Background: Extracellular matrix (ECM) alterations during aging contribute to various pathological phenotypes. Results: Collagen fibrils, fibers, and bone alter their structural integrity and susceptibility toward degradation by cathepsin K when age-modified. Conclusion: Age-related modifications of collagen affect its biomechanics and proteolytic stability. Significance: Our research reveals how matrix modifications may increase the risk of ECM disorders.
Hydroxyapatite/alginate nanocomposite fibrous scaffolds were fabricated via electrospinning and a novel in situ synthesis of hydroxyapatite (HAp) that mimics mineralized collagen fibrils in bone tissue. Poorly crystalline HAp nanocrystals, as confirmed by X-ray diffractometer peak approximately at 2θ = 32° and Fourier transform infrared spectroscopy spectrum with double split bands of PO4(v 4) at 564 and 602 cm(-1), were induced to nucleate and grow at the [-COO(-)]-Ca(2+)-[-COO(-)] linkage sites on electrospun alginate nanofibers impregnated with PO4 (3-) ions. This novel process resulted in a uniform deposition of HAp nanocrystals on the nanofibers, overcoming the severe agglomeration of HAp nanoparticles processed by the conventional mechanical blending/electrospinning method. Preliminary in vitro cell study showed that rat calvarial osteoblasts attached more stably on the surface of the HAp/alginate scaffolds than on the pure alginate scaffold. In general, the osteoblasts were stretched and elongated into a spindle-shape on the HAp/alginate scaffolds, whereas the cells had a round-shaped morphology on the alginate scaffold. The unique nanofibrous topography combined with the hybridization of HAp and alginate can be advantageous in bone tissue regenerative medicine applications.
Summary
Multigrid methods are popular solution algorithms for many discretized PDEs, either as standalone iterative solvers or as preconditioners, due to their high efficiency. However, the choice and optimization of multigrid components such as relaxation schemes and grid‐transfer operators is crucial to the design of optimally efficient algorithms. It is well known that local Fourier analysis (LFA) is a useful tool to predict and analyze the performance of these components. In this article, we develop a local Fourier analysis of monolithic multigrid methods based on additive Vanka relaxation schemes for mixed finite‐element discretizations of the Stokes equations. The analysis offers insight into the choice of “patches” for the Vanka relaxation, revealing that smaller patches offer more effective convergence per floating point operation. Parameters that minimize the two‐grid convergence factor are proposed and numerical experiments are presented to validate the LFA predictions.
Summary
Multigrid methods that use block‐structured relaxation schemes have been successfully applied to several saddle‐point problems, including those that arise from the discretization of the Stokes equations. In this paper, we present a local Fourier analysis of block‐structured relaxation schemes for the staggered finite‐difference discretization of the Stokes equations to analyze their convergence behavior. Three block‐structured relaxation schemes are considered: distributive relaxation, Braess–Sarazin relaxation, and Uzawa relaxation. In each case, we consider variants based on weighted‐Jacobi relaxation, as is most suitable for parallel implementation on modern architectures. From this analysis, optimal parameters are proposed, and we compare the efficiency of the presented algorithms with these parameters. Finally, some numerical experiments are presented to validate the two‐grid and multigrid convergence factors.
In this paper, we employ local Fourier analysis (LFA) to analyze the convergence properties of multigrid methods for higher-order finite-element approximations to the Laplacian problem. We find that the classical LFA smoothing factor, where the coarse-grid correction is assumed to be an ideal operator that annihilates the low-frequency error components and leaves the high-frequency components unchanged, fails to accurately predict the observed multigrid performance and, consequently, cannot be a reliable analysis tool to give good performance estimates of the two-grid convergence factor. While two-grid LFA still offers a reliable prediction, it leads to more complex symbols that are cumbersome to use to optimize parameters of the relaxation scheme, as is often needed for complex problems. For the purposes of this analytical optimization as well as to have simple predictive analysis, we propose a modification that is "between" two-grid LFA and smoothing analysis, which yields reasonable predictions to help choose correct damping parameters for relaxation. This exploration may help us better understand multigrid performance for higher-order finite element discretizations, including for Q 2 -Q 1 (Taylor-Hood) elements for the Stokes equations. Finally, we present two-grid and multigrid experiments, where the corrected parameter choice is shown to yield significant improvements in the resulting two-grid and multigrid convergence factors.
K E Y W O R D Sfinite-element method, higher-order elements, Jacobi iteration, local Fourier analysis, multigrid
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