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 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.

scholarly journals Analysis of electromagnetic non-destructive evaluation modelling using Stratton-Chu formulation-based fast algorithm

2020 ◽  
Vol 378 (2182) ◽  
pp. 20190583
Author(s):  
Yang Bao ◽  
Jiming Song
Keyword(s):  
Fast Algorithm ◽  
Normal Component ◽  
Solution Process ◽  
Low Frequency ◽  
Iterative Solution ◽  
Basis Functions ◽  
Linear System Of Equations ◽  
Non Destructive Evaluation ◽  
Multi Stage ◽  
Non Destructive

The eddy current non-destructive evaluation (NDE) modelling using Stratton-Chu formulation-based fast algorithm is analysed. Stratton-Chu formulations, which have no low frequency breakdown issue, are selected for modelling electromagnetic NDE problems with low frequency and high conductivity approximations. As the main contribution of this article, the robustness and efficiency of the approximations, which result in big savings in both memory and CPU time, are validated and analysed using examples from practical EC testing. The boundary element method (BEM) is used to discretize the integral equations into a linear system of equations: the first order Rao-Wilton-Glisson (RWG) vector basis functions with the flat triangle meshes of the object and pulse basis functions are selected to expand the equivalent surface currents and the normal component of magnetic fields, respectively. Then the multilevel adaptive cross approximation (MLACA) algorithm is applied to accelerate the iterative solution process. The performance and efficiency of adaptively applying a multi-stage (level) algorithm based on the criteria concluded for the operators are shown. This article is part of the theme issue ‘Advanced electromagnetic non-destructive evaluation and smart monitoring’.

3D controlled-source electromagnetic edge-based finite element modeling of conductive and permeable heterogeneities

Geophysics ◽  
2011 ◽  
Vol 76 (4) ◽  
pp. F215-F226 ◽  
Author(s):  
Souvik Mukherjee ◽  
Mark E. Everett
Keyword(s):  
Finite Element ◽  
Magnetic Permeability ◽  
Numerical Solutions ◽  
Governing Equations ◽  
Fe Method ◽  
Controlled Source ◽  
Continuity Equations ◽  
Potential Formulation ◽  
Edge Based ◽  
Further Development

A new 3D controlled-source electromagnetic finite element (FE) modeling algorithm is presented which is capable of handling local inhomogeneities in the magnetic permeability and electrical conductivity distribution of buried geologic and anthropogenic structures. An ungauged, coupled-potential formulation of the governing electromagnetic vector diffusion and scalar continuity equations is used. The formulation introduces magnetic reluctivity, the inverse of magnetic permeability, to facilitate a separation of secondary and primary potentials. The governing equations are solved using a tetrahedral edge-based FE method. The postprocessing steps to obtain electromagnetic fields are outlined. The code is validated for non-magnetic and permeable conductive structures by comparisons against analytic and previously published numerical solutions. Some limitations of the implementation are explored and directions are proposed for its further development.

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.

Three-dimensional inversion of controlled-source audio-frequency magnetotelluric data based on unstructured finite-element method

2020 ◽  
Vol 17 (3) ◽  
pp. 349-360
Author(s):  
Xiang-Zhong Chen ◽  
Yun-He Liu ◽  
Chang-Chun Yin ◽  
Chang-Kai Qiu ◽  
Jie Zhang ◽  
...  
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Three Dimensional ◽  
Audio Frequency ◽  
Magnetotelluric Data ◽  
Controlled Source ◽  
Element Method

scholarly journals Self-updated four-node finite element using deep learning

2021 ◽  
Author(s):  
Jaeho Jung ◽  
Hyungmin Jun ◽  
Phill-Seung Lee
Keyword(s):  
Finite Element ◽  
Deep Learning ◽  
Iterative Solution ◽  
Solution Procedure ◽  
Element Formulation ◽  
Zero Energy ◽  
Energy Mode ◽  
Bending Modes ◽  
Solution Accuracy ◽  
Iterative Solution Procedure

AbstractThis paper introduces a new concept called self-updated finite element (SUFE). The finite element (FE) is activated through an iterative procedure to improve the solution accuracy without mesh refinement. A mode-based finite element formulation is devised for a four-node finite element and the assumed modal strain is employed for bending modes. A search procedure for optimal bending directions is implemented through deep learning for a given element deformation to minimize shear locking. The proposed element is called a self-updated four-node finite element, for which an iterative solution procedure is developed. The element passes the patch and zero-energy mode tests. As the number of iterations increases, the finite element solutions become more and more accurate, resulting in significantly accurate solutions with a few iterations. The SUFE concept is very effective, especially when the meshes are coarse and severely distorted. Its excellent performance is demonstrated through various numerical examples.

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