Developments in the application of the generalized finite element method to thick shell problems

Abstract

This paper develops and analyzes two techniques to extend the use of generalized finite element method techniques to structural shell problems. The first one is a procedure to define local domains for enrichment functions based on the use of pseudo-tangent planes. The second one is a procedure for imposing homogeneous essential boundary conditions and treatment of boundary layer problems by utilizing special functions. The main idea supporting the pseudo-tangent proposition is the separation of the geometric description, with its intrinsical distortions with respect to the physical domain, from the approximation space, which is defined in a locally undistorted domain. The treatment of essential boundary conditions allows an adequate enrichment in the boundary vicinity, preserving the completeness of the polynomials defining the basis functions. A set of numerical cases are tested in order to show the behavior of the proposed strategies, and a number of observations are drawn from the results, as follows. First, the technique of constructing the enrichment functions on a pseudo-tangent plane shows good results, even with strongly curved shell surfaces. With respect to the locking problem, the method behaves in a similar way as the classical hierarchical finite element methods, avoiding locking for appropriate levels of p-refinements. The procedure considered to impose essential boundary conditions in strong form appears to be more accurate than with the penalty or Lagrange multiplier methods. The inclusion of exponential modes for the treatment of boundary layers in shells provided extremely good results, even with integration elements much larger than the shell thickness.

Large deflection and initial instability analysis of anisotropic plates by the generalized finite element method

“…In this way, the enriched function is preserved throughout the domain. For arbitrarily shaped domains, where the boundary segments are neither straight nor parallel to a given coordinate axis, the procedure proposed in Garcia et al (2009) can be used, where the partition of unity associated to the boundary node is substituted by it multiplied by a ramp function which is zero at the boundary.…”

Analysis of piezoelectric laminates by generalized finite element method and mixed layerwise-HSDT models

“…Barcellos et al (2009) develops a C k continuous finite element formulation for the Kirchhoff laminate model. A procedure to define richer approximate subspaces for shell structures and the treatment of the boundary layer phenomenon is addressed by Garcia et al (2009).…”

“…Therefore, the Dirichlet boundary conditions cannot be directly imposed, making special procedures necessary to impose them. One efficient procedure to impose these conditions is through the so-called boundary functions, as developed by Garcia (2003) and Garcia et al (2009).…”

Effects of the smoothness of partitions of unity on the quality of representation of singular enrichments for GFEM/XFEM stress approximations around brittle cracks

“…Extrinsic enrichment is characterized by adding unknowns to the nodes, such that each additional unknown is associated with a new applied enrichment functions (see, for instance, [24][25][26][27][28]). Therefore, the PoU functions can be called "pasting functions" as they bring about implicit domain discretization (defining compact supports), merge various local approximations by performing an unbiased average, and determine their order or continuity across element boundaries [29].…”