Numerical analysis of a FE/SAV scheme for a Caginalp phase field model with mechanical effects in stereolithography

TL;DR

The paper develops a multiphysics phase-field model for stereolithography based on a $\mathcal{Caginalp}$-type system that couples phase evolution with heat conduction and quasi-static elasticity. It introduces a fully discrete numerical scheme using finite elements in space and a scalar auxiliary variable (SAV) approach in time, proving unconditional stability and convergence of discrete solutions to a weak solution. It also establishes regularity and optimal error estimates for the phase-field submodel, along with detailed inductive arguments to obtain global error bounds and rates $\mathcal{O}(h^2+\tau)$ for $\varphi$ and $\theta$ in $l^\infty(L^2)$, and $\mathcal{O}(h+\tau)$ for gradients. Numerical simulations in 2D demonstrate gel–sol transitions, heat diffusion due to laser irradiation, and mechanical deformations consistent with temperature and phase changes, including moving heat-source scenarios that form characteristic shapes. Overall, the work provides a rigorous mathematical foundation and efficient computational framework for simulating stereolithography with mechanical effects using a SAV-based FE discretization.

Abstract

In this work we propose a phase field model based on a Caginalp system with mechanical effects to study the underlying physical and chemical processes behind stereolithography, which is an additive manufacturing (3D printing) technique that builds objects in a layer-by-layer fashion by using an ultraviolet laser to solidify liquid polymer resins. Existence of weak solutions is established by demonstrating the convergence of a numerical scheme based on a first order scalar auxiliary variable temporal discretization and a finite element spatial discretization. We further establish uniqueness and regularity of solutions, as well as optimal error estimates for the Caginalp system that are supported by numerical simulations. We also present some qualitative two-dimensional simulations of the stereolithography processes captured by the model.

Figures (10)

• Figure 1: Section view of the printing process involved in stereolithography. An object is printed layer by layer on an adjustable platform submerged in a vat of liquid resin. The resin hardens when struck by an ultraviolet laser positioned outside the vat, and the platform lowers in order to harden the next layer of resin directly on top of the previous one.
• Figure 2: Errors of the phase field $\varphi$: (a) the x-axis corresponds to the mesh size $h$; (b) the x-axis corresponds to the time step size $\tau$.
• Figure 3: Errors of the temperature field $\theta$: (a) the x-axis corresponds to the mesh size $h$; (b) the x-axis corresponds to the time step size $\tau$.
• Figure 4: Errors of the displacement field $\bm{u}$: (a) the x-axis corresponds to the mesh size $h$; (b) the x-axis corresponds to the time step size $\tau$.
• Figure 5: In the fixed source heat source simulation, the four subplots display the phase field $\varphi$ at $t=0.01$, $0.05$, $0.10$, and $0.20$.

Theorems & Definitions (20)

• Remark 2.1
• Remark 3.1
• Proposition 3.1
• proof
• Remark 3.2
• Lemma 3.1
• proof
• Proposition 3.2: Compactness
• proof
• Theorem 3.1: Convergence