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3. It follows that a small difference between the melt-fronts is permissible. That allowance can be implemented by time allowance control or preferable location allowance control of the melt-front nodes.
4. The boundaries of the flow field expand by each melt-front advancement, so it is necessary to check the condition Eq. 10 after each change in the melt-front.
5. In view of above-mentioned analysis, the physical parameters at the nodes of the same section should be compared and adjusted, so the information describing finite elements of the same section should be prepared before simulation, that is, the matching operation among the elements should be preformed.
Fig. 2a,b. Illustrative of boundary conditions in the gap-wise direction a of the middle-plane model b of the surface model
2.2.2 Numerical implementation
Pressure field. In modeling viscosity η, which is a function of shear rate, temperature and pressure of melt, the shear-thinning behavior can be well represented by a cross-type model such as:
where n corresponds to the power-law index, and τ∗ characterizes the shear stress level of the transition region between the Newtonian and power-law asymptotic limits. In terms of an
Arrhenius-type temperature sensitivity and exponential pressure dependence, η0(T, P) can be represented with reasonable accuracy as follows:
Equations 11 and 12 constitute a five-constant (n, τ∗, B, Tb, β) representation for viscosity. The shear rate for viscosity calculation is obtained by:
Based on the above, we can infer the following filling pressure equation from the governing Eqs. 1–4:
where S is calculated by S = b0/(b−z)2 η dz. Applying the Galerkin method, the pressure finite-element equation is deduced as:
where l_ traverses all elements, including node N, and where I and j represent the local node number in element l_ corresponding to the node number N and N_ in the whole, respectively. The D(l_) ij is calculated as follows:
where A(l_) represents triangular finite elements, and L(l_) i is the pressure trial function in finite elements.
Temperature field. To determine the temperature profile across the gap, each triangular finite element at the surface is further pided into NZ layers for the finite-difference grid.
The left item of the energy equation (Eq. 4) can be expressed as:
where TN, j,t represents the temperature of the j layer of node N at time t. The heat conduction item is calculated by:
where l traverses all elements, including node N, and i and j represent the local node number in element l corresponding to the node number N and N_ in the whole, respectively.
The heat convection item is calculated by:
For viscous heat, it follows that:
Substituting Eqs. 17–20 into the energy equation (Eq. 4), the temperature equation becomes:
2.3 Structural analysis of the mold
The purpose of structural analysis is to predict the deformation occurring in the photopolymer mold due to the thermal and mechanical loads of the filling process. This model is based on a three-dimensional thermoelastic boundary element method (BEM). The BEM is ideally suited for this application because only the deformation of the mold surfaces is of interest. Moreover, the BEM has an advantage over other techniques in that computing effort is not wasted on calculating deformation within the mold.
The stresses resulting from the process loads are well within the elastic range of the mold material. Therefore, the mold deformation model is based on a thermoelastic formulation. The thermal and mechanical properties of the mold are assumed to be isotropic and temperature independent.
Although the process is cyclic, time-averaged values of temperature and heat flux are used for calculating the mold deformation. Typically, transient temperature variations within a mold have been restricted to regions local to the cavity surface and the nozzle tip [8]. The transients decay sharply with distance from the cavity surface and generally little variation is observed beyond distances as small as 2.5 mm. This suggests that the contribution from the transients to the deformation at the mold block interface is small, and therefore it is reasonable to neglect the transient effects. The steady state temperature field satisfies Laplace’s equation 2T = 0 and the time-averaged boundary conditions. The boundary conditions on the mold surfaces are described in detail by Tang et al. [9]. As for the mechanical boundary conditions, the cavity surface is subjected to the melt pressure, the surfaces of the mold connected to the worktable are fixed in space, and other external surfaces are assumed to be stress free.