Efficient and Adaptive Orthogonal Finite Element Representation of the Geopotential

John L. Junkins, Ahmad Bani Younes, Robyn M. Woollands, Xiaoli Bai

Research output: Contribution to journalArticlepeer-review


We unify and extend classical results from function approximation theory and consider their utility in astrodynamics. Least square approximation, using the classical Chebyshev polynomials as basis functions, is reviewed for discrete samples of the to-be-approximated function. We extend the orthogonal approximation ideas to n-dimensions in a novel way, through the use of array algebra and Kronecker operations. Approximation of test functions illustrates the resulting algorithms and provides insight into the errors of approximation, as well as the associated errors arising when the approximations are differentiated or integrated. Two sets of applications are considered that are challenges in astrodynamics. The first application addresses local approximation of high degree and order geopotential models, replacing the global spherical harmonic series by a family of locally precise orthogonal polynomial approximations for efficient computation. A method is introduced which adapts the approximation degree radially, compatible with the truth that the highest degree approximations (to ensure maximum acceleration error < 10−9 m s−2, globally) are required near the Earth’s surface, whereas lower degree approximations are required as radius increases. We show that a four order of magnitude speedup is feasible, with efficiency optimized using radial adaptation.

Original languageEnglish (US)
Pages (from-to)118-155
Number of pages38
JournalJournal of the Astronautical Sciences
Issue number2
StatePublished - Jun 1 2017
Externally publishedYes

ASJC Scopus subject areas

  • Aerospace Engineering
  • Space and Planetary Science


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