Proposta de metodologia para aplicação generalizada do método das soluções fundamentais

Abstract

Numerical modeling of physical phenomena governed by the Laplace equation is a central topic in several engineering fields, including hydrogeological applications, in which limited data availability and geometric complexity of the domains impose additional challenges. Traditional mesh-based methods, such as Finite Difference and Finite Element methods, are highly robust but relieve an extensive set of physical parameters and mesh-related choices, which may restrict their applicability in exploratory studies and preliminary diagnostics. In this context, this thesis proposed and evaluated a generalized methodology for applying the Method of Fundamental Solutions (MFS) to two-dimensional problems governed by the Laplace operator, with subsequent application to groundwater flow simulation. The methodology is based on the adoption of a circular fictitious boundary with uniformly distributed points and on the systematic sweeping of its radius (dvalue), combined with objective criteria of stability, convergence, and accuracy. Numerical performance was assessed in an integrated manner using the root mean square error (RMSE), the inverse of the condition number of the main matrix (rcond), and pointwise evaluations inside the domain, complemented by a successiveerror convergence test. This approach transformed the selection of MFS operational parameters into a reproducible procedure, avoiding empirical decisions and numerical instabilities. The methodology was first applied to theoretical examples with known analytical solutions, involving rectangular and circular geometries, enabling validation under controlled conditions. The results evidenced the existence of well-defined operational ranges of dvalue, which depend on the number of boundary points (N) and the domain geometry. For the unit circular boundary, approximate practical limits of dvalue were identified as 1211.5 for N = 8, 25.1 for N = 16, and 3.58 for N = 32, while applications with N = 64 proved infeasible without additional numerical treatment due to severe degradation of matrix conditioning. Although increasing N improved solution accuracy, it significantly narrowed the operational window of the method, particularly for circular domains. Subsequently, the methodology was applied to a groundwater flow case study in an aquifer using observed piezometric data. In this application, the procedure proved capable of diagnosing operational limits of the MFS and guiding the interpretation of discrepancies between computed and observed values, suggesting the influence of uncertainties associated with boundary conditions and/or the presence of heterogeneities and anisotropies not explicitly represented. Comparison with traditional numerical procedures indicated that, under assumptions of homogeneous domains and steady-state conditions, the MFS produced numerically consistent and statistically comparable results, with reduced dependence on additional hydrogeological data. It was concluded that the proposed methodology enhances the reproducibility and applicability of the Method of Fundamental Solutions, positioning it as a robust tool for screening, diagnostics, and modeling support in generic Laplacian problems and in groundwater flow simulations under limited data availability. By establishing objective and quantifiable operational ranges for the method parameters, this thesis contributes to filling a gap identified in the literature and reinforces the potential of the MFS as an alternative or complementary approach to traditional numerical models in specific contexts.