Solution of a Second Order Elliptic Partial Differential Equation with Varying Complex Coefficients: An Application for Computing Effective Complex Electrical Properties of Materials represented by 3D Images

Abstract

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 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.