Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding
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 d...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/7/6/537 |
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| author | Hunter Johnston Carl Leake Yalchin Efendiev Daniele Mortari |
| author_facet | Hunter Johnston Carl Leake Yalchin Efendiev Daniele Mortari |
| author_sort | Hunter Johnston |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-13T06:17:40Z |
| format | Article |
| id | doaj.art-efc5e671eeca4eca916c74fcf72659d0 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-13T06:17:40Z |
| publishDate | 2019-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-efc5e671eeca4eca916c74fcf72659d02022-12-21T23:56:56ZengMDPI AGMathematics2227-73902019-06-017653710.3390/math7060537math7060537Selected Applications of the Theory of Connections: A Technique for Analytical Constraint EmbeddingHunter Johnston0Carl Leake1Yalchin Efendiev2Daniele Mortari3Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USADepartment of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USADepartment of Mathematics, Texas A&M University, College Station, TX 77843, USADepartment of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USAIn 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.https://www.mdpi.com/2227-7390/7/6/537linear constraint optimizationcalculus of variationover-constrained differential equationsinequality constraintstriangular domainsTheory of Connections |
| spellingShingle | Hunter Johnston Carl Leake Yalchin Efendiev Daniele Mortari Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding Mathematics linear constraint optimization calculus of variation over-constrained differential equations inequality constraints triangular domains Theory of Connections |
| title | Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding |
| title_full | Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding |
| title_fullStr | Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding |
| title_full_unstemmed | Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding |
| title_short | Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding |
| title_sort | selected applications of the theory of connections a technique for analytical constraint embedding |
| topic | linear constraint optimization calculus of variation over-constrained differential equations inequality constraints triangular domains Theory of Connections |
| url | https://www.mdpi.com/2227-7390/7/6/537 |
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