Digital terrain models of geological information occasionally require smooth data in domains with complex irregular boundaries due to its data distribution. Traditional thin plate splines produce visually pleasing surfaces, but they are too computationally expensive for data of large sizes. Finite element thin plate spline smoother (TPSFEM) is an alternative that uses first-order finite elements to efficiently interpolate and smooth large data sets. Previous studies focused on regular square domains, which are insufficient for real-world applications. This article builds on prior work and investigates the performance of the TPSFEM and adaptive mesh refinement for real-world data sets in irregular domains. The Dirichlet boundaries are approximated using the thin plate spline and data-dependent weights are applied to prevent over-refinement near boundaries. Three geological surveys (aerial, terrestrial and bathymetric) with distinct data distribution patterns were tested in the numerical experiments. We found that irregular domains with adaptive mesh refinement significantly improve the efficiency of the interpolation. While the inconsistency in approximated boundary conditions, we may prevent it using additional constraints like weights. This finding is also applicable to other finite element-based smoothers.

翻译：数字地形模型在处理地质信息时，常因其数据分布特性需要在具有复杂不规则边界的区域内获得平滑数据。传统薄板样条虽能生成视觉上令人满意的曲面，但处理大规模数据时计算成本过高。有限元薄板样条平滑器（TPSFEM）作为一种替代方案，采用一阶有限元对大数据集进行高效插值与平滑。此前研究多集中于规则矩形区域，难以满足实际应用需求。本文在前人工作基础上，探究了TPSFEM及自适应网格细化在不规则区域实际数据集上的性能。采用薄板样条对狄利克雷边界进行近似，并引入数据相关权重防止边界附近过度细化。数值实验测试了三种具有不同数据分布模式的地质勘测数据（航空、陆地及海底地形）。研究表明，采用自适应网格细化的不规则区域能显著提升插值效率。尽管近似边界条件存在不一致性，但可通过权重等附加约束加以规避。该结论同样适用于其他基于有限元的平滑器。