The Bathe time integration method with controllable spectral radius: The ρ∞-Bathe method

A‐stable two‐step time integration methods with controllable numerical dissipation for structural dynamics

“…Moreover, the eigenvalues solved from Equation are conjugate with those solved from Equation , which leads to two conjugate pairs of eigenvalues. The complex conjugate eigenvalues can lead to numerical results being similar to the exact solution, as shown in a later section, and may result from the identical coefficients in the approximations of displacement and velocity in Equations and , which is also indicated in the works of Zhang et al and Noh and Bathe …”

“…The distribution of ρ for the OALTS time integration method with typical ρ ∞ in all frequency ranges is shown in Figure . For the physically undamped case, the results of some typical algorithms in the literature, including the TPOS (G‐α) method, the BDF‐α‐1 method, and the ρ ∞ ‐Bathe method, are compared. As there are two substeps involved in the ρ ∞ ‐Bathe method, to compare the performances of the algorithms under an equivalent computational cost, we use an equivalent time step concept for the ρ ∞ ‐Bathe method in the present study.…”

“…Implicit algorithms tend to be numerically stable, permit a large time step, and are most effective for structural dynamics problems where the response is controlled by a relatively small number of low‐frequency modes . A number of implicit integration algorithms have been proposed (see review articles by Subbaraj and Dokainish, Tamma et al, and Fung as well as the references therein); representative members of these algorithms are, among others, the Newmark method, the Wilson‐ θ method, the HHT‐ α method, the WBZ‐ α method, the three‐parameters optimal schemes (TPOS) by Shao et al or, equivalently, the generalized‐ α (G‐ α ) method, the generalized single‐step single‐solve (GSSSS) method, the η method, and the composite implicit time integration schemes . It is important to note that many typical algorithms can be unified under the umbrella of the well‐known GSSSS framework .…”

“…As the solutions will be destroyed by the slowly damped artifact high‐frequency responses when using the methods with weak numerical dissipation and the accuracy of solutions will be decreased by the significantly decayed low‐frequency responses when using the methods with strong numerical dissipation, it is desirable that a method has a controllable numerical dissipation property, where stability and accuracy can be performed in an optimal manner. It had been shown that a certain amount of numerical dissipation is enough to stabilize the numerical solution in many cases, and the controllable numerical dissipation concept had been successfully developed in many algorithms, such as the HHT‐ α method, the WBZ‐ α method, the TPOS (G‐ α ) method, the GSSSS method, the β 1 / β 2 ‐Bathe method, the ρ ∞ ‐Bathe method, and so on. Moreover, the controllable numerical dissipation can effectively improve accuracy without additional computation efforts.…”

An accurate two‐stage explicit time integration scheme for structural dynamics and various dynamic problems

“…The generalized composite method of Kim and Choi could control the full range of numerical dissipation and always provide identical effective stiffness matrix for the first and second substeps regardless of the level of numerical dissipation. Recently, Noh and Bathe introduced the ρ ∞ ‐Bathe method whose spectral properties were identical to those of the method proposed by Kim and Reddy …”