The effect of landscape fragmentation on Turing-pattern formation

<abstract><p>Diffusion-driven instability and Turing pattern formation are a well-known mechanism by which the local interaction of species, combined with random spatial movement, can generate stable patterns of population densities in the absence of spatial heterogeneity of the underlying medium. Some examples of such patterns exist in ecological interactions between predator and prey, but the conditions required for these patterns are not easily satisfied in ecological systems. At the same time, most ecological systems exist in heterogeneous landscapes, and landscape heterogeneity can affect species interactions and individual movement behavior. In this work, we explore whether and how landscape heterogeneity might facilitate Turing pattern formation in predator–prey interactions. We formulate reaction-diffusion equations for two interacting species on an infinite patchy landscape, consisting of two types of periodically alternating patches. Population dynamics and movement behavior differ between patch types, and individuals may have a preference for one of the two habitat types. We apply homogenization theory to derive an appropriately averaged model, to which we apply stability analysis for Turing patterns. We then study three scenarios in detail and find mechanisms by which diffusion-driven instabilities may arise even if the local interaction and movement rates do not indicate it.</p></abstract>

Combining Macro- and Mesoscale Optimization: A Case Study of the General Electric Jet Engine Bracket

Topology optimization (TO) is a mathematical method that optimizes the material layout in a pre-defined design domain. Its theoretical background is widely known for macro-, meso-, and microscale levels of a structure. The macroscale TO is now available in the majority of commercial TO software, while only a few software packages offer a mesoscale TO with the design and optimization of lattice structures. However, they still lack a practical simultaneous macro–mesoscale TO. It is not clear to the designers how they can combine and apply TO at different levels. In this paper, a two-scale TO is conducted using the homogenization theory at both the macro- and mesoscale structural levels. In this way, the benefits of the existence and optimization of mesoscale structures were researched. For this reason, as a case study, a commercial example of the known jet engine bracket from General Electric (GE bracket) was used. Different optimization workflows were implemented in order to develop alternative design concepts of the same mass. The design concepts were compared with respect to their weight, strength, and simulation time for the given load cases. In addition, the lightest design concept among them was identified.

Determination of anisotropic elastic parameters from morphological parameters of cancellous bone for osteoporotic lumbar spine

In biomechanics, large finite element models with macroscopic representation of several bones or joints are necessary to analyze implant failure mechanisms. In order to handle large simulation models of human bone, it is crucial to homogenize the trabecular structure regarding the mechanical behavior without losing information about the realistic material properties. Accordingly, morphology and fabric measurements of 60 vertebral cancellous bone samples from three osteoporotic lumbar spines were performed on the basis of X-ray microtomography (μCT) images to determine anisotropic elastic parameters as a function of bone density in the area of pedicle screw anchorage. The fabric tensor was mapped in cubic bone volumes by a 3D mean-intercept-length method. Fabric measurements resulted in a high degree of anisotropy (DA = 0.554). For the Young’s and shear moduli as a function of bone volume fraction (BV/TV, bone volume/total volume), an individually fit function was determined and high correlations were found (97.3 ≤ R2 ≤ 99.1,p < 0.005). The results suggest that the mathematical formulation for the relationship between anisotropic elastic constants and BV/TV is applicable to current μCT data of cancellous bone in the osteoporotic lumbar spine. In combination with the obtained results and findings, the developed routine allows determination of elastic constants of osteoporotic lumbar spine. Based on this, the elastic constants determined using homogenization theory can enable efficient investigation of human bone using finite element analysis (FEA).

Electric Transverse Emissivity of Sinusoidal Surfaces Determined by a Differential Method: Comparison with Approximation of Geometric Optics

We present in this paper a numerical study of the validity limit of the optics geometrical approximation in comparison with a differential method which is established according to rigorous formalisms based on the electromagnetic theory. The precedent studies show that this method is adopted to the study of diffraction by periodic rough surfaces. For periods much larger than the wavelength, the mechanism is analog to what happens in a cavity where a ray is trapped and undergoes a large number of reflections. For gratings with a period much smaller than the wavelength, the roughness essentially behaves as a transition layer with a gradient of the optical index. Such a layer reduces the reflection there by increasing the absorption. The code has been implemented for TE polarization. We determine by the two methods such as differential method and the optics geometrical approximation the emissivity of gold and tungsten cylindrical surfaces with a sinusoidal profile, for a wavelength equal to 0.55 microns. The obtained results for a fixed height of the grating allowed us to delimit the validity domain of the optic geometrical approximation for the treated cases. The emissivity calculated by the differential method and that given on the basis of the homogenization theory are satisfactory when the period is much smaller than the wavelength.

The interplay between Fano and Fabry–Pérot resonances in dual-period metagratings

We study theoretically and numerically the occurrence of Fano resonances in a metagrating made of slits with some symmetry breaking resulting in a dual period. At low frequency, a grating composed of long enough slits supports Fabry–Pérot resonances on which Fano resonances superimpose when the grating acquires dual period. The resulting spectrum exhibits flat-banded peaks interrupted by sharp dips with successions of perfect and zero transmissions. To model these scattering properties, homogenization theory is used resulting in an effective problem governing the solutions in the two, non-identical, slits, which are coupled through jump conditions at the grating interfaces. These jumps efficiently encode the effect of the evanescent field able to resonate in the radiative region due to the folding of the spoof plasmon polaritons branches. The model is validated with direct numerics and a local analysis allows us to characterize the resonant mechanisms.