The workshop is centered around homogenization and upscaling in materials science. Topics include the motion of grain boundaries, phase boundaries in shape memory alloys, domain walls in ferromagnets, propagating fronts in heterogeneous media. The workshop brings together experimental and theoretical physicists and engineers with mathematicians in analysis, numerical simulation and probability theory.
Please see also the poster overview at Poster blitz
The magnetoelasticity of BCC Fe-Ga,Al,Ge solid solutions has been widely discussed during the last decade. In this talk we summarize the magnetoelastic properties of these alloys and show how the unique linear, reversible and isotropic magnetic characteristics represent the response of a magnetoelastically adaptive state.
Interfaces sometimes contain functionalities that do not exist in the bulk. This discovery is fairly recent and has triggered a systematic search for materials where the functionalities such as superconductivity, ferroelectricity and ferromagnetism are the key desired physical properties of the interface. During phase transitions, interfaces nucleate and the functionality, such as the polarity in the Fig.1, co-nucleate and strongly interact with the adjacent bulk material to form specific vortex structures. This enables us to apply external fields, which induce novel features such as wall switching, Bloch line switching and high conductivity pathways. Several results from experimental observations and computer simulations will be discussed.
Imposing a shape with Gaussian curvature on a solid sheet, generates in it elastic stress. This coupling between geometry (curvature) and mechanics (stress) is a basic demonstration of Gauss’ theorema Egregium, and underlies the morphological richness observed in solid sheets, and their nontrivial response to exerted forces. In this talk I will attempt to provide a unifying framework for morphological transitions in elastic sheets with imposed curvature, by using asymptotic analysis around ``tension field theory”. This singular limit assumes a sheet with zero bending modulus under finite tensile load. Considering crystalline sheets with small, finite bending modulus, we predict a transition between patterns of wrinkles (shape deformation) and grain boundaries. Considering a vanishing tensile load, we predict a transition between two types of “asymptotic isometry”: a developable type (which repels Gaussian curvature) and a non-developable on (which accommodates the imposed Gaussian curvature). The predicted transitions will be demonstrated through examples from a few realistic systems.
This is joint work with Xavier Blanc (University Denis Diderot, Paris) and Pierre-Louis Lions (College de France, Paris). We present a general approach to approximate at the fine scale the solution to an elliptic equation with oscillatory coefficient when this coefficient consists of a ”nice” (in the simplest possible case say periodic) function which is, in some sense to be made precise, perturbed. The approach is based on the determination of a local profile, solution to an equation similar to the corrector equation in classical homogenization. We prove that this equation has a unique solution, in various functional settings depending upon the perturbation: local perturbation, two different periodic structures separated by a common interface, etc.
For the past decades, the method of choice for the determination of atomic structure in structural biology and nanoscience has been X-ray diffraction. This exploits the spectacular Bragg/Von Laue phenomenon that plane waves scattered at crystals yield discrete diffraction patterns. The catch is that a native assembly of proteins has to be broken and the protein needs to be crystallized, which is difficult and may lead to non-native forms. In my talk I discuss alternatives such as fiber diffraction which avoid crystallization at the expense of lower resolution in angular direction (Cochran, Crick, Vand 1952), and a recent theoretical advance which proposes incoming waveforms which exhibit fully discrete diffraction patterns when scattered at helices (Friesecke, James, Juestel, arXiv 1506.04240, 2015).
Shock waves in composites and ferroic materialsVinamra AgrawalCalifornia Institute of TechnologyDislocation-mediated relaxation in nanograinedcolumnar palladium films revealed by on-chiptime-resolved HRTEM testingBehnam AminahmadiUniversity of AntwerpCauchy-Born rule for incommensurate systems of coupled atomic chainsPaul CazeauxUniversity of MinnesotaOne of the many exciting perspectives opened by the recent experimental discovery of 2D materials, such as graphene or boron nitride, is the possibility of stacking a few layers of such materials. In principle, one can tune the properties of these heterostructures [Geim et. al., 2013], such as elasticity, conductivity, electronic and optical properties... To explore the possibilities of this atomic-scale Lego game, it is necessary to develop multiscale models and adapted numerical methods.
A particular difficulty encountered in this study is the incommensurate character resulting from the coupling of periodic lattices associated with each monolayer. The difference between cristalline structure of the materials, or the rotation of one layer relative to another results in a quasi-periodic structure, sometimes presenting a Moire pattern or a super-lattice.
In this poster, we present a first step towards the mathematical understanding and analysis of such structures and methods. We study the derivation of a Cauchy-Born rule for the elastic response to stretching a system of two periodic coupled atomic chains. We will in particular focus on the case where the respective periods are incommensurate. The two chains interact through an interatomic potential such as Lennard-Jones. When the length of the system grows to infinity, we show that the elastic response converges uniformly to a homogenized response. We further study the impact of the incommensurate character on the convergence rate of numerical approaches using the tools of discrepancy theory.
Two cases are studied in particular: first, two rigid, parallel chains, and next a relaxed system allowing in-plane ripples. In this second case, the equations of the model can be seen as an innovating application of the generalized Frenkel-Kontorova model. In particular, the minimum energy configuration can be analytically and numerically studied by the tools of Aubry-Mather theory.Reduced models for domain walls in soft ferromagnetic filmsLukas DöringRWTH Aachen UniversityIn soft ferromagnetic films of moderate thickness, the transition layers ("domain walls") that separate domains of constant magnetization may favor a two scale-structure, combining a stray-field free domain wall in the core of the transition with logarithmically decaying tails that complete the rotation between neighboring domains.
On this poster, reduced models for an isolated domain wall in an infinitely extended film (joint work with R. Ignat and F. Otto) and for periodic systems of parallel walls with potentially interacting tails are presented that quantify the above-mentioned splitting of domain walls into a stray-field free core and logarithmic tails (both are obtained from the Landau-Lifshitz energy via Gamma convergence).
In the case of periodic, interacting walls, also a prediction of the average hard-axis magnetization in the domains (as a function of the domain width and the strength of an external hard-axis magnetic field) is available.
For CoFeB films, provided the film thickness is not too large, the predicted hard-axis magnetization agrees well with corresponding experimental data (joint work with C. Hengst, R. Schäfer and F. Otto).Isometric Immersions and Self Similar Buckling in Non-Euclidean Elastic SheetsJohn GemmerBrown UniversityThe edge of torn elastic sheets and growing leaves often form a hierarchical buckling pattern. Within non-Euclidean plate theory this complex morphology can be understood as low bending energy isometric immersions of Riemannian metrics. In this poster we show that for a large class of growth profiles there exist periodic and self-similar deformations of the sheet with vanishing in-plane strain. The construction of these surfaces consists of gluing together local solutions to the isometric immersion problem along "lines of inflection" and "branch points" in such a manner that the resulting surface has finite bending energy. For hyperbolic non-Euclidean sheet, complex wrinkling patters are thus possible and our results identify the key the regularity of the isometric immersion plays in determining the global structure of a non-Euclidean elastic sheet.Quantitative stochastic homogenization: Green's function estimates for elliptic systemsArianna GiuntiMax Planck Insitute for Mathematics in the SciencesHomogenization of layered materials with rigid components in single-slip finite plasticityCarolin KreisbeckUniversität RegensburgThresholding schemes for geometric flowsTim LauxMax Planck Institute for Mathematics in the SciencesThe thresholding scheme by Merriman, Bence and Osher is an algorithm to construct solutions to the mean-curvature flow equation. We prove convergence of two variants of the scheme, one for multi-phase mean-curvature flow and one for volume-preserving mean-curvature flow. The results establish convergence towards a weak formulation in the
BV-framework. The proofs are based on the interpretation of the schemes as minimizing movement schemes by Esedoglu and Otto. This interpretation means that the schemes preserve the gradient flow structure. In this sense, the proofs are similar to the convergence results of Almgren, Taylor and Wang and Luckhaus and Sturzenhecker.
This is based on joint works with Felix Otto and Drew Swartz.Locality of the TFW equationsFaizan NazarUniversity of WarwickIn this poster I will discuss the existence and uniqueness of a coupled system of partial differential equations that arises from minimising the Thomas-Fermi-von Weizsäcker energy functional for general infinite nuclear arrangements. This gives rise to stability estimates, which give pointwise control of the electron density in terms of a local nuclear defect. We then discuss the applications of this result, including the neutrality of local defects in TFW theory and the lattice relaxation problem.Rate of Convergence and Correctors for a multi-scale Model of dilute EmulsionsGrigor NikaWorcester Polytechnic InstituteIn this study we are interested in a problem of dilute emulsions of two immiscible viscous fluids, in which one is distributed in the other in the form of droplets of arbitrary shape, with non-uniform surface tension due to surfactants. The problem includes an essential kinematic condition on the droplets. In the periodic homogenization framework, it can be shown using Mosco-convergence that, as the size of the droplets converges to zero faster than the distance between the droplets, the emulsion behaves in the limit like the continuous phase. Here we determine the rate of convergence of the velocity fifield for the emulsion to that of the velocity for the one fluid problem and in addition, we determine the corrector in terms of the bulk and surface polarization tensors.Twisting of an Elastic RibbonEthan O'BrienCourant Institute (NYU)An elastic ribbon, when twisted and held with low tension, will form wrinkles in the center. We analyze this by proving an upper and lower bound for an energy functional in terms of the ribbon's thickness. We also find some information about the low energy states (an in particular the minimizer): we get a rigidity result for the outer edges of the ribbon and estimates on the amplitude of the wrinkles in the interior. This work uses an energy functional reminiscent of von Kármán plate theory, derived formally by considering small deflections from a helicoid. Chopin, Démery and Davidovitch studied the relaxed problem using a closely related energy.
This is joint work with Robert V. Kohn.Study of polarisation patterns in ferroelectrics using a phase-field modelAnanya Renuka BalakrishnaUniversity of OxfordHomogenization of interfacial energies defined on random latticesMatthias RufTechnical University MunichRigidity of Shape Memory AlloysThilo SimonMPI MISWe study a geometrically linear model for shape memory alloys undergoing cubic-to-tetragonal transformations. The aim is to explain the alignment of habit planes by energy minimization.Microstructural Origin of Magnetostriction in FeGa and FePdJake SteinerUniversity of Maryland, College ParkBoth FePd and FeGa show large magnetostriction in the cubic state: ≈200 and ≈400 ppm, respectively. These large values are difficult to reconcile with classical Joulian magnetostriction. We used magnetostriction and magnetic torque measurements as well as microstructural imaging to show that both large values are of microstructural origin. In FePd, the magnetostriction is a property of the well-known tweed structure and hence isotropic; in FeGa, coherent magnetically and elastically self-sufficient microcells form whose reorientation under applied field dictates magnetostrictive behavior. The unique microstructures also lead to linear and completely reversible magnetization characters for both alloys.Dynamics of a Second Order Gradient Model for Phase TransitionsDrew SwartzPurdue UniversityIn 2000 Fonseca and Mantegazza introduced a second order gradient theory for phase transitions. The model is similar in spirit to the Ginzburg-Landau model, which is a first order model. In their work, Fonseca and Mantegazza showed that their energy Gamma-converges to a perimeter functional. The current project examines the gradient flow dynamics for the Fonseca-Mantegazza energy. The corresponding evolution equation is fourth order, thus creating some interesting difficulties in its analysis. We analyze properties of the optimal transition profile through a combination of analytical and numerical techniques. Then in the radially symmetric setting, we use this to demonstrate that the gradient flow dynamics converges to motion by mean curvature. This is joint work with Prof. Aaron Yip.Reaction-diffusion equations with hysteresisSergey TikhomirovMax Planck Institute for Mathematics in the ScienceWe consider reaction-diffusion equations with discontinuous hysteretic nonlinearities. We connect these equations with free boundary problems and introduce a related notion of spatial transversality for initial data and solutions. We assert that the equation with transverse initial data possesses a unique solution, which remains transverse for some time. At a moment when the solution becomes nontransverse, we discretize the spatial variable and analyze the resulting lattice dynamical system with hysteresis. In particular, we discuss a new pattern formation mechanism — rattling, which indicates how one should reset the continuous model to make it well posed.Energy scaling laws for an axially compressed thin elastic cylinderIan TobascoCourant Institute of Mathematical SciencesA longstanding open problem in elasticity is to identify the minimum energy scaling law of a crumpled sheet of paper whose thickness tends to zero. Though much is known about scaling laws for thin sheets in tensile settings, the compressive regime is mostly unexplored. In the poster, we discuss an axially confined thin elastic cylinder which is prevented from inward displacement by a hard mandrel core. Our focus is on the dependence of the minimum energy on the thickness and confinement of the cylinder in the Foppl-von Karman model. We prove upper and lower bounds for this scaling.Atomistic-to-continuum coupling: the quasi-non-local approachHuan WuUniversity of WarwickAtomistic-to-continuum (a/c) coupling methods are a class of computational multi-scale schemes for simulating crystal defects. It combines the accuracy of atomistic models and the efficiency of continuum models. Among a few popular a/c methods, the quasi-non-local coupling exhibits the advantage of eliminating ghost-force which otherwise presents great difficulty in error analysis. My work focuses on the rigorous error analysis of qnl-type models in 2D with high-order finite elements.On generalized Poisson-Nernst-Planck equationsAnna ZubkovaUniversity of GrazA time-dependent Poisson-Nernst-Planck system of nonlinear partial differential equations is considered. It is modeled in terms of the Fickian multiphase diffusion law coupled with electrostatic and quasi-Fermi electrochemical potentials. The model describes a plenty of electrokinetic phenomena in physical and biological sciences. The generalized model is supplemented by a positivity and volume constraints, by quasi-Fermi electrochemical potentials depending on the pressure, and by inhomogeneous transmission boundary conditions representing reactions at the micro-scale level. We aim at a proper variational modeling, optimization, and asymptotic analysis as well as homogenization of the model at the macro-scale level. The work is supported by the Austrian Science Fund (FWF) in the framework of the research project P26147-N26: PION.
Functional shape memory alloys need to operate reversibly and repeatedly. This is especially crucial for many future applications such as artificial heart valves or elastocaloric cooling, where more than ten million transformation cycles will be required. Here we report on the discovery of an ultra-low fatigue shape memory alloy film system based on TiNiCu that allows at least ten million transformation cycles. We found these films contain Ti2Cu precipitates embedded in the base alloy that serve as sentinels to ensure complete and reproducible transformation in the course of each memory cycle. In addition we found an almost perfect fulfillment of the compatibility conditions in TiNiCuCo thin films that also show ultra low fatigue.
Martensitic phases of shape memory alloys can be found in a broad variety of geometrically ordered microstrucutres; at the macro-scale, these microstructures often exhibit strongly anisotropic, temperature-dependent, and possibly also strongly non-linear elastic responses to the external mechanical loads, depending on the actual micromorphology.
The lecture will summarize the experimental approaches for studying this homogenized elastic responses with the focus laid on the relations between theoretical models of the microstructure and the macro-scale behavior.
The martensitic microstructure is decisive for most functionalities of shape memory alloys, including pseudoelastic, pseudoplastic, magnetic shape memory and caloric effects. Here we examine the formation of a hierarchical microstructure by a combination of high resolution ex-situ and in-situ experiments and theoretical models. As model system we select epitaxial Ni-Mn-Ga films as the high fraction of surface to volume makes surface experiments representative. Moreover, the high aspect ratio of a film minimizes mechanical interactions between different regions, making films a statistically relevant ensemble of many nucleation events.
The need to form phase boundaries requires introducing nanotwin boundaries. Interaction energy results in an ordered arrangement, appearing as a modulated phase and forming a-b twin boundaries. Nucleation and growth within the volume requires mesoscopic type I and II twin boundaries. As a consequence of different nucleation sites also macroscopic twin boundaries form. To conclude, the resulting hierarchical microstructure is not result of a global minimization of energy but determined by the most easy transformation path, ending in a metastable configuration containing a well-defined arrangement of many different types of twin boundaries.
This work is supported by DFG through SPP 1599 www.FerroicCooling.de.
Shock waves in composites and ferroic materialsVinamra AgrawalCalifornia Institute of TechnologyDislocation-mediated relaxation in nanograinedcolumnar palladium films revealed by on-chiptime-resolved HRTEM testingBehnam AminahmadiUniversity of AntwerpCauchy-Born rule for incommensurate systems of coupled atomic chainsPaul CazeauxUniversity of MinnesotaOne of the many exciting perspectives opened by the recent experimental discovery of 2D materials, such as graphene or boron nitride, is the possibility of stacking a few layers of such materials. In principle, one can tune the properties of these heterostructures [Geim et. al., 2013], such as elasticity, conductivity, electronic and optical properties... To explore the possibilities of this atomic-scale Lego game, it is necessary to develop multiscale models and adapted numerical methods.
A particular difficulty encountered in this study is the incommensurate character resulting from the coupling of periodic lattices associated with each monolayer. The difference between cristalline structure of the materials, or the rotation of one layer relative to another results in a quasi-periodic structure, sometimes presenting a Moire pattern or a super-lattice.
In this poster, we present a first step towards the mathematical understanding and analysis of such structures and methods. We study the derivation of a Cauchy-Born rule for the elastic response to stretching a system of two periodic coupled atomic chains. We will in particular focus on the case where the respective periods are incommensurate. The two chains interact through an interatomic potential such as Lennard-Jones. When the length of the system grows to infinity, we show that the elastic response converges uniformly to a homogenized response. We further study the impact of the incommensurate character on the convergence rate of numerical approaches using the tools of discrepancy theory.
Two cases are studied in particular: first, two rigid, parallel chains, and next a relaxed system allowing in-plane ripples. In this second case, the equations of the model can be seen as an innovating application of the generalized Frenkel-Kontorova model. In particular, the minimum energy configuration can be analytically and numerically studied by the tools of Aubry-Mather theory.Reduced models for domain walls in soft ferromagnetic filmsLukas DöringRWTH Aachen UniversityIn soft ferromagnetic films of moderate thickness, the transition layers ("domain walls") that separate domains of constant magnetization may favor a two scale-structure, combining a stray-field free domain wall in the core of the transition with logarithmically decaying tails that complete the rotation between neighboring domains.
On this poster, reduced models for an isolated domain wall in an infinitely extended film (joint work with R. Ignat and F. Otto) and for periodic systems of parallel walls with potentially interacting tails are presented that quantify the above-mentioned splitting of domain walls into a stray-field free core and logarithmic tails (both are obtained from the Landau-Lifshitz energy via Gamma convergence).
In the case of periodic, interacting walls, also a prediction of the average hard-axis magnetization in the domains (as a function of the domain width and the strength of an external hard-axis magnetic field) is available.
For CoFeB films, provided the film thickness is not too large, the predicted hard-axis magnetization agrees well with corresponding experimental data (joint work with C. Hengst, R. Schäfer and F. Otto).Isometric Immersions and Self Similar Buckling in Non-Euclidean Elastic SheetsJohn GemmerBrown UniversityThe edge of torn elastic sheets and growing leaves often form a hierarchical buckling pattern. Within non-Euclidean plate theory this complex morphology can be understood as low bending energy isometric immersions of Riemannian metrics. In this poster we show that for a large class of growth profiles there exist periodic and self-similar deformations of the sheet with vanishing in-plane strain. The construction of these surfaces consists of gluing together local solutions to the isometric immersion problem along "lines of inflection" and "branch points" in such a manner that the resulting surface has finite bending energy. For hyperbolic non-Euclidean sheet, complex wrinkling patters are thus possible and our results identify the key the regularity of the isometric immersion plays in determining the global structure of a non-Euclidean elastic sheet.Quantitative stochastic homogenization: Green's function estimates for elliptic systemsArianna GiuntiMax Planck Insitute for Mathematics in the SciencesHomogenization of layered materials with rigid components in single-slip finite plasticityCarolin KreisbeckUniversität RegensburgThresholding schemes for geometric flowsTim LauxMax Planck Institute for Mathematics in the SciencesThe thresholding scheme by Merriman, Bence and Osher is an algorithm to construct solutions to the mean-curvature flow equation. We prove convergence of two variants of the scheme, one for multi-phase mean-curvature flow and one for volume-preserving mean-curvature flow. The results establish convergence towards a weak formulation in the
BV-framework. The proofs are based on the interpretation of the schemes as minimizing movement schemes by Esedoglu and Otto. This interpretation means that the schemes preserve the gradient flow structure. In this sense, the proofs are similar to the convergence results of Almgren, Taylor and Wang and Luckhaus and Sturzenhecker.
This is based on joint works with Felix Otto and Drew Swartz.Locality of the TFW equationsFaizan NazarUniversity of WarwickIn this poster I will discuss the existence and uniqueness of a coupled system of partial differential equations that arises from minimising the Thomas-Fermi-von Weizsäcker energy functional for general infinite nuclear arrangements. This gives rise to stability estimates, which give pointwise control of the electron density in terms of a local nuclear defect. We then discuss the applications of this result, including the neutrality of local defects in TFW theory and the lattice relaxation problem.Rate of Convergence and Correctors for a multi-scale Model of dilute EmulsionsGrigor NikaWorcester Polytechnic InstituteIn this study we are interested in a problem of dilute emulsions of two immiscible viscous fluids, in which one is distributed in the other in the form of droplets of arbitrary shape, with non-uniform surface tension due to surfactants. The problem includes an essential kinematic condition on the droplets. In the periodic homogenization framework, it can be shown using Mosco-convergence that, as the size of the droplets converges to zero faster than the distance between the droplets, the emulsion behaves in the limit like the continuous phase. Here we determine the rate of convergence of the velocity fifield for the emulsion to that of the velocity for the one fluid problem and in addition, we determine the corrector in terms of the bulk and surface polarization tensors.Twisting of an Elastic RibbonEthan O'BrienCourant Institute (NYU)An elastic ribbon, when twisted and held with low tension, will form wrinkles in the center. We analyze this by proving an upper and lower bound for an energy functional in terms of the ribbon's thickness. We also find some information about the low energy states (an in particular the minimizer): we get a rigidity result for the outer edges of the ribbon and estimates on the amplitude of the wrinkles in the interior. This work uses an energy functional reminiscent of von Kármán plate theory, derived formally by considering small deflections from a helicoid. Chopin, Démery and Davidovitch studied the relaxed problem using a closely related energy.
This is joint work with Robert V. Kohn.Study of polarisation patterns in ferroelectrics using a phase-field modelAnanya Renuka BalakrishnaUniversity of OxfordHomogenization of interfacial energies defined on random latticesMatthias RufTechnical University MunichRigidity of Shape Memory AlloysThilo SimonMPI MISWe study a geometrically linear model for shape memory alloys undergoing cubic-to-tetragonal transformations. The aim is to explain the alignment of habit planes by energy minimization.Microstructural Origin of Magnetostriction in FeGa and FePdJake SteinerUniversity of Maryland, College ParkBoth FePd and FeGa show large magnetostriction in the cubic state: ≈200 and ≈400 ppm, respectively. These large values are difficult to reconcile with classical Joulian magnetostriction. We used magnetostriction and magnetic torque measurements as well as microstructural imaging to show that both large values are of microstructural origin. In FePd, the magnetostriction is a property of the well-known tweed structure and hence isotropic; in FeGa, coherent magnetically and elastically self-sufficient microcells form whose reorientation under applied field dictates magnetostrictive behavior. The unique microstructures also lead to linear and completely reversible magnetization characters for both alloys.Dynamics of a Second Order Gradient Model for Phase TransitionsDrew SwartzPurdue UniversityIn 2000 Fonseca and Mantegazza introduced a second order gradient theory for phase transitions. The model is similar in spirit to the Ginzburg-Landau model, which is a first order model. In their work, Fonseca and Mantegazza showed that their energy Gamma-converges to a perimeter functional. The current project examines the gradient flow dynamics for the Fonseca-Mantegazza energy. The corresponding evolution equation is fourth order, thus creating some interesting difficulties in its analysis. We analyze properties of the optimal transition profile through a combination of analytical and numerical techniques. Then in the radially symmetric setting, we use this to demonstrate that the gradient flow dynamics converges to motion by mean curvature. This is joint work with Prof. Aaron Yip.Reaction-diffusion equations with hysteresisSergey TikhomirovMax Planck Institute for Mathematics in the ScienceWe consider reaction-diffusion equations with discontinuous hysteretic nonlinearities. We connect these equations with free boundary problems and introduce a related notion of spatial transversality for initial data and solutions. We assert that the equation with transverse initial data possesses a unique solution, which remains transverse for some time. At a moment when the solution becomes nontransverse, we discretize the spatial variable and analyze the resulting lattice dynamical system with hysteresis. In particular, we discuss a new pattern formation mechanism — rattling, which indicates how one should reset the continuous model to make it well posed.Energy scaling laws for an axially compressed thin elastic cylinderIan TobascoCourant Institute of Mathematical SciencesA longstanding open problem in elasticity is to identify the minimum energy scaling law of a crumpled sheet of paper whose thickness tends to zero. Though much is known about scaling laws for thin sheets in tensile settings, the compressive regime is mostly unexplored. In the poster, we discuss an axially confined thin elastic cylinder which is prevented from inward displacement by a hard mandrel core. Our focus is on the dependence of the minimum energy on the thickness and confinement of the cylinder in the Foppl-von Karman model. We prove upper and lower bounds for this scaling.Atomistic-to-continuum coupling: the quasi-non-local approachHuan WuUniversity of WarwickAtomistic-to-continuum (a/c) coupling methods are a class of computational multi-scale schemes for simulating crystal defects. It combines the accuracy of atomistic models and the efficiency of continuum models. Among a few popular a/c methods, the quasi-non-local coupling exhibits the advantage of eliminating ghost-force which otherwise presents great difficulty in error analysis. My work focuses on the rigorous error analysis of qnl-type models in 2D with high-order finite elements.On generalized Poisson-Nernst-Planck equationsAnna ZubkovaUniversity of GrazA time-dependent Poisson-Nernst-Planck system of nonlinear partial differential equations is considered. It is modeled in terms of the Fickian multiphase diffusion law coupled with electrostatic and quasi-Fermi electrochemical potentials. The model describes a plenty of electrokinetic phenomena in physical and biological sciences. The generalized model is supplemented by a positivity and volume constraints, by quasi-Fermi electrochemical potentials depending on the pressure, and by inhomogeneous transmission boundary conditions representing reactions at the micro-scale level. We aim at a proper variational modeling, optimization, and asymptotic analysis as well as homogenization of the model at the macro-scale level. The work is supported by the Austrian Science Fund (FWF) in the framework of the research project P26147-N26: PION.
I will present a notion of kinetic/entropy solutions for multi-diemsional scalar conservation laws with rough time dependence, a particular case being Brownian motion and I will discuss its well posedness. In the special case of the stochastic conservation laws I will also present results about long time dependence and error estimates.
One of the obstructions to truly microscopic statistical physics theory of elasticity lies in the fact that one should consider a metastable system of particles. While metastability is reasonably well understood for lattice systems, particles in continuum are more difficult to analyse. Even a rigorous proof of just the existence of a phase transition has been achieved only for few simple models. We will discuss the need for considering a metastable system as well as formulate/explain the results concerning metastability for the particular case of the Widom-Rowlinson model.
Based on a joint work with F. den Hollander, S. Jansen, and E. Pulvirenti.
As the link between the atomic and continuum scales, microstructural models and simulations are a critical element of computational materials science. However, while standardized computational tools have become widely accepted at the electronic, atomic, and continuum scales, mesoscale simulation methods remain diverse and are still evolving. In part, this stems from the breadth of phenomena included under the microstructural umbrella, but it is also due to intrinsic limitations in the prevailing methods. Discrete microstructural evolution simulations (i.e. the Monte Carlo Potts model, probabilistic cellular automata) are computationally efficient and easy to implement; they suffer, however, from various artifacts of the underlying computational lattice. This talk will introduce a lattice-‐free, discrete kinetic Monte Carlo method. The Material Point Monte Carlo (MPMC) method uses randomly placed material points to overcome the unphysical effects of lattice anisotropy on interfacial and volumetric energies and to enable the correct evolution of systems that undergo shape distortion. MPMC simulations retain most of the computational benefits of other discrete methods and provably reproduce the physics of interface motion by surface and bulk driving forces. One goal of mesoscale simulations is to understand rare events, such as failure initiation, hot spot formation, and abnormal grain nucleation. Because these phenomena typically arise from the localization of long-‐range interactions, identifying an incipient rare event is challenging. This talk will present a network theory approach to analyzing long-‐range grain neighborhoods using the random walk graph kernel. Utilizing data from microstructural evolution simulations, a machine-‐learning system can be trained to classify potential abnormal growth events within the grain network. By operating beyond a nearest-‐neighbor or mean-‐field interaction distance, this method has promise for characterizing a number of long-‐range microstructural phenomena.
Elizabeth A. Holm, Philip Goins, and Brian DeCost
Threshold dynamics is a very efficient algorithm for moving an interface (e.g. a surface in 3D) by mean curvature motion. It was proposed by Merriman, Bence, and Osher in 1989, and also extended to networks of surfaces in the same paper. This dynamics arises as gradient flow for the sum of the areas of the surfaces in the network, and plays a prominent role in materials science applications where it describes the motion of grain boundaries in polycrystals (such as most metals) under heat treatment.
Further extension of the algorithm to weighted mean curvature flow of networks, where the surface tension of each interface in the network may be distinct (unequal) and may depend on the direction of the normal, is of great interest for applications, but has remained elusive. We describe how to extend threshold dynamics to unequal and anisotropic (normal dependent) surface tensions. Joint work with Matt Elsey and Felix Otto.
A fundamental aspect of 2D cellular networks with isotropic line tension is the Mullins-von Neumann $n-6$ rule: the rate of change of the area of a (topological) $n$-gon is proportional to $n-6$. As a consequence, cells with fewer than $6$ sides vanish in finite time, and the network coarsens. Numerical and physical experiments have revealed a form of statistical self-similarity in the long time dynamics.We propose a kinetic description for the evolution of such networks. The ingredients in our model are an elementary $N$ particle system that mimics essential features of the von Neumann rule, and a hydrodynamic limit theorem for population densities when $N \rightarrow \infty$. This model is compared with a set of models derived in the physics and materials science communities, as well as extensive numerical simulations by applied mathematicians. This is joint work with Joe Klobusicky (Brown University and Geisinger Health Systems) and Bob Pego (Carnegie Mellon University).
Shock waves in composites and ferroic materialsVinamra AgrawalCalifornia Institute of TechnologyDislocation-mediated relaxation in nanograinedcolumnar palladium films revealed by on-chiptime-resolved HRTEM testingBehnam AminahmadiUniversity of AntwerpCauchy-Born rule for incommensurate systems of coupled atomic chainsPaul CazeauxUniversity of MinnesotaOne of the many exciting perspectives opened by the recent experimental discovery of 2D materials, such as graphene or boron nitride, is the possibility of stacking a few layers of such materials. In principle, one can tune the properties of these heterostructures [Geim et. al., 2013], such as elasticity, conductivity, electronic and optical properties... To explore the possibilities of this atomic-scale Lego game, it is necessary to develop multiscale models and adapted numerical methods.
A particular difficulty encountered in this study is the incommensurate character resulting from the coupling of periodic lattices associated with each monolayer. The difference between cristalline structure of the materials, or the rotation of one layer relative to another results in a quasi-periodic structure, sometimes presenting a Moire pattern or a super-lattice.
In this poster, we present a first step towards the mathematical understanding and analysis of such structures and methods. We study the derivation of a Cauchy-Born rule for the elastic response to stretching a system of two periodic coupled atomic chains. We will in particular focus on the case where the respective periods are incommensurate. The two chains interact through an interatomic potential such as Lennard-Jones. When the length of the system grows to infinity, we show that the elastic response converges uniformly to a homogenized response. We further study the impact of the incommensurate character on the convergence rate of numerical approaches using the tools of discrepancy theory.
Two cases are studied in particular: first, two rigid, parallel chains, and next a relaxed system allowing in-plane ripples. In this second case, the equations of the model can be seen as an innovating application of the generalized Frenkel-Kontorova model. In particular, the minimum energy configuration can be analytically and numerically studied by the tools of Aubry-Mather theory.Reduced models for domain walls in soft ferromagnetic filmsLukas DöringRWTH Aachen UniversityIn soft ferromagnetic films of moderate thickness, the transition layers ("domain walls") that separate domains of constant magnetization may favor a two scale-structure, combining a stray-field free domain wall in the core of the transition with logarithmically decaying tails that complete the rotation between neighboring domains.
On this poster, reduced models for an isolated domain wall in an infinitely extended film (joint work with R. Ignat and F. Otto) and for periodic systems of parallel walls with potentially interacting tails are presented that quantify the above-mentioned splitting of domain walls into a stray-field free core and logarithmic tails (both are obtained from the Landau-Lifshitz energy via Gamma convergence).
In the case of periodic, interacting walls, also a prediction of the average hard-axis magnetization in the domains (as a function of the domain width and the strength of an external hard-axis magnetic field) is available.
For CoFeB films, provided the film thickness is not too large, the predicted hard-axis magnetization agrees well with corresponding experimental data (joint work with C. Hengst, R. Schäfer and F. Otto).Isometric Immersions and Self Similar Buckling in Non-Euclidean Elastic SheetsJohn GemmerBrown UniversityThe edge of torn elastic sheets and growing leaves often form a hierarchical buckling pattern. Within non-Euclidean plate theory this complex morphology can be understood as low bending energy isometric immersions of Riemannian metrics. In this poster we show that for a large class of growth profiles there exist periodic and self-similar deformations of the sheet with vanishing in-plane strain. The construction of these surfaces consists of gluing together local solutions to the isometric immersion problem along "lines of inflection" and "branch points" in such a manner that the resulting surface has finite bending energy. For hyperbolic non-Euclidean sheet, complex wrinkling patters are thus possible and our results identify the key the regularity of the isometric immersion plays in determining the global structure of a non-Euclidean elastic sheet.Quantitative stochastic homogenization: Green's function estimates for elliptic systemsArianna GiuntiMax Planck Insitute for Mathematics in the SciencesHomogenization of layered materials with rigid components in single-slip finite plasticityCarolin KreisbeckUniversität RegensburgThresholding schemes for geometric flowsTim LauxMax Planck Institute for Mathematics in the SciencesThe thresholding scheme by Merriman, Bence and Osher is an algorithm to construct solutions to the mean-curvature flow equation. We prove convergence of two variants of the scheme, one for multi-phase mean-curvature flow and one for volume-preserving mean-curvature flow. The results establish convergence towards a weak formulation in the
BV-framework. The proofs are based on the interpretation of the schemes as minimizing movement schemes by Esedoglu and Otto. This interpretation means that the schemes preserve the gradient flow structure. In this sense, the proofs are similar to the convergence results of Almgren, Taylor and Wang and Luckhaus and Sturzenhecker.
This is based on joint works with Felix Otto and Drew Swartz.Locality of the TFW equationsFaizan NazarUniversity of WarwickIn this poster I will discuss the existence and uniqueness of a coupled system of partial differential equations that arises from minimising the Thomas-Fermi-von Weizsäcker energy functional for general infinite nuclear arrangements. This gives rise to stability estimates, which give pointwise control of the electron density in terms of a local nuclear defect. We then discuss the applications of this result, including the neutrality of local defects in TFW theory and the lattice relaxation problem.Rate of Convergence and Correctors for a multi-scale Model of dilute EmulsionsGrigor NikaWorcester Polytechnic InstituteIn this study we are interested in a problem of dilute emulsions of two immiscible viscous fluids, in which one is distributed in the other in the form of droplets of arbitrary shape, with non-uniform surface tension due to surfactants. The problem includes an essential kinematic condition on the droplets. In the periodic homogenization framework, it can be shown using Mosco-convergence that, as the size of the droplets converges to zero faster than the distance between the droplets, the emulsion behaves in the limit like the continuous phase. Here we determine the rate of convergence of the velocity fifield for the emulsion to that of the velocity for the one fluid problem and in addition, we determine the corrector in terms of the bulk and surface polarization tensors.Twisting of an Elastic RibbonEthan O'BrienCourant Institute (NYU)An elastic ribbon, when twisted and held with low tension, will form wrinkles in the center. We analyze this by proving an upper and lower bound for an energy functional in terms of the ribbon's thickness. We also find some information about the low energy states (an in particular the minimizer): we get a rigidity result for the outer edges of the ribbon and estimates on the amplitude of the wrinkles in the interior. This work uses an energy functional reminiscent of von Kármán plate theory, derived formally by considering small deflections from a helicoid. Chopin, Démery and Davidovitch studied the relaxed problem using a closely related energy.
This is joint work with Robert V. Kohn.Study of polarisation patterns in ferroelectrics using a phase-field modelAnanya Renuka BalakrishnaUniversity of OxfordHomogenization of interfacial energies defined on random latticesMatthias RufTechnical University MunichRigidity of Shape Memory AlloysThilo SimonMPI MISWe study a geometrically linear model for shape memory alloys undergoing cubic-to-tetragonal transformations. The aim is to explain the alignment of habit planes by energy minimization.Microstructural Origin of Magnetostriction in FeGa and FePdJake SteinerUniversity of Maryland, College ParkBoth FePd and FeGa show large magnetostriction in the cubic state: ≈200 and ≈400 ppm, respectively. These large values are difficult to reconcile with classical Joulian magnetostriction. We used magnetostriction and magnetic torque measurements as well as microstructural imaging to show that both large values are of microstructural origin. In FePd, the magnetostriction is a property of the well-known tweed structure and hence isotropic; in FeGa, coherent magnetically and elastically self-sufficient microcells form whose reorientation under applied field dictates magnetostrictive behavior. The unique microstructures also lead to linear and completely reversible magnetization characters for both alloys.Dynamics of a Second Order Gradient Model for Phase TransitionsDrew SwartzPurdue UniversityIn 2000 Fonseca and Mantegazza introduced a second order gradient theory for phase transitions. The model is similar in spirit to the Ginzburg-Landau model, which is a first order model. In their work, Fonseca and Mantegazza showed that their energy Gamma-converges to a perimeter functional. The current project examines the gradient flow dynamics for the Fonseca-Mantegazza energy. The corresponding evolution equation is fourth order, thus creating some interesting difficulties in its analysis. We analyze properties of the optimal transition profile through a combination of analytical and numerical techniques. Then in the radially symmetric setting, we use this to demonstrate that the gradient flow dynamics converges to motion by mean curvature. This is joint work with Prof. Aaron Yip.Reaction-diffusion equations with hysteresisSergey TikhomirovMax Planck Institute for Mathematics in the ScienceWe consider reaction-diffusion equations with discontinuous hysteretic nonlinearities. We connect these equations with free boundary problems and introduce a related notion of spatial transversality for initial data and solutions. We assert that the equation with transverse initial data possesses a unique solution, which remains transverse for some time. At a moment when the solution becomes nontransverse, we discretize the spatial variable and analyze the resulting lattice dynamical system with hysteresis. In particular, we discuss a new pattern formation mechanism — rattling, which indicates how one should reset the continuous model to make it well posed.Energy scaling laws for an axially compressed thin elastic cylinderIan TobascoCourant Institute of Mathematical SciencesA longstanding open problem in elasticity is to identify the minimum energy scaling law of a crumpled sheet of paper whose thickness tends to zero. Though much is known about scaling laws for thin sheets in tensile settings, the compressive regime is mostly unexplored. In the poster, we discuss an axially confined thin elastic cylinder which is prevented from inward displacement by a hard mandrel core. Our focus is on the dependence of the minimum energy on the thickness and confinement of the cylinder in the Foppl-von Karman model. We prove upper and lower bounds for this scaling.Atomistic-to-continuum coupling: the quasi-non-local approachHuan WuUniversity of WarwickAtomistic-to-continuum (a/c) coupling methods are a class of computational multi-scale schemes for simulating crystal defects. It combines the accuracy of atomistic models and the efficiency of continuum models. Among a few popular a/c methods, the quasi-non-local coupling exhibits the advantage of eliminating ghost-force which otherwise presents great difficulty in error analysis. My work focuses on the rigorous error analysis of qnl-type models in 2D with high-order finite elements.On generalized Poisson-Nernst-Planck equationsAnna ZubkovaUniversity of GrazA time-dependent Poisson-Nernst-Planck system of nonlinear partial differential equations is considered. It is modeled in terms of the Fickian multiphase diffusion law coupled with electrostatic and quasi-Fermi electrochemical potentials. The model describes a plenty of electrokinetic phenomena in physical and biological sciences. The generalized model is supplemented by a positivity and volume constraints, by quasi-Fermi electrochemical potentials depending on the pressure, and by inhomogeneous transmission boundary conditions representing reactions at the micro-scale level. We aim at a proper variational modeling, optimization, and asymptotic analysis as well as homogenization of the model at the macro-scale level. The work is supported by the Austrian Science Fund (FWF) in the framework of the research project P26147-N26: PION.
We consider a Schroedinger equation with a weakly random time-independent potential. When the correlation function of the potential is, roughly speaking, of the Schwartz class, it has been shown by Spohn (1977), and Erdos and Yau (2001) that the kinetic limit holds -- the expectation of the phase space energy density of the solution converges to the solution of a kinetic equation. We "extend" this result to potentials whose correlation functions satisfy (in some sense) "sharp" conditions, and also prove a parallel homogenization result for slowly varying initial conditions. I will explain the quotation marks above and make some speculations on the genuinely sharp conditions on the random potential that separate various regimes. This talk is a joint work with T. Chen and T. Komorowski
We consider nonlinear diffusion equations that arise as scaling limit of certain interacting particle systems, the zero range process. We show that they are the gradient flow of the thermodynamic entropy of the process with respect to a weighted Wasserstein metric, where the weight is related to the diffusion of a tagged particle.
(Joint work with Mark Peletier, Marios Stamatakis and Johannes Zimmer)
A review will be given on the role of grains and grain boundaries for the formation and behavior of magnetic domains and domains walls, based on experimental imaging by Kerr microscopy. The examples range from sintered and nanostructured permanent magnets to coarse grained- and nanocrystalline soft magnets. The role of grain boundary orientation for flux propagation as well a homogenization effects in nanocrystalline material will be addressed.
We consider a macroscopic limit for the Ohta-Kawasaki energy. This model has been used to described to describe phase separation for diblock-copolymers. We first investigate existence and shape of minimizers of the energy with prescribed volume (of the one phase) in the full space setting. We then consider situation of periodic configurations with prescribed density of the minority phase. We show that in a certain regime, the energy Gamma-converges to a homogenized problem. This is joint work with C. Muratov and M. Novaga.
Experimental findings suggest that an important parameter controlling the width of the thermal hysteresis loop in certain martensitic transformations is the crystallographic compatibility of the highly symmetric austenite phase and the martensitic variants. Based on this observation, a theory of hysteresis has been proposed that assigns an important role to the energy of the transition layer (Zhang, James, Mueller, Acta mat. 57(15):4332-4352, 2009). In this talk, I will report analytical results on microstructures in such low-hysteresis shape memory alloys. This talk is partly based on joint work with Sergio Conti (Bonn).
In this talk I present and analyze a two-well Hamiltonian on a 2D atomic lattice for low energy states. The two wells of the Hamiltonian consist of the SO(2) orbit of two rank-one connected matrices. Seeking to obtain an understanding of the origin of surface energies, I focus on special $(1+\epsilon)-dimensional$ states, certain "atomic chains". For these I explain the formation of the twinning and boundary layers. This is joint work with G. Kitavtsev and S. Luckhaus.
I will describe the Objective Structures framework introduced by James, and further developed by in our collaborations with him. The framework has enabled a systematic approach to a broad variety of problems relevant to the design and characterization of materials at the atomic level. I will describe some of these activities from the research groups at University of Minnesota and Carnegie Mellon. These applications include efficient new computational methods for the study of electron structure and transport in low dimensional materials, theoretical insights and computational methods for non-equilibrium molecular dynamics and Boltzmann equation, and the self-assembly of complex nanostructures.
Participants
Hala A.H. Shehadeh
James Madison University
Vinamra Agrawal
California Institute of Technology
Behnam Aminahmadi
University of Antwerp
Thomas Blesgen
FH Bingen
Paul Cazeaux
University of Minnesota
Thomas Chen
University of Texas at Austin
Samuel Cohn
Carnegie Mellon University
Benny Davidovitch
University of Massachusetts Amherst
Kaushik Dayal
Carnegie Mellon University
Laurent Dietrich
Univ. Toulouse III
Nicolas Dirr
Cardiff University
Lukas Döring
RWTH Aachen University
Peter Embacher
Cardiff University
Selim Esedoglu
University of Michigan Ann Arbor
Sebastian Fähler
Leibniz Institute for Solid State and Materials Research Dresden
Benjamin Fehrman
University of Chicago
Julian Fischer
Max Planck Institute for Mathematics in the Sciences
Irene Fonseca
Carnegie Mellon University
Gero Friesecke
Technical University of Munich
Peter Friz
TU Berlin
John Gemmer
Brown University
Arianna Giunti
Max Planck Insitute for Mathematics in the Sciences
Michael Goldman
Paris 7-CNRS
Elena Griniari
Springer Verlag
Yu Gu
Stanford University
TRAN Hoang Son
University of Liege
Elizabeth Holm
Carnegie Mellon University
Emanuel Indrei
Carnegie Mellon University
Richard James
University of Minnesota
Marc Josien
Université Pierre et Marie Curie
Benedikt Jost
University of Muenster
Eun Heui Kim
California State University Long Beach
Georgy Kitavtsev
University of Bristol
Joe Klobusicky
Geisinger Health Systems
Hans Knüpfer
Heidelberg University
Robert Kohn
Courant Institute, New York University
Roman Kotecký
University of Warwick
Noa Kraitzman
Michigan State University
Carolin Kreisbeck
Universität Regensburg
Leonard Kreutz
Gran Sasso Science institut
Tim Laux
Max Planck Institute for Mathematics in the Sciences
Claude Le Bris
CERMICS - ENPC
Giovanni Leoni
Carnegie Mellon University
Xin Yang Lu
Carnegie Mellon University
Stephan Luckhaus
Universität Leipzig
Reza Malek-Madani
US Naval Academy
Jason Marshall
California Institute of Technology
Daniel Massatt
University of Minnesota twin-cities
Govind Menon
Brown University
Markus Mittnenzweig
WIAS Berlin
Marco Morandotti
SISSA
Ryan Murray
Carnegie Mellon University
Faizan Nazar
University of Warwick
Stefan Neukamm
TU Dresden
Robert Niemann
IFW Dresden
Grigor Nika
Worcester Polytechnic Institute
Ethan O'Brien
Courant Institute (NYU)
Heiner Olbermann
University of Bonn
Michael Ortiz
CALTECH
Felix Otto
Max Planck Institute for Mathematics in the Sciences
Matthäus Pawelczyk
TU Dresden
Jacobus Portegies
MPI MiS
Henning Pöttker
Universität Bonn
Eckhard Quandt
Christian-Albrechts-Universität zu Kiel
Claudia Raithel
MPI MIS (Leipzig)
Ananya Renuka Balakrishna
University of Oxford
Matteo Rinaldi
Carnegie Mellon University
Matthias Ruf
Technical University Munich
Angkana Rüland
University of Oxford
Lenya Ryzhik
Stanford University
Deividas Sabonis
Technical University Munich
Ekhard Salje
University of Cambridge
Rudolf Schäfer
Leibniz Institute for Solid State and Materials Research Dresden
Anja Schlömerkemper
University of Würzburg
Benjamin Seeger
University of Chicago
Hanuš Seiner
Institute of Thermomechanics AS CR
Thilo Simon
MPI MIS
Takis Souganidis
University of Chicago
Emanuele Spadaro
Max-Planck-Institut Leipzig
Jake Steiner
University of Maryland, College Park
Angela Stevens
University of Münster
Daniel Sutton
University of Bath
Drew Swartz
Purdue University
Shlomo Ta'asan
Carnegie Mellon University
Sergey Tikhomirov
Max Planck Institute for Mathematics in the Science
Ian Tobasco
Courant Institute of Mathematical Sciences
Tat Dat Tran
Max Planck Institute for Mathematics in the Sciences
Manuel Villanueva Pesqueira
Universidad Complutense de Madrid
Stephen Watson
University of Glasgow
Peter Weidemaier
Fraunhofer Society, Ernst-Mach-Institute
Huan Wu
University of Warwick
Manfred Wuttig
University of Maryland
Anna Zemlyanova
Kansas State University
Anna Zubkova
University of Graz
Barbara Zwicknagl
University of Bonn
Organizers
Irene Fonseca
Carnegie Mellon University
Richard James
University of Minnesota
Stephan Luckhaus
Universität Leipzig
Felix Otto
Max-Planck-Institut für Mathematik in den Naturwissenschaften
Peter Smereka
University of Michigan
Administrative Contact
Valeria Hünniger
Max Planck Institute for Mathematics in the Sciences
Contact via Mail
Saskia Gutzschebauch
Max-Planck-Institut für Mathematik in den Naturwissenschaften
Contact via Mail
Katja Heid
Max Planck Institute for Mathematics in the Sciences
Contact via Mail
