example concerned verifying the method for a large conductivity contrast where a magnetic dipole transmitter-receiver pair moves over a graphite cube immersed in brine. Solutions from the numerical approach were in good agreement with the data from physical scale modeling of this scenario. The last example verified the solution for a resistive disk model buried in marine conductive sediments. For all examples, convergence of the solution that used potentials were significantly quicker than that using the electric field.

Three dimensional controlled-source electromagnetic forward modeling by edge-based finite element with a divergence correction

Geophysics ◽  
2021 ◽  
pp. 1-71
Author(s):  
Wenwu Tang ◽  
Yaoguo Li ◽  
Jianxin Liu ◽  
Juzhi Deng
Keyword(s):  
Finite Element ◽  
Electric Field ◽  
Normal Component ◽  
Solution Process ◽  
Iterative Solution ◽  
Controlled Source ◽  
Correction Technique ◽  
Speed Up ◽  
Edge Based ◽  
Minimal Residual

We present an edge-based finite element modeling algorithm with a divergence correction for calculating controlled-source electromagnetic (CSEM) responses of a 3D conductivity earth model. We solve a curl-curl equation to directly calculate the secondary electric field in order to eliminate the source singularity. The choice of the edge-based finite element method enables us to properly handle the discontinuity of the normal component of electric fields across conductivity boundaries. Although we can solve the resulting complex-symmetric linear system of equations efficiently by a quasi-minimal residual method preconditioned with an incomplete Cholesky decomposition for the high frequency band, the iterative solution process encounters a common problem in the field formulation and does not converge within a practically feasible number of iterations for low frequencies. To overcome this difficulty and to accelerate the iterative solution process in general, we combine a divergence correction technique with the secondary field solution using the quasi-minimal residual solver. We have found that applying the divergence correction intermittently during the iterative solution process ensures the calculation of sufficiently accurate electric and magnetic fields and can significantly speed up the solution process by more than an order of magnitude. We have tested the efficiency and accuracy of the proposed algorithm with 1D and 3D models, and have found that the divergence correction technique is able to guide the electric field to satisfy the boundary conditions across conductivity interfaces. Although there is a computational overhead required for applying the divergence correction, that cost is significantly offset by the substantial gains in the solution accuracy and speed-up. The work makes the field-based curl-curl formulation using edge elements an efficient and practical method for CSEM simulations.

3D vector finite-element electromagnetic forward modeling for large loop sources using a total-field algorithm and unstructured tetrahedral grids

Geophysics ◽  
2017 ◽  
Vol 82 (1) ◽  
pp. E1-E16 ◽  
Author(s):  
Jianhui Li ◽  
Colin G. Farquharson ◽  
Xiangyun Hu
Keyword(s):  
Finite Element ◽  
Half Space ◽  
Complex Geometry ◽  
Forward Modeling ◽  
Coefficient Matrix ◽  
Total Field ◽  
Direct Solver ◽  
Homogeneous Half Space ◽  
Tetrahedral Grids ◽  
Vector Finite Element

Unstructured tetrahedral grids with local refinement facilitate the use of total-field solution approaches to geophysical electromagnetic (EM) forward problems. These approaches, when combined with the vector finite-element (FE) method and with refinement near transmitters and receivers, can give accurate solutions and can easily handle realistic models with complex geometry and topography. We have applied this approach to 3D forward modeling for fixed- and moving-loop configurations. MUMPS, a direct solver, was used to solve the linear system of equations generated by FE analysis. A direct solver is particularly suited to the moving-loop configuration for which the right side is different for every transmitter loop, but for which the coefficient matrix is unchanged. Therefore, the coefficient matrix need only be factorized once, and then the system can be solved efficiently for all different right sides. We compared our results with several typical scenarios from the literature: a conductive brick in a homogeneous half-space and a complex conductor at a vertical contact both for fixed-loop configurations, and a homogeneous half-space for a moving-loop configuration. We also evaluated results for the massive sulfide ore deposit of the Ovoid Zone at Voisey’s Bay, Labrador, Canada, for which we considered fixed- and moving-loop configurations. This model also provides an illustration of the complex vortex current systems that are generated by time-domain EM methods within highly conductive ore bodies in a resistive host.

A parallel adaptive finite element solver for global electromagnetic induction modeling using hierarchal tetrahedral meshes 

2021 ◽  
Author(s):  
Hongbo Yao ◽  
Zhengyong Ren ◽  
Jingtian Tang
Keyword(s):  
Finite Element ◽  
Electric Field ◽  
Adaptive Mesh Refinement ◽  
Electromagnetic Induction ◽  
Numerical Solutions ◽  
Mesh Refinement ◽  
High Order ◽  
Error Estimator ◽  
Computational Domain ◽  
Tetrahedral Meshes

<p>We present an accurate and fast finite element solver for global electromagnetic induction forward modeling problems in spherical Earth. We solve for the electric field equation using the first-order Nedelec elements. The magnetic field is then obtained by computing the curl of the electric field. The computational domain composed of the air space and the conductive Earth is discretized by disjoint unstructured tetrahedral elements. To improve the accuracy with an optimal number of unknowns, we propose a simple two-step goal-oriented adaptive mesh refinement (AMR) strategy. In the first step, an h-type AMR procedure is used to obtain an optimal finite element mesh. The mesh refinement is accomplished by bisection to generate a set of hierarchal tetrahedral meshes. The AMR procedure is driven by a goal-oriented error estimator, which is based on face jumps of normal components of current density. In the second step, we adopt the high-order finite elements at the last iteration to update the accuracy of final numerical solutions. This simple two-step adaptive strategy takes advantage of both h-type AMR and high-order basis functions, and in the meanwhile, it is also computationally economical. To improve efficiency, the solver is parallelized with an MPI-based domain decomposition technique. The sophisticated auxiliary space preconditioned linear solver is adopted to efficiently solve the linear system of equations. This new solver is verified on both semi-analytic and realistic 3-D Earth models. It can be used as a core to derive the inversion of global electromagnetic induction data.</p><p> </p>

A finite element multifrontal method for 3D CSEM modeling in the frequency domain

Geophysics ◽  
2012 ◽  
Vol 77 (2) ◽  
pp. E101-E115 ◽  
Author(s):  
Nuno Vieira da Silva ◽  
Joanna V. Morgan ◽  
Lucy MacGregor ◽  
Mike Warner
Keyword(s):  
Finite Element ◽  
Electric Field ◽  
Frequency Domain ◽  
Closed Form Solution ◽  
Form Solution ◽  
Computational Domain ◽  
Order Partial Differential Equation ◽  
Primary Field ◽  
Direct Solver ◽  
Linear System Of Equations

There has been a recent increase in the use of controlled-source electromagnetic (CSEM) surveys in the exploration for oil and gas. We developed a modeling scheme for 3D CSEM modeling in the frequency domain. The electric field was decomposed in primary and secondary components to eliminate the singularity originated by the source term. The primary field was calculated using a closed form solution, and the secondary field was computed discretizing a second-order partial differential equation for the electric field with the edge finite element. The solution to the linear system of equations was obtained using a massive parallel multifrontal solver, because such solvers are robust for indefinite and ill-conditioned linear systems. Recent trends in parallel computing were investigated for their use in mitigating the computational overburden associated with the use of a direct solver, and of its feasibility for 3D CSEM forward modeling with the edge finite element. The computation of the primary field was parallelized, over the computational domain and the number of sources, using a hybrid model of parallelism. When using a direct solver, the attainment of multisource solutions was only competitive if the same factors are used to achieve a solution for multi right-hand sides. This aspect was also investigated using the presented methodology. We tested our proposed approach using 1D and 3D synthetic models, and they demonstrated that it is robust and suitable for 3D CSEM modeling using a distributed memory system. The codes could thus be used to help design new surveys, as well to estimate subsurface conductivities through the implementation of an appropriate inversion scheme.

scholarly journals Downscaled Finite Element Modeling of Metal Targets for Surface Roughness Level under Pulsed Laser Irradiation

2021 ◽  
Vol 11 (3) ◽  
pp. 1253
Author(s):  
Evaggelos Kaselouris ◽  
Kyriaki Kosma ◽  
Yannis Orphanos ◽  
Alexandros Skoulakis ◽  
Ioannis Fitilis ◽  
...  
Keyword(s):  
Surface Roughness ◽  
Finite Element ◽  
Surface Acoustic Waves ◽  
Pulsed Laser ◽  
Multiscale Simulation ◽  
Experimental Methodology ◽  
Computational Domain ◽  
Area Of Interest ◽  
Depth Analysis ◽  
Laser Solid Interactions

A three-dimensional, thermal-structural finite element model, originally developed for the study of laser–solid interactions and the generation and propagation of surface acoustic waves in the macroscopic level, was downscaled for the investigation of the surface roughness influence on pulsed laser–solid interactions. The dimensions of the computational domain were reduced to include the laser-heated area of interest. The initially flat surface was progressively downscaled to model the spatial roughness profile characteristics with increasing geometrical accuracy. Since we focused on the plastic and melting regimes, where structural changes occur in the submicrometer scale, the proposed downscaling approach allowed for their accurate positioning. Additionally, the multiscale simulation results were discussed in relation to experimental findings based on white light interferometry. The combination of this multiscale modeling approach with the experimental methodology presented in this study provides a multilevel scientific tool for an in-depth analysis of the influence of heat parameters on the surface roughness of solid materials and can be further extended to various laser–solid interaction applications.

scholarly journals An Online Generalized Multiscale Finite Element Method for Unsaturated Filtration Problem in Fractured Media

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1382
Author(s):  
Denis Spiridonov ◽  
Maria Vasilyeva ◽  
Aleksei Tyrylgin ◽  
Eric T. Chung
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Model Reduction ◽  
Basis Functions ◽  
Computational Domain ◽  
Filtration Problem ◽  
Spectral Problems ◽  
Multiscale Finite Element Method ◽  
Multiscale Finite Element ◽  
Element Method

In this paper, we present a multiscale model reduction technique for unsaturated filtration problem in fractured porous media using an Online Generalized Multiscale finite element method. The flow problem in unsaturated soils is described by the Richards equation. To approximate fractures we use the Discrete Fracture Model (DFM). Complex geometric features of the computational domain requires the construction of a fine grid that explicitly resolves the heterogeneities such as fractures. This approach leads to systems with a large number of unknowns, which require large computational costs. In order to develop a more efficient numerical scheme, we propose a model reduction procedure based on the Generalized Multiscale Finite element method (GMsFEM). The GMsFEM allows solving such problems on a very coarse grid using basis functions that can capture heterogeneities. In the GMsFEM, there are offline and online stages. In the offline stage, we construct snapshot spaces and solve local spectral problems to obtain multiscale basis functions. These spectral problems are defined in the snapshot space in each local domain. To improve the accuracy of the method, we add online basis functions in the online stage. The construction of the online basis functions is based on the local residuals. The use of online bases will allow us to get a significant improvement in the accuracy of the method. We present results with different number of offline and online multisacle basis functions. We compare all results with reference solution. Our results show that the proposed method is able to achieve high accuracy with a small computational cost.

The Finite Element Analysis of the Large-Scale Converter Transformer Valve Side of the Electric Field

2013 ◽  
Vol 325-326 ◽  
pp. 476-479 ◽  
Author(s):  
Lin Suo Zeng ◽  
Zhe Wu
Keyword(s):  
Finite Element ◽  
Electric Field ◽  
Large Scale ◽  
Calculation Model ◽  
Simulation Software ◽  
Field Calculation ◽  
Element Analysis ◽  
Ansys Simulation ◽  
The Finite Element Analysis ◽  
Electric Field Calculation

This article is based on finite element theory and use ANSYS simulation software to establish electric field calculation model of converter transformer for a ±800kV and make electric field calculation and analysis for valve winding. Converter transformer valve winding contour distribution of electric field have completed in the AC, DC and polarity reversal voltage.

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