Axially variable-length solid element of absolute nodal coordinate formulation

  • Jialiang Sun
  • Qiang Tian
  • Haiyan HuEmail author
  • Niels L. Pedersen
Research Paper


An axially variable-length solid element with eight nodes is proposed by integrating the arbitrary Lagrangian–Eulerian (ALE) formulation and the absolute nodal coordinate formulation (ANCF). In addition to the nodal positions and slopes of eight nodes, two material coordinates in the axial direction are used as the generalized coordinates. As a consequence, the nodes in the ALE–ANCF are not associated with any specific material points and the axial length of the solid element can be varied over time. These two material coordinates give rise to a variable mass matrix and an additional inertial force vector. Computationally efficient formulae of the additional inertial forces and elastic forces, as well as their Jacobians, are also derived. The dynamic equation of a flexible multibody system (FMBS) with variable-length bodies is presented. The maximum and minimum lengths of the boundary elements of an FMBS have to be appropriately defined to ensure accuracy and non-singularity when solving the dynamic equation. Three numerical examples of static and dynamic problems are given to validate the variable-length solid elements of ALE–ANCF and show their capability.


Flexible multibody dynamics Arbitrary Lagrangian–Eulerian formulation Absolute nodal coordinate formulation Variable-length solid element 



This work was supported in part by the National Natural Science Foundation of China (Grants 11521062, 11722216) and the 111 China Project (Grant B16003). It was also supported in part by Postgraduate Research and Practice Innovation Program of Jiangsu Province (Grant KYCX17_0226), and by China Scholarship Council.


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Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Jialiang Sun
    • 1
  • Qiang Tian
    • 2
  • Haiyan Hu
    • 1
    • 2
    Email author
  • Niels L. Pedersen
    • 3
  1. 1.State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace EngineeringNanjing University of Aeronautics and AstronauticsNanjingChina
  2. 2.MOE Key Laboratory of Dynamics and Control of Flight Vehicle, School of Aerospace EngineeringBeijing Institute of TechnologyBeijingChina
  3. 3.Department of Mechanical Engineering, Solid MechanicsTechnical University of DenmarkKongens LyngbyDenmark

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