@PHDTHESIS{Le07a,
author = {Martin Lenz},
title = {{M}odellierung und {S}imulation des effektiven {V}erhaltens von {G}renzfl{\"{a}}chen
in {M}etalllegierungen},
school = {University Bonn},
year = {2007},
type = {Dissertation},
abstract = {This thesis is concerned with the modeling and the numerical simulation
of phase transitions during the Ostwald ripening of metal alloys
and in magnetic shape memory materials. The phase transition is modeled
on a continuum level with methods of elasticity theory. The coarsening
of particles in a metal alloy after spinodal decomposition can be
seen as a gradient flow: The set of particles moves on the manifold
of all possible particle configurations in the direction of steepest
descent of an energy functional containing interface energy and elasticity,
with respect to a metric tensor describing the diffusion mechanism.
The restriction of this evolution onto the submanifold of rectangular
particles aligned to the coordinate axes, as they are preferred by
the anisotropy of the elasticity tensor, gives a reduced model that
describes the evolution of such particles. The numerical simulation
of both models employs the boundary element method. The integral
operators occurring are approximated by hierarchical matrices, this
approximation also gives an appropriate preconditioner. To avoid
the coupling of the time step size to the side length of the smallest
particle, one uses localized timesteps close to small particles,
where the screening effect makes it possible to restrict to small
neighbourhoods of the respective particle. In this way one constructs
an efficient method to simulate both models; in the reduced model
accordingly the simulation of larger particle ensembles is possible.
Comparative computations verify that the reduced model reproduces
many important qualitative and quantitative properties of the full
model. Magnetic shape memory materials can be modeled on a continuum
scale using a combination of elasticity and micromagnetism. Here,
a discrete phase parameter couples the variants of the elastic strain
to the magnetic anisotropy. The anisotropy prefers a magnetization
in the direction of contraction. This model can be applied to the
description of several types of microstructured material: composites
with a non-magnetic background matrix and polycrystalline structures.
To compute the effective behaviour of the micro structure, one considers
cell problems in the spirit of homogenization theory. The numerical
solution of these cell problems uses again the boundary element method,
here embedded in a descent algorithm for energy minimization. Thereby
the influence of parameters of the microscopic structure of the material,
such as form, distribution and shape of particles or the elasticity
of the background matrix, on the macroscopic behaviour, especially
the observed strain and the work output, can be quantified.},
pdf = {http://numod.ins.uni-bonn.de/research/papers/public/Le07a.pdf},
url = {http://nbn-resolving.de/urn:nbn:de:hbz:5N-12741}
}