746 research outputs found
Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces
The aim of this manuscript is to present for the first time the application of the finite element method for solving reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces. Furthermore we present pattern formation generated by the reaction-diffusion systemwith cross-diffusion on evolving domains and surfaces. A two-component reaction-diffusion system with linear cross-diffusion in both u and v is presented. The finite element method is based on the approximation of the domain or surface by a triangulated domain or surface consisting of a union of triangles. For surfaces, the vertices of the triangulation lie on the continuous surface. A finite element space of functions is then defined by taking the continuous functions which are linear affine on each simplex of the triangulated domain or surface. To demonstrate the role of cross-diffusion to the theory of pattern formation, we compute patterns with model kinetic parameter values that belong only to the cross-diffusion parameter space; these do not belong to the standard parameter space for classical reaction-diffusion systems. Numerical results exhibited show the robustness, flexibility, versatility, and generality of our methodology; the methodology can deal with complicated evolution laws of the domain and surface, and these include uniform isotropic and anisotropic growth profiles as well as those profiles driven by chemical concentrations residing in the domain or on the surface
Coupling nonpolar and polar solvation free energies in implicit solvent models
Recent studies on the solvation of atomistic and nanoscale solutes indicate
that a strong coupling exists between the hydrophobic, dispersion, and
electrostatic contributions to the solvation free energy, a facet not
considered in current implicit solvent models. We suggest a theoretical
formalism which accounts for coupling by minimizing the Gibbs free energy of
the solvent with respect to a solvent volume exclusion function. The resulting
differential equation is similar to the Laplace-Young equation for the
geometrical description of capillary interfaces, but is extended to microscopic
scales by explicitly considering curvature corrections as well as dispersion
and electrostatic contributions. Unlike existing implicit solvent approaches,
the solvent accessible surface is an output of our model. The presented
formalism is illustrated on spherically or cylindrically symmetrical systems of
neutral or charged solutes on different length scales. The results are in
agreement with computer simulations and, most importantly, demonstrate that our
method captures the strong sensitivity of solvent expulsion and dewetting to
the particular form of the solvent-solute interactions.Comment: accpted in J. Chem. Phy
Application of the level-set method to the implicit solvation of nonpolar molecules
A level-set method is developed for numerically capturing the equilibrium
solute-solvent interface that is defined by the recently proposed variational
implicit solvent model (Dzubiella, Swanson, and McCammon, Phys. Rev. Lett. {\bf
104}, 527 (2006) and J. Chem.\Phys. {\bf 124}, 084905 (2006)). In the level-set
method, a possible solute-solvent interface is represented by the zero
level-set (i.e., the zero level surface) of a level-set function and is
eventually evolved into the equilibrium solute-solvent interface. The evolution
law is determined by minimization of a solvation free energy {\it functional}
that couples both the interfacial energy and the van der Waals type
solute-solvent interaction energy. The surface evolution is thus an energy
minimizing process, and the equilibrium solute-solvent interface is an output
of this process. The method is implemented and applied to the solvation of
nonpolar molecules such as two xenon atoms, two parallel paraffin plates,
helical alkane chains, and a single fullerene . The level-set solutions
show good agreement for the solvation energies when compared to available
molecular dynamics simulations. In particular, the method captures solvent
dewetting (nanobubble formation) and quantitatively describes the interaction
in the strongly hydrophobic plate system
The thermodynamics and roughening of solid-solid interfaces
The dynamics of sharp interfaces separating two non-hydrostatically stressed
solids is analyzed using the idea that the rate of mass transport across the
interface is proportional to the thermodynamic potential difference across the
interface. The solids are allowed to exchange mass by transforming one solid
into the other, thermodynamic relations for the transformation of a mass
element are derived and a linear stability analysis of the interface is carried
out. The stability is shown to depend on the order of the phase transition
occurring at the interface. Numerical simulations are performed in the
non-linear regime to investigate the evolution and roughening of the interface.
It is shown that even small contrasts in the referential densities of the
solids may lead to the formation of finger like structures aligned with the
principal direction of the far field stress.Comment: (24 pages, 8 figures; V2: added figures, text revisions
The Effect of Neutral Atoms on Capillary Discharge Z-pinch
We study the effect of neutral atoms on the dynamics of a capillary discharge
Z-pinch, in a regime for which a large soft-x-ray amplification has been
demonstrated. We extended the commonly used one-fluid magneto-hydrodynamics
(MHD) model by separating out the neutral atoms as a second fluid. Numerical
calculations using this extended model yield new predictions for the dynamics
of the pinch collapse, and better agreement with known measured data.Comment: 4 pages, 4 postscript figures, to be published in Phys. Rev. Let
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Efficient segmentation based on Eikonal and diffusion equations
Segmentation of regions of interest in an image has important applications in medical image analysis, particularly in computer aided diagnosis. Segmentation can enable further quantitative analysis of anatomical structures. We present efficient image segmentation schemes based on the solution of distinct partial differential equations (PDEs). For each known image region, a PDE is solved, the solution of which locally represents the weighted distance from a region known to have a certain segmentation label. To achieve this goal, we propose the use of two separate PDEs, the Eikonal equation and a diffusion equation. In each method, the segmentation labels are obtained by a competition criterion between the solutions to the PDEs corresponding to each region. We discuss how each method applies the concept of information propagation from the labelled image regions to the unknown image regions. Experimental results are presented on magnetic resonance, computed tomography, and ultrasound images and for both two-region and multi-region segmentation problems. These results demonstrate the high level of efficiency as well as the accuracy of the proposed methods
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