## 1 Introduction  
Small intersecting holes are a common structure in large engine nozzles, hydraulic valves, and other parts [1, 2]. When the fluid medium flows through the pipeline, in order to make the medium flow evenly and smoothly, avoid incrustation, and ensure the stable operation of the system, it is not only necessary to remove the burrs from the intersecting lines, but also more precise rounding is needed for some high-performance parts, to reduce the blocking at the intersecting line of the intersecting hole, reduce the energy loss, and improve the reliability and accuracy of the fluid system [3, 4]. Restricted by the structure and size of the inner intersecting hole, accurate and controllable rounding of the intersecting line is a difficult problem. At present, the methods for improving the shape of the intersecting line mainly include mechanical machining [5, 6], abrasive waterjet machining [7, 8], ultrasonic machining [9], magnetic abrasive machining, and electrochemical machining [10, 11]. Electrochemical machining is non-contact machining, which has the advantages of not being restricted by material hardness, a good surface quality after machining, and high machining efficiency, and it is more suitable for the machining of difficult structures, such as the intersecting line of pinholes, thin tube inner walls, and blades [12, 13, 14]. [PERSON] et al. [15] used electrochemical finishing on the inner surface of deep holes after additive manufacturing. The \(R_{\text{a}}\) value of the inner surface roughness of the deep holes decreased from 15.991 \(\upmu\)m to 10.396 \(\upmu\)m, and the \(R_{\mathrm{z}}\) value of the surface roughness decreased from 84.226 \(\upmu\)m to 54.962 \(\upmu\)m. [PERSON] et al. [16] conducted a multi-physical field coupling simulation on electrochemical machining in film cooling holes. The results showed that pulsed electrochemical machining improved the temperature distribution and current density of the machining area, which improved the accuracy and stability of electrochemical machining in the holes. [PERSON] et al. [17] used a new pull downstream cathode structure to process a large-diameter special-shaped inner spiral tube, effectively improving the uniformity of the electrolyte flow field and improving the surface quality. [PERSON] et al. [18] improved the flow field and processing positioning using a synchronous pulse current and low-frequency oscillations, thus reducing the stray corrosion on the side wall of rhomboid holes. [PERSON] et al. [19] optimized the cathode shape of electrochemical machining for small inner-walled ring grooves of 1J116 material, and adopted an optimized cathode shape to effectively change the flow field distribution during the machining process and improved the surface quality. The above research shows that electrochemical machining can effectively improve the surface quality and shape accuracy of special-shaped parts, but it has strict requirements for the design of the cathode structure and the distribution of the flow field in the machining [20,21]. The electrochemical machining processes of intersecting holes, deep holes, and through-holes are affected by physical fields, such as the flow field and electric field [22]. The finite element simulation method has been used to explore the physical field characteristics in the machining process, to predict the workpiece formation condition, which can shorten the cathode development cycle and reduce the manufacturing cost [23, 24, 25]. [PERSON] et al. [26] used finite element simulation to optimize the design of a cathode for machining complex parts with multi-stage internal cone holes and verified through experiments that the gap and flow field of the cathode with a cone angle of 2\({}^{\circ}\) were evenly distributed during the machining process. [PERSON] et al.