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A tensor-based formulation of hetero-functional graph theory.

Amro M Farid1,2, Dakota J Thompson3, Wester Schoonenberg1

  • 1Thayer School of Engineering at Dartmouth, Hanover, NH, USA.

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Summary

Hetero-functional graph theory (HFGT) now uses tensors for mathematical modeling of complex engineering systems. This tensor-based approach strengthens the link between HFGT and model-based systems engineering (MBSE).

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Area of Science:

  • Complex Systems Engineering
  • Network Science
  • Mathematical Modeling

Background:

  • Hetero-functional graph theory (HFGT) models complex engineering systems by integrating network science and model-based systems engineering (MBSE).
  • Existing HFGT uses graph-based structures and matrix analysis, but struggles with the multi-dimensional relationships inherent in MBSE's heterogeneous constructs (system form, function, concept).

Purpose of the Study:

  • To introduce the first tensor-based formulation of hetero-functional graph theory (HFGT).
  • To enhance the mathematical rigor and ontological grounding of HFGT within MBSE.
  • To explore the analytical capabilities of tensor representations for complex systems.

Main Methods:

  • Developed a tensor-based treatment of HFGT, focusing on the 'system concept' and adjacency matrix.
  • Introduced the hetero-functional incidence tensor as a novel data structure.
  • Utilized tensor algebra to analyze relationships within HFGT and with multi-layer networks.

Main Results:

  • The tensor-based formulation strengthens the connection between HFGT and its MBSE ontological foundations.
  • New analytical insights into the relationships between HFGT and multi-layer networks were facilitated by the tensor approach.
  • Demonstrated the suitability of tensors for representing multi-dimensional relationships in complex systems.

Conclusions:

  • Tensor-based HFGT provides a more robust mathematical framework for modeling complex, flexible engineering systems.
  • This approach bridges the gap between network science methodologies and the rich conceptual models of MBSE.
  • The tensor formulation opens new avenues for quantitative analysis in systems engineering.