3-D dc resistivity modelling based on spectral element method with unstructured tetrahedral grids

Abstract

SUMMARY In this paper, we propose a spectral element method (SEM) based on unstructured tetrahedral grids for direct current (dc) resistivity modelling. Unlike the tensor-product of 1-D Gauss–Lobatto–Legendre (GLL) quadrature in conventional SEM, we use Proriol–Koornwinder–Dubiner (PKD) polynomials to form the high-order basis polynomials on tetrahedral grids. The final basis functions are established by using Vandermonde matrix. Compared to traditional SEM, our method truly takes into account the high precision of spectral method and the flexibility of finite element method with unstructured grids for modelling the complex underground structures. After addressing the theory on the construction of basis functions and interpolation and integration nodes, we validate our algorithm using the analytical solutions for a layered earth model and the results from other methods for multiple geoelectrical models. We further investigate a dual-track scheme for improving the accuracy of our SEM by increasing the order of interpolation polynomials or by refining the grids.

Characteristic Research on Electromagnetic Response of Two‐Dimensional Complex Structure Targets Based on Meshless Method

“…There are mainly two kinds of models, one is the actual geological target, and the other is composed of several geological targets with regular shape. In general, the model is established based on unstructured grids or discrete nodes, and the high‐precision numerical simulation is realized by using the finite element method (FEM) (Cao et al., 2022; Key & Weiss, 2006; Li et al., 2017; Ye et al., 2018; Zhu et al., 2020), finite volume method (Ismagilov, 2015; Jahandari & Farquharson, 2015; Liu et al., 2019; Lu et al., 2021; Wang et al., 2016) or meshless method (Ji et al., 2018; Long & Farquharson, 2019; Nicomedes et al., 2021; Ortiz et al., 2021; Shams et al., 2022; Wittke & Tezkan, 2014). In these methods, the meshless method has been widely used in the modeling of complex structures in recent years due to its advantages such as no mesh generation and strong adaptability.…”

Spectral element method for 3-D controlled-source electromagnetic forward modelling using unstructured hexahedral meshes

“…Their approach represents an alternative to deal with complicated spatial distributions of material properties. For 3-D direct current (DC) forward modelling, Zhu et al (2020) enabled application of the SEM to unstructured tetrahedral meshes by transforming between tetrahedra and hexahedra. Instead of using 1-D GLL quadrature as commonly used in the spectral element method, they use Proriol-Koornwinder-Dubiner polynomials to construct the arbitrary order nodal basis functions to represent the electric potential.…”

“…The elemental matrix computation relies on mapping relations between the physical domain, the tetrahedral reference domain and the hexahedral domain. Zhu et al (2022) developed a forward modelling algorithm for frequency-domain airborne EM applications based on EM potentials and the nodal unstructured tetrahedral spectral element method (Zhu et al 2020). Owing to the necessity to transform between tetrahedra and hexahedra and to interpolate, Zhu et al (2022) method imposes additional complexity to the implementation.…”

Comparison of terrain corrections based on the point source and line source DC methods

“…During geoelectric exploration in a complex environment, to eliminate the influence of terrain fluctuations on measurements, terrain corrections can be conducted using diverse numerical simulation methods (Wang et al, 2015;McCubbine et al, 2017;Rimary et al, 2019;Saber et al, 2020;. Numerical simulation methods for solving DC exploration problems include the following: boundary element, finite difference, and finite element (Ma et al, 2019;Zhu et al, 2020;. The finite element method exhibits advantages relative to other numerical simulation methods.…”