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2. Because the two parts have respective flow fields with respect to the nodes at point A and point C (as shown in Fig. 2b) at the same section, it is possible to have either both filled or one filled (and one empty). These two cases should be handled separately, averaging the operation for the former, whereas assigning operation for the latter.
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.