Luca Berardocco defends his Ph.D. thesis on " Hybridizable Discountinous Galerkin Methods for time-domain electromagnetic diffusion in underground strata formation"


The simulation of electromagnetic phenomena in geophysical applications has benefited
from the continued evolution of computers and numerical methods. However, despite the
great success that high-order methods experienced in other engineering fields over the last
decades, their application in the field of geophysics remains largely unexplored.
It is the goal of this thesis to lay the groundwork for the application of high-order meth-
ods in the numerical investigation of electromagnetics in geophysical applications. The
focus is on the spatial discretization of Maxwell’s equations by means of the Hybridizable
Discontinuous Galerkin (HDG) method.
In the first part of the thesis, an HDG method based on a mixed-equation formulation,
for the solution of electromagnetic wave propagation problems in dielectric and conductive
media, is proposed. Perfect Electric Conductor (PEC) boundary conditions and Absorb-
ing Boundary Conditions (ABC) are developed and implemented to allow the simulation
of wave and diffusion phenomena. The spatial and temporal convergence properties of the
proposed method are assessed and the problem of electromagnetic diffusion in a unit cube
containing a current source is used as a validation scenario. The method is then tested
with a benchmark problem for wave propagation and scattering by a dielectric obstacle.
Finally, the problem of scattering by means of a conductive sphere is solved.
In the second part, an HDG method for the solution of electromagnetic diffusion phe-
nomena is proposed. This novel method is based on the electromagnetic diffusion equation
and is obtained neglecting the displacement current term with respect to the conduction
current term in the full-wave equation. Additionally, an HDG method for the solution
of the direct current resistivity problem is proposed. This method can be used to obtain
the initial condition for the electromagnetic diffusion initial value problem. To improve
the efficiency of the novel method, the properties of the linear system of equations are
investigated. The proposed formulation is modified such that the condition number of the
linear system’s matrix improves in the range of parameters typically found in geophysical
applications. Finally, the method is shown to achieve optimal convergence rates of the
primary variables and an improved convergence rate of the post-processed variable.
Next, a real-world application scenario, for which an analytical solution is available,
is used to validate the proposed method. The scenario models a controlled-source elec-
tromagnetic survey in which the transient response of the electromagnetic field to a sub-
merged current source is measured by surface receivers.
Finally, the method is used to simulate the transmission of an electromagnetic signal in
an example of a Measure-While-Drilling application. In this model, an arbitrarily oriented
current source is buried in a conductive stratum and the effects of different simulation
parameters on the received signal are explored.