Mathematical analysis of a solution method for finite-strain holonomic plasticity of Cosserat materials

Meccanica - This article deals with the mathematical derivation and the validation over benchmark examples of a numerical method for the solution of a finite-strain holonomic (rate-independent)...     

A Variational Framework for Spectral Approximations of Kohn–Sham Density Functional Theory

We reformulate the Kohn–Sham density functional theory (KSDFT) as a nested variational problem in the one-particle density operator, the electrostatic potential and a field dual to the electron density. The corresponding functional is linear in the density operator and thus amenable to spectral representation. Based on this reformulation, we introduce a new approximation scheme, termed spectral binning, which does not require smoothing of the occupancy function and thus applies at arbitrarily low temperatures. We prove convergence of the approximate solutions with respect to spectral binning and with respect to an additional spatial discretization of the domain.     

The dynamics of transition layers in solids with discontinuous chemical potentials

We derive a two‐phase segregation model in solids under isothermal conditions where due to plastic effects the number of vacancies changes when crossing a transition layer, i.e. a reconstitutive phase transition. We show the thermodynamic correctness of the model and review the existence of weak solutions in suitable spaces. By a formal asymptotic analysis we study the dynamics of the interface and its dependence on the unsymmetric vacancy distribution. Copyright © 2006 John Wiley & Sons, Ltd.     

Non-periodic finite-element formulation of Kohn-Sham density functional theory

We present a real-space, non-periodic, finite-element formulation for Kohn-Sham density functional theory (KS-DFT). We transform the original variational problem into a local saddle-point problem, and show its well-posedness by proving the existence of minimizers. Further, we prove the convergence of finite-element approximations including numerical quadratures. Based on domain decomposition, we develop a parallel finite-element implementation of this formulation capable of performing both all-electron and pseudopotential calculations. We assess the accuracy of the formulation through selected test cases and demonstrate good agreement with the literature. We also evaluate the numerical performance of the implementation with regard to its scalability and convergence rates. We view this work as a step towards developing a method that can accurately study defects like vacancies, dislocations and crack tips using density functional theory (DFT) at reasonable computational cost by retaining electronic resolution where it is necessary and seamlessly coarse-graining far away.     

A sharp interface model for phase transitions in crystals with linear elasticity

A model describing phase transitions coupled with diffusion and linear elasticity in crystals under isothermal conditions is introduced. The elastic deformation as well as the phase parameter are obtained directly by the minimization of the free energy. After stating the model, the existence of strong solutions is proved. Copyright © 2004 John Wiley & Sons, Ltd.     

Diffusion induced segregation in the case of the ternary system sphalerite,chalcopyrite and cubanite

A mathematical model for describing natural and experimental diffusion induced segregation (DIS) in the case of a (Zn,Fe)S single crystal with three coexisting phases is derived. As main result, a new and quite general segregation principle for ternary systems is discovered where one phase has a flat free energy density and serves as catalyst for the segregation of the other two phases. The model includes also a stochastic noise term to represent fluctuations of the copper concentration. Numerical simulations in 2‐d underline the physical significance of the model and allow to make quantitative predictions. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A theory for elastic properties of single crystals with microstructure and its application to diffusion induced segregation

In this article a general theory for elastically stressed single crystals in the presence of microstructure is presented and an explicit formula for the resulting non‐linear stored mechanical energy is obtained. The optimal microstructure under applied stress is characterised and the optimal laminates are identified in 2D. The analysis is based on a sharp lower estimate of the energy that relies on relaxation. The new theory is then used to extend existing models for diffusion induced segregation (DIS) in the case of (Zn,Fe)S single crystals. Numerical simulations based on finite elements are carried out and the results are compared with former computations of the homogeneous case. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the competition of elastic energy and surface energy in discrete numerical schemes

The Γ-limit of certain discrete free energy functionals related to the numerical approximation of Ginzburg–Landau models is analysed when the distance h between neighbouring points tends to zero. The main focus lies on cases where there is competition between surface energy and elastic energy. Two discrete approximation schemes are compared, one of them shows a surface energy in the Γ-limit. Finally, numerical solutions for the sharp interface Cahn–Hilliard model with linear elasticity are investigated. It is demonstrated how the viscosity of the numerical scheme introduces an artificial surface energy that leads to unphysical solutions.