Possibilities of node location optimization and the quest isoparametric versus isogeometric in the collocation boundary element method

Resumo

We have recently laid down the theoretical basis for the consistent formulation of the collocation boundary element method, as it should have been conceived from the beginning. We proved a convergence theorem for two- and three-dimensional problems of elasticity and potential, which applies to arbitrarily curved elements in the frame of an isoparametric analysis. We also showed that arbitrarily high precision and accuracy may be achieved limited only by the machines capacity to represent numbers. On the other hand, there still is the cost-benefit question considering that the physical phenomenon is mathematically adequately idealized of how to improve a real problems simulation without refining too much a discretization mesh. The first possibility of doing this is optimizing the geometric location of the primary parameters (as for displacements and tractions, in elasticity) for the problem's mechanical description. This primarily consists in locally refining a mesh. We may also attach the problems parameters to optimal locations inside the boundary element. A second issue is that an isoparametric formulation (generally in terms of polynomial interpolations along the boundary segment) may fail to reproduce the exact geometry of the idealized physical problem as the isogeometric approach does. Since, for two-dimensional problems, we have the boundary element formulation under control regarding all numerical evaluations, we assess how an isoparametric analysis with the introduced elegancy of a convergence theorem compares to a formulation that preserves the problems idealized geometry, but to which the theorem no longer applies. We present the conceptual formulation, code implementation, and numerical illustrations.