More accurate and stable time integration scheme

Abstract

A novel second-order time integration algorithm for solving dynamic problems is presented. To demonstrate the abilities of this formulation, several linear and geometrical nonlinear structures are solved. The outcomes of authors’ technique are compared with the ones obtained by other researchers. The findings not only confirm the improved accuracy of the new scheme but also indicate that the suggested method remains stable when the trapezoidal rule fails to produce a stable solution. Effectively maximizing high-frequency numerical dissipation, without inducing excessive algorithmic damping in the important low-frequency region, is one of the proposed tactic merits. Spurious oscillations are removed, and small numerical dispersion error is achieved, when using this approach.

Appendix A: Coefficients

The amplification matrix for the proposed formulation has the following form:

$$ A = \left[ {\begin{array}{*{20}c} {A_{11} } && {A_{12} } && {A_{13} } \\ {A_{21} } && {A_{22} } && {A_{23} } \\ {A_{31} } && {A_{32} } && {A_{33} } \\ \end{array} } \right] $$

(A1)

where

$$ \lambda = \left( { ( 1 / ( ( 0. 5\times \omega \times \Delta {\text{t)}}^{ 2} ) )+ 0. 2 5} \right)^{( -1 )} $$

(A2)

$$ A_{1} = \left( { - 4 \times \lambda /(\Delta t^{2} )} \right) + (0.25 \times \omega \times \omega \times \lambda ) \quad {A}_{2} = - 2 \times \lambda /\Delta t $$

(A3)

$$ b_{11} = 1 + \left( {\omega^{2} \times (\Delta t^{2} ) \times (1 - \alpha ) \times (1 - \alpha )} \right) $$

(A4)

$$ A_{11} = - \left( {(\omega^{2} \times (\Delta t^{2} ) \times (1 - \alpha ) \times (1 - \alpha )} \right) + \left( {\omega^{2} \times (\Delta t^{2} ) \times \beta )} \right)/b_{11} $$

(A5)

$$ \, A_{12} = - \left( {(\omega^{2} \times (\Delta t)) + ((\omega^{2} \times (\Delta t^{2} ) \times (1 - \alpha ) \times \; (2 \times \alpha - 1) \times A_{2} )) + \; (\omega^{2} \times (\Delta t^{2} ) \times (\alpha - \beta - 0.5) \times A_{2} )} \right)/b_{11} $$

(A6)

$$ A_{13} = - \left( {(\omega^{2} ) + ((\omega^{2} \times (\Delta t^{2} ) \times (1 - \alpha ) \times (2 \times \alpha - 1) \times \; A_{1} )) + (\omega^{2} \times (\Delta t^{2} ) \times \; (\alpha - \beta - 0.5) \times A_{1})} \right)/b_{11} $$

(A7)

$$ A_{21} = \Delta t \times (1 - \alpha ) \times (1 + A_{11} ) \quad A_{22} = 1 + \left( {\Delta t \times (2 \times \alpha - 1) \times A_{2} } \right) + \left( {\Delta t \times (1 - \alpha ) \times A_{12} } \right) $$

(A8)

$$ A_{23} = \left( {\Delta t \times (2 \times \alpha - 1) \times A_{1} } \right) + \left( {\Delta t \times (1 - \alpha ) \times A_{13} } \right) $$

(A9)

$$ A_{31} = \left( {\Delta t \times (1 - \alpha ) \times A_{21} } \right) + (\Delta t^{2} \times \beta ) $$

(A10)

$$ A_{32} = (\Delta t \times \alpha ) + \left( {\Delta t \times (1 - \alpha ) \times A_{22} } \right) + \left( {(\Delta t^{2} ) \times (\alpha - \beta - 0.5) \times A_{2} } \right) $$

(A11)

$$ \begin{aligned} A_{{33}} & = 1 + \left( {\Delta t \times (1 - \alpha ) \times A_{{23}} } \right) \\ & \quad + {\text{( }}(\Delta t^{2} ) \times (\alpha - \beta - 0.5) \times A_{1} {\text{)}} \\ \end{aligned} $$

(A12)