Silvia Preda

Silvia Preda, Matteo Semplice

We propose a level-set-based semi-Lagrangian method on graded adaptive Cartesian grids to address the problem of surface reconstruction from point clouds. The goal is to obtain an implicit, high-quality representation of real shapes that can subsequently serve as computational domain for partial differential equation models. The mathematical formulation is variational, incorporating a curvature constraint that minimizes the surface area while being weighted by the distance of the reconstructed surface from the input point cloud. Within the level set framework, this problem is reformulated as an advection-diffusion equation, which we solve using a semi-Lagrangian scheme coupled with a local high-order interpolator. Building on the features of the level set and semi-Lagrangian method, we use quadtree and octree data structures to represent the grid and generate a mesh with the finest resolution near the zero level set, i.e., the reconstructed surface interface. The complete surface reconstruction workflow is described, including localization and reinitialization techniques, as well as strategies to handle complex and evolving topologies. A broad set of numerical tests in two and three dimensions is presented to assess the effectiveness of the method.

Silvia Preda, Gabriella Bretti, Francesco Freddi et al.

We present a complete workflow for predicting stone degradation phenomena, such as marble sulfation, in works of art. The main challenge is to accurately acquire the geometry of the artwork and then use it to perform simulations based on a mathematical model of the degradation process, typically formulated as a system of partial differential equations (PDEs). To address this, we generate a point cloud of the object surface using photogrammetric techniques and subsequently post-process it to obtain a level-set description of the three-dimensional geometry. This representation is then incorporated into the numerical discretization of the PDE system. Combined with suitable time-stepping and preconditioning strategies, the resulting framework enables the prediction of degradation evolution, such as the growth of gypsum crust thickness on marble, under different scenarios.