Numerical Integration in Meshless Galerkin Methods

In meshless Galerkin methods, numerical integration serves as a crucial component primarily used for computing integral terms in weak-form equations. Unlike traditional finite element methods that rely on element meshing, integration must be performed either on background grids or directly over nodal distributions due to the absence of predefined mesh structures. Numerical integration typically employs multi-point Gaussian quadrature schemes, whose core principle involves transforming continuous integrals into discrete summations by selecting appropriate integration points and corresponding weight coefficients. Gaussian quadrature exhibits high algebraic precision characteristics and generally achieves excellent approximation results for smooth integrand functions. In code implementation, this involves creating integration point arrays with associated weights, often implemented through precomputed Gauss-Legendre quadrature tables stored in lookup arrays. In practical implementations, users can flexibly choose the number of integration points, which directly influences both computational accuracy and efficiency. Increasing integration points enhances precision but simultaneously increases computational burden. For shape functions in meshless methods, integration transformations are typically performed in local coordinate systems due to potential variations in support domain sizes. Programmatically, this requires Jacobian matrix computations for coordinate transformations and careful handling of integration domain boundaries. The method of weighted residuals forms the theoretical foundation of meshless Galerkin methods, while numerical integration acts as the essential tool for realizing this theory. Through appropriate selection of integration schemes, stability and convergence of the discrete system can be ensured. Algorithm implementation often involves validation checks for integration accuracy and adaptive quadrature techniques for handling irregular domains.