The generalization of diagonally implicit Runge–Kutta–Nyström method with controllable numerical dissipation for structural dynamics

Abstract

This paper strictly focuses upon novel designs of the time-integration algorithms as applied to structural dynamics systems with or without physical damping. The significant advances and contributions are summarized as follows: (1) the identity between the composite time-integration algorithms and the diagonally implicit Runge–Kutta family of algorithms are specifically established and demonstrated in order to clarify the originality, development, contribution, and pros/cons of the composite time-integration algorithms developed over the recent decades; (2) then, it is pointed out that the design of potential next-generation multi-stage time-integration algorithms with improved numerical properties can directly emanate from and already exist within the diagonally implicit Runge–Kutta–Nyström (DIRKN) computational framework itself, unlike composite-type time-integration methods paying efforts and attempting to design new algorithmic structures, although they are identical to and pertain primarily to the existing RK-type variants; (3) one- and two-stage DIRKN family of new algorithms and novel designs are taken into consideration for the first time, leading to novel sets of parameters with different numerical properties, which not only encompass existing methods by assigning two identical principal roots, but also produce new and novel designs by employing altogether distinctive principal roots; and finally, (4) the much coveted BN-stability feature and condition are additionally achieved and taken into consideration in order to optimize the design of parameters, which is competitive for nonlinear structural dynamics. Numerical examples are demonstrated to validate the analysis, new designs and the proposed overall efforts.

Appendices

Appendices

Appendix A: Mathematical formulation in the two-stage DIRKN method

The components of the amplification matrix in the proposed two-stage DIRKN method are written as Eq. (A.1), and the local error is written as Eq. (A.2).

$$\begin{aligned} {A_{11}} =&\frac{1}{D}\left[ \left( \overline{a}_{11}\overline{a}_{22} - \overline{a}_{11}\overline{b}_{2} + \overline{a}_{21}\overline{b}_{2} - \overline{a}_{22}\overline{b}_{1}\right) \varOmega ^4 \right. \nonumber \\&+ 2\xi \left( a_{11}\overline{a}_{22} + a_{22}\overline{a}_{11} - a_{11}\overline{b}_{2} + a_{21}\overline{b}_{2} - a_{22}\overline{b}_{1}\right) \varOmega ^3 \nonumber \\&+ \left( 4a_{11}a_{22}\xi ^2 + \overline{a}_{11} + \overline{a}_{22} - \overline{b}_{1} - \overline{b}_{2}\right) \varOmega ^2 \nonumber \\&\left. + 2\xi \left( a_{11} + a_{22}\right) \varOmega + 1 \right] \end{aligned}$$

(A.1a)

$$\begin{aligned} {A_{12}} =&\frac{1}{D}\left[ \left( \overline{a}_{11}\overline{a}_{22} - \overline{a}_{11}\overline{b}_{2}c_2 + \overline{a}_{21}\overline{b}_{2}c_1 - \overline{a}_{22}\overline{b}_1c_1\right) \varOmega ^4 \right. \nonumber \\&+ 2\xi \left( a_{11}\overline{a}_{22} + a_{22}\overline{a}_{11} - \overline{a}_{11}\overline{b}_{2} + \overline{a}_{21}\overline{b}_{2} \right. \nonumber \\&\left. - \overline{a}_{22}\overline{b}_1 - a_{11}\overline{b}_{2}c_2 + a_{21}\overline{b}_{2}c_1 - a_{22}\overline{b}_1c_1\right) \varOmega ^3 \nonumber \\&+ \left( 4\xi ^2(a_{11}a_{22} - a_{11}\overline{b}_{2} + a_{21}\overline{b}_{2} - a_{22}\overline{b}_1)\right. \nonumber \\&\left. +\overline{a}_{11} + \overline{a}_{22} - \overline{b}_1c_1 - \overline{b}_{2}c_2\right) \varOmega ^2 \nonumber \\&\left. + 2\xi \left( a_{11} + a_{22} - \overline{b}_1 - \overline{b}_{2}\right) \varOmega + 1 \right] \end{aligned}$$

(A.1b)

$$\begin{aligned} {A_{21}} =&\frac{1}{D}\left[ \left( \overline{a}_{21} b_{2} - \overline{a}_{11} b_{2} - \overline{a}_{22} b_1\right) \varOmega ^4 + 2\xi \left( a_{21} b_{2} \right. \right. \nonumber \\&\left. \left. - a_{11} b_{2} - a_{22} b_1\right) \varOmega ^3 - \left( b_1+b_{2}\right) \varOmega ^2 \right] \end{aligned}$$

(A.1c)

$$\begin{aligned} {A_{22}} =&\frac{1}{D}\left[ \left( \overline{a}_{11} \overline{a}_{22} - \overline{a}_{11} b_{2} c_2 + \overline{a}_{21} b_{2} c_1 - \overline{a}_{22} b_1 c_1\right) \varOmega ^4 \right. \nonumber \\&+ 2\xi \left( a_{11} \overline{a}_{22} + a_{22} \overline{a}_{11} - \overline{a}_{11} b_{2} + \overline{a}_{21} b_{2} - \overline{a}_{22} b_1 \right. \nonumber \\&\left. - a_{11} b_{2} c_2 + a_{21} b_{2} c_1- a_{22} b_1 c_1\right) \varOmega ^3 \nonumber \\&+ \left( 4\xi ^2(a_{11} a_{22} - a_{11} b_{2} + a_{21} b_{2} - a_{22} b_1)\right. \nonumber \\&\left. +\overline{a}_{11} + \overline{a}_{22} - b_1 c_1 -b_{2} c_2\right) \varOmega ^2 \nonumber \\&\left. + 2\xi \left( a_{11} + a_{22} - b_1 - b_{2}\right) \varOmega + 1 \right] \end{aligned}$$

(A.1d)

where $D = \left( \overline{a}_{11}\varOmega ^2 + 2\xi a_{11}\varOmega + 1\right) \left( \overline{a}_{22}\varOmega ^2 + 2\xi a_{22}\varOmega \right. \left. + 1\right)$.

$$\begin{aligned} e_u =&\left( 2\overline{b}_1 + 2\overline{b}_2 - 1\right) \frac{2\xi \omega \dot{u}_n + \omega ^2 u_n - 1}{2\omega ^2}\varOmega ^2 \nonumber \\&+\left[ \left( 1-6a_{11}\overline{b}_1-6a_{21}\overline{b}_2-6a_{22}\overline{b}_2\right) \left( \frac{2\xi ^2\dot{u}_n}{3\omega }\right. \right. \nonumber \\&\left. +\frac{\xi (u_n \omega ^2-1)}{3\omega ^2}\right) \nonumber \\&\left. + \frac{\omega ^2\dot{u}_n-1}{6\omega ^3}\left( 6\overline{b}_1c_1 + 6\overline{b}_2 c_2 - 1\right) \right] \varOmega ^3 + \mathcal {O}(\varOmega ^4) \end{aligned}$$

(A.2a)

$$\begin{aligned} e_{\dot{u}} =&\left( b_1+b_2-1\right) \frac{2\xi \omega \dot{u}_n+\omega ^2 u_n-1}{\omega }\varOmega \nonumber \\&+ \left[ \left( 1-2a_{11}b_1-2a_{21}b_2-2a_{22}b_2\right) \left( 2\xi ^2\dot{u}_n\right. \right. \nonumber \\&\left. +\frac{\xi (u_n\omega ^2-1)}{\omega }\right) \nonumber \\&\left. + \frac{\omega ^2\dot{u}_n-1}{2\omega ^2}\left( 2b_1c_1 + 2b_2c_2 - 1\right) \right] \varOmega ^2 \nonumber \\&+ \left[ \left( 6b_1a_{11}^2 + 6a_{21}b_2a_{11} + 6b_2a_{22}^2 + 6a_{21}b_2a_{22} - 1\right) \right. \nonumber \\&\left( \frac{4\xi ^3\dot{u}_n}{3} + \frac{2\xi ^2(u_n\omega ^2-1)}{3\omega }\right) \nonumber \\&+ \left( 1-3\overline{a}_{11}b_1 - 3\overline{a}_{21}b_2-3\overline{a}_{22}b_2-3a_{11}b_1c_1\right. \nonumber \\&\left. -3a_{21}b_2c_1-3a_{22}b_2c_2\right) \frac{2\xi \dot{u}_n}{3}+ \nonumber \\&+\left( 6a_{11}b_1c_1+6a_{21}b_2c_1+6a_{22}b_2c_2-1\right) \frac{\xi }{3\omega ^2}\nonumber \\&+ \left( 6\overline{a}_{11}b_1 + 6\overline{a}_{21}b_2 + 6\overline{a}_{22}b_2 - 1\right) \frac{-\omega ^4 u_n + \omega ^2}{6\omega ^3} \nonumber \\&\left. + \left( 1 - 3b_1 c_1^2 - 3b_2c_2^2\right) \frac{1}{6\omega ^3} \right] \varOmega ^3 + \mathcal {O}(\varOmega ^4) \end{aligned}$$

(A.2b)