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Presented results validate the simulation of the blow and blow forming process using a three-dimensional model. This shows the capabilities of the simulation tool in the development of glass bottles with complex geometries, such as those used in the perfume industry. At the same time, the transition from 2 Da to 3D models brought with it new difficulties and opportunities.
Axisymmetric models are usually written in terms of cylindrical coordinates, making use of simplifications in the third coordinate (θ). This approximation greatly reduces the associated computational costs. However, it also oversimplifies the application, being limited to only modeling glass bottles with circular cross-sections or axisymmetric geometries. In case of willing to model any perfume geometry, Cartesian coordinates in a three-dimensional space are required. Therefore, the mathematical model must be rewritten (governing equations, boundary conditions, free surfaces, contact detection equations, etc.). In addition, three-dimensional simulations are typically characterized by excessive computational times, thus requiring highly computationally efficient solutions as the element count drastically increases.
Hence, the fact that a three-dimensional model needs to be developed, implemented, and validated may have been a real limitation by many authors. In the present study, as the model is based on commercial software such as ANSYS Polyflow, axisymmetric and three-dimensional formulations are already implemented. Furthermore, since in previous studies [13, 19] the three-dimensional model was initially validated, many of the numerical difficulties concerning the transition from 2 Da to 3D models are already resolved. On the other hand, a three-dimensional model is much more tedious to work with. Planes of symmetry are generally used whenever possible to reduce the computational cost of the simulations. With the axisymmetrical model only one section of the domain was simulated. However, three-dimensional simulations require at least ¼ of the domain. Furthermore, material properties should be characterized for the right finite element type such as damping  crating experiments for validation.
The fact that axisymmetric simplifications can no longer be used has brought with it new experimental difficulties that must be successfully addressed. As the size and complexity of the glass domain increases, the preparation of the three-dimensional model becomes much more detailed and time-consuming, as additional experimental data under industrial manufacturing conditions is required. This is a critical task because simulations require many input values sensitive to glass conditions to obtain an accurate prediction (i.e., process temperatures and heat transfer phenomena at the boundary conditions). For example, Luo et al.  included in their simulation anisotropic properties for polymer bottles but using axisymmetric geometry with software Abaqus. At the same time, a larger amount of experimental data is also required to validate the results predicted by the three-dimensional model. This implies extended experimental infrared thermal measurements of glass at different stages of the blow and blow forming process. Finally, more section profiles obtained from the manufactured glass bottles are also required to compare the predicted glass thickness distributions in different horizontal planes.
On top of that, a previously accepted assumption needs to be discussed again. This affects something as important as the initial conditions of the glass domain. As stated above, the gob forming, loading, and settle blow stages were not modeled in the blow and blow container forming simulations. Thus, at the initial simulation step, the glass gob is already loaded inside the blank mold (Fig. 4A). Initial conditions of the glass domain, geometry, and temperature were experimentally defined during the delivery and once the gob is loaded. This simplification was successfully used in previous simulations of axisymmetric bottles.
Two factors influence the amount of error introduced with the definition of the initial geometry of the glass domain. First, the shape of the funnel and blank mold sections. In the present non-axisymmetrical case, the section of the extruded glass gob is cylindrical, but the sections of the funnel and blank mold cavity are rounded squares. Therefore, when the gob is loaded into the blank mold cavity, gob sections lose its axial symmetry due to low glass viscosity. Second, the length ratio of the extruded to the loaded gob. In this case, due to the shape of the blank mold cavity, the gob does not suffer a significant variation in length during loading, maintaining a similar aspect ratio in both geometries. Therefore, the amount of error introduced into the glass gob geometry with this simplification is considered correct.
On the other hand, the initial thermal conditioning of the glass domain is also influenced by its initial geometry. Before the counter blow, a significant heat transfer takes place at the loaded gob areas in contact with the walls of the blank cavity (Fig. 4B). Again, extruded glass gobs have circular sections, but the funnel and blank mold used to produce PAPA 100 bottles have rounded square sections. Due to the low viscosity of the glass, the sections of the loaded gobs tend to adapt to the geometry of the cavity. However, it may be the case that, in some areas, not the entire glass section is in contact with the blank cavity walls, conditioning the initial glass-mold contact and its heat transfer.
For all these reasons, in addition to validation of the predicted numerical results, extensive experimentations of the glass gob were conducted in the past at the delivery and loading stages [15, 19]. The obtained results allow the author to be confident in the presented simplifications. However, since the loading stage can cause big deformations to the glass gob geometry, it becomes clear that as gob loading and blank cavity three-dimensional geometries become more complex, the definition of the loaded glass gob also does. To perform future simulations with more complex blank geometries, it may be convenient to simulate the gob loading stage.
The design of the blank mold cavity of a square-based prism bottle is approached as the design of two axisymmetric bottles. The two most relevant sections of the blank mold cavity are defined by the apothem and the diagonal of the square. The square’s apothem would be approximated as the radius of the circle inscribed within the square, and the square’s diagonal would resemble the diameter of the circumscribed circle. Therefore, defining the blank cavity profiles for these two axisymmetric bottles, whose sections are the inscribed and circumscribed circles of the square, is the most common approach to designing blank mold cavities for square-based prism perfume bottles. Muijsenberg  in 2018 described how industrial revolution 4.0 will impact the glass industry. With the validation of 3D simulation approach and capabilities of machine learning, this iteration of designs should be optimized. This means that we expect to predict in the future how to design a blank mold that delivers the desired wall thickness of bottle without doing iterations shown in Fig. 1A, B, and C.
As stated, square-based prism bottles are generally well understood by glassmakers. However, there are increasing levels of complexity within the wide variety of non-axisymmetrical bottle geometries. At the same time, the geometry of the glass bottle has a strong impact on the development of its mold equipment as it can complicate a proper glass thickness distribution. Therefore, as the complexity of the non-axisymmetrical glass perfume bottles increase, the use of simulation tools gains more prominence to minimize trial and error iterations and reduce the number of production tests required for the development of new containers.
Avoiding the axisymmetric simplifications and using at least ¼ of the domain instead of just one section also has important benefits for the three-dimensional model. This fact expands the capabilities of the model as far more complex and interesting geometries can be studied with the three-dimensional simulations of the blow and blow forming process. In this sense, the study of horizontal sections in the glass domain allows obtaining a better understanding of how glass geometry evolves throughout the blowing stages. That is, in the case of the PAPA 100 bottles, to determine how the blank mold geometry influences the glass transition from a circular to a squared section, and how this relates to the achievement of a final glass perfume bottle with constant glass thickness distributions.
Figure 7 shows various horizontal section profiles of the glass domain at representative time steps (columns) and at determined heights (rows) of the numerical simulation. That figure helps to track the evolution of the glass geometry throughout the blowing stages of the forming process. Cross-section profiles at the same planes of the glass perfume bottle are also presented. Thickness distributions predicted by the three-dimensional model compared to the manufactured bottle show a very close result, not only in dimensions, but especially in the definition of the differentiated shape of the glass geometry in each of the three horizontal sections.Fig. 7
Horizontal section profiles of the glass domain at representative time steps (columns) and determined heights (rows) obtained from the numerical simulations with PAPA 100 bottle. Cross-section profiles of manufactured PAPA 100 bottles are presented on the right
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These results strongly highlight how useful this numerical tool is to understand how glass behaves in the blowing stages and to clarify the influence of the blank mold cavity on the distribution of the glass thicknesses of the manufactured container. From this simple example, it can be concluded that already known facts, previously based on empirical knowledge, such as the approach followed in the design of the blank mold cavity profiles for a square-based prism bottle, can be revised by analyzing the results predicted by the three-dimensional model.
Due to space limitations, only sections of the upper, middle, and lower body at the start and end of the blank and blow side simulations are presented. At the same time, a detailed explanation of blank mold design methodologies is out of the scope and only a simple idea is presented: glass reaching the corners (circumscribed circle) must expand farther than glass reaching the square’s apothem (inscribed circle). As glass expands, it loses thickness. Therefore, to achieve constant glass thickness distributions, non-constant thicknesses in the parison are required. The expansion is obviously influenced by glass temperatures, but this principle is still valid when designing the blank mold cavities. These profiles allow understanding the glass expansion as a function of the blank mold cavity design of complex non-axisymmetrical glass perfume bottles.
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