Isogeometric Analysis Applied to Nonlinear Elastodynamics

A numerical investigation is carried out in this work in order to evaluate the computational performance of a numerical model based on isogeometric analysis for applications on geometrically nonlinear elastodynamics. In the FEM (finite element method) practice, it is well known that the Newmark’s method is unconditionally stable for linear structural systems, but this characteristic is frequently lost when problems with nonlinear behavior are analyzed. Since numerical instability is induced owing to the lack of energy balance within every time step of the time integration process, numerical schemes with energy-conserving and controllable numerical dissipation properties are usually adopted to stabilize the time integration procedure. Therefore, the main objective of the present work is to determine if isogeometric analysis can handle dynamic problems more efficiently and more accurately than finite element models, especially for applications where the nonlinear response presents some drawbacks. In this sense, geometrically nonlinear dynamic problems are analyzed employing an isogeometric model based on NURBS (non-uniform rational B-splines), which is obtained considering the Bubnov-Galerkin weighted residual method. The kinematical description is performed using the corotational approach formulated in the context of isogeometric analysis and a transformation matrix given by the classical polar decomposition. The constitutive equation is written in terms of corotational variables according to the hypoelastic theory, where the small strain hypothesis is adopted. Large displacements and large rotations are also considered in the present scheme. Some numerical examples are analyzed to determine the behavior of the present formulation for nonlinear dynamic problems.