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Additive Manufacturing (AM) conventionally necessitates an intermediary slicing procedure using the standard tessellation language (STL) data, which can be computationally burdensome, especially for intricate microcellular architectures. This study aims to propose a direct slicing method tailored for digital light processing-type AM processes for the efficient generation of slicing data for microcellular structures.

The authors proposed a direct slicing method designed for microcellular structures, encompassing micro-lattice and triply periodic minimal surface (TPMS) structures. The sliced data of these structures were represented mathematically and then convert into 2D monochromatic images, bypassing the time-consuming slicing procedures required by 3D STL data. The efficiency of the proposed method was validated through data preparations for lattice-based nasopharyngeal swabs and TPMS-based ellipsoid components. Furthermore, its adaptability was highlighted by incorporating 2D images of additional features, eliminating the requirement for complex 3D Boolean operations.

The direct slicing method offered significant benefits upon implementation for microcellular structures. For lattice-based nasopharyngeal swabs, it reduced data size by a factor of 1/300 and data preparation time by a factor of 1/8. Similarly, for TPMS-based ellipsoid components, it reduced data size by a factor of 1/60 and preparation time by a factor of 1/16.

The direct slicing method allows for bypasses the computational burdens associated with traditional indirect slicing from 3D STL data, by directly translating complex cellular structures into 2D sliced images. This method not only reduces data volume and processing time significantly but also demonstrates the versatility of sliced data preparation by integrating supplementary features using 2D operations.

We present an approach for pricing American put options with a regime-switching volatility. Our method reveals that the option price can be expressed as the sum of two components: the price of a European put option and the premium associated with the early exercise privilege. Our analysis demonstrates that, under these conditions, the perpetual put option consistently commands a higher price during periods of high volatility compared to those of low volatility. Moreover, we establish that the optimal exercise boundary is lower in high-volatility regimes than in low-volatility regimes. Additionally, we develop an analytical framework to describe American puts with an Erlang-distributed random-time horizon, which allows us to propose a numerical technique for approximating the value of American puts with finite expiry. We also show that a combined approach involving randomization and Richardson extrapolation can be a robust numerical algorithm for estimating American put prices with finite expiry.