## 2 Materials and Methods  
### Finite Element Simulation Model  
The pull-out of the mandrel in the cold expansion strengthening process of compression bushings is a relatively slow process, which can be treated as a quasi-static loading and unloading process in establishing the finite element model. Therefore, Standard, a static analysis module in ABAQUS 2019 software, can be the problem solver to establish a 3D finite element model for the cold expansion of compression bushings. To maximize the accuracy of the finite element analysis, hexahedral 20-node elements were selected. In the finite element analysis, SOLID186 solid elements were adopted to mesh the constructed model. The bushing component is the focus of this study, so the meshing of the bushing part is more dense. The mesh of the simulation model has been tested several times to avoid the sensitivity and dependence problem of mesh density in the simulation results. In order to improve the accuracy of mesh delineation and enhance the computational efficiency, a double symmetric model of \(1/4\) structure is established based on the symmetric structure of lug plates and compression bushings, as shown in Figure 1. First, the right-handed right-angle coordinate system is established. Finally, the assembly and mutual positions of the components before the initial expansion are defined. The intersection of the axis of the lug plate hole with the expansion surface is set to the origin of the coordinate system. The intersection of the symmetry plane in the length direction of the lug plates with the expansion plane is the X-axis, and the hole axis is the Z-axis. The loading variations are accomplished through the various extrusion amounts between the mandrel and the compression bushing. In this research, general extrusion amounts of 1.5% to 4% were ex amined and the higher extrusion amount of 4% was applied on the models. Before the cold extrusion simulation, the mandrel and lug sheet holes were set as master surfaces, while the inner and outer surfaces of the compression bushing were set as slave surfaces. Since most mandrels are pre-lubricated, the friction between the mandrels and the compression bushing could be omitted in the cold extrusion simulation. Instead, the installation of the compression bushing relied on the friction between the compression bushing and lug plate hole, so the friction coefficient of the contact pair between the compression bushing and lug plate hole was set to 0.1. During the simulation, all the degrees of freedom of the parts except the mandrel were fixed. This was to prevent the failure caused by the extrusion of the mandrel resulting in the follow-through of the other parts. The degrees of freedom of the mandrel along the axis were released, and the other degrees of freedom were fixed. As the mandrel moved along the axis, the tapered section of the mandrel gradually squeezed the bushing and lugs to implement the loading process.  
Since the most important process in the cold expansion strengthening process of compression bushings is the axial pull-out process of the mandrel, the initial analysis step (initial) and the expansion analysis step (expansion) are mainly set up to establish the cold expansion strengthening model. In the initial analysis step, the positioning and initial contact of each component is completed. In the expansion analysis step, the mandrel expansion and gradual unloading of the bushing are carried out to realize the interference fit between the bushing and the lug plate. In addition, to further study the stress redistribution law of the reinforced structure under external loads, it is also necessary to set up the loading and unloading analysis step.