Uma comparação entre elementos de contorno contínuos e descontínuos na solução de problemas de Laplace - Cruzeiro, Filipe Lopes (2024)
The Boundary Element Method (BEM) is one of the most powerful techniques for solving continuum mechanics problems and, along with the Finite Element Method (FEM) and the Finite Difference Method (FDM), has completely transformed engineering. The Boundary Element Method emerged as a powerful alternative among other numerical methods, particularly for problems requiring high precision, such as stress concentration problems and problems with infinite domains. The main characteristic of the Boundary Element Method is the integration only at the boundaries of the problem, meaning the elements are located solely on the boundary. In some cases, it is necessary to place source points within the domain, either to introduce degrees of freedom, as in problems related to membrane vibrations, or to obtain specific properties at that point, such as a heat source. However, in self-adjoint equations, such as the Laplace equation, it is not necessary to place points within the domain, with only the boundary being discretized. For this type of problem, the quality of the solution will depend on how the elements are constructed. In its simplest form, BEM has an element with only one source point at its center, and the properties of this point are adopted for the entire element. Another formulation involves continuous linear elements, which have two functional points positioned on geometric points, and these functional points are shared with neighboring elements. This formulation evolved to include high-order elements, with quadratic and cubic elements being quite common. These elements have some deficiencies, such as the need for special treatment at corners, as well as the difficulty of generating meshes for the subdomain technique. These problems led to the creation of a type of element called discontinuous. This type of element is characterized by discontinuity between element i and the adjacent elements, meaning there is no sharing of functional points between elements. The use of this element is a specific feature of BEM, and it cannot be applied in other classical methods. This work will analyze the efficiency of the discontinuous Boundary Element Method in solving the Laplace equation, evaluating the quality and the order of convergence of this formulation compared to the continuous formulation.