ABSTRACT. Lattice kinetic Monte Carlo simulations involving charged particles pose quite some challenges in terms of computational time. The origin is the long range Coulomb interactions which cause the transition rates to depend on the configuration of the whole system. As a consequence, the computational costs per kMC step behave as O(N) in the system size N - even if in every step the change in the configuration is only local, i.e. computational costs of the corresponding system without Coulomb interaction would have a complexity per step of O(1). I will present a multilevel kinetic Monte Carlo strategy to reduce the computational costs. For this, we split the rate updates into short range and long range contributions. The costly long range updates are then conducted only every M steps. Different values for M are then exploited in a multilevel Monte Carlo fashion. This allows to conveniently balance the bias due to approximate updates with the sampling error. Further, we can conduct most of the sampling at larger values of $M$, thereby reducing the costs further and achieving close to optimal convergence rates. The approach will be employed to shine some light on the diffusion of lithium ions in graphite electrodes.

ABSTRACT. In this presentation, we will present two examples of KMC hybrid methods / couplings. The first one, applied to study the formation of a C Cottrell atmosphere around a screw dislocation in bcc Fe, couples the usual rigid-lattice approximation KMC with the soft-lattice KMC offered by k-ART. Near the dislocation where the energy landscape is unknown, k-ART is called to search for the possible transitions and compute the C migration energies. On the contrary, the diffusion of C atoms far from the dislocation is well described by using the rigid-lattice approximation, which is much faster than k-ART. The second example is applied to the modelling of the reactor pressure vessel steel and is more precisely a hybrid between atomistic and object KMC. In this method, point-defect clusters whose sizes are less than a value set by the user are considered as ”objects”.

ABSTRACT. The temperature-programmed molecular dynamics (TPMD) method [1] is a rare-event acceleration technique that provides a convenient way of estimating the effective rate constants and the Arrhenius parameters of kinetic pathways even in situations where the underlying landscape is rugged. In this talk, I will present some of the new advances in the TPMD method. In particular, I will talk about a new variation of the TPMD method that employs finite time MD calculations in order to speed-up thermally activated events from a particular state of the system. Kinetic pathways are sought from a collection of states without any prior knowledge of these pathways. Since kinetic pathways are selected with a probability that is proportional to their rate constants, we find that slow pathways with small pre-exponential factors and large activation barriers are rarely sampled with TPMD. We introduce a procedure to overcome this limitation by employing bias potentials. This additional feature in the TPMD method dramatically improves its ability to estimate Arrhenius parameters. Examples of the variation of the TPMD method are provided for metal surface diffusion in presence of solvent.

1. S. Shivpuje, M. Kumawat, A Chatterjee, Computer Physics Communications, 262, 107828 (2021).

ABSTRACT. A desired energy landscape can achieve ease of actuation and precise actuation response for a morphing structure. This presentation considers morphing structures consisting of (1) pin-jointed rigid bars and hinge-joined plates and (2) torsional and translational springs. A prescribed energy landscape during motion is targeted for such a structure through optimizing the spring parameters, i.e., spring rest positions and stiffness. The method proposed in this talk first analyzes the mechanism numerically and then solves for the spring parameters to yield an energy landscape closely matching the target. This general approach is applicable to complex structures where the analytical solution of kinematics is not available. Various examples are provided in the presentation to demonstrate the design process and the effectiveness of the method.

ABSTRACT. A creased thin disk can be bent to a second stable state with a vertex on the crease. Recently, we investigated this bistable behavior by cutting a hole around the vertex and found a critical hole size that destroys the bistability. Here, we consider several additional geometric factors, which include an angle deficit that determines the amount of materials being cut/inserted circumferentially, the number of evenly distributed creases, and the eccentricity that measures the position of the hole on the radius. We find both in experiments and with numerical continuation of an inextensible strip model that: (1) By cutting out an annular sector, the critical hole size increases significantly and could be as large as the disk, and bistability holds (the structure becomes an annular strip); while inserting a sector quickly decreases the critical hole size to zero. (2) With a single crease, the hole could be as large as the disk. (3) A family of stable states could be obtained by bending almost anywhere on the crease, and the critical hole size depends on the eccentricity. Our results shed new light on the mechanics of creased thin structures.

ABSTRACT. Multistable structures have multiple stable configurations for different functions and have been used to design deployable structures and artificial materials. Recently, rigid origami, which can be regarded as a mechanism by considering the facets and creases as rigid parts and hinges, finds its way as multistable structure by adding compliant segment or torsional springs to creases. To achieve more possibilities of rigid origami, cutting are introduced to traditional origami which is kirigami. In this research, a kirigami structure created by cutting faces of an origami cuboid is firstly designed, which corresponds to multiple degrees of freedom mechanism unit, and its kinematics and geometries are discussed based on mechanism theory. With torsional springs delicately set-in, the kirigami structure realises three stable states including closed, opened and compact folding configurations. Then the transformation between states under uniaxial forces is studied. Finally, a multistable structure was fabricated by tessellating the tristable structures.

ABSTRACT. Rigid origami is developed as a tool for effectively transforming a two-dimensional material into a three-dimensional structure, which has been broadly applied to robots, deployable structures, metamaterials, architectures, etc. These successful applications inspire a great number of research on the kinematics and mechanical properties of rigid-foldable creased papers. However, here we want to consider the statics and stability of a creased paper: Is it statically rigid under a certain equilibrium load? If it is, what is the internal force inside the creased paper? Is a creased paper stable even if it is not statically rigid? How would a pre-stress change the stability? The answer to these questions on statics are linked with the kinematics: first-order rigidity and second-order rigidity. The theoretical results presented here will be instructive in the design of rigid origami structures that carry load.

ABSTRACT. I will present a correspondence between a spectral flow -- i.e. eigenvalues of an operator that transit from bands to bands when sweeping a parameter -- and the Chern numbers of the symbol of such operators. These Chern numbers are associated to U(1) fiber bundles defined from each eigenstate of the symbol that share a degeneracy point in a three-dimensional parameter space. I will show how such a correspondence yields a powerful tool to predict the existence of unidirectional waves in two-dimensions, with various applications from electromagnetic waves to quantum walks and condensed matter systems, and in particular to geophysical waves where this work was performed in collaboration with Antoine Venaille.

ABSTRACT. In this talk, I will apply tools from topological insulators to a fluid dynamics problem: the rotating shallow-water wave model with odd viscosity. The bulk-edge correspondence explains the presence of remarkably stable waves propagating towards the east along the equator and observed in some Earth oceanic layers. The odd viscous term is a small-scale regularization that provides a well defined Chern number for this continuous model where momentum space is unbounded. Equatorial waves then appear as interface modes between two hemispheres with a different topology. However, in presence of a sharp boundary there is a surprising mismatch in the bulk-edge correspondence: the number of edge modes depends on the boundary condition. I will explain the origin of such a mismatch using scattering theory and Levinson’s theorem. This talk is based on a series of joint works with Pierre Delplace, Antoine Venaille, Gian Michele Graf and Hansueli Jud.

ABSTRACT. We investigate some foundational issues in the quantum theory of spin transport, in the general case when the unperturbed Hamiltonian operator does not commute with the spin operator in view of Rashba interactions. A gapped periodic one-particle Hamiltonian is perturbed by adding a constant electric field of small intensity and the linear response, w.r.t. the strength of the electric field, in terms of a spin current is computed. We derive a general formula for the spin conductivity that covers both the choice of the conventional and of the proper spin current operator. We investigate the independence of the spin conductivity from the choice of the fundamental cell (unit cell consistency), and we isolate a subclass of discrete periodic models where the conventional and the proper spin conductivity agree, thus showing that the controversy about the choice of the spin current operator is immaterial as far as models in this class are concerned. The method relies on the characterization of a non-equilibrium almost-stationary state (NEASS), which well approximates the physical state of the system. This seminar is based on joint work with G. Panati and S. Teufel.

ABSTRACT. Applying time-periodic forcing is a common technique to effectively change materials properties. A well-known example is the transformation of graphene from a conductor to an insulator ("Floquet topological insulator'') by applying to it a time-periodic magnetic potential. While it is a well-known property of certain approximate models, it is not at all obvious whether the insulation property can be derived from the first-principle, continuum, model of graphene. We will introduce the notion of an "effective gap", or low-oscillations gap, and prove its existence in forced graphene. This new notion distinguishes a part of the energy-spectrum in a quantitative way. It implies that the medium is effectively insulating for a class of physically-likely wavepackets.

ABSTRACT. We will survey on the regularity theory of minimizers of the Mumford-Shah functional, focusing in particular on that of the corresponding singular sets. Starting with nowadays classical results, we will finally discuss more recent developments.

ABSTRACT. In the realization of electronic device structures, thin layers of crystals are deposited onto a crystalline substrate with different elastic stiffness, inducing an elastic strain in the deposited crystal. This process is said epitaxial growth of the film on the substrate. It is governed by the competition of two forms of energy, the bulk elastic energy and the surface energy, proportional to the free surface area.

I will present a work with M. Friedrich (University of Münster), in which we prove existence and approximation of equilibrium configurations for the corresponding energy introduced by Bonnetier and Chambolle in 2002, in any dimension and without any a priori regularity assumption on the displacement of the thin film. The analysis extends the available 2-dimensional results.

ABSTRACT. I will discuss recent results on the equilibrium and dynamical behavior of phase transitions in heterogeneous media. In spatially homogeneous media, it is well known that the Allen-Cahn functional of phase transitions behaves, to leading order and at large scales, like the perimeter functional on the associated interfaces. In the first part of the talk, I will describe how to use averaging to uncover a similar picture in stationary ergodic media. The dynamical picture, that is, the macroscopic behavior of the associated gradient flow, is less clear. The second part of the talk will detail some results in this direction in periodic media.

ABSTRACT. Consider a tiling T of the Euclidean plane E^2 whose symmetry group is a plane crystallographic group G≤Isom(E^2). Suppose G has a subgroup L of type pg. We may select a generating set for L that consists of a translation x with vector x and a glide reflection g with axis perpendicular to the direction of x .Since L is a discrete subgroup that acts freely on E^2, it defines the orbit space E^2/L={Lu∶u∈E^2} called a Klein bottle. This space is the collection of all L-orbits of points in E^2. More specifically, a point v∈E^2 belongs to the orbit Lu of u∈E^2 if and only if v=lu for some l∈L. Under this setting, the tiling T gives rise to the tiling T^*=T/L on the Klein bottle whose set of vertices, edges, faces are L-orbits of vertices, edges, faces, respectively, of T.

This talk describes a geometric approach in characterizing symmetry groups of tilings embedded on a Klein bottle using the notion of crystallographic group diagrams. A realization of one of the tilings in three dimensional Euclidean space E^3will be discussed.

ABSTRACT. Viral capsids are protein containers with icosahedral symmetry that contain the genome and self-assemble from copies of a single building block. The general classification paradigm for the relative position of the proteins in viral capsids is the Caspar-Klug scheme, similar to the Goldberg approach to construct polyhedra with high symmetry.

Much interest has recently arisen in self-assembling protein nanoparticles (SAPNs): however, these nanoparticles do not adhere to the Caspar-Klug’s scheme, and an original approach is needed for the experimental determination of their structure.

In this talk I will present a mathematical approach to the prediction and classification of the structure of a family of artificial nanoparticles that self-assemble from multiple copies of a single polypeptide, and that have been engineered to act as antigen display systems for vaccines. Such a theoretical approach is necessary because current experimental methods alone are not capable to determine the structure of these SAPNs.

I will show that graph-theoretical tools, combined with symmetry methods based on the Goldberg scheme, allow to study the topology of the protein networks. This approach is based on an unexpected relation with fullerene geometries and enables a full classification of the high and low symmetry particles observed in experiments.

ABSTRACT. Nanoporous materials are of great interest in applications ranging from gas separation and storage, to catalysis. The chemistry of these materials allows us to obtain an essentially unlimited number of new materials by combining different molecular building blocks, which exceeds the growth of synthesized nanoporous materials published in the recent experimental works. This sheer abundance of structures requires novel computational techniques to shed light on the existing or even unexplored libraries, as well as to facilitate the search for materials with optimal properties. In this talk, I will discuss our recent efforts into the discovery and design of novel nanoporous materials using computational modeling combined with topological data analysis.

ABSTRACT. During the past decade persistent homology has been successfully used to extract insights on particulate materials. We focus on frameworks where the only information available is the position and size of the particles. Physical evolution of particulate systems produces situations where different physical states cannot be distinguished by traditional methodologies. We will show a collection of geometric-topological measures based on persistent homology that can be used to distinguish these physical states.

ABSTRACT. We apply a quasistatic nonlinear model for nonsimple viscoelastic materials at a finite-strain setting in the Kelvin's-Voigt's rheology to derive a viscoelastic plate model of von Kármán type. We start from solutions to a model of three-dimensional viscoelasticity for 2nd-grade materials where the viscosity stress tensor complies with the principle of time-continuous frame-indifference. Combining the derivation of nonlinear plate theory by Friesecke, James and Müller and the abstract theory of gradient flows in metric spaces we perform a dimension-reduction from 3D to 2D and identify weak solutions of viscoelastic form of von Kármán plates. Based on joint work with Martin Kruzik.

ABSTRACT. Starting from the seminal work of Ambrosio and Tortorelli, singularly perturbed elliptic functionals have become a powerful tool in the variational approximation of free-discontinuity problems. In this talk we will consider two general classes of such functionals: One part of the talk is devoted discretizations of the original Ambrosio-Tortorelli functionals on stochastic lattices and their asymptotic behaviour in terms of Gamma-convergence. A second part focuses on the asymptotic analysis of general elliptic functionals of Ambrosio-Tortorelli type in the continuum setting. The analysis is carried out in the context of stochastic homogenization, which provides a common framework for the two classes of functionals presented here.

This talk is based on joint works with Marco Cicalese (München) and Matthias Ruf (Lausanne) and with Roberta Marziani (Münster) and Caterina Ida Zeppieri (Münster).

ABSTRACT. We obtain a measure representation for certain functionals, arising in the context of optimal design problems, under p-q growth conditions and a perimeter penalisation. We show that one of the functionals under consideration only admits a weak measure representation, whereas for the other a strong measure representation holds. Under some convexity assumptions, we provide a partial characterisation of the corresponding measures, a full representation is obtained in the one-dimensional setting. We further identify some conditions under which the relaxation process gives rise to no concentration effects. In this case, we show that the integral representation in question is composed of a term which is absolutely continuous with respect to the Lebesgue measure, and a perimeter term, but has no additional singular term.

ABSTRACT. Computing the homogenized properties of random materials is often very expensive. A standard approach consists in considering a large domain, and solving the so-called corrector problem on that domain, submitted to e.g. periodic boundary conditions. Because the computational domain is finite, the approximate homogenized properties are random, and fluctuate from one realization of the microstructure to another. We have recently introduced several efficient numerical approaches to reduce the statistical noise. These approaches allow to compute the expectation of the homogenized coefficients in a more efficient manner than brute force Monte Carlo methods.

Beside the (averaged) behavior of the material response on large space scales (which is given by its homogenized limit), another question of interest is to understand how much this response fluctuates around its coarse approximation, before the homogenized regime is attained. More generally, we aim at understanding which parameters of the distribution of the material coefficients affect the distribution of the response, and whether it is possible to compute that latter distribution without resorting to a brute force Monte Carlo approach.

This talk, based on joint works with P.-L. Rothe, will review the recent progresses made on these questions, both from the theoretical and numerical viewpoints.

ABSTRACT. This contribution provides an a priori error analysis of a localized orthogonal decomposition method (LOD) for the numerical stochastic homogenization of a model random diffusion problem. If the uniformly elliptic and bounded random coefficient field of the model problem is stationary and satisfies a quantitative decorrelation assumption in form of the spectral gap inequality, then the expected $L^2$ error of the method can be estimated, up to logarithmic factors, by $H + (\varepsilon/H)d/2$; $\varepsilon$ being the small correlation length of the random coefficient and H the width of the coarse finite element mesh that determines the spatial resolution. The proof bridges recent results of numerical homogenization and quantitative stochastic homogenization.

ABSTRACT. We are interested in computing the electrical field generated by a charge distribution localized on scale $\ell$ in an infinite heterogeneous medium, in a situation where the medium is only known in a box of diameter $L\gg\ell$ around the support of the charge. We propose an artificial boundary condition that with overwhelming probability is (near) optimal with respect to scaling in terms of $\ell$ and $L$, in the setting where the medium is a sample from a stationary ensemble with a finite range of dependence (set to be unity and in the regime $1\ll\ell$). The boundary condition is motivated by stochastic homogenization that allows for a multipole expansion [Bella, Giunti, Otto 2020]. This work extends [Lu, Otto] from two to three dimensions, which requires to take quadrupoles, next to dipoles, into account. This in turn relies on stochastic estimates of second-order, next to first-order, correctors. These estimates are provided for finite range ensembles under consideration, based on an extension of the semi-group approach of [Gloria, Otto 2015].

ABSTRACT. We shall revisit the error analysis for some classical variants of the multi-scale finite element method (MsFEM), such as MsFEM with linear boundary conditions or « oversampling » [1, 2]. We shall present the proofs in the settings not covered by the numerical analysis in the existing literature. In the case of the diffusion equation with highly oscillating periodic coefficient, we can deal with non-simplicial meshes and to work under the minimal regularity assumptions on the coefficients. Furthermore, we shall look into some cases beyond the periodic setting, such as inclusion of the local defects, the quasiperiodic coefficients etc. We shall finally discuss possible applications to the stochastic homogenization. This communication is partly based on the joint work with Rutger Biezemans, Claude Le Bris, and Frédéric Legoll (ENPC and Inria, France).

References: [1] T. Hou and X.-H. Wu, "A multiscale finite element method for elliptic problems in composite materials and porous media," Journal of Computation Physics, 134:69-189 (1997). [2] T. Hou, X.-H. Wu, and Z. Cai, "Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coeffcients," Mathematics of Computation, 68:913-943 (1999).

ABSTRACT. We propose a Nitsche type method for multiscale partial differential equations, which retrieves the macroscopic information and the local microscopic information at one stroke, by contrast to the classical global local methods, we prove the convergence of the method for problem with bounded and measurable coefficients. The rate of convergence may be derived for coefficients with further structure such as periodicity and ergodicity. Extensive numerical results confirm the theoretical predictions.

ABSTRACT. In material science and engineering, density functional theory occasionally be used in multiscale modeling. Due to the singularity in Hamiltonian, physical constraints for the solution, the size of the system, etc., the efficiency of numerical simulation is a key point for the computation of those multiscale models. In this talk, numerical issues in density functional theory will be described, and our works towards efficient algorithms will be given. In particular, a numerical library AFEABIC will be introduced, based on which a variety of numerical experiments will be demonstrated.

ABSTRACT. In this talk, we introduce a new phase field model for Ni-Al alloy. The total free energy in the phase field model is constructed by taking the limit from moleculor model. A second-order unconditional energy stable finite difference scheme is given in the talk. Some numerical simulations are done based on the constructed phase field model and the total free energy.

ABSTRACT. Grain boundaries are surface defects in crystalline materials. Dynamic properties of grain boundaries play vital roles in the mechanical and plastic behaviors of polycrystalline materials. The properties of grain boundaries strongly depend on their microscopic structures. We present continuum models for the dynamics of grain boundaries based on the continuum distribution of the line defects (dislocations or disconnections) on them. The long-range elastic interaction between the line defects is included in the continuum models to maintain stable microstructure on grain boundaries during the evolution. The continuum models is able to describe both normal motion and tangential translation of the grain boundaries due to both coupling and sliding effects that were observed in atomistic simulations and experiments.

ABSTRACT. Advanced statistical methods are rapidly impregnating many scientific fields, offering new perspectives on long-standing problems. When coupling machine learning (ML) to physical systems, many problems of interest display dauntingly-large interpolation spaces, limiting their immediate application without undesired artefacts (e.g., extrapolation). The incorporation of physical information, such as conserved quantities, symmetries, and constraints, can play a decisive role in reducing the interpolation space. I will illustrate some of these aspects in the context of efficient kernel-based machine learning models for structure-based coarse-grained force fields that are both covariant and energy-conserving. Going back down the multiscale ladder, I will describe a backmapping strategy that learns the atomistic distribution of atoms conditional on the coarse-grained degrees of freedom. Our generative adversarial network is capable of accurately reproducing the Boltzmann distribution beyond pairwise statistics.

ABSTRACT. Molecular Dynamics (MD) simulations have proven to be a very useful complementary tool, and sometimes even an alternative to experiments. Despite their wide use to capture fast occurring phenomena, there are still many cases where the time scales accessible to MD simulations are far smaller than the time scales needed for the observation of important conformational changes of the system. Many Enhanced Sampling methods have emerged to accelerate the observation of such changes, but most of these methods rely on the knowledge of low-dimensional slow degrees of freedom, i.e. Collective Variables (CV). For large and complex dynamical systems, Machine Learning and Dimensionality Reduction techniques can be used to identify CVs. In particular, new methods have emerged for iterative learning of CVs using adaptive biasing data: at each step, the learned CV is used to perform enhanced sampling to generate new data and learn a new CV. Here, we introduce AE-ABF, a new iterative method for CV learning that includes a reweighting scheme to ensure the learning model optimizes the same loss, and thus achieves CV convergence.

ABSTRACT. The development of systematic coarse-graining (CG) of molecular systems, using detailed atomistic data, is a very intense and diverse research field. Traditional approaches for developing rigorous CG models are based on statistical mechanics. Methods such as inverse Boltzmann, inverse Monte Carlo, force matching, and relative entropy, provide parameterizations of the free energy at the CG level (many-body potential of mean force) coarse-grained models at equilibrium, by minimizing a fitting functional over a parameter space. During the last few years, the perspective of developing data-driven methodologies to derive (atomistic and CG) molecular models is attracting much attention.

Here we first give an overview of physics-based methods for obtaining optimal parametrized coarse-grained models, starting from detailed atomistic representation for high dimensional molecular systems. Then, we propose a new data-driven approach to approximate efficiently the many-body free energy surface (FES). The method is based on a detailed analysis of the atomistic simulation results, via machine learning techniques.

All the above approaches are applied to realistic molecular models: from simple liquids, such as water, up to macromolecules, which are prototype examples for multi-scale modelling.

ABSTRACT. The data-based discovery of effective, coarse-grained (CG) models of high-dimensional dynamical systems presents a unique challenge in computational physics and particularly in the context of multiscale problems. In this talk, a data-based, probabilistic perspective is presented that enables the quantification of predictive uncertainties. One of the outstanding problems has been the introduction of physical constraints in the probabilistic machine learning objectives. The primary utility of such constraints stems from the undisputed physical laws such as conservation of mass, energy etc that they represent. Furthermore and apart from leading to physically realistic predictions, they can significantly reduce the requisite amount of training data which for high-dimensional, multiscale systems are expensive to obtain (Small Data regime). We formulate the coarse-graining process by employing a probabilistic state-space model and account for the aforementioned (in)equality constraints as virtual observables in the associated densities. We demonstrate how probabilistic inference tools can be employed to identify the coarse-grained variables and their evolution model without ever needing to define a fine-to-coarse (restriction) projection and without needing time-derivatives of state variables.

ABSTRACT. Polymerization processes are known to be at the core of the replication of prions, the proteins involved in the mad cow or Creutzfeldt-Jakob diseases. The so-called "prion equation" is a coagulation-fragmentation system which models this polymerization. It is composed of an ODE for the evolution of the amount of monomers coupled to a growth-fragmentation PDE for the polymers. The difficulty of the mathematical analysis lies in the fact that the nonlinear coupling is on the transport term of the PDE. In this talk, we will explain how we can take advantage of self-similar techniques to study the cases where the individual growth of polymers is exponential. It makes it possible to prove the convergence of the solutions to a steady state or the existence of a limit cycle, depending of the coefficients of the model.

ABSTRACT. In this talk I will discuss the stability properties of a special class of solutions to a coagulation-fragmentation equation. We assume that the coagulation kernel is close to the diagonal kernel, and that the fragmentation kernel is diagonal: in this setting, we construct a two-parameter family of stationary solutions concentrated in Dirac masses, and we precisely characterise the asymptotic decay of their tails. For initial data which are sufficiently concentrated, we show that the corresponding solutions to the equation approach one of these stationary solutions for large times. This is a joint work with B. Niethammer and J. Velázquez.

ABSTRACT. We introduce a coagulation-fragmentation-death system of ordinary differential equations modelling silicosis. Recent work on existence, uniqueness, and basic properties of solutions will be reported. The structure of the set of equilibria for certain classes of coefficients relevant in applications will be presented, and a preliminary study of the stability properties of the equilibria will be addressed.

ABSTRACT. The fragmentation equation is used to describe several physical or biological phenomena. In that equation the two important parameters are the fragmentation rate, and the fragmentation kernel κ. Estimating these parameters is important for applications. The application we have in mind is to explore the mechanical stability of differente amyloid fibrils. A reconstruction formula for κwas previously provided, based on the knowledge of the long time asymptotic profile of the solution. The formula involved the Mellin transform of the. asymptotic profile.

Our goal here is to provide an alternative method to estimate the kernel from measurable data.

ABSTRACT. Advances in nanofabrication offer an unprecedented degree of precision and control in manufacturing novel types of magnetic materials and heterostructures. As the dimensions of the constituent components go down to the atomic scale, the magnetic properties of these materials begin to be dominated by the interfaces between the adjacent material layers. This leads to the emergence of a plethora of new phenomena, including those that give rise to topological magnetism, whereby the observed spin textures acquire non-trivial topolgical characteristics. Examples of such textures include chiral domain walls and magnetic skyrmions, with promising applications in spintronics --- an emergent field of microelectronics that takes advantage of both the charge and the spin degrees of freedom of an electron. This talk will overview the chalenges and opportunities offered by modeling and analysis of the magnetic properties of these novel materials, focusing on applications of rigrorous asymptotic techniques of calculus of variations and PDE analysis.

ABSTRACT. The advent of advanced simulation techniques such as the accelerated molecular dynamics (AMD) methods have provide the capability for understanding physical processes that were heretofore inaccessible. Here, we use AMD methods to examine the problem of mass transport in complex oxides, such as pyrochlores and spinels, as it relates to radiation damage and ionic conductivity. The AMD methods allow us to examine a wider range of scenarios than possible with conventional molecular dynamics (MD), such as the role of cation chemistry and cation distribution on the kinetics of the various point defects that drive mass transport under non-equilibrium conditions. These simulations often reveal counter-intuitive behavior. For example, we find that only some species of interstitials are stable within the oxide matrix, but that stability is very sensitive to not only the chemistry of the oxide, but also the distribution of the cations within the crystal structure. These results offer a more complete perspective regarding the nature of transport in these complex systems and have ramifications beyond these specific materials, providing new insight into transport in chemical disordered systems, such as high entropy alloys and entropy stabilized oxides, more generally.

ABSTRACT. We introduce a "trajectory fragment" framework, reminiscent of "ParSplice," that can be used to design and prove consistency of parallel replica ("ParRep") algorithms for generic Markov processes. We use our framework to formulate a novel condition that guarantees an asynchronous ParRep algorithm is consistent. Exploiting this condition and the trajectory fragment framework, we present synchronous and asynchronous ParRep algorithms for piecewise deterministic Markov processes.

ABSTRACT. We consider the Fleming-Viot particle system associated with a continuous-time Markov chain in a finite space. Assuming irreducibility, it is known that the particle system possesses a unique stationary distribution, under which its empirical measure converges to the quasistationary distribution of the Markov chain. We complement this Law of Large Numbers with a Central Limit Theorem. Our proof essentially relies on elementary computations on the infinitesimal generator of the Fleming-Viot particle system, and involves the so-called π-return process in the expression of the asymptotic variance. Our work can be seen as an infinite-time version, in the setting of finite space Markov chains, of results by Del Moral and Miclo, and Cérou, Delyon, Guyader and Rousset.

This is a joint work with Tony Lelièvre and Loucas Pillaud-Vivien.

ABSTRACT. In this talk I will present on a variant of Gamow's liquid drop model where we consider the mass-constrained minimization of an energy functional given by the sum of a density perimeter and a nonlocal interaction of Riesz type. In particular, I will show that for a wide class of density functions this energy admits a minimizer for any choice of parameters, and that for monomial densities the unique minimizer is given by the ball of fixed volume when the nonlocal effects are sufficiently small.

ABSTRACT. We consider a non-local free energy functional, modelling a competition between entropy and pairwise interactions, reminiscent of the Maier-Saupe theory of nematic liquid crystals. We fucus on the large-domain limit, where minimisers converge to minimisers of a quadratic elastic energy with manifold-valued constraint, analogous to harmonic maps. We extend previous convergence results to establish Hölder bounds for minimisers on bounded domains, and demonstrate uniform convergence of minimisers away from the singular set of the limit solution. The talk is based on a joint work with Jamie M. Taylor (BCAM, Bilbao).

ABSTRACT. Liquid crystals are found in many technological and biological applications, making control of such systems a tempting goal. Especially in thin domains, surface effects can dominate, and thus their structure, or design, is a key to influencing the behaviour of the bulk of the sample. In this talk we will discuss an approach to understanding how fine scale oscillations on a liquid crystal/homogeneous solid interface (rugosity) can give rise to "effective" surface conditions, phrased mathematically as a homogenisation problem. Our results apply to more general systems, and we can produce stronger, explicit results in concrete systems with simpler geometries. We propose that the results may be interpreted as a design problem for liquid crystalline systems.

ABSTRACT. We study the asymptotic behavior of the minimisers of the Landau-de Gennes model for nematic liquid crystals in a two-dimensional domain in the regime of small elastic constant. At leading order in the elasticity constant, the minimum-energy configurations can be described by the simpler Oseen-Frank theory. Using a refined notion of Gamma-development we recover Landau-de Gennes corrections to the Oseen-Frank energy.

ABSTRACT. The object of our investigation are surfaces in Euclidean space of constant mass density that are suspended from a closed curve and hangs under its weight. These surfaces are models of the shape of a `hanging roof' for the construction of perfect domes according to the ideas of the architect F. Otto. In this talk, we give an approach to the study of the shape of these surfaces obtained by the author. In the topics to consider, we classify the surfaces invariant by a uniparametric group of translation and rotations, we solve the Dirichlet problem under suitable conditions and we study how the geometry of the boundary curve of the surface imposes geometric restrictions to the global shape of the very surface.

ABSTRACT. For a graph model of several stacked layers of graphene, the dispersion function of wave vector and energy is shown to be a polynomial in the dispersion function of the single layer. This leads to the reducibility of the Fermi surface, at any energy, into several components. Each component corresponds to hybrid states in the multi-layer structure that contribute a sequence of bands to the spectrum. Both AA- and AB-stacking are allowed. The reducibility allows the creation of local defects that engender embedded eigenvalues. Distributing defects at a mesoscale and homogenizing leads to materials with narrow "embedded spectral bands". This construction can be generalized to a class of periodic graph operators.

ABSTRACT. Glaciers and ice sheets are essential elements of Earth’s climate system. They interact with the atmosphere, hydrosphere, lithosphere and biosphere through multiple processes, which comprise the Structure–Form–Environment Interplay (SFEI): environmental change (e.g. climate change) acts on form and structure; deformation (e.g. glacier flow) affects structure and environment; structural development (e.g. ice recrystallization and phase change) influences form and environment. In this work we focus on the multiscale modelling of the structural development of glaciers, as well as their implications for glacier flow (form) and the environment. The main structural features of glaciers and ice sheets are related to their stratigraphy, which has its roots in the polycrystalline microstructure of natural ice, including dislocations, micro-and macroscopic inclusions, grains and subgrains. We will present how to incorporate such microscopic structural elements into a general theory of polycrystalline glacier dynamics, which includes evolving anisotropy and recrystallization, by using the rigorous continuum thermodynamic approach of continuous diversity. Finally, we will compare the resulting theory with current models of glacier dynamics and discuss the potential implications for climate change and the integrity of ice-core palaeoclimate records.

ABSTRACT. This talk will investigate the relationship between processing parameters and microstructures formed during additive manufacturing of Titanium-based alloys. Grain size, morphology and orientation as well as mesoscale statistics such as entropy and correlation with nearby defects will be investigated in relation to the macroscopic performance indicators. An integrated ML-based approach to determining processing maps for additive manufacturing aimed at failure life prediction will be presented, based on a combination of DEM and KMC simulations coupled with novel mesoscale descriptions.

ABSTRACT. We present a model for curvature driven interface propagation through a homogeneous medium with random obstacles. The energy is fully nonlinear and the dissipation is mixed, capturing both viscous dissipation as well as dry friction. If the interface passes over an obstacle it incurs additional dry friction. This model is relevant for the study of dislocations. Under an applied force, we investigate the pinning (i.e., a solution becomes stuck) and depinning behavior. We show that the model obeys Taylor Scaling, i.e., the critical pinning force scales like the square root of the concentration of the obstacles. Joint work with Patrick Dondl (Freiburg) and Michael Ortiz (Pasadena).

ABSTRACT. Thin liquid films are ubiquitous in natural phenomena and technological applications. They have been extensively studied via deterministic hydrodynamic equations, but thermal fluctuations often play a crucial role that needs to be understood. An example of this is dewetting, which involves the rupture of a thin liquid film and the formation of droplets. Such a process is thermally activated and requires fluctuations to be taken into account self-consistently. In this work we present an analytical and numerical study of a stochastic thin-film equation derived from first principles. Following a brief review of the derivation, we scrutinise the behaviour of the equation in the limit of perfectly correlated noise along the wall-normal direction, as opposed to the perfectly uncorrelated limit studied by Grün et al. (J Stat Phys 2006). We also present a numerical scheme based on a spectral collocation method, which is then utilised to simulate the stochastic thin-film equation. With our numerical scheme we explore the fluctuating dynamics of the thin film and the behaviour of its free energy in the vicinity of rupture. Finally, we study the effect of the noise intensity on the rupture time, using a large number of sample paths as compared to previous studies.

ABSTRACT. Relaxation rates for the Cahn-Hilliard equation are of interest because they characterize the macroscopic behaviour of the modeled coarsening process. It is natural to consider perturbations whose distance to (meta-)stable states is controlled in L^1 since this distance is linked to the conserved quantity.

I will report on recent progress in the L^1 framework for the 1D Cahn-Hilliard equation. I will then go on to discuss differences and similarities to the two-dimensional case, where the geometry of the interface plays a prominent role in the evolution. In particular I will consider the Stefan problem as a sharp interface approximation of the Cahn-Hilliard equation and will present approaches to obtain relaxation rates for it.

ABSTRACT. We analyze Floquet topological insulators resulting from the time-harmonic irradiation of electromagnetic waves on two dimensional materials such as graphene. We analyze the bulk and edge topologies of approximations to the evolution of the light-matter interaction. Topologically protected interface states are created by spatial modulations of the drive polarization across an interface. In the high-frequency modulation regime, we obtain a sequence of topologies that apply to different time scales. Bulk-difference invariants are computed in detail and a bulk-interface correspondence is shown to apply.

ABSTRACT. I will present some recent results concerning the rate of decay of Wannier functions in quantum Hall systems and Chern insulators. More specifically: - I will show how optimally-localized Wannier functions forming an orthonormal basis for the valence states of the insulator are subject to a localization dichotomy, dictated by the value of the Chern number: either they are exponentially localized (vanishing Chern number), or the second moment of the position operator diverges (non-vanishing Chern number); - I will then explain how exponential localization can be achieved for any value of the Chern number if one relaxes the orthonormality condition to that of forming a Parseval frame. The talk will be based on joint works with G. Panati, A. Pisante and S. Teufel, and with H. Cornean and M. Moscolari.

ABSTRACT. Topological materials have generated great interest over the past few years due their remarkable physical properties. Mathematically speaking, we can understand the properties of periodic topological insulators by performing the Bloch-Floquet transform which maps electronic states to a fiber bundle over the Brillouin torus. This type of analysis has lead to the ``localization-topology correspondence'' which states that the Fermi projector for an insulator admits an orthonormal basis with finite second moment if and only if it is topologically trivial and an insulator is topologically trivial if and only if its Fermi projector admits an exponentially localized orthonormal basis. In contrast, in non-periodic systems it is unknown whether a localization-topology correspondence still holds since the Bloch-Floquet transform cannot be applied.

In this talk, I will discuss some recent work which establishes a weaker version of the localization-topology correspondence for both periodic and non-periodic insulators in two dimensions. In particular, we show by construction that if the Fermi projector admits an orthonormal basis with sufficiently fast algebraic decay then it also admits an orthonormal basis with exponentially fast decay. This result is based on an algorithm for constructing exponentially localized bases using projected position operators (i.e. operators of the form PXP).

ABSTRACT. We present equilibrium configurations of hexagonal columnar chromonic liquid crystals in the context of characterizing packaging structures of bacteriophage viruses in a protein capsid. These are viruses that infect bacteria and are currently the focus of intense research efforts, with the goal of finding new therapies for bacteria resistant antibiotics. The energy that we propose consists of the Oseen-Frank free energy of nematic liquid crystals that penalizes bending of the columnar directions, in addition to the cross-sectional elastic energy accounting for distortions of the transverse hexagonal structure; we also consider the isotropic contribution of the disordered inner core. The problem becomes of free boundary type, subject to constraints, with an energy of polyconvex type. We show that the concentric, spool-like configuration is the absolute minimizer. The optimality of the packaging is nature’s way to ensure successful DNA delivery, in bacteria infection events, process that starts with a phase transition from the hexagonal phase to the cholesteric nematic, that we shall also investigate. The role of the motor in filling the capsid will also be briefly discussed.

ABSTRACT. Liquid crystalline materials have fascinated scientists for over 150 years, having both fluidity and properties associated with solids (i.e., birefringence, elasticity, piezoelectricity). The presence of Liquid Crystal Displays (LCDs) in almost all modern electronic devices also means that this intriguing state of matter is now essential to our everyday life. One aspect of liquid crystals that was key to the original identification of these materials, but is now often unwanted in LCDs, is their ability to maintain permanent defect structures. These topological structures can move within the liquid crystal, interact and annihilate with other defects, and disappear through solid walls. In this talk we consider this last effect, as well as their emergence out of solid walls, and consider the possible benefits of either maintaining defects within the liquid crystal, or their (temporary) removal together with the creation of external virtual defects. Numerical solutions and analytical results indicate that by tailoring surface properties (anchoring strengths) and applying orienting fields (electric/magnetic) allows close control of these defects.

ABSTRACT. We study suspensions of magnetic nanoparticles in a nematic host, with nematic and magnetic order, in one-dimensional and two-dimensional reduced settings. These suspensions are examples of dilute ferronematics with much enhanced magnetic susceptibilities, compared to conventional nematic systems. The physically observable equilibria are stable critical points of an appropriately defined free energy, including a nemato-magnetic coupling energy. We study intricate ferronematic solution landscapes in terms of the nemato-magnetic coupling parameter, and geometrical characteristics. We present novel results on (i) co-existence of multiple stable ferronematic states with magnetic domain walls and states with interior nematic defects; and (ii) results on how the multiplicty of the solutions depends on the geometrical characteristics and the nemato-magnetic coupling parameter. These results have potential applications to novel nematic composite systems with unusual nemato-magnetic responses, for use in photonics, sensors and nano-technologies.

ABSTRACT. Boundaries are an essential mechanism to control liquid crystalline order: they may promote a local orientation or order and, globally, can enforce topological constraints that promote or inhibit defect formation. In many emerging applications of liquid crystals, the boundary of the system is not fixed and stationary states of the system must be determined by extremizing the free energy both with respect to the order and the overall shape of the system. Results obtained with an explicit finite element method are presented for a selection of applications, from tactoids to patterned films.

ABSTRACT. Microstructure reconstruction over large domains from small-scale experimental data is an efficient strategy to predict the microstructure evolution. However, such prediction is influenced by the effects of the uncertainties in computations and experiments. These uncertainties propagate into the grain shape of synthesized microstructures which affects the material properties determined from it. While the uncertainty quantification (UQ) of crystallographic texture and grain size is addressed with state-of the-art methods, the UQ of grain shapes is still an unexplored research challenge. We address this in our present work by proposing a new methodology for UQ of grain topology and texture of metallic microstructures. We utilize the concept of shape moment invariants in physics to quantify the grain shape and its uncertainty propagation to the material properties. To generate sufficient statistical information, synthetic microstructures are reconstructed from the experimental data using Markov Random Field (MRF) method on Titanium-7wt%- Aluminum (Ti-7Al) microstructures. The texturing of the synthetic microstructures is described with the Orientation Distribution Function (ODF), while the shape moment invariants are used to represent the grain shapes. The propagation of the microstructural uncertainty on the stress-strain response is investigated by performing the crystal plasticity simulations.

ABSTRACT. Reducing uncertainties and therefore risks in structural design implies determining accurate statistically-based properties of the material. IRT Saint-Exupéry has been developing a software solution called VIMS that uses the GEMS open source python library to generate material allowables. VIMS offers a framework to integrate, evaluate and use advanced composites models in association with experimental data post-processing, decision-making support and an innovation-friendly environment that facilitates the deployment within design offices. Models developed by Onera, University of Porto and University of Girona have been wrapped into VIMS. Toolboxes are provided to facilitate model exploration through sensitivity analyses and surrogate modelling. An automated workflow was implemented to assist the user in the calibration phase, i.e. the identification of the model parameters from standard tests. The model validation phase provides the user with error maps and guidance on the complementary experimental tests that could improve the confidence in the model. The generation of B-basis strength values can be based on different methods including the Composite Materials Handbook-17 (CMH-17) method and a new efficient Uncertainty Quantification and Management (UQ&M) methodology based on adaptive surrogate models and active learning techniques.

ABSTRACT. The validation of simulation models is typically achieved through the calibration of a set of unknown but fixed modeling parameters. However, such a perspective neglects the functional dependence of the calibration parameters on the modeling variables. Being able to learn this functional dependence would bring about more generalizable models and enable inference into unknown physical phenomenon. Learning these functional dependencies involves a nonisometric surface matching of a low-fidelity model to a lower-dimensional high-fidelity response surface. Available methods require modelers to specify the functional form of the calibration functions a priori or adopt a Bayesian approach that scales poorly with an increased number of input dimensions. In this work, we propose an adaptive sampling scheme that makes sequential inference into the eigenvectors of the covariance matrix of the calibration function parameters through a singular value decomposition by optimization of the likelihood function with a constant learning rate. The quantified uncertainty provides modeling support when selecting the functional form of the calibration functions. The validity of the method is demonstrated by calibrating a coarse-grained epoxy model with eight responses (Debye-Waller factor, Density, Modulus and Yield strength for two curing agents), one design variable (degree of crosslinking), and 14 unknown forcefield calibration functions.

ABSTRACT. The exploration of automated fiber placement technologies has evolved the aerospace material industry in designing composite fiber paths to avoid material failure and improve mechanical performance. The objective of this study is to optimize the spatially varying fiber paths of composite materials to minimize the critical stresses under compressive loads as opposed to utilizing widely used fiber configurations, such as uni-directional fiber alignments. The methodology is applied on a rectangular, symmetric, orthotropic composite plate with a central circular cutout that is fixed from the right and left edges to depict an area of airplane’s fuselage. The composite fiber paths are mathematically quantified and modeled using Non-Uniform Rational Basis Spline (NURBS). Of particular interest is optimizing the NURBS parameters that control the fiber paths to enhance the mechanical performance of the composite. A finite element method is applied to optimize a set of 25 control points that dictate the fiber path alignment on an evenly spaced grid with respect to discretized constitutive fiber angles. Next, the influence of the uncertainties that arise from the manufacturing imperfections on the optimized mechanical properties is modeled through the NURBS representation of the fibers.

ABSTRACT. Many aspects of renewable energy, from its generation (e.g., high-efficiency photovoltaic cells) and storage (e.g., large-capacity battery electrodes) to its utilization (e.g., lightweight alloys), are profoundly associated with materials discovery. Energy materials, like battery electrodes, generally function in a complex chemical environment involving various processes. The design of better energy materials, therefore, requires a comprehensive understanding of the material behaviors upon operation. Traditionally, the discovery of new materials with targeted properties involves a large number of experimental trials, while these efforts are far from adequate considering the well-recognized near-infinite chemical space. The efficient investigation of the unexplored chemical space calls for automated techniques with smart navigation. In this talk, I will first give an overview of how simulations, cheminformatics, deep generative models, and robotics have been accelerating the materials design process. Followingly, I will demonstrate the realization of the generative design of reticular frameworks (e.g., MOFs and COFs) using a supramolecular variational autoencoder empowered automated nanoporous materials discovery platform.

ABSTRACT. The range of potential chemical compositions grows infinitely large when accounting for non-stoichiometric compositions of increasing fractional resolution. These increasing dimensions impose steep computations costs on traditional methods of materials discovery that predict the properties of a sampled compositions space. We propose an approach to materials discovery that utilizes gradient ascent optimize fractional compositions for extraordinary properties. Using the learned weights on a property trained neural network, it is possible to predict fractional composition components such that they satisfy a maximize a starting property. The method is still in development, but it is expected that the approach will be computationally inexpensive and expand the range of compositions that can be sampled in materials discovery approaches. Ten predicted compositions will be synthesized and characterized to demonstrate the new architectures capabilities.

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ABSTRACT. Halides hplay a significant role in achieving shape controlled growth of metal nanocrystals, but the exact mechanisms are not well understood. In this study, we use a combination of quantum density functional theory (DFT) and the theory of absorbing Markov chains to understand the chloride-mediated growth of Cu nanowires. We used DFT calculations in the framework of ab initio thermodynamics to identify the chlorine-covered Cu(100) and Cu(111) surfaces with the lowest energies as a function of the solution-phase concentration (chemical potential) of chloride. We find that the surface energy of Cl-Cu(111) is higher than Cl-Cu(100) that a Cu atom binds more strongly to Cl-Cu(111) - opposite to the trend for bare Cu surfaces. We also find that absorbed Cl facilitates the diffusion of Cu adatoms on Cl-Cu(100) by decreasing the diffusion-energy barrier, but hinders the diffusion on Cl-Cu(111) by increasing the barrier. Furthermore,{100}-{111} inter-facet barrier favors transport from Cu(100) to Cu(111). All of these factors promote the growth of Cu into fivefold-twinned nanowires. Using a model based on the theory of absorbing Markov chains, we predict nanowires with high aspect ratios, comparable to those in experiment.

ABSTRACT. Diamond nanocrystals are complicated materials, presenting a diverse range of sizes, shapes, speciation and defects that cannot easily be controlled during synthesis. Each of these features contribute to structure/property relationships, but in different ways. Attempts to purify samples to tune the properties often fail due to lack of specificity, and polydispersivity usually persists. An alternative approach is to separate classes of nanodiamonds based on their properties, rather than seeking monodispersed samples based on their structure. This would be a more industrially relevant approach, since manufacturers need to be able to accommodate a fault tolerance. To investigate if this approach has merit we have used machine learning to classify nanodiamonds and identify class-dependent properties and characteristics. Using a fully reconstructed data set that spans the experimentally observed size range from 1 nm to 4.5 nm we find that there are multiple classes of particles, based on their similarity in the high dimensional feature space. By using an interpretable classifier, with excellent accuracy, precision and recall, we also identify possible purification pathways based on the important features ranked by the AI.

ABSTRACT. We have analyzed the physical mechanisms responsible for the formation of an ordered sequence of nanoclusters synthesized on a nanowire in the diffusion mode of deposition of free atoms. The results were obtained using a kinetic Monte Carlo Model, which takes into account only the interaction between the nearest atoms of the crystal lattice. Nevertheless, this model describes the correlations between the elements of the system at distances significantly exceeding their sizes.

We show that the long-range spatial correlation between the synthesized clusters is due to two factors; first, the surface diffusion in the metastable system of deposited atoms which leads to the formation of primary nuclei, and second, the shadow effect arising when growing nanoclusters become rather large. It is these processes that are the "tools" through which individual clusters suppress the development of their neighboring nuclei in the competition for survival and form high self-ordering at the final stage of synthesis. Numerical experiments were carried out for one-dimensional systems with a diamond-like lattice structure. The observed variety of morphologies of one-dimensional systems at the final stage of synthesis is in good agreement with numerous experimental data obtained during the synthesis of nanoclusters on silicon and germanium nanowires.

ABSTRACT. This talk focuses on how to model the formation process of metallic and bimetallic nanoparticles as it is supposed to occur inside inter-gas aggregation sources. Based on classical molecular dynamics simulations, three models are discussed, namely the atom-by-atom growth, annealing, and coalescence. These processes are likely to lead to the formation of the out-of-the-equilibrium shapes due to strong memory effects kinetically trap the nanoparticle in certain shapes[1]. Similarly we discuss when structural rearrangements of whole structures can occur [2]. In other words, we can generate various and heterogeneous nanosamples, for which we need robust classification and characterisation tools. Following a multiscale approach [nanoCHE, 3], we show how the aforementioned formation processes change the adsorption sites distribution available per each nanoparticle, affecting significantly their catalytic performance [4].

[1] F. Baletto J Phys Cond Matt. 31 (2019) 113001; K. Rossi et al. Sc. Rep 8 (2018) 1; L. Delgado et al., Nanoscale 13 (2021) 1172. [2] K. Rossi and F. Baletto, PCCP 19 (2017) 11057; K. Rossi et al. EPJB 91 (2018) 1 [3] K. Rossi, GG Asara, F. Baletto, ACS Cat. 10, (2020) 3911-3920 [4] E. Gazzarrini, K. Rossi, F. Baletto, Nanoscale 2021, just accepted.

ABSTRACT. The growth-fragmentation equation describes a system of growing and dividing particles that arises in some models in population dynamics. Several important questions about the equation concern the asymptotic behavior of solutions at large times: at what rate do they grow, and what does the asymptotic profiles of solutions look like? Does the rescaled solution converge to its asymptotic profile at an exponential speed? These questions have traditionally been studied using analytic techniques such as entropy methods and spectral theory of operators. In this talk, we present a probabilistic approach via a Feynman–Kac formula that enables us to express the solution of the growth-fragmentation equation in terms of a certain Markov process. The asymptotic behavior of solutions of growth-fragmentation equations can then be determined in terms of that Markov process. We shall also study particle systems with growth-fragmentation dynamics, and again determine their asymptotic behaviors in terms of the latter.

(partly based on joint works with Alex Watson, University College London).

ABSTRACT. The growth-fragmentation equation models systems of particles that grow and reproduce as time passes. An important question concerns the asymptotic behaviour of its solutions. This question has traditionally been addressed using analytic techniques such as entropy methods or splitting of operators.

Bertoin and Watson (2018) developed a probabilistic approach relying on the Feynman-Kac formula, that enabled them to answer to this question in the case in which the growth rate is sublinear and the mass is conserved at fragmentation events. This assumption on the growth ensures that microscopic particles remain microscopic.

In this talk, we present a recent work of the speaker, in which we go further in the analysis, assuming bounded fragmentations and allowing arbitrarily small particles to reach macroscopic mass in finite time. Moreover, we drop the hypothesis of conservation of mass when a fragmentation occurs. With the Feynman-Kac approach, we establish necessary and sufficient conditions on the coefficients of the equation that ensure the so-called Malthusian behaviour with exponential speed of convergence to the asymptotic profile. Furthermore, we provide an explicit expression of the latter.

ABSTRACT. In this talk, we give a brief explanation of Harris's Theorem and its precursor Doeblin's Theorem which is developed for the study of discrete-time Markov chains. This probabilistic approach is based on quantitatively verifying a minorisation condition and a drift condition in order to obtain quantitative estimates on the long-time behaviour of time evolution for some linear and non-local integro-PDEs. We will see how this method applies to achieve spectral gap for the growth-fragmentation equation which is a linear equation describing a system of growing and dividing particles and may be used as a model for many processes in different contexts like ecology, neuroscience, telecommunications and cell biology.

ABSTRACT. In our work we consider the Lifschitz-Slyozov-Wagner (LSW) equations with subcritical initial data. The LSW equations can be interpreted as an evolution equation for the probability density function of random variables. We show that there is a convergence in distribution of the normalized process to the appropriate process corresponding to the self-similar solution. These results extend the local asymptotic stability results of Niethammer and Velazquez. Our method follows earlier work of Conlon and Niethammer, who proved a global asymptotic stability result for a quadratic analog of the LSW equation. The asymptotic stability of a family of nonlinear PDEs containing a nonlocal functional is the essential component of this work. We will prove convergence of the nonlocal functional on a certain collection of functions to obtain our result.

ABSTRACT. We give an overview of local boundary conditions (BC) in nonlocal problems as presented in [1]. We explain methodically how to construct forcing functions to enforce local BC and their relationship to initial values.

We elaborate in great detail how to implement BC when a collocation method is utilized. We put a special emphasis on how to implement the local Neumann BC. In particular, a critical interpolation strategy is employed to find the appropriate value of the forcing function from its derivative. We present numerical results and computed displacement and strain fields with local Dirichlet and Neumann BC.

[1] Burak Aksoylu George A. Gazonas, On Nonlocal Problems with Inhomogeneous Local Boundary Conditions, Journal of Peridynamics and Nonlocal Modeling (2020) 2:1-25, https://doi.org/10.1007/s42102-019-00022-w

ABSTRACT. Peridynamics is a nonlocal reformulation of classical continuum mechanics suitable for material failure and damage simulation. Governing equations in peridynamics are based on spatial integration rather than spatial differentiation, allowing natural representation of material discontinuities such as cracks. A meshfree method where the domain is represented by a set of nodal volumes has been demonstrated to be an effective discretization approach for large-scale engineering simulations, particularly those involving large deformation and complex fractures. However, this approach has been shown to suffer from convergence issues in static simulations and simulations of dynamic wave propagations. For this class of problems mitigation strategies such as the use of smoothly-decaying influence functions and the so-called partial volume approach lead to improved convergence behaviour. Nonetheless, a robust quantitative assessment of the performance of such approaches, particularly in fracture scenarios, is lacking. In this talk, we will discuss recent convergence studies of wave propagation and extensions to dynamic crack propagation in meshfree peridynamic simulations, under different choices of influence functions and integration weights.

ABSTRACT. This work explores material stability in peridynamics from the perspective of potential energy minimization. In classical linear elasticity, with an absence of external forces, material stability takes the form of restrictions on the elasticity tensor. In [1] it was shown that positivity of the micromodulus function is a sufficient but not necessary condition for material stability. We consider an anisotropic bond-based peridynamic model which agrees with the classical linear elasticity model up to third-order terms in a Taylor expansion. We determine what constraints positivity of the corresponding micromodulus places on the elasticity tensor and prove the constraints are more restrictive than those imposed by material stability in classical linear elasticity. We then numerically explore necessary and sufficient conditions for peridynamic material stability for select symmetry classes.

References [1] Silling, Stewart. Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces. Journal of the Mechanics and Physics of Solids. (2000)48:175-209.

SNL is managed and operated by NTESS under DOE NNSA contract DE-NA0003525. SAND2021-0551 A

ABSTRACT. In recent years, we have studied locomotion and shape control in several biological systems using tools ranging from theoretical and computational mechanics, to experiment and observations at the microscope, to manufacturing of prototypes. A particularly interesting case study is provided by Euglena gracilis. This unicellular protist is particularly intriguing because it can adopt different motility strategies: swimming by flagellar propulsion, or crawling thanks to large amplitude shape changes of the whole body (a behavior known as “metaboly”, or “amoeboid motion”). These shape changes are achieved by Gaussian morphing, the paradigm by which curvature can be produced by differential in-plane stretches of the mid-surface (the tubular shell of the flagellum in one case, the body envelope in another). Understanding the way the shape-shifting mechanism is grounded in the body architecture opens the way to the design of new, bio-inspired, morphable structures. This is being explored using additive manufacturing techniques. We will survey our most recent findings in this stream of research, obtained in collaboration with M. Arroyo, A. Beran, G. Cicconofri, G. Noselli and D. Riccobelli.

ABSTRACT. In pursuit of modeling springy wires we study framed curves, assuming that their evolution is driven by a linear combination of bending energy and twisting energy. The latter tracks the rotation of the frame about the centerline of the curve.

In order to incorporate impermeability which precludes topology changes throughout the evolution, we add a self‐avoiding term, namely the tangent‐point potential. We discuss the discretization of this model and present some numerical simulations.

ABSTRACT. The methods of stress-functions are an important analytical tool in classical elasticity. They are motivated by the observation that in two-dimensional systems, the fact that the stress tensor at equilibrium is symmetric and divergence-free leads to its representation as the “curl-curl” of a scalar function. By combining this with so-called “constitutive relations”, one obtains a biharmonic equation for this potential.Recent work by physicists, motivated by problems in incompatible elasticity, suggested a generalization of the stress function approach to nonlinear and non-Euclidean systems with arbitrary constitutive relations. In that respect, mathematical difficulties arose in both understanding the mechanisms that allow the stress function approach to be extended thus, and furtherly in establishing the analytic foundations beyond the two-dimensional case. In this talk, I present a rigorous framework for obtaining stress potentials in such a generalized setting, with emphasis on suitable choice of gauge for solving problems. I shall also describe the line of reasoning which allows a representation for the stress to be produced, based on newly developed geometric-analytical tools – namely, the regular ellipticity of a boundary value problem for a bilaplacian operator acting on “double forms”.

This talk will be based on elements of a joint work with Raz Kupferman:

https://arxiv.org/abs/2103.16823 and https://arxiv.org/abs/2104.05794 

ABSTRACT. In this talk, we derive an effective bending plate model via simultaneous homogenization and dimension reduction. Our starting point is a 3d nonlinear elasticity model describing a composite whose components are prestrained with magnitude that scales with the thickness of the plate. We assume that both the composite as well as the prestrain feature a periodic microstructure. The limiting energy features a spontaneous curvature term that is linked to the prestrain and the composite's microstructure by means of a homogeneous formula. Our derivation invokes flexibility results for the Monge-Ampere equation. This is joint work with S. Neukamm.