Edge-based Finite Element Research Articles

Three-Dimensional MT Conductive Anisotropic and Magnetic Modeling Using A − ϕ Potentials Employing a Mixed Nodal and Edge-Based Element Method

Magnetotelluric (MT) sounding is a geophysical technique widely utilized in mineral resource surveys, where conductivity and magnetic permeability serve as essential physical parameters for forward modeling and inversion. However, the effects of conductive anisotropy and non-zero magnetic susceptibility are usually ignored. In this study, we present a three-dimensional (3D) MT modeling algorithm using Coulomb-gauged electromagnetic potentials, incorporating a mixed nodal and edge-based finite element method capable of simulating MT responses for conductive anisotropic and magnetic anomalies. Subsequently, the algorithm’s accuracy was validated in two steps: first, it was compared with analytical solutions for a 1D magnetic model; then, a comparison was made with previously published numerical results for a 3D generalized conductive anisotropic model. The results of two tests show that the maximum relative error is below 0.5% for both models. Furthermore, representative models were computed to comprehensively analyze the responses of MT. The findings illustrate the relationship between anisotropic parameters and electric fields and emphasize the significance of considering the impact of magnetic susceptibility in magnetite-rich regions.

Free-burning arc discharge simulation: The self-induced magnetic field analysis and its effect on arc plasma characteristics

Free-burning arc discharges play important roles in physical processes such as cutting, welding, arc furnaces, and switchgear. Therefore, in this paper, a combination of node-based and edge-based finite-element methods with the finite-volume scheme is developed to investigate the dynamics of these arc discharges. Considering the significant effect of self-induced magnetic fields on the dynamics of the thermal plasma arcs, accurate analysis of these magnetic fields is essential, especially for 3D geometries describing realistic conditions. Accordingly, the edge-based finite-element module is utilized to study the Ampere law in its vector form for estimating the vector potential and the corresponding magnetic field. Furthermore, the current conservation equation is solved using the node-based finite-element technique. The fluid dynamics are also investigated with the well-known finite-volume method. This hybrid model gives more accurate magnetic fields and Lorentz forces. Electromagnetic forces create high-speed streams of thermal plasma and increase the pressure in the near regions of the electrodes. As a result, the pressure and velocity profiles are closer to the predicted results. In addition, the fluid flow changes the temperature distribution in a way that agrees with experimental measurements.

Open-source complex-geometry 3D fluid dynamics for applications with unpredictable heterogeneous dynamic high-performance-computing loads

We discuss the software implementation of a finite element method, intended for the simulation of complex-geometry 3D flows, using hardware resources effectively, including problems with a priori unknown, heterogeneous, and dynamic parallel computational load. Our fundamental choices of data structures, algorithm, and software design specifically target engineering resolution and accuracy requirements. Some of these choices are: unstructured grids (to explicitly resolve complex 3D geometries), tetrahedra-only computational elements (to enable automatic mesh generation), edge-based finite element scheme (to reduce indirect addressing for increased performance), distributed-memory parallel computing paradigm (to enable large problems), and Charm++ (https://charmplusplus.org) as the runtime system (to effectively use computing resources even in the presence of hardware heterogeneities and dynamic application requirements). We discuss aspects of the implementation that enable exercising unique features of the runtime system, e.g., the single Charm++ programming abstraction, overdecomposition, asynchronous execution, latency-hiding parallel communication and computation, task-parallelism, and dynamic load balancing via object migratability. Multiple test problems are used to verify and validate the numerical solutions and computational performance and scalability to high-performance computing environments are discussed. For maximum transparency and reproducibility, and to encourage future research, development, and use, the full source code, together with regression tests and documentation, is publicly available at https://xyst.cc.

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.

An efficient hybrid direct-iterative solver for three-dimensional higher-order edge-based finite element simulation for magnetotelluric data in anisotropic media

This work investigates the use of the high-order finite element method (FEM) for 3D magnetotelluric (MT) modeling in anisotropic media. The high-order FE is more accurate but also more difficult than the linear elements. Numerically, the resulting linear system is larger and denser than the linear case with the same grid. In 3D cases, finite element discretization of Maxwell's equations often leads to large but sparse linear systems of equations, requiring effective methods to solve them. However, for the domain-frequency Maxwell's equations, the discrete rotor operator with a large kernel hampers the convergence of common iterative solvers. Robust and effective solvers are needed to solve 3D electromagnetic (EM) problems. In this study, we presented a robust direct-iterative solver based on a block-diagonal preconditioner. The proposed solver is efficient for high-order FE modeling with arbitrary anisotropy. We implemented the algorithm in a high-level Julia language, which is easy to read, maintain, and extend. Numerical examples show that the high-order elements produce a more accurate solution on a coarse grid than the linear case with the same grid. Compared with the direct solver, the hybrid direct-iterative solver reduces the computation time and memory usage. In addition, the solver only needs a few iterations to converge compared with the traditional BCISGTAB solver with an SSOR preconditioner. Our modeling results suggested the FGMRES solver with the block-diagonal preconditioner is quite effective for 3D MT high-order FE modeling.

Analyzing the effect of dipping anisotropy on 3D marine controlled-source electromagnetic response based on a goal-oriented adaptive finite element method

This paper presents a goal-oriented adaptive edge-based finite element (FE) algorithm for modelling marine controlled-source electromagnetic (CSEM) responses in three-dimensional (3-D) dipping anisotropic conductive media. We adopt the secondary field formulation for quasi-static Maxwell's equations to avoid the source singularity. The algorithm is implemented on unstructured tetrahedral grids, which allow for the simulation of complex model geometries, including tilted electrical interfaces and bathymetry. We validate the adaptive finite element code against a one-dimensional (1-D) layered model with vertical anisotropy. Further, we compare the 3-D FE numerical solutions with those obtained by a 2-D FE code for a two-dimensional dipping anisotropic model. The modelling results from the new 3-D FE algorithm are in good agreement with both the analytical solutions derived from the 1-D code and the numerical results from the anisotropic 2-D FE algorithm. We then use the algorithm to simulate the marine CSEM responses over different anisotropic 3-D models, including an anisotropic reservoir, anisotropic surrounding sediments, and an anisotropic overburden, and performed a detailed analysis of the effect of VTI, HTI, and TTI anisotropy on 3-D marine CSEM response. The modelling results illustrate that marine CSEM responses are considerably influenced by various anisotropic parameters, but to different extents. The research on different types of anisotropy within different media can help us avoid the misinterpretation of marine CSEM data when processing and interpreting of the measured geophysical electromagnetic data.

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.

HALO3D: An All-Mach Approach to Hypersonic Flows Simulation

This paper presents HALO3D, a CFD code simulating the wide range of Mach numbers a hypervelocity vehicle experiences. HALO3D is composed of (a) a RANS edge-based Finite Element (FE) solver modelling high-velocity thermo-chemical non-equilibrium flows, (b) an electromagnetic interactions solver, (c) an ablation module, (d) a Direct Simulation Monte Carlo (DSMC) solver for the rarefied regime, seamlessly linked to the continuum one, and (e) a powerful automatic mesh optimiser finely capturing the many singular hypersonic flow phenomena. Verification and validation simulations for two- and three-dimensional hypersonic flows over a flat plate, a cylinder, a blunt conical body, re-entry capsules, and a waverider geometry demonstrate accurate aerothermodynamic predictions in a wide range of Knudsen numbers, complex geometries and multiphysics.

Investigation of electromagnetic wave propagation in the bicomplex 3D-FEM using a wavenumber Whitney Hodge operator

PurposeThe purpose of this paper is to show the applicability of a discrete Hodge operator in the context of the De Rham cohomology to bicomplex-valued electromagnetic wave propagation problems. It was applied in the finite element method (FEM) to get a higher accuracy through conformal discretization. Therewith, merely the primal mesh is needed to discretize the full system of Maxwell equations.Design/methodology/approachAt the beginning, the theoretical background is presented. The bicomplex number system is used as a geometrical algebra to describe three-dimensional electromagnetic problems. Because we treat rotational field problems, Whitney edge elements are chosen in the FEM to realize a conformal discretization. Next, numerical simulations regarding practical wave propagation problems are performed and compared with the common FEM approach using the Helmholtz equation.FindingsDifferent field problems of three-dimensional electromagnetic wave propagation are treated to present the merits and shortcomings of the method, which calculates the electric and magnetic field at the same spatial location on a primal mesh. A significant improvement in accuracy is achieved, whereas fewer essential boundary conditions are necessary. Furthermore, no numerical dispersion is observed.Originality/valueA novel Hodge operator, which acts on bicomplex-valued cotangential spaces, is constructed and discretized as an edge-based finite element matrix. The interpretation of the proposed geometrical algebra in the language of the De Rham cohomology leads to a more comprehensive viewpoint than the classical treatment in FEM. The presented paper may motivate researchers to interpret the form of number system as a degree of freedom when modeling physical effects. Several relationships between physical quantities might be inherently implemented in such an algebra.

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.

Calculation of Dispersion Characteristics of Finlines Loaded with Anisotropic Left and Right Hand Materials

In this paper, the edge-based finite element formula for calculating the dispersion characteristics of transmission lines loaded with anisotropic left handed materials and anisotropic right handed materials with electric field as working variable is derived, and the calculation is carried out by taking unilateral finline, bilateral finline and antipodal overlapping finline as examples. Firstly, the dispersion characteristics of three modes in three kinds of finlines are compared when the relative dielectric constant and permeability are fixed; Secondly, taking the unilateral fin line as an example, the changes of dispersion characteristics with the changes of relative dielectric constant and relative permeability in each direction are calculated. The calculation method and results derived in this paper further expand the application of anisotropic left handed and right handed materials in transmission lines, and also have great guiding significance for the use of finlines in microwave devices such as mixers, filters and resonators.

High order edge-based finite elements for 3D magnetotelluric modeling with unstructured meshes

We present a 3D magnetotelluric (MT) forward modeling program which computes the distribution of the electric field on an unstructured mesh. We employ high order edge elements and implement the Edge Finite Element method to accurately simulate magnetotelluric data in the form of impedance tensor. The program is open-source and optimized for both high-performance computing (HPC) architectures and non-HPC architectures. The paper details the theory, method and code work-flow. We evaluate the accuracy of our algorithm with various synthetic test models and compare our results with other Finite Element (FE) and Finite Difference (FD) algorithms. The experiments show that the implemented high order elements are effective in providing highly accurate solutions for multiple frequencies with few elements independently of the model’s geometry. Hence, leading to reduced computing times.

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.

Calculation of Transmission Characteristics of Shielded Microstrip Lines Filled with Anisotropic Left-Handed and Right-Handed Materials

In this paper, the edge-based finite element formula for calculating the transmission characteristics of shielded microstrip lines filled with anisotropic materials with magnetic field as working variable is derived in detail. The filling materials include anisotropic left-handed materials and anisotropic right-handed materials, and the transmission characteristics include dominant mode cutoff wavelength and electric field structure. We mainly take Rectangular-shaped, V-shaped and Trapezoidal-shaped shielded microstrip lines as examples to calculate the transmission characteristics. The formulas and calculation examples derived in this paper have strong guiding significance for the application of anisotropic left-handed and right-handed materials in inhomogeneous transmission lines, which will further expand the use of shielded microstrip lines in optical devices such as electro-optic waveguide modulators.

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

We have developed an edge-based finite-element (FE) modeling algorithm with a divergence correction for calculating the controlled-source electromagnetic (CSEM) responses of a 3D conductivity earth model. We solve a curl-curl equation to directly calculate the secondary electric field to eliminate the source singularity. The choice of the edge-based FE 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 quasiminimal residual (QMR) 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 QMR 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 our algorithm with 1D and 3D models, and we 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.

Efficient goal-oriented mesh refinement in 3-D finite-element modelling adapted for controlled source electromagnetic surveys

SUMMARY We developed a 3-D forward modelling code, which simulates controlled source electromagnetic problems in frequency domain using edge-based finite elements and a total electric field approach. To evaluate electromagnetic data acquired across complex subsurface structures, software performing accurate 3-D modelling is required, especially for incorporation in inversion approaches. Our modelling code aims at finding a good compromise between the necessary solution accuracy at the points of interest and the general problem size by using a goal-oriented mesh refinement strategy designed for models of variable electric conductivity and magnetic permeability. To formulate an improved error estimator suitable for controlled source electromagnetic problems, we developed literature approaches of mesh refinement further targeting three aspects. First, to generate a roughly homogeneously fine mesh discretization around all receiver sites, our new error estimator weights the adjoint source term by the approximate decay of the electric field with increasing distance from the primal source using the expression for a homogeneous half-space. This causes almost no additional computational cost. Second, the error estimator employed in the refinement approach can be optimized for models with pronounced conductivity and magnetic permeability contrasts as often encountered in, for example, mineral prospecting scenarios by optionally including terms that measure the continuity of the normal component of current flow and the tangential component of the magnetic field across interfaces of abutting elements. Third, to avoid amplitude-dependent over-refining of the mesh, we formulate our element-wise error estimators relative to the local amplitude of the electromagnetic field. In this work, we evaluate the implemented adaptive mesh refinement approach and its solution accuracy comparing our solutions for simple 1-D models and a model with 3-D anomalies to semi-analytic 1-D solutions and a second-order finite-element code, respectively. Furthermore, a feasibility study for controlled-source electromagnetic measurements across ferrous mineral deposits is conducted. The numerical experiments demonstrate that our new refinement procedure generates problem-specific finite-element meshes and yields accurate solutions for both simple synthetic models and realistic survey scenarios. Especially for the latter, characteristics of our code, such as the possibility of modelling extended sources as well as including arbitrary receiver distributions and detailed subsurface anomalies, are beneficial.