Photopolymerization is a technique for producing parts in additive manufacturing. It involves solidifying a polymer material layer by layer into a predetermined shape using a light source controlled by G-code instructions. Finite Element Method (FEM) is commonly used to simulate the process, but it can be slow due to the time-stepping algorithm used to calculate each time increment. In this study, we propose using the Laplace Transform Finite Element Method (LTFEM) with a clever numerical Laplace inversion algorithm to accelerate photopolymerization simulations. By transforming the equations from time to frequency domain, we can compute solutions in parallel, in a single step, and for any desired printing time without using the time-stepping algorithm. We first discretize the parabolic heat equation and the mechanical balance equation in space by FEM to model the thermomechanical system driven by a heat source following a G-code path. Then, we split the G-code path into instructions, each with its own heat source that turns on or off to mimic the original path. After applying the Laplace transform to the model and the heat sources, we use the superposition principle to obtain the thermal history for the whole path in a single step. We then model the behavior of the photopolymer by using a fractional calculus reaction kinetics model in Laplace domain. Finally, we incorporate chemical shrinkage and thermal expansion deformation mechanisms into the mechanical balance to determine the residual stresses in the polymer. Afterwards, we provide examples and compare our method to conventional time-stepping FEM to further demonstrate and assess its performance. Our results show that compared to classical FEM, the LTFEM is faster by over 5 times for long time steps and over 100 times for small ones. Our method can provide accurate predictions in a fraction of the time required by classical FEM, making it a promising tool for real-time simulation and control of additive manufacturing.