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Implicit-explicit (IMEX) time integration schemes are well suited for non-linear structural dynamics because of their low computational cost and high accuracy. However, the stability of IMEX schemes cannot be guaranteed for general non-linear problems. In this article, we present a scalar auxiliary variable (SAV) stabilization of high-order IMEX time integration schemes that leads to unconditional stability. The proposed IMEX-BDFk-SAV schemes treat linear terms implicitly using kth-order backward difference formulas (BDFk) and non-linear terms explicitly. This eliminates the need for iterations in non-linear problems and leads to low computational costs. Truncation error analysis of the proposed IMEX-BDFk-SAV schemes confirms that up to kth-order accuracy can be achieved and this is verified through a series of convergence tests. Unlike existing SAV schemes for first-order ordinary differential equations (ODEs), we introduce a novel SAV for the proposed schemes that allows direct solution of the second-order ODEs without transforming them into a system of first-order ODEs. Finally, we demonstrate the performance of the proposed schemes by solving several non-linear problems in structural dynamics and show that the proposed schemes can achieve high accuracy at a low computational cost while maintaining unconditional stability.
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This content will become publicly available on January 1, 2026
Coupling composite schemes with different time steps for multi-scale structural dynamics
Simulating the dynamics of structural systems containing both stiff and flexible parts with a time integration scheme that uses a uniform time-step for the entire system is challenging because of the presence of multiple spatial and temporal scales in the response. We present, for the first time, a multi-time-step (MTS) coupling method for composite time integration schemes that is well-suited for such stiff-flexible systems. Using this method, the problem domain is divided into smaller subdomains that are integrated using different time-step sizes and/or different composite time integration schemes to achieve high accuracy at a low computational cost. In contrast to conventional MTS methods for single-step schemes, a key challenge with coupling composite schemes is that multiple constraint conditions are needed to enforce continuity of the solution across subdomains. We develop the constraints necessary for achieving unconditionally stable coupling of the composite ρ∞-Bathe schemes and prove this property analytically. Further, we conduct a local truncation error analysis and study the period elongation and amplitude decay characteristics of the proposed method. Lastly, we demonstrate the performance of the method for linear and nonlinear stiff-flexible systems to show that the proposed MTS method can achieve higher accuracy than existing methods for time integration, for the same computational cost.
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- Award ID(s):
- 2229136
- PAR ID:
- 10574496
- Publisher / Repository:
- Begell House
- Date Published:
- Journal Name:
- International Journal for Multiscale Computational Engineering
- ISSN:
- 1543-1649
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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