Enforcement of nonhomogeneous Dirichlet boundary conditions in GFEM using preprocessed nodal coefficients on the boundary
André Rudnytskyj1; Diego Amadeu Furtado Torres2; Paulo de Tarso Rocha Mendonça1; Clóvis Sperb de Barcellos1
1 Federal University of Santa Catarina - UFSC; 2 Federal Technological University of Paraná - UTFPR
doi:10.20906/CPS/CILAMCE2015-0476
Resumo
The Generalized Finite Element Method (GFEM) has as its essence the augmentation of the approximation subspace by adding special functions which may reflect the known information about the boundary value problem and the input data. Since the enforcement of nonhomogeneous essential boundary conditions is an aspect of concern, as in higher-order Finite Element Method (FEM), it is presented a procedure inspired on the fact that the solution of a linear boundary value problem can be decomposed into a sum of a homogeneous solution and a particular one. The coefficients for the nodes on the Dirichlet boundary are computed through the solution of local systems of algebraic equations built from the interpolation condition of the a priori known Dirichlet data, considering all available functions, as all of them are easily computed on the boundary. Such local systems of equations are built and solved during the pre-processing step, as are the local stiffness and load contributions. Two strategies for the construction of such local systems are presented: element-wise and cloud-wise. These pre-processed coefficients define a particular solution non-null only over narrow bands formed by the union of elements near the Dirichlet boundary. The coefficients thus obtained can be enforced through standard manipulation of the global system of algebraic equilibrium equations. A classic problem of two-dimensional linear elasticity is tested to investigate the influence of the way such local systems are built, the effects of the parameters used for solving the local systems, the polynomial degree of the enrichment on the boundary and in the interior neighborhood, as well as the orientation of the boundary segments. Convergence analysis were performed comparing the performance of the conventional C^0-GFEM and the C^k-GFEM, the smooth version, in a global perspective considering the strain energy, and in a local perspective in terms of the L^2-norm of the displacement on the Dirichlet boundary for the studied case. In the C^0-GFEM the Pa
Palavras-chave: Numerical methods; Solid mechanics; enforcement of nonhomogeneous boundary conditions; GFEM; higher-order approximations; two-dimensional elasticity; convergence analysis; Dirichlet boundary conditions; L-shaped domain; Smooth GFEM