This research presents novel methodologies for reference form estimation in triangular-mesh-based freeform surfaces, which is the first step in surface characterisation for complex geometries. Freeform surfaces, widely employed in optics, biomedical engineering, and aerospace applications, exhibit complex underlying forms with a range of Gaussian curvatures that make traditional surface characterisation methods inadequate. The study proposes the application of mesh smoothing techniques as an alternative to conventional mathematical fitting approaches, which require prior knowledge of the surface form. Through systematic investigation of smoothing parameters including nesting index settings, distance metrics, and neighbour range selection, this research establishes optimised frameworks for mesh convolution and Laplacian smoothing. Key contributions include the development of novel weighting functions for the discrete Laplace-Beltrami operator, demonstrating superior smoothing performance and computational efficiency compared to traditional methods. The proposed angle weighting function and dual distance weighting function maintain the linearity and simplicity of Laplace smoothing while mitigating its inherent drawbacks, with significant computation time reduction particularly evident in meshes with high vertex and face counts. The research further introduces robust improvements to handle outliers through generalised Rk filters and curvature-based methods, alongside dimension preservation techniques to counteract the shrinkage effects common in traditional mesh smoothing. Feature preservation is enhanced through improved bilateral mesh smoothing with dynamic parameters, adaptive weighting schemes and new predictors. Validation through case studies on advanced manufacturing products, including an element from a gyroid structures, a blade from a turbine, and an artificial tooth. These examples demonstrate the practical applicability of the proposed methods. The results show substantial reduction in form deviation and improved preservation of geometric features, providing an effective framework for reference form estimation in freeform surface characterisation without requiring nominal models. This work bridges the gap between mesh smoothing techniques and precision surface metrology in high-precision manufacturing, delivering robust, efficient methods for characterising complex freeform surfaces on industrial components.
| Date of Award | 15 Dec 2025 |
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| Original language | English |
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| Supervisor | Jane Jiang (Main Supervisor) & Wenhan Zeng (Co-Supervisor) |
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