Smooth digital terrain modeling in irregular domains using finite element thin plate splines and adaptive refinement

Lishan Fang; Lishan Fang

doi:10.3934/math.20241450

AIMS Mathematics

2024, Volume 9, Issue 11: 30015-30042. doi: 10.3934/math.20241450

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Research article

Smooth digital terrain modeling in irregular domains using finite element thin plate splines and adaptive refinement

Lishan Fang ^,

School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China

Received: 18 August 2024 Revised: 07 October 2024 Accepted: 09 October 2024 Published: 22 October 2024
MSC : 65D10, 65N30, 68U05

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 investigated 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 were approximated using the TPS with a subset of sampled points. Numerical experiments were conducted on aerial, terrestrial, and bathymetric surveys. It was shown that the TPSFEM works well in square and irregular domains for modeling 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 cost.
- smoothing,
- thin plate spline,
- mixed finite element,
- adaptive mesh refinement,
- digital terrain model
Citation: Lishan Fang. Smooth digital terrain modeling in irregular domains using finite element thin plate splines and adaptive refinement[J]. AIMS Mathematics, 2024, 9(11): 30015-30042. doi: 10.3934/math.20241450

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Abstract

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 investigated 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 were approximated using the TPS with a subset of sampled points. Numerical experiments were conducted on aerial, terrestrial, and bathymetric surveys. It was shown that the TPSFEM works well in square and irregular domains for modeling 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 cost.

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