Fully and semi-automated shape differentiation in NGSolve

In this paper, we present a framework for automated shape differentiation in the finite element software . Our approach combines the mathematical Lagrangian approach for differentiating PDE-constrained shape functions with the automated differentiation capabilities of . The user can decide which degree of automatisation is required, thus allowing for either a more custom-like or black-box–like behaviour of the software. We discuss the automatic generation of first- and second-order shape derivatives for unconstrained model problems as well as for more realistic problems that are constrained by different types of partial differential equations. We consider linear as well as nonlinear problems and also problems which are posed on surfaces. In numerical experiments, we verify the accuracy of the computed derivatives via a Taylor test. Finally, we present first- and second-order shape optimisation algorithms and illustrate them for several numerical optimisation examples ranging from nonlinear elasticity to Maxwell’s equations.

Comparing shape along growth trajectories in two marine snail ecotypes of Littorina saxatilis: a test of evolution by paedomorphosis

ABSTRACT Two sympatric ecotypes (‘crab’ and ‘wave’) of Littorina saxatilis are adapted to different microhabitats. It has been claimed, based on the comparison of proteomic differentiation across ontogeny, that the wave ecotype may have evolved by paedomorphosis from an ancestor more similar to the crab ecotype. Here, we test the paedomorphosis hypothesis at a morphological level by comparing crab and wave specimens from two localities using the pattern of shell shape differentiation across ontogeny. The results obtained show a pattern of shell shape differentiation similar to the one observed in previous proteomic studies, but such a pattern could be caused by different modes of evolution, and not necessarily by paedomorphosis. This study emphasizes that in addition to studying the pattern of differentiation, the direction of the evolutionary change across ontogeny has to be analysed before conclusions can be drawn on particular developmental modes of evolution.

A new route for the shape differentiation of cesium lead bromide perovskite nanocrystals with near-unity photoluminescence quantum yield

Engineering the chemical approach for obtaining cesium lead bromide perovskite nanocrystals allows their shape modulation while maintaining their near-unity photoluminescence quantum yield.

Linear stability of inviscid vortex rings to axisymmetric perturbations

We consider the linear stability to axisymmetric perturbations of the family of inviscid vortex rings discovered by Norbury (J. Fluid Mech., vol. 57, 1973, pp. 417–431). Since these vortex rings are obtained as solutions to a free-boundary problem, their stability analysis is performed using recently developed methods of shape differentiation applied to the contour-dynamics formulation of the problem in the three-dimensional axisymmetric geometry. This approach allows us to systematically account for the effects of boundary deformations on the linearized evolution of the vortex ring. We investigate the instantaneous amplification of perturbations assumed to have the same the circulation as the vortex rings in their equilibrium configuration. These stability properties are then determined by the spectrum of a singular integro-differential operator defined on the vortex boundary in the meridional plane. The resulting generalized eigenvalue problem is solved numerically with a spectrally accurate discretization. Our results reveal that while thin vortex rings remain neutrally stable to axisymmetric perturbations, they become linearly unstable to such perturbations when they are sufficiently ‘fat’. Analysis of the structure of the eigenmodes demonstrates that they approach the corresponding eigenmodes of Rankine’s vortex and Hill’s vortex in the thin-vortex and fat-vortex limit, respectively. This study is a stepping stone on the way towards a complete stability analysis of inviscid vortex rings with respect to general perturbations.

Variability of Otolith Morphology and Morphometry in Eight Juvenile Fish Species in the Coastal Eastern Adriatic

Sagittae otoliths of eight juvenile species: Boops boops, Diplodus vulgaris, Diplodus puntazzo, Sarpa salpa (family Sparidae), Liza ramada, Liza aurata (family Mugilidae), Atherina boyeri, Atherina hepsetus (family Atherinidae) were analysed and compared using descriptive morphological characters and morphometric indices. The noticeable differences among the otoliths of the investigated species are in their overall shape, margins (i.e. irregular, sinuate or crenate) and anterior region. Otolith shape varied from elliptic to pentagonal in sparids, elliptic to rectangular in mugilids and elliptic in two atherinids. Aspect ratio (OW/OL), ratio of the sulcus length occupied by the cauda length (CL/SL) and ratio of the sulcus length occupied by the ostium length (OSL/SL) were calculated for all species. The otolith contour was described using wavelets. The Canonical Analysis of Principal Coordinates (CAP) gave an overview of the otolith shape differentiation between eight juveniles. Using the Wavelet coefficients, the first principal component (CAP1) explained 58.1% of the variation among species and the second principal component (CAP2) 25.2%.