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

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Title: Multiple Time-Weighted Residual Methodology for Design and Synthesis of Time Integration Algorithms.
Authors: Wang, Yazhou1 (AUTHOR) yazhou.wang@tum.de, Maxam, Dean2 (AUTHOR) maxam010@umn.edu, Adams, Nikolaus1 (AUTHOR) nikolaus.adams@tum.de, Tamma, Kumar2 (AUTHOR) ktamma@umn.edu
Source: Archives of Computational Methods in Engineering. Oct2025, Vol. 32 Issue 7, p4225-4264. 40p.
Subjects: Time integration scheme, Computational mechanics, Multibody systems, Heat transfer, Numerical analysis, Structural dynamics, 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 , GS4-II p , q , and GS4-II p , q , r computational frameworks, offering analysts a wide bandwidth of design options. Based on the proposed theory, we introduce the V0 TSS ∗ schemes, which exhibit numerical properties comparable to those of the existing V0 ∗ and traditional schemes, while offering the added benefit of the truly self-starting feature. The much coveted ZOO m 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. [ABSTRACT FROM AUTHOR]
Copyright of Archives of Computational Methods in Engineering is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Multiple Time-Weighted Residual Methodology for Design and Synthesis of Time Integration Algorithms.
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  Data: <searchLink fieldCode="JN" term="%22Archives+of+Computational+Methods+in+Engineering%22">Archives of Computational Methods in Engineering</searchLink>. Oct2025, Vol. 32 Issue 7, p4225-4264. 40p.
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  Data: <searchLink fieldCode="DE" term="%22Time+integration+scheme%22">Time integration scheme</searchLink><br /><searchLink fieldCode="DE" term="%22Computational+mechanics%22">Computational mechanics</searchLink><br /><searchLink fieldCode="DE" term="%22Multibody+systems%22">Multibody systems</searchLink><br /><searchLink fieldCode="DE" term="%22Heat+transfer%22">Heat transfer</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Structural+dynamics%22">Structural dynamics</searchLink><br /><searchLink fieldCode="DE" term="%22Algorithms%22">Algorithms</searchLink>
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  Data: 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 , GS4-II p , q , and GS4-II p , q , r computational frameworks, offering analysts a wide bandwidth of design options. Based on the proposed theory, we introduce the V0 TSS ∗ schemes, which exhibit numerical properties comparable to those of the existing V0 ∗ and traditional schemes, while offering the added benefit of the truly self-starting feature. The much coveted ZOO m 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. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Archives of Computational Methods in Engineering is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s11831-025-10262-3
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        Text: English
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        PageCount: 40
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      – SubjectFull: Time integration scheme
        Type: general
      – SubjectFull: Computational mechanics
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      – SubjectFull: Multibody systems
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      – SubjectFull: Heat transfer
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      – SubjectFull: Numerical analysis
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      – SubjectFull: Structural dynamics
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      – SubjectFull: Algorithms
        Type: general
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      – TitleFull: Multiple Time-Weighted Residual Methodology for Design and Synthesis of Time Integration Algorithms.
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            NameFull: Maxam, Dean
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              M: 10
              Text: Oct2025
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              Y: 2025
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