A Statistical Approach to Surface Metrology for 3D-Printed Stainless Steel

Surface metrology is the area of engineering concerned with the study of geometric variation in surfaces. This article explores the potential for modern techniques from spatial statistics to act as generative models for geometric variation in 3D-printed stainless steel. The complex macro-scale geometries of 3D-printed components pose a challenge that is not present in traditional surface metrology, as the training data and test data need not be defined on the same manifold. Strikingly, a covariance function defined in terms of geodesic distance on one manifold can fail to satisfy positive-definiteness and thus fail to be a valid covariance function in the context of a different manifold; this hinders the use of standard techniques that aim to learn a covariance function from a training dataset. On the other hand, the associated covariance differential operators are locally defined. This article proposes to perform inference for such differential operators, facilitating generalization from the manifold of a training dataset to the manifold of a test dataset. The approach is assessed in the context of model selection and explored in detail in the context of a finite element model for 3D-printed stainless steel.

表面形貌测量学（Surface metrology）是工程学领域中研究表面几何变异的分支方向。本文探讨了将空间统计学的现代技术应用于3D打印不锈钢表面几何变异生成模型的潜力。3D打印构件的复杂宏观尺度几何结构带来了传统表面形貌测量学中未曾面临的挑战：训练数据与测试数据无需定义于同一流形（manifold）之上。值得注意的是，在某一流形上基于测地线距离（geodesic distance）定义的协方差函数（covariance function），在另一流形的语境下可能无法满足正定性（positive-definiteness），因此无法成为合法的协方差函数；这一问题阻碍了旨在从训练数据集学习协方差函数的标准技术的应用。另一方面，与之相关的协方差微分算子（covariance differential operators）具备局部定义的特性。本文提出对这类微分算子开展推断研究，以实现从训练数据集所属流形到测试数据集所属流形的泛化。本文在模型选择的场景下对所提方法进行了评估，并针对3D打印不锈钢的有限元模型（finite element model）展开了详细的探讨。