BDF-like methods for nonlinear dynamic analysis

We present several time integration algorithms of second-order accuracy that are numerically simple and effective for nonlinear elastodynamic problems. These algorithms are based on a general four-step scheme that has a resemblance to the backward differentiation formulas. We also present an extension to the composite strategy of the Bathe method. Appropriate values for the algorithmic parameters are determined based on considerations of stability and dissipativity, and less dissipative members of each algorithm have been identified. We demonstrate the convergence characteristics of the proposed algorithms with a nonlinear dynamic problem having analytic solutions, and test these algorithms with several three-dimensional nonlinear elastodynamic problems involving large deformations and rotations, employing St. Venant-Kirchhoff and compressible Neo-Hookean hyperelastic material models. These tests show that stable computations are obtained with the proposed algorithms in nonlinear situations where the trapezoidal rule encounters a well-known instability.