Modellierung und Simulation des effektiven Verhaltens von
Grenzflächen in Metalllegierungen.
Dissertation, University Bonn, 2007.
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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.