A Dissipation-Based Algorithm With Energy Control For Geometrically Nonlinear Elastodynamics Using Eight-Node Finite Elements And One-Point Quadrature

Abstract

A formulation for the geometrically nonlinear dynamic analysis of elastic structures is
presented in this paper. It is well known that the Newmark`s method, which is considered the most
popular time-stepping scheme for structural dynamics, exhibits unconditional stability in the case of
linear dynamical systems. However, this characteristic is lost in the nonlinear regime owing to the lack
of an energy balance within each time step of the integration process. In order to obtain a numerical
scheme with energy-conserving and controllable numerical dissipation properties a new algorithm is
proposed in this work. The formulation is based on the Generalized-α method, adjusting optimized
time integration parameters and the addition of an algorithmic control of the energy balance
restriction, which is introduced in the Newton-Raphson iterative process within each time step of the
time marching. The Finite Element Method (FEM) is employed in the present model for spatial
discretizations using an eight-node hexahedral isoparametric element with one-point quadrature. In
order to avoid the excitation of spurious modes an efficient hourglass control technique is used and
therefore volumetric locking as well as shear locking are not observed. Some examples are analyzed in
order to validate the present algorithm.