Author: Daniel Mejía Parra

Directors: Jorge Posada Velásquez (Vicomtech) Oscar E. Ruiz Salguero (University)

University: EAFIT

Date: 20.05.2020

This Doctoral Thesis develops novel articulations of Differential Operators on Manifolds for applications on Computer Aided Design, Manufacture and Computer Graphics, as follows: (1) Mesh Parameterization and Segmentation. Development and application of Laplace-Beltrami, Hessian, Geodesic and Curvature operators for topology and geometry – driven segmentations and parameterizations of 2-manifold triangular meshes. Applications in Reverse Engineering, Manufacturing and Medicine.
(2) Computing of Laser-driven Temperature Maps in thin plates. Spectral domain - based analytic solutions of the transient, non-homogeneous heat equation for simulation of temperature maps in multi-laser heated thin plates, modeled as 2-manifolds plus thickness.
(3) Real-time estimation of dimensional compliance of hot outof-forge workpieces. A Special Orthogonal SO(3) transformation between 2-manifolds is found, which enables a distance operator between 2-manifolds in R^3 (or m-manifolds in R^n). This process instruments the real-time assessment of dimensional compliance of hot workpieces, in the factory floor shop.
(4) Slicing or Level-Set computation for 2-manifold triangular meshes in Additive Manufacturing. Development of a classification of non-degenerate (i.e. non-singular Hessian) and degenerate (i.e. singular Hessian) critical points of non-Morse functions on 2-manifold objects, followed by computation of level sets for Additive Manufacturing.
Most of the aforementioned contributions have been screened and accepted by the international scientific community (and published). Non-published material corresponds to confidential developments which are commercially exploited by the sponsors and therefore banned from dissemination.


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