ABSTRACT. In 2012, experimental measurements of electric resistance of nanowires in Si doped with phosphorus atoms demonstrate that quantum effects on charge transport almost disappear for nanowires of lengths larger than a few nanometers, even at very low temperature (4.2K). Such experiments suggest an exponentially fast convergence of microscopic quantities towards macroscopic ones in real fermion systems. In Mathematics, it corresponds to the existence of so-called large deviations for the corresponding quantities. We will discuss this mathematical issue for fermion systems on the lattice.

ABSTRACT.        

ABSTRACT. One important approach for the description of quantum many-body systems is to design functionals of relatively simple quantities, such as the density or a one-body Green’s function, instead of calculating expectation values with many-body wavefunctions. However, many such functionals still lack precision, and some observables cannot be accessed at all in a satisfactory way.

One way of designing approximations is to use results of model systems. Here, we will show how model results can be used in an in principle exact way, called “Connector Theory”, in order to describe observables and systems of interest. Within this approach, a quantity of interest is calculated for a model system as function of a parameter once and forever, and the results are stored. Under certain conditions, the model result for an appropriate choice of parameter can then be used to replace a value of interest in a given real system. This choice of parameter is called “connector”.

We will discuss the principles and general properties of such an approach, and show that it is a convenient starting point for approximations.

[1] M. Vanzini, et al., arXiv:1903.07930v4 [2] A. Aouina, M. Gatti, and L. Reining, Faraday Discussions 224, 27 (2020)

ABSTRACT. The Strictly Correlated Electrons (SCE) limit of the Levy-Lieb functional in Density Functional Theory (DFT) gives rise to a symmetric multimarginal optimal transport problem with Coulomb cost, where the number of marginal laws is equal to the number of electrons in the system, which can be very large in relevant applications. We investigate a relaxation of optimal transport problems when marginal constraints are replaced by some moment constraints (MCOT problem), and show the convergence of the latter towards the former. Using Tchakaloff's theorem, we show that the MCOT problem is achieved by a finite discrete measure. For multimarginal optimal transport problems, the number of points weighted by this measure scales linearly with the number of marginal law. We leverage the sparsity of those minimizers in the design of a numerical method and prove that any local minimizer to the resulting problem is actually a global one. We illustrate the performance of the proposed method by numerical examples which solves MCOT relaxations of 3D systems with up to 100 electrons.

ABSTRACT. A grand challenge problem in engineering of polycrystals is to develop prescriptive process technologies capable of producing an arrangement of grains that provides for a desired set of materials properties. One method by which the grain structure is engineered is through grain growth or coarsening of a starting structure. Grain growth can be viewed as the evolution of a large metastable network, and can be mathematically modeled by a set of deterministic local evolution laws for the growth of an individual grain combined with stochastic models to describe the interaction between grains. Thus, to develop a predictive theory, investigation of a broad range of statistical measures of microstructure are needed and must be obtained using experiments, simulations, data analytics and mathematical modeling. We will present results of both in-situ and ex-situ experiments of grain growth in thin films, which are used as the experimental test bed. One important advantage of thin films is the ability to combine in-situ examination of the motion of boundaries with periodic freezing of the structure to map and quantify statistically significant populations, before reheating to promote continued grain growth. The experimental results will be compared and contrasted with results of sharp and diffuse interface simulations.

ABSTRACT. Cellular networks are ubiquitous in nature. Most technologically useful materials arise as polycrystalline microstructures, composed of a myriad of small monocrystalline cells or grains, separated by interfaces, or grain boundaries. Grain boundaries play an essential role in determining the properties of materials across a wide range of scales. During grain growth (also termed coarsening), an initially random grain boundary arrangement reaches a steady state that is strongly correlated to the interfacial energy density. In this talk, we will discuss recent progress on modeling, simulation and analysis of the evolution of the grain boundary network in polycrystalline materials.

ABSTRACT. The boundary effects and thermal effects play crucial roles in real life applications. These problem give formidable challenges to the mathematical analysis and numerical simulations. In this talk, I will employ the energetic variational approaches (EnVarA) in rigorously establishing the dynamical systems that are include all the basic laws and constraints of thermodynamics. I will demonstrate the difficulties and subtleties by looking at these effects in various general diffusion dynamics and gradient dynamics.

ABSTRACT. The systematic tuning of the lattice parameters to achieve improved kinematic compatibility between phases is an effective strategy for improving the reversibility, and lowering the hysteresis, of solid-solid phase transformations. We present an apparently paradoxical example in which tuning to near perfect compatibility in oxides of ZrHfYNb leads to a high degree of irreversibility, as manifested by violently explosive behavior. Tuning lattice parameters slightly away from near perfect compatibility gives an intermediate “weeping” behavior, in which the polycrystal slowly and steadily falls apart at the grain boundaries. Finally, tuning to satisfy a previously unrecognized condition we term the “equidistance condition” results in reversible behavior with the lowest hysteresis in this system. We give evidence that these observations are explained by a more careful mathematical analysis of compatibility of the polycrystal. These results show an extreme diversity of behavior, from reversible to explosive, is possible in a chemically homogeneous system by manipulating conditions of compatibility. Joint work with Eckhard Quandt, Justin Jetter, Jascha Rohmer, Hanlin Gu

ABSTRACT. In this talk we review a series of results obtained in collaboration with E. Runa and A. Kerschbaum concerning one-dimensionality and periodicity of minimizers for a class of local/nonlocal interactions functionals in general dimension. The competition between a local attractive and a nonlocal repulsive term is at the base of the emergency of periodic patterns. In particular, while the functionals we consider are symmetric under permutation of coordinates, in a suitable regime minimizers are one-dimensional, thus incurring in the phenomenon of symmetry breaking. We prove such a characterization of minimizers for functionals which model pattern formation in colloidal systems or in generalized antiferromagnetic interactions, both in the sharp and in the diffuse interface setting.

ABSTRACT. In this talk I will discuss the minimization of an energy functional given by the sum of a crystalline perimeter and a nonlocal interaction of Riesz type, under volume constraint. I will show that, in the small mass regime, if the polygonal Wulff shape of the anisotropic perimeter has certain symmetry properties, then it is the unique global minimizer of the total energy. I will also present a rigidity result for the structure of (local) minimizers in two dimensions. This is a joint work with M. Bonacini and R. Cristoferi.

ABSTRACT. We consider the edge-isoperimetric problem in a periodic weighted graph in Euclidean space. Basic motivation is to understand shapes of crystals beyond lattice cases. The aim is to find cases where the precise isoperimetric inequality can be deduced. By this we mean that the inequality must be achieved by specific shapes, which we also aim to determine. To treat this problem we take inspiration from the PDE approach introduced in the continuum isoperimetric inequality by Cabre, based on early results of Trudinger, and from the Optimal Transport approach to the isoperimetric inequality described by Gromov and by Figalli-Maggi-Pratelli. These two approaches become strongly intertwined in our discrete setting, and our problem becomes related to a Semidiscrete Optimal Transport problem. The precise geometry of cells in the optimal transportation are determined based on a result related to Minkowski's theorem. Several interesting questions appear as next steps in the classification. This is work is based on a project in collaboration with Matias Gomez, who recently completed his master at Universidad Tecnica Federico Santa Maria.

ABSTRACT. Microstructure and shape memory property of TiNi-based alloys not satisfying the cofactor condition were investigated to understand the relation among the incompatibility of martensite microstructure, dislocation structure and shape memory properties. Lattice parameters of the parent and martensite phases were controlled by additional elements. Some alloys exhibited abnormal martensite microstructure in which there is no remnant of parent-martensite interface (habit plane). Shift of martensite transformation temperatures, accumulation of dislocation and shape memory behavior upon cyclic thermal transformation are discussed based on the geometry of the abnormal martensite microstructure.

ABSTRACT. First-principle density functional theory calculations are one of the cornerstones of theoretical condensed matter theory. These first-principles methods allow for the prediction of the physical properties of a wide variety of materials, even those unknown to experiment. Recent advances in first-principle calculations and high-throughput methods allow for the advancement of these calculations as tools for accurately predicting the physical properties of materials. Here, we will summarize recent advances in improving the accuracy of first-principles calculations and describe our high-throughput methods to find the optimal phase-change material. Finally, we will discuss open challenges and questions that arose during our pursuit of our goal and discuss some possible answers to these challenges.

ABSTRACT. (20-1) twins are often observed in B19‘ martensite in the Ni-Ti shape memory alloys deformed beyond the reversibility limit. The mechanism of formation of these twins is assumed to be a combination of twinning and shuffle, which was found to be significantly more energy demanding than all other (pseudo)plastic straining mechanisms in NiTi, in particular the transformation twinning and the [100](001) plastic slip. The (20-1) twins are typically found forming quasi-regular wedge-like patterns, clearly resembling kink patterns observed in systems with extremely anistropic plastic behavior. In this contribution, we will show that the (20-1) twinning can be obtained by combining the plastic kinking and the classical martensitic twinning. This mechanism does not require additional shuffle and might be, thus, energetically favorable. We explore the theoretical framework for describing the coupled kink-twin structure (suggesting the name kwink for it) at the continuum mechanics level, and suggest possible approaches for mathematical modelling of kwinks.

[1] M. Nishida et al. Scripta Materialia 39 (1998) 1749–1754. [2] T. Ezaz et al. Materials Science & Engineering A 558 (2012) 422-430

ABSTRACT. We examine the discrete-to-continuum variational limit of the $J_1$-$J_3$ spin model on the square lattice in the vicinity of the helimagnet/ferromagnet transition point as the lattice spacing vanishes. Carrying out the Gamma-convergence analysis of proper scalings of the energy, we prove the emergence and characterize the geometric rigidity of the chirality phase transitions.

ABSTRACT. We are analyzing the influence of stochastic perturbations on the spectrum of the diffusion operator. The operator has certain defficiencies (there are holes in the domain of radius $\varepsilon$ where the coefficients are of order $\varepsilon^2$). This is in the literature known as high-contrast and the purely periodic case is analyzed by V.V. Zhikov. We are separately analyzing finite and infinite domains, which show different behavior with respect to stochastic homogenization. This is a joint work with M. Cherdantsev (University of Cardiff) and Kirill Cherednichenko (University of Bath).

ABSTRACT. We study weak limits of Sobolev homeomorphisms. It turns out that these mappings are injective almost everywhere if Sobolev exponent p>n−1 both in the image and in the domain. For p≤n−1 we construct a strong limit of homeomorphisms such that the preimage of a point is a continuum for every point in a set of positive measure in the image and a topological image of a point is a continuum for every point in a set of positive measure in the domain. In the critical case p=n−1 injectivity almost everywhere for limits of Sobolev homeomorphisms follows under certain additional assumptions on the distortion of mappings. The first part of the research is joint research with O. Bouchala and S. Hencl (Charles University, Prague, Czech Republic) and the second part is joint research with S. Vodopyanov (Sobolev Institute of Mathematics, Novosibirsk, Russia).

ABSTRACT. This presentation deals with experimentally observed three dimensional solitons that develop in flexoelectric nematic liquid crystals, with negative dielectric and conduction anisotropies, when subject to an alternating electric field. The liquid crystal is confined in a thin region between two plates, perpendicular to the applied field. The horizontal, uniformly aligned director field is at equilibrium due to the negative anisotropy of the media. However, this state is unstable to perturbations that manifest themselves as confined, bullet-like, director distortions traveling up and down the sample at a speed of several hundred microns per second. We develop a variational model that couples the Ericksen-Leslie equation of director field distortion and the Poisson-Nearnst-Planck equations governing the diffusion and transport of electric charge, and the electrostatic potential. We perform a stability analysis of the equilibrium state to determine the threshold wave numbers and speed of the disturbance. The linear system obtained by taking the Fourier transform of the original one is analyzed by combining methods of the Floquet theory, together with time and space averaging tools as well as arguments based on asymptotic analysis. The predictions based on the size, phase-shift and speed of the soliton show good agreement with the experimentally measured ones.

ABSTRACT. We propose a novel free energy functional for modelling Smectic-A liquid crystals. The functional is written in terms of a Q-tensor and a real-valued density variable. The model can reproduce important and partially-understood structures in smectics, such as focal conic domains and oily streaks. The model is nonconvex and supports many solutions; calculating the ground state is therefore a difficult numerical challenge. We discuss the computation of multiple solutions using a nonlinear deflation technique.

ABSTRACT. A variational model for the delamination of polymer gel thin films from rigid substrates is presented. A formal asymptotic analysis of a simplified 2D version of the underlying governing equations show that, as the film grows thinner, the absorption of the moisture of its surroundings tends to produce a homogeneous vertical stretch in the part of the film that remains bonded to the substrate and a state of free swelling in the debonded part. A transition layer, with a width comparable to the film thickness, is developed in order to connect the two swelling modes. This work is joint with Carme Calderer (U. Minnesota), Carlos Garavito-Garzon (U. Minnesota), Suping Lyu (Medtronic, Inc.), and Lorenzo Tapia (PUC - Chile).

ABSTRACT. Liquid crystals with molecules constrained to the tangent bundle of a curved surface show interesting phenomena resulting from the tight coupling of the elastic and bulk-free energies of the liquid crystal with geometric properties of the surface. We derive a thermodynamically consistent Landau-de Gennes- Helfrich model which considers the simultaneous relaxation of the Q-tensor field and the surface. The resulting system of tensor-valued surface partial differential equation and geometric evolution laws is numerically solved to tackle the rich dynamics of this system and to compute the resulting equilibrium shape. The results strongly depend on the intrinsic and extrinsic curvature contributions and lead to unexpected asymmetric shapes.

ABSTRACT. In direct simulations using Born-Oppenheimer Molecular dynamics (BOMD), the underlying electron structures have to be computed at each time step, which involves the computation of the eigenvalues and eigenstates. In addition, the calculation of the forces on the nuclei also requires access to the density-matrix and the energy density-matrix. We present a reformulation of the BOMD with a stochastic relaxation, which is free of the diagonalization steps. In addition, we show the convergence of the relaxed model.

ABSTRACT. High entropy alloys (HEAs) are single phase crystals that consist of random solid solutions of multiple elements in approximately equal proportions. This class of novel materials have exhibited superb mechanical properties, such as high strength combined with other desired features. The strength of crystalline materials is associated with the motion of dislocations. In this talk, we present a stochastic continuum model based on the Peierls--Nabarro framework for interlayer dislocations in a bilayer HEA from an atomistic model that incorporates the atomic level randomness. We use asymptotic analysis and limit theorem in the convergence from the atomistic model to the continuum model. The total energy in the continuum model consists of a stochastic elastic energy in the two layers, and a stochastic misfit energy that accounts for the interlayer nonlinear interaction. The obtained continuum model can be considered as a stochastic generalization of the classical, deterministic Peierls--Nabarro model for the dislocation core and related properties. This derivation also validates the stochastic model adopted by Zhang et al. [Acta Mater., 166 (2019), pp. 424--434].

ABSTRACT. We propose a novel numerical algorithm for computing the electronic structure related eigenvalue problems of incommensurate systems. Unlike the conventional practice that approximates the system by a large commensurate supercell, our algorithm directly discretizes the eigenvalue problems under the framework of a plane wave method. The emerging ergodicity and the interpretation from higher dimensions give rise to many unique features compared to what we have been familiar with in the periodic systems. The numerical results of 1D and 2D quantum eigenvalue problems are presented to show the reliability and efficiency of our algorithm. Furthermore, the extension of our algorithm to full Kohn-Sham density functional theory calculations is discussed.

ABSTRACT. This talk will give an overview on Density Functional Theory (DFT), its successes and open challenges.

ABSTRACT. This will be an overview talk on the mathematical models of electronic structure. We will mainly focus on the density functional theory with semilocal exchange-correlation functionals and the corresponding mathematical structure. In particular, we will discuss the localized representation of the occupied space for insulating systems, including density matrices and Wannier functions.

ABSTRACT. Noncovalent van der Waals (vdW) or dispersion forces are ubiquitous in nature and influence the structure, stability, dynamics, and function of molecules and materials throughout chemistry, biology, physics, and materials science. These forces are quantum mechanical in origin and arise from electrostatic interactions between fluctuations in the electronic charge density. This lecture will discuss the mathematical and physical concepts, methodology, and practice for treating the ubiquitous vdW interactions in materials modeling. We will discuss a common framework [Chem. Rev. 117, 4714 (2017); Science 351, 1171 (2016)] for understanding all existing vdW-enabled density functionals, show their strengths and limits, and conclude with many challenges that remain towards the development of a universal vdW methodology, which will be able to deliver accurate and robust insights into physical, chemical, and biological systems.

ABSTRACT. The talk provides an overview of the application of density functional theory (DFT) to study the mechanical properties of crystalline solids. Of particular interest are defects that play a critical role in the determination of mechanical properties even at small concentrations. These are challenging because they call for the solution of the expensive equations of DFT on very large domains. So we introduce a spectral reformulation of DFT that enables the computation of quantities of interest locally. We then take advantage of this local property for sub-grid sampling which leads to a sub-linear scaling method that enables calculations at realistic scales. Finally, the talk describes how one can exploit machine learning approximations for preconditioning. We illustrate these ideas using selected examples, and describe open mathematical questions.

ABSTRACT. Variational phase-field models of fracture have been at the center of a large multidisciplinary effort over the last 25 years or so. In this talk, I will start with a modern interpretation of Griffith's classical criterion as a variational principle for a free discontinuity energy and will recall some of the milestones in its analysis. Then, I will introduce the phase-field approximation per se and describe its numerical implementation. I will show how this approach has led to a paradigm shift in the predictive understanding of fracture in brittle solids in a broad range of applications, including some multi-scale and multi-physics problems, the design of meta-materials with extreme fracture properties, and advances in manufacturing and material testing. Finally, I will conclude with open questions and challenges.

ABSTRACT. We study de Gennes and Chen-Lubensky free energies for smectic A liquid crystals over S^2 valued vector fields to understand the chevron (zigzag) pattern formed in the presence of an applied magnetic field. As the applied field increases well above the critical field, the sinusoidal shape of the smectic layer at the onset of undulation will change into the chevron patterns with a longer period. We consider a square domain to represent the cross section of a three dimensional smectic A liquid crystal sample. Well above the instability threshold, we show via Gamma-convergence that a chevron structure where the director connects two minimum states of the sphere is favored. Numerical simulations illustrating the chevron structures for both models will be presented. If times permits, We will also discuss the switching mechanism in smectic tilted columnar phases using electric fields. This work was done in collaboration with Prof. Tiziana Giorgi and Prof. Sookyung Joo.

ABSTRACT. We present a scalable and matrix-free eigensolver for studying nearest-neighbor Heisenberg spin chain plus random on-site disorder models that undergo a many-body localization (MBL) transition. This type of problem is computationally challenging because its dimension grows exponentially with the physical system size, and the solve must be iterated many times to average over different configurations of the random disorder. For each eigenvalue problem, eigenvalues from different regions of the spectrum and their corresponding eigenvectors need to be computed. Traditionally, the interior eigenstates for a single eigenvalue problem are computed via the shift-and-invert Lanczos algorithm. Due to the extremely high memory footprint of the LU factorizations, this technique is not well suited for large number of spins, e.g., one needs thousands of compute nodes on modern high performance computing infrastructures to go beyond 24 spins. We propose a new matrix-free approach that does not suffer from this memory bottleneck and even allows for simulating spin chains up to 24 spins on a single compute node. The efficiency and effectiveness of the proposed algorithm is demonstrated by computing eigenstates in a massively parallel fashion, and analyzing their entanglement entropy to gain insight into the MBL transition beyond 26 spins.

ABSTRACT. In quantum embedding theories, a quantum many-body system is divided into localized clusters of sites which are treated with an accurate `high-level' theory and glued together self-consistently by a less accurate `low-level' theory at the global scale. We introduce variational embedding, which combines the insights of semidefinite relaxation and quantum embedding theory to provide a lower bound on the ground-state energy that improves as the cluster size is increased. The variational embedding method is formulated as a semidefinite program (SDP), which suffers from suboptimal computational scaling when treated with black-box solvers. We exploit the interpretation of this SDP as an embedding method to develop an algorithm which alternates parallelizable local updates of the high-level quantities with updates that enforce the low-level global constraints.

ABSTRACT. Solving the Schroedinger equation numerically for N electrons suffers from the curse of dimension as an electronic wavefunction is a function on R^3N. So for just a single water molecule you are already dealing with a function on R^30. I discuss rigorous mathematical results on sparsity of electronic wavefunctions in special model systems or asymptotic scaling regimes which are either weakly or strongly correlated. Both the Tucker format as associated with the CI (Configuration-Interaction) and MCSCF (Multi-Configuration Self Consistent Field) method and in the more recent tensor-train/matrix product state format associated with the QC-DMRG (Quantum Chemistry Density Matrix Renormalization Group) method are considered.

The talk is based on joint work with various co-authors over the years: Benjamin Goddard (Edinburgh), Huajie Chen (Beijing Normal University), Mi-Song Dupuy (TU Munich), and Benedikt Graswald (TU Munich).

ABSTRACT. Part of the high computational cost of quantum Monte Carlo methods comes from the necessity of using small time steps to keep the error coming from the Trotter breakup of the projectos small. Some recent developments have enabled the use of considerably larger time steps than previously possible. One development pertains to the use of pseudopotentials. One of two approximations are usually made when using pseudopotentials in diffusion Monte Carlo, the locality approximation and the T-moves approximation. Each has its advantages and disadvantages relative to the other. We describe a method that combines the advantages of both methods. The other development enables large time steps in both pseudpotential and all-electron calculations.

ABSTRACT. Cellular networks are ubiquitous in nature. Most engineered materials are polycrystalline microstructures composed of a myriad of small grains separated by grain boundaries, thus comprising cellular networks. A central problem is to develop technologies capable of producing an arrangement, or ordering, of the material, in terms of mesoscopic parameters like geometry and crystallography, appropriate for a given application. Is there such an order in the first place? We describe the emergence of the grain boundary character distribution (GBCD), a statistic that details texture evolution, and illustrate why it should be considered a material property. Its identification as a gradient flow by our method is tantamount to exhibiting the harvested statistic as the iterates in a mass transport JKO implicit scheme, which we found astonishing. Consequently the GBCD is the solution of a Fokker-Planck Equation. The development exposes the question of how to understand the circumstances under which a harvested empirical statistic is a property of the underlying process.

(joint work with P. Bardsley, K. Barmak, E. Eggeling, M. Emelianenko, Y. Epshteyn and S. Ta'asan)

ABSTRACT. It has been argued for quite a while that there is one more law of thermodynamics controlling the transition from mesoscale to macroscale: entropy of mesoscale microstructures decays in isolated systems. The talk aims to describe this concept and present recent experimental results confirming entropy decay for the process of grain growth [1]. It will be discussed also the problem of constitutive relations in grain growth.

1. P. Vedanti, X. Wu, V. Berdichevsky, 2020, Entropy decay during grain growth, Scientific Reports, 10 :11912

ABSTRACT. Mathematical analysis of the evolution of grain boundaries is one of the main topics in material sciences. In my talk, I explain the modeling of planar grain boundaries together with dynamic lattice orientations and triple junction drag. By the maximal dissipation principle, we obtain three equations; a curve shortening equation, the evolution of the orientations, and the triple junctions. We take the relaxation limit for the curve shortening equation in order to study the effect of the misorientations and the triple junction drag. Next, I show mathematical analysis of the system, such as the existence and long time asymptotics of a solution. Finally, we discuss the extension of the model to grain boundary networks and show numerical calculations.

ABSTRACT. We present a sixth-order phase field crystal equation which is a phase field model describing the dynamical formation of crystalline structures. The numerical simulation of this equation presents challenges such as the spatial discretization of the nonlinear higher-order differential operators that appear in the equation and the approximation of dynamic interfaces that travel over the entire material. To cope with these challenges, we propose an unconditionally stable and uniquely solvable convergent numerical scheme and prove these theoretical properties. We close the talk by presenting the numerical results of some benchmark problems to verify the practical performance of the proposed approach.

ABSTRACT. Randomness is ubiquitous in natural sciences and modern engineering. The uncertainty is often modeled as a random coefficient in the differential equations that describe the physics. The resulted stochastic problems are often solved by the polynomial chaos (PC) approach that expands the weak solution in the polynomial chaos basis. In this work, we recast the stochastic problem as a Dirichlet energy minimization problem using the variational characterization. This formulation naturally leads us to apply the stochastic gradient descent (SGD) to solve the stochastic optimization problem. Since the SGD often has very slow convergence rate, we propose a control variate type variance reduction method to speed up the convergence of SGD. We provide numerical evidence to demonstrate that our SGD based solver can significantly outperform the vanilla PC solver in some cases.

ABSTRACT. Pattern formation is a widely observed phenomenon in diverse fields including materials physics, developmental biology and many others. The physics underlying the patterns is specific to the mechanisms, and is encoded by partial differential equations (PDEs). With the aim of discovering hidden physics, we have previously presented a variational approach to identifying such systems of PDEs in the face of noisy data at varying fidelities (Comp Meth App Mech Eng, 353:201 2019). We extend our methods to address the challenges presented by image data on microstructures in materials physics. PDEs are posed as initial and boundary value problems over combinations of time intervals and spatial domains whose evolution is either fixed or can be tracked. However, micrographs of pattern evolution in materials are over domains that are unrelated at different time instants, and come from different physical specimens. The temporal resolution can rarely capture the fastest time scales, and noise abounds. We exploit the variational framework to choose weighting functions and identify PDE operators from such dynamics. A consistency condition arises for parsimonious inference of spatial operators at steady state, complemented by a confirmation test for acceptance of the inferred operators. The framework is demonstrated on synthetic and real data.

ABSTRACT. We demonstrate a new data-driven approach to physics discovery with predictive abilities on a variety of multiscale material problems. We rely on a probabilistic learning on manifolds (PLoM) framework. First, we infer an intrinsic structure that describes the physical quantities using diffusions maps (DMAPS). Data used for the physics discovery is sampled from the observation fields using stencils adapted to each application. The shape and size of the stencils encode our beliefs about physics and relationships that govern the different quantities. Once a low-dimensional manifold embedding is discovered, we augment the training dataset by using a sampling on manifolds technique that combines the solution to an Îto stochastic differential equation with a high-dimensional Gaussian Kernel Density Estimate. Finally, non-parametric density estimation and conditioning are performed on the augmented dataset to predict physical quantities of interest. We also explore the sensitivity of the learning and prediction to various DMAPS parameters, as well as the structure and size of the stencil used to draw samples from the observation data.

ABSTRACT. We introduce twisted trilayer graphene (tTLG) with two independent twist angles as an ideal system for the precise tuning of electronic interlayer coupling to maximize the effect of correlated behaviors. As established by experiment and theory in the related twisted bilayer graphene system, van Hove singularities (VHS) in the density of states can be used as a proxy of the tendency for correlated behaviors. To explore the evolution of VHS in the twist-angle phase space of tTLG, we present a general low-energy electronic structure model for any pair of twist angles. We show that the basis of the model has infinite dimensions even at a finite energy cutoff and that no Brillouin zone exists even in the continuum limit. Using this model, we demonstrate that the tTLG system exhibits a wide range of magic angles at which VHS merge at the charge neutrality point through two distinct mechanisms: the incommensurate perturbation of twisted bilayer graphene's flat bands or the equal hybridization between two bilayer moiré superlattices. Joint work with Ziyan Zhu, Stephen Carr, Daniel Massatt, and Efthimios Kaxiras

ABSTRACT. This talk will discuss two recent developments that can pave the way for large-scale quantum accuracy materials simulations using density functional theory (DFT). Firstly, we present the development of computational methods and numerical algorithms for conducting fast and accurate large-scale DFT calculations using adaptive higher-order finite-element discretization, which forms the basis for the recently released DFT-FE open-source code. The computational efficiency, scalability and performance of DFT-FE will be presented, which will demonstrate a significant outperformance of widely used plane-wave DFT codes. Secondly, we will discuss the ongoing efforts in improving the exchange-correlation description in DFT, leveraging the recent breakthrough in computing exact exchange-correlation potentials from ab-initio correlated densities for polyatomic systems.

ABSTRACT. Applications in energy harvesting, soft robotics, wearable and prosthetic devices call for the development of new soft responsive materials that can be easily synthesized and integrated into functional devices. One approach to doing so is to incorporate responsive molecules or particles into a solid matrix, but this requires a careful balance to optimize the overall response. We provide a variational formulation for this problem, and this naturally leads to phase transforming solids. We then turn to the problem of integrated functional devices, and discuss the mathematical difficulties of optimal design of such devices. We illustrate both topics with selected examples

ABSTRACT. We study energy-driven pattern formation in shape memory alloys. No matter how shape memoryalloys are deformed at a low temperature, they return to their original shape by heating them above acritical temperature. The cause of this phenomenon is a first order solid-to-solid phase transformation.Microstructures close to phase boundaries of martensitic nuclei can be modeled variationally. Weconsider a model by Kohn and M ̈uller (1992 & ’94), and prove asymptotic self-similarity of minimizers.This generalizes results by Conti (2000) to various physically relevant boundary conditions, moregeneral domains, and arbitrary volume fractions, including low-hysteresis shape memory alloys. The proof relies on pointwise estimates and local energy scaling laws for minimizers.

Based on a joint work with Sergio Conti, Johannes Diermeier and Barbara Zwicknagl.

ABSTRACT. Situated between liquid and crystalline phases, liquid crystals are phases with various physical applications including their utilization in optical and display devices. The molecular shape of a liquid crystal material has profound effects on the macroscopic physics of the phase, causing some material to produce more efficient and less expensive optical devices than others. One example is a recently discovered phase made up of bow-shaped molecules, a characteristic that endows them with spontaneous ferroelectricity. Under the effect of an applied electric field, two competing mechanisms of switching can be detected in the tilted structure of these materials. An important question in this setup is how switching can be affected by specific system parameters. We formulate the model as an energy minimization problem allowing us to use several variational tools in its analysis. We emphasize how we can deal with challenges that arise from mixed constraints and nonlinearities peculiar to this problem. Our results address existence and uniqueness of solutions to the ensuing partial differential equations, which in turn shed light on the physical mechanisms observed.

ABSTRACT. The determination of the effective properties of viscoelastic composites is a long-standing problem. Many of the results proposed in the literature have been obtained by applying variational methods and by considering the response of the material to cyclic loadings. On the other hand, very few results have been provided for the case of transient loadings. In this talk, we will highlight the challenges of using variational formulations and we will provide some alternative solutions.

ABSTRACT. Rigidity results in elasticity are powerful statements that allow deriving global properties of a deformation from local ones. The classical Liouville theorem states that every local isometry of a domain corresponds to a rigid body motion. If the connectedness of the set fails, clearly, global rigidity can no longer be true.

In this talk, we will discuss two new asymptotic rigidity theorems, showing that if an elastic body contains sufficiently stiff connected components arranged into fine parallel layers or fibers, then strict global constraints of anisotropic nature occur in the limit of vanishing layer or cross-section thickness. Besides their theoretical interest, these findings facilitate the homogenization of variational problems modeling high-contrast bilayered and fiber-reinforced composite materials. We will address two types of models in nonlinear elasticity and finite crystal plasticity and show how to determine their homogenized Gamma-limits in terms of explicit formulas.

ABSTRACT. Bacteriophages densely pack their long dsDNA genome inside a proteinic capsid. The conformation of the viral genome inside the capsid is consistent with a hexagonal liquid crystalline structure, and experimental results have confirmed that it depends on environmental ionic conditions. In this work, we propose a biophysical model to describe the dependence of DNA configurations inside bacteriophage capsids on ions types and concentrations. The total free energy of the system combines the liquid crystal free energy, the electrostatic energy and the Lennard--Jones energy. The equilibrium points of this energy solve a highly nonlinear, second order partial differential equation (PDE) that defines the distributions of DNA and the ions inside the capsid. We develop a computational approach to simulate predictions of our model. The numerical results show good agreement with existing experiments and molecular dynamics simulations.

ABSTRACT. In the Landau-de Gennes $\mathbf{Q}$-tensor model for uniaxial nematic liquid crystals with variable degree of orientation, molecule distribution is given by a rank-one tensor and its degree of orientation by a scalar field. We present a structure-preserving discretization of the liquid crystal energy with piecewise linear finite elements that can handle the degenerate elliptic resulting problem without regularization, and show that it is consistent and stable. We present simulations in two and three dimensions to illustrate the method's ability to handle non-trivial defects as well as colloidal and electric field effects.

ABSTRACT. The sliding friction is important for the motion of soft materials like rubber and gel on substrates. We develop a mathematical model for the sliding of a gel sheet adhered to a moving substrate. The sliding takes place by the motion of detached region between the gel sheet and the substrates, i.e., the propagation of a Schallamach wave. The model is based on a reduced elastic energy for a thin film and the Onsager Principle for its dynamical sliding. Efficient numerical methods are developed to solve the problem. Numerical examples illustrate that the model can describe the Schallamach wave and are consistent with the existing experiments qualitatively.

ABSTRACT. The genome of some viruses, such as bacteriophages or human herpes, is a double stranded DNA (dsDNA) molecule that is stored inside a viral protein capsid at a concentration of 200 mg/ml-800mg/ml and an osmotic pressure of 60 atmospheres. The organization of the viral genome under these extreme physical conditions is believed to be liquid crystalline. Cryoelectron microscopy experiments suggest that dsDNA near the viral capsid is in a chromonic state, and it is found in an isotropic state at the center of the capsid. Topological experiments of DNA extracted from viral capsids show a wide variety of knotted molecules and a knot distribution suggests a chiral organization of the genome. In this talk we will discuss the emergent picture that suggests that DNA is a chirally organized liquid crystalline phase in which knots may be the product of liquid crystal defects.

ABSTRACT. Monopoles, dubbed dowsons, are point singularities of the unitary complex order parameter exp(iphi) characterizing the so-called dowser texture in a thin nematic layer with homeotropic boundary conditions. Using setups called dowsons colliders, pairs of dowsons are generated and set into motion on counter-rotating trajectories leading to collisions. In a first approximation, the velocity of dowsons is orthogonal and proportional to the local phase gradient. The outcome of collisions depends on the distance of dowsons' trajectories in terms of the phase: for Dphi< Pi, a collision of a pair of dowsons leads to their annihilation while for Dphi > Pi the colliding dowsons are passing by. This rule is valid only for quasi-static stationary wound up phase fields and can be easily broken by application of a Poiseuille flow in an appropriate direction.

Pairs of disclinations are generated in a reproducible manner by the isotropic-nematic transition in twisted nematic cells submitted to magnetic fields of appropriate geometries. Their collision, driven by an electric field, leads to a rewiring process that can be fully controlled by means of the magnetic and electric fields.

ABSTRACT. We investigate minimizers for the Landau-de Gennes energy for liquid crystals in a slab, D x (-a,a) where the cross section D is a bounded simply connected domain in two dimensions. We require that the liquid crystal is tangential on the top and bottom and prescribe boundary conditions on the lateral surfaces. Our results describe the defect structures and provide estimates on the energies of minimizers. We also show how these depend on the relative sizes of the Landau-de Gennes order parameter and the slab thickness 2a. This is joint work with Dan Phillips.

ABSTRACT. We investigate the structure of nematic liquid crystal thin films described by the Ball Majumdar, tensor-valued order parameter model with Dirichlet boundary conditions on the sides of nonzero degree. We prove that as the elastic modulus goes to zero in the energy a limiting uniaxial nematic texture forms, locally minimizing a planar Frank energy trapping a finite number of defects, all of degree 1/2 or all of degree -1/2, corresponding to vertical disclination lines at those locations.

ABSTRACT. In smectic liquid crystals, rod-like molecules tend to align parallel to each other along a common director n and form parallel layers with equal thickness of the order of a few nm.

These materials may exhibit defects, i.e., regions of rapid changes in the molecular orientation, that occur along a surface. Thus, it might be reasonable to consider variational models which, unlike most classical theories for liquid crystals, are based on discontinuous order parameters, as proposed in a recent paper by Ball and Bedford.

In this talk, I'll discuss some preliminary results towards the mathematical analysis of defect patterns in thin film of smectic A phases.

ABSTRACT. Tensor network states and specifically matrix-product states have proven to be a powerful tool for simulating ground and excited states of strongly correlated spin and fermionic models. In this contribution, we overview tensor network states techniques, like Hierarchical Tucker tensor (HT) format and Tensor Trains (TT), that can be used for the treatment of high-dimensional optimization tasks used in many-body quantum physics with long range interactions, ab initio quantum chemistry and nuclear structure theory. We will also discuss the controlled manipulation of the entanglement, which is in fact the key ingredient of such methods, and which provides relevant information about correlations. We will present recent developments on fermionic orbital optimization, tree-tensor network states, multipartite entanglement, externally corrected coupled cluster density matrix renormalization group (TCCSD-DMRG). Finally, new results will be shown for Wigner crystals.

[1] Szalay, Pfeffer, Murg, Barcza, Verstraete, Schneider, Legeza, INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 115:(19) pp. 1342-1391. (2015)

[2] Krumnow, Veis, Legeza, Eisert, Phys. Rev. Lett. 117, 210402 (2016)

[3] Veis, Antalik, Neese, Legeza, Pittner, Journal of Physical Chemistry Letters, 7 (20) 4072-4078 (2016)

[4] Shapir, Hamo, Pecker, Moca, Legeza, Zaránd, Ilani, Science 364(6443):870-875 (2019).

ABSTRACT. Strongly-correlated quasi-1d quantum many-body systems can in many cases be efficiently described by matrix product states (tensor trains). In this talk, I will discuss the relation between the scaling of entanglement entropies and the scaling of tensor dimensions required to achieve a certain approximation accuracy. Field-theoretical results for the entanglement suggest that ground states and equilibrium finite-temperature states can be simulated with polynomial cost. I will also describe new schemes that allow us to efficiently compute dynamic response functions, which inform about excitations and are central experimental quantities. An important frontier of current research is the simulation of nonequilibrium dynamics in quantum matter. Interesting questions concern equilibration and thermalization, dynamical phase transitions, decoherence effects, quantum transport etc. A major obstacle for corresponding matrix product state simulations is the growth of entanglement with time. I will show how this problem can be mitigated through a new trick that we call entanglement decomposition.

ABSTRACT. We consider tensor-structured methods for numerical modeling of the electrostatic potentials in many-particle systems [1]. The method of grid-based assembled tensor summation of the electrostatic potentials on 3D finite lattices exhibits the computational complexity of the order of $O(L)$ compared with $O(L^3)$ for traditional Ewald-type summation. The canonical tensor rank of the collective potential of a finite 3D rectangular lattice is proven to be as low as a rank for a single reference tensor for a Newton kernel. Recent range-separated (RS) tensor format applies to many-particle systems of general type. Main advantage of the RS tensor format is that the rank of the canonical tensor representing the sum of long range contributions from all particles in the collective potential depends only logarithmically on the number of particles $N$. The interaction energies and forces of the many-particle system are computed by using only the long-range part of the collective potential, with representation complexity $O(n\log N)$, where $n$ is the univariate grid size. Basic tool for calculation of the RS tensor representation is the reduced higher order SVD (RHOSVD) [1]. The numerical examples are presented.

[1] Venera Khoromskaia and Boris N. Khoromskij. Tensor Numerical Methods in Quantum Chemistry. De Gruyter, Berlin, 2018.

ABSTRACT. Weak van der Waals interactions between 2D materials do not impose limitations on integrating highly disparate atomically thin layers such as graphene, phosphorene, boron-nitride... This is both a blessing, allowing the realization of many stable assemblies, but also a computational curse due to the loss of periodicity.

We will discuss how the computation of mechanical and electronic quantities of interest can be efficiently implemented within a unified framework: the space of configurations. This allows us in particular to introduce a novel spectral element discretization of the tight-binding generating functions which reduces the complexity of evaluating Kubo formulae and compute macroscopic observables such as conductivity.

ABSTRACT. The last decade has seen a renaissance in the geometrically-nonlinear buckling of sheets, which is central to the mechanics of synthetic skins, biological tissues, textiles, and 2D materials like graphene. Significant effort has been devoted to understanding and exploiting buckled structures on nearly-planar substrates. We show that smooth wrinkles do not survive to large curvatures in general geometries, giving way to a generic “crumpled” [1] response that we characterize in a suite of experiments across scales [2]. We probe the wrinkle-to-crumple transition using polymer films confined on spherical, hyperbolic, and cylindrical surfaces, and we reproduce the same behaviors in macroscopic membranes inflated with gas. These varied setups reveal robust morphological features of the crumpled phase, and they allow us to disentangle the competing effects of curvature and compression. Our work highlights the need for a theoretical understanding of this ubiquitous elastic building block, and we lay out concrete directions for such studies. [1] King et al., PNAS 109, 9716 (2012). [2] Timounay et al., PRX 10, 021008 (2020).

ABSTRACT. Gauss' Theorema Egregium implies that changes in Gaussian curvature of a surface must involve stretching of the surface. For thin elastic sheets, bending deformations are favourable over stretching and so Gaussian curvature is generally preserved. As a consequence, curvature imposed in the transverse direction induces a resistance to bending in the longitudinal direction: the sheet would have to stretch to accommodate a deformation that requires a change of the Gaussian curvature, which is prohibitively expensive. The inhibition of bending in one principal direction that results from an imposed curvature in the orthogonal direction is referred to as 'curvature-induced rigidity'. Here, we investigate the behaviour of a rectangular strip of finite thickness held horizontally in a gravitational field, with a transverse curvature imposed at one end. There are two ways that the finite thickness of the sheet affects the efficacy of curvature-induced rigidity: the finite bending stiffness acts to `uncurve' the sheet, at the expense of some stretching energy; the finite weight deforms the strip downwards, eventually causing buckling. We identify the dimensionless parameters that control this problem and determine the critical imposed curvature required to prevent buckling before describing the buckled shape of the strip well beyond the buckling threshold.

ABSTRACT. We revisit the problem of buckling of cylindrical shells under axial compression, considering cylinders with convex cross sections. We prove that if the cross section has a uniformly apart from zero curvature, then the critical buckling load scales like the thickness to the power of 1, and if it contains at least one zero curvature point, then the buckling load scales at most like the thickness to the power 1.5. This provides another evidence to the famous sensitivity to imperfections phenomenon for the cylindrical shell buckling problem. This is joint work with Andre Martins Rodrigues (UCSB).

ABSTRACT. We consider a thin elastic sheet in the shape of a sector that is clamped along the curved part of the boundary, and left free at the remainder. On the curved part, the boundary conditions agree with those of a conical deformation. We prove upper and lower bounds for the Föppl-von-Kármán energy under the assumption that the out-of-plane component of the deformation is convex. The lower bound is optimal in the sense that it matches the upper bound in the leading order with respect to the thickness of the sheet. As a corollary, we obtain a new estimate for the Monge-Ampère equation in two dimensions. Joint work with Peter Gladbach (Bonn).

ABSTRACT. Nonlocal theories have been developed around successful implementation of nonlocal models in different areas of science (continuum mechanics, biology, image processing). Theoretical advances have been made by introducing and investigating integral operators and associated systems of equations. In this talk I will present some recent results on nonlocal frameworks systems based on some existing, as well as some newly introduced, nonlocal operators. An in-depth study of properties of the operators includes a series of results on nonlocal versions of integration by parts theorems, Helmholtz-Hodge type decompositions, as well as convergence of operators to their classical equivalents as the interaction horizon vanishes.

ABSTRACT. We present a real-space computational method called treecode-accelerated Green Iteration (TAGI) for all-electron Kohn-Sham Density Functional Theory. In this approach the Kohn-Sham differential eigenvalue problem is converted into an integral fixed-point problem by convolution with the modified Helmholtz Green's function. The fixed-point problem is solved by self-consistent field iteration and Green Iteration, where the discrete convolution sums are evaluated by a GPU-accelerated barycentric Lagrange treecode. TAGI also uses adaptive mesh refinement, Fejer quadrature, singularity subtraction, gradient-free eigenvalue update, and Anderson mixing. Ground state energy computations for several atoms and small molecules demonstrate TAGI's ability to achieve chemical accuracy.

ABSTRACT. A problem for a nanosized penny-shaped fracture in an infinite homogeneous isotropic elastic medium is considered. The fracture is opened by applying an axisymmetric normal traction to its surface. The surface energy in the Steigmann-Ogden form is acting on the boundary of the fracture. The problem is solved by using the Boussinesq potentials represented by the Hankel transforms of certain unknown functions. With the help of these functions, the problem can be reduced to a system of two singular integro-differential equations. The numerical solution of this system can be obtained by expanding the unknown functions into the Fourier-Bessel series. Then the approximations of the unknown functions can be obtained by solving a system of linear algebraic equations. Accuracy of the numerical procedure is studied. Various numerical examples for different values of the surface energy parameters are considered. Parametric studies of the dependence of the solutions on the mechanical and the geometric parameters of the system are undertaken. It is shown that the surface parameters have a significant influence on the behavior of the material system. In particular, presence of surface energy leads to the size-dependency of the solutions and increased toughness of the material.

ABSTRACT. In this talk, we discuss the homogenization of a variational boundary value problem that governs the small vibrations of a periodic mixture of an elastic solid and a slightly viscous fluid with a slip boundary condition on their interface. We use two-scale convergence to obtain the macroscopic behaviour of the solution and identify the role played by the connectedness of the phases and the interface boundary condition.