## 4 Parametric Analysis on Creep Deformation of PMMA Frustum Observation Windows
### Finite Element Model and Calculation Methodology for the Observation Window  
Due to the symmetry of the geometric shape, constraint conditions, and external forces with respect to the central axis, the distribution of displacement, stress, and strain on any rotational section passing through the central axis is identical. Consequently, the spatial axisymmetric problem can be simplified and transformed into a plane problem. The simulation and calculation of the observation window structure are performed using a two-dimensional axisymmetric plane model, as depicted in Figure 4.  
Given the absence of precise material performance data, it is recommended to set simulation parameters by considering two sets of combinations for an elastic modulus and Poisson's ratio: (2000 MPa, 0.38) and (2400 MPa, 0.35) [22]. In this paper, the material properties of the observation window are established using an elastic modulus of 2000 MPa and a Poisson's ratio of 0.38 for simulation. The external loading input is applied as a pressure of 115 MPa to the model. For the two-dimensional model, the element type selects the four-node bilinear axisymmetric quadrilateral element, reduces the integration, and the hourglass control divides the mesh, as shown in Figure 5a. The interaction between the window and the window seat is defined as surface-to-surface contact. The rigid window seat surface serves as the main surface, while the observation window is designated as the secondary surface. Contact is established between the window seat and the window body, and friction is introduced with a friction coefficient set at 0.1. The contact definition and interaction relationship of the observation window are shown in Figure 5b. During the finite element simulation, the contact direction is consistently aligned with the normal direction of the main surface. This ensures that nodes on the slave surface do not penetrate the main surface, while nodes on the main surface may cross the slave surface. A fixed constraint is applied to the bottom of the window seat, and a symmetrical constraint is established on the section, as shown in Figure 5c. Based on the test loading conditions, the finite element analysis steps are configured, with two analysis steps. The first step involves quasi-static loading, and the pressurization time is set to be 4173 s. In the second step, a viscosity analysis is conducted. When the pressure reaches 115 MPa, the loading state is sustained, and the holding time is set to 18,000 s.  
Figure 4: Two-dimensional geometric model of the window. In the finite element contact nonlinear analysis, different mesh sizes may result from variations in the contact master and slave surfaces, leading to potential penetration issues in the contact calculations. Typically, the mesh size is set and divided differently. The rigid window seat is used as the main surface, and the mesh size can be slightly larger, while the flexible window is used as the slave surface. The mesh should be denser and the mesh size should be smaller. To prevent escalation in the computational cost resulting from a high number of meshes in subsequent calculations, a convergence test is conducted with different sizes for the window and window seat meshes. Initially, a convergence study is conducted with mesh sizes of 1 mm, 2 mm, 3 mm, 4 mm, 5 mm, 6 mm, and 8 mm. The results, as depicted in Figure 6, reveal that as time increases, the \(y\)-axis displacement of the center of the lower surface of the window stabilizes. The \(y\)-axis displacement tends to converge as the mesh size of the window is set to 5 mm. Then, the convergence of the mesh size of the window seat is 10 mm, 12 mm, 14 mm, 15 mm, 16 mm, 18 mm, and 20 mm. The results are illustrated in Figure 7. As time increases, the \(y\)-axis displacement of the center of the inner surface remains stable. The \(y\)-axis displacement tends to converge when the mesh size of the window seat is set to the intermediate size of 15 mm.