Structure preserving discretisations of gradient flows for axisymmetric two-phase biomembranes

The form and evolution of multi-phase biomembranes is of fundamental importance in order to understand living systems. In order to describe these membranes, we consider a mathematical model based on a Canham--Helfrich--Evans two-phase elastic energy, which will lead to fourth order geometric evolution problems involving highly nonlinear boundary conditions. We develop a parametric finite element method in an axisymmetric setting. Using a variational approach, it is possible to derive weak formulations for the highly nonlinear boundary value problems such that energy decay laws, as well as conservation properties, hold for spatially discretised problems. We will prove these properties and show that the fully discretised schemes are well-posed. Finally, several numerical computations demonstrate that the numerical method can be used to compute complex, experimentally observed two-phase biomembranes

Finite element methods for fourth order axisymmetric geometric evolution equations

Fourth order curvature driven interface evolution equations frequently appear in the natural sciences. Often axisymmetric geometries are of interest, and in this situation numerical computations are much more efficient. We will introduce and analyze several new finite element schemes for fourth order geometric evolution equations in an axisymmetric setting, and for selected schemes we will show existence, uniqueness and stability results. The presented schemes have very good mesh and stability properties, as will be demonstrated by several numerical examples

Numerical approximation of curve evolutions in Riemannian manifolds

We introduce variational approximations for curve evolutions in two-dimensional Riemannian manifolds that are conformally flat, i.e.\ conformally equivalent to the Euclidean space. Examples include the hyperbolic plane, the hyperbolic disk, the elliptic plane as well as any conformal parameterization of a two-dimensional surface in ${\mathbb R}^d$, $d\geq 3$. In these spaces we introduce stable numerical schemes for curvature flow and curve diffusion, and we also formulate a scheme for elastic flow. Variants of the schemes can also be applied to geometric evolution equations for axisymmetric hypersurfaces in ${\mathbb R}^d$. Some of the schemes have very good properties with respect to the distribution of mesh points, which is demonstrated with the help of several numerical computations

Cahn-Hilliard-Brinkman systems for tumour growth

A phase field model for tumour growth is introduced that is based on a Brinkman law for convective velocity fields. The model couples a convective Cahn-Hilliard equation for the evolution of the tumour to a reaction-diffusion-advection equation for a nutrient and to a Brinkman-Stokes type law for the fluid velocity. The model is derived from basic thermodynamical principles, sharp interface limits are derived by matched asymptotics and an existence theory is presented for the case of a mobility which degenerates in one phase leading to a degenerate parabolic equation of fourth order. Finally numerical results describe qualitative features of the solutions and illustrate instabilities in certain situations

Finite Element Approximation for the Dynamics of Fluidic Two-Phase Biomembranes

Biomembranes and vesicles consisting of multiple phases can attain a multitude of shapes, undergoing complex shape transitions. We study a Cahn--Hilliard model on an evolving hypersurface coupled to Navier--Stokes equations on the surface and in the surrounding medium to model these phenomena. The evolution is driven by a curvature energy, modelling the elasticity of the membrane, and by a Cahn--Hilliard type energy, modelling line energy effects. A stable semidiscrete finite element approximation is introduced and, with the help of a fully discrete method, several phenomena occurring for two-phase membranes are computed

Beam test studies with silicon sensor module prototypes for the CMS Phase-2 Outer Tracker

The Large Hadron Collider (LHC) at CERN will be upgraded to the High-Luminosity LHC (HL-LHC) by 2029. In order to fully exploit the physics potential of the high luminosity era the experiments must undergo major upgrades. In the context of the upgrade of the Compact Muon Solenoid (CMS) experiment the silicon tracker will be fully replaced. The outer part of the new tracker (Outer Tracker) will be equipped with about 13,000 double-layer silicon sensor modules with two different flavors: PS modules consisting of a macro-pixel and a strip sensor and 2S modules using two strip sensors. These modules can discriminate between trajectories of charged particles with low and high transverse momentum. The different curvature of the trajectories in the CMS magnetic field leads to different hit signatures in the two sensor layers. By reading out both sensors, matching hits in the seed and correlation layer "stubs" are identified. This stub information is generated at the LHC bunch crossing frequency of 40 MHz and serves as input for the first stage of the CMS trigger. In order to quantify the hit and stub detection efficiency, beam tests have been performed. This article comprises selected studies from measurements gathered during two beam tests at the DESY test beam facility with 2S prototype modules assembled in 2021, featuring the Low Power Gigabit Transceiver (lpGBT). In order to compare the module performance at the beginning and end of the CMS runtime, a module with irradiated components has been built and intensively tested

Numerical computations of facetted pattern formation in snow crystal growth

Facetted growth of snow crystals leads to a rich diversity of forms, and exhibits a remarkable sixfold symmetry. Snow crystal structures result from diffusion limited crystal growth in the presence of anisotropic surface energy and anisotropic attachment kinetics. It is by now well understood that the morphological stability of ice crystals strongly depends on supersaturation, crystal size and temperature. Until very recently it was very difficult to perform numerical simulations of this highly anisotropic crystal growth. In particular, obtaining facet growth in combination with dendritic branching is a challenging task. We present numerical simulations of snow crystal growth in two and three space dimensions using a new computational method recently introduced by the authors. We present both qualitative and quantitative computations. In particular, a linear relationship between tip velocity and supersaturation is observed. The computations also suggest that surface energy effects, although small, have a larger effect on crystal growth than previously expected. We compute solid plates, solid prisms, hollow columns, needles, dendrites, capped columns and scrolls on plates. Although all these forms appear in nature, most of these forms are computed here for the first time in numerical simulations for a continuum model.Comment: 12 pages, 28 figure

Late Quaternary Distribution of the Cycladophora davisiana Radiolarian Species: Reflection of Possible Ventilation of the North Pacific Intermediate Water during the Last Glacial Maximum

A comparison of micropaleontological data on the distribution of the Cycladophora davisiana radiolarian species in the surface sediment layer and the Late Quaternary sediments from the Subarctic Pacific and Far East marginal seas allowed conclusions concerning the possible conditions and occurrence of intermediate waters during the last glacial maximum. We used the modern data on the C. davisiana species, which is a micropaleontological indicator of the cold oxygen-rich upper intermediate water mass, which is now forming only in the Sea of Okhotsk. The high amount of C. davisiana in sediments of the last glacial maximum may point to the possible formation and expansion of the ventilated intermediate water in the most part of the Subarctic paleo-Pacific: the Bering Sea, the Sea of Okhotsk, within the NW Gyre, and in the Gulf of Alaska