Johann von Neumann Lectures
Modelling & Simulation of Microstructure Evolution
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Monday, July 11th, 2005, 9:30 a.m
Friday, July 15th, 2005, 16:00 p.m
Humboldt-Universität zu Berlin, Institute of Mathematics
Rudower Chaussee 25, D-10099 Berlin, Germany
The summer school 2005 “Johann von Neumann Lectures” at the Humboldt University Berlin shall provide graduate and postgraduate students of different backgrounds with important techniques to describe microstructures, the evolution of different morphologies, the motion of multiple interfaces and to study phase transition phenomena. The considered topics are modeling, computational methods, numerical analysis, simulations and applications in materials science.
Description of the topics:
· Phase-field modeling for solidification:
The derivation of a general class of diffuse interface models for microstructure evolution such as crystal growth and solidification in complex alloy systems is discussed. The set of parabolic differential equations is derived by minimization of a non-convex Ginzburg-Landau type entropy functional in a thermodynamically consistent way. The fairly new modeling methodology is capable to simultaneously describe diffusion processes of multiple components, phase transitions between multiple phases, the formation of a polycrystalline grain structure and the development of the temperature field. We relate the model to classical sharp interface models by formally matched asymptotic expansions.
· Multiscale modeling for grain growth:
For most physical processes, scientists and engineers are interested in gaining understanding (or, even better, control) not on one but rather on many scales of detail. The same process can be considered and described as a microscopic (in form of local atomic interactions), as a macroscopic (this is when some global properties are an object/ subject of observation/study), or as a mesoscale (a very rich scale that accommodates everything in between the first two). It is very natural to assume that all scales are connected. Moreover, it is reasonable to expect that a good understanding of the process on the fine detailed scale should be sufficient for an accurate prediction of its features on coarser (less detailed) scales. In practice, however, such connection is often difficult to establish. In this lecture, we will give an example of a multiscale modeling by presenting an hierarchy of models for grain growth, the process of crystall evolution in polycrystalline materials. We will start by introducing a mesoscale model of curvature driven growth. This is a detailed PDE model that describes grain growth as an evolution of grain boundaries. The first part of the lecture will be devoted to the description of this model, both mathematical and programming aspects of it. The second part of the lecture will focus on how to use the results of the PDE simulations (i.e., statistical information observed there) in order to construct other reliable but much less expensive (computationally) models of grain growth that show the same average properties as the initial detailed model.
· Computational methods:
A collection of numerical methods such as finite difference and finite element methods is introduced aiming to study the motion of multiple interfaces in microstructures at different time and length scales by numerical computations. The investigation of multiscale aspects and of the interaction of different physical fields during the microstructure evolution requires the use of efficient computational algorithms optimized with respect to computation time and memory usage. Modern parallelization techniques such as MPI and OpenMP for high performance computing on personal computer clusters, adaptive non-uniform mesh discretization as well as multigrid methods will be discussed. Furthermore, we give an introductory and practical example of coding a prototype non-linear, parabolic PDE.
· Numerical analysis:
The lectures on numerical analysis are addressed to minimization problems of free energy density functionals modeling phase transitions in solids and fluids. In particular, we will consider model examples such as the non-negative and non-convex quartic polynomial and related forms to study oscillatory microstructure phenomena. Quantities such as the macroscopic displacement field, the Young measure for the description of oscillations and the stress field are identified and the corresponding discrete counterparts are selected.
· Multigrid methods:
Developing fast numerical solvers is one of the main challenges of computational mathematics. In this lecture, we will discuss one of the recent (it is only about thirty years old) and efficient trends in computing - multigrid methods. These methods have been used to solve numerous applications arising in differential and integral equations, inverse problems, pattern recognition, Monte-Carlo simulations, and many others. We, however, will concentrate on what happened to be the first application of multigrid methods: elliptic partial differential equations. Local Fourier analysis will be used to give a motivation to use multigrid methods. We will discuss main components of a standard multigrid solver (called V-cycle), such as relaxation, restriction, and interpolation. We will compare the costs of the multigrid solvers with the ones of the traditional one-grid methods.
· Multigrid algorithms for the Helmholtz equation:
In this lecture, we will develop a multigrid algorithm for solving the Helmholtz equation accompanied by radiation boundary condition and considered on large computational domains. This equation describes a wide range of processes arising in acoustics, electromagnetics, and other areas that involve wave propagation. It also gives an excellent example of the problem for which standard multigrid methods (such as a V-cycle we discussed in the previous lecture) fail. This happens for several reasons, including special type of discretization error (phase errors) typical for problems that describe wave propagation, non-smooth form of the slow to converge components, global boundary conditions. Using Fourier analysis, we will identify the problematic (slow to converge) components and will establish that they need a special representation and treatment. We will also find out that such representation is beneficial for an efficient introduction of the radiation boundary conditions. By the end of the lecture we will develop an algorithm that has the same efficiency as the multigrid V-cycle for the Laplace equation at the very competitive costs.
· Simulations and applications to real materials:
In real materials, e.g. in metallic alloys, the formation of microstructures involves different length and time scales and can experimentally not in-situ be observed due to the high temperatures and the non-transparency. The characteristics of the microstructure such as grain size distributions or dendritic arm spacings are strongly responsible for the resulting mechanical properties of the materials. Hence, computer modeling provides valuable information in order to gain insights into the fundamental mechanisms and influences of crystallization processes. In a series of lectures, we show a variety of applications of the developed 3D simulator to real metallic alloys. The presented simulation results visualize the motion of multiple interfaces (phase and grain boundaries) in complex alloy systems. In particular, phenomena such as anisotropic curvature flow, grain growth and coarsening are described. Another special emphasis of the computations is the modeling of phase transformations and solidification processes in multi-component alloys. The specific phase diagrams of the alloys are incorporated in the diffuse interface model via the free energies. Within this context, complex dendritic and eutectic structures are simulated in 2D and 3D.
Cosponsored by the:
German Research Foundation (DFG) within the Priority Research Program 1095: “Analysis, Modelling and Simulation of Multiscale Problems”
The participation is free of charge.
For registration, please send an email to Mrs. Ramona Klaass-Thiele firstname.lastname@example.org at your earliest convenience, but at the latest until June 30th, 2005.
For members of the Priority Program "Analysis, Modeling and Simulation of
Multiscale Problems", the travel costs are covered by the coordinator fund.
Other participants are welcome, but they shall cover their expenses on their