Investigation of dynamic characteristics in the hydraulic shock absorber based on different FSI FEA methods

Two three-dimensional fluid-structure interaction (FSI) finite element analysis (FEA) models with different fluid cells and coupling solution methods of a gas-pressured hydraulic shock absorber is constructed, and the results are partially verified experimentally. Models with FCBI fluid cell and direct FSI solution method or FCBI-C fluid cell and iterative FSI solution method provide good simulation results of the FSI system of the hydraulic damper.


Introduction
There are many studies of shock absorbers based on FSI method at home and abroad [1][2][3][4][5][6], and the interaction calculation procedures include direct and iterative methods in solving the finite unit equations for fluid-structure coupled systems.
The main numerical methods for solving the fluid control equations in fluid-structure coupled systems include finite difference, finite volume and finite element methods.The finite volume method is the most widely used, the FCBI-C fluid cell based on the finite volume method defines all the variables in the centre of the grid cell, and the coupling between velocity and pressure is solved by iterative method, so in the fluid-structure coupled model using the FCBI-C fluid cell, only the iterative coupling solution algorithm can be chosen.The application of finite element method in fluid problems is later developed than finite difference and finite volume methods, but it has certain advantages in dealing with complex boundary problems, and the solution of solid problems is mostly based on the finite element method, so it has a greater advantage in the solution of flow-solid coupled problems.Bathe et al [7] proposed a Control Volume Finite Element Method (CVFEM) based on flow-condition-based interpolation (FCBI) to solve the system of N-S equations without pressure correction.
If FCBI fluid cells are used, the FSI model can be solved by direct or iterative method.If FCBI-C fluid cells are used, the FSI model can only be solved by iterative method.For some problems, such as VOF algorithms, slipping mesh flow-flow coupling algorithms, model considering thermal coupling effects, etc., which can only be used in conjunction with the FCBI-C fluid cell and iterative coupling solution methods.Therefore, it is important to investigate the effectiveness of the two fluid cells and the two FSI methods for practical engineering applications.

Model Construction
Figure 1(a) shows the structure of a gas-pressured hydraulic shock absorber, and the finite element mesh models of the structure and fluid are shown in Figure 1(b) and Figure 1(c).[8] In the FEA models, surfaces of the floating piston, the valve plates, the flow channels, and the main piston are defined as FSI boundaries.Surfaces of the main piston and the valve plates are defined as contact pairs.The oil is defined as a kind of slightly-compressible fluid, the structures' material are steel, the material parameters are shown in Table 1.

Model Solution
In this section, FCBI fluid cell with direct FSI solution method and FCBI-C fluid cell with iterative FSI solution method are used to solve the FSI dynamics model of the same gas-pressured hydraulic damper, in order to explore the computational accuracy and differences between the two models: (1) Definition of the gas chamber is different.Model using FCBI fluid cells and direct FSI solution methods define the gas FEA mesh model directly in the fluid model, gas in the chamber is defined by the material parameters and a low velocity compressible fluid model.While in the model using FCBI-C fluid cells and iterative FSI solution method, gas characteristics in the chamber is defined by constructing non-linear spring cells in the structural model, which is consistent with the pressure-volume change of ideal gas under adiabatic conditions.
(2) The preloading approach is different.Model using FCBI fluid cells and direct FSI solution method defines the initial pressure field directly in the fluid model.Model using FCBI-C fluid cells and iterative FSI solution method achieves loading of the initial fluid pressure field by compressing the nonlinear spring cells representing the gas chamber.
(3) The fluid control equations' solution method is different.In the FCBI finite element method, the momentum equations are solved in conjunction with the continuum equation.The non-linear equations formed by velocity and pressure are solved by Newtonian methods.In the FCBI-C finite volume method, the variables are defined at the centre of the cells and the coupling between velocity and pressure is handled by iteration.
(4) The method for solving the control equations of FSI systems is different.The direct FSI solution method couples the fluid variables and structural variables in a single finite element equation, which is more efficient and accurate, but occupies more memory.The iterative FSI solution method solves the fluid and structural variables separately, which is slightly less accurate but takes up less memory.

Model Validation
A simple harmonic excitation with an excitation frequency of 6 Hz, a piston displacement amplitude of 40 mm and a piston velocity amplitude of 1.5 m/s was applied to the end of the piston rod.By extracting the end face load of the piston rod from the calculation results, the damping characteristics of the hydraulic damper can be obtained, as shown in Figure 2. As can be seen in Figure 2, the results show good consistency, which verifies the correctness of the simulation models, and the damping force values obtained by the model with the FCBI fluid cells and direct FSI method are slightly larger than those obtained by the model with the FCBI-C fluid cells and iterative FSI method.

Dynamic Characteristics
The pressure distribution clouds in the flow field near the damper's valvetrain at different times using the FCBI fluid cells and the direct FSI method are shown in Figure 3 and the velocity vector diagrams are shown in Figure 4.The pressure distribution clouds at different times using the FCBI-C fluid cells and the iterative FSI method are shown in Figure 5 and the velocity vector diagrams are shown in Figure 6.As can be seen in Figure 3 and Figure 4, the pressure distributions obtained from both models are generally consistent at the moment of maximum velocity of the compressive stroke piston, while the minimum pressure value at the moment of maximum velocity of the extension stroke piston is smaller when the FCBI-C fluid cells and the iterative FSI method are used.As can be seen in Figure 5 and Figure 6, the velocity distributions obtained from the two models are generally consistent, while the maximum velocity values are smaller when the FCBI-C fluid cells and the iterative FSI method are used.
Cloud maps of pressure distribution of the fluid near the valve system at the slit exit section under different FSI methods can be obtained, as shown in Figure 7 and Figure 8.It can be obtained from the figures, pressure distributions at the cross-section obtained by the two FSI methods are basically the same.Moreover, the average values of the pressure at all nodes in the cross-sections are calculated to obtain the pressure change history curves, as shown in Figure 9.It can be found that the pressure at the outlet of the flow channel obtained by the two methods is basically the same, which is consistent with the variation regulation of the damping characteristics of the damper.

Conclusion
This paper discusses the construction and solution methods of a FSI FEA dynamic model for a gaspressured hydraulic shock absorber.In the simplified 1/6 part model of the damper, the FSI surfaces, contact surfaces, elastic properties of the compression springs of the valve plate, material parameters of the structure and fluid, flow field dynamic mesh control techniques, fluid cells and coupling solution methods are defined.The results show that, models with FCBI fluid cell and direct FSI solution method or FCBI-C fluid cell and iterative FSI solution method provide good simulation results of the FSI system of the hydraulic damper.
But considering that the direct FSI method reflects the latest progress in the field of FSI and realizes the coupled solution of fluid variables and structural variables, it is recommended that subsequent research work be based on direct FSI methods as far as possible.

Figure 1 .
Figure 1.Simplified structure and FEA mesh models of the shock absorber

Figure 3 .
Figure 3. Pressure Distribution Cloud with FCBI fluid cells and direct FSI method

Figure 4 .
Figure 4. Velocity vector diagram with FCBI fluid cells and direct FSI method

Figure 5 .Figure 6 .
Figure 5. Pressure Distribution Cloud with FCBI-C fluid cells and iterative FSI method

Figure 7 .
Figure 7. Cross-sectional pressure distribution with FCBI fluid cells and direct FSI method

Figure 8 .
Figure 8. Cross-sectional pressure distribution with FCBI-C fluid cells and iterative FSI method

Figure 9 .
Figure 9. Pressure history curves with different models

Table 1 .
Material parameters