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Hp-adaptive formulation for the global-local generalized finite element method and applications

Grant number: 19/00435-3
Support type:Scholarships in Brazil - Doctorate
Effective date (Start): May 01, 2019
Effective date (End): March 31, 2022
Field of knowledge:Engineering - Civil Engineering - Structural Engineering
Principal Investigator:Sergio Persival Baroncini Proenca
Grantee:Murilo Henrique Campana Bento
Home Institution: Escola de Engenharia de São Carlos (EESC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

The use of numerical methods to solve complex computational mechanics problems has become essential, since the existence of exact solutions to most problems, which often involve coupled formulations, is limited. The Generalized Finite Element Method (GFEM) is the alternative explored in this investigation. It is a Galerkin method that proposes the generation of numerical approximations by increasing the approximation spaces of the standard low order finite elements by enrichment functions that best represent the local behavior of the solutions. On the other hand, from an overall viewpoint, an important aspect for any numerical method is the control over the accuracy of the resulting numerical solution. To carry out this control in the GFEM, a posteriori error estimates are being explored. A fairly accurate and low computational cost estimate was recently proposed, consisting of a new version of Zienkiewicz-Zhu's classical recovery-based methodology (ZZ). This version, referenced by the acronym ZZ-BD, differs from the classical one by the use of a local weighted projection for the calculation of the recovered stress field. The central theme of this research is the investigation and development of original adaptive procedures that explore the ZZ-BD estimator as an indicator of the regions where solution improvement is necessary in plane and three-dimensional analyzes. In terms of the applications of the adaptive resources to be developed, it is proposed to use it as an instrument coupled with the so-called global-local version of the GFEM, referenced as GFEM-gl. This version can be used advantageously to treat localized singularities at different scales, analyzed in the form of the so-called local problems. The estimator can be used initially for the identification of regions where strictly local analyzes are required. Then, adaptive analyzes of local problems can be conducted to obtain sufficiently precise solutions. It is observed that the initial studies will be conducted in two-dimensional scope, but the formulation allows a consistent extension to three-dimensional field, and this possibility should be investigated. It is intended that the adaptive version resulting from GFEM-gl be efficient and robust, especially in nonlinear problems involving analyzes at different scales with localized nonlinearities. (AU)