Using differential algebra to compute Lagrangian coherent structures for mission design and analysis

Tyler, Jack and Wittig, Alexander (2022) Using differential algebra to compute Lagrangian coherent structures for mission design and analysis. 73rd International Astronautical Congress, Paris Convention Centre, Paris, France. 18 - 22 Sep 2022. 10 pp .

Abstract

Recent mission designs have exploited dynamical phenomena such as invariant manifolds to achieve fuel-efficient low-energy transfers. To aid the mission design process, methods have emerged in the literature to profile system dynamics and identify regions of different dynamical behaviour. In time-independent approximations to motion the classical invariant manifolds partition phase space and separate dynamical behaviour, but in systems with arbitrary time-dependence invariant manifolds are not guaranteed to exist. Lagrangian Coherent Structures (LCS) are a generalisation of the invariant manifold to dynamical systems with arbitrary time dependence and are defined with respect to the derivatives of the leading eigenvector of the deformation tensor for the flow. Unfortunately, these derivatives can be numerically difficult and expensive to compute.

We present the results of DA-LCS, a new numerical method for the identification of three-dimensional LCS in time-dependent astrodynamical systems. We use Differential Algebra (DA) to automatically construct polynomial expansions of the flow with respect to its initial conditions, giving access to derivatives accurate to machine precision. A modified power law iteration is then used to construct similar polynomial expansions of the leading eigenvector of the deformation tensor as a function of the position.

We demonstrate the effectiveness of DA-LCS by applying it to a numerically challenging three-dimensional astrodynamics problem, where previous analysis has thus far largely been limited to two-dimensions. How LCS overcomes some of the false positives present in other methods, such as those based on the Finite-Time Lyapunov Exponent, is also elaborated. The structures
obtained will be explained both in terms of the dynamical phenomena and how they can be used in the design of space missions, even when there is no a priori knowledge of system dynamics.

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Contributors

Author: Jack Tyler

Author: Alexander Wittig

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