چكيده لاتين
The analysis and monitoring of ground deformations play a fundamental role in geotechnical hazard assessment, infrastructure monitoring, and land resource management. Accurate analysis of these phenomena requires precise modeling of ground behavior, which is often performed by solving partial differential equations. One such equation commonly used to investigate ground behavior is the Antiplane strain equation, which is applicable in homogeneous environments without body forces to study surface elevation changes. Given the complexity of most governing physical equations of the Earth, analytical solutions are often difficult or impossible to obtain; thus, numerical methods have become essential tools for modeling ground deformation. Numerical approaches such as the FEM, a mesh-based method, and meshfree methods like MLS and RBF have been widely used to solve partial differential equations. In this research, the RMLS method—a novel extension of the MLS method—is applied for the first time to solve partial differential equations. By gradually expanding the support domain and implementing boundary conditions in stages, the RMLS method enables both local and global modeling capabilities. One of the key advantages of RMLS in solving partial differential equations is its ability to use arbitrary-shaped support domains at each step and it is not limited to a specific shape. Furthermore, the method can solve the problem at each stage using only new data, without the need for previous-step information. To evaluate the numerical accuracy of these four methods, three case studies are examined: two synthetic scenarios with known analytical solutions and one real case study based on time-series data from Sentinel-1A radar imagery, representing cumulative ground displacement in the Porterville region of California from 2015 to 2024. For the Antiplane strain equation under the first type of boundary conditions with synthetic data, the RMLS method showed a 94% improvement over FEM and about 7% over MLS and RBF. In the second type of boundary conditions, RMLS improved accuracy by approximately 15% over FEM and MLS, and by 24% over RBF. Moreover, the RMLS method yielded accurate and reliable results for problems with localized support domains. To assess performance under different noise levels, various types of noise were added to the synthetic data, and RMLS still outperformed the other methods. In addition to its accuracy, the computational complexity of each method was analyzed. Results showed that RMLS reduced the computational time by 15%, 37.5%, and 19% compared to FEM, RBF, and MLS, respectively.
