Tetrahedral Grids Research Articles

Massively parallel FDTD full-band Monte Carlo simulations of electromagnetic THz pulses in p-doped silicon at cryogenic temperatures

In semiconductor samples with a low density of free charge carriers the current response to an intense electromagnetic pulse is weak and can be treated as a perturbation in the finite-difference-time-domain calculation of the electromagnetic field. Up to first order the electric field that drives the particles does not depend on their density and the particles can be simulated independently. This opens the door for massively parallel Monte Carlo simulations that can be processed efficiently on a computer cluster and a large number of particles can be simulated resulting in a very low noise level. By this full-band Monte Carlo finite-difference-time-domain approach the generation of higher harmonics in p-doped silicon at cryogenic temperatures is investigated. It is found that the higher harmonics are mostly due to the warped valence bands, which are discretized on adaptive unstructured tetrahedral grids in the k-space to avoid numerical artifacts.

A Multi‐Resolution Finite‐Element Approach for Global Electromagnetic Induction Modeling With Application to Southeast China Coastal Geomagnetic Observatory Studies

AbstractWe present a multi‐resolution finite‐element approach for three‐dimensional (3D) electromagnetic (EM) induction modeling in spherical Earth. First, the secondary electric field approach is employed so that both magnetospheric and ionospheric current sources are naturally considered. Second, the multi‐resolution tetrahedral grids are used to approximate the heterogeneous crust and mantle, so that the local ocean effects at coastal and island observatories can be accurately simulated. Furthermore, a parallel goal‐oriented hp‐adaptive finite‐element method with Nédélec vector elements is employed to guarantee the accuracy of solutions for arbitrary 3D conductivity distributions. Finally, two synthetic models are used to verify the accuracy and efficiency of our newly developed forward modeling solver. Results show that accurate solutions can be obtained for problems with several million to hundreds of millions of unknowns in a few minutes using 128 cores on a cluster. We apply this approach to correct the near‐surface ocean effects for several unused Chinese coastal observatories by performing multi‐resolution 3D modeling. The corrected data are inverted for the subsurface layered mantle conductivity structures. The conductivity model beneath southeast China is more resistive than that beneath northeast China by more than half an order of magnitude. By comparing the inverse models with the latest laboratory conductivity‐depth profiles, the estimated transition zone water content is less than 0.01 wt% beneath southeast China irrespective of which laboratory data is used. Considering the low‐velocity anomalies in this region, which suggest high‐temperature structures, less water is expected. We, therefore, infer that the mantle transition zone beneath southeast China is dry.

Three‐Dimensional Magnetotelluric Inversion for Arbitrarily Anisotropic Earth Using Unstructured Tetrahedral Discretization

AbstractWe propose a new three‐dimensional anisotropic inversion scheme for magnetotelluric (MT) data. In this method, the earth is discretized into unstructured tetrahedral grids that can fit complex structures well, such as the earth topography and coastline. We use a 3 × 3 tensor to describe the anisotropic conductivity in the governing equation for MT inversions that can be simplified to three principal conductivities and three rotation angles. The edge‐based finite‐element (FE) method is adopted to do the forward and adjoint modeling to compute full impedance and sensitivity. To make the inversion flexible in the restoration of anisotropic and isotropic structures tendentiously, we add an additional regularization in the objective function, with its weight controlled by a distribution function. Before solving the inverse problem using L‐BFGS optimization, we add upper‐bound and lower‐bound constraints to the parameters to suppress the solution space. Four synthetic experiments are designed to test the resolvability and resolution for the six anisotropic parameters in the inversion. The results show that our inversion method can well recover the resistivity in the principal x and y directions and the strike angle γ. Finally, we apply our inversion algorithm to the USArray MT data with the aim to reveal anisotropic structure under Oregon State.

A flux correction for finite-volume discretizations: Achieving second-order accuracy on arbitrary polyhedral grids

In this paper, we propose a flux correction technique generally applicable to practical finite-volume discretizations of a single flux evaluation per face for achieving second-order accuracy on arbitrary polyhedral grids involving non-planar faces. The proposed technique is derived from the k-exact finite-volume discretization approach originally introduced by Brenner. We take it as a general methodology for constructing a second-order discretization on arbitrary polyhedral grids, and identify a term considered as missing in other practical finite-volume discretizations; it is thus considered as a reason for the loss of second-order accuracy in many of such methods: e.g., a cell-centered discretization loses second-order accuracy on grids with non-planar faces as typical in polyhedral grids. In particular, we write the term as a vector involving flux gradients and show that it can be easily implemented as a correction to a numerical flux at each face. Also, we will show that a consistent definition of a control volume is critically important for achieving second-order accuracy on general polyhedral grids. We then discuss its general applicability and impact on accuracy and efficiency with simple but illustrative examples: the edge-based solver is made to achieve faster iterative convergence on irregular tetrahedral grids with adjustable centroids while preserving second-order accuracy; both the edge-based and cell-centered discretizations are made second-order accurate on irregular prismatic grids having non-planar faces.

A reconstructed discontinuous Galerkin method based on variational formulation for compressible flows

A new reconstructed discontinuous Galerkin (rDG) method based on variational formulation is developed for compressible flows. In the presented method, a higher-order piece-wise polynomial is reconstructed based on the underlying discontinuous Galerkin (DG) solution. This reconstruction is done by using a newly developed variational formulation. The variational reconstruction (VR) can be seen as an extension of the compact finite difference (FD) schemes to unstructured grids. The reconstructed variables are obtained by solving an extreme-value problem, which minimizes the jumps of the reconstructed piece-wise polynomials across the cell interfaces, and therefore maximizes the smoothness of the reconstructed solution. Intrinsically, the stencils of the presented reconstruction are the entire mesh, so this method is robust even on tetrahedral grids. A variety of benchmark test cases are presented to assess the accuracy, efficiency and robustness of this rDG method. The numerical experiments demonstrate that the developed rDG method based on variational formulation can maintain the linear stability, obtain the designed high-order accuracy, and outperform the rDG counterpart based on the least-squares reconstruction for both inviscid and viscous compressible flows.

Robust computation of implicit surface networks for piecewise linear functions

Implicit surface networks, such as arrangements of implicit surfaces and materials interfaces, are used for modeling piecewise smooth or partitioned shapes. However, accurate and numerically robust algorithms for discretizing either structure on a grid are still lacking. We present a unified approach for computing both types of surface networks for piecewise linear functions defined on a tetrahedral grid. Both algorithms are guaranteed to produce a correct combinatorial structure for any number of functions. Our main contribution is an exact and efficient method for partitioning a tetrahedron using the level sets of linear functions defined by barycentric interpolation. To further improve performance, we designed look-up tables to speed up processing of tetrahedra involving few functions and introduced an efficient algorithm for identifying nested 3D regions.

A two-grid simulation framework for fast monitoring of fault stability and ground deformation in multiphase geomechanics

We present a simulation framework for fast monitoring of fault stability and ground deformation in multiphase geomechanics and demonstrate its efficacy for a joint CO2 sequestration and enhanced oil recovery case study. Fault stability is estimated by tracking the time evolution of the change in Coulomb Failure Function at critical points on the fault. The staggered solution algorithm for the coupled problem is augmented with a feature that allows for the flow and geomechanics sub-problems to be solved on two different unstructured tetrahedral grids. We demonstrate the accuracy of the two-grid method on the classical Mandel's problem. For the field scale problem, the geomechanics grid extends to the ground surface while the flow grid is truncated at a depth above which no well activity occurs and the flow field can be assumed to remain unperturbed. This framework reduces the computational burden associated with initialization of pressure and stress fields in the overburden, allows for a study of the critical interaction between overburden and faults, allows for fast renditions of flow-induced ground deformation, and allows choosing flow and geomechanics grid resolutions independently to capture disparate length scales of the underlying physics.

Conforming Finite Element DIVDIV Complexes and the Application for the Linearized Einstein--Bianchi System

This paper presents the first family of conforming finite element divdiv complexes on tetrahedral grids in three dimensions. In these complexes, finite element spaces of $H(\text{divdiv},\Omega;\mathbb{S})$ are from a current preprint [Chen and Huang, arXiv: 2007.12399, 2020] while finite element spaces of both $H(\text{symcurl},\Omega;\mathbb{T})$ and $H^1(\Omega;\mathbb{R}^3)$ are newly constructed here. It is proved that these finite element complexes are exact. As a result, they can be used to discretize the linearized Einstein-Bianchi system within the dual formulation.

Conservative Finite Element Modeling of EEG and MEG on Unstructured Grids.

For interpretation of electroencephalography (EEG) and magnetoencephalography (MEG) data, multiple solutions of the respective forward problems are needed. In this paper, we assess performance of the mixed-hybrid finite element method (MHFEM) applied to EEG and MEG modeling. The method provides an approximate potential and induced currents and results in a system with a positive semi-definite matrix. The system thus can be solved with a variety of standard methods (e.g. the preconditioned conjugate gradient method). The induced currents satisfy discrete charge conservation law making the method conservative. We studied its performance on unstructured tetrahedral grids for a layered spherical head model as well as a realistic head model. We also compared its accuracy versus the conventional nodal finite element method ( P1 FEM). To avoid modeling singular sources, we completed our computations with a subtraction approach; the derived expression for the MEG response different from earlier published and involves integration of finite quantities only. We conclude that although the MHFEM is more computationally demanding than the P1 FEM, its use is justified for EEG and MEG modeling on low-resolution head models where P1 FEM loses accuracy.

Stable Spectral Difference Approach Using Raviart-Thomas Elements for 3D Computations on Tetrahedral Grids

In this paper, the Spectral Difference approach using Raviart-Thomas elements (SDRT) is formulated for the first time on tetrahedral grids. To determine stable formulations, a Fourier analysis is conducted for different SDRT implementations, i.e. different interior flux points locations. This stability analysis demonstrates that using interior flux points located at the Shunn-Ham quadrature rule points leads to linearly stable SDRT schemes up to the third order. For higher orders of accuracy, a significant impact of the position of flux points located on faces is shown. The Fourier analysis is then extended to the coupled time-space discretization and stability limits are determined. Additionally, a comparison between the number of interior FP required for the SDRT scheme and the Flux Reconstruction method is proposed and shows that the two approaches always differ on the tetrahedron. Unsteady validation test cases include a convergence study using the Euler equations and the simulation of the Taylor-Green vortex.

A Cut-Cell Polyhedral Finite Element Model for Coupled Fluid Flow and Mechanics in Fractured Reservoirs

Summary Mechanical deformation induced by injection and withdrawal of fluids from the subsurface can significantly alter the flow paths in naturally fractured reservoirs. Modeling coupled fluid flow and mechanical deformation in fractured reservoirs relies on either sophisticated gridding techniques or enhancing the variables (degrees of freedom) that represent the physics to describe the behavior of fractured formation accurately. The objective of this study is to develop a spatial discretization scheme that cuts the “matrix” grid with fracture planes and utilizes traditional formulations for fluid flow and geomechanics. The flow model uses the standard low-order finite volume method with the compartmental embedded discrete fracture model (cEDFM). Due to the presence of nonstandard polyhedra in the grid after cutting/splitting, we use numerical harmonic shape functions within a polyhedral finite element (PFE) formulation for mechanical deformation. To enforce fracture-contact constraints, we use a penalty approach. We provide a series of comparisons between the approach that uses conforming unstructured grids and an unstructured discrete fracture model (uDFM) with the new cut-cell PFE formulation. The manuscript validates and compares both methods for linear elastic, single-fracture slip, and Mandel’s problems with tetrahedral, Cartesian, and perpendicular-bisectional (PBI) grids. Finally, the paper presents a fully coupled 3D simulation with multiple inclined intersecting faults activated in shear by fluid injection, which caused an increase in effective reservoir permeability. Our approach allows for great reduction in the complexity of the (gridded) model construction while retaining the solution accuracy together with great savings in the computational cost compared with uDFM. The flexibility of our model with respect to the types of grid polyhedra allows us to eliminate mesh artifacts in the solution of the transport equations typically observed when using tetrahedral grids and two-point flux approximation.

Total Variation Constrained Graph Manifold Learning Strategy for Cerenkov Luminescence Tomography.

Harnessing the power and flexibility of radiolabeled molecules, Cerenkov luminescence tomography (CLT) provides a novel technique for non-invasive visualisation and quantification of viable tumour cells in a living organism. However, owing to the photon scattering effect and the ill-posed inverse problem, CLT still suffers from insufficient spatial resolution and shape recovery in various preclinical applications. In this study, we proposed a total variation constrained graph manifold learning (TV-GML) strategy for achieving accurate spatial location, dual-source resolution, and tumour morphology. TV-GML integrates the isotropic total variation term and dynamic graph Laplacian constraint to make a trade-off between edge preservation and piecewise smooth region reconstruction. Meanwhile, the tetrahedral mesh-Cartesian grid pair method based on the k-nearest neighbour, and the adaptive and composite Barzilai-Borwein method, were proposed to ensure global super linear convergence of the solution of TV-GML. The comparison results of both simulation experiments and in vivo experiments further indicated that TV-GML achieved superior reconstruction performance in terms of location accuracy, dual-source resolution, shape recovery capability, robustness, and in vivo practicability. Significance: We believe that this novel method will be beneficial to the application of CLT for quantitative analysis and morphological observation of various preclinical applications and facilitate the development of the theory of solving inverse problem.

Application of Optimal Control to Inversion of Self-Potential Data: Theory and Synthetic Examples

Last decades, there has been an increased interest in the use of the self-potential (SP) method in hydrogeophysics. In response to this strong interest, we develop a novel approach to the inversion of SP data. Mathematically, the SP inverse problem is the source identification problem for the Poisson equation. Our approach substantially differs from the standard regularization approach, which explicitly includes the forward-problem operator into the cost functional. We formulated the inverse problem is as an optimal control problem and then translate it into a variational system. The system is approximated in suitable finite-element spaces giving rise to an algebraic problem with the saddle-point structure. In contrast to the standard approach, which leads to a dense linear system, our method results in a system with a sparse block matrix. It can be efficiently solved by either direct sparse solvers or preconditioned iterative solvers. In this paper, we present the formulation of the problem and its finite-element approximation. We discuss the iterative solution and preconditioning strategies. Our software implementation is based on an industrial finite-element package. We also present a numerical experiment with node-based linear finite elements on tetrahedral grids. Our results suggest that the proposed approach may serve as a rapid and reliable tool for large-scale SP inverse problems. Moreover, the same technique can easily be extended to a wide range of geophysical linear inverse problems, such as inversions of magnetic and gravity data.

3D Unstructured Spectral Element Method for Frequency-Domain Airborne EM Forward Modeling Based on Coulomb Gauge

In this article, to solve the frequency-domain airborne electromagnetic (EM) modeling problem, we express the electric and magnetic field by the vector magnetic potential and the scalar electric potential and derive the governing equation using the Galerkin weighted residual method under the Coulomb gauge. To avoid the strong singularity of the solution near the transmitting source, we directly solve the relatively slow-changing secondary potential. By introducing the spectral element method (SEM) based on unstructured tetrahedral grids, in which the EM field in each element is characterized by high-order Proriol–Koornwinder–Dubiner (PKD) orthogonal polynomials, we can obtain stable and accurate numerical solutions. By establishing the mapping relationship between the physical domain, the right-angled tetrahedral reference domain, and the orthogonal hexahedral domain, we can easily accomplish the element matrix analysis and calculation. The numerical examples show that, when the SEM is used to simulate the airborne EM response, a high accuracy can be obtained even if a very coarse grid is used. Furthermore, since the tetrahedral grids can easily model the complex boundaries, the SEM based on unstructured grids has significant advantages for simulating complex underground structures.

О сходимости разностных схем метода опорных операторов для ротационных операций векторного анализа на тетраэдрических сетках

One approach to the description of the metric properties of an irregular grid is proposed for discretizing repeated rotational operations of vector analysis (rot rot) as applied to modeling the magnetic and electric fields. On the basis of the support operator method and the used grid metric, integrally consistent vector analysis operations rot, div, grad are constructed that are necessary to obtain convergence estimates for difference schemes for repeated rotational operations, as well as solving physical problems with discontinuous magnetic viscosity, dielectric permeability or thermal resistance of a medium. On smooth solutions of the model magnetostatic problem on a tetrahedral grid with first order of accuracy in the mean-square sense, the convergence of the constructed difference schemes with zero eigenvalue of the spectral problem is proved. At the same time, no restrictions are imposed on an irregular tetrahedral mesh, except for its non-degeneracy.

3-D Inversion of Z-Axis Tipper Electromagnetic Data Using Finite-Element Method With Unstructured Tetrahedral Grids

<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula> -axis tipper electromagnetic (ZTEM) technique is an airborne electromagnetic method that detects anomalies in the deep earth that are induced by natural sources. Conventional ZTEM forward modeling is generally conducted using structured grids that have limited accuracy and cannot be used to invert complex underground structures and topography. However, unstructured grids can accurately model complex underground structures with fewer cells. Therefore, to effectively model and recover the complex underground structures, we developed a 3-D framework with unstructured tetrahedral grids for ZTEM forward modeling and inversions. The forward problem was formulated using the finite-element method with unstructured tetrahedral grids. To solve the inverse problem, we used a limited-memory quasi-Newton algorithm (L-BFGS) with a parallel direct solver for optimization in order to avoid the explicit calculation of the Hessian matrix and save the memory and computational time. To validate our forward and inversion algorithms, we conducted numerical experiments on three synthetic models and inverted a survey dataset acquired for a tunnel project in Tibet, China. The experimental results demonstrate the effectiveness of our unstructured finite-element and L-BFGS method for modeling and inverting ZTEM data.

다리우스와 사보니우스를 결합한 하이브리드 타입의 조류발전 터빈 설계

This study aimed to design a vertical axis turbine of tidal energy convertor by solving the unsteady Reynolds-averaged Navier-Stokes equation. Precise flow analysis was carried out to design a hybrid vertical dual turbine that combines the darrieus type and savonius type. The hexa- and tetra-hedral grids were used to construct the computational domain and the grid dependency test was performed to find the optimum grid system. Hybrid vertical turbine has been shown to further improve operational stability and power performance compared to the conventional darrieus turbine.

USM3D Simulations for the Third AIAA Sonic Boom Prediction Workshop

The NASA USM3D flow solver was used to compute test cases for the Third AIAA Sonic Boom Prediction Workshop (SBPW3). The test cases included a near-field biconvex shock–plume interaction wind tunnel model and the C608 low boom flight demonstrator. Numerical simulations were conducted on the mixed-element and tetrahedral grids provided by the workshop committee, as well as a family of grids generated by an in-house approach known as BoomGrid. The near-field pressure signatures were extracted and propagated to the ground, and the perceived loudness levels on the ground were computed. The USM3D near-field pressure signatures, corresponding ground signatures, and loudness levels on the ground were compared with that of mean values from other workshop participants. The effect of three flux-limiters on the accuracy of near-field pressure signature prediction was investigated and results were compared with that of mean values from other workshop participants. The effect of using wall-function models for predicting near-field pressure signatures was also evaluated. Results showed that the overpressure signatures extracted from the wall-modeled simulations and the wall-resolved simulations are in good agreement. The use of wall functions (wall-modeled simulations) allowed for approximately 23% savings in the time to solution and yielded comparable perceived loudness levels on the ground to the wall-resolved Reynolds-averaged Navier–Stokes simulations.

A goal-oriented adaptive finite-element approach for 3-D marine controlled-source electromagnetic problems with general electrical anisotropy

SUMMARY We present a goal-oriented adaptive finite-element algorithm for accurately modelling marine controlled-source electromagnetic responses in 3-D media with general electrical anisotropy. We formulate the primal boundary value problem in terms of the total electric field for general applications. Following the goal-oriented adaptivity concept, the dual problem is derived from a functional designed to measure the data errors of interest in light of the data quality. We approximate the solutions to the primal and dual formulations using the edge finite-element method on tetrahedral grids for a flexible treatment of complex geological settings and survey geometries. To control the mesh adaptation, we develop a reliable residual-type a posteriori error estimation that takes account of the volumetric residual and the numerical discontinuity of the normal component of the electrical current density and that of the tangential component of the magnetic field with respect to non-smooth and anisotropic coefficients. We demonstrate the proposed modelling solver on 1-D MCSEM scenarios with varying degrees of electrical anisotropy. The comparison with goal-oriented adaptivity results obtained from other three commonly used error indicators shows that our approach is robust in dealing with both moderate and strong electrical anisotropy. After that, we constructed a 3-D hydrocarbon-bearing reservoir model with slope seafloor topography and tilted transverse isotropy in the background to examine our algorithm for the case of multiple sources. Finally, we implement a sensitivity analysis procedure to evaluate the resolution of the electrical anisotropy. The quantitative results indicate limitations and preferences of conventional MCSEM data in resolving anisotropic models, providing fundamental insights for inversion based data interpretation.

3‐D Magnetic Unstructured Inversion

AbstractMagnetic inversion methods based on structured grids have been used extensively in geomagnetic research. Nevertheless, structured grids limit the modeling capability and accuracy of models with arbitrary geometries. To address these challenges, 3‐D magnetic numerical forward modeling and inversion methods using an unstructured tetrahedral grid based on a partial differential equation (PDE) framework are proposed. The methods are derived from Maxwell's partial equations and constructed using the finite element method. Arbitrary undulating topographies, complex geometries, and demagnetization effects can be represented exactly, leading to high‐accuracy forward modeling and inversion solutions. The proposed methods are suitable for both local‐ and global‐scale magnetic data interpretation, for example, in mineral exploration and tectonic research. A synthetic example and real airborne magnetic survey data collected from Mount Iliamna, Alaska, USA, are tested. The results demonstrate that the novel methods significantly improve the capability and accuracy of magnetic data interpretation.