Discontinuous Galerkin time-domain method for GPR simulation of conducting objects

In this paper we describe the discontinuous Galerkin time-domain method and apply it to the simulation of ground-penetrating radar (GPR) problems in 3D. The method is first validated with analytical solutions and we show its superior behaviour when compared to the classical finite-difference time-domain method, widely employed in GPR simulation. In the discontinuous Galerkin time-domain the solution is allowed to be discontinuous at the boundaries between adjacent elements (unlike in finite elements) and continuous numerical fluxes are employed at the interface to connect the solution between them. The resulting algorithm is quasi-explicit in space, only requiring the inversion of M square matrices of Q × Q elements (with Q the number of basis functions). A two-dimensional discontinuous Galerkin time-domain approach has been successfully applied to GPR simulations involving buried objects in a lossy half-space (Lu et al. 2005). In this paper, we present a general description of a three-dimensional discontinuous Galerkin time-domain method including both nodal and vector formulations and show an application to the simulation of a full GPR scenario. Validations of the method with benchmark problems serve to prove the superior accuracy of this technique compared to the classical FDTD, outperforming the later in computer requirements. DISCONTINUOUS GALERKIN TIME-DOMAIN THEORY Vector elements formulation Let us assume Maxwell’s curl equations for linear isotropic homogeneous media in Cartesian coordinates. Now, let us divide the space in M non-overlapping elements Vm, each bounded by Sm and enforce a weak form of them by performing the inner product of each equation with a basis of local continuous vector test functions. The term ‘weak’ here means that we no longer require the equation to hold absolutely and we search for ‘weak’ solutions with respect to certain test functions to be defined later (Hesthaven and Warburton 2007). (1) INTRODUCTION Numerical techniques are an indispensable tool in the analysis and design of all kinds of electromagnetic systems. In particular, they have been successfully applied to the simulation and optimization of ground-penetrating radar (GPR) systems (Fernández Pantoja et al. 2000; López et al. 2001; Diamanti and Giannopoulos 2009). Among them, time-domain methods are especially suitable for GPR simulation, since they are able to provide the full transient response of the system on a single run, allowing the user to analyze the system response in a causal way. The finitedifference time-domain (FDTD) method has been the most employed one, mainly because of its simplicity, ease of implementation and simulation speed (Giannopoulos 2005). However, FDTD has severe drawbacks related to the staircased approximation it employs for curved boundaries. A recent alternative of FDTD is given by the discontinuous Galerkin timedomain method, which is experimenting an increasing development in computational electromagnetics (Hesthaven and Warburton 2002; Gedney et al. 2007; Pebernet et al. 2008; García et al. 2008; Alvarez et al. 2010). Discontinuous Galerkin time-domain employs a discontinuous Galerkin weighting procedure to handle the spatial part of time-domain Maxwell’s curl equations. Like in the finite elements method, the space is divided into M non-overlapping elements (e.g., curvilinear tetrahedra), in each of which the solution is expanded in a set of nodal (Bernacki et al. 2006) or vector (Gedney et al. 2007) basis functions of arbitrary order. The temporal part of Maxwell curl equations can be handled by finite differences or by any other finite differentiation technique. Near Surface Geophysics, 2011, 9, 257-263 doi:10.3997/1873-0604.2011004 L. Diaz Angulo et al. 258 © 2011 European Association of Geoscientists & Engineers, Near Surface Geophysics, 2011, 9, 257-263 Notice, that boundary conditions between different dielectric/ magnetic media are naturally handled in a weak manner in the discontinuous Galerkin time-domain formulation, thanks to taking the same tangential components of the fields and in the flux integrals for two adjacent elements. Perfect electric conductor boundary conditions are also enforced in a weak manner by requiring the tangential electric field employed in the flux integrals to be null and the tangential magnetic field to be continuous (Alvarez et al. 2010)