Efficient and stable formulation for the generalized finite element method and applications

• Grant number: 19/00434-7
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: May 01, 2019
• End date: April 30, 2023
• Field of knowledge: Engineering - Civil Engineering - Structural Engineering
• Principal Investigator: Sergio Persival Baroncini Proenca
• Grantee: Caio Silva Ramos
• Host Institution: Escola de Engenharia de São Carlos (EESC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated scholarship(s): 20/14605-5 - A well-conditioned global-local extended/generalized finite element method for three-dimensional fracture mechanics: an approach employing phase-field, BE.EP.DR

Abstract

This research project is related to the development of robust formulations in the field of numerical methods for the analysis of complex problems that include non-linear effects at different scales, time dependent responses and multi-physics coupled phenomena. The methodology adopted is aimed at the development of a formulation for the Generalized Finite Element Method (GFEM) that is efficient and stable, in this sense, presenting optimal order of convergence and conditioning comparable to the conventional Finite Element Method (FEM). Such aspects can be essentially contemplated by guaranteeing the linear independence of the shape functions, typically generated by enrichment of the partition of unit. Basically the alternative, with content of originality, that will be adopted to achieve the objectives of efficiency and overall stability avoiding its deterioration: the use of appropriate bases for the generation of the shape functions (for example, the flat-top partitions of unit in the set of enriched shape functions). Different aspects of implementation will be addressed, in addition to convergence studies and a-posteriori analysis of the error in order to evaluate the optimal convergence rates and the computational efficiency gains. Applications should include two and three-dimensional analyzes. In this particular, it is intended to expand the formulation of the flat-top partition of unit attached to the three-node triangular finite element for the tetrahedral and hexahedron finite element. The predicted applications include static and time-dependent nonlinear analyzes of problems with singularities and material interfaces (in solids or solid-fluid interaction), typically situations where a robust GFEM formulation should have clear advantages over the conventional FEM formulation. (AU)