In this paper, we consider multiscale methods for nonlinear elasticity. In particular, we investigate the Generalized Multiscale Finite Element Method (GMsFEM) for a strain-limiting elasticity problem. Being a special case of the naturally implicit constitutive theory of nonlinear elasticity, strain-limiting relation has presented an interesting class of material bodies, for which strains remain bounded (even infinitesimal) while stresses can become arbitrarily large. The nonlinearity and material heterogeneities can create multiscale features in the solution, and multiscale methods are therefore necessary. To handle the resulting nonlinear monotone quasilinear elliptic equation, we use linearization based on the Picard iteration. We consider two types of basis functions, offline and online basis functions, following the general framework of GMsFEM. The offline basis functions depend nonlinearly on the solution. Thus, we design an indicator function and we will recompute the offline basis functions when the indicator function predicts that the material property has significant change during the iterations. On the other hand, we will use the residual based online basis functions to reduce the error substantially when updating basis functions is necessary. Our numerical results show that the above combination of offline and online basis functions is able to give accurate solutions with only a few basis functions per each coarse region and updating basis functions in selected iterations.
Acute respiratory distress syndrome (ARDS) is a common cause of mortality in patients with acute kidney injury (AKI). Inflammatory crosstalk from the kidney to the lung has been shown to contribute to lung injury after AKI, but anti-inflammatory therapies have not been proven beneficial in human studies. Recently, AKI was shown to alter mitochondria and related metabolic pathways in the heart, but the impact of AKI on lung metabolism has not been investigated to our knowledge. In this study, we evaluated the metabolomic profile of the lung following renal ischemia and reperfusion to identify novel pathways that may be modifiable. We randomized C57BL/6 mice to 20 minutes of bilateral renal arterial clamping or sham operation under ketamine/xylazine anesthesia. At 4 hours after reperfusion, we found a significant increase in markers of lung injury, as well as significant metabolomic changes across lung, kidney, plasma and bronchoalveolar lavage fluid (BALF) compared to shams. Comparative analyses revealed that the fatty acid oxidation pathway was the most significantly altered metabolic pathway, a finding which is consistent with mitochondrial dysfunction systemically and in the lung. These metabolomic changes correlated with the extracellular accumulation of the mitochondrial damage associated molecular patterns (mtDAMPs), mitochondrial DNA (mtDNA) and transcription factor A, mitochondria (TFAM). Finally, we found that intraperitoneal injection of renal mtDAMPs caused metabolomic changes consistent with mitochondrial dysfunction in the lung in vivo. Mitochondrial function and mtDAMPs warrant further investigation as potential therapeutic targets in preventing lung injury due to AKI.
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