Fig. 7. Flowchart of the MNMS algorithm. Red notes are parts for
revision, the blue arrow shows the schematic of the double minimum
check, a perturb point is randomly put to adjacent domain.
Fig. 8 shows the changes in function value with the re-checks for the
global minimum. After comparing the values of the random points next to
the search area with the values of the simplex points numerous times, no
superior configuration has been discovered. Thus, we can ensure the
validity of the minima. It should be noted that this is not a true
global minimum, the globality of the result has not been proven
strictly.
Fig. 8. Function value and various electrodes voltages as functions of
the iteration number of the algorithm.
4.4. Parameter setting of
MNMS algorithm
Correctly setting algorithm
parameters is critical as it ensures the correctness of the solutions
and the robustness of the optimization procedure. A number of parameters
influence the optimization process. To decide the optimal values for
these parameters, the performance with different parameter choices was
investigated. Four basic algorithm parameters (consisting of
coefficients of reflection, expansion, contraction, and shrinkage) that
define a complete NMS algorithm are chosen to use the default values.
Other optional parameters in simulations, such as maxcalls and
minradius, are also set to use the default values. Because the default
values of the parameters are suitable for most of the general problems,
changing the values did not show much benefit but instead affected
algorithm stability.
A problem encountered in this procedure was that the size of the initial
simplex had an impact on the speed of convergence and globality of the
searching process. The step size is a crucial algorithm parameter that
requires careful consideration as it determines the shape of the simplex
at startup. Therefore, several simulations are performed to explore the
relationship between the step value and the effect of optimizations.
Fig. 9 shows the changes in the searching range in two variable
dimensions with the step value increasing from 0.5 to 20. A tendency
towards a larger searching area for the two variables was observed as
the step value increased until 13, after which an opposite trend was
observed. As previously indicated, a variable transformation was
introduced to the NMS algorithm. This transformation imposes a periodic
change on the variables, making excessively large variable values
meaningless. To avoid the optimization result getting trapped in a local
minimum prematurely, broad search range for the solution space and a big
starting simplex are required. Taking into account the periodicity of
the variables, the optimal step value might be selected from 5 to 13. In
the following simulation, a step value of 5 is adopted considering the
time-consuming of the calculations.
Fig. 9. Distribution in two dimensions under different Step value
setting, variable 1 and 4 are first and third drift tube voltages,
respectively.
Results and discussion
In this study, we focus on optimizing ion transport in an MRTOF-MS. By
utilizing an MNMS algorithm in the simulation analysis, we optimized
eight electrodes with significant influence on the transportation of ion
beams starting from the flat ion trap. After a series of iterative
optimizations, the ion trajectories have been improved, and the
objective function value has been reduced from the tens level to
10–2.
In applications, MRTOF works either in straight mode (Fig. 10) for full
spectral or in reflection mode (Fig. 11, reflected with 50 laps) for
high resolution. The ion trajectory simulation after extraction from the
FTL is presented in Figs. 10 and 11. is the optimizations in straight
mode, and Fig. 11 is the reflection flight optimizations with
50 laps. The objective of these
simulations is to minimize spatial and energy distributions in the
radial direction. For each of the figures, the screenshots marked A to
D, from top to bottom, show the changes in simulation results during the
optimization process. Herein, A1, A2, A3, DT1, DT2, DT3, and two lens
electrodes in MRTOF were
optimized. Comparing the
trajectories of the ions, we can see that constant optimizations produce
good results, whether in the full spectral mode or high-resolution mode.
The algorithm has apparent optimization effects compared to the initial
state. Trajectories become so accordant that they almost overlap in a
line. This situation keeps similar in a reflective state. Moreover, in
reflection mode, the transmission efficiency also improves. Ion losses
before optimization can be clearly observed in Fig. 11(A), where ions
are lost due to collision on the lens. While a well-adjusted ion beam
appears with the use of appropriate electrical parameters for
optimization and a 100% transmission efficiency is achieved as shown in
Figs. 11(B), (C), and (D).
Significant improvements in ion transmission were observed after running
the MNMS algorithm, indicating that the new MNMS algorithm is effective
in MRTOF simulation. It is quite efficient in quickly finding the
optimal solution (within a few hours for each optimization). Moreover,
it requires very little storage and information for execution. Because
of its simplicity and robustness, it seldom fails in the optimization
process.