# Vortex-Induced Micro-Cantilever Vibrations with Small and Large Amplitudes in Rarefied Gas Flow
## 4 Numerical Setup
### DSMC Numerical Parameters and Mesh Setup  
The relation between Knudsen, Mach, and Reynolds numbers can be expressed as follows (see [26]):  
\[Re=\frac{16}{5\pi}\sqrt{\frac{\gamma^{8}\pi}{2}}\frac{Ma}{Kn}\approx\sqrt{\frac{ \gamma^{8}\pi}{2}}\frac{Ma}{Kn} \tag{21}\]  
The Reynolds number is 47, which ensures the formation of a stable and regular Karman vortex street behind the cantilever (see Figure 4a). The initial boundary conditions are as follows: the velocities are equal to inlet velocities, the temperature equals the reference temperature, and the pressure equals the reference pressure. At the commencement of the simulation, a transitional interval precedes the full establishment of the nonstationary regime. In this work, the DSMC method utilized a hard-sphere model for a monoatomic gas coupled with the SBT collision scheme [27], following recommendations provided in [28]. Air was represented with a reference viscosity of 18.2 \(\times\) 10\({}^{-6}\) Pa s, an average molar mass of 29 g/mol, and a corresponding particle diameter of 3.7 \(\times\) 10\({}^{-10}\) m. A diffuse reflection boundary condition was set at the elastic micro-cantilever walls. The DSMC simulations employed an adaptive mesh (see Figure 4b) divided into two types. A Cartesian uniform mesh is referred to as a \"basic\" mesh. Each \"basic\" cell is dynamically divided into subcells, a well known approach named Transient Adaptive Sub-cell (TAS) [29]. We used the following criteria:  
\[level=\left\lfloor\sqrt{PPC^{(basic)}/PPC_{\min}}\right\rfloor, \tag{22}\]  
where \(level\) is the TAS subdivision factor and represents the number of subcells in each direction. Thus, a basic cell is divided into \(level\times level\) subcells in two dimensions. Here, \(PPC^{(basic)}\) specifies the number of particles within each basic cell and \(PPC_{min}\) is the minimum average number of particles per subcell. This approach maintains the particle population within each cell at an approximately constant level. The collision algorithm is performed independently within each subcell, considering only the particles therein. In accordance with the validity recommendations for the SBT collision scheme [28], we used \(PPC_{min}=3\)  
Figure 4: (**a**) Mach number field and streamlines; (**b**) DSMC basic mesh is Cartesian uniform mesh with TAS and mesh near cantilever is unstructured adaptive mesh. Additionally, an unstructured adaptive mesh was used in the region around the elastic micro-cantilever (depicted in blue in Figure 4b), generated via the Delaunay algorithm in Gmsh [30] through C++ API functions integrated into our parallel C++ DSMC code. In contrast, the elastic beam code can complete all calculations on a single core in minutes. Most of the time is consumed by DSMC, making the beam calculation time negligible in comparison. Therefore, code maintenance and easy model modifications are of great importance. A novel aspect of our approach is the use of a molecular model (DSMC) for rarefied gas, which captures nonequilibrium microfluidic phenomena. The coupled FSI approach, combining DSMC and an elastic micro-element, is described in detail in [4]. DSMC is always numerically stable regardless of time step [33]. DSMC accuracy is the most important part here. In our case, DSMC accuracy was ensured by using a sufficient number of particles, adaptive meshing, a sufficiently small time step, and following recommendations in [28], as was described in the previous section. Recording the impact of each particle on the beam surfaces leads to an accurate determination of the gas force on the beam. The beam was divided into 20 elements along its length. The forces from particles hitting each beam element from the front and the back sides are accumulated. In DSMC, we used one trajectory and time-averaging of 20 time steps (\(\Delta t^{DSMC}\)) to obtain macroscopic properties, which is the time lag between the two models (\(20\times\Delta t^{DSMC}\)).