The Semi Implicit Gradient Augmented Level Set Method

Here a semi-implicit formulation of the gradient augmented level set method is presented. By tracking both the level set and it's gradient accurate subgrid information is provided,leading to highly accurate descriptions of a moving interface. The result is a hybrid Lagrangian-Eulerian method that may be easily applied in two or three dimensions. The new approach allows for the investigation of interfaces evolving by mean curvature and by the intrinsic Laplacian of the curvature. In this work the algorithm, convergence and accuracy results are presented. Several numerical experiments in both two and three dimensions demonstrate the stability of the scheme.Comment: 19 Pages, 14 Figure

Editorial: Schools as enabling environments

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Linear stability analysis of capillary instabilities for concentric cylindrical shells

Motivated by complex multi-fluid geometries currently being explored in fibre-device manufacturing, we study capillary instabilities in concentric cylindrical flows of $N$ fluids with arbitrary viscosities, thicknesses, densities, and surface tensions in both the Stokes regime and for the full Navier--Stokes problem. Generalizing previous work by Tomotika (N=2), Stone & Brenner (N=3, equal viscosities) and others, we present a full linear stability analysis of the growth modes and rates, reducing the system to a linear generalized eigenproblem in the Stokes case. Furthermore, we demonstrate by Plateau-style geometrical arguments that only axisymmetric instabilities need be considered. We show that the N=3 case is already sufficient to obtain several interesting phenomena: limiting cases of thin shells or low shell viscosity that reduce to N=2 problems, and a system with competing breakup processes at very different length scales. The latter is demonstrated with full 3-dimensional Stokes-flow simulations. Many $N > 3$ cases remain to be explored, and as a first step we discuss two illustrative $N \to \infty$ cases, an alternating-layer structure and a geometry with a continuously varying viscosity

Dewetting-controlled binding of ligands to hydrophobic pockets

We report on a combined atomistic molecular dynamics simulation and implicit solvent analysis of a generic hydrophobic pocket-ligand (host-guest) system. The approaching ligand induces complex wetting/dewetting transitions in the weakly solvated pocket. The transitions lead to bimodal solvent fluctuations which govern magnitude and range of the pocket-ligand attraction. A recently developed implicit water model, based on the minimization of a geometric functional, captures the sensitive aqueous interface response to the concave-convex pocket-ligand configuration semi-quantitatively

Some flows in shape optimization

Geometric flows related to shape optimization problems of Bernoulli type are investigated. The evolution law is the sum of a curvature term and a nonlocal term of Hele-Shaw type. We introduce generalized set solutions, the definition of which is widely inspired by viscosity solutions. The main result is an inclusion preservation principle for generalized solutions. As a consequence, we obtain existence, uniqueness and stability of solutions. Asymptotic behavior for the flow is discussed: we prove that the solutions converge to a generalized Bernoulli exterior free boundary problem

Characterizing Width Uniformity by Wave Propagation

This work describes a novel image analysis approach to characterize the uniformity of objects in agglomerates by using the propagation of normal wavefronts. The problem of width uniformity is discussed and its importance for the characterization of composite structures normally found in physics and biology highlighted. The methodology involves identifying each cluster (i.e. connected component) of interest, which can correspond to objects or voids, and estimating the respective medial axes by using a recently proposed wavefront propagation approach, which is briefly reviewed. The distance values along such axes are identified and their mean and standard deviation values obtained. As illustrated with respect to synthetic and real objects (in vitro cultures of neuronal cells), the combined use of these two features provide a powerful description of the uniformity of the separation between the objects, presenting potential for several applications in material sciences and biology.Comment: 14 pages, 23 figures, 1 table, 1 referenc

A new ghost cell/level set method for moving boundary problems:application to tumor growth

In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasi-steady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics—an effect observed in real tumor growth

A Replica Inference Approach to Unsupervised Multi-Scale Image Segmentation

We apply a replica inference based Potts model method to unsupervised image segmentation on multiple scales. This approach was inspired by the statistical mechanics problem of "community detection" and its phase diagram. Specifically, the problem is cast as identifying tightly bound clusters ("communities" or "solutes") against a background or "solvent". Within our multiresolution approach, we compute information theory based correlations among multiple solutions ("replicas") of the same graph over a range of resolutions. Significant multiresolution structures are identified by replica correlations as manifest in information theory overlaps. With the aid of these correlations as well as thermodynamic measures, the phase diagram of the corresponding Potts model is analyzed both at zero and finite temperatures. Optimal parameters corresponding to a sensible unsupervised segmentation correspond to the "easy phase" of the Potts model. Our algorithm is fast and shown to be at least as accurate as the best algorithms to date and to be especially suited to the detection of camouflaged images.Comment: 26 pages, 22 figure