Mixed convolved action for classical and fractional-derivative dissipative dynamical systems

The principle of mixed convolved action provides a new rigorous weak variational formalism for a broad range of initial value problems in mathematical physics and mechanics. Here, the focus is initially on classical single-degree-of-freedom oscillators incorporating either Kelvin-Voigt or Maxwell dissipative elements and then, subsequently, on systems that utilize fractional-derivative constitutive models. In each case, an appropriate mixed convolved action is formulated, and a corresponding weak form is discretized in time using temporal shape functions to produce an algorithm suitable for numerical solution. Several examples are considered to validate the mixed convolved action principles and to investigate the performance of the numerical algorithms. For undamped systems, the algorithm is found to be symplectic and unconditionally stable with respect to the time step. In the case of dissipative systems, the approach is shown to be robust and to be accurate with good convergence characteristics for both classical and fractional-derivative based models. As part of the derivations, some interesting results in the calculus of Caputo fractional derivatives also are presented.