An Introduction to Isogeometric Analysis Applied to Solid Mechanics

The basic concepts related to Isogeometric Analysis applied to Solid Mechanics are introduced in this paper. Isogeometric Analysis is mainly concerned with exact discretization of the spatial field by using the same technologies adopted in Computer Aided Design (CAD) for the computational representation of geometric entities, which are also utilized to approximate the solution field. In addition, the method preserves the same framework employed by numerical models based on the Finite Element Method (FEM). In the present work, fundamental aspects on computational representation of geometric entities using B-Splines are briefly reviewed as a background for the introduction of Non Uniform Rational B-Splines (NURBS). A numerical model for linear and geometrically nonlinear Elasticity is formulated using the Bubnov-Galerkin weighted residual method and the isoparametric approach, where a corrotational formulation is adopted for the kinematical description of the motion. Element assembly procedures and evaluation of the stiffness matrix and load vector at local and global levels are described utilizing analogies with the FEM. Gauss quadrature is adopted for numerical integration of element matrices and relations among physical, parametric and parent fields are established. Classical examples are analyzed using NURBS and the displacement formulation. Results obtained with the present model are compared to benchmark predictions obtained from numerical schemes based on the FEM.