SIM-AM 2023

Fast Photopolymerization Simulation using the Laplace Transform Finite Element Method

  • Sanchez Martinez, Nelson Martin (Eindhoven University of Technology)
  • van der Sluis, Olaf (Eindhoven University of Technology)
  • Remmers, Joris (Eindhoven University of Technology)

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Photopolymerization is a popular technique for producing parts in additive manufacturing. It involves solidifying a polymer material layer by layer using a light source controlled by G-code instructions to achieve the desired shape. Finite Element Method (FEM) is commonly used to simulate the process [1], 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 an efficient numerical Laplace inversion algorithm to accelerate the simulation of photopolymerization [2]. 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. Once the thermal field is known, we predict the mechanical response of the photopolymer resin by estimating its degree of curing using a simple reaction kinetics model in Laplace domain. Finally, we incorporate the polymer's chemical shrinkage and thermal expansion deformation mechanisms into the mechanical balance to determine the residual stresses in the material. Our results show that compared to classical FEM, the LTFEM is faster by over 5 times for large time steps and over 50 times for small ones. We provide examples and compare our method to conventional time-stepping FEM to demonstrate and assess its performance. Our study is able to accelerate the simulation of photopolymerization using LTFEM and a numerical Laplace inversion algorithm. 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.