Geometric deviation modeling for single surface tolerancing using Laplace-Beltrami Operator

ISO standards on geometrical product specifications and verification (GPS) categorize the geometrical features partitioned from shapes based on their kinematic invariance classes. As a key issue to achieve predictable and interchangeable shape, geometrical tolerancing defines the tolerance zone within which the partitioned features shall be contained for dimensional and geometrical accuracy. However, most of the current research activities consider the geometrical features as orientation and position deviations of substitute ideal surfaces and neglect form errors within the developed models. In order to enhance the tolerance analysis and simulation, the concept of Skin Model Shapes (SMS) is therefore proposed to represent surface deviations in accordance with random nature of geometric deviations. In this paper, a new approach based on spectral analysis is proposed for SMS generation on mechanical parts. The Eigen-system of Laplace-Beltrami Operator (LBO) enables shape analysis with an unambiguous mathematical language due to its invariance of the position and orientation. The SMS generation process is completed with LBO by developing a modeling methodology for geometric deviation representation and generation on single surfaces in the design process. The obtained spatial frequency LBO decomposition on the single surface of invariant class is used to represent the statistical characteristics of the deviations and further used to reproduce deviations with similar distributions. A comparison is presented in the paper to illustrate the difference between a conventional Discrete Cosine Transform (DCT) based method and the proposed LBO-based method. Another case study is investigated to explain the implementation of the proposed method on different types of single surfaces of invariant class generation based on SMS.