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Mixed convolved action.

Physical review. E, Statistical, nonlinear, and soft matter physics·2012
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Mixed convolved action for classical and fractional-derivative dissipative dynamical systems.

G F Dargush1

  • 1Department of Mechanical and Aerospace Engineering, University at Buffalo, State University of New York Buffalo, New York 14260, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 2, 2013
PubMed
Summary

A new mixed convolved action principle offers a rigorous numerical method for solving initial value problems in physics and mechanics. This approach is stable and accurate for both classical and fractional-derivative models.

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Area of Science:

  • Mathematical Physics
  • Mechanical Engineering
  • Computational Mechanics

Background:

  • Classical and fractional-derivative models are used to describe dissipative systems.
  • Existing numerical methods may face challenges with stability and accuracy for these models.

Purpose of the Study:

  • To introduce a novel weak variational formalism based on the principle of mixed convolved action.
  • To develop and validate a numerical algorithm for solving initial value problems in classical and fractional-derivative systems.

Main Methods:

  • Formulation of mixed convolved action for single-degree-of-freedom oscillators and fractional-derivative systems.
  • Discretization of the weak form in time using temporal shape functions.
  • Numerical solution and validation through several examples.

Main Results:

  • The developed algorithm is symplectic and unconditionally stable for undamped systems.
  • The approach demonstrates robustness, accuracy, and good convergence for dissipative systems, including fractional models.
  • Novel results in the calculus of Caputo fractional derivatives were obtained.

Conclusions:

  • The mixed convolved action principle provides a powerful and versatile tool for analyzing complex mechanical systems.
  • The numerical algorithm is effective for both classical and advanced fractional-derivative constitutive models.
  • The study contributes to the advancement of computational methods in mathematical physics and mechanics.