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A matrix framework is developed for single and multispan micro-cantilevers Timoshenko beam models of use in atomic force microscopy (AFM). They are considered subject to general forcing loads and boundary conditions for modeling tipsample interaction. Surface effects are considered in the frequency analysis of supported and cantilever microbeams. Extensive use is made of a distributed matrix fundamental response that allows to determine forced responses through convolution and to absorb non-homogeneous boundary conditions. Transients are identified from intial values of permanent responses. Eigenanalysis for determining frequencies and matrix mode shapes is done with the use of a fundamental matrix response that characterizes solutions of a damped second-order matrix differential equation. It is observed that surface effects are influential for the natural frequency at the nanoscale. Simulations are performed for a bi-segmented free-free beam and with a micro-cantilever beam actuated by a piezoelectric layer laminated in one side.
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Bayesian Analysis for Robust Synthesis of Nanostructures

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Nanomaterials, because of their unique properties such as extremely small size and increased ratio of surface area to volume, have a great potential in many industrial applications that involve electronics, sensors, solar cells, super-strong materials, coatings, drug delivery, and nanomedicine. They have the potential also to improve the environment by direct applications of these materials to detect, prevent and remove pollutants. While nanomaterials present seemingly limitless possibilities, they bring with them new challenges. Among them is the precise control of the morphology of nanomaterials, which is extremely critical to the development of advanced nanodevices with various functionalities. The one-dimensional nanostructures of Cadmium Selenide (CdSe) have been found to represent morphologies of nanowires, nanobelts, and nanosaws, however, their synthesis is by trial and error. Predictive modeling and control methods are essential to process yield and productivity improvement. The process yield (response) is a vector whose elements correspond to the number of appearances of the different types of nanostructures, namely nanosaws, nanowires, and nanobelts. The goal in this paper is to apply existing Bayesian methodologies to describe the growths of these nanostructures in terms of process variables and to predict the probability of transition from one nanostructure to another when changes are made to one or more process variables. We also propose a Bayesian algorithm to identify the optimal process conditions that maximize the predicted probability of each type of nanostructure.
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This editorial provides an overview of both fundamental and applied research areas covered by the journal of Nanoscale Systems: Mathematical Modeling, Theory and Applications (NanoMMTA), as well as of articles published in the journal inaugural volume. The unique feature of NanoMMTA is its focus on the interface between the study, development, and application of systems at the nanoscale with theoretical methods and experimental techniques on the one hand and mathematical, statistical, and computational tools on the other. NanoMMTA is the first international, interdisciplinary, peer-reviewed journal focusing specifically on this interface. This emerging multidisciplinary field at the interface of mathematical modeling, nanoscience and nanotechnology includes applications and advancements of these tools in all of the disciplines facing the challenges associated with the nanoscale systems.
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