elfe3D_GPR Theory and Computational Framework

Note

Much of the rigorous mathematical derivation of the:

  1. finite element formulation of the GPR wave equation,

  2. resulting discrete system of equations, and

  3. PML formulation that is implemented for the unstructured tetrahedral mesh of elfe3D_GPR

is discussed and developed in the thesis work [SIN2025]. Please refer to it for complete details.

Following is a snippet of the theoretical background that elfe3D_GPR is based on. It will be completed soon.

Physics of Electromagnetic Waves for GPR

elfe3D_GPR solves the full Maxwell’s equations for wave-regime electromagnetism. This enables it to solve GPR problems in heterogeneous subsurface models. Refer to [GRI2023] and optionally [JIN2015] for complete treatment of the wave-electromagnetism problem.

First-Order Edge-Based Finite Elements

The software uses first-order edge-based FE to discretize the 3D model. Refer to [JIN2008] and [JIN2015] for complete analysis of the finite element method for electromagnetism, specifically for wave-regime problems. For elfe3D_GPR, the key FE implementation details are as follows:

  1. Order refers to the degree of interpolation. First-order means linear interpolation. This has been proven adequate for many physical problems using tetrahedral mesh with a suitable level of discretization.

  2. Edge-Based refers to where the unknowns are formulated in the finite element mesh. A tetrahedron has 6 edges, which means per-element, there will be 6 unknowns in the system of linear equations. Multiply that by 6 for each of the electromagnetic field components \(E_x, E_y, E_z, H_x, H_y, H_z\), and you have 36 unknowns per element. Worth noting is that since a majority of elements are usually connected to each other, the total number of unknowns does not scale linearly to the number of elements in the model (especially including the boundary conditions).

  3. The meshes elfe3D_GPR uses are unstructured and produced by tetgen. Unstructured means they use irregularly connected elements to fit complex geometries. Unlike the regular grids of structured meshes, unstructured ones naturally favor modelling complex geometry with a high degree of accuracy.

Absorbing Boundary: Perfectly Matched Layers

Our finite element meshes will have a finite length in all three axes. As such, Perfectly Matched Layers (PML) are applied around the domain boundaries to absorb outgoing waves. This ensures that they do not reflect waves back from the truncated computational domain [BER1994]. The specific type of PML is the Uniaxial-PML [BER2007], [PEK1995], [PLE2022]. Unlike traditional U-PMLs, elfe3D_GPR implements an exact decay function that has been extensively studied over the years [BER2004], [OZG2023] and proven efficient for Finite Element meshes [SIN2025], [DIN2025], [FEN2019].