Concurrent Multiscale Computing of Deformation Microstructure by Relaxation and Local Enrichment with Application to Single-Crystal Plasticity

This paper is concerned with the effective modeling of deformation microstructures within a concurrent multiscale computing framework. We present a rigorous formulation of concurrent multiscale computing based on relaxation; we establish the connection between concurrent multiscale computing and enhanced-strain elements; and we illustrate the approach in an important area of application, namely, single-crystal plasticity, for which the explicit relaxation of the problem is derived analytically. This example demonstrates the vast effect of microstructure formation on the macroscopic behavior of the sample, e.g., on the force/travel curve of a rigid indentor. Thus, whereas the unrelaxed model results in an overly stiff response, the relaxed model exhibits a proper limit load, as expected. Our numerical examples additionally illustrate that ad hoc element enhancements, e.g., based on polynomial, trigonometric, or similar representations, are unlikely to result in any significant relaxation in general

Incompatible sets of gradients and metastability

We give a mathematical analysis of a concept of metastability induced by incompatibility. The physical setting is a single parent phase, just about to undergo transformation to a product phase of lower energy density. Under certain conditions of incompatibility of the energy wells of this energy density, we show that the parent phase is metastable in a strong sense, namely it is a local minimizer of the free energy in an $L^1$ neighbourhood of its deformation. The reason behind this result is that, due to the incompatibility of the energy wells, a small nucleus of the product phase is necessarily accompanied by a stressed transition layer whose energetic cost exceeds the energy lowering capacity of the nucleus. We define and characterize incompatible sets of matrices, in terms of which the transition layer estimate at the heart of the proof of metastability is expressed. Finally we discuss connections with experiment and place this concept of metastability in the wider context of recent theoretical and experimental research on metastability and hysteresis.Comment: Archive for Rational Mechanics and Analysis, to appea

Secure distributed detection in wireless sensor networks via encryption of sensor decisions

We consider the problem of binary hypothesis testing using a distributed wireless sensor network. Identical binary quantizers are used on the sensor\u27s observations and the outputs are encrypted using a probabilistic cipher. The third party (enemy) fusion centers are unaware of the presence of the probabilistic encipher. We find the optimal (minimum-probability-of-error) fusion rule for the ally (friendly) fusion center subject to a lower bound on the the probability of error for the third-party fusion centers. To obtain the minimum probability of error, we first prove the quasi-convexity of error probability with respect to the sensor\u27s threshold for a given cipher and show the existence of a unique positive minimum for error probability of the ally fusion center. The threshold corresponding to the minimum error-probability is evaluated numerically and the appropriate cipher that deteriorates the performance of the third-party fusion center below the required limits is obtained. Our results show that, by adjusting the sensor threshold and the encryption parameters, it is possible to achieve acceptable performance for the ally fusion center while causing significant degradation to the performance of the third party fusion center

Gradient Young measures generated by quasiconformal maps in the plane

In this contribution, we completely and explicitly characterize Young measures generated by gradients of quasiconformal maps in the plane. By doing so, we generalize the results of Astala and Faraco \cite{AstalaFaraco} who provided a similar result for quasiregular maps and Bene\v{s}ov\'a and Kru\v{z}\'ik \cite{bbmk2013} who characterized Young measures generated by gradients of bi-Lipschitz maps. Our results are motivated by non-linear elasticity where injectivity of the functions in the generating sequence is essential in order to assure non-interpenetration of matter