Open Access

ARTICLE

Multibody Dynamics Using Quasi Energy and Momentum Conservative Algorithm Implemented with a Novel Four-Node Co-Rotational Quadrilateral Shell Element

Zhongxue Li^1,*, Yu Peng¹, Jin Xu², Loc Vu-Quoc³, Bassam A. Izzuddin⁴

1 Department of Civil Engineering, Zhejiang University, Hangzhou, China
2 Simulation & Design Development, Altair Engineering Inc., Shanghai, China
3 Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, USA
4 Department of Civil and Environmental Engineering, Imperial College London, London, UK

* Corresponding Author: Zhongxue Li. Email:

(This article belongs to the Special Issue: Advances in Modeling and Analysis of Complex Dynamics in Nonlinear Systems)

Computer Modeling in Engineering & Sciences 2026, 148(1), 7 https://doi.org/10.32604/cmes.2026.081240

Received 26 February 2026; Accepted 30 June 2026; Issue published 27 July 2026

Abstract

This paper proposes a computational theory for multibody dynamics based on a novel co-rotational formulation of a four-node quadrilateral shell element, designed to address nonlinear dynamic problems in flexible multibody systems undergoing arbitrarily large displacements and rotations. To circumvent the numerical inefficiency caused by the asymmetric tangent stiffness matrix in conventional co-rotational approaches, an incrementally additive vectorial rotational variable is introduced, which ensures the symmetry of the element’s tangent stiffness matrix in both global and local coordinate systems. The adoption of this vectorial rotational variable substantially improves computational efficiency and numerical stability. Hamilton’s principle is employed to derive the system’s dynamic equilibrium differential equation. For time integration, a generalized midpoint scheme coupled with a quasi energy–momentum conservative algorithm is used. This scheme ensures nearly exact conservation of the system’s total energy, linear momentum, and angular momentum in long-term simulations. The accuracy and stability of the proposed methodology are verified through four benchmark cases: an L-shaped plate, a ruler-shaped plate, a hemispherical shell with an 18° top opening, and a non-smooth shell-three intersecting plates. The results indicate that after the external loads are removed, the system’s total energy, total linear momentum, and total angular momentum are conserved with near-exact precision. The numerical results show excellent agreement with reference solutions from prior publications, and the computational process remains robustly stable.

---

Keywords

---