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Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding

Author

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  • Hunter Johnston

    (Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA)

  • Carl Leake

    (Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA)

  • Yalchin Efendiev

    (Department of Mathematics, Texas A&M University, College Station, TX 77843, USA)

  • Daniele Mortari

    (Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA)

Abstract

In this paper, we consider several new applications of the recently introduced mathematical framework of the Theory of Connections (ToC). This framework transforms constrained problems into unconstrained problems by introducing constraint-free variables. Using this transformation, various ordinary differential equations (ODEs), partial differential equations (PDEs) and variational problems can be formulated where the constraints are always satisfied. The resulting equations can then be easily solved by introducing a global basis function set (e.g., Chebyshev, Legendre, etc.) and minimizing a residual at pre-defined collocation points. In this paper, we highlight the utility of ToC by introducing various problems that can be solved using this framework including: (1) analytical linear constraint optimization; (2) the brachistochrone problem; (3) over-constrained differential equations; (4) inequality constraints; and (5) triangular domains.

Suggested Citation

  • Hunter Johnston & Carl Leake & Yalchin Efendiev & Daniele Mortari, 2019. "Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding," Mathematics, MDPI, vol. 7(6), pages 1-19, June.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:6:p:537-:d:239218
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    References listed on IDEAS

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    1. Shirazi, Ali & Najafi, Behzad & Aminyavari, Mehdi & Rinaldi, Fabio & Taylor, Robert A., 2014. "Thermal–economic–environmental analysis and multi-objective optimization of an ice thermal energy storage system for gas turbine cycle inlet air cooling," Energy, Elsevier, vol. 69(C), pages 212-226.
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    Cited by:

    1. Daniele Mortari, 2022. "Theory of Functional Connections Subject to Shear-Type and Mixed Derivatives," Mathematics, MDPI, vol. 10(24), pages 1-16, December.

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