An Efficient and Accurate Method for Modeling Nonlinear Fractional Viscoelastic Biomaterials

Computational biomechanics plays an important role in biomedical engineering: using modeling to understand pathophysiology, treatment and device design. While experimental evidence indicates that the mechanical response of most tissues is viscoelasticity, current biomechanical models in the computation community often assume only hyperelasticity. Fractional viscoelastic constitutive models have been successfully used in literature to capture the material response. However, the translation of these models into computational platforms remains limited. Many experimentally derived viscoelastic constitutive models are not suitable for three-dimensional simulations. Furthermore, the use of fractional derivatives can be computationally prohibitive, with a number of current numerical approximations having a computational cost that is $\mathcal{O} ( N_T^2)$ and a storage cost that is $\mathcal{O}(N_T)$ ($N_T$ denotes the number of time steps). In this paper, we present a novel numerical approximation to the Caputo derivative which exploits a recurrence relation similar to those used to discretize classic temporal derivatives, giving a computational cost that is $\mathcal{O} (N)$ and a storage cost that is fixed over time. The approximation is optimized for numerical applications, and the error estimate is presented to demonstrate efficacy of the method. The method is shown to be unconditionally stable in the linear viscoelastic case. It was then integrated into a computational biomechanical framework, with several numerical examples verifying accuracy and computational efficiency of the method, including in an analytic test, in an analytic fractional differential equation, as well as in a computational biomechanical model problem.