## 3 Structural Parameter Optimization of the MRE Isolator
### Parameter Optimization Method  
(1) Optimization module  
COMSOL provides general simulation software based on advanced numerical simulation methods that can perform single-physics or multi-physics coupling simulations. COMSOL also includes various additional modules to realize its application and analysis in the fields of electromagnetics, structure, acoustics, fluid, and chemical engineering. In this paper, the electromagnetic field module and optimization module in COMSOL are used to optimize the parameters of the MRE isolator. The optimization module of COMSOL is mainly used to solve two kinds of optimization analysis problems. One is a single-objective optimization design problem, which achieves the best performance by finding the appropriate control variable value. The other is the inverse problem, by determining a series of parameter values so that the simulated data can match the measured data, such as the curve fitting problem. In the optimization module, the objective function, control variables, upper and lower limits, and constraints need to be established. The objective function is the quantitative representation of the best performance, and any model output can be used as the objective function. In order to make the MRE isolator have a larger magnetic control range, the average magnetic induction intensity at the MRE is selected as the design goal of parameter optimization. The design variables can be any model input, such as geometric size, geometric shape, material properties, material distribution, etc., and some geometric size parameters are selected as design variables. The optimization module of COMSOL supports gradient-based optimization algorithms and gradient-free optimization algorithms. Gradient-based optimization algorithms include MMA, IPOPT, SNOPT, Levenberg-Marquardt, and so on, which are widely used in various objective functions and constraints and can calculate all analytical derivatives at the same time. Compared with gradient-free optimization algorithms, they are more suitable for shape optimization and topology optimization. Gradient-free algorithms include BOBYQA, COBYLA, Nelder-Mead, and coordinate search, which are suitable for parameter optimization. Therefore, the Nelder-Mead algorithm is selected as the optimization algorithm when the size parameters of the MRE are optimized. (2) Variables  
The optimization problem of the size parameters of the MRE isolator is abstracted into a mathematical model, which mainly includes variables, objective functions, and constraints. In the calculation of the stiffness of the elastic element in Section 2.2.3, the values of the parameters r2, r3, and h3 in the magnetic circuit structure of the MRE isolator have been determined. In addition, due to the existence of the geometric relationship h4 = h2 + h3, h2 is set to 13.5 mm, and h4 can be obtained from h2 and h3. Therefore, the variables of size parameter optimization of the MRE isolator can be determined as r1, r4, and h4. The parameter settings for parameter optimization of the MRE isolator and the upper and lower limits of the variable parameters are shown in Table 1.  
As shown in Figure 3, when the injected fluid is transported from ground to the main channel of the eccentric pipe, the ground control motor drives the screw to rotate forward, and the sliding assembly is threaded with the lead screw. In this case, it starts to move to the left along the direction of the lead screw, the first channel is slowly opened, the injected liquid enters the first channel through the flow channel and leads to the formation through the connecting pipe of the first channel. The first layer of water injection is completed. (3) Optimization objective  
The objective of size parameter optimization is to maximize the magnetic control range at the MRE, and the objective function is the average magnetic induction intensity at the MRE. The optimization function can be expressed by the following formula:  
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