Edge-based Finite Element Method Research Articles

Fast 3D simulation of magnetotelluric data in anisotropic media using a rational Krylov method

This study introduces a computational method aimed at accelerating multifrequency 3D magnetotelluric (MT) forward modeling in 3D conductivity structures with general anisotropy. The approach combines an edge-based finite-element (FE) method with a rational Krylov subspace method. In this method, the MT source term is expressed as a planar current source, and the frequency-dependent electric field response is the product of a transfer function and a constant vector of the current source. Consequently, the rapid approximate electric field for multifrequency calculations can be obtained by constructing the orthogonal basis of the rational Krylov subspace. By applying this technique, the large sparse matrix resulting from FE discretization can be projected onto a much lower-order matrix, significantly reducing its size to tens or hundreds. Incorporating the direct solver PARDISO to construct the orthogonal basis enables a considerably faster solution compared with the previous FE method. Moreover, the algorithm is implemented using Julia, a high-level programming language known for its readability, maintainability, and extensibility. The validity of the algorithm is demonstrated through a 1D anisotropic model, and two additional 3D models are designed to further evaluate the applicability and effectiveness of our algorithm. Numerical experiments illustrate a notable increase in modeling speed compared with the traditional edge-based FE method.

3-D anisotropic modelling of geomagnetic depth sounding based on unstructured edge-based finite-element method

SUMMARY Geomagnetic depth sounding (GDS) is a geophysical electromagnetic (EM) method that studies the deep structure and composition of the Earth by using long-period EM signals from geomagnetic observatories and satellites. In this paper, a 3-D anisotropic GDS modelling algorithm is developed. The curl–curl equation is discretized using the edge-based finite-element method on unstructured tetrahedral grids. In order to solve the computationally demanding problem of EM modelling on a global scale, the complex linear system is first separated into the equivalent real linear systems and then the real system is iteratively solved by the flexible generalized minimum residual method with a block diagonal pre-conditioner. This will greatly reduce the condition number of the linear system and thus speed up the solution process. We verify the accuracy of the proposed algorithm by comparing our results with the existing methods. After that, we design a subduction zone model to simulate the EM responses under isotropic and anisotropic environments, respectively. The numerical results show the high efficiency of the proposed algorithm and the response differences between isotropic and anisotropic models. This research can provide theoretical and technical support for the high-accuracy and efficient inversion of GDS data for the geo-dynamic study.

Efficient Solution Scheme for Large-Scale Anisotropic Forward Modeling of 3-D Magnetotelluric Data

Efficient three-dimensional magnetotelluric anisotropy forward modeling is one of the key research techniques used for inversion interpretation. We propose an improved multi-level down-sampling scheme to reduce the degrees of freedom of the stiffness matrix derived from the edge-based finite element method to improve the computational efficiency, saving on memory usage and calculation time for the forward modeling. Then, to further reduce the memory requirements and speed up the solution of the discretized electric system, we develop a multiple right-hand direct–iterative hybrid solver based on a block rational Krylov preconditioner. The solver we propose can further save computational costs and time based on the multi-level down-sampling scheme. Moreover, the convergence performance of the direct–iterative solver is less affected by the frequency, which solves the problem of slow convergence of the electric field control equation at low frequencies. We also use the high-level language Julia, which is easy to load into third-party packages, to ensure the stability and efficiency of the program. Finally, the validity and advantages of the two schemes are analyzed in detail using three examples. The example results show that the multiple right-hand direct–iterative hybrid solver and improved multi-level down-sampling can significantly reduce the computational memory and save computational time.

Study on Transmission Characteristics in Three Kinds of Deformed Finlines Based on Edge-Based Finite Element Method

Abstract The influence of model deformation on the transmission characteristics in unilateral finline, antipodal finline and bilateral finline is discussed using edge-based finite element method (FEM). The deformation is considered with respect to the eight boundaries of the model, and their amplitude is set to 0.01–0.05; the transmission characteristics include the cutoff wavelength characteristics of the dominant mode and the single-mode bandwidth characteristics. The results show a decreasing trend when the vacuum area is deformed, while they show a variety of possibilities when the loading region is deformed. These numerical results have strong guiding significance for the influence of the deformation of finlines on the overall device.

Three-Dimensional Magnetotelluric Inversion for Triaxial Anisotropic Medium in Data Space

The interpretation of three-dimensional (3-D) magnetotelluric (MT) data is usually based on the isotropic assumption of the subsurface structures, and this assumption could lead to erroneous interpretation in the area with considerable electrical anisotropy. Although arbitrary anisotropy is much closer to the ground truth, it is generally more challenging to recover full anisotropy parameters from 3-D inversion. In this paper, we present a 3-D triaxial anisotropic inversion framework using the edge-based finite element method with a tetrahedral mesh. The 3-D inverse problem is solved by the Gauss-Newton (GN) method which shows fast convergence behavior. The computation cost of the data-space method depends on the size of data, which is usually smaller than the size of model; therefore, we transform the inversion algorithm from the model space to the data space for memory efficiency. We validate the effectiveness and applicability of the developed algorithm using several synthetic model studies.

Investigating the Magnetotelluric Responses in Electrical Anisotropic Media

When interpreting magnetotelluric (MT) data, because of the inherent anisotropy of the earth, considering electrical anisotropy is crucial. Accordingly, using the edge-based finite element method, we calculated the responses of MT data for electrical isotropic and anisotropic models, and subsequently used the anisotropy index and polar plot to depict MT responses. High values of the anisotropy index were mainly yielded at the boundary domains of anomalous bodies for isotropy cases because the conductive differences among isotropic anomalous bodies or among anomalous bodies and background earth can be regarded as macro-anisotropy. However, they only appeared across anomalous bodies in the anisotropy cases. The anisotropy index can directly differentiate isotropy from anisotropy but exhibits difficulty in reflecting the azimuth of the principal conductivities. For the isotropy cases, polar plots are approximately circular and become curves with a big ratio of the major axis to minor axis, such as an 8-shaped curve for the anisotropic earth. Furthermore, the polar plot can reveal the directions of principal conductivities. However, distorted by anomalous bodies, polar plots with a large ratio of the major axis to minor axis occur in isotropic domains around the anomalous bodies, which may lead to the misinterpretation of these domains as anisotropic earth. Therefore, combining the anisotropy index with a polar plot facilitates the identification of the electrical anisotropy.

3D marine controlled-source electromagnetic modeling using an edge-based finite element method with a block Krylov iterative solver

Abstract An edge-based finite element method for the numerical modeling of electromagnetic fields in complex media is presented. We used the analytical solution on an electric field in a homogeneous half space to develop a source correct factor to reduce the influence of source singularity and boundary conditions on the numerical accuracy, so that we can minimize the time required to construct the field source term in the scattered field formula. The modeling domain was discretized using an unstructured tetrahedral mesh. We transformed the complex equations of the electrical field into two real-valued equations by decomposing the field into real and imaginary components. Thereafter, we adopted a block conjugate orthogonal conjugate gradient (BL_COCR) iterative solver with an incomplete LU decomposition preconditioner, which was robust for ill-conditioned systems and efficient for multiple source electromagnetic modeling to solve the real-valued equation systems. Using the analytical solution on an electric field in a homogeneous layer model, we evaluated the accuracy of our numerical forward solution and the results showed that the source correct factor can reduce forward modeling errors associated with boundary effects and source singularities. We also applied the developed algorithm to compute the CSEM responses for typical 3D marine geo-electric models with different number of sources and compared with different iterative solvers, and the results showed that the BL_COCR solver has high computational efficiency when solving multiple right-hand term problems.

Three-dimensional regularized inversion of magnetotelluric data with a minimum support gradient constraint

Regularization is an effective technique to obtain stable solutions for 3D magnetotelluric (MT) inversion. Smooth constraints are commonly used in traditional inversion algorithms. It is unsuitable if anomalies occur in block patterns. In some cases, it is necessary to obtain sharp boundaries. Some studies have proposed stabilizing functionals to obtain clear and sharp boundaries, such as minimum support (MS) and minimum gradient support (MGS) functionals; however, the optimal value of the focusing parameter should be selected to produce reliable inversion results. In this study, we expanded a new stabilizing functional that we called minimum support gradient (MSG) stabilizing functional as a stabilizer in 3D MT inversion. We developed a Gauss-Newton (GN) algorithm based on the edge-based finite-element method (FEM) to invert the MT data. In the inversion process, the forward modeling and the adjoint problem are solved by the Math Kernel Library (MKL) PARDISO parallel direct solver. we also parallelized the inversion program over frequencies by using Message Passing Interface (MPI) to further speed up the inversion process. Two synthetic models are tested to compare the inversion results for different stabilizing functionals. Compared to traditional smooth inversion, the MSG inversion can clearly distinguish the boundaries of the anomalies. The inverted resistive targets are closer to the true test model. Additionally, the MSG inversion results are stable and robust for a wide range of focusing parameter values, indicating that our method is effective and easy to implement. Finally, the proposed algorithm is applied to the long MT data from the USArray component of EarthScope project in the United States.

3-D Inversion of CSEM Data With Hexahedral Mesh in the Multinary Model Space

The controlled-source electromagnetic (CSEM) method is a crucial tool for near-surface investigations and hydrocarbon exploration because of its economic benefits. Limited resolution is one of the inherent defects of the CSEM method, and to obtain high contrast results a multinary transform function was introduced to CSEM 3-D inversion. The multinary transform function was constituted by superposing several error functions, and the transform function transformed the model parameters from continuously distributed space into a semistep distributed multinary space. To deal with complex geometries, the edge-based finite element (FE) method with an irregular hexahedral grid was applied to the modeling and inverse problem. We used the Gauss-Newton optimization method to minimize the Tikhonov parametric function. The complex chessboard model and the marine CSEM model with complex geometry were used to validate the ability of the new method in improving the resolution of the CSEM method. By comparing the results of conventional, focusing, and multinary inversion methods, the effectiveness of the multinary inversion method in depicting sharp boundaries of different physical properties was proved. Additionally, the comparisons proved that multinary inversion method can, to some extent, overcome the insensitivity to low conductors of the CSEM method.

3D edge-based and nodal finite element modeling of magnetotelluric in general anisotropic media

Research on the magnetotelluric (MT) forward modeling response in anisotropic media has been important since the discovery of anisotropic electric fields in the Earth's interior. In this study, we presented an edge-based and nodal finite-element method (FEM) for three-dimensional (3D) MT forward modeling problems in general anisotropic media. Partial differential equations are derived using the Coulomb-gauged approach from Maxwell's equation and discretized by the edge-based and nodal FEM. The magnetic vector A was discretized by an edge-based FEM and the electric scalar ψ was discretized by nodal FEM. The computational domain was refined using a hexahedral mesh. The Galerkin variant of the weighted residuals method was used to obtain a sparse linear system for magnetic vector A and electric scalar ψ. Three models were designed to validate and analyze our forward modeling code by comparing it with a one-dimensional analytical solution and with the results from codes developed by other scholars. A direct solver and iterative solvers with preprocessors were used to evaluate the performance of our code. Numerical experiments showed that our forward modeling code is accurate and robust and converges faster. In addition, the stiffness matrix assembled by our algorithm was smaller than that assembled by the nodal FEM, which further demonstrates the advantage of our algorithm in terms of computational memory.

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

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.

Edge-Based Finite Element Formulation of Magnetohydrodynamics at High Mach Numbers

In the context of the development of HALO3D (High Altitude Low Orbit 3D), an edge-based Finite Element Method multidisciplinary solver for hypersonic flows, this paper presents the simulation of flows with an imposed magnetic field. Applying the low-magnetic Reynolds number approximation, a current-continuity equation replaces Maxwell's equations with the Hall effect introduced through an electrical conductivity tensor. The approach is validated via the simulation of ionized flow through singly-paired segmented electrodes. Viscous, ionized flow simulations over a hemisphere at Mach 21.38 and over the OREX (Orbital Re-entry Experiment) capsule at Mach 17.61 are also presented. The 3D unstructured hybrid meshes used for these cases are optimized with a highly anisotropic methodology based on the Hessians of the solution, combining multi-physics adaptation criteria such as density, pressure, temperature, velocity, turbulent viscosity and electric potential. Unlike uniform mesh refinement or gradient-based mesh enrichment, this approach greatly improves accuracy without increasing mesh size.

3-D Magnetotelluric Inversion and Application Using the Edge-Based Finite Element With Hexahedral Mesh

Three-dimensional (3-D) inversion technique has become an important and practical approach for magnetotelluric (MT) data interpretation. In this article, we developed a 3-D parallelized MT inversion scheme using the edge-based finite element method and applied the developed method to the newly collected MT data in the Xinjiang Luntai area. The distorted hexahedral element is adopted to incorporate topography into the forward modeling and inversion for complicated scenarios. We use the Gauss–Newton optimization method to minimize the objective functional for MT inversion. The developed algorithm is parallelized using MPI over frequencies and parallel direct solvers when solving the forward and adjoint problems for each frequency. We compare the performance of the least-square QR (LSQR) factorization and preconditioned conjugate gradient (PCG) solvers for the model update within each Gauss–Newton iteration and found that the LSQR solver is more stable. The developed inversion algorithm is validated using several synthetic models. Finally, we applied the inversion algorithm to the subsurface resistivity imaging in the Luntai area. The recovered geoelectric model from full 3-D inversion fits well with the known geological and geophysical information. The recovered model shows a low resistivity layer which may be caused by the salt strata. Besides, the inversion results reveal the movement tectonic in this survey area within a depth of 9 km.

Three-dimensional edge-based finite element modeling of magnetotelluric data in anisotropic media with a divergence correction

When simulating a 3-D magnetotelluric (MT) response, the convergence rate slows at low frequencies. Previous works have shown that by implementing divergence correction, the convergence rate of the iterative solution can be accelerated, the computational accuracy can be enhanced. However, traditional correction schemes of MT are mostly based on isotropic modeling. Electrical anisotropy strongly affects the interpretation of MT data and anisotropic influence cannot be ignored. To our best knowledge, there has been no open literature to apply divergence correction in 3-D MT anisotropic modeling. In this study, based on the divergence-free property of the current density, a divergence correction technique is incorporated into the edge-based finite element (FE) method to speed up the modeling process for 3-D MT problems with axial anisotropic earth. By applying the divergence scheme, the computation time and iteration number for MT numerical simulation are largely reduced. The effectiveness of the anisotropic divergence correction technique on MT modeling is verified by numerical experiments.

High precision interval analysis of the frequency response of structural-acoustic systems with uncertain-but-bounded parameters

The recently developed edge-based smoothed finite element method (ES-FEM) is an efficiency method for solving frequency response of structural-acoustic systems with nominally deterministic parameters. In order to further dealing with the unavoidable uncertainties, both the interval perturbation techniques and the subinterval perturbation techniques are introduced and embedded into the hybrid edge-based smoothed finite element method for structural-acoustic systems with uncertain-but-bounded uncertainties in this work. Firstly, the structural subsystems are described by using 2D edge-based smoothed finite element method; meanwhile, the acoustic subsystems are established by using 3D edge-based finite element method. Then the interval perturbation technique is introduced to establish the interval perturbation equations of structural-acoustic coupled system with small uncertain level. For parameters with large uncertain level, the subinterval perturbation technique is further embedded to improve the computational accuracy (named SIPES-FEM/ES-FEM). The results obtained by IPES-FEM/ES-FEM and SIPES-FEM/ES-FEM are compared with results obtained by Monte-Carlo method. The higher computational accuracy and efficiency of the proposed IPES-FEM/ES-FEM and SIPES-FEM/ES-FEM are verified by two numerical examples.

An efficient parallel algorithm for 3D magnetotelluric modeling with edge-based finite element

Three-dimensional magnetotelluric modeling algorithm of high accuracy and high efficiency is required for data interpretation and inversion. In this paper, edge-based finite element method with unstructured mesh is used to solve 3D magnetotelluric problem. Two boundary conditions—Dirichlet boundary condition and Neumann boundary condition—are set for cross-validation and comparison. We propose an efficient parallel algorithm to speed up computation and improve efficiency. The algorithm is based on distributed matrix storage and has three levels of parallelism. The first two are process level parallelization for frequencies and matrix solving, and the last is thread-level parallelization for loop unrolling. The algorithm is validated by several model studies. Scalability tests have been performed on two distributed-memory HPC platforms, one consists of Intel Xeon E5-2660 microprocessors and the other consists of Phytium FT2000 Plus microprocessors. On Intel platform, computation time of our algorithm solving Dublin Test Model-1 with 3,756,373 edges at 21 frequencies is 365 s on 2520 cores. The speedup and efficiency are 1609 and 60% compared to 100 cores. On Phytium platform, scalability test shows that the speedup from 256 cores to 86,016 cores has been increased to 11,255.

A moment-based stochastic edge-based smoothed finite element method for electromagnetic forming process

In this paper, a novel stochastic method named as the moment-based stochastic edge-based finite element method (MSES-FEM) is proposed to deal with the uncertain electromagnetic problems. First, electromagnetic and mechanical field are formulated by smoothed Galerkin Weak Form under edge-based smoothed finite element method (ES-FEM) scheme. The moment analysis is then applied to obtain the first four moments of the responses and to observe the effects of each random variable on electromagnetic field responses. The maximum entropy theory is employed to calculate the probability density functions (PDFs) of the responses. A quasi-static electromagnetic problem and a practical electromagnetic forming problem (EMF) are performed. The proposed method successfully solves stochastic electromagnetic forming analysis under the uncertain parameters. Numerical results obtained by the proposed MSES-FEM are quite satisfactory with the ones by the Monte Carlo simulation (MCS).

Multiparameter Full-Waveform Inversion of 3-D On-Ground GPR With a Modified Total Variation Regularization Scheme

Ground-penetrating radar (GPR) full-waveform inversion (FWI) is an emerging near-surface high-resolution imaging technology that has been widely used in GPR 2-D imaging to better quantitatively evaluate the permittivity and conductivity of the subsurface. However, the 2-D imaging method limits the applicability and accuracy of FWI for the 3-D object cases. In this letter, we develop a 3-D FWI algorithm involving permittivity and conductivity for GPR in the frequency domain. The method employs the edge-based finite element method to solve the forward problem and the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method to solve the inverse problem, which avoids the calculation of the Hessian matrix. In particular, by using the parameter scaling and scaling factor, it is convenient to carry out the simultaneous inversion of permittivity and conductivity with high efficiency. In addition, a modified total variation regularization scheme is utilized for ensuring the stability of the inversion, as well as identifying the abnormal body boundary more effectively. As a demonstration, a synthetic 3-D model experiment is presented to test the proposed algorithm. The results show that the proposed inversion strategy can accurately reconstruct the permittivity structure together with the conductivity distribution from the 3-D on-ground GPR data.

Comparison of response characteristics between the electromagnetic method of “Earth-ionosphere” mode and traditional magnetotellurics

ABSTRACTAs an emerging electromagnetic exploration method, the electromagnetic method of “Earth-ionosphere” mode utilises a fixed high-power long-source antenna to emit extremely low frequency electromagnetic waves. Signals can be received with a large source–receiver offset to perform resource exploration. As a novel exploration method, it is necessary to compare the results with traditional electromagnetic exploration methods to verify its feasibility and analyse its response characteristics. We first introduce the basic principle of the electromagnetic method of “Earth-ionosphere” mode. Then, one-dimensional forward simulations are developed to determine the similarities and differences between the response of this electromagnetic method in the near, far, and waveguide zones with that from the magnetotellurics. The responses of the three-dimensional model are simulated using the edge-based vector finite element method. The forward simulation responses show that the electromagnetic method of “Earth-ionosphere” mode has a significant abnormal response to geological targets, which is similar to other electromagnetic exploration methods. At large source–receiver offsets, the tensor response of the proposed electromagnetic method is consistent with the magnetotellurics. At small source–receiver offsets, there is a near-field response that is similar to the controlled source audio-frequency magnetotellurics. There are some differences between the scalar response and the tensor response when the antennas have certain azimuth angles. Finally, we present the field measured data for the electromagnetic method of “Earth-ionosphere” mode and compare the field measured data with the magnetotelluric data. We deepen our understanding of the electromagnetic method of “Earth-ionosphere” mode by comparing the results with the traditional magnetotellurics, which provides guidance for further development.

Edge-Based Finite Element Modeling of Magnetogasdynamic-Based Propulsion Systems

The paper presents an edge-based finite element method formulation for the coupled magnetohydrodynamic equations used for the simulation of magnetogasdynamic propulsion systems. The flow is modeled by the Reynolds-averaged Navier–Stokes equations along with the electromagnetic Lorentz force and joule heating resulting from the magnetic/electric interactions represented by source terms, whereas the electromagnetic phenomena are modeled through the magnetic induction equation. Turbulence is treated through the one-equation Spalart–Allmaras model. Both the flow and the electromagnetic problems are transformed into their pseudo-unsteady forms, and a segregated solution strategy is employed for the coupled system. The algebraic systems are solved by applying the generalized minimum residual with an incomplete lower upper factorization as a right preconditioner. After using the Hartmann flow as a validation test, results have been obtained for an magnetohydrodynamic converging channel accelerator, a two-dimensional inviscid flow in a scramjet, and a three-dimensional viscous flow in the same scramjet. The numerical results are compared against published numerical ones and show good agreement.