Dimension reduction for thin films prestrained by shallow curvature

We are concerned with the dimension reduction analysis for thin three-dimensional elastic films, prestrained via Riemannian metrics with weak curvatures. For the prestrain inducing the incompatible version of the Föppl–von Kármán equations, we find the Γ -limits of the rescaled energies, identify the optimal energy scaling laws, and display the equivalent conditions for optimality in terms of both the prestrain components and the curvatures of the related Riemannian metrics. When the stretching-inducing prestrain carries no in-plane modes, we discover similarities with the previously described shallow shell models. In higher prestrain regimes, we prove new energy upper bounds by constructing deformations as the Kirchhoff–Love extensions of the highly perturbative, Hölder-regular solutions to the Monge–Ampere equation obtained by means of convex integration.

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On the Kirchhoff-Love Hypothesis (Revised and Vindicated)

Journal of Elasticity ◽

10.1007/s10659-021-09819-7 ◽

2021 ◽

Author(s):

Olivier Ozenda ◽

Epifanio G. Virga

Keyword(s):

Free Energy ◽

Dimension Reduction ◽

Three Dimensional ◽

Energy Functional ◽

The Other ◽

Variational Model ◽

Two Dimensional ◽

Elastic Plates ◽

Kinematic Constraint ◽

Free Energy Functional

AbstractThe Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its paternity has been questioned, and recent rigorous dimension-reduction tools (based on standard $\varGamma $ Γ -convergence) have proven to be incompatible with it. We find that an appropriately revised version of the Kirchhoff-Love hypothesis is a valuable means to derive a two-dimensional variational model for elastic plates from a three-dimensional nonlinear free-energy functional. The bending energies thus obtained for a number of materials also show to contain measures of stretching of the plate’s mid surface (alongside the expected measures of bending). The incompatibility with standard $\varGamma $ Γ -convergence also appears to be removed in the cases where contact with that method and ours can be made.

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Small covers, infra-solvmanifolds and curvature

Forum Mathematicum ◽

10.1515/forum-2013-0084 ◽

2015 ◽

Vol 27 (5) ◽

Cited By ~ 1

Author(s):

Shintarô Kuroki ◽

Mikiya Masuda ◽

Li Yu

Keyword(s):

Sectional Curvature ◽

Riemannian Metrics ◽

Flat Torus ◽

Combinatorial Structures ◽

Simple Polytope ◽

Simple Polytopes ◽

Small Cover ◽

Equivalent Conditions

AbstractIt is shown that a small cover (resp. real moment-angle manifold) over a simple polytope is an infra-solvmanifold if and only if it is diffeomorphic to a real Bott manifold (resp. flat torus). Moreover, we obtain several equivalent conditions for a small cover to be homeomorphic to a real Bott manifold. In addition, we study Riemannian metrics on small covers and real moment-angle manifolds with certain conditions on the Ricci or sectional curvature. We will see that these curvature conditions put very strong restrictions on the topology of the corresponding small covers and real moment-angle manifolds and the combinatorial structures of the underlying simple polytopes.

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Simulations of the neutral structure within the dusk side aurora

Annales Geophysicae ◽

10.5194/angeo-24-2519-2006 ◽

2006 ◽

Vol 24 (10) ◽

pp. 2519-2532

Author(s):

H. F. Parish ◽

L. R. Lyons

Keyword(s):

Electric Fields ◽

Three Dimensional ◽

Meridional Wind ◽

Recovery Phase ◽

Limited Area ◽

Neutral Winds ◽

E Region ◽

Linear Interactions ◽

Simulated Wind ◽

Equivalent Conditions

Abstract. Observations of neutral winds from rocket release experiments within the premidnight and postmidnight substorm recovery phase aurora, show very large E-region neutral winds of several hundred m/s, where winds measured on the dusk side are even larger than those on the dawn side. These large winds are also associated with strong shears, and there is evidence that some of the regions below these shears may be unstable. The mechanisms which generate this strong vertical structure are not well understood. It is also not known whether the acceleration conditions in the pre and post midnight sectors of the aurora may produce significantly different neutral responses on the dawn and dusk sides. Simulations have been performed using a three-dimensional high resolution limited area thermosphere model to try to understand the neutral structure within the dawn and dusk side aurora. When simulations are performed using auroral forcing alone, for equivalent conditions within the dawn and dusk sectors, differences are found in the simulated response on each side. When measured values of auroral forcing parameters, and background winds and tides consistent with recent observations, are used as model inputs, some of the main features of the zonal and meridional wind observations are reproduced in the simulations, but the magnitude of the peak zonal wind around 140 km tends to be too small and the maximum meridional wind around 130 km is overestimated. The winds above 120 km altitude are found to be sensitive to changes in electric fields and ion densities, as was the case for the dawn side, but the effects of background winds and tides on the magnitudes of the winds above 120 km are found to be relatively small on the dusk side. The structure below 120 km appears to be related mainly to background winds and tides rather than auroral forcing, as was found in earlier studies on the dawn side, although the peak magnitudes of simulated wind variations in the 100 to 120 km altitude range are smaller than those observed. The source of the strong shears measured around 110 km altitude on the dusk side is uncertain, but may be related to different kinds of oscillations, such as gravity waves, non migrating semidiurnal tides, or secondary oscillations produced by non linear interactions between waves.

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Extraction of mechanical properties of materials through deep learning from instrumented indentation

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.1922210117 ◽

2020 ◽

Vol 117 (13) ◽

pp. 7052-7062 ◽

Cited By ~ 14

Author(s):

Lu Lu ◽

Ming Dao ◽

Punit Kumar ◽

Upadrasta Ramamurty ◽

George Em Karniadakis ◽

...

Keyword(s):

Mechanical Properties ◽

Deep Learning ◽

Scaling Laws ◽

Three Dimensional ◽

Instrumented Indentation ◽

Multilayered Structures ◽

Mechanical Properties Of Materials ◽

Properties Of Materials ◽

Indentation Problem ◽

And Function

Instrumented indentation has been developed and widely utilized as one of the most versatile and practical means of extracting mechanical properties of materials. This method is particularly desirable for those applications where it is difficult to experimentally determine the mechanical properties using stress–strain data obtained from coupon specimens. Such applications include material processing and manufacturing of small and large engineering components and structures involving the following: three-dimensional (3D) printing, thin-film and multilayered structures, and integrated manufacturing of materials for coupled mechanical and functional properties. Here, we utilize the latest developments in neural networks, including a multifidelity approach whereby deep-learning algorithms are trained to extract elastoplastic properties of metals and alloys from instrumented indentation results using multiple datasets for desired levels of improved accuracy. We have established algorithms for solving inverse problems by recourse to single, dual, and multiple indentation and demonstrate that these algorithms significantly outperform traditional brute force computations and function-fitting methods. Moreover, we present several multifidelity approaches specifically for solving the inverse indentation problem which 1) significantly reduce the number of high-fidelity datasets required to achieve a given level of accuracy, 2) utilize known physical and scaling laws to improve training efficiency and accuracy, and 3) integrate simulation and experimental data for training disparate datasets to learn and minimize systematic errors. The predictive capabilities and advantages of these multifidelity methods have been assessed by direct comparisons with experimental results for indentation for different commercial alloys, including two wrought aluminum alloys and several 3D printed titanium alloys.

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On inertial-range scaling laws

Journal of Fluid Mechanics ◽

10.1017/s0022112096001279 ◽

1996 ◽

Vol 306 ◽

pp. 167-181 ◽

Cited By ~ 29

Author(s):

John C. Bowman

Keyword(s):

Steady State ◽

High Resolution ◽

Scaling Laws ◽

State Energy ◽

Three Dimensional ◽

Inertial Range ◽

Energy Spectra ◽

Two Dimensional ◽

Unified Framework ◽

Asymptotic Nature

Inertial-range scaling laws for two- and three-dimensional turbulence are re-examined within a unified framework. A new correction to Kolmogorov's k−5/3 scaling is derived for the energy inertial range. A related modification is found to Kraichnan's logarithmically corrected two-dimensional enstrophy-range law that removes its unexpected divergence at the injection wavenumber. The significance of these corrections is illustrated with steady-state energy spectra from recent high-resolution closure computations. Implications for conventional numerical simulations are discussed. These results underscore the asymptotic nature of inertial-range scaling laws.

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Multiple‐input–multiple‐output radar super‐resolution three‐dimensional imaging based on a dimension‐reduction compressive sensing

IET Radar Sonar & Navigation ◽

10.1049/iet-rsn.2015.0345 ◽

2016 ◽

Vol 10 (4) ◽

pp. 757-764 ◽

Cited By ~ 19

Author(s):

Xiaowei Hu ◽

Ningning Tong ◽

Yongshun Zhang ◽

Guoping Hu ◽

Xingyu He

Keyword(s):

Compressive Sensing ◽

Dimension Reduction ◽

Multiple Input Multiple Output ◽

Three Dimensional ◽

Super Resolution ◽

Three Dimensional Imaging ◽

Multiple Input ◽

Input Multiple Output

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Three-dimensional scaling laws of cetacean propulsion characterize the hydrodynamic interplay of flukes' shape and kinematics

Journal of The Royal Society Interface ◽

10.1098/rsif.2019.0655 ◽

2020 ◽

Vol 17 (163) ◽

pp. 20190655 ◽

Cited By ~ 1

Author(s):

Fatma Ayancik ◽

Frank E. Fish ◽

Keith W. Moored

Keyword(s):

Aspect Ratio ◽

Scaling Laws ◽

Three Dimensional ◽

Force Production ◽

Numerical Data ◽

Added Mass ◽

Cetacean Species ◽

Propulsive Efficiency ◽

Fluid Environment ◽

Pitch Ratio

Cetaceans convert dorsoventral body oscillations into forward velocity with a complex interplay between their morphological and kinematic features and the fluid environment. However, it is unknown to what extent morpho-kinematic features of cetaceans are intertwined to maximize their efficiency. By interchanging the shape and kinematic variables of five cetacean species, the interplay of their flukes morpho-kinematic features is examined by characterizing their thrust, power and propulsive efficiency. It is determined that the shape and kinematics of the flukes have considerable influence on force production and power consumption. Three-dimensional heaving and pitching scaling laws are developed by considering both added mass and circulatory-based forces, which are shown to closely model the numerical data. Using the scaling relations as a guide, it is determined that the added mass forces are important in predicting the trend between the efficiency and aspect ratio, however, the thrust and power are driven predominately by the circulatory forces. The scaling laws also reveal that there is an optimal dimensionless heave-to-pitch ratio h * that maximizes the efficiency. Moreover, the optimal h * varies with the aspect ratio, the amplitude-to-chord ratio and the Lighthill number. This indicates that the shape and kinematics of propulsors are intertwined, that is, there are specific kinematics that are tailored to the shape of a propulsor in order to maximize its propulsive efficiency.

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ON THE REGULARITY OF THREE-DIMENSIONAL ROTATING EULER–BOUSSINESQ EQUATIONS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s021820259900049x ◽

1999 ◽

Vol 09 (07) ◽

pp. 1089-1121 ◽

Cited By ~ 16

Author(s):

A. BABIN ◽

A. MAHALOV ◽

B. NICOLAENKO

Keyword(s):

Initial Data ◽

Three Dimensional ◽

Fluid Flows ◽

Boussinesq Equations ◽

Stable Stratification ◽

Primitive Equations ◽

Time Interval ◽

Regular Solutions ◽

Time Intervals ◽

Long Time

The 3-D rotating Boussinesq equations (the "primitive" equations of geophysical fluid flows) are analyzed in the asymptotic limit of strong stable stratification. The resolution of resonances and a nonstandard small divisor problem are the basis for error estimates for such fast singular oscillating limits. Existence on infinite time intervals of regular solutions to the viscous 3-D "primitive" equations is proven for initial data in Hα, α≥ 3/4. Existence on a long-time interval T*of regular solutions to the 3-D inviscid equations is proven for initial data in Hα, α > 5/2 (T*→∞ as the frequency of gravity waves →∞).

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