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Towards a Unified Single Analysis Framework Embedded with Multiple Spatial and Time Discretized Methods for Linear Structural Dynamics

David Tae, Kumar K. Tamma*
Department of Mechanical Engineering, University of Minnesota, Twin Cities, Minneapolis, 55455, USA
* Corresponding Author: Kumar K. Tamma. Email:

Computer Modeling in Engineering & Sciences 2023, 135(2), 843-885.

Received 07 April 2022; Accepted 01 July 2022; Issue published 27 October 2022


We propose a novel computational framework that is capable of employing different time integration algorithms and different space discretized methods such as the Finite Element Method, particle methods, and other spatial methods on a single body sub-divided into multiple subdomains. This is in conjunction with implementing the well known Generalized Single Step Single Solve (GS4) family of algorithms which encompass the entire scope of Linear Multistep algorithms that have been developed over the past 50 years or so and are second order accurate into the Differential Algebraic Equation framework. In the current state of technology, the coupling of altogether different time integration algorithms has been limited to the same family of algorithms such as the Newmark methods and the coupling of different algorithms usually has resulted in reduced accuracy in one or more variables including the Lagrange multiplier. However, the robustness and versatility of the GS4 with its ability to accurately account for the numerical shifts in various time schemes it encompasses, overcomes such barriers and allows a wide variety of arbitrary implicit-implicit, implicit-explicit, and explicit-explicit pairing of the various time schemes while maintaining the second order accuracy in time for not only all primary variables such as displacement, velocity and acceleration but also the Lagrange multipliers used for coupling the subdomains. By selecting an appropriate spatial method and time scheme on the area with localized phenomena contrary to utilizing a single process on the entire body, the proposed work has the potential to better capture the physics of a given simulation. The method is validated by solving 2D problems for the linear second order systems with various combination of spatial methods and time schemes with great flexibility. The accuracy and efficacy of the present work have not yet been seen in the current field, and it has shown significant promise in its capabilities and effectiveness for general linear dynamics through numerical examples.


Time integration; structural dynamics; multiple scale and multiple methods; ordinary differential equations; differential algebraic equations

Cite This Article

Tae, D., Tamma, K. K. (2023). Towards a Unified Single Analysis Framework Embedded with Multiple Spatial and Time Discretized Methods for Linear Structural Dynamics. CMES-Computer Modeling in Engineering & Sciences, 135(2), 843–885.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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