Nesting and degeneracy of Mie resonances of dielectric cavities within zero-index materials

Resonances in optical cavities are used to manipulate light propagation, enhance light-matter interaction, modulate quantum states, and so on. However, the index contrast between the traditional cavities and the host is generally not high, which to some extent limited their performances. By putting dielectric cavities into a host of zero-index materials, index contrast in principle can approach infinity. Here, we analytically deduced Mie resonance conditions at this extreme circumstance. Interestingly, we discovered a so-called resonance nesting effect, in which a set of cavities with different radii can possess the same type of resonance at the same wavelength. We also revealed previously unknown degeneracy between the 2l-TM (2l-TE) and 2l+1-TE (2l+1-TM) modes for " 0 ( 0) material, and the 2l-TM and 2l-TE for both " 0 and 0. Such extraordinary resonance nesting and degeneracy provide additional principles to manipulate cavity behaviors.

Robust Wake Rollup Modelling Using Dyes

This project report gives details on a modification of VAPTOR, a program that can predict the aerodynamic performance of aircrafts using a potential flow method with a relaxed wake model. In VAPTOR the wake is modelled using distributed vorticity elements (DVEs). DVEs can induce velocities at certain points used to relax the wake. A DVE has inbuilt singularity protections i.e. prevents the calculated velocity to approach infinity, but when two adjacent DVEs have a very low relative angle, these protections lead to an error in the calculation of the velocity at its shared midpoint during the relaxation process. In most cases these errors are negligible until a rotor is analysed during hover or vortex ring state. In these special cases the wake rollup is more intense leading to relatively small angles. The subsequent errors caused by the singularity protections cannot be ignored since they cause the solutions to be erratic and not smooth. It also causes the wake DVEs to deform disproportionally which is a visual indication of the errors. The modification uses a method that involves splitting the DVE in order to eliminate the errors when calculating the velocity at the junction of two adjacent DVEs. The splitting is temporary and only applied during the calculation of the velocity at the junction. The algorithm for the splitting of the DVE and its implementation into MATLAB is provided in this report. The implementation is tested by ensuring that all conditions are kept the same except when splitting is enabled or disabled. A number of test runs were conducted, and an index called the Smoothness Index was created in order to quantify the improvements of the DVE splitting method. The results shown are promising as the solution with splitting enabled is twice as smooth as when the splitting is disabled. There is also a noticeable improvement during visual comparison of the wake diagrams when splitting is enabled and disabled. The results combined with the fact that the extra computation required to execute the DVE splitting method is negligible, the author recommends it be enabled in all cases. Having said that, the end user has full control whether he or she would like to use it or not. They can also change the parameters of splitting to suit their needs.

Robust Wake Rollup Modelling Using Dyes

This project report gives details on a modification of VAPTOR, a program that can predict the aerodynamic performance of aircrafts using a potential flow method with a relaxed wake model. In VAPTOR the wake is modelled using distributed vorticity elements (DVEs). DVEs can induce velocities at certain points used to relax the wake. A DVE has inbuilt singularity protections i.e. prevents the calculated velocity to approach infinity, but when two adjacent DVEs have a very low relative angle, these protections lead to an error in the calculation of the velocity at its shared midpoint during the relaxation process. In most cases these errors are negligible until a rotor is analysed during hover or vortex ring state. In these special cases the wake rollup is more intense leading to relatively small angles. The subsequent errors caused by the singularity protections cannot be ignored since they cause the solutions to be erratic and not smooth. It also causes the wake DVEs to deform disproportionally which is a visual indication of the errors. The modification uses a method that involves splitting the DVE in order to eliminate the errors when calculating the velocity at the junction of two adjacent DVEs. The splitting is temporary and only applied during the calculation of the velocity at the junction. The algorithm for the splitting of the DVE and its implementation into MATLAB is provided in this report. The implementation is tested by ensuring that all conditions are kept the same except when splitting is enabled or disabled. A number of test runs were conducted, and an index called the Smoothness Index was created in order to quantify the improvements of the DVE splitting method. The results shown are promising as the solution with splitting enabled is twice as smooth as when the splitting is disabled. There is also a noticeable improvement during visual comparison of the wake diagrams when splitting is enabled and disabled. The results combined with the fact that the extra computation required to execute the DVE splitting method is negligible, the author recommends it be enabled in all cases. Having said that, the end user has full control whether he or she would like to use it or not. They can also change the parameters of splitting to suit their needs.

An Exact Axisymmetric Solution in Anisotropic Plasticity

A hollow cylinder of incompressible material obeying Hill’s orthotropic quadratic yield criterion and its associated flow rule is contracted on a rigid cylinder inserted in its hole. Friction occurs at the contact surface between the hollow and solid cylinders. An axisymmetric boundary value problem for the flow of the material is formulated and solved, and the solution is in closed form. A numerical technique is only necessary for evaluating ordinary integrals. The solution may exhibit singular behavior in the vicinity of the friction surface. The exact asymptotic representation of the solution shows that some strain rate components and the plastic work rate approach infinity in the friction surface’s vicinity. The effect of plastic anisotropy on the solution’s behavior is discussed.

Far-red Fraction: An Improved Metric for Characterizing Phytochrome Effects on Morphology

Phytochrome, a well-studied photoreceptor in plants, primarily absorbs in the red (R) and far-red (FR) regions and is responsible for the perception of shade and subsequent morphological responses. Experiments performed in controlled environments have widely used the R:FR ratio to simulate the natural environment and used phytochrome photoequilibrium (PPE) to simulate the activity of phytochrome. We review why PPE may be an unreliable metric, including differences in weighting factors, multiple phytochromes, nonphotochemical reversions, intermediates, variations in the total pool of phytochrome, and screening by other pigments. We suggest that environmental signals based on R and FR photon fluxes are a better predictor of plant shape than the more complex PPE model. However, the R:FR ratio is nonintuitive and can approach infinity under electric lights, which makes it difficult to extrapolate from studies in controlled environments to the field. Here we describe an improved metric: the FR fraction (FR/R+FR) with a range from 0 to 1. This is a more intuitive metric both under electric lights and in the field compared with other ratios because it is positively correlated with phytochrome-mediated morphological responses. We demonstrate the reliability of this new metric by reanalyzing previously published data.

Uniqueness theorem for Fourier transformable measures on LCA groups

We show that if points of supports of two discrete ”not very thick” Fourier transformable measures on locally compact abelian (LCA) groups tend to one another at infinity and the same is true for the masses at these points, then these measures coincide. The result is valid for discrete almost periodic measures on LCA groups too. Also, we show that the result is false for some discrete ”thick” measures. To do this, we construct a discrete almost periodic measure on the real axis, whose masses at the points of support tend to zero as these points approach infinity.

Solution Behavior in the Vicinity of Characteristic Envelopes for the Double Slip and Rotation Model

The boundary conditions significantly affect solution behavior near rough interfaces. This paper presents general asymptotic analysis of solutions for the rigid plastic double slip and rotation model in the vicinity of an envelope of characteristics under plane strain and axially symmetric conditions. This model is used in the mechanics of granular materials. The analysis has important implications for solving boundary value problems because the envelope of characteristics is a natural boundary of the analytic solution. Moreover, an envelope of characteristics often coincides with frictional interfaces. In this case, the regime of sticking is not possible independently of the friction law chosen. It is shown that the solution is singular in the vicinity of envelopes. In particular, the profile of the velocity component tangential to the envelope is described by the sum of the constant and square root functions of the normal distance to the envelope in its vicinity. As a result, some components of the strain rate tensor approach infinity. This finding might help to develop an efficient numerical method for solving boundary value problems and provide the basis for the interpretation of some experimental results.

An efficient numerical approach for simulating contact in origami assemblages

Origami-inspired structures provide novel solutions to many engineering applications. The presence of self-contact within origami patterns has been difficult to simulate, yet it has significant implications for the foldability, kinematics and resulting mechanical properties of the final origami system. To open up the full potential of origami engineering, this paper presents an efficient numerical approach that simulates the panel contact in a generalized origami framework. The proposed panel contact model is based on the principle of stationary potential energy and assumes that the contact forces are conserved. The contact potential is formulated such that both the internal force vector and the stiffness matrix approach infinity as the distance between the contacting panel and node approaches zero. We use benchmark simulations to show that the model can correctly capture the kinematics and mechanics induced by contact. By tuning the model parameters accordingly, this methodology can simulate the thickness in origami. Practical examples are used to demonstrate the validity, efficiency and the broad applicability of the proposed model.

Solution Behavior near Envelopes of Characteristics for Certain Constitutive Equations Used in the Mechanics of Polymers

The present paper deals with plane strain deformation of incompressible polymers that obey quite a general pressure-dependent yield criterion. In general, the system of equations can be hyperbolic, parabolic, or elliptic. However, attention is concentrated on the hyperbolic regime and on the behavior of solutions near frictional interfaces, assuming that the regime of sliding occurs only if the friction surface coincides with an envelope of stress characteristics. The main reason for studying the behavior of solutions in the vicinity of envelopes of characteristics is that the solution cannot be extended beyond the envelope. This research is also motivated by available results in metal plasticity that the velocity field is singular near envelopes of characteristics (some space derivatives of velocity components approach infinity). In contrast to metal plasticity, it is shown that in the case of the material models adopted, all derivatives of velocity components are bounded but some derivatives of stress components approach infinity near the envelopes of stress characteristics. The exact asymptotic expansion of stress components is found. It is believed that this result is useful for developing numerical codes that should account for the singular behavior of the stress field.

Singular values of large non-central random matrices

We study largest singular values of large random matrices, each with mean of a fixed rank [Formula: see text]. Our main result is a limit theorem as the number of rows and columns approach infinity, while their ratio approaches a positive constant. It provides a decomposition of the largest [Formula: see text] singular values into the deterministic rate of growth, random centered fluctuations given as explicit linear combinations of the entries of the matrix, and a term negligible in probability. We use this representation to establish asymptotic normality of the largest singular values for random matrices with means that have block structure. We also deduce asymptotic normality for the largest eigenvalues of a random matrix arising in a model of population genetics.