## 2 Materials and Methods
### Computational Model for Photopolymerization Process
#### 2.2.2 Modeling of Photopolymerization in COMSOL Multiphysics  
The photopolymerization model is solved by commercial finite element analysis (FEA) software, COMSOL Multiphysics. We created a 2D axisymmetric domain representing a cross-section of cylindrical volume of resin in the vat of the PuSL system, as shown in Figure 3a. Here \(z\) represents the depth direction along which UV light travels and r is the radial coordinate. The domain is selected in such a way that the light incident on the surface will allow enough area for resin components to diffuse, i.e., the equations for parameter concentrations will be able to converge within the refined mesh area of the domain. The computational domain is divided into two sub regions (higher and lower mesh densities) for computational efficiency. Adaptive time step is used for stability and consistency of the algorithm. Figure 3b represents the 2D axisymmetric domain and boundary conditions. As the partial differential equations (PDEs) of the photopolymerization process are strongly coupled, COMSOL Multiphysics solves the PDEs iteratively to obtain converged solutions in each time-step. Table 1 lists the initial and boundary conditions. Following assumptions are made while setting up the simulation. * Thermal properties are considered to be constant during polymerization reactions. This assumption is made based on the fact that when polymerization on micro scale is limited on a small area, surrounding resin acts a heat sink. * The rate constants \(k_{t}\) and \(k_{p}\) are kept constant for simplification of system. [PERSON] et al. [24] explained more detailed information regarding the change of \(k_{t}\) and \(k_{p}\) with respect to monomer conversion \(C\). * Optical effects such as refraction or reflection are not considered. Table 2 lists constants and their values. When computation is complete, it gives a continuous field of conversion ratio, defined as Equation (10). Interface between cured solid and remaining uncured \begin{table}
\begin{tabular}{c c c} \hline \hline
**Equation** & **Initial Condition** & **Boundary Condition** \\ \hline Light intensity \(I\) & - & \(I(\text{r},\ \text{z}=0)=I_{0}\times\frac{\text{r}}{\text{s}}\text{e}^{\frac{-2(t-z-z )\text{s}}{\text{s}^{2}}}\) \\ Photoinitiator [PI] & [PI] (t = 0, r, z) = [PI]\({}_{0}\) & - \\ Free radical [R] & [R] (t = 0, r, z) = 0 & [R] (r, z = 0) = 0 \\ Oxygen [O] & [O] (t = 0, r, z) = [O]\({}_{1}\) & [O] (r, z = 0) = [O]\({}_{0}\) \\ Monomer [M] & [M] (t = 0, r, z) = [M]\({}_{0}\) & - \\ \hline \hline \end{tabular}  
* r = radial direction, z = depth direction, and t = time.  
\end{table} 
Table 1: Initial and boundary conditions. Figure 3: (a) 2D axisymmetric computational domain representation of UV exposed resin region in the vat. 2D axisymmetric domain consists of two domains that have different mesh sizes due to computational efficiency. (b) Computational domain with boundary conditions. liquid resin is determined by a cut-off conversion ratio contour, from which curing depth, curing width, and surface profile can be extracted. Detailed method is given in Appendices C and D.  
Meta-Heuristic Optimization Technique for Multiple Objectives: Multi-Objective Particle Swarm Optimization (MOPSO)  
Improving surface roughness of additively manufactured parts while keeping printing time as small as possible is considered as finding sub-optimal solutions in a parameter search space that satisfies multiple objectives. Because the photopolymerization process involves many parameters and printing variables and a system of PDEs, this problem has highly nonlinear and multivariate response surface. In addition, it is often under various complex constraints. Traditional gradient-based optimization algorithms often fail to find an optimizer of this kind of problems or may be computationally too expensive to calculate derivatives.