Moving Least Squares (MLS) as the Foundation of Meshfree Approximation Methods

Moving Least Squares (MLS) serves as a fundamental technique widely applied in meshfree approximation methods. The core concept of MLS involves constructing approximation functions through locally weighted least squares fitting, thereby eliminating the dependency on structured grids required by traditional finite element methods. In implementation, this typically requires defining a weight function that decays with distance from the evaluation point and selecting appropriate polynomial basis functions for local approximations.

The MLS method is particularly suitable for high-dimensional problems since it doesn't require fixed grid structures, instead building approximation functions through local sampling points. This approach proves especially valuable for irregular geometries or dynamic boundary problems, such as fluid dynamics simulations and deformation analysis in solid mechanics. Algorithmically, MLS operates by solving a series of local least squares problems at each evaluation point, where the solution involves constructing and inverting a moment matrix composed of weighted basis functions.

The key to MLS algorithm lies in selecting appropriate weight functions and basis functions to maintain both local smoothness and global numerical stability. Common implementation choices include Gaussian or cubic spline weight functions with compact support, combined with linear or quadratic polynomial bases. While MLS excels in multiphysics coupling problems and large deformation analysis due to its meshfree nature, computational efficiency can be influenced by local node density and the support radius selection.

During implementation, MLS typically performs least squares fitting for sampling points surrounding each computational point to construct continuous field variable approximations. This approach provides greater flexibility than traditional mesh-based methods when handling high-dimensional problems, though it demands higher computational resources to maintain accuracy, particularly in matrix assembly and inversion operations at each evaluation point.