2086 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
in the precision of conductivity distributions within two-
phase flow systems. This progress opens avenues for more
accurate assessments in two-phase flows and lays the
groundwork for potential implementation in three-phase
flows. This work applies polynomial optimization (RBO)
and Gauss-Newton-one step algorithms to analyse the
air-core phase boundary reconstruction in two-phase and
three-phase flows.
IMAGE RECONSTRUCTION
ALGORITHMS
This section explains the process of reconstructing bound-
ary voltage using GN-one step and RBO algorithms,
aiming to compute the matrix conductivity distribution.
Initially, the Finite Element Method (FEM) in MATLAB
EIDORS is used to generate a circular mesh with 6400 tri-
angular elements. The mesh shown in Figure 2 corresponds
to a unit radius circular shape, with each element’s conduc-
tivity determined based on water (conductivity value of 1)
and air (conductivity value of 10–9). ERT system applies
currents through adjacent electrodes, and measured volt-
ages for each current pattern are rearranged into an L ×
L matrix denoted as an V. The ITS Z8000 ERT system,
following the Sheffield protocol, provides L(L – 3)/2 inde-
pendent voltage measurements, resulting in a total of n =
16 *13/2 =104 independent voltages for a setup with L =
16 electrodes. The reconstruction of the conductivity pro-
file from ERT data involves the difference of voltages ΔV =
V – V0, where V and V0 represent true and reference voltage
matrices, respectively. The reference voltage matrix V0 cor-
responds to the domain filled with water.
Relaxed Binary Optimization (RBO)
This paper investigates the utilization of binary optimiza-
tion for the ERT reconstruction problem. The objective is
to find binary solutions,
.t. argminl1 s Y Jl 0,1,n 1 !h, l -
3 "(1)
where is a parameter. However, it is well known that binary
programming is NP-Complete. The binary programming
problem can be cast as a polynomial optimization problem
as
.t. argmin s Y Jl
0 1
i
n
i i
1
2 1
##
l l h,
l
--
3 l
=
/_i
(2)
Recent developments in polynomial optimization have
enabled constructing converging SDP relaxations to prob-
lems such as above (Lasserre 2001). (Fosson and Abuabiah
2019) reported results from the first-order SDP relax-
ation of the above problem. In the current work, we have
Figure 2. Mesh
in the precision of conductivity distributions within two-
phase flow systems. This progress opens avenues for more
accurate assessments in two-phase flows and lays the
groundwork for potential implementation in three-phase
flows. This work applies polynomial optimization (RBO)
and Gauss-Newton-one step algorithms to analyse the
air-core phase boundary reconstruction in two-phase and
three-phase flows.
IMAGE RECONSTRUCTION
ALGORITHMS
This section explains the process of reconstructing bound-
ary voltage using GN-one step and RBO algorithms,
aiming to compute the matrix conductivity distribution.
Initially, the Finite Element Method (FEM) in MATLAB
EIDORS is used to generate a circular mesh with 6400 tri-
angular elements. The mesh shown in Figure 2 corresponds
to a unit radius circular shape, with each element’s conduc-
tivity determined based on water (conductivity value of 1)
and air (conductivity value of 10–9). ERT system applies
currents through adjacent electrodes, and measured volt-
ages for each current pattern are rearranged into an L ×
L matrix denoted as an V. The ITS Z8000 ERT system,
following the Sheffield protocol, provides L(L – 3)/2 inde-
pendent voltage measurements, resulting in a total of n =
16 *13/2 =104 independent voltages for a setup with L =
16 electrodes. The reconstruction of the conductivity pro-
file from ERT data involves the difference of voltages ΔV =
V – V0, where V and V0 represent true and reference voltage
matrices, respectively. The reference voltage matrix V0 cor-
responds to the domain filled with water.
Relaxed Binary Optimization (RBO)
This paper investigates the utilization of binary optimiza-
tion for the ERT reconstruction problem. The objective is
to find binary solutions,
.t. argminl1 s Y Jl 0,1,n 1 !h, l -
3 "(1)
where is a parameter. However, it is well known that binary
programming is NP-Complete. The binary programming
problem can be cast as a polynomial optimization problem
as
.t. argmin s Y Jl
0 1
i
n
i i
1
2 1
##
l l h,
l
--
3 l
=
/_i
(2)
Recent developments in polynomial optimization have
enabled constructing converging SDP relaxations to prob-
lems such as above (Lasserre 2001). (Fosson and Abuabiah
2019) reported results from the first-order SDP relax-
ation of the above problem. In the current work, we have
Figure 2. Mesh