Page 1 Next

Displaying 1 – 20 of 70

Showing per page

A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems

Michael Ortiz, Alexander Mielke (2008)

ESAIM: Control, Optimisation and Calculus of Variations

This work is concerned with the reformulation of evolutionary problems in a weak form enabling consideration of solutions that may exhibit evolving microstructures. This reformulation is accomplished by expressing the evolutionary problem in variational form, i.e., by identifying a functional whose minimizers represent entire trajectories of the system. The particular class of functionals under consideration is derived by first defining a sequence of time-discretized minimum problems and subsequently...

A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems

Alexander Mielke, Michael Ortiz (2007)

ESAIM: Control, Optimisation and Calculus of Variations

This work is concerned with the reformulation of evolutionary problems in a weak form enabling consideration of solutions that may exhibit evolving microstructures. This reformulation is accomplished by expressing the evolutionary problem in variational form, i.e., by identifying a functional whose minimizers represent entire trajectories of the system. The particular class of functionals under consideration is derived by first defining a sequence of time-discretized minimum problems and...

A computational approach to fractures in crystal growth

Matteo Novaga, Emanuele Paolini (1999)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In the present paper, we motivate and describe a numerical approach in order to detect the creation of fractures in a facet of a crystal evolving by anisotropic mean curvature. The result appears to be in accordance with the known examples of facet-breaking. Graphical simulations are included.

A simple and efficient scheme for phase field crystal simulation

Matt Elsey, Benedikt Wirth (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We propose an unconditionally stable semi-implicit time discretization of the phase field crystal evolution. It is based on splitting the underlying energy into convex and concave parts and then performing H-1 gradient descent steps implicitly for the former and explicitly for the latter. The splitting is effected in such a way that the resulting equations are linear in each time step and allow an extremely simple implementation and efficient solution. We provide the associated stability and error...

A two well Liouville theorem

Andrew Lorent (2005)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller. Let H = σ 0 0 σ - 1 for σ > 0 . Let 0 < ζ 1 < 1 < ζ 2 < . Let K : = S O 2 S O 2 H . Let u W 2 , 1 Q 1 0 be a C 1 invertible bilipschitz function with Lip u < ζ 2 , Lip u - 1 < ζ 1 - 1 . There exists positive constants 𝔠 1 < 1 and 𝔠 2 > 1 depending only on σ , ζ 1 , ζ 2 such that if ϵ 0 , 𝔠 1 and u satisfies the...

A Two Well Liouville Theorem

Andrew Lorent (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller.
Let H = σ 0 0 σ - 1 for σ > 0 . Let 0 < ζ 1 < 1 < ζ 2 < . Let K : = S O 2 S O 2 H . Let u W 2 , 1 Q 1 0 be a invertible bilipschitz function with Lip u < ζ 2 , Lip u - 1 < ζ 1 - 1 . 
There exists positive constants 𝔠 1 < 1 and 𝔠 2 > 1 depending only on σ, ζ 1 , ζ 2 such that if ϵ 0 , 𝔠 1 and u satisfies...

A variational problem modelling behavior of unorthodox silicon crystals

J. Hannon, M. Marcus, Victor J. Mizel (2003)

ESAIM: Control, Optimisation and Calculus of Variations

Controlling growth at crystalline surfaces requires a detailed and quantitative understanding of the thermodynamic and kinetic parameters governing mass transport. Many of these parameters can be determined by analyzing the isothermal wandering of steps at a vicinal [“step-terrace”] type surface [for a recent review see [4]]. In the case of o r t h o d o x crystals one finds that these meanderings develop larger amplitudes as the equilibrium temperature is raised (as is consistent with the statistical mechanical...

A Variational Problem Modelling Behavior of Unorthodox Silicon Crystals

J. Hannon, M. Marcus, Victor J. Mizel (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Controlling growth at crystalline surfaces requires a detailed and quantitative understanding of the thermodynamic and kinetic parameters governing mass transport. Many of these parameters can be determined by analyzing the isothermal wandering of steps at a vicinal [“step-terrace”] type surface [for a recent review see [4]]. In the case of orthodox crystals one finds that these meanderings develop larger amplitudes as the equilibrium temperature is raised (as is consistent with the statistical...

Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics

Xavier Blanc, Claude Le Bris, Frédéric Legoll (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have recently been proposed. They aim at coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We provide here a theoretical ground for such a coupling in a one-dimensional setting. We briefly study the general case of a convex energy, and next concentrate on a specific example of a nonconvex energy, the Lennard-Jones case....

Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics

Xavier Blanc, Claude Le Bris, Frédéric Legoll (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have recently been proposed. They aim at coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We provide here a theoretical ground for such a coupling in a one-dimensional setting. We briefly study the general case of a convex energy, and next concentrate on a specific example of a nonconvex energy, the Lennard-Jones case....

Approximation of a Martensitic Laminate with Varying Volume Fractions

Bo Li, Mitchell Luskin (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We give results for the approximation of a laminate with varying volume fractions for multi-well energy minimization problems modeling martensitic crystals that can undergo either an orthorhombic to monoclinic or a cubic to tetragonal transformation. We construct energy minimizing sequences of deformations which satisfy the corresponding boundary condition, and we establish a series of error bounds in terms of the elastic energy for the approximation of the limiting macroscopic deformation and...

Asymptotic behavior of solutions to an area-preserving motion by crystalline curvature

Shigetoshi Yazaki (2007)

Kybernetika

Asymptotic behavior of solutions of an area-preserving crystalline curvature flow equation is investigated. In this equation, the area enclosed by the solution polygon is preserved, while its total interfacial crystalline energy keeps on decreasing. In the case where the initial polygon is essentially admissible and convex, if the maximal existence time is finite, then vanishing edges are essentially admissible edges. This is a contrast to the case where the initial polygon is admissible and convex:...

Asymptotic behaviour for a phase-field model with hysteresis in one-dimensional thermo-visco-plasticity

Olaf Klein (2004)

Applications of Mathematics

The asymptotic behaviour for t of the solutions to a one-dimensional model for thermo-visco-plastic behaviour is investigated in this paper. The model consists of a coupled system of nonlinear partial differential equations, representing the equation of motion, the balance of the internal energy, and a phase evolution equation, determining the evolution of a phase variable. The phase evolution equation can be used to deal with relaxation processes. Rate-independent hysteresis effects in the strain-stress...

Comparative Study of a Solid Film Dewetting in an Attractive Substrate Potentials with the Exponential and the Algebraic Decay

M. Khenner (2008)

Mathematical Modelling of Natural Phenomena

We compare dewetting characteristics of a thin nonwetting solid film in the absence of stress, for two models of a wetting potential: the exponential and the algebraic. The exponential model is a one-parameter (r) model, and the algebraic model is a two-parameter (r, m) model, where r is the ratio of the characteristic wetting length to the height of the unperturbed film, and m is the exponent of h (film height) in a smooth function that interpolates the system's surface energy above and below...

Currently displaying 1 – 20 of 70

Page 1 Next