A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces

We present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is computed with a regularized kernel and corrections are added for regularization and discretization, which are found from analysis near the singular point. The surface integrals are computed from a new quadrature rule using surface points which project onto grid points in coordinate planes. The method does not require coordinate charts on the surface or special treatment of the singularity other than the corrections. The accuracy is about $O(h^3)$, where $h$ is the spacing in the background grid, uniformly with respect to the point of evaluation, on or near the surface. Improved accuracy is obtained for points on the surface. The treecode of Duan and Krasny for Ewald summation is used to perform sums. Numerical examples are presented with a variety of surfaces.Comment: to appear in Commun. Comput. Phy

Molecular diffusion and slip boundary conditions at smooth surfaces with periodic and random nanoscale textures

The influence of periodic and random surface textures on the flow structure and effective slip length in Newtonian fluids is investigated by molecular dynamics (MD) simulations. We consider a situation where the typical pattern size is smaller than the channel height and the local boundary conditions at wetting and nonwetting regions are characterized by finite slip lengths. In case of anisotropic patterns, transverse flow profiles are reported for flows over alternating stripes of different wettability when the shear flow direction is misaligned with respect to the stripe orientation. The angular dependence of the effective slip length obtained from MD simulations is in good agreement with hydrodynamic predictions provided that the stripe width is larger than several molecular diameters. We found that the longitudinal component of the slip velocity along the shear flow direction is proportional to the interfacial diffusion coefficient of fluid monomers in that direction at equilibrium. In case of random textures, the effective slip length and the diffusion coefficient of fluid monomers in the first layer near the heterogeneous surface depend sensitively on the total area of wetting regions.Comment: 30 pages, 11 figure

How round is a protein? Exploring protein structures for globularity using conformal mapping.

We present a new algorithm that automatically computes a measure of the geometric difference between the surface of a protein and a round sphere. The algorithm takes as input two triangulated genus zero surfaces representing the protein and the round sphere, respectively, and constructs a discrete conformal map f between these surfaces. The conformal map is chosen to minimize a symmetric elastic energy E S (f) that measures the distance of f from an isometry. We illustrate our approach on a set of basic sample problems and then on a dataset of diverse protein structures. We show first that E S (f) is able to quantify the roundness of the Platonic solids and that for these surfaces it replicates well traditional measures of roundness such as the sphericity. We then demonstrate that the symmetric elastic energy E S (f) captures both global and local differences between two surfaces, showing that our method identifies the presence of protruding regions in protein structures and quantifies how these regions make the shape of a protein deviate from globularity. Based on these results, we show that E S (f) serves as a probe of the limits of the application of conformal mapping to parametrize protein shapes. We identify limitations of the method and discuss its extension to achieving automatic registration of protein structures based on their surface geometry

Molecular dynamics simulations of oscillatory Couette flows with slip boundary conditions

The effect of interfacial slip on steady-state and time-periodic flows of monatomic liquids is investigated using non-equilibrium molecular dynamics simulations. The fluid phase is confined between atomically smooth rigid walls, and the fluid flows are induced by moving one of the walls. In steady shear flows, the slip length increases almost linearly with shear rate. We found that the velocity profiles in oscillatory flows are well described by the Stokes flow solution with the slip length that depends on the local shear rate. Interestingly, the rate dependence of the slip length obtained in steady shear flows is recovered when the slip length in oscillatory flows is plotted as a function of the local shear rate magnitude. For both types of flows, the friction coefficient at the liquid-solid interface correlates well with the structure of the first fluid layer near the solid wall.Comment: 31 pages, 11 figure

On the Origin of Frictional Adhesion in Geckos: Small Morphological Changes Lead to a Major Biomechanical Transition in the Genus \u3cem\u3eGonatodes\u3c/em\u3e

The evolutionary history of vertebrate locomotion is punctuated by innovations that have permitted expansion into novel ecological niches. Frictional adhesion of geckos is an innovation renowned for enabling locomotion on vertical and inverted smooth surfaces. Much is known about the microstructure and function of the fully-expressed gekkotan adhesive apparatus, although how it originated is poorly understood. Therefore, identifying species that exhibit the earliest stages of expression of frictional adhesion will provide significant insights into the evolution of this trait. Our previous investigation of digital proportions, shape, scalation, skeletal form, and subdigital epidermal micro-ornamentation in the genus Gonatodes led us to hypothesize that Gonatodes humeralisexpresses incipient frictional adhesion. To test this, we first conducted a phylogenetic analysis of Gonatodes and related sphaerodactyl genera to clarify the historical context of the evolution of frictional adhesive capability in the genus. We then measured the ability of G. humeralis and its close relatives to generate frictional adhesive force, examined their locomotor capabilities on low-friction surfaces, and observed animals in their natural habitat. After accounting for body mass and phylogenetic relationships, we found that G. humeralis generates frictional adhesive force essentially equivalent to that of Anolis, and can scale vertical smooth surfaces. Gonatodes vittatus, a species that lacks elaborated epidermal setae, generates negligible frictional adhesive force and can only ascend smooth inclined surfaces with a pitch of ≤ 40°. We conclude that the ostensibly padless G. humeralis, with feet lacking the musculoskeletal, tendinous, and vascular modifications typical of pad-bearing geckos, nevertheless can employ frictional adhesive contact to assist locomotion. As in Anolis, the release of frictional adhesive contact occurs when the foot is plantar flexed after the heel has lifted from the surface. Our findings indicate that the origin of frictional adhesion was likely gradual but that, ultimately, this led to major shifts in ecology and function