Adaptive discretization of fractional order viscoelasticity using sparse time history

An efficient numerical method to integrate the constitutive response of fractional order viscoelasticity is developed. The method can handle variable time steps. To overcome the problem of the growing amount of data that has to be stored and used in each time step we introduce sparse quadrature. We use an internal variable formulation of the viscoelastic equations where the internal variable is of stress type. The rate equation that governs the evolution of the internal variable involves a fractional integral and can be identified as a Volterra integral equation of the second kind with a weakly singular kernel. For the numerical integration of the rate equation we adopt the finite element method in time, in particular the discontinuous Galerkin method with piecewise constant basis functions is used. A priori and a posteriori error estimates are proved. An adaptive strategy based on the a posteriori error estimate is developed. Finally, the precision and effectiveness of the method are demonstrated by comparing the numerical solutions with analytical solutions.

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