Stable phase field approximations of anisotropic solidification

We introduce unconditionally stable finite element approximations for a phase field model for solidification, which take highly anisotropic surface energy and kinetic effects into account. We hence approximate Stefan problems with anisotropic Gibbs{Thomson law with kinetic undercooling, and quasi-static variants thereof. The phase field model is given by #wt + � %(') 't = r: (b(')rw) ; c a � %(')w = " � � �(r') '

Eliminating spurious velocities with a stable approximation of incompressible two-phase flow

We present a parametric finite element approximation of two-phase flow. This free boundary problem is given by the Stokes equations in the two phases, which are coupled via jump conditions across the interface. Using a novel variational formulation for the interface evolution gives rise to a natural discretization of the mean curvature of the interface. In addition, the mesh quality of the parametric approximation of the interface does not deteriorate, in general, over time; and an equidistribution property can be shown for a semidiscrete continuous-in-time variant of our scheme in two space dimensions. Moreover, on using a simple XFEM pressure space enrichment, we obtain exact volume conservation for the two phase regions. Furthermore, our fully discrete finite element approximation can be shown to be unconditionally stable. We demonstrate the applicability of our method with some numerical results which, in particular, demonstrate that spurious velocities can be avoided in the classical test cases

On stable parametric finite element methods for the Stefan problem and the Mullins-Sekerka problem with applications to dendritic growth

We introduce a parametric finite element approximation for the Stefan problem with the Gibbs–Thomson law and kinetic undercooling, which mimics the underlying energy structure of the problem. The proposed method is also applicable to certain quasi-stationary variants, such as the Mullins–Sekerka problem. In addition, fully anisotropic energies are easily handled. The approximation has good mesh properties, leading to a well-conditioned discretization, even in three space dimensions. Several numerical computations, including for dendritic growth and for snow crystal growth, are presented

Finite-Element Approximation of One-Sided Stefan Problems with Anisotropic, Approximately Crystalline, Gibbs--Thomson Law

We present a finite-element approximation for the one-sided Stefan problem and the one-sided Mullins--Sekerka problem, respectively. The problems feature a fully anisotropic Gibbs--Thomson law, as well as kinetic undercooling. Our approximation, which couples a parametric approximation of the moving boundary with a finite element approximation of the bulk quantities, can be shown to satisfy a stability bound, and it enjoys very good mesh properties which means that no mesh smoothing is necessary in practice. In our numerical computations we concentrate on the simulation of snow crystal growth. On choosing realistic physical parameters, we are able to produce several distinctive types of snow crystal morphologies. In particular, facet breaking in approximately crystalline evolutions can be observed.Comment: 50 pages, 32 figures, 14 tables. Corrected typo

Stable approximations for axisymmetric Willmore flow for closed and open surfaces

For a hypersurface in ${\mathbb R}^3$, Willmore flow is defined as the $L^2$--gradient flow of the classical Willmore energy: the integral of the squared mean curvature. This geometric evolution law is of interest in differential geometry, image reconstruction and mathematical biology. In this paper, we propose novel numerical approximations for the Willmore flow of axisymmetric hypersurfaces. For the semidiscrete continuous-in-time variants we prove a stability result. We consider both closed surfaces, and surfaces with a boundary. In the latter case, we carefully derive weak formulations of suitable boundary conditions. Furthermore, we consider many generalizations of the classical Willmore energy, particularly those that play a role in the study of biomembranes. In the generalized models we include spontaneous curvature and area difference elasticity (ADE) effects, Gaussian curvature and line energy contributions. Several numerical experiments demonstrate the efficiency and robustness of our developed numerical methods.Comment: 52 pages, 19 figure

Global distribution of a chlorophyll f cyanobacterial marker

Some cyanobacteria use light outside the visible spectrum for oxygenic photosynthesis. The far-red light (FRL) region is made accessible through a complex acclimation process that involves the formation of new phycobilisomes and photosystems containing chlorophyll f. Diverse cyanobacteria ranging from unicellular to branched-filamentous forms show this response. These organisms have been isolated from shaded environments such as microbial mats, soil, rock, and stromatolites. However, the full spread of chlorophyll f-containing species in nature is still unknown. Currently, discovering new chlorophyll f cyanobacteria involves lengthy incubation times under selective far-red light. We have used a marker gene to detect chlorophyll f organisms in environmental samples and metagenomic data. This marker, apcE2, encodes a phycobilisome linker associated with FRL-photosynthesis. By focusing on a far-red motif within the sequence, degenerate PCR and BLAST searches can effectively discriminate against the normal chlorophyll a-associated apcE. Even short recovered sequences carry enough information for phylogenetic placement. Markers of chlorophyll f photosynthesis were found in metagenomic datasets from diverse environments around the globe, including cyanobacterial symbionts, hypersaline lakes, corals, and the Arctic/Antarctic regions. This additional information enabled higher phylogenetic resolution supporting the hypothesis that vertical descent, as opposed to horizontal gene transfer, is largely responsible for this phenotype’s distribution

Renal fibrosis, immune cell infiltration and changes of TRPC channel expression after unilateral ureteral obstruction in Trpc6-/- mice

Background/Aims: The transient receptor potential cation channel subfamily C member 6 (TRPC6) is a Ca-permeable nonselective cation channel and has received recent attention because of its capability to promote chronic kidney disease (CKD). The aims of this study were (i) to examine whether deletion of TRPC6 impacts on renal fibrosis and inflammatory cell infiltration in an early CKD model of unilateral ureter obstruction (UUO) in mice; and (ii) whether TRPC6-deficiency as well as UUO affect the regulation of TRPC expression in murine kidneys. Methods: Wild-type (WT), Trpc6-knockout (Trpc6) and New Zealand obese (NZO) mice underwent sham operation or unilateral ureteral obstruction (UUO). The kidneys were harvested 7 days after surgery. We examined renal fibrosis and inflammatory cell infiltration by histological and immunohistochemical staining. The mRNA expression of TRPC members and markers of fibrosis and inflammation in kidney were assessed by using real-time quantitative reverse transcription PCR. Results: Histological and immunohistochemical analyses revealed less inflammatory cell infiltration (F4/80 and CD3) in UUO kidneys of Trpc6 mice compared to UUO kidneys of WT mice as well as less fibrosis. Genomic deletion of TRPC6 also affected the expression of pro-fibrotic genes in UUO Trpc6 kidneys compared to UUO WT kidneys while the expression of pro-inflammatory genes did not differ. UUO caused marked up-regulation of Trpc6 and down-regulation of Trpc1 mRNA in kidneys of WT and NZO mice. Trpc3 mRNA expression was significantly elevated in kidneys of Trpc6 mice underwent UUO while the levels did not change in kidneys of neither WT nor in NZO mice underwent UUO. Conclusion: TRPC6 contributes to renal fibrosis and immune cell infiltration in the UUO mouse model. Therefore, inhibition of TRPC6 emerges as a promising novel therapeutic strategy for treatment of chronic kidney failure in chronic obstructive nephropathy. However, confounding genomic and non-genomic effects of other TRPC channels should be taken into consideration to fully comprehend the renoprotective potential of targeting TRPC6 therapeutically under chronic kidney damaging conditions