Adaptive multilevel space-time-stepping scheme for transport in heterogeneous porous media (ADM-LTS)
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Multilevel Parareal Algorithm with Averaging for Oscillatory Problems
The present study is an extension of the work done by Peddle, Haut, and Wingate and Haut and Wingate, where a two-level Parareal method with mapping and averaging is examined. The method proposed in this paper is a multilevel Parareal method with arbitrarily many levels, which is not restricted to the two-level case. We give an asymptotic error estimate which reduces to the two-level estimate for the case when only two levels are considered. Introducing more than two levels has important consequences for the averaging procedure, as we choose separate averaging windows for each of the different levels, which is an additional new feature of the present study. The different averaging windows make the proposed method especially appropriate for nonlinear multiscale problems, because we can introduce a level for each intrinsic scale of the problem and adapt the averaging procedure such that we reproduce the behavior of the model on the particular scale resolved by the level. The method is applied to nonlinear differential equations. The nonlinearities can generate a range of frequencies in the problem. The computational cost of the new method is investigated and studied on several examples.
97 MATHEMATICS AND COMPUTING↗
A novel approach for tetrahedral-element-based finite element simulations of anisotropic hyperelastic intervertebral disc behavior
Intervertebral discs are microstructurally complex spinal tissues that add greatly to the flexibility and mechanical strength of the human spine. Attempting to provide an adjustable basis for capturing a wide range of mechanical characteristics and to better address known challenges of numerical modeling of the disc, we present a robust finite-element-based model formulation for spinal segments in a hyperelastic framework using tetrahedral elements. We evaluate the model stability and accuracy using numerical simulations, with particular attention to the degenerated intervertebral discs and their likely skewed and narrowed geometry. To this end, 1) annulus fibrosus is modeled as a fiber-reinforced Mooney-Rivlin type solid for numerical analysis. 2) An adaptive state-variable dependent explicit time step is proposed and utilized here as a computationally efficient alternative to theoretical estimates. 3) Tetrahedral-element-based FE models for spinal segments under various loading conditions are evaluated for their use in robust numerical simulations. For flexion, extension, lateral bending, and axial rotation load cases, numerical simulations reveal that a suitable framework based on tetrahedral elements can provide greater stability and flexibility concerning geometrical meshing over commonly employed hexahedral-element-based ones for representation and study of spinal segments in various stages of degeneration.
59 BASIC BIOLOGICAL SCIENCES↗