Desenvolvimento de um framework para métodos sem malha

Abstract

Mesliless methods are an alternative to traditional numerical methods such as the Finite Element Method (FEM) and the Finite Difference Method (FDM). The main feature of meshless methods is the absence of a mesh to construct their shape functions, which are used to generate approximations for the solution. Thus, the domain is represented by a set of scattered nodes instead of a mesh or a grid, as in FEM and FDM. This work presents a framework for solving electromagnetic problems using meshless methods, called MFree Framework. The framework is developed in C++ programming language under the generic programming paradigm using the template mechanism. Generic programming aims to represent algorithms and data structures in the most generic form as possible, with the abstractions described by concepts. ln MFree Framework, the data structures are parameterized, allowing methods, shape functions, integration procedures, formulations and other components to be combined in various ways to solve a particular problem. As data types are implemented to meet one or a set of concepts, new types can be developed and used with the previously available data structures in a direct manner, which makes the framework very extensible. ln the current version, the framework instantiates applications able to solve problems using the Element»Free Galer/ctn Method (EFG), the Meshless Local Petroo»Galerktn Method (MLPG) and Point Interpolation Method (PIM) and its variations. However, other methods can be easily implemented due to the generic characteristic with which it is built. To test the framework instantiations, electrostatic and magnetostatic problems are solved and the MFree Framework shows to be a tool with great flexibility, utility and importance for future developments in the field of meshless methods.