## 1 Introduction
The model could correctly predict the dispensed liquid volume in dependence of the main parameters like pressure and opening time. Another multi-physics modeling method was introduced by [PERSON] et al. [8]. They successfully predicted the dynamic characteristics of a proportional solenoid valve and used it for valve design. [PERSON] et al. [9] optimized the structure of a large flow solenoid valve by multi-physics modeling method. [PERSON] et al. [10] extended the dynamic model of the high-response dual proportional solenoid valve discontinuous projection-based adaptive robust control to synthesize controller to deal with the parametric uncertainties and uncertain nonlinearities. Both the simulation and experimental results showed the proposed controller was effective. The nonlinear characteristic, the orifice area and the dynamic model of the solenoids were also analyzed using Matlab by [PERSON] and [PERSON] [11]. Meanwhile, [PERSON] et al. [12] analyzed the influences of throttle nozzles' diameter and other parameters on the dynamic and static characteristics of a low-pressure large-flow pilot-operated solenoid valve via AMESim simulation software. And optimization was conducted in the model. [PERSON] et al. [13] provided a convenient method of design verification of solenoid operated valve. [PERSON] et al. [14] found that firing current, holding current, spring pre-tightening force and spring stiffness had great effects on the dynamic response characteristics of solenoid valve. An electromagnetic mathematical model of a high-speed solenoid valve was developed in Fortran language. [PERSON] et al. [15] discovered that the electromagnetic energy conversion characteristics of the HSV were affected by the drive current and the total reluctance, consisting of the gap reluctance and the reluctance of the iron core and armature soft magnetic materials. [PERSON] et al. [16] established mathematical models of the integrated electrohydraulic solenoid valve and wet clutch piston assembly in the Simulink environment of Matlab. The dependency of the system to system variables on input pressure and the influence of air content on dynamic response of the valve were investigated. Though an amount of work has been done on the direct operating solenoid valve, there still exist several shortcomings in previous studies. The magnetic force obtained from the tests is input directly into the model rather than real-time computing which would result in the capability of analyzing the magnetic force. Forces, including the magnetic force, play a very important role in a direct operating valve because they have strong relation to both the pressure response and the leakage. Therefore, it is essential to introduce a multi-physics method which could compute all forces in real time and predict the dynamic characteristics of the direct operating solenoid valve. In this paper, a multidisciplinary approach was conducted to obtain the dynamic characteristics of the direct operating solenoid valve. According to the approach, the simulation model of the valve was developed using the minimum element method. Secondly, experiments have been conducted on the test rig in order to verify the numerical results. The experimental results showed good agreement with the simulated ones, which confirm the accuracy of the simulation model. Thirdly, based on the model, the influencing factors including forces, critical size and leakage flow on the performance of the valve were detailed and analyzed. Finally, suggestions are proposed to improve the performance of the direct operating solenoid valve, which could be effective for the valve design.