ABSTRACT. Failure in amorphous solids arises from local rearrangements in the material's atomic configurations and how these respond to stress. To this end, the Local Yield Stress (LYS) method was recently developed to probe local regions and quantify their susceptibilities to plasticity by measuring the incremental stress required to trigger a local rearrangement. While this methodology shows great potential for enhancing predictive plastic theories, it is not a practical means for characterizing materials structure due to its computationally intensive nature. We propose a manifold learning-based framework to extract microstructural descriptors from atomistic configurations. More specifically, we deploy Diffusion Maps (a nonlinear manifold learning technique) to systematically extract the structural information features from the high-dimensional data (cartesian coordinates of local clusters of atoms and relate this structure to LYS). Due to the nature of the problem, a “point” corresponds to a geometric conformation of atoms, and thus meaningful similarity measure between configurations must be devised for capturing the actual distance between two points. Therefore, we utilize the Grassmann manifold to measure the similarity between the cluster of atoms. Our application is a two-dimensional binary glass-forming system, deformed with an incremental Athermal Quasistatic Shear (AQS) method.

ABSTRACT. Moment invariants define a shape quantification concept in physics. According to this concept, physical shapes can be represented with the values of the nonlinear moment functions that are invariant to affine and/or similarity transformations. While this technique has been utilized in other fields, its use in materials science is limited. In this study, we present a numerical methodology that is based on shape moment invariants to quantify the grain topology of polycrystalline microstructures. The topological quantification of polycrystals in micro-scale is still a challenging problem due to a wide variety and complexity of possible grain shapes, including highly non-convex or concave grains, as a result of the strong microstructure formation during thermo-mechanical processing. The state-of-the-art quantification methods are based on two-point correlation functions that can only provide implicit information of the grain shapes; however, they do not explicitly detect the sizes and shapes of individual grains. To overcome this fundamental drawback, the moment invariants are used as topology descriptors in this study to quantitively model the grain shapes of polycrystalline microstructures. The methodology is tested on several anisotropic microstructure samples that are processed with conventional forging and additive manufacturing techniques.

ABSTRACT. Polycrystalline materials are made up of aggregates of crystal grains. The electron-backscatter diffraction (EBSD) is a scanning electron microscopy (SEM) based technique used to obtain the grain orientation data. This data is instrumental to quantitative modeling and understanding of materials. EBSD data often come with many misoriented pixels, which have the appearance of noise, and may have regions of missing data. To alleviate these issues, we have developed PDE-based denoising and inpainting algorithms, which produce high quality reconstructions of the EBSD images. Our algorithms build on weighted total variation flow and minimal surface equation. We compare our PDE-based algorithms with other denoising and data reconstruction algorithms commonly used for EBSD images. We observe that our algorithms outperform others and produced better reconstructions.

ABSTRACT. A major objective of integrated computational materials engineering (ICME) is to simulate microstructure evolution during thermomechanical processing at the grain scale. Explicitly triangulating the grain boundaries is useful when evaluating properties, e.g. grain boundary energy and mobility, but makes changing the underlying stratification and maintaining the finite element mesh difficult. Several existing simulations of this system cannot be used to predict the behavior of real materials because they do not allow for anisotropic grain boundary properties, have unphysical anisotropy from the underlying numerical model, or allow only a restricted set of topological events that bias the grain boundary network evolution.

Recent progress will be reported in developing a finite element simulation that (1) uses a volumetric mesh to eventually allow the inclusion of arbitrary material physics, (2) significantly expands the set of topological events to allow for general grain boundary network dynamics, and (3) proposes an energy dissipation criterion to identify the physically most plausible of these events. In addition, several topological events not usually considered in such simulations will be shown to occur even for the simplest case of homogeneous grain boundary energy to emphasize the importance of correctly identifying and allowing the required changes in the topology.

ABSTRACT. The purpose of this article to study the interaction between two moving Griffith cracks under anti-plane shear and electro-mechanical loadings. These cracks are located between two dissimilar layers of functionally graded piezoelectric materials (FGPMs). It is assumed that the FGPMs' layers are connected weak-discontinuously and their properties are varying continuously along the thickness. For the propagation of cracks, a constant velocity Yoffe-type model is considered. The Fourier cosine transform is used to solve the dynamic anti-plane equations which are used to obtain a system of Fredholm integral equations of the second kind. These equations are solved numerically to obtain dynamic intensity factors namely, dynamic stress intensity factors, dynamic strain intensity factors, dynamic electric displacement intensity factors, and dynamic electric field intensity factors. Dynamic energy release rates (DERRs) are calculated numerically using dynamic intensity factors. For the numerical purpose, it is assumed that material properties follow PZT-5H at the interface. The salient feature of this article is the pictorial presentations of dynamic energy release rates and dynamic stress intensity factors at the cracks' tips under electro-mechanical loading.

ABSTRACT. The Poisson-Nernst-Planck-Lennard-Jones (PNP-LJ) model is a mathematical model for ionic solution with Lennard-Jones interactions between the ions. Due to the singular nature of the Lennard-Jones interaction kernel, however, the PNP-LJ model gives rise to an intractable analytic problem and a highly demanding computational problem. Previous works tackled this problem by replacing the LJ potential with a leading order approximation of the LJ potential giving rise to the steric PNP model. The steric PNP proved to be a successful model for the transport of ions in biological ionic channels. However, in important parameter regimes, it is ill- posed. In this talk, we present a study that goes beyond the leading order approximation of the Lennard-Jones (LJ) interaction kernel, and develops a new class of high-order steric PNP equations. Surprisingly, we show that the introduction of high-order terms does not regularize the steric PNP model. Namely at the limit of vanishing ionic sizes, high-order steric PNP equations, at all orders including the PNP-LJ model, are ill-posed at important parameters regimes. Further analysis shows that this is an inherent property of PNP equations which accounts for steric effects and is related to pattern formation in these systems.

ABSTRACT. Bistability, i.e. the coexistence of different states at identical external parameters, is a frequently encountered phenomenon in electrode reactions. If the reactions occur on individual catalytically active areas that are all electrically connected, each active area can be considered as a bistable component, and the entire ensemble as a many particle system of such interconnected bistable components. Examples range from (micro-)electrode arrays, where the individual electrodes constitute the bistable components, to insertion battery cathodes, where each of the billions of nano-particulate storage particles can be considered a bistable component.

In this talk, we will first introduce a general mathematical description of bistable many particle systems and compile key results [1]: (1) All steady states are composed of one, two or three clusters. (2) Stable steady states possess at most one electrode on the intermediate, autocatalytic branch. (3) Ensembles of globally coupled bistable components might exhibit oscillations. We will derive a necessary condition of when the system might become oscillatory. In the second part, the model will be applied to different electrochemical systems, such as the oxidation of CO on coupled Pt electrodes, or Li insertion batteries. [1] M. Salman, Chr. Bick and K. Krischer, Phys. Rev. Research 2, 043125 (2020)

ABSTRACT. We present a reduction of self-consistent mean field theory models for blends of amphiphilic molecules that reprises earlier work of Choksi and Ren [2003] and Uneyama and Doi [2005]. The standard reduction provides a bilinear form of the free energy, and we extrapolate by replacing the mean field with a convolution of the local field. This leads to higher order phase field models. We derive reductions scalar models that are similar to the Functionalized Cahn Hilliard free energy, and hence to the models of Gommper and Goos [1995]. This energy is not in the 'nearly perfect square' format of the FCH, with a primary result that merging events are inhibited and bilayer structures generate an 'adhesion' energy. We apply this to model the thylakoid membranes that enclose the active parts of chloroplasts -- these self assemble into stacks that act as photo-receptors. We examine the role of adhesion energy in this process.

ABSTRACT. The circular front of a viscous fluid that displaces a less viscous fluid in shear- dominated flows is known to be stable. This is the case when the viscous fluid is confined between two parallel plates (Hele-Shaw geometry), and when it has a free surface as in viscous gravity currents propagating on horizontal planes. In a set of laboratory experiments we show that a similar front in predominantly extensional flows of strain-rate-softening fluids can become unstable and evolve tearing patterns consisting of rifts and tongues. These patterns tend to coarsen over time through the closure of some rifts and the merging of adjacent tongues. We model the emergence of these patterns as a fluid mechanical instability, which predicts that the number of rifts and tongues declines with time and is selected by competition between interfacial hoop stress, geometric stretching, momentum dissipation and spatial curvature. Our results elucidate fracture dynamics in complex fluids under extension and are applicable to a variety of systems ranging from polymer solutions in laboratory scale to ice shelves on a planetary scale.

ABSTRACT. We will discuss the problem of the best approximation of the three-dimensional Lebesgue measure on a domain of volume 1 by a discrete probability measure supported on a Bravais lattice. This is work in collaboration with David Bourne (Heriot-Watt University) and Steven Roper (University of Glasgow).

ABSTRACT. Linear elasticity can be rigorously derived from finite elasticity under the assumption of small loadings in terms of Gamma-convergence. This was first done in the case of one-well energies with super-quadratic growth [Dal Maso et al. 2002], and later generalised to different settings, in particular to the case of multi-well energies where the distance between the wells is very small (comparable to the size of the load)[Schmidt 2008], [Agostiniani et al. 2015]. The case when the distance between the wells is independent of the size of the load was studied in [Dal Maso et al. 2018]. In this context linear elasticity can be derived by adding to the multi-well energy a singular higher order term which penalises jumps from one well to another. In this talk we will discuss the derivation of linear elasticity from multi-well discrete models, under different assumptions on the geometry of the crystals and on the scaling of the coefficients. The role of the singular perturbation term is played in this setting by interactions beyond nearest neighbours.

ABSTRACT. The determination of minimizing structures for pairwise interaction energies is a challenging crystallization problem. The goal of this talk is to present recent optimality results among charges and lattice structures obtained with Markus Faulhuber (Vienna) and Hans Knüpfer (Heidelberg). The central object of these works is the heat kernel associated to a lattice, also called lattice theta function. Several connections will be showed between interaction energies and theta functions for the following problems: - Born’s Conjecture: how to distribute charges on a fixed lattice in order to minimize the associated Coulombian energy? In the simple cubic case, Max Born conjectured that the alternation of charges +1 and -1 (i.e. the rock-salt structure of NaCl) is optimal. The proof of this conjecture and its generalizations obtained with H. Knüpfer will be briefly discussed. - maximality of the triangular lattices among lattices with alternation of charges: we will present this new universal optimality among lattices obtained with M. Faulhuber. - stability of the rock-salt structure: what could be conditions on interaction potentials such that the minimal energies among charges and lattices has a rock-salt structure? Many results, both theoretical and numerical and obtained with M. Faulhuber and H.Knüpfer, will be presented.

ABSTRACT. We study the 'best' approximation of an absolutely continuous probability measure by a discrete probability measure in two dimensions, where 'best' means with respect to the 2-Wasserstein metric. This is an example of an optimal location problem (or an optimal sampling or quantization problem) and it has applications in economics, signal compression, numerical integration, and in generating initial conditions for particle methods for PDEs. We consider a 1-parameter family of constraints on the discrete measure (on the size of the atoms). Our main result is to establish the asymptotic minimum value of the (rescaled) approximation error as the number of particles goes to infinity. In particular, we show that for the constrained best approximation of the Lebesgue measure by a discrete measure, a discrete measure supported on a triangular lattice is asymptotically optimal. Our work generalises the crystallization result of Bourne, Peletier and Theil (Communications in Mathematical Physics, 2014) from a single particle system to a class of particle systems, and proves a case of a conjecture by Bouchitté, Jimenez and Mahadevan (Journal de Mathématiques Pures et Appliquées, 2011).

ABSTRACT. The Peierls-Nabarro (PN) model for dislocations is a hybrid model that incorporates the atomistic information of the dislocation core structure into the continuum theory. In this paper, we study the convergence from a full atomistic model to the PN model with gamma-surface for the dislocation in a bilayer system. We prove that the displacement field and the total energy of the dislocation solution of the PN model are asymptotically close to those of the full atomistic model. Our work can be considered as a generalization of the analysis of the convergence from atomistic model to Cauchy-Born rule for crystals without defects.

ABSTRACT. We examine phase-field modeling coupled to elasticity in two settings of interest to current modeling. First, we examine the behavior of phase-field modeling in settings with inertia, and find that current models are unable to model supersonic interface motion correctly.  We propose an augmented phase-field model that adds linear viscous dissipation to the momentum balance and show that this resolves the issue of supersonic interface motion. Second, we examine the behavior of phase-field models of fracture under general (compressive and tensile) loading across the crack. We propose the use of a QR decomposition of the deformation gradient rather than the standard polar decomposition, and show this is able to model correctly the assymetric tension-compression behavior of a crack.

Joint work with Janel Chua, Vaibhav Agarwal, Maryam Hakimzadeh, Timothy Breitzman, George Gazonas, Carlos Mora-Corral.

ABSTRACT. Modeling the domain switching processes in ferroelectric ceramics by phase-field approaches has classically relied on the simple gradient flow model of Allen and Cahn. While this is suitable for generating equilibrium domain structures, its applicability to the generally rate-dependent domain evolution and the intricate kinetic switching mechanisms in ferroelectrics is highly limited. We here introduce a new phase field formulation that is suitable for general kinetic evolution laws, based on the sharp-interface formulation of continuum mechanics. The model separates domain nucleation from domain wall motion, allowing for independent kinetic laws to be defined for each and also accounting for anisotropic domain mobility. We demonstrate the performance of this phase-field framework by numerically simulating ferroelectric switching in a representative volume element, and we discuss the shortcomings of available techniques and the benefits of this new approach, which is generally applicable beyond ferroelectrics with potential for, among others, phase transitions and deformation twinning.

ABSTRACT. In this talk I will present a notion of irreversibility for the evolution of cracks in presence of cohesive forces, which allows for different responses in the loading and unloading processes, motivated by a variational approximation with damage models. I will also discuss its applicability to the construction of a quasistatic evolution in a one-dimensional setting. The cohesive fracture model arises naturally via Gamma-convergence from a phase-field model of the generalized Ambrosio-Tortorelli type. This is a joint work with S. Conti and F. Iurlano.

ABSTRACT. We analyze the Gamma-convergence of sequences of free-discontinuity functionals arising in the modeling of linear elastic solids with surface discontinuities, including phenomena as fracture, damage, or material voids. We prove compactness with respect to Gamma-convergence and represent the Gamma-limit in an integral form defined on the space of generalized special functions of bounded deformation (GSBD^p). We identify the integrands in terms of asymptotic cell formulas and prove a non-interaction property between bulk and surface contributions. Eventually, we investigate sequences of corresponding boundary value problems and show convergence of minimum values and minimizers. In particular, our techniques allow to characterize relaxations of functionals on GSBD^p, and cover the classical case of periodic homogenization.

ABSTRACT. Antiferromagnetic spin systems are magnetic lattice systems in which the exchange interaction between two spins favors anti-alignment. Such systems are said to be geometrically frustrated if, due to the geometry of the lattice, there is no orientation of spins that simultaneously minimizes all pairwise interactions. As a consequence, the system has two families of ground states which can be distinguished one from the other by what is called their chirality. This is the case for the antiferromagnetic XY model on the two-dimensional triangular lattice, whose asymptotic behaviour in terms of Gamma-convergence is the subject of the present talk. Namely, we characterise the discrete-to-continuum Gamma-limit of the XY-model energy in a regime which detects chirality transitions between the two admissible chirality phases at the surface scaling

This is joint work with Marco Cicalese (München), Leonard Kreutz (Münster), and Gianluca Orlando (München).

ABSTRACT. We derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator with stationary law (i.e. spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale ε>0, we establish homogenization error estimates of the order ε in case of three or more spatial dimensions, respectively of the order ε|logε|^{1/2} in the two-dimensional case. Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence. We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes. Our results also hold in the case of systems for which a (small-scale) gradient Hoelder regularity theory is available.

ABSTRACT. Microtubule-based systems are viewed as minimal in vitro reconstitutions of the cytoskeleton. They are made active by mixing micron-size microtubules with kinesin proteins fueled with ATP. When this material is two-dimensionally interfaced with oil, it conforms nematic textures pervaded by topological defects and active flows. I will start by briefly reviewing the field of active systems, singularly those in relation to liquid crystals. In the central part of the talk I will present experimental results corresponding to different scenarios of the microtubule/kinesin active nematic system. First, I will introduce recent observations relative to the onset dynamics and full characterization of a turbulent-like regime, identifying the basic length scales involved in the instability mechanism [1,2,]. Later, a strategy of control of these active flows will be commented, based on preparing an interface between the active nematic and a passive smectic [3]. Finally, I will refer to situations of active nematics either constrained within channels or encapsulated in droplets dispersed in nematic liquid crystals [4]. References [1] B. Martinez et al., Nature Physics 15:362, 2019 [2] P. Guillamat et al. Nat. Comm. 8:564, 2017 [3] P. Guillamat et al., PNAS 114, 5498, 2016 [4] P. Guillamat et al., Sci. Adv. 4:eeao1470, 2018

ABSTRACT. This presentation addresses modeling, analysis and simulation of active flow of bacteria swimming in a chromonic nematic liquid crystal (LC), in confined channels. The bacteria behave collectively as a liquid crystal, whose interactions with the chromonic medium renders an effective LC system, revealing new flow and pattern behavior not previously observed in passive liquid crystals or in bacteria swimming in isotropic fluid. An external oxygen supply provides energy to the bacteria, activating the swimming dipole forces. We construct a biphasic model of the combined LC and bacteria components by coupling the Ericksen-Leslie equations of liquid crystal flow with the relative anchoring between chromonic and bacteria particles and their friction forces. We perform a stability analysis of the rest state, and classify the emerging unstability pattern due to activity, and conclude with analyzing the dislocation flow pattern that forms as the distance to the nutrient source increases. We show good agreement with the laboratory experiments by the group of Professor Lavrentovich at Kent State University.

ABSTRACT. Hybrid quantum/molecular mechanics models (QM/MM methods) are widely used in material and molecular simulations when MM models do not provide sufficient accuracy but pure QM models are computationally prohibitive. Adaptive QM/MM coupling methods feature on-the-fly classification of atoms during the simulation, allowing the QM and MM subsystems to be updated as needed. In this work, we propose such an adaptive QM/MM method for material defect simulations based on a new residual based a posteriori error estimator, which provides both lower and upper bounds for the true error. We validate the analysis and illustrate the effectiveness of the new scheme on numerical simulations for material defects.

ABSTRACT. We derive a nonlinear elasticity model for elastostatic problems from the atomistic description of a crystal lattice in one dimension. The elasticity model is of higher order compared with the well-known Cauchy- Born model in the sense that it utilizes higher order derivatives of the deformation gradient and is thus also called the higher order continuum model. We present a sharp convergence analysis for such higher order continuum model and we show that, compared to the second order accuracy of the Cauchy-Born model, the higher order continuum model is of forth oder accuracy with respect to the interatomic spacing in the thermal dynamic limit. The theoretical results are illustrated by our numerical experiments.

ABSTRACT. We study the 2+1 dimensional continuum model for long-range elastic interaction on stepped epitaxial surface, proposed by Xu and Xiang (SIAM J. Appl. Math. 69, 1393–1414, 2009). The long-range interaction term and two length scales in this model present challenges in the analysis of the instability phenomena of the surface. We obtain energy scaling law for bunching and undulation of steps and prove the existence of the energy minimizer. The minimum energy comparison between straight steps and undulation steps shows a transition from step bunching instability to step undulation instability as the adjacent distance increases.

ABSTRACT. The multiple scattering theory (MST), also called Korringa-Kohn-Rostoker (KKR) method, is widely used in electronic structure calculations of solid materials. In the MST method, a perfect separation between the atomic potentials and configuration geometries can be achieved. This can be exploited in the simulations to reduce the computational costs for many large-scale systems, including defected and disordered systems. This work studies the MST method by a rigorous numerical analysis and derives a spectral convergence rate for the numerical approximations, which justifies the reliability and efficiency of the MST method.

ABSTRACT. Open-shell systems are molecules and materials in which some electrons are single, in the sense that they do not go in pairs. Such systems, which include radicals, functional proteins (e.g. hemoglobin-iron complexes), and magnetic systems, are of major importance in chemistry, biology and materials science. They cannot be handled by standard density-functional theory (DFT) nor standard post-Hartree-Fock methods such as single-reference coupled cluster. In this talk, I will give a brief overview of the mathematical models used to simulate open-shell systems and discuss their numerical simulation. This is joint work with Robert Benda, Emmanuel Giner and Laurent Vidal.

ABSTRACT. In this talk, I will present and analyse a method to solve self-adjoint eigenvalue problems in electronic structure calculations. This method is based on the Feshbach-Schur map (i.e. a Schur complement) to recast the infinite-dimensional problem as a finite-dimensional one, combined with a perturbative formulation to obtain explicit bounds on the eigenvalues and eigenfunctions. I will then illustrate this method on two different examples: first, for the estimation of the ground state energy of Helium-type atoms, and second, to compute the eigenvectors and eigenvalues of Schrödinger operators in the context of planewave discretisation. Finally, I will present some numerical results that underline the theoretical findings.

ABSTRACT. In this talk I will report on recent results on the derivation of the macroscopic Poisson-Boltzmann equation in dielectrics from the (microscopic) density functional theory (DFT). I will also review some mathematical background on the DFT and mention some open problems. This is a joint work with Ilias Chenn.

ABSTRACT. Substructure analysis reduces the computational order of a discretized structural component from the full set of degrees of freedom needed to solve a boundary value problem (e.g., the displacements of all nodes in an FEA mesh) to a predefined and much smaller set of retained degrees of freedom. Given only one initial analysis considering all degrees of freedom, this technique reduces the computational cost associated with subsequent analyses of the same component by eliminating degrees of freedom, usually internal to the body, which are not essential for interfacing the component with a larger system/assembly. For linear multiscale problems (e.g., small-deformation linear elasticity), substructure analysis can predict the exact static and dynamic responses of larger assemblies consisting of one or more of these previously analyzed and dimensionally-reduced bodies.

In this work, we extend traditional substructure analysis to consider general nonlinear responses by leveraging the mathematical framework developed for computational plasticity, including the decomposition of deformations, criteria for nonlinearity initiation, and evolution equations. While computational plasticity provides nonlinear constitutive relationships between six independent stress and strain components, we show that the same mathematical formulation can capture similar relations between an arbitrary number of forces and displacements (i.e., the retained degrees of freedom).

ABSTRACT. The polyurethane foam is a porous heterogeneous material composed of pure polyurethane and air. It is mainly used in thermal insulation of buildings thanks to its low apparent thermal conductivity. For this reason, it is important to properly characterize its thermal properties.

The objective of this poster is to use the theory of homogenization in order to compute the effective thermal conductivity of polyurethane foam. First, a simplified geometry is considered along with a periodic structure. We can thus use the theory of periodic homogenization. The preliminary results are compared with a numerical approach using the Hill-Mandel assumption as well as with classical bounds on the effective thermal conductivity. Periodic homogenization gives results that are consistent with the former results, while being numerically much more efficient. Second, in order to take account of the realistic structure of polyurethane foam which is composed of a fraction of closed cells and open cells, stochastic homogenization is implemented and discussed in terms of computation time and performance. Finally, these two homogenization approaches – periodic or stochastic – are carried out on a more realistic geometry of the polyurethane foam, which has a tetrakaidecahedron shape, and again a mixed-cell microstructure.

ABSTRACT. In recent years, the development of coarse-grained models for studying large-scale processes that cannot be practically studied with atomically detailed molecular dynamics simulations is an active research field. Defining the new effective coarse-grained system, which reduces the dimensionality, accounts for finding the model best representing the reference system both in structure and dynamic properties. In the present work, we approximate the dynamics of coarse-grained systems at the transient regime. Under the assumption that it is possible to perform molecular dynamics simulations of the atomistic system only in a short time interval corresponding to the transient regime, we propose a Langevin equation model characterized by time-depended pair potential accounting the combined interaction forces of the system. We present the application of the path-space force matching method to retrieve the coarse space parametrized drift. At a long time limit, the time-dependent pair potential can reproduce the classical force matching potential of the mean force. In contrast, at transient -short time- regimes, we generate time-dependent drift coefficients describing the coarse-grained systems' dynamics. The model's effectiveness is examined by comparing its structural and dynamical properties with the corresponding reference system. The methodology is illustrated for various molecular systems such as liquid methane and water.

ABSTRACT. The electron-backscatter diffraction (EBSD) is a scanning electron microscopy (SEM) based technique used to obtain the grain orientation data. EBSD data often come with many misoriented pixels, which have the appearance of noise, and may have regions of missing data. We have developed partial differential equation-based denoising and inpainting algorithms, which produce high-quality reconstructions of the EBSD data. Our algorithms build on weighted total variation flow and minimal surface equation. In this poster, we compare the results obtained using our algorithms with various other denoising and data reconstruction algorithms commonly used for EBSD images. To this effect, we produce synthetic EBSD data with misorientation noise and missing data. We compare our PDE-based approach with other reconstruction algorithms such as mean filter, median filter, spline filter, Kuwahara filter, half-quadratic filter, and infimal convolution.

ABSTRACT. Among the various predictors used to study plasticity in amorphous materials, the Local Yield Stress (LYS) method has been proven to correlate highly with the plastic rearrangements in 2D model glasses. As a direct measurement of the local elasto-plastic response, the LYS method probes local regions of atoms in order to calculate the incremental stresses required to trigger plasticity. The resulting local yield stress is measured with respective to a particular loading orientation. Here in 3D, we sample 432 distinct deformations to characterize the local yield surface. From this yield surface we can derive the projected local yield stress with respect to the loading on the boundary. Since the direction of the local plastic rearrangement may not align perfectly with the boundary loading, we quantify how knowledge of the full yield surface enhances the predictive capability of the LYS, in comparison with predictions derived from a single local probing aligned with the loading on the boundary.

ABSTRACT. The process of scientific inquiry involves observing a signal (data) and interpreting it to generate information (knowledge). Artificial intelligence (AI) – a broad term comprising data science, machine learning (ML), neural network computing, computer vision, and other technologies – opens new avenues for extracting information from high-dimensional materials data. This presentation will focus on AI applications in the context of multimodal image-based data, including experimental and simulated atomic structures, defect structures, and microstructures. The visual information contained in these images is numerically encoded using black box computer vision (CV) methods as well as feature-based representations. ML tools are then selected based on the characteristics of the data set and the desired outcome. Case studies will present results that range from advanced methods for microstructural segmentation and characterization to prediction of microstructural evolution and material properties. The ultimate goal is to develop AI as a new tool for information extraction and knowledge generation in materials science.

ABSTRACT. I will present several examples in which materials informatics can be used to elucidate and quantify complex correlations linking structure, properties and processing of materials. In the first example, I consider the case of high-entropy (HE) (or multi-principal element) alloys, typically comprising five or more elements. The study of these alloys is a relatively new area of materials research that has attracted intense interest in recent years as, in many cases, these systems possess unexpected and superior mechanical (and other) properties relative to those of conventional alloys. However, the identification of promising HE alloys presents a daunting challenge given the associated vastness of the chemistry/composition space. I will describe a supervised learning strategy for the efficient screening of HE alloys that combines two complementary tools, namely: (1) a canonical-correlation analysis (CCA) and (2) a genetic algorithm (GA) with a CCA-inspired fitness function. In the second example, I consider the ubiquitous phenomenon of grain abnormality in a microstructure that involves the unusually rapid growth of a minority of constituent grains, with the resulting bimodal structure often having a deleterious impact on the thermomechanical properties of a system.

ABSTRACT. I will present recent developments in predicting quasiparticle energies using the combination of stochastic computational techniques and many-body theory. The methodology relies on operators' decomposition via random vectors, recasting expectation values as statistical estimators, and real-time and real-space sampling.[1] This formalism leads to substantial computational savings and reduced scaling. In practice, quasiparticle energies can be computed for systems with thousands of atoms. I will describe our recent work on simulating nanoscale condensed systems within the linear scaling stochastic GW approximation.[2,3,4] Further, I will describe an efficient route to go beyond GW using the stochastic formulation of non-local vertex corrections.[5,6]

[1] V Vlcek, W Li, R Baer, E Rabani, D Neuhauser, Physical Review B 98 (7), 075107 (2018) [2] J Brooks, G Weng, S Taylor, V Vlcek, Journal of Physics: Condensed Matter 32 (23), 234001 (2020) [3] G Weng, V Vlcek The Journal of Physical Chemistry Letters 11 (17), 7177-7183 (2020) [4] M Romanova, V Vlcek The Journal of Chemical Physics 153 (13), 134103 (2020) [5] V Vlcek, Journal of Chemical Theory and Computation 15 (11), 6254-6266 (2019) [6] C Mejuto-Zaera, et al., arXiv preprint arXiv:2009.0240122020

ABSTRACT. In this talk, we present some results about the well-posedness of the reduced Hartree-Fock model with self-generated magnetic Fields for molecules and crystals. In particular, we exhibit a critical value a_c > 0 such that, if the fine structure constant a is smaller than a_c, then the corresponding system is stable, whereas if a is greater than a_c, it is unstable. We give an explicit characterization of a_c as a minimization problem over the set of zero-modes, and we prove that the critical values for the molecular case and the periodic case coincide. Finally, we give some results about the existence of minimizers.

ABSTRACT. The existence of zero modes for Dirac operators with magnetic fields is the cause of break down of stability of matter for charged systems.

However the known examples are geometrically complex, and a complete classification of zero modes is unknown. In particular, one does not know the characteristics of the magnetic fields which produce the zero modes.

To better understand them, we studied the particular case of magnetic fields with finitely many field lines which form a link. These singular fields can be seen as generalizations of the Aharonov-Bohm solenoids.

Tuning one flux from $0$ to $2\pi$ gives rise to a loop of Dirac operators for which we can study the spectral flow, a non-trivial spectral flow indicating the occurence of zero modes. It turns out that this number depends on the geometry of the field lines.

(Joint work with F. Portmann and J.P. Solovej)

ABSTRACT. During abnormal grain growth (AGG), a very few grains grow at a much faster rate than the others and eventually consume the microstructure. Unfortunately, the origins and mechanisms of AGG have remained an enigma for at least the past seventy years. One such long-standing mystery concerns AGG in the presence of particles. Our traditional understanding is that particles retard and eventually "pin" the grain boundaries. Yet, and paradoxically so, abnormal grain growth is most frequently seen in particle-containing systems, such as transformer steels. Here, we shine new light on particle-assisted abnormal grain growth by making use of new strides in laboratory- and synchrotron-based 4D (i.e., 3D space plus time) imaging. Following the in situ experiments, advances in data science methods and high-performance computing open the doors to a wealth of information on the collection of grains undergoing AGG. For instance, we have extracted grain boundary characteristics and velocities together with particle locations, as a function of time during heat treatments. These metrics provide direct evidence on the kinetic pathways leading to the localized growth of abnormally large grains. They also serve as benchmark data that can be used to validate theories and simulations of grain growth, e.g., phase field.

ABSTRACT. This talk surveys recent progress in developing a fully-parallelizable matrix-free GPU-based algorithm for implementing a two-dimensional vertex model of recrystallization based on the stored energy formalism. Nucleation is assumed to take place at triple junctions and obeys a Metropolis-type criterion. Conditions under which nucleation is successful are derived and stability analysis of the dynamics of a triple junction under the presence of bulk energy is provided. A polling system for handling topological transitions to ensure robust GPU implementation is described and a set of numerical experiments for large scale systems is presented to explore the effect of initial distributions of stored energy on several statistical characteristics.

ABSTRACT. Recent advances in transmission and scanning transmission electron microscopy (TEM/STEM) sample holders and MEMS chips technology have enabled the acquisition of in situ imaging of grain boundary dynamics during grain growth of polycrystalline materials. While TEM/STEM imaging provides fine scale detail, automated grain boundary detection in bright-field TEM images is a challenge due to the so-called diffraction contrast. This results in different intensities assigned to different orientations within single grains (crystallites) as well as the same intensities assigned to similar orientations of neighboring grains, leading to both false and missing grain boundaries. In this work we present an automated grain boundary detection algorithm that is designed to overcome the challenges caused by diffraction contrast. A key piece of the algorithm involves a well-posed adaptive anisotropic diffusion for noise removal and edge preservation, with a specific focus on avoiding the introduction of false boundaries.

ABSTRACT. I will discuss recent work on microstructure formation in graphene. Mathematical modeling predicts formation of network of wall-like structures that have also been observed in experiments.

ABSTRACT. In this talk, I will emphasize the need for stronger coupling between different scales (from quantum to continuum) in a multiscale approach to materials modeling. In addition, the increasing availability of reliable quantum calculations calls for novel data-driven approaches to multiscale modeling. These two points will be illustrated by two applications: (i) blind prediction of molecular crystal structures for pharmaceutical and photovoltaic applications (Science Adv. 5, eaau3338 (2019)); and (ii) understanding ultra-long-range adhesive forces in the delamination of atomically thin layers from substrates (Nature Commun. 11, 1651 (2020)). Both applications demonstrate the power of modern multiscale modeling techniques but also highlight many existing challenges that will drive the field of data-driven multiscale modeling for years to come.

ABSTRACT. Gaussian processes are a well established Bayesian machine learning algorithm with significant merits, despite a strong limitation: lack of scalability. Clever solutions address this issue by inducing sparsity through low-rank approximations, often based on the Nystrom method. Here, we propose a different approach to achieve unprecedented scalability using quantum computing. Unlike most recent quantum machine learning research, we do not recommend just replacing the computationally expensive linear algebra operations with their quantum counterparts. Instead, we induce a low-rank approximation using Quantum Phase Estimation, in order to achieve a simultaneous improvement of scalability and accuracy. We provide evidence that quantum Gaussian processes can address the "curse of dimensionality", where each additional input parameter no longer leads to an exponential growth of the computational cost. We include numerical tests, error estimation and complexity analysis to support our claims. The algorithmic developments, however, require significant quantum computing hardware improvements that are expected within the next decades in order to achieve quantum advantage.

ABSTRACT. The need for improved exchange-correlation (XC) functionals in DFT that can provide quantum accuracy can hardly be over-emphasized. To that end, we envisage a data-driven approach to constructing XC functionals by using accurate ground-state densities ($\rho(\br)$) from many-body calculations. The key idea is to use the $\rho(\br)$ and its corresponding exchange-correlation potential ($\vxc(\br)$), from multiple molecules, as training data to model the XC functional---$\vxc[\rho]$. As a crucial first step, we present an accurate and robust approach to solve the inverse DFT problem of obtaining the exact $\vxc(\br)$ from a given $\rho(\br)$, which had, heretofore, remained an open challenge owing to various numerical instabilities in previous attempts. We present the exact $\vxc$ for five molecules---hydrogen molecule at three different bond-lengths, lithium hydride, and water---using accurate ground-state densities from configuration interaction (CI) calculations. Moreover, we present the model $\vxc$'s for the same set of molecules using ground-state densities from DFT calculations with two highly used non-local exchange-correlation functionals---B3LYP and HSE06. The availability of these exact $\vxc$'s, along with its comparison with model $\vxc$'s, forms the basis of our ongoing efforts at a neural network (NN) based design of XC functionals.

ABSTRACT. Complex chemical processes, such as the decomposition of energetic materials and the chemistry of planetary interiors, are typically studied using large-scale molecular dynamics simulations that run for weeks on high performance parallel machines. These computations may involve thousands of atoms forming hundreds of molecular species and undergoing thousands of reactions. It is natural to wonder whether this wealth of data can be utilized to build more efficient, interpretable, and predictive models. In this talk, we will use techniques from statistical learning to develop a framework for constructing Kinetic Monte Carlo (KMC) models from molecular dynamics data. We will show that our KMC models can not only extrapolate the behavior of the chemical system by as much as an order of magnitude in time, but can also be used to study the dynamics of entirely different chemical trajectories with a useful degree of fidelity. Then, we will discuss three different methods for reducing our learned KMC models, including a new and efficient data-driven algorithm using L1-regularization. We demonstrate our framework throughout on a system of high-temperature high-pressure hydrocarbons, thought to be a major component of gas giant planetary interiors.

ABSTRACT. Palladium-hydrogen (Pd-H) is a prototypical system for studying solute-induced phase transformation in various energy conversion and storage applications. While the behaviors of bulk Pd-H have been studied extensively, the detailed atomic picture of hydride phase transformation within individual Pd nanoparticles is still unclear. In this work, we employ a novel atomistic computational model, referred to as Diffusive Molecular Dynamics (DMD), to characterize the H absorption dynamics in Pd nanoparticles of spherical, octahedral and cubic shapes. The DMD model couples a non-equilibrium thermodynamic model with a discrete diffusion law, which allows it to reach diffusive time scales with atomic resolution. The model is capable of capturing the propagation of an atomistically sharp hydride phase boundary. A remarkable feature of the phase boundary structure that is predicted by the calculations is the emergence of misfit dislocations distributed over the interface. These dislocations relieve the elastic residual stresses induced by the change of volume that accompanies the phase transformation. We also identify the mechanisms that enable the movement of the stacking faults as they track the propagation of the phase boundary. Finally, we find that the rate of H absorption is reduced by the formation and movement of the stacking faults.

ABSTRACT. We present novel optimal control problems motivated by the search for photovoltaic materials with high power-conversion efficiency. The material must perform the first step: convert light (photons) into electronic excitations. We formulate various desirable properties of the excitations as mathematical control goals at the Kohn-Sham-DFT level of theory, with the control being given by the nuclear charge distribution. We prove that nuclear distributions exist which give rise to optimal HOMO-LUMO excitations, and present illustrative numerical simulations for 1D finite nanocrystals. We observe pronounced goal-dependent features such as large electron-hole separation, and a hierarchy of length scales: internal HOMO and LUMO wavelengths << atomic spacings << (irregular) fluctuations of the doping profiles << system size.

Reference: G. Friesecke, M. Kniely, Multiscale Model. Simul., 17(3), 926–947, 2019

ABSTRACT. We focus on a one-dimensional particle system with convex-concave potentials that have non-standard growth conditions and allow for fracture. We are interested in the continuum limit and study this in the context of $\Gamma$-convergence methods. This requires a new technique for deriving compactness results which allows to overcome the different role of global minimizers in the context of external potentials.

ABSTRACT. Most dynamic homogenization techniques are performed for an infinite periodic composite and the results are applied to finite domains. This approximation is not always true. At an interface between 2 composites, the displacement and traction continuity need to be satisfied, and evanescent waves play an important role. We consider the problem of transmission and reflection of waves from an interface between a bilayer composite and an homogenous medium. Our aim is to determine the scattering coefficients that satisfy the conservation of energy flux and find the appropriate jump conditions to be satisfied at the interface. We use a rational function approximation of the exact dispersion relation, and then invert it to the space-time domain to obtain a dynamic higher-order pde and interface jump conditions. The scattering coefficients are obtained by imposing the higher order jump conditions at the interface. Numerical calculations are performed to test performance of the higher order jump conditions, by comparing with the results obtained from solving the exact wave equation. We present results for both 1d and 2d cases. Furthermore, we suggest a method to capture the band gap between the optical and acoustic branches of the dispersion curve and discuss the emergence of space-time non-locality.

ABSTRACT. Heat conduction depends on the Knudsen number Kn = lth/L, where lth the mean free path of phonons and L a characteristic length. If Kn >> 1, the type of transport is diffusive, and we use Fourier Law. When Kn ≃ 1, the transport is non-diffusive, the regime is named mesoscopic, and we can model the heat transport using the Guyer-Krumhansl Equation. This model takes into account time delay and no-locality, needed at these length scales. We work with a composite system made of a matrix with thermal conductivity κ with N inclusions of thermal conductivity κ1 embedded in the matrix. We compute the effective thermal properties within an effective medium theory in the mesoscale regime for inclusions of arbitrary shape. In particular, we present the case study when the inclusions are cylinders of elliptical cross-section and when the inclusions are toroidal particles. Both of these shapes will introduce anisotropic effects in the thermal conductivity. The effective thermal conductivity κeff is calculated as a function of the volume fraction f, the shape factor of inclusions, and Knudsen number Kn. In summary, we obtain a homogenization technique that can be useful in the non-diffusive regime for composites with arbitrarily shaped inclusions.

ABSTRACT. The main motivation of this problem is the non-destructive testing of elastic structures for material inclusions. We consider the inverse problem of recovering the shape of the inclusions from the Neumann-to-Dirichlet map. In doing so, we deal with the rigorously proven theory of the monotonicity methods developed for linear elasticity with the explicit application of the methods, i.e. the implementation and simulation of the reconstruction of inclusions in elastic bodies for both artificial and experimental data. More specifically, we give an insight into the monotonicity-based methods and take a look at a lab experiment. Finally, we present our reconstructions based on experimental data and compare them with the simulations obtained from artificial data, where we want to highlight that all inclusions can be detected from the noisy experimental data, thus, we obtain accurate results.

ABSTRACT. Hard disk systems are the prototypical simple fluids, and as such they constitute perhaps the simplest system to study the origins of phase transitions and the glass transition. A phase transition is conventionally identified by a discontinuous change in the weighted average of thermodynamic quantities over the accessible portion of the configuration space. If this is true, then the topology of the configuration space as a function of disk radius should be intimately related to the occurrence of these phenomena. The critical points of a regularized energy function defined on the configuration space correspond to configurations where the topology changes. These critical points and a large number of other sampled configurations are used to construct an explicit triangulation of the configuration space using techniques from topological data analysis. The distance between two points in the triangulation is then defined as the sum of the Euclidean distances between corresponding disks (modulo symmetries), and the geometry of the configuration space is studied as a function of disk radius.

ABSTRACT. Given two analytic functions, representing extrapolants of the same experimental data, we examine how much they can differ at an extrapolation point outside of the experimentally accessible frequency band. We give a sharp upper bound on the worst case extrapolation error, in terms of a solution of an integral equation of Fredholm type. We show that this bound exhibits a power law precision deterioration as one moves further away from the frequency band containing measurement data.

ABSTRACT. Machine-learning interatomic potentials are models providing the energy of interatomic interaction as a function of atomic coordinates (of the nuclei). This ten-year-old field has been booming in the past five years and has already made pathways towards making a tangible impact on computational materials studies. A number of communities, nucleated within materials science, chemistry, and physics, have been actively working on this topic, however, some of the important outstanding problems require mathematical tools.

Unlike classical interatomic potentials, the machine-learning ones are built with the goal of approximating a quantum-mechanical energy with an arbitrary accuracy through fitting to a large amount of quantum-mechanical data. In practice, the accuracy of potentials is limited by a number of factors such as nonlocality of interaction or non-smoothness of the quantum-mechanical potential energy surface. Nevertheless, in many applications the use of machine-learning potentials yields an acceleration by several orders of magnitude with an accuracy better than the model error of density functional theory calculations.

In my talk I will give a short review of machine-learning potentials, present some of successful applications of machine learning potentials, and formulate some of the outstanding problems in mathematical language.

ABSTRACT. A central goal of computational physics and chemistry is to predict material properties using first-principles methods based on quantum mechanics. However, the high computational costs of these methods typically prevent rigorous predictions of macroscopic quantities at finite temperatures, such as chemical potential, heat capacity and thermal conductivity.

In this talk, I will first discuss how to enable such predictions by combining statistical mechanics with data-driven machine learning interatomic potentials. I will give example applications on the systems of water and high-pressure hydrogen. Besides thermodynamic properties, I will also talk about how to compute the heat conductivities of liquids just from equilibrium molecular dynamics trajectories.

During the second part of the talk, I will rationalize why machine learning potentials work at all, and in particular, the locality argument.

ABSTRACT. Accurate molecular simulation requires computationally expensive ab initio models models that makes simulating complex material phenomena or large molecules intractable. However, if no chemistry is required, but only interatomic forces then it should in principle be possible to construct much cheaper surrogate models, interatomic potentials, that capture full QM accuracy. This talk will aim to explain why this is possible, focusing on analysis and numerical analysis perspectives. Specifically, I will explore whether we can rigorously justify the extremely low-dimensional functional forms proposed for interatomic potentials, and whether we can construct practical approximation schemes that can, in principle at least, close the complexity gap between density functional theory and interatomic potentials.

ABSTRACT. Constitutive and closure models play important roles in computational mechanics and computational physics in general. Classical constitutive models for solid and fluid materials are typically local, algebraic equations or flow rules describing the dependence of stress on the local strain and/or strain-rate. Closure models such as those describing Reynolds stress in turbulent flows and laminar-turbulent transition can involve transport PDEs (partial differential equations). Such models play similar roles to constitutive relations, but they are often more challenging to develop and calibrate as they describe nonlocal mappings and often contain many submodels. Those nonlocal mappings have many similarities with the interatomic potential and force fields. Inspired by the remarkable success of machine learning-based interatomic potential, we propose a neural network representing a region-to-point mapping to describe such nonlocal constitutive models. Numerical experiments demonstrate the proposed method's predictive capability and flexibility, making it a promising alternative to traditional nonlocal constitutive models.

ABSTRACT. The classical hard-particle system, despite its apparent simplicity and nearly a century of study, still leaves open a wealth of questions, particularly those concerning its behaviour in dense regimes. Within this work we consider a general and exact formalism for providing the free energy and equation of state of a single species system, which can be complemented by various approximations and ansatzes to provide some qualitative and quantitative results on the behaviour of such systems across both their dilute anddense regimes.

ABSTRACT. In this work we use the concept of free volume to extend cell theories of crystalline materials to the intermediate-density regime. To compute these free volumes we introduce a leaky cell model, in which the accessible space accounts for the possibility that spheres may escape from the local cage of lattice neighbors.This model predicts phase transitions in ordered matter, including cubic-cubic phase transitions which have been observed experimentally, and provides general principles to understand crystalline materials that may be applicable to many systems.

ABSTRACT. Transferring structural information from the nanoscale to the macroscale is a promising strategy for developing adaptive and dynamic materials. Here we demonstrate that the knotting and unknotting of a molecular strand can be used to control, and even invert, the handedness of a helical organization within a liquid crystal. An oligodentate tris(2,6-pyridinedicarboxamide) strand with six point-chiral centres folds into an overhand knot of single handedness upon coordination to lanthanide ions, both in isotropic solutions and in liquid crystals. In achiral liquid crystals, dopant knotted and unknotted strands induce supramolecular helical organizations of opposite handedness, with dynamic switching achievable through in situ knotting and unknotting events. Tying the molecular knot transmits information regarding asymmetry across length scales, from Euclidean point chirality (constitutional chirality) via molecular entanglement (conformation) to liquid-crystal (centimetre-scale) chirality. The magnitude of the effect induced by the tying of the molecular knots is similar to that famously used to rotate a glass rod on the surface of a liquid crystal by synthetic molecular motors.

ABSTRACT. The density functional theory (DFT) states that the ground-state energy of a molecule, defined as the lowest eigenvalue of a high-dimensional PDE, is also given by a functional of the electronic density. The functional is however unknown and approximations have been designed in the last decades based on asymptotics for specific systems and empirical formulas.

The particle-hole random phase approximation (phRPA) is an approximation of this functional which correctly dissociates the H2 molecule, unlike many other density functional approximations.

The aim of the talk is to present a proof of this exact dissociation in a finite and infinite-dimensional setting. This is a joint work with Gero Friesecke and Kyle Thicke.

ABSTRACT. We present a low-scaling algorithm for the computation of the particle-particle Random Phase Approximation (ppRPA) correlation energy in the context of density functional theory. We utilize the Hutchinson algorithm; but, the real reduction in computational cost comes from creatively inserting the identity operator and structuring the randomness to match the structure of the problem. We end up with a Hutchinson algorithm whose cost per iteration grows merely quadratically in the system size.

ABSTRACT. This talk follows Paola Gori-Giorgi's overview and relies on the connection between Optimal Transport and Density Functional Theory introduced by M. Seidl, P. Gori-Giorgi, C. Cotar, G. Friesecke, G.Buttazzo and others. Particular emphasis will be given on computational algorithms and novel insights to build (kinetic) exchange-correlation functionals via Entropic regularized Strictly-Correlated-Electron functional.

ABSTRACT. In this talk a model for excited electronic states based on the Bethe-Salpeter Hamiltonian is presented. In this model excited states is described by excitions: quasi particles consisting of an electron interacting with an electron hole. This model is widely used by physicists in the computation of optical absorption spectra. I will explain how in a strong screening limit the model reduces to the Wannier-Mott model for excitons, how this can be exploited in the diagonalization of the ab initio Bethe-Salpeter Hamiltonian, and discuss how the approach can be extended to include lattice interaction.

ABSTRACT. \begin{abstract} At the outset we shall discuss a particular problem class, which is closely linked to the classical concept of Friedrichs systems. We shall re-consider Friedrichs systems from a operator theoretic perspective by initially studying operator equations of the form \[ \left(1+A\right)U=F, \] where $A$ is maximal accretive, i.e. $A,A^{*}$ accretive. Starting out with the structural assumption that $A$ is an extension of a skew-symmetric operator $\overset{\,\,\,\,\circ}{A}$, we are interested in describing maximal accretive extensions of $\overset{\,\,\,\,\circ}{A}$. Complex materials can be addressed by combining such maximal accretive extensions with suitable material laws, which allows to go beyond the classical Friedrichs type systems framework. Such materials are distinguished by resulting in maximal accretive space-time equations. We shall illustrate the setting by inspecting its utility in the context of electrodynamics in metamaterials.\end{abstract}

ABSTRACT. A Steklov-type problem for Maxwell's equations will be discussed, which is related to an interior Calderon operator and an appropriate Dirichlet-to-Neumann type map. The corresponding Neumann-to-Dirichlet map turns out to be compact and this provides a Fourier basis of Steklov eigenfunctions for the associated energy spaces. We provide natural spectral representations for the appropriate trace spaces, for the Calderon operator itself and for the solutions of the corresponding boundary value problems subject to electric or magnetic boundary conditions on a cavity.

ABSTRACT. Scattering of electromagnetic waves by a particulate media is addressed. In general, the non-intersecting particles can be of arbitrary form, material and shape with number density n0 (number of scatterers per volume). The main aim of this paper is to calculate the coherent reflection and transmission characteristics for a collection of spherical particles confined within two parallel planes. Typical applications of the results are found at a wide range of frequencies (radar up to optics), such as attenuation of electromagnetic propagation in rain, fog, and clouds etc. The integral representation constitutes the underlying framework of the solution of the deterministic problem, which then serves as the starting point for the solution of the stochastic problem. Conditional averaging and the employment of the Quasi-Crystalline Approximation lead to a system of integral equations in the unknown expansion coefficients. The planar geometry and normal plane wave incidence imply a system of integral equations in the depth variable. The system of integral equations is solved by the Nyström’s method and Gauss-Legendre quadrature. Numerical examples are presented and the results are compared with the tenuous and low frequency approximations as well as the reflection and transmission of a homogenized slab.

ABSTRACT. I will describe a class of metamaterial arrays, whose designs are based on the behavior of hypoelliptic PDE of Hormander sums of squares type. These are valid for any waves modeled by the Helmholtz equation, including scalar optics and acoustics. These resonators behave superdimensionally, with a higher local density of eigenvalues in parts of the spectrum, and greater concentration of waves (giant focusing), than expected from the physical dimension. Thus, planar resonators can function as 3- or higher-dimensional media, and bulk material are effectively of dimension 4 or higher. Applications, which include antennas, are potentially broadband.

ABSTRACT. Confine a thin elastic shell to an incompatibly shaped substrate and a complex wrinkle pattern can form. We present a method, uncovered in [1] and developed and simplified in [2], to predict the layout of the resulting wrinkle peaks and troughs. Our method is based on the use of certain Airy-like potentials, which exist in the infinitesimally-wrinkled and zero thickness limit. These potential functions are required to be convex, but in general will fail to be smooth. Their subdifferentials encode the wrinkle peaks and troughs, reminiscent of the way in which the level sets of stream functions encode the particle paths of ideal fluid flows. Applying the tools of convex analysis, we derive via our encoding a simple yet complete set of diagrammatic rules for the wrinkles of positively and negatively curved confined shells. We highlight a previously unnoticed "reciprocal rule" relating the wrinkles of positively and negatively curved shells: their peaks and troughs form isosceles triangles when appropriately paired. Systematic application of this and other such rules recovers the wrinkle patterns of hundreds of experiments and simulations of confined shells.

[1] Tobasco, to appear in ARMA (2021). [2] Tobasco, Timounay, Todorova, Leggat, Paulsen, and Katifori, submitted.

ABSTRACT. Thin elastic structures are difficult to stretch but easy to bend; their deformations are therefore normally understood via comparison with known isometries of the structure. However, local isometry is well known to prevent changes of Gaussian curvature. Here, I will discuss examples where macroscopic changes in the apparent Gaussian curvature do take place, but without significant material stretching. I will show examples in which the apparent Gaussian curvature in the deformed state can be chosen (and hence controlled) as well as other scenarios in which the system spontaneously chooses a particular apparent Gaussian curvature.

ABSTRACT. Natural materials are highly organized from the molecular scale to the macroscale, thereby achieving superior properties. In the wake of biomimicry, there is a growing interest in designing architected materials in the laboratory. While such structures could enable a myriad of different functions for engineering applications, their fabrication remains challenging. Here we demonstrate a new methodology to assemble pixelated soft materials. Our approach is conceptually analogous to the watershed transform used in image analysis and allows us to pas- sively assemble discrete materials tile by tile through the capillary mediated flow and subsequent curing of liquid elastomers in confined geometries. Rationalizing the fluid mechanics at play allows us to program the exact structural arrangement of each component in the newly formed material.

ABSTRACT. A classical problem in mechanics is the following: given a set of forces applied at some points in space, is there any discrete network with all the wires under tension that supports such a system of forces? In usual wire or cable networks, such as in a bridge or bicycle wheel, one distributes the forces by adjusting the tension in the wires. Here our discrete networks provide an alternative way of distributing the forces through the geometry of the network. In particular the network can be chosen so that it is uniloadable, i.e. supports only one set of forces at the terminal nodes. Such uniloadable networks are relevant as the tension in one element determines the tension in the joint wires and, by extension, uniquely determines the stress in the entire network.

ABSTRACT. A graphene sheet is a single-atom thick macromolecule of carbon atoms arranged in a honeycomb hexagonal lattice. When observing a graphene sheet suspended over a substrate, moiré patterns appear driven by lattice and orientation mismatches. In this talk, we start by presenting a formal discrete-to-continuum procedure to derive a continuum variational model for two chains of atoms with slightly incommensurate lattices. The chains represent a cross-section of a three-dimensional system consisting of a graphene sheet suspended over a substrate. We show that the continuum model recovers both qualitatively and quantitatively the behavior observed in the corresponding discrete model. We then extend the discrete-to-continuum procedure to square lattices and honeycomb hexagonal lattices. In all cases, we observe the presence of large commensurate regions separated by localized incommensurate domain walls, in agreement with experiments.

ABSTRACT. We consider the propagation of acoustic, electromagnetic, and elastic waves in a one-dimensional periodic two-component material. Accurate asymptotic formulas are provided for the group velocity as a function of the material parameters when the concentration of scatterers is small or the characteristic impedances of the two media differ substantially. In the latter case, it is shown that the minimum group velocity occurs when the volume fractions of the components of the material are equal. In both asymptotic cases, we show that the leading terms of the group velocity do not depend on frequency. Thus slowdown is frequency-independent and is not related to the resonance phenomena.

ABSTRACT. We study an inverse problem of antiplane elasticity for n curvilinear inclusions having constant stresses inside when the body at infinity is subjected to antiplane shear. The elastic medium is a half-plane, and its boundary is free of traction. To determine the shape of inclusions, we employ a conformal mapping that transforms an (n+1)-connected parametric slit domain onto the elastic half-plane with n inclusions. The map is expressed through the solution of a vector Hilbert problem on a hyperelliptic Riemann surface of genus n. In a particular case of loading, it is possible to reduce the problem to two consequently solvable scalar Riemann-Hilbert problems on the surface. An analytic solution of these problems are derived in closed form in terms of singular integrals on the Riemann surface. In the general case we show how the problem may be converted into a system of singular integral equations on n+1 loops of the hyperelliptic surface. We study this new class of integral equations and discuss possible approaches for their solution.

ABSTRACT. The dielectrophoretic separation of infected cells from healthy cells in a microfluidic channel with insulating constrictions is numerically investigated. In this work, electrophysiological properties of cells were iteratively extracted through the fitting of a single-shell model with the frequency-conductivity data obtained from AC dielectrophoretic microwell experiments. In the numerical computation, the gradient of the electric field required to generate the necessary dielectrophoretic force within the constriction zone was provided through the application of electric potential across the whole fluidic channel. By adjusting the difference in potentials between the global inlet and outlet of the fluidic device, the minimum (effective) potential difference with the optimum particle transmission probability was found. The geometry of the constrictions at which the effective potential difference was swept to obtain the optimum constriction size was also found. Independent particle discretization analysis was also made to underscore the accuracy of the numerical solution. The numerical results, which was obtained by the integration of fluid flow, electric current, and particle tracing module in COMSOL v5.6, reveals the required electric conditions to maximally separated one type of cell from another. This result is the first step towards the production of a supplementary or confirmatory point-of-care diagnostic device.