## 1 Introduction  
To shift from one gear to another in an automatic transmission, one clutch needs to be released and another needs to be applied. The shifting control system of the automatic transmission controls the clutch pressure and the corresponding transmitted torque during shifting process. The dynamic characteristics of the shifting control system are directly related to the shifting quality. Figure 1 is the conventional structure of a shifting control system consisting of a relieve valve, a pilot solenoid valve and a clutch control valve. In this structure, the automatic transmission fluid passes through several valves and complicated oil lines from the main line to the clutch. The long passage causes the delayed pressure response, low control accuracy and big leakage. With the development of the industry, a new pressure valve called direct operating solenoid valve could take place of the conventional two-stage structure for its small leakage, large flow, high reliability and high control accuracy. Figure 2 is the structure of the shifting control system applying the direct operating solenoid valve. It is obvious that simple structure and light weight of the system can be obtained by utilizing this kind of structure which has been more and more adopted in the latest automatic transmission. The direct operating valves are produced by many well-known worldwide companies such as Tosok, Bosch, Highlight and Narchi. Improving the characteristics of the valve can directly contribute to the performance of the shifting control system. Hence, a growing interest in direct operating valves emerges from scientific centers. Presently, many studies on the solenoid valve are focused on a specific physical field such as the magnetic field or the flow field. [PERSON] et al. [1] proposed a new direct drive valve which has high frequency voice coil motor and advanced digital controller. A digital controller and a hybrid controller were developed to control the valve motion and were indicated to improve the control accuracy and robustness of the system through test. The static characteristics of the electromagnetic linear actuators were analyzed by [PERSON] [2]. 3D electromagnetic fields analysis of the electromagnet was carried out using ANSYS. Influence factors such as working area, coil turns, core number, armature depth and iron material that affects the magnetic force were discussed in detail. The influencing law to the magnetic force was obtained [3]. [PERSON] et al. [4] reported that the electromagnetic force of the direct action solenoid valve was mostly influenced by the current. Optimization of the design parameters of the valve was achieved by complex method. The control accuracy of the magnetic force could be well improved based on the studies. [PERSON] et al. [5] adapted Ansys Maxwell electromagnetic analysis software to model the electromagnetic dynamics and selected the best optimization model using the verified approximation model. However, the dynamic characteristic of the valve could not be adequately enhanced if we only study the magnetic force because a sensitive pressure response is caused by the interaction of the resultant force. Multi-physics coupling models were reported in many studies. The valves were usually designed as compact constructions, which contain a mechanical part, a hydraulic part and an electronic controller in a single module. [PERSON] et al. [6] carried out a three-dimensional numerical  
Figure 1: Conventional structure of the shifting control system. Figure 2: Structure of the shifting control system applying the direct operating solenoid valve. simulation employing a moving mesh with dynamic layering meshes for varying boundary conditions to investigate the flow dynamic behavior and pressure characteristics of a variable force solenoid valve. [PERSON] et al. [7] presented an alternative approach using network simulation methods to model a dispensing valve using a simulation software SABER.