@PhDThesis{	  Arndt:2004b,
  author	= {M.~Arndt},
  title		= {Upscaling from Atomistic Models to Higher Order Gradient
		  Continuum Models for Crystalline Solids},
  school	= {Institute for Numerical Simulation, University of Bonn},
  year		= {2004},
  annote	= {IAMdiss,thesis,C4,multi},
  type		= {Dissertation},
  pdf		= {http://wissrech.ins.uni-bonn.de/research/pub/arndt/diss.pdf}
		  ,
  abstract	= {In this work a new upscaling scheme for the derivation of
		  a continuum mechanical model from an atomistic model for
		  crystalline solids is developed. The scheme, called the
		  inner expansion technique, is based on a Taylor series
		  expansion of the deformation function and leads to a
		  continuum mechanical model which involves higher order
		  derivatives. It provides an approximation of the atomistic
		  model within the quasi-continuum regime and allows to
		  capture the microscopic material properties and the
		  discreteness effects of the underlying atomistic system up
		  to an arbitrary order.
		  
		  The quality of approximation is investigated for the model
		  problem of an atomic chain with different types of
		  potentials, including many-body potentials. The outcome of
		  the inner expansion technique is numerically compared to
		  other upscaling techniques, namely the classical
		  thermodynamic limit and the direct expansion technique. It
		  is shown that our technique carries over certain properties
		  such as convexity from the atomistic to the continuum
		  mechanical level, which results in well-posed problems on
		  the continuum mechanical level. Furthermore, macroscopic
		  approximation techniques are discussed to reduce the
		  complexity of the continuum model.
		  
		  The upscaling technique is applied to the Stillinger-Weber
		  potential for crystalline silicon and to the potential
		  given by the Embedded-Atom Method (EAM) for shape memory
		  alloys (SMA). Numerical simulations of the dynamic response
		  of a silicon crystal and of one-way and two-way SMA
		  micro-actuators are performed.}
}