Publication

Analysis Of An Energy-Based Atomistic/Continuum Approximation Of A Vacancy In The 2D Triangular Lattice

2013
Journal Articles

Abstract

We present an a priori error analysis of a practical energy based atomistic/continuum coupling method (A. V. Shapeev, Multiscale Model. Simul., 9(3): 905-932, 2011) in two dimensions, for finite-range pair-potential interactions, in the presence of vacancy defects. We establish first-order consistency and stability of the method, from which we obtain a priori error estimates in the H-1-norm and the energy in terms of the mesh size and the "smoothness" of the atomistic solution in the continuum region. From these error estimates we obtain heuristics for an optimal choice of the atomistic region and the finite element mesh, as well as convergence rates in terms of the number of degrees of freedom. Our analytical predictions are supported by extensive numerical tests.

Official source

https://infoscience.epfl.ch/entities/publication/6168b934-f8d0-41f5-98ba-fa97480e33d9

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2013
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/6168b934-f8d0-41f5-98ba-fa97480e33d9

About this result

Related concepts (32)

Mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the actual value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive (and not zero) is because of randomness or because the estimator does not account for information that could produce a more accurate estimate.

Reduced chi-squared statistic

In statistics, the reduced chi-square statistic is used extensively in goodness of fit testing. It is also known as mean squared weighted deviation (MSWD) in isotopic dating and variance of unit weight in the context of weighted least squares. Its square root is called regression standard error, standard error of the regression, or standard error of the equation (see ) It is defined as chi-square per degree of freedom: where the chi-squared is a weighted sum of squared deviations: with inputs: variance , observations O, and calculated data C.

Vacancy defect

In crystallography, a vacancy is a type of point defect in a crystal where an atom is missing from one of the lattice sites. Crystals inherently possess imperfections, sometimes referred to as crystallographic defects. Vacancies occur naturally in all crystalline materials. At any given temperature, up to the melting point of the material, there is an equilibrium concentration (ratio of vacant lattice sites to those containing atoms). At the melting point of some metals the ratio can be approximately 1:1000.

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Analysis Of An Energy-Based Atomistic/Continuum Approximation Of A Vacancy In The 2D Triangular Lattice

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