For direct time integrations: A comparison of the Newmark and ρ∞-Bathe schemes. (December 2019) - Noh, Gunwoo; Bathe, Klaus-Jürgen
Highlights: We consider the Newmark and ρ ∞ -Bathe time integrations for dynamic solutions. New insights into the Newmark and ρ ∞ -Bathe methods are presented. We show that the Newmark method with α = 0.25 ( δ + 0.5 ) 2 and δ ⩾ 0.5 is a special case of the ρ ∞ -Bathe method. Some example solutions of structural dynamics and wave propagations are presented to illustrate the theoretical findings. Abstract: We consider the unconditionally stable Newmark and ρ ∞ - Bathe methods for the direct time integration of the finite element equations in structural dynamics and wave propagations. In our evaluation of the Newmark method we consider the parameters δ and α, and in the ρ ∞ - Bathe method we consider the parameters γ and ρ ∞, with 0 < γ < ∞, γ ≠ 1 and ρ ∞ ∈ [ - 1, + 1 ] . We show that the Newmark method as usually used with its δ and α parameters, α = 0.25 ( δ + 0.5 ) 2 and δ ⩾ 0.5, is a special case of the ρ ∞ - Bathe method. We also show that the β 1 / β 2 -Bathe method is a special case of the ρ ∞ - Bathe scheme. The study of the curves of numerical dissipation and dispersion shows that the ρ ∞ -Bathe method provides effective dissipation and dispersion whereas the Newmark method lacks in that regard. To illustrate our theoretical findings we give the results of some example solutions of structural dynamics and wave propagations. Our study also shows that further research is needed to identify the optimal use of the ρ ∞ - Bathe scheme and other implicit methods in wave propagation analyses.