Computational Homology and Materials Science Workshop

Georgia Tech, Atlanta, GA

February 2-4, 2006

Abstracts

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Donald Estep, Fast Deterministic Methods for Ascertaining the Evolution of Uncertain Parameters in Differential Equations

A very common problem in science and engineering is the determination of the effects of uncertainty or variation in parameters and data on the output of a nonlinear operator. For example, such variations may describe the effect of experimental error or may arise as part of a sensitivity analysis of the model. The Monte-Carlo Method is a widely used tool for understanding such effects that employs random sampling of the input space in order to describe the output. It is a robust and easily implemented tool. Unfortunately, it generally requires sampling the operator very many times at a significant cost. In this talk, I will present an alternative approach for ascertaining the effects of variations and uncertainty in parameters in a differential equation that is based on techniques borrowed from a posteriori error analysis for finite element methods. The generalized Green's function is used to describe how variation propagates into the solution around localized points in the parameter space. This information can be used either to create a higher order method or to create an adaptive sampling procedure. Both approaches are generally orders of magnitude faster than Monte-Carlo methods in a variety of situations.

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Edwin R. Fuller, Jr., Microstructure Response Isosurfaces and Their Metrics

The response of a homogeneous medium to the gradient of an applied field is a uniform flux, corresponding to that field (for example, a uniform stress in response to a strain, or a uniform heat flow in response to a gradient in temperature). When a medium is heterogeneous (i.e., it has a microstructure), the response is likewise heterogeneous, and is a characteristic of the microstructure. For example, when polycrystalline materials with crystalline thermal expansion anisotropy are cooled (or heated), producing an internal thermal strain, internal residual stresses develop within the microstructure. Moreover, these stresses can develop a network structure, which has a length-scale encompassing many grains. As recently demonstrated, the length scale of this network structure with respect to the grain size depends strongly upon the intergranular misorientation distribution function (MDF), but is also influenced by the grain orientation distribution function (ODF). This phenomenon is readily apparent in two-dimensional, microstructure-based finite-element simulations, but is not so clear in similar three-dimensional simulations. To elucidate the microstructure-induced response, response isosurfaces are generated by defining surfaces at specific threshold levels of the response. The resulting distribution of residual stresses (or heat flux for a thermal field) throughout the microstructure is considered as a geometric object itself: a microstructure response isosurface. These microstructure response isosurfaces provide new information regarding the microstructure. Computational techniques, such as computing topological invariants of homology groups (e.g., Betti numbers) and the fractal character, are being explored for characterizing and quantifying these microstructure response isosurfaces.

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Marcio Gameiro Homological Characterization of Patterns in Phase Separation and Spiral Defect Chaos

We use homology to characterize patterns arising in simulations of phase separation processes in compound materials as well as patterns arising in the weakly turbulent state of spiral defect chaos in experiments on Rayleigh-Bénard convection.

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Arun M. Gokhale, Utility of Metric and Topological Microstructural Attributes for Computer Simulations of Realistic Microstructures: Implications for Materials Design

Current methodologies for microstructure simulations mostly involve idealized simple particle/feature shapes; uniform-random spatial distribution of microstructural features; and isotropic feature orientations. Further, the simulated microstructures do not mimic the topology of the corresponding real microstructures. On the other hand, the corresponding "real" microstructures often have complex feature shapes/morphologies; non-random/non-uniform spatial distributions; complex topological connectivity, and partially anisotropic feature orientations. Consequently, such simulations do not capture these aspects of microstructural reality. In this contribution, we present a methodology that enables simulations of "realistic" microstructures where metric and topologicstr microstructural attributes, feature shapes/morphologies, spatial arrangement, and feature orientations are statistically similar to those in the corresponding real microstructures. The methodology as applied for simulations of numerous microstructures such that simulated microstructures have the same complex particle morphologies, volume fraction, size distribution, spatial clustering, and anisotropy of the particles as in the corresponding real microstructures. A small set of simulation parameters are used for generation of these microstructures, which are then correlated to the process parameters. These correlations permit generation of a set of "virtual" microstructures that cover a wide range of process conditions. The virtual microstructures can be then implemented in computational models of materials behavior to predict the properties of the "virtual" materials. The methodology is useful for design of new materials as well as for optimizing the properties of existing materials.

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Yasumasa Nishiura, Entropy and Sensitivity of Particle Patterns in Dissipative Systems

Though typically observed at many stages in biological systems, self-organization is ubiquitous in nature over wide range of spectrum in space and in time. Traditional knowledge defines it as an organizing process of matter without human manipulation. However, self-organization is no longer limited to the issue of pure science, but of engineering also. Definitely, this trend was triggered when the self-assembly of tailor-made molecules was highlighted as one of promising technologies in so-called bottom-up nanotechnology. On the other hand, researchers' interests have been focused on a novel approach for self-organization of hierarchy, where new initial and boundary conditions emerge in the course of non-equilibrium processes in an open system. In this talk, we discuss two things: first we present our numerical results on the self-organization process of 2-dimensional patterns composed of many spatially localized dots and lines. We deal with the reversible version of the Gray-Scott model that generates both the Turing structures and chaotic patterns. The model enables us to calculate the important quantity of modern thermodynamics, the rate of entropy production, in the course of hierarchic pattern formation in the reaction-diffusion system. Results are discussed on the entropy production rate in the motion of components (dots or lines), dynamics of the entire system, and pattern selection in the system. Second we discuss how the moving spot behaves when there occurs heterogeneity in its environments, namely when the distribution of some kinetic coefficients or diffusivities depend on the space. It turns out that spots are very sensitive to the tiny change of environments and show a variety of dynamics depending on special profiles of environments. The first part of my talk is a joint work with H. Mahara, T. Yamaguchi, and M. Shimomura.

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Gregory S. Rohrer, Fingerprinting the structure and properties of polycrystalline materials with the grain boundary character distribution

Grain boundary networks within polycrystalline materials are structurally complex. Five independent parameters are required to distinguish one grain boundary from another and the different types of grain boundaries are connected in non-random configurations. To establish control over the structure of the grain boundary network and the macroscopic properties it influences, it is first necessary to define a metric or "fingerprint" that uniquely describes the network's structural characteristics. It is proposed here that this fingerprint is a five-dimensional quantity referred to as the grain boundary character distribution (the relative areas of different boundary types, distinguished by lattice misorientation and grain boundary plane orientation). We have developed techniques to measure this quantity and based on observations in a range of metals and ceramics (Al, grain boundary engineered Ni and +-brass, Fe-1%Si, WC, MgO, SrTiO3, TiO2, MgAl2O4, and Al2O3), we are beginning to understand how the grain boundary character distribution evolves with time and is influenced by impurities and processing conditions. One general observation that will be described in the seminar is that grains within polycrystals have preferred habit planes that correspond to the same low energy, low index planes that dominate the external growth forms and equilibrium shapes of isolated crystals of the same phase. This finding amounts to a paradigm shift in our understanding of grain boundaries, which has been dominated for decades by consideration of crystal lattice orientation relationships instead of the grain surface relationships that now appear to be more influential. A second topic will be the probable existence of a steady state grain boundary character distribution that is correlated to grain boundary energies and is established in the early stages of growth. Finally, spatial correlations in the connectivity of the distribution will be discussed.

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A. D. (Tony) Rollett, The Significance of Connectivity in Microstructures

It is generally straightforward to analyze microstructures in terms of their constituent parts such as grains in single-phase materials, two-phase systems where there is a small volume fraction of well-dispersed second phase particles. In some cases, however, the connections between the constituents cannot be ignored. Examples will be given of specific cases such as grain boundary networks influencing fracture and corrosion, non-random crystallographic relationships between recrystallized and unrecrystallized regions, and contiguity in two-phase systems with high volume fractions of regularly shaped particles. The classical analytical approach is to measure contiguity and this can be easily elaborated to include crystal orientation (texture, misorientation). Where appropriate the classical measures will be related to computational homology.

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David M. Saylor, Structure Evolution and Release Behavior in Controlled Drug Delivery Devices

To maintain optimal levels of drug in the body over time, devices have been developed that allow for controlled release of the drug. A popular method of control is to incorporate the drug into a polymer matrix, which acts as a diffusion barrier, slowing the rate at which the drug is released. The effects of changes in the manufacturing process on the final microstructure and subsequent drug release rates are not well-understood. Thus, we have developed a model that enables us to predict the complex morphologies that form under different manufacturing conditions, as well as the release behavior from these devices. Further, we have developed a robust metric based on computational homology for the composite structures that allows us to quantitatively map the processing effects on structure, as well as the structure-sensitivity of release behavior of these systems.

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Takashi Teramoto, Morphological Characterization of Diblock Copolymer Problem and Topological Computation

Micro-phase separation of diblock copolymer melts have become an excellent model system for studying fundamental phenomena associated with molecular self-assembly. We deal with the gradient system derived from the free-energy functional of nonlocal type and focus on the morphological analysis of the periodic 3D structures obtained by the simulations. We have numerically confirmed that "balanced scaling law", i.e., a morphology is attained where two competing terms are balanced, is valid for the functional in the singular limit which is consistent with the theoretical results by Ren and Wei. This view point allows us to demonstrate the mechanism behind the appearance of the gyroid minimizers in terms of some geometrical measures. We also apply computational homology to characterize the complex morphology during the phase transition dynamics. Our topological characterization points to the transient perforated lamellar state in the lamellar-hexagons transition and the t-1 law of the Betti number in the late stage of phase-ordering process. This is a joint work with Prof. Yasumasa Nishiura.

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Peter Voorhees, Coarsening of Topologically Complex Systems: Experiments and Simulations

Recent advances in experimental and numerical methods allows for routine visualization and computation of the three-dimensional microstructure of materials. It is now possible to quantify the morphology of complex microstructures using measurements of the interfacial shape distribution, the probability of finding a patch of interface with a given pair of principle curvatures and the genus of the microstructure. We have examined coarsening in dendritic sold-liquid mixtures in the Al-Cu and Sn-Pb systems using three-dimensional reconstructions. We have also used phase field simulations to follow the evolution of the three-dimensional interfacial morphology of the dendritic solid-liquid mixtures that were measured experimentally. The manner in which these morphologically complex two-phase mixtures evolve during coarsening will be discussed.

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