Valbuena Soler, Johnny Ramon

### Description

Materials may be characterised by using their electrical
properties which establish how they interact when an electric
field is applied at various frequency ranges. This interaction
is used to determine properties of materials such as moisture
content, bulk density, bio-content, chemical concentration and
stress-strain. In the case of the physical characteristics of
rocks, the response of the minerals under the influence of an
electric field is different at...[Show more] distinct frequencies due to their
chemical compounds. It affects the electrical properties.
The computing of complex effective permittivity and complex
effective conductivity of materials plays an important role due
to its applications in different fields. The response of these
properties under the influence of an alternating current field is
used to characterise materials. The development of an approach to
calculate these properties involves the solution of the second
order elliptic partial differential equation as ∇ [Q(w)∇ u(x, y, z)] = 0, where Q(w)ϵC3˟3 represents the physical parameters of
the different phases in the material, and u(x,y,z) is the
electric potential. The main difficulty in solving this equation
comes from the high contrast of the coefficients in the distinct
phases of the material.
There is an efficient approach that is used to compute the
effective electrical conductivity of material under the influence
of a static field. The material is represented in a 3D image. The
Finite Element method and periodic boundary conditions are used
to build an energy function which is minimised using the
Conjugate Gradient algorithm. It allows to obtain the electric
potentials, and then the computation of the conductivity is
carried out. Moreover, this approach can also be used to
calculate effective permittivity of material when a static field
is applied, just by making a few changes in the approach.
However, it is not possible to modify this approach to compute
complex effective permittivity and complex effective conductivity
because if one wants to obtain the electric potentials, a
complex energy function has to be minimised.
This research is focused on developing a numerical scheme that
allows to solve the second order elliptic partial differential
equation with varying complex coefficients in order to obtain the
electrical potentials. In the initial stage, a few tools of
functional analysis are used to transfer the strong formulation
into the variational or weak formulation in the appropriate
functional space. A demonstration is made to prove that the
sesquilinear form is bounded and $V$-elliptic. These conditions
are necessary to use the Lax-Milgram theorem which guarantees
that there is a solution, and that it is unique. In order to
find the best approximation uh to the solution u of the
variational problem, the Galerkin method and the orthogonality
condition between u and uh are used to produce the best
uh in a given approximating subspace in a finite-dimensional
space. The process of construction of the finite-dimensional
subspace is carried out using the Finite Element method.
The first stage of the numerical scheme consists in constructing
a complex system of linear equations that arises from the second
order elliptic partial differential equation. This is carried out
by using the physical parameters of the material represented in a
3D image, the frequency where an electric field is applied, the
employment of the Finite Element method, and the application of
the Dirichlet and the Neumann boundary conditions. The second
phase in the numerical scheme focuses on solving the complex
system. The solution is computed using the technique of
Hierarchical Matrices in combination with a Linear Method and the
Generalised Minimal Residual Method algorithm. A C code was
written to implement the scheme. The code uses the NetCDF library
to read the 3D image and the H-Lib(Pro) library
to work with the Hierarchical Matrices.
The scheme was evaluated using three artificial materials and
three types of rocks with their 3D images, their electrical
parameters, and their ranges of frequencies where the electric
fields are applied. A complex system of linear equations is
generated by each frequency within the range of each sample. In
total, there are 199 complex systems of linear equations
generated from the six different samples that were used to assess
the scheme. The performance of the scheme is measured in terms of
the convergence rate and the frequency. The numerical results
show that the scheme is a robust tool to solve the second order
elliptic partial differential equation to obtain the electric
potentials, which are needed to compute the complex effective
electrical properties.

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