Multiple Time-Weighted Residual Methodology for Design and Synthesis of Time Integration Algorithms

Abstract This paper proposes a novel multiple time-weighted residual methodology with new insights to enable the design of generalized linear multi-step algorithms in computational dynamics. Leveraging single, double, and triple time-weighted residuals in single, two, and three-field forms, respectively, we develop a new generation of Generalized Single-Step Single-Solve algorithms for second-order time-dependent systems. This approach yields the GS4-II $$_{p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mi>p</mml:mi> <mml:mrow/> </mml:mmultiscripts> </mml:math> , GS4-II $$_{p,q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> <mml:mrow/> </mml:mmultiscripts> </mml:math> , and GS4-II $$_{p,q,r}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> <mml:mrow/> </mml:mmultiscripts> </mml:math> computational frameworks, offering analysts a wide bandwidth of design options. Based on the proposed theory, we introduce the V0 $$^*_{\text {TSS}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mtext>TSS</mml:mtext> <mml:mo>∗</mml:mo> </mml:mmultiscripts> </mml:math> schemes, which exhibit numerical properties comparable to those of the existing V0 $$^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mmultiscripts> </mml:math> and traditional schemes, while offering the added benefit of the truly self-starting feature. The much coveted ZOO $$_m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mi>m</mml:mi> <mml:mrow/> </mml:mmultiscripts> </mml:math> schemes (zero-order overshooting with m roots) are also synthesized to achieve second-order time accuracy in all variables, unconditional stability, zero-order overshooting, controllable numerical dissipation/dispersion, and minimal computational complexity. The relationship between the newly proposed computational frameworks and existing methods is analyzed via a comprehensive overview to date, most of which are included as subsets in the newly proposed methodology. Therefore, the multiple time-weighted residual methodology provides a new insight and in-depth understanding of the advances in the literature, showcasing the significance of the proposed theory. Finally, numerical examples from multidisciplinary applications, encompassing multi-body dynamics, structural dynamics, and heat transfer, are presented to substantiate the proposed methodology.