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

Abstract

Summary In this article, a two‐stage explicit time integration method is presented for the numerical analysis of various dynamic problems described by second‐order ordinary differential equations in time. The newly proposed method have negligible damping and period errors in the important low‐frequency range and can introduce moderate numerical damping into the spurious high‐frequency range. The computational effort per step required in the new method is the same as the effort needed in the existing two‐stage explicit methods. The proposed method can provide fourth‐order accurate solutions for velocity independent problems and second‐order accurate solutions for velocity dependent problems, which is not possible in the existing methods. The proposed new method is expected to be an excellent tool for numerical analyses of various dynamic problems including structural dynamics.

Higher-order explicit time integration methods for numerical analyses of structural dynamics

“…By using � +∆ 3 ⁄ and �̇+ ∆ 3 ⁄ given in Eqs. (15) and (16), the acceleration vector at + ∆ 3 ⁄ is computed as…”

“…As more sophisticated spatial finite element models [1][2][3][4] are developed constantly, demands for more accurate time integration methods are also increasing to take full advantage of improved spatial models in transient analyses. Recently, numerous implicit [5][6][7][8][9] and explicit [10][11][12][13][14][15][16] time integration methods have been introduced to effectively analyze challenging dynamic problems.…”

A high‐order accurate explicit time integration method based on cubic b‐spline interpolation and weighted residual technique for structural dynamics

“…Then the maximum angle max = 3.139847324337799 (ie, 179.9 • ) and the period T = 33.72102056485366 are obtained. 16 In Figure 15A, two non-dissipative cases of the proposed method are compared with the Bathe method. In Figure 15A, when t = T 1000 , the Bathe method gives divergent numerical results, while the proposed method renders stable results.…”

“…More details of advantages and disadvantages of explicit and implicit methods can be found in In recent years, a large number of researches were conducted to propose new explicit time integration methods where desirable algorithmic properties such as high calculation efficiency, small numerical dispersion, or dissipation errors are harvested. [15][16][17][18] For instance, to introduce controllable numerical dispersion, an explicit method evolved from the central difference (CD) method was presented by Tchamwa and Wielgosz (TW). 19,20 This method is proved to be very effective for wave propagation problems, but only first-order accurate.…”

An identical second‐order single step explicit integration algorithm with dissipation control for structural dynamics

“…Recently, some novel explicit integration algorithms 10‐15 using the composite sub‐step technique have been proposed to enhance the stability and accuracy. For example, Bathe and co‐workers 11 proposed a composite two sub‐step explicit method with desired numerical dissipations.…”