ABSTRACT. Grain boundaries are the interfaces between grains with different orientations in polycrystalline materials. Energetic and dynamic properties of grain boundaries play essential roles in the mechanical and plastic behaviors of the materials. These properties of grain boundaries strongly depend on their microscopic structures. We present continuum models for the energy and dynamics of grain boundaries based on continuous distributions of the line defects (dislocations or disconnections) on them. The long-range elastic interaction between these line defects is included in the continuum models. The continuum models are able to describe the shear coupling motions of grain boundaries, which were observed in atomistic simulations and experiments and cannot be explained by the classical motion by mean curvature models.

ABSTRACT. Owing to their many extraordinary physical properties, metal halide perovskite semiconductors offer exciting opportunities to create efficient, next-generation single and multi-junction photovoltaic (PV) devices. Here, we demonstrate a simulation model that well describes efficient p-i-n type perovskite solar cells and a range of experiments using experimental input from electro-optical, impedance and transient and steady-state photoluminescence (PL) measurements. We discuss the role of important device and material parameters with a particular focus on the carrier mobilities, mobile ions, doping, energy-level alignment and the built-in potential (V_BI). Using only parameters that have been already demonstrated in recent literature, we demonstrate that an efficiency regime of 30% can be unlocked by optimizing the built-in potential across the perovskite layer by using either doped (10^19 cm^-3) or ultrathin undoped transport layers (TLs), e.g. self-assembled monolayers. Finally, will address the important question to what extent the presence of mobile ions influences the built-in field and with that the device performance of our triple cation based p-i-n type perovskite devices and optimized cells in the future.

ABSTRACT. The recent emergence of lead-halide perovskite solar cells (PSCs) has motivated the development of several new device models that include the effects of mobile ionic charge within the perovskite active layer. In this talk I will present two models recently developed in our group: Driftfusion, a one-dimensional mixed ionic-electronic drift-diffusion device simulation, and a rationally designed equivalent circuit model for PSCs. Key to the device physics of PSCs is the role of ionic charge accumulation at the perovskite-contact layer interfaces in modulating electronic carrier currents. Here I will use simulations from Driftfusion to illustrate this phenomenon and to show how PSCs can be modelled using an equivalent circuit in which the interfaces are described by ionically-gated bipolar transistors.

ABSTRACT. From Maxwell-Stefan diffusion and general electrostatics, we derive a drift- diffusion model for charge transport in perovskite solar cells (PSCs) where any ion in the perovskite layer may flexibly be chosen to be mobile or immobile. Unlike other models in the literature, our model is based on quasi Fermi potentials instead of densities. This allows to easily include nonlinear diffusion (based on Fermi-Dirac, Gauss-Fermi or Blakemore statistics for example) as well as limit the ion depletion (via the Fermi-Dirac integral of order −1). The latter will be motivated by a grand-canonical formalism of ideal lattice gas. Furthermore, our model allows to use different statistics for different species. We discuss the thermodynamic equilibrium, electroneutrality as well as generation/recombination. Finally, we present numerical finite volume simulations to underline the importance of limiting ion depletion.

ABSTRACT. Low hystheresis in the current voltage characteristics of perovskite solar cells is often understood as a low ionic and interface charging activity in the device. However, even cells having very low hystheresis can show strong ionic effects seen in the transient response to excitation events. Here, we explore by numerical simulations the interplay between purely electronic parameters and ionic parameters determining the transient response of perovskite solar cells with mobile ions. Two transient types are investigated: the evolution of open-circuit voltage upon illumination and the evolution of injected current upon biasing in the dark. In agreement with experimental results, the simulations show that solar cells exhibiting low-hystheresis under normal voltage sweep rates, may display slow transient responses to light and biasing events. The results shown here show that, to a large extent, the shape of the transients is determined by ionic and interface parameters, while the stationary values obey bulk and interface recombination parameters. Furthermore, links are established between the transient response of the radiative component of the injected current in the dark and the open circuit voltage transients, hinting to interpretation in terms of optoelectronic reciprocity.

ABSTRACT. The main focus will be on a method for analyzing localized spot patterns on general surfaces. Past analytic frameworks have been restricted to analyses on flat and spherical surfaces; we discuss a new addition to the analytic framework that allows us to obtain results on more general (and perhaps more realistic) surfaces. We also discuss briefly recent results obtained for the narrow escape problem inside an arbitrary bounded three-dimensional domain with a small target on the boundary. Both problems require detailed knowledge of local properties of Green's functions in curved geometries.

ABSTRACT. The Functionalised Cahn-Hilliard (FCH) is a higher-order free energy for blends of amphiphilic polymers and solvent which balances solvation energy of ionic groups against elastic energy of the underlying polymer backbone. Its gradient flows describe the formation of solvent network structures which are essential to ionic conduction in polymer membranes. The FCH possesses stable, coexisting network morphologies, and we characterise their geometric evolution, bifurcation and competition through a centre-stable manifold reduction which encompasses a broad class of coexisting network morphologies. The stability of the different networks is characterised by the meandering and pearling modes associated to the linearized system. For the H^{-1} gradient flow of the FCH energy, using functional analysis and asymptotic methods, we drive a sharp-interface geometric motion which couples the flow of co-dimension 1 and 2 network morphologies, through the far-field chemical potential. In particular, we derive expressions for the pearling and meander eigenvalues for a class of far-from-self-intersection co-dimension 1 and 2 networks and show that the linearization is uniformly elliptic off of the associated centre stable space.

ABSTRACT. The multicomponent functionalized free energies characterize the low-energy packings of amphiphilic molecules within a membrane through their proximity to connecting orbits within a reduced dynamical system. We develop a criteria under which the resulting interfaces are robustly stable and present a class of examples that arise naturally from geometric singular perturbation techniques. These examples include a model that characterizes the role of cholesterol as a stabilizing agent within phospholipid membranes. We present rigorous analysis of the associated interfacial motion as a quasi-steady motion in the tangent plane of a submanifold, illuminating the roles of bulk and surface diffusion in the underlying interfacial motion.

ABSTRACT. In this talk, We discuss several numerical schemes and solvers for the functionalized Cahn-Hilliard equation (FCH) and the phase field crystal equation (PFC) with periodic boundary conditions. As sixth order parabolic equations, they share common numerical difficulties. For some fully implicit schemes, we build nonlinear solvers featuring preconditioned Nesterov's accelerated gradient descent method (PAGD). We also discuss some semi-implicit schemes and compare their performances on several benchmark problems. 

ABSTRACT. In this talk we present the emergence of rigid polycrystalline structures from atomistic particle systems. The atomic interaction is governed by a suitably normalized pair interaction energy, where the "sticky disk" interaction potential models the atoms as hard spheres that interact when they are tangential. The discrete energy is frame invariant and no underlying reference lattice on the atomistic configurations is assumed. By means of Gamma-convergence, we characterize the asymptotic behavior of configurations with finite surface energy scaling in the infinite particle limit. The effective continuum theory is described in terms of a piecewise constant field delineating the local orientation and micro-translation of the configuration. The limiting energy is local and concentrated on the grain boundaries, i.e., on the boundaries of the zones where the underlying microscopic configuration has constant parameters. The corresponding surface energy density depends on the relative orientation of the two grains, their microscopic translation misfit, and the normal to the interface. We further provide a fine analysis of the surface energies at grain boundaries both for vacuum-solid and solid-solid phase transitions.

ABSTRACT. In this seminar we will investigate the relationship between the N-clock model and the XY model (at zero temperature) through a Gamma-convergence analysis as both the number of particles and N diverge. The N-clock model is a two-dimensional nearest neighbors ferromagnetic spin system, in which the values of the spin field are constrained to lie in a set of N equispaced points of the unit circle. For N large enough, it is usually considered as an approximation of the XY model, for which the spin field is allowed to attain all the values of the unit circle. By suitably renormalizing the energy of the N-clock model, we will illustrate how its thermodynamic limit strongly depends on the rate of divergence of N with respect to the number of particles. We shall see that the N-clock model turns out to be a good approximation of the XY model only for N sufficiently large; in other regimes of N, we will show with the aid of cartesian currents that its asymptotic behavior can be described by an energy which may concentrate on geometric objects of different dimensions.

ABSTRACT. Plasticity of metals is the emergent behaviour of many crystallographic defects (dislocations) interacting on the micro-scale. The limited understanding of plasticity stems from a largely incomplete, rigorous theory of this emergent behaviour. In my talk I present the next building block of this theory.

We model dislocation dynamics as particles moving on the real line. Then, dislocations interact by the electrostatic potential. Depending on their 'charge', dislocations either repel or attract each other. When two dislocations of opposite charge collide, they are taken out of the system (annihilation).

The aim is to pass to the limit as the number of dislocations tends to infinity (upscaling). The main challenge is that annihilation is new in the literature on upscaling, and that the available methods on upscaling do not apply. The main reason for this is that prior to collision, the velocity of the colliding dislocations blows up. In my talk I will show how we succeeded in establishing an upscaling result which includes annihilation.

This is joint work with M. A. Peletier and N. Pozar.

ABSTRACT. We develop discrete geometry methods to resolve the data ambiguity problem for periodic point sets to accelerate materials discovery. In any high-dimensional Euclidean space, a periodic point set is obtained from a finite set (motif) of points in a parallepiped (unit cell) by periodic translations of the motif along basis vectors of the cell. An important equivalence of periodic sets is a rigid motion or an isometry that preserves interpoint distances.

Crystals are still compared by descriptors that are either not isometry invariants or depend on manually chosen tolerances or cut-off parameters. All discrete invariants including symmetry groups can easily break down under atomic vibrations, which are always present in real crystals. We introduce a complete isometry invariant for all periodic sets of points, which can additionally carry labels such as chemical elements. The main classification theorem says that any two periodic sets are isometric if and only if their proposed complete invariants (called isosets) are equal. A potential equality between isosets can be checked by an algorithm, whose computational complexity is polynomial in the number of motif points. The key advantage of isosets is continuity under perturbations, which allows us to quantify similarities between any periodic point sets.

ABSTRACT. We model a periodic crystal structure C as a periodic set of points at centres of atoms. The natural equivalence relation on solid crystals is a rigid motion or an isometry that preserves distances between points. Hence crystals can be efficiently distinguished only by isometry invariants that remain unchanged under any rigid motion and a change of a lattice basis. To classify periodic crystals C up to isometries, we propose new invariants coming from a filtration of motif graphs MG(C;s), where s is a distance threshold. We have designed an efficient algorithm to compute motif graphs and will present experimental results on large crystal datasets.

ABSTRACT. As the atoms in periodic crystals are arranged periodically, such a crystal can be modeled by a periodic point set, i.e. by the union of several translates of a lattice. Two periodic point sets are considered equivalent if there is a rigid motion from one to the other. A periodic point set can be represented by a finite cutout such that copying this cutout infinitely often in all directions yields the periodic point set. The fact that these cutouts are not unique creates problems when working with them. Therefore, material scientists would like to work with a complete, continuous invariant instead. We conjecture that a tool from topological data analysis, namely the sequence of order k persistence diagrams for all positive integers k, is such a complete, continuous invariant of equivalence classes of periodic point sets.

ABSTRACT. Machine learning has emerged as a powerful approach in materials discovery. Its main challenge is selecting representations and features that enable universal and interpretable materials representations across multiple prediction tasks. We introduce an end-to-end machine learning model that automatically generates descriptors that capture a complex representation of a material’s structure and chemistry. This approach expands on computational topology applications (namely, persistent homology) and word embeddings from natural language processing by encapsulating chemical information in an automatic fashion from only the initial material system. We demonstrate our approach on multiple nanoporous metal–organic framework datasets by predicting methane and carbon dioxide adsorption across different conditions. Our results show considerable improvement in both accuracy and transferability across targets compared to models constructed from current commonly used manually curated features, consistently achieving an average 25–30% decrease in root-mean-squared-deviation, and an average increase of 40–50% in R2 scores. A key advantage of our approach is avoiding the “black–box” and providing interpretability: Our model allows us to locate the pores that correlate best to adsorption at different pressures, contributing to understanding atomic level structure-property relationships for materials design.

ABSTRACT. We propose a novel approximation scheme within the variational theory of fracture which approximates the energy functional of a body by a family of non-local functionals depending on a small parameter and two fields: the displacement field and an eigendeformation field which, in particular, describes damaged regions of the body. This extends previous work (joint with F. Fraternali, M. Ortiz) on brittle materials to materials undergoing cohesive fracture.

ABSTRACT. We derive a complete hierarchy of one-dimensional incompressible string and rod theories from fully three-dimensional variational models in nonlinear elasticity with the help of Gamma-convergence. To this end, we tailor the methodology introduced in the 3D-2D analysis of incompressible membranes and plates to our 3D-1D reduction setting. A technical difficulty in all scaling regimes is the construction of recovery sequences, which have to satisfy a nonlinear differential inclusion arising from local volume preservation. We overcome this issue with a suitable reparametrization argument in one of the two cross-section variables. Looking into the reduced models, we find that the limit energies differ from their compressible counterparts, whereas the incompressibility constraint does not affect the set of admissible deformations in the limit model in all physically relevant scaling regimes. We compare the Euler-Lagrange equations of the limit functional for isotropic rods with the corresponding compressible case to illustrate our findings. This is joint work with Carolin Kreisbeck (KU Eichstätt-Ingolstadt).

ABSTRACT. The continuum model related to the Winterbottom problem, i.e., the problem of determining the equilibrium shape of crystalline drops resting on substrates, is derived by means of a rigorous discrete-to-continuum passage performed by Gamma-convergence from atomistic models. Such atomistic models are introduced by taking into account both the interactions of the drop particles among themselves and with the fixed substrate atoms.

In particular, previous results in the literature are generalized to the presence of a half-plane substrate and, as a byproduct of the analysis, effective expressions for the drop anisotropy at the free surface and the drop wettability at the contact region with the substrate are characterized in terms of the atomistic potentials, which are chosen of Heitmann-Radin type.

Furthermore, a threshold condition only depending on such potentials is determined distinguishing the wetting regime, where discrete minimizers are explicitly characterized as configurations contained in a one-atom thick layer on the substrate, from the dewetting regime. In the latter regime, also in view of a proven conservation of mass in the limit as the number of atoms tends to infinity, proper scalings of the discrete minimizers converge to a bounded minimizer of the Winterbottom continuum model satisfying a nonzero volume constraint.

ABSTRACT. In the simplest case an Objective Structure is an atomic structure for which each atom “sees the same environment”. Familiar examples are buckyballs, single-walled carbon nanotubes (all chiralities), phosphorene, and the parts of many viruses. This key organizing principle can be used in many ways. As examples, we explain a new X-ray method of structure determination for noncrystalline structures, and we show how the principle can be used to design unusual phased arrays and macroscopic deployable origami structures.

ABSTRACT. We demonstrate a new class of elastic waves in the bulk: When longitudinal and transverse components propagate at the same speed, rolling waves with a spin that is not parallel to the wave vector can emerge. First, we give a general definition of spin for traveling waves. Then, since rolling waves cannot exist in isotropic solids, we derive conditions for anisotropic media and proceed to design architected materials capable of hosting rolling waves. Numerically, we show spin manipulations by reflection. Structures reported in this work can be fabricated using available techniques, opening new possibilities for spin technologies in acoustics, mechanics and phononics.

ABSTRACT. Controlling how waves propagate, attenuate and amplify is a daunting challenge for science and technology. In this talk, I will discuss how active --- or so-called non-Hermitian --- metamaterials can be used to steer mechanical waves in unprecedented ways. Using analogies between classical and quantum mechanics, I will discuss the emergence of unidirectionally amplified waves, of topological waves and of one-way solitons in non-Hermitian media. I will further show how these non-Hermitian waves can be used to create unusual responses to impacts.

ABSTRACT. We develop a multiscale continuum model to describe the interface structure in crystalline material such as FCC metals. The interface structure for twist, tilt and misfit grain boundaries are described by the dislocation network. The model incorporates both the anisotropy elasticity of each grain in crystalline materials and the molecular dynamics calculation informed interaction between two bulks, i.e., the nonlinear generalized stacking-fault energy. The equilibrium structures are obtained from the numerical simulations of the force balance differential equations. We apply this approach to determine the structure and energetics of twist, tilt and general grain boundaries. We also investigated the dislocation structure in heterogeneous crystalline material. Our model agrees well with the atomistic results. An analytical description is developed based on the obtained structural features.

ABSTRACT. TBA

ABSTRACT. We develop a continuum model for the dynamics of grain boundaries in three dimensions that incorporates the motion and reaction of the constituent dislocations. The continuum model includes evolution equations for both the motion of the grain boundary and the evolution of dislocation structure on the grain boundary. The critical but computationally expensive long-range elastic interaction of dislocations is replaced by a projection formulation that maintains the constraint of the Frank’s formula describing the equilibrium of the strong long-range interaction. This continuum model is able to describe the grain boundary motion and grain rotation due to both coupling and sliding effects, to which the classical motion by mean curvature model does not apply. Comparisons with atomistic simulation results show that our continuum model is able to give excellent predictions of evolutions of low angle grain boundaries and their dislocation structures.

ABSTRACT. The movement of dislocations and the corresponding crystal plastic deformation are highly influenced by the interaction between dislocations and nearby free surfaces. The boundary condition for inclination angle θ_{inc} which indicates the relation between a dislocation line and the surface is one of the key ingredients in the dislocation dynamic simulations. In this talk, we first present a systematical study on θ_{inc} by molecular static simulations in BCC-irons samples. We also study the inclination angle by using molecular dynamic simulations. A continuum description of inclination angle in both static and dynamic cases is derived based on Onsager’s variational principle. We show that the results obtained from continuum description are in good agreement with the molecular simulations. These results can serve as boundary con- ditions for dislocation dynamics simulations.

ABSTRACT. We will present a method to build effective dynamics along a few degrees of freedom (called reaction coordinates in the context of molecular dynamics), starting from high dimensional stochastic dynamics. The approach is very much related to the Mori-Zwanzig or projection operator method. Mathematical analysis of the quality of the effective dynamics compared to the original dynamics can be conducted, using function inequalities associated with the invariant measure sampled by the original dynamics conditioned to fixed values of the reaction coordinates. We will also present how to build good reaction coordinates relying on this mathematical understading, using a statistical estimation of the Poincaré constant using reproducing kernel Hilbert spaces.

References: F. Bach, T. Lelièvre, L. Pillaud-Vivien, A. Rudi and G. Stoltz, Statistical Estimation of the Poincaré constant and Application to Sampling Multimodal Distributions, Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR, 2020. F. Legoll and T. Lelièvre, Effective dynamics using conditional expectations, Nonlinearity, 2010. F. Legoll, T. Lelièvre and S. Olla, Pathwise estimates for an effective dynamics, Stochastic Processes and their Applications, 2017. F. Legoll, T. Lelièvre and U. Sharma, Effective dynamics for non-reversible stochastic differential equations: a quantitative study, Nonlinearity, 2019.

ABSTRACT. Extracting insight from the enormous quantity of data generated from molecular simulations requires the identification of a small number of collective variables whose corresponding low-dimensional free-energy landscape retains the essential features of the underlying system. Data-driven techniques provide a systematic route to constructing this landscape, without the need for extensive a priori intuition into the relevant driving forces. In this talk, I will introduce two methods for generating such collective variables, along with coarse configuration-space representations that can then be used to construct discrete-space kinetic models, e.g., Markov state models. The first approach leverages a hidden Markov model framework to overcome the noisy behavior of low-dimensional descriptors of the system by treating the descriptors as indicators of a latent Markov process. In the second approach, a variational autoencoder is employed to perform dimensionality reduction, while incorporating a Gaussian mixture model as a prior distribution on the latent space. This architecture enforces physical constraints onto the reduced space and allows for simultaneous determination of metastable states through an implicit clustering algorithm. The methods are demonstrated on model systems as well as more challenging molecular trajectories of disordered peptides and glassy liquids.

ABSTRACT. Coarse-graining high-dimensional molecular systems is an intense area of research in the past decades. Τhe exact coarse dynamics are described by the Mori-Zwanzing formalism leading to the generalized Langevin equation. Data-driven estimations of approximations for the force field and memory kernel of the generalized Langevin equation require an extensive effort and amount of all-atom simulations to achieve adequate statistics.

In the current work, we elaborate with high-dimensional systems for which the available observations are limited to short-time intervals and at transient regimes. We model the dynamics in such transient regimes by Markovian dynamics with time-dependent force-fields. Our approach is based on path-space variational inference reduced to a path-space force matching method under assumptions. We present the application of the path-space force matching method to retrieve the coarse space parametrized drift. At equilibrium, we can reproduce the classical force matching pair interaction potential. At transient -short time- regimes, we generate time-dependent drift coefficients, which can reproduce the all-atom dynamics. We present results for liquid water and a methane system at transient and equilibrium dynamical regimes.

ABSTRACT. Many problems in materials science and biology involve particles interacting with strong, short-ranged bonds, that can break and form on experimental timescales. Treating such bonds as constraints can significantly speed up sampling their equilibrium distribution, and there are several methods to sample subject to fixed constraints. We introduce a Monte Carlo method to handle the case when constraints can break and form. Abstractly, the method samples a probability distribution on a stratification: a collection of manifolds of different dimensions, where the lower-dimensional manifolds lie on the boundaries of the higher-dimensional manifolds. We show several applications in polymer physics, self-assembly of colloids, and high-dimensional volume calculation.

ABSTRACT. This talk presents a short overview of new probabilistic coupling methods for Hamiltonian Monte Carlo for people who are not familiar with the topic.

ABSTRACT. We propose a new sampling method based on the Metropolis Adjusted Langevin Algorithm (MALA), by introducing an adaptive diffusion function allowing a more efficient exploration of the parameter space. This diffusion function is obtained by optimizing the convergence rate of the sampling algorithm, which boils down to solving a spectral problem. The properties of this optimization problem are studied, and a numerical method is developed to compute the corresponding optimal diffusion. In addition, explicit solutions are derived in the homogenized asymptotic setting. Numerical experiments are also performed to evaluate the gain in terms of convergence speed, in comparison to the original MALA with constant diffusion.

ABSTRACT. We study the convergence to equilibrium of an underdamped Langevin equation that is controlled by a linear feedback force. Specifically, we are interested in sampling the possibly multimodal invariant measure at low temperature that is characterised by metastability and slow convergence to equilibrium. We follow an approach proposed by Pavon and co-workers (J. Math. Phys. 56:113302, 2015) and consider a Langevin equation that is simulated at high temperature, with the control balancing the additional noise so as to restore the original invariant measure at low temperature. We discuss different limits as the temperature ratio goes to infinity and prove convergence to a limit dynamics. It turns out that, depending on whether the lower ("target") or the higher ("simulation") temperature is fixed, the controlled dynamics converges either to the overdamped Langevin equation or to a deterministic gradient flow. This implies that (a) the ergodic limit and the large temperature separation limit do not commute in general, and that (b) it is not possible to accelerate the speed of convergence to the ergodic limit by making the temperature separation larger and larger. We discuss the implications of these observation from the perspective of stochastic optimisation and enhanced sampling schemes in molecular dynamics.

ABSTRACT. Enhanced sampling techniques generate trajectories at a biased potential, such that the exploration of the molecular state space and transitions across barriers is sped up. Path reweighing techniques recover the transition rates of the unbiased system from the biased trajectories by calculating the path probability ratio. Path reweighing requires that the trajectory has been generated using an integration scheme for stochastic dynamics, and that the formula for the path probability ratio has been tailored for that specific integration scheme. Thus, a separate reweighing factor for each stochastic integration scheme is needed. Most published path probability ratios are derived for overdamped Langevin dynamics. Yet, overdamped Langevin dynamics is less suited to model molecular dynamics than Langevin dynamics.Simply applying the path probability ratio for overdamped Langevin dynamics to a Langevin trajectory introduces a sizeable error. Here we derive the path probability ratio for the integration scheme of Langevin dynamics implemented in OpenMM. By comparing this path probability ratio to path probability ratio for overdamped Langevin dynamics, we then derive an approximate and general path probability ratio for Langevin dynamics. We show that the approximate path probability ratio yields highly accurate results, and discuss the limits of the approximation.

ABSTRACT. We present a new method to sample conditioned trajectories of a system evolving under Langevin dynamics. The trajectories are conditioned to end at a certain point or in a certain region of phase space, with the aim of simulating rare events such as barrier crossings. The bridge equation can be recast exactly in the form of a non linear stochastic integro-differential equation. At low temperature (low noise), the trajectories can be expanded around the zero temperature one. In the general case, the equation can be solved iteratively (fixed point method). We discuss how to choose the initial trajectory and show some examples of how the method works on some simple problems.

ABSTRACT. In this talk, we introduce parareal algorithms in the context of molecular dynamics, where we couple a fine propagator based on the reference potential energy landscape with a coarse propagator based on a surrogate potential. Although the parareal algorithm, in its original formulation, always converges, it suffers from various limitations in the context of MD. In particular, the gain of the algorithm is observed to converge to one when the time-horizon of the simulation increases. This numerical observation is backed up with theoretical discussions. We then introduce a modified version of the parareal algorithm wherein the algorithm adaptively divides the entire time-horizon into smaller time slabs. We then numerically show that the adaptive algorithm overcomes the various limitations of the standard parareal algorithm, thereby allowing for significantly improved gains.

Joint work with T. Lelievre and U. Sharma (ENPC).

ABSTRACT. To sample high-dimensional distributions, one may use the Hybrid Monte Carlo (HMC) algorithm, which generates nonlocal, nonsymmetric moves in the state space, alleviating random-walk behavior. In exploring the target distribution, the effort usually grows with the number of dimensions of the state space. Here, I build on an already defined HMC framework, Hybrid Monte Carlo on Hilbert spaces [A. Beskos, F.J. Pinski, J.-M. Sanz-Serna, A.M. Stuart, Stoch. Proc. Applic. 121, 2201 - 2230 (2011); doi:10.1016/j.spa.2011.06.003] that provides finite-dimensional approximations of measures, pi, which have density with respect to a Gaussian measure on path space, an infinite-dimensional Hilbert space. One has some freedom to choose the mass operator which is the novel feature of the algorithm described in this talk. With the original choice of the mass operator, the phase space pi was augmented with Brownian bridges. In the novel method, the phase space is augmented with Ornstein-Uhlenbeck (OU) bridges. In the former, the covariance of a proposed path grows with its length with concomitant negative effects on the acceptance rate, which is in contrast to the mass operator used here where the covariance is independent of the path length. The advantages of this HMC algorithm are demonstrated using computer experiments.

ABSTRACT. We consider a one-dimensional variational problem arising in connection with a model for cholesteric liquid crystals. The principal feature of our study is the assumption that the twist deformation of the nematic director incurs much higher energy penalty than other modes of deformation. The appropriate ratio of the elastic constants then gives a small parameter epsilon entering an Allen-Cahn-type energy functional augmented by a twist term. We consider the behavior of the energy as epsilon tends to zero. We demonstrate existence of local energy minimizers classified by their overall twist, find the Gamma-limit of these energies and show that it consists of twist and jump terms.

ABSTRACT. We consider minimizers of an energy of Ginzburg-Landau type with weak anchoring, on a bounded smooth domain in two dimensions. This functional was previously derived as a thin-film limit of the Landau-de Gennes energy, assuming weak anchoring on the boundary favoring a nematic director lying along a cone of fixed aperture, centered at the normal vector to the boundary. In a regime where boundary vortices are present, the phase jumps by either $2\alpha$ (light boojums) or $2(\pi-\alpha)$ (heavy boojums). Our main result is the fine-scale description of the light boojums.

ABSTRACT. The mathematical analysis of liquid crystal models poses many challenging questions, as can be seen by their close relationship to the study of singularities for harmonic maps. In this talk, I will discuss the structure of defects in the context of different models of nematic liquid crystals, and their connection to classical results on harmonic maps into the sphere. To illustrate, I will present the physically fundamental problem of defects created by a colloid particle immersed in a nematic, and present recent results using the Landau-de Gennes energy. We find that the Landau-de Gennes model allows for a greater variety of types of singularity than the (harmonic map-based) Oseen-Frank energy, including line singularities such as the “Saturn Ring” defect. This work is joint with S. Alama, D. Golovaty and X. Lamy.

ABSTRACT. Lipid membranes present in all domains of life, acting as defensive barriers that separate cell from the external environment, differentiating intracellular compartments, and controlling the flow of materials and information between those. This presentation will discuss our recent efforts on understanding the interactions between biomembranes with small molecules and nanoparticles using multiscale numerical simulations. On the atomistic scale, we have exploited all-atom molecular dynamics to identify the specific interactions between Pseudomonas quinolone signal (PQS) with a model Pseudomonas outer membrane (OM). The simulations confirmed the spontaneous insertion of PQS into the lipid A leaflet of OM aided by hydrogen bonds. A prominent conformational change of PQS was observed to facilitate its insertion. On the mesoscale, we developed a coarse-grained model in the framework of dissipative particle dynamics to examine the cellular uptake of nanoplastics. We have revealed the long-time translational and rotational dynamics of hydrophobic nanotetrahedrons embedded in a lipid bilayer. We found that particles with size comparable to or larger than membrane thickness exhibits different configurations. The translation dynamics of these anisotropic particles are diffusive and independent of size while the out-of-plane rotation of large particles are arrested and show subdiffusive behavior.

ABSTRACT. We investigate the spatio-temporal dynamics of a mass-conserved bulk-surface coupled model for Cdc42 intracellular oscillations. In 1-D the model captures the dynamics of a cytosolic bulk diffusion field with nonlinear binding kinetics near two opposite pointlike cellular membranes, and is formulated as a coupled PDE-ODE system. When assuming a 2-D circular bulk domain, surface or membrane diffusion is added to the model and contributes to the overall protein pattern-forming dynamics. Our analysis of the 1-D case reveals the existence of symmetric and asymmetric steady states, as well as pole-to-pole relaxation oscillations typical of slow-fast systems, whereas in 2-D symmetry-breaking instabilities cause the formation of stationary Turing patterns and rotating waves.

ABSTRACT. The Functionalized Cahn-Hilliard equation has been proposed as a model for the interfacial energy of phase-separated mixtures of amphiphilic molecules. We study the existence of a nonnegative weak solutions of a gradient flow of the Functionalized Cahn-Hilliard equation subject to a degenerate mobility $M(u)$ that is zero for $u\leq 0$. Assuming the initial data $u_0(x)$ is positive, we construct a weak solution as the limit of solutions corresponding to non-degenerate mobilities and verify that it satisfies an energy dissipation inequality

ABSTRACT. A coupled opto-electrical model for a multi-layer methylammonium lead iodide (MAPI) perovskite device with mobile ionic charge carriers, charge trapping, Shockley-Read-Hall (SRH) recombination and doped transport layers is validated with multiple DC, AC and transient experiments [1]. We show the influence of certain physical ingredients such as e.g. the ions on the simulated experiments. Parameter extraction from a single experiment is often difficult as model parameters might be correlated. The combination of time-dependent and steady-state measurements can reduce the correlation between the extracted parameters [2]. At the same time finding a set of parameters that optimally describes multiple experiments is a challenge [1,3]. Finally, we discuss a study where the parameter space is explored by a machine learning model.

[1] M.T. Neukom et al., ACS Appl. Mater. Interfaces (2019), 11, 26, 23320-23328.] [2] M.T. Neukom et al., Organic Electronics (2012), 13, 2910-2916. [3] S. Jenatsch, S. Altazin, P.-A. Will, M. T. Neukom, E. Knapp, S. Züfle, S. Lenk, S. Reineke, and B. Ruhstaller, Journal of Applied Physics 124, 105501 (2018); doi: 10.1063/1.5044494.

ABSTRACT. The effects of transport layers on perovskite solar cell performance, in particular anomalous hysteresis, are investigated. A model for coupled ion vacancy motion and charge transport is formulated and solved, via both asymptotic and numerical methods, in a three-layer planar perovskite solar cell. Its results are used to demonstrate that the replacement of standard transport layer materials (spiro-OMeTAD and TiO2) by materials with lower permittivity and/or doping leads to a shift in the scan rates at which hysteresis is most pronounced to rates higher than those commonly used in experiment. These results provide a cogent explanation for why organic electron transport layers can yield seemingly ‘‘hysteresis-free’’ devices but which nevertheless exhibit hysteresis at low temperature. In these devices the decrease in ion vacancy mobility with temperature compensates for the increase in hysteresis rate with use of low permittivity/doping organic transport layers. Finally, features of the steady-state potential profile for a device held near the maximum power point are used to suggest ways in which interfacial recombination can be reduced, and performance enhanced, via tuning transport layer properties.

ABSTRACT. Device level modeling of Perovskite based solar cells is challenging on several levels. For example, the active layer is crystalline and the blocking layers are amorphous so what is actually the meaning of band alignment? The crystalline layer is actually poly-crystalline with the recombination losses taking place mostly at grain boundaries - so how can one use a 1D model, that "runs" perpendicular to these boundaries, account for anything?Ion motion under electric field is actually electrochemistry. What about all the other electrochemical reactions that are more difficult to model? We'll address such questions and try to answer most of them.

ABSTRACT. Impedance spectroscopy (IS) is a simple measurement technique that yields useful results for established photovoltaic technologies. However, the results from IS for perovskite solar cells (PSCs) are not fully understood. We present an analytic model, derived from drift-diffusion theory, that includes the coupled electronic-ionic dynamics of a PSC. Close to open-circuit, spectra calculated from our analytic model show excellent agreement with those determined via numerical solution to the full drift-diffusion equations. We identify how to calculate an ideality factor for PSCs, that diagnoses the dominant recombination mechanism taking place. This factor, that we term the ‘electronic ideality factor’, can be determined experimentally via high frequency impedance measurements. This analytic model relates the resistances and capacitances associated with the high and low frequency features to cell properties and physical mechanisms. This includes peculiar features, such as the giant low frequency capacitance and inductive responses, which emerge naturally from the model.

ABSTRACT. Smectic liquid crystals enjoy both dislocations and disclinations. However, the two types of defects are not independent -- disclinations can alter not only the texture but also the charge of dislocations. Using an approach based on Morse theory we develop the language to study defect motion and interactions in smectics.

ABSTRACT. A nematic liquid-crystal cell subject to an electric field created by electrodes held at constant potential is modeled as a variable capacitor in an RC circuit. The state of the system is characterized in terms of the director field in the cell and the charge on the electrodes. A dynamical system is developed that couples director dynamics in the cell and charge dynamics in the circuit. The dynamical equations are derived from expressions for the total potential energy of the system and a dissipation involving a single rotational viscosity for the director plus Joule heating associated with current in the circuit. An effort is made to quantify effects, in particular the widely varying time scales for the processes involved. The exercise illuminates aspects of the modeling of equilibrium states of such a system.

ABSTRACT. In this talk I review work on the theory of liquid crystal (LC) films sandwiched between two flat, parallel substrates, and its validation by Monte Carlo computer simulation. For generality we consider a purely steric microscopic model of uniaxial particles of length-to-breadth ratio κ, either prolate, (κ>1) or oblate (0<κ<1), represented by the hard Gaussian overlap (HGO) potential. Each substrate sees a prolate particle as an infinitely thin hard needle, or an oblate particle as an infinitely thin hard disc. The needle length or the disc diameter can be chosen to be less than or equal to the actual length of the prolate particle or diameter of the oblate particle: homeotropic (planar) anchoring is achieved for small (large) needle lengths of prolate particles or large (small) disc diameters of oblate particles. We used classical density-functional theory, implemented at the level of Onsager's second-virial approximation with Parsons-Lee rescaling to calculate the structure of both symmetric and hybrid HGO films. Specifically we looked at the effects of varying anchoring strengths, as well as film thickness. Our theory provides a reasonable description of the non-uniform density and orientational order, including layering effects, for prolate particles, but is somewhat less reliable for oblate particles.

ABSTRACT. Twist cells are characterized by a director field that rotates perpendicularly to the cell axis in the absence of an applied voltage, but also leans into the axis direction as the voltage is increased. This poses two modeling challenges.

The first is that the director field is not constrained to a plane and so must be described by two coordinate angles. Unfortunately, the spherical coordinate system is degenerate so that it is not possible to write evolution equations for this representation of the director field. We overcome this problem by representing the director field using two mutually orthogonal spherical coordinate systems and switching seamlessly between them. The resulting equations are then integrated using a Chebyshev collocation method.

The second challenge is that light propagating through the cell experiences neither a simple phase lag, as in planar cells, or a polarization rotation, as in pure twist cells. We use an efficient beam propagation method [Oldano, Phys. Rev. A 40. 6014 (1989)] to represent the field propagation as a boundary value problem. The accurate representation of the field oscillations requires a fine resolution that we obtain by resampling the grid used to solve the alignment equations.

ABSTRACT. We systematically engineer a series of square and rectangular phononic crystals to create experimental realizations of complex topological phononic circuits. The exotic topological transport observed is wholly reliant upon the underlying structure which must belong to either a square or rectangular lattice system and not to any hexagonal-based structure. The phononic system chosen consists of a periodic array of square steel bars which partitions acoustic waves in water over a broadband range of frequencies (∼0.5MHz). An ultrasonic transducer launches an acoustic pulse which propagates along a domain wall, before encountering a nodal point, from which the acoustic signal partitions towards three exit ports. Numerical simulations are performed to clearly illustrate the highly resolved edge states as well as corroborate our experimental findings. To achieve complete control over the flow of energy, power division and redirection devices are required. The tunability afforded by our designs, in conjunction with the topological robustness of the modes, will result in their assimilation into acoustical devices.

ABSTRACT. Cavity design is crucial for single-mode semiconductor lasers such as the ubiquitous distributed feedback and vertical-cavity surface-emitting lasers. By recognizing that both of these optical resonators feature a single mid-gap mode localized at a topological defect in the one-dimensional lattice, we upgrade this topological cavity design concept into two dimensions using a honeycomb photonic crystal with a vortex Dirac gap by applying the generalized Kekulé modulations. We theoretically predict and experimentally show on a silicon-on-insulator platform that the Dirac-vortex cavities have scalable mode areas, arbitrary mode degeneracies, vector-beam vertical emission and compatibility with high-index substrates. Moreover, we demonstrate the unprecedentedly large free spectral range, which defies the universal inverse relation between resonance spacing and resonator size. We believe that our topological micro-resonator will be especially useful in applications where single-mode behaviour is required over a large area, such as the photonic-crystal surface-emitting laser.

Similarly, we propose a topological bandgap fibre whose bandgaps along in-plane directions are opened by generalised Kekulé modulation of a Dirac lattice with a vortex phase and the number of guiding modes equals the winding number of the spatial vortex. The single-vortex design provides a single-polarisation single-mode for a bandwidth as large as one octave.

ABSTRACT. In this talk we describe recent progress on achieving non-reciprocal wave motion in elastic waveguides. In the first part, we describe an experimental platform consisting on an electromechanical beam with an array of piezoelectric patches that are used to achieve real-time control over the beam’s stiffness. We illustrate how topological pumping is achieved by the careful scheduling of the beam’s stiffness modulation, which causes a transition of a topological edge state from one end of the waveguide to the other. Such transition is experimentally achieved at controllable speeds, suggesting intriguing opportunities for robust transferring of information through elastic waves. In the second part, we investigate non-Hermitian elastic lattices with feedback control interactions. We show that the family of lattices exhibits entirely non-reciprocal bands presenting directional wave amplification. Furthermore, we illustrate the non-Hermitian skin effect, whereby the majority of the bulk modes of both 1D and 2D lattices are found to be localized at the boundaries. Finally, we present numerical investigations demonstrating how feedback interactions of this type may be implemented on elastic waveguides with piezoelectric constituents, which may pave the way to future experimental studies.

ABSTRACT. Recent interest in metamaterials has led to a renewed study of wave mechanics in different branches of physics. Elastodynamics is special owing to the coupling between the volumetric and shear parts of the elastic waves. Through a study of in-plane waves traversing periodic laminates, we show that this coupling results with unusual energy transport. We find that the frequency spectrum contains modes which simultaneously attenuate and propagate, and demonstrate that these modes coalesce to purely propagating modes at exceptional points in the spectrum. These non-Hermitian degeneracies with propagating modes, which were realized by balancing gain and loss in the system, are reported in a purely elastic setting. We show that the laminate exhibits metamaterial features near these points, such as negative refraction. While negative refraction in laminates has been demonstrated, here we realize it for coupled waves impinging on a simple single-layer interface. This feature, together with the appearance of exceptional points, are absent from the model problem of anti-plane shear waves which have no volumetric part, and hence from the mathematically identical electromagnetic waves in materials with positive refractive index. Thereby, our work paves the way for future applications such as asymmetric mode switches using a tangible elastic apparatus.

ABSTRACT. For large dimensional and stiff diffusion problems, explicit stabilized integrators are an efficient alternative to implicit or semi-implicit methods to avoid the severe timestep restriction faced by standard explicit time integrators. We present a family of explicit stabilized integrators for ergodic and stiff stochastic problems, based on second kind Chebyshev polynomials, that yield an optimal size of extended mean-square stability domain that grows at the same quadratic rate as the optimal family for deterministic problems. We also show that the new explicit stabilized schemes converge in the strong sense when applied to stochastic semilinear diffusion partial differential equations.

Based on joint works with A. Abdulle (Lausanne), I. Almuslimani (Rennes), and C.-E. Bréhier (Lyon).

ABSTRACT. We study numerical methods for dissipative particle dynamics (DPD), which is a system of stochastic differential equations and a popular stochastic momentum-conserving thermostat for simulating complex hydrodynamic behavior at mesoscales. We propose a new splitting method that is able to substantially improve the accuracy and efficiency of DPD simulations in a wide range of the friction coefficients, particularly in the extremely large friction limit that corresponds to a fluid-like Schmidt number, a key issue in DPD. Various numerical experiments on both equilibrium and transport properties are performed to demonstrate the superiority of the newly proposed method over popular alternative schemes in the literature.

ABSTRACT. In statistical mechanics, linear response is the measure to quantify the average response of a steady state statistical system to the small external perturbation. In this talk, we discuss numerical schemes for the linear response computations of invariant measures from fluctuations at equilibrium. The schemes are based on Girsanov's change-of-measure theory and apply reweighting of trajectories by factors derived from a linearization of the Girsanov weights leading to the class of estimators we call martingale product estimators. We investigate both the discretization error and the finite time approximation error. The designed numerical schemes are shown to be of bounded variance with respect to the integration time, which is a desirable feature for long time simulations. We also show how the discretization error can be improved to second order accuracy in the time step by modifying the weight process in an appropriate way. The resulted methodology provides an alternative computational approach to classical linear response methods such as the Green-Kubo approach. We provide some numerical evidences to demonstrate the efficiency and accuracy of the method. (joint work with Gabriel Stoltz and Ting Wang).

ABSTRACT. We present the results of an extensive numerical study of the macroscopic steady state of the rotor chain, performed by numerically integrating the system of partial differential equations which describes it. We study various properties of the profiles of temperature and angular momentum solutions to these equations with boundary conditions determined by the thermo-mechanical forcing. This allows to characterize the regime of parameters leading to uphill energy diffusion -- a situation in which the energy flows in the direction of the gradient of temperature -- and to identify regions of parameters corresponding to a negative energy conductivity (i.e. a positive linear response of the energy current to a gradient of temperature). The macroscopic equations we derive are consistent with some previous results obtained by numerical simulation of the microscopic physical system. (Joint work with S.Olla and G.Stoltz)

ABSTRACT. Nonequilibrium (or irreversible) processes arise in all areas of science whenever external forces or flows act on a system. Efficient calculation of the steady state of a nonequilibrium process tends to be difficult. In particular, for nonequilibrium processes, the density of the steady state is typically not known, and therefore conventional methods of variance reduction such as importance sampling do not apply. We present and analyze a method that adapts the principle of stratified survey sampling to the efficient calculation of nonequilibrium steady states. Our methods also apply to the calculation of certain dynamical quantities such as reaction rates.

ABSTRACT. We explore whether splitting and killing methods can improve the accuracy of Markov chain Monte Carlo (MCMC) estimates of rare event probabilities, and we make three contributions. First, we prove that "weighted ensemble" is the only splitting and killing method that provides asymptotically consistent estimates when combined with MCMC. Second, we prove a lower bound on the asymptotic variance of weighted ensemble's estimates. Third, we give a constructive proof and numerical examples to show that weighted ensemble can approach this optimal variance bound, in many cases reducing the variance of MCMC estimates by multiple orders of magnitude.

ABSTRACT. We recently showed that Markov state models (MSMs) constructed from molecular dynamics trajectory data could be improved by including a minimal amount of history information. Stratifying trajectories into two sets based on the last macrostate visited leads to history-augmented MSMs (haMSMs) which provide accurate kinetics at arbitrary lag times and also yield mechanistic information which MSMs typically cannot capture accurately due to the need for lag times which may be longer than the transition path time (event duration). In other recent work, we have developed a framework for self-consistently using all history information in discretized trajectories even when the last-macrostate label is not available. The new approach uses stationarity (first-step analysis) for the self-consistent computation of equilibrium or non-equilibrium steady distributions, committors (splitting probabilities) and first-passage times.

ABSTRACT. Construction of solution to hyperbolic equations is challenging because the solution does not smooth out with time and discontinuities persist such as a shock wave. They should be properly approximated during the solution. Also, the solution method should ideally preserve the conservation of energy. Furthermore, the knowledge of characteristic directions is essential prior to the solution process. Therefore, the solution procedure is problem dependent and becomes more of an art. This study presents a peridynamic (PD) approach to solve linear and nonlinear hyperbolic equations. It specifically employs the PD differential operator to recast the nonlocal form of these equations by introducing an internal length parameter (horizon) that defines association among the points within a finite distance. It enables their computational solution without special treatments through simple discretization. The capability of this approach is demonstrated by considering various linear advection-diffusion equations and the nonlinear Eikonal equation with a complex velocity field (Marmousi). It also presents the dependence of dispersion, dissipation and stability on the horizon size.

ABSTRACT. We formulate a nonlocal phase-field modeling framework for better understanding the failure processes in material. To this end, we employ a multi-scale simulation and modeling (from molecular dynamics to continuum mechanics) approach in order to understand the anomalous sub-grid scales' dynamics in our finite-element analyses. We develop and introduce a new mathematical-statistical-computational framework which allows one to propagate the nonlocal effects across the scales up to the observable continuum failure modeling, where the nucleation of fractional operators occurs after the onset of failure.

ABSTRACT. Phase-field models are a popular choice in computational physics to describe complex dynamics of substances with multiple phases and are widely applied in various applications including solidification or fracture mechanics. Usually, diffuse interface models that are governed by local differential operators are employed, such as Cahn-Hilliard or Allen-Cahn. In contrast, we analyze models where the interface evolution is represented by a nonlocal operator. While the classical local phase-field models always lead to a diffuse interface, we demonstrate that a careful choice of the nonlocal operator can allow for a sharp interface in the solution. In particular, we analyze a Cahn-Hilliard model with a nonsmooth double-well potential of an obstacle type. Here, the lack of smoothness of the potential is essential to guarantee the aforementioned sharp-interface property. Mathematically, this introduces additional inequality constraints that, in a weak form, lead to a coupled system of variational inequalities. We prove the well-posedness and regularity of the semi-discrete and continuous in time weak solutions and derive conditions under which pure phases are admitted. We also present space-time discretizations that can be realized efficiently. Finally, we discuss extensions to more complex nonlocal phase-field models arising in the context of solidification of pure materials.

ABSTRACT. Deducing the emergent behavior of materials from the properties of their molecular or atomic constituents is one of the greatest challenges of condensed matter theory. Considering many-body systems with highly cooperative ground states renders this task even more challenging. Geometrically frustrated assemblies are comprised of ill-fitting constituents whose locally preferred arrangement cannot be globally realized and thus lack a stress-free rest state. The ground state of frustrated assemblies is highly cooperative, leading them to exhibit super-extensive energy growth, filamentation, size limitation, and exotic response properties. Such systems arise in naturally occurring structures in biology and organic chemistry as well as in manmade synthetic materials. In this talk, I will discuss how the intrinsic approach, in which matter is described only through local properties available to an observer within the material, overcomes the lack of a stress-free rest state for frustrated assemblies and leads to a general framework. This newly devised framework simultaneously accounts for the internal interaction in the material and its underlying geometric structure. In particular, it allows predicting the super-extensive energy exponent for sufficiently small systems.  I will discuss its application to several specific systems exhibiting geometric frustration: growing elastic bodies, frustrated liquid crystals, and twisted molecular crystals.

ABSTRACT. Mechanical and electrical metamaterials exhibiting gapped oscillation spectra or topologically protected modes enable precise control of structural, acoustic or transport functionalities. We present a flexible computational inverse-design framework that allows the efficient tuning of one or more gaps at nearly arbitrary positions in the spectrum of discrete metamaterial structures. The underlying algorithm, which directly optimizes the linear network response, is applicable to ordered and disordered structures, scales efficiently in two and three dimensions, and can be combined with a wide range of numerical optimization schemes. We illustrate the broad practical potential of this approach by designing mechanical band-gap switches that open and close preprogrammed spectral gaps in response to an externally applied stimulus such as shear or compression. The spectra of nonlinear active mechanical and electric circuits can be designed similarly.

ABSTRACT. We investigate rods made of nonlinearly elastic, composite–materials that feature a micro‐heterogeneous prestrain that oscillates on a scale that is small compared to the length of the rod. As a main result we derive a homogenized bending–torsion theory for rods as Γ‐limit from 3D nonlinear elasticity by simultaneous homogenization and dimension reduction under the assumption that the prestrain is of the order of the diameter of the rod. The limit model features a spontaneous curvature‐torsion tensor that captures the macroscopic effect of the micro‐heterogeneous prestrain. We device a formula that allows to compute the spontaneous curvature‐ torsion tensor by means of a weighted average of the given prestrain, with weights depending on the geometry of the composite encoded by correctors. We observe a size‐effect: For the same prestrain a transition from flat minimizers to curved minimizers occurs by just changing the value of the ratio between microstructure‐scale and thickness.

This joint work with S. Neukamm and R. Bauer (TU Dresden)

ABSTRACT. Thin elastic sheets exhibit a rich variety of morphologies and instabilities, all through the interplay of geometry and boundary constraints. In the extreme limit of atomically thin membranes, such a graphene, thermal fluctuations play an important role by dramatically renormalizing the elastic moduli in a scale dependent fashion. Motivated by the influence of geometry in the mechanics of ultrathin nanodevices, I will first discuss how the humble buckling instability acquires novel features at finite temperature as an unusual size dependent critical phase transition. Boundary conditions surprisingly lead to inequivalent ensembles and distinct universality classes of thermalized buckling, allowing for temperature dependent control of stresses in the sheet. Post buckling, geometry is once again key. The curvature of the buckled sheet allows for localized deformations that can be patterned and manipulated from the boundary. In the context of acoustics, either in electromechanical resonators or macroscale musical instruments such as the singing saw, I will demonstrate how curvature profiles of bent plates can localize sound modes to inflection points in a robust fashion. This is akin to similar protected states in electronic topological insulators and it allows for an attractive geometric route to designing mechanoacoustics in thin sheets.

ABSTRACT. The interplay between geometry and topology is of fundamental importance throughout many disciplines. In this respect, the investigation of physical effects governing the responses of curved magnetic nanoobjects to electric and magnetic fields is of strong fundamental interest but is also technologically appealing. Owing to intense theoretical and experimental efforts, an emerging area of curvilinear magnetism has relevantly expanded during past few years, demonstrating that it can encompass a range of fascinating geometry-induced effects in the magnetic properties of materials.

Here I focus on the peculiarities emerging from geometrically curved magnetic wires and films. Emergent interactions, induced by the curvilinear geometry manifest themselves in topological magnetization patterning and magnetochiral effects in conventional magnetic materials. These curvature-induced interactions can be not only local (when they stem from the exchange energy) but also non-local (when they are due to magnetostatics). As a consequence, family of novel curvature-driven effects emerges, resulting in theoretically predicted unlimited domain wall velocities, chirality symmetry breaking etc. Current and future challenges of the curvilinear magnetism will be discussed.

ABSTRACT. We consider the numerical approximation of the inertial Landau-Lifshitz-Gilbert equation (iLLG), which describes the dynamics of the magnetization in ferromagnetic materials at subpicosecond time scales. We present two fully discrete numerical schemes, both implicit and based on first-order finite elements, which preserve the inherent unit-length constraint of iLLG at the vertices of the underlying mesh, and generate approximations that converge towards a weak solution of the problem.

ABSTRACT. We analyse two variants of a nonconvex variational model from micromagnetics with a nonlocal energy functional, depending on a small parameter epsilon > 0. The model gives rise to transition layers, called Néel walls, and we study their behaviour in the limit epsilon -> 0. The analysis has some similarity to the theory of Ginzburg-Landau vortices. In particular, it gives rise to a renormalised energy that determines the interaction (attraction or repulsion) between Néel walls to leading order. But while Ginzburg-Landau vortices show attraction for degrees of the same sign and repulsion for degrees of opposite signs, the pattern is reversed in this model. First, we show that the Néel walls stay separated from each other and we determine the renormalised energy for one of the models. The theory gives rise to an effective variational problem for the positions of the walls, encapsulated in a Gamma-convergence result. Second, we turn our attention to another, more physical model, including an anisotropy term. We show that it permits a similar theory, but the anisotropy changes the renormalised energy in unexpected ways and requires different methods to find it. This is a joint work with R. Moser (Univ. of Bath).

ABSTRACT. The study of magnetization oscillations of ferromagnetic systems around equilibrium configurations is fundamental for understanding magnetodynamics driven by microwave fields[Brown1963,Suhl1957], spin currents[Demidov2016], thermal fluctuations[Perzlmeier2005] and has lately become relevant for analyzing magnonic waveguides[Kruglyak2010]. The normal oscillation problem was originally tackled with analytical techniques[Brown1963] limited to saturated particles of special shapes. Here a frequency-domain setting for small magnetization oscillations in ferromagnets with generic shapes around arbitrary stable micromagnetic equilibria (e.g. vortex, skyrmion) is presented[dAquino2009]. This formulation has several advantages over time-domain techniques, leads to a matrix-free numerical eigenmode solver with no need to assemble the effective field operator avoiding unpractical O(N^2) storage and computational cost (N=number of discretization cells) and allows finite-difference/finite element eigenmode analysis for micron-sized ferromagnets composed of several hundred thousand computational cells in reasonably short time. Moreover, it yields semi-analytical computation of thermal power spectra[Bruckner2019] orders of magnitude faster than conventional techniques. Finally, this approach is amenable of extension to reduced-order description of nonlinear dynamics around micromagnetic equilibria.

References W.F.Brown Jr, Micromagnetics, Wiley (1963). H.Suhl, J.Phys.Chem.Solids 1,209 (1957). V.E.Demidov et al, Nat.Commun. 7,10446 (2016). K.Perzlmeier et al, PRL. 94,057202 (2005). V.V.Kruglyak et al, J.Phys.D:Appl.Phys. 43:264001 (2010) M.d'Aquino et al, J.Comput.Phys. 228,6130 (2009). F.Bruckner et al, J.Magn.Magn.Mater. 475,408 (2019).