A finite element method for computational full two-body problem: I. The mutual potential and derivatives over bilinear tetrahedron elements

  • Yang YuEmail author
  • Bin Cheng
  • Masatoshi Hayabayashi
  • Patrick Michel
  • Hexi Baoyin
Original Article


A finite element method (FEM) for computing the gravitational interactions between two arbitrarily shaped celestial bodies is proposed. Expressions for the gravitational potential, attraction and torques are derived in terms of the finite element mesh division and mass density distribution. This method is implemented to a parallel-simulation package on a local cluster. Benchmarking tests are performed to confirm the convergence properties and to measure the computational costs. For a representative application, we construct the FEM model of the binary Near-Earth asteroid 65803 Didymos and simulate the coupled spin–orbit motion of its two components. The results show our method propagates the binary motion precisely, which is significantly dependent on the primary’s internal structure. In this numeric example, we show the finite element method is capable of modeling complex geometry and dissimilar material properties, which is useful to address questions in predicting the evolution of actual binary asteroids.


Finite element Heterogeneous media Spin–orbit coupling 



Y.Y. acknowledges support from Natural Science Foundation of China (Grants No. 11702009 and No. 11525208). For mesh subdivision of the finite element models, the authors made use of the MATLAB FEM toolbox. The freeware, multi-platform software ParaView, is applied for the visualization of the asteroid’s internal structures.

Supplementary material

10569_2019_9930_MOESM1_ESM.mp4 (7.9 mb)
Supplementary material 1 (mp4 8058 KB)


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Beihang UniversityBeijingChina
  2. 2.Tsinghua UniversityBeijingChina
  3. 3.Department of Aerospace EngineeringAuburn UniversityAuburnUSA
  4. 4.CNRS, Observatoire de la Côte d’AzurUniversity of Nice Sophia AntipolisNice Cedex 4France
  5. 5.Tsinghua UniversityBeijingChina

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