采用有限元薄板样条与自适应细化的不规则域光滑数字地形建模
Smooth digital terrain modelling in irregular domain using finite element thin plate splines and adaptive refinement
AI总结：
针对大数据量下薄板样条地形建模计算成本高的问题，提出结合边界近似的有限元薄板样条与自适应细化方法，在多类测量数据上验证了其在精度和效率上的竞争力。
摘要：
数字地形模型（DTM）是利用地质调查中通过机载激光雷达、水深测深等多种采样技术采集的高程数据构建的。这通常会形成具有不同分布模式的大型数据集，可能需要在具有复杂边界的不规则域中进行光滑数据逼近。薄板样条（TPS）可对散乱数据进行插值并生成视觉效果良好的曲面，但对于大数据量而言计算成本过高。有限元薄板样条（TPSFEM）具有与TPS相似的光滑特性，且能高效插值大型数据集。本文研究了TPSFEM与自适应网格细化在不规则域中的性能。在具有任意形状边界的域中，边界条件对求解精度至关重要，本文采用TPS结合采样点子集来近似边界条件。研究在航空、陆地和水深测量数据上开展了数值实验。结果表明，TPSFEM在正方形域和不规则域中均能很好地用于地形表面建模，且自适应细化显著提升了其效率。通过将TPSFEM与TPS、紧支撑径向基函数进行对比，证明了其在精度和成本方面的竞争力。
英文摘要：
Digital terrain models (DTMs) are created using elevation data collected in geological surveys using varied sampling techniques like airborne lidar and depth soundings. This often leads to large data sets with different distribution patterns, which may require smooth data approximations in irregular domains with complex boundaries. The thin plate spline (TPS) interpolates scattered data and produces visually pleasing surfaces, but it is too computationally expensive for large data sizes. The finite element thin plate spline (TPSFEM) possesses smoothing properties similar to those of the TPS and interpolates large data sets efficiently. This article investigates the performance of the TPSFEM and adaptive mesh refinement in irregular domains. Boundary conditions are critical for the accuracy of the solution in domains with arbitrary-shaped boundaries and are approximated using the TPS with a subset of sampled points. Numerical experiments are conducted on aerial, terrestrial and bathymetric surveys. It is shown that the TPSFEM works well in square and irregular domains for modelling terrain surfaces and adaptive refinement significantly improves its efficiency. A comparison of the TPSFEM, TPS and compactly supported radial basis functions indicates its competitiveness in terms of accuracy and costs.
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