Construction of finite element spaces in Hdiv for three- dimensional geometries

• Grant number: 13/21959-4
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: May 01, 2014
• End date: April 30, 2015
• Field of knowledge: Engineering - Civil Engineering - Structural Engineering
• Principal Investigator: Philippe Remy Bernard Devloo
• Grantee: Douglas Azevedo Castro
• Host Institution: Faculdade de Engenharia Civil, Arquitetura e Urbanismo (FEC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil

Abstract

The postdoctoral project considers a new construction of finite element spaces in Hdiv for three-dimensional meshes formed by tetrahedral or exahedra, having in mind applications to the simulation of porous media flows. For this kind of problem, one alternative is the use of mixed formulations, characterized by simultaneous calculations of pressure and velocity fields. However, an appropriate use of mixed finite element formulations requires a compatibility of the employed approximation spaces. The approximation space suitable for the velocity variable is of Hdiv type, where the vectorial functions are not necessarily continuous, but their normal components are continuous at the interface between the elements of the domain partition, which is crucial for mass conservation, a fundamental property of this type of application. The methodology to be adopted for the construction of Hdiv bases consists in using hierarchical scalar H1 bases multiplied by vectors that should be properly chosen over the geometrical elements. This methodology has already been successful applied to bi-dimensional triangular and quadrilateral partitions. The implementation and validation of the Hdiv spaces will be performed in the scientific computation environment named NeoPZ. This environment consists of finite element libraries, using computing knowledge of object oriented programming. The required H1 bases, for a variety of three-dimensional geometries, and bidimensional Hdiv bases are alredy implemented in the NeoPZ

Scientific publications

(4)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

FARIAS, AGNALDO M.; DEVLOO, PHILIPPE R. B.; GOMES, SONIA M.; DE SIQUEIRA, DENISE; CASTRO, DOUGLAS A.. Two dimensional mixed finite element approximations for elliptic problems with enhanced accuracy for the potential and flux divergence. COMPUTERS & MATHEMATICS WITH APPLICATIONS, v. 74, n. 12, p. 3283-3295, DEC 15 2017. (13/21959-4)

DEVLOO, P. R. B.; DURAN, O.; GOMES, S. M.; SHAUER, N.. Mixed finite element approximations based on 3-D hp-adaptive curved meshes with two types of H(div)-conforming spaces. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, v. 113, n. 7, p. 1045-1060, FEB 17 2018. (16/13637-5, 13/21959-4)

CASTRO, DOUGLAS A.; DEVLOO, PHILIPPE R. B.; FARIAS, AGNALDO M.; GOMES, SONIA M.; DURAN, OMAR. Hierarchical high order finite element bases for H(div) spaces based on curved meshes for two-dimensional regions or manifolds. Journal of Computational and Applied Mathematics, v. 301, p. 241-258, AUG 1 2016. (13/21959-4)

CASTRO, DOUGLAS A.; DEVLOO, PHILIPPE R. B.; FARIAS, AGNALDO M.; GOMES, SONIA M.; DE SIQUEIRA, DENISE; DURAN, OMAR. Three dimensional hierarchical mixed finite element approximations with enhanced primal variable accuracy. COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, v. 306, p. 479-502, JUL 1 2016. (13/21959-4)