Potential problems with polygonal boundaries by a BEM with parametric linear functions

Triangular Bézier surface patches in modeling shape of boundary geometry for potential problems in 3D

“…Presented briefly, the modification of traditional BIE is considered as a generalization of the modification applied to 2D problems as in [10][11][12][13][14][15][16][17]. In general, it consists in analytically defining curvilinear boundary geometry in traditional BIE with the help of triangular Bézier surface patches.…”

“…To find a solution in the domain, we need to obtain an integral identity known for BIE that makes use of the solution on the boundary obtained by PIES. After using similar modifications as in the case of 2D problems [10,11], we have had an integral identity which used solutions (23) and (24) at the boundary, previously obtained by the PIES solution. The modified identity takes the following form: …”

“…So far, however, PIES has been mainly used to solve 2D potential boundary value problems modeled by partial differential equations such as: Laplace [10,11], Poisson [12], Helmholtz [13] and Navier-Lame [14]. These equations have been written in the alternative form, with the help of PIES, which takes into account in its mathematical formalism the shape of the boundary modeled by curves known from computer graphics.…”

Parametric integral equation system (PIES) for 2D elastoplastic analysis

“…The author of the paper and the research team to which she belongs develop a method whose main objective is to completely get rid of the classical discretization. The characteristic of the method and its application to the Laplace equation were first published by prof. E. Zieniuk in [9]. Then, the method was successfully used to solve two-dimensional elastic [10], elastic with body forces [11] or acoustic [12] problems.…”

Formulação do MEC considerando efeitos microestruturais e continuidade geométrica G1: tratamento de singularidade e análise de convergência

“…Neste caso, os autores afirmam que o contorno pode ser descrito por uma quantidade relativamente pequena de pontos de controle necessários para a criação da superfície de Bézier. Análogo ao problema 2D analisado em Zieniuk (2001), a proposta de não utilizar elementos para representar a forma e, sim, inserir a representação do contorno dentro do formalismo das PIES (sistema de equações integrais paramétricas), a qual tem sido usada para resolver problema de valor de contorno em 3D foi utilizada. Os autores focaram em analisar problemas potencial 3D modelado por equações diferenciais parcial de Laplace.…”