## 1 Introduction  
Additive manufacturing (AM) based on vat photopolymerization is one of the most promising 3D printing technologies for polymer-based products fabrication [1, 2, 3]. Among the various vat photopolymerization AM technologies, constrained-surface stereolithography (SL) is the most widely used one. It allows mass production in a wide size range (from millimeters to meters) and product commercialization. Compared to other polymer 3D printing technologies, such as fused deposition modeling (FDM), constrained surface SL has many advantages, such as high speed [4] and surface finish [5, 6, 7], multi-material fabrication capability [8, 9], and high accuracy [6, 10]. It has been successfully implemented in multiple industrial applications, varying from medicine and education to aerospace and consumer goods production. Similar to other 3D printing technologies, constrained-surface SL starts from a three-dimensional (3D) computer-aided-design (CAD) model, which is then sliced into a set of two-dimensional (2D) layers. As its name implies, the constrained (liquid) surface is confined between the platform or the part being fabricated and the bottom (or top) surface of the liquid vat. The ultraviolet (UV) light beam can easily penetrate this transparent surface, focusing on the emerging part, and then curing the layer of the liquid polymer constrained in the region (Figure 0(a)). To separate the cured layer from the constrained surface and reconfine a new layer of a liquid resin for curing, the platform is elevated with a speed \(V\). The process repeats until a 3D object is built. Despite rapid development of constrained-surface SL, successful separation of the newly cured layer from the constrained surface is still the major challenge, limiting its applications. The extreme values of the separation force were primarily responsible for manufacturing failures [3, 4, 11, 12, 13, 14]. Many researchers addressed this problem experimentally, to identify the key factors that affect the separation [4, 13, 14]. It turned out to be a complicated process, influenced by multiple parameters and geometry factors. Hence, modeling and simulation could be beneficial to tackle this problem resulting in lower labor costs and fabrication time. [PERSON] et al. [11] used a mechanical approach to develop a cohesive zone model (CZM) for the separation force. The authors employed bilinear cohesive law and considered the material loaded with pure normal tensile stress only. They also proposed an optimization scheme for constitutive cohesive stiffness parameters estimation. [PERSON] et al. [15] proposed another CZM-based approach and demonstrated the efficiency of combining bilinear (for resonant values of force) and non-linear (for force values in the rest of the time interval) traction separation laws in predicting separation force-time profiles. Another approach, namely tilting separation, has been studied by [PERSON] et al. in [12, 16]. The authors developed a CZM-based approach for analytical prediction of the tilting separation force and verified it experimentally. This study revealed the critical importance of the cohesive curve in understanding the separation process under different conditions. Moreover, they modeled the key parameters of separation: cohesive stiffness, fracture energy, and maximal traction stress. It provided insights into the crucial differences between tilting and pulling-up separation. The vibration-assisted approach was proposed by [PERSON] et al. [17, 18] and also resulted in the separation force reduction. Most recently, [PERSON] and co-workers developed a rotation-assisted model for constrained-surface SL and demonstrated its potential [19]. Several noticeable studies also considered the development of composites with improved performance [20, 21, 22, 23, 24]. However, approaches rather than mechanical to model the separation force remain mostly undeveloped.