A simple and efficient adaptive time stepping technique for low‐order operator splitting schemes applied to cardiac electrophysiology

Abstract

We present a simple, yet efficient adaptive time stepping scheme for cardiac electrophysiology (EP) simulations based on standard operator splitting techniques. The general idea is to exploit the relation between the splitting error and the reaction's magnitude—found in a previous one‐dimensional analytical study by Spiteri and Ziaratgahi—to construct the new time step controller for three‐dimensional problems. Accordingly, we propose to control the time step length of the operator splitting scheme as a function of the reaction magnitude, in addition to the common approach of adapting the reaction time step. This conforms with observations in numerical experiments supporting the need for a significantly smaller time step length during depolarization than during repolarization. The proposed scheme is compared with classical proportional–integral–differential controllers using state‐of‐the‐art error estimators, which are also presented in details as they have not been previously applied in the context of cardiac EP with operator splitting techniques. Benchmarks show that choosing the time step as a sigmoidal function of the reaction magnitude is highly efficient and full cardiac cycles can be computed with precision even in a realistic biventricular setup. The proposed scheme outperforms common adaptive time stepping techniques, while depending on fewer tuning parameters.

Unique identification of stiffness parameters for nonlinear, anisotropic textile fabrics based on full‐field measurements on a single experiment

“…A suitable strain energy function describing the nonlinear anisotropic behavior of the textile membrane is illustrated in [1] and given as: ψ • • = ψ ti warp + ψ ti fill + ψ orth warp + ψ orth fill , where the terms ψ ti i and ψ orth i are contributions coming from each fiber family and from the interaction between the fibers, respectively. By analogy with [2], the free energy function is reformulated to be linear in material-stiffness related parameters as follows: ψ • • = α ti warp Φ ti warp + α ti fill Φ ti fill + α orth warp Φ orth warp + α orth fill Φ orth fill , with Φ ti i and Φ orth i being polyconvex functions defined as [3,4]: According to the Japanese guideline MSAJ/M-02-1995, the loading profile for the characterization of textile membranes consists of three biaxial stress ratios warp:fill 1:1, 2:1 and 1:2, and two uniaxial stress ratios 1:0 and 0:1. The model parameters appearing in the functions Φ ti i and Φ orth i have been attained in a classical fashion by fitting the five stress-strain data resulting from the standard tests simultaneously as shown in Fig.…”