How to Characterize Amorphous Shapes: The Tale of a Reverse Micelle

Aerosol-OT reverse micelles represent a chemical construct where surfactant molecules self-assemble to stabilize water nanodroplets ~1-10 nm in diameter. Although commonly assumed to adopt a spherical shape, all-atom molecular dynamics simulations and some experimental studies predict a non-spherical shape. If these aggregates are not spherical, then what shape do they take? Because the tools needed to evaluate the shape of something that lacks regular structure, order, or symmetry are not well developed, we present a set of three intuitive metrics- coordinate-pair eccentricity, convexity, and the curvature distribution- that estimate the shape of an amorphous object and we demonstrate their use on a simulated Aerosol-OT reverse micelle. These metrics are all well-established methods and principles in mathematics, and each provides unique information about the shape. Together, these metrics provide intuitive descriptions of amorphous shapes, facilitate ways to quantify those shapes, and follow their changes over time.

Clamped-end effect on static detection signals of DNA-microcantilever

Boundary constraint induced inhomogeneous effects are important for mechanical responses of nano/micro-devices. For microcantilever sensors, the clamped-end constraint induced inhomogeneous effect of static deformation, so called the clamped-end effect, has great influence on the detection signals. This paper is devoted to developing an alternative mechanical model to characterize the clamped-end effect on the static detection signals of the DNA-microcantilever. Different from the previous concentrated load models, the DNA adsorption is taken as an equivalent uniformly distributed tangential load on the substrate upper surface, which exactly satisfies the zero force boundary condition at the free-end. Thereout, a variable coefficient differential governing equation describing the non-uniform deformation of the DNA-microcantilever induced by the clamped-end constraint is established by using the principle of minimum potential energy. By reducing the order of the governing equation, the analytical solutions of the curvature distribution and static bending deflection are obtained. By comparing with the previous approximate surface stress models, the clamped-end effect on the static deflection signals is discussed, and the importance of the neutral axis shift effect is also illustrated for the asymmetric laminated microcantilever.

Quasi-Newtonian methods for modeling of plan curve

The paper is devoted to the methods of geometric modeling of plane curves given in the natural parameterization. The paper considers numerical modeling methods that make it possible to find the equation of curvature of the desired curve for different cases of the input data. The unknown curvature distribution coefficients of the required curve are determined by solving a system of nonlinear integral equations. Various numerical methods are considered to solve this nonlinear system. The results of computer implementation of the proposed methods for modeling two curvilinear contours with different initial data are presented. For the first curve, the input data are the coordinates of three points, the angles of inclination of the tangents at the extreme points and the linear law of curvature distribution. The second example considers an S-shaped curve with a quadratic law of curvature distributi.

Unfolding the multiscale structure of networks with dynamical Ollivier-Ricci curvature

Describing networks geometrically through low-dimensional latent metric spaces has helped design efficient learning algorithms, unveil network symmetries and study dynamical network processes. However, latent space embeddings are limited to specific classes of networks because incompatible metric spaces generally result in information loss. Here, we study arbitrary networks geometrically by defining a dynamic edge curvature measuring the similarity between pairs of dynamical network processes seeded at nearby nodes. We show that the evolution of the curvature distribution exhibits gaps at characteristic timescales indicating bottleneck-edges that limit information spreading. Importantly, curvature gaps are robust to large fluctuations in node degrees, encoding communities until the phase transition of detectability, where spectral and node-clustering methods fail. Using this insight, we derive geometric modularity to find multiscale communities based on deviations from constant network curvature in generative and real-world networks, significantly outperforming most previous methods. Our work suggests using network geometry for studying and controlling the structure of and information spreading on networks.

Redesigning a compressor inlet guide vane for 0 to 60 degree stagger angles

This report's objective is to reduce the total pressure loss coefficient of an inlet guide vane (IGV) at high stagger angles and to therefore reduce the overall fuel consumption of an aircraft engine. IGVs are usually optimized for cruise where the stagger angle is approximately 0 degrees. To reduce losses, four different methodologies were tested: increasing the leading edge radius, increasing the camber, creating a "drooped nose", and creating an "S" curvature distribution. A baseline IGV was chosen and modified using these methodologies to create 10 new IGV designs. CFX was used to perform a CFD analysis on all 11 IGV designs at 5 stagger angles from 0 to 60 degrees. Typical missions were analyzed and it was discovered that the new designs decreased the fuel consumption of the engine. The IGV with the "S" curvature and thicker leading edge was the best and decreased the fuel consumption by 0.24%.

Redesigning a compressor inlet guide vane for 0 to 60 degree stagger angles

This report's objective is to reduce the total pressure loss coefficient of an inlet guide vane (IGV) at high stagger angles and to therefore reduce the overall fuel consumption of an aircraft engine. IGVs are usually optimized for cruise where the stagger angle is approximately 0 degrees. To reduce losses, four different methodologies were tested: increasing the leading edge radius, increasing the camber, creating a "drooped nose", and creating an "S" curvature distribution. A baseline IGV was chosen and modified using these methodologies to create 10 new IGV designs. CFX was used to perform a CFD analysis on all 11 IGV designs at 5 stagger angles from 0 to 60 degrees. Typical missions were analyzed and it was discovered that the new designs decreased the fuel consumption of the engine. The IGV with the "S" curvature and thicker leading edge was the best and decreased the fuel consumption by 0.24%.

$$\epsilon \kappa $$-Curves: controlled local curvature extrema

The $$\kappa $$ κ -curve is a recently published interpolating spline which consists of quadratic Bézier segments passing through input points at the loci of local curvature extrema. We extend this representation to control the magnitudes of local maximum curvature in a new scheme called extended- or $$\epsilon \kappa $$ ϵ κ -curves.$$\kappa $$ κ -curves have been implemented as the curvature tool in Adobe Illustrator® and Photoshop® and are highly valued by professional designers. However, because of the limited degrees of freedom of quadratic Bézier curves, it provides no control over the curvature distribution. We propose new methods that enable the modification of local curvature at the interpolation points by degree elevation of the Bernstein basis as well as application of generalized trigonometric basis functions. By using $$\epsilon \kappa $$ ϵ κ -curves, designers acquire much more ability to produce a variety of expressions, as illustrated by our examples.

Development of a method for geometric modeling of centrifugal compressor impellers

The practice of designing centrifugal compressors in which impellers are the main components shows that there are reserves for their further improvement. One of the main reserves consists in improving flow conditions for the compressed medium in the compressor setting and, above all, in the impeller. A method of geometric modeling settings of the impellers of centrifugal compressors was proposed which involves the construction of meridional boundaries of impellers and the blade profile on an involute of the cylindrical surface of the outer radius of the impeller. The blade is represented by ruled surfaces. The outer boundary of the impeller is described by a curve in natural parameterization using cubic dependence of curvature on the arc length. Dependences and length of the arc are determined in the process of modeling the boundary based on the set source data. The problem is solved by minimizing deviations of intermediate curves from the boundary endpoint. The inner boundary is obtained as an envelope of circles inscribed in the meridional channel of the impeller. Radii of the circles are determined taking into account the flow areas of the channel. The midline of the blade profile on an involute of the cylindrical surface of the outer radius of the impeller is modeled using a curve that is presented in natural parameterization with quadratic law of curvature distribution. A computer code was developed in the Fortran Power Station programming environment that visualizes the obtained numerical results graphically on a computer display in addition to digital information on the modeled boundaries and the blade profile. Graphical results were presented. They confirmed the efficiency of the proposed method of modeling the settings of centrifugal compressor impellers. The method can be useful to offices involved in the design of centrifugal compressors