ABSTRACT. We are interested in the connection between a metastable continuous state space Markov process (satisfying e.g. the Langevin or overdamped Langevin equation) and a jump Markov process in a discrete state space. We use the notion of quasi-stationary distribution within a metastable state for the continuous state space Markov process to parametrize the exit event from the state. This approach is useful to analyze and justify methods which use the jump Markov process underlying a metastable dynamics as a support to efficiently sample the state-to-state dynamics (accelerated dynamics techniques à la Arthur Voter). Moreover, it is possible by this approach to quantify the error on the exit event when the parametrization of the jump Markov model is based on the Eyring-Kramers formula. This therefore provides a mathematical framework to justify the use of transition state theory and the Eyring-Kramers formula to build kinetic Monte Carlo or Markov state models.

References: - G. Di Gesu, T. Lelièvre, D. Le Peutrec and B. Nectoux, Jump Markov models and transition state theory: the Quasi-Stationary Distribution approach, Faraday Discussion, 2016. - G. Di Gesu, T. Lelièvre, D. Le Peutrec and B. Nectoux, Sharp asymptotics of the first exit point density, Annals of PDE, 2019.

ABSTRACT. Metallic nanoclusters are functional materials with many applications owing to their unique physical and chemical properties, which are sensitively controlled by their shapes and structures. An in­depth understanding of their morphology stability is therefore of crucial importance. It has been well documented by transmission electron microscopy (TEM) studies that metallic nanoclusters can interconvert between different isomers. However, the relevant mechanisms remain elusive because the timescales of such shape fluctuations are too short to be resolved experimentally and yet too long for conventional atomistic simulations. By employing a recently introduced Accelerated Molecular Dynamics method, Parallel Trajectory Splicing, we present simulations that reached timescales of milliseconds and thus provide a clear description of the dynamic process of the experimentally observed shape fluctuation in metallic nanoclusters. We observe transformations back and forth between face-­centered-­cubic (fcc) and structures with five­fold symmetry (decahedron or icosahedron). These transitions occur following either by a partial­dislocation­mediated twinning mechanism or by a surface­reconstruction driven process. The identified pathway is in remarkable agreement with the existing microscopy results and serves as further evidence that shape fluctuation can occur directly through thermal activation, without involving melting or other external factors.

ABSTRACT. Defect transport is a key process in materials science, but atomistic mechanisms are often too complex to enumerate a priori, complicating uncertainty analysis and observable convergence. I will describe an asynchronous, massively parallel accelerated sampling scheme[1], autonomously controlled by rigorous Bayesian estimators of statewise sampling completeness, which builds atomistic kinetic Monte Carlo models on a state space irreducible under exchange and space group symmetries. A combined Monte Carlo and analytic procedure is outlined which provides a novel convergence metric for defect transport coefficients, via a Kullback-Leiber divergence across the ensemble of diffusion processes consistent with the sampling uncertainty. The autonomy and efficacy of the method is demonstrated on a range of challenging materials science problems. [1] "Automated calculation and convergence of defect transport tensors" TD Swinburne and D Perez, NPJ Computational Materials, December 2020

ABSTRACT. Quasi-stationary distributions can be seen as the first eigenvector associated with the generator of the stochastic differential equation at hand, on a domain with Dirichlet boundary conditions (which corresponds to absorbing boundary conditions at the level of the underlying stochastic processes). Many results on the quasi-stationary distribution hold for non degenerate stochastic dynamics, whose associated generator is elliptic. The case of degenerate dynamics is less clear. In this work, together with T. Lelièvre and J. Reygner (Ecole des Ponts, France) we generalize well-known results on the probabilistic representation of solutions to parabolic equations on bounded domains to the so-called kinetic Fokker Planck equation on bounded domains in positions, with absorbing boundary conditions. Furthermore, a Harnack inequality, as well as a maximum principle, is provided for solutions to this kinetic Fokker-Planck equation, as well as the existence of a smooth transition density for the associ- ated absorbed Langevin dynamics. The continuity of this transition density at the boundary is studied as well as the compactness, in various functional spaces, of the associated semigroup. This work is a cornerstone to prove the consistency of some algorithms used to simulate metastable trajectories of the Langevin dynamics, for example the Parallel Replica algorithm.

ABSTRACT. A quad-surface is made of rigid quadrilateral plates connected by rotational joints in the square grid pattern. Generic quad-surfaces are rigid, so that flexible instances are exceptional. A widely known flexible example is Miura-ori. In this talk I will present an approach which has led to a classification of all flexible quad-surfaces and some ongoing research into the design methods.

ABSTRACT. Quadrilateral origami and kirigami with rigid panels connected by flexible hinges are generically non-deployable due to the geometrical constraints. Special patterns with two-dimensional symmetry are found to satisfy these constraints to achieve rigid deployability. Examples include the famed Miura origami and the twisting-square kirigami. Here we formulate the geometrical compatibility conditions for the deployability of quadrilateral origami and kirigami by composing the folding angle functions and the opening angle functions respectively. Solving these conditions leads to the designs of general deployable patterns capable of achieving the desired functionality such as approximating curved surfaces and boundaries.

ABSTRACT. Many symmetric origami structures can be obtained as the orbit of a partially folded unit cell under a certain discrete group of isometries; for instance, the classical Miura-ori can be obtained by applying a translation group to a unit cell, and recent work of Feng, Plucinsky, and James investigates a miura-ori structure obtained by applying a helical group to a unit cell.

Firstly, we consider a generalisation of the above: origami obtained by applying conformal affine linear transformations of R^n to a unit cell. For our purposes, we develop a classification of all groups of such transformations that are "semi-discrete": where in the orbit of any point, there is a fixed positive distance between accumulation points. We will use these groups to create conformal Miura-ori and waterbomb origami. Secondly, we will consider the problem of creating generalised waterbomb origami tubes with helical symmetry, with a view towards applications in medical stents. We will use an inverse approach which makes use of several necessary symmetries in these objects.

ABSTRACT. Folding is a systematic method that transforms planar material into three-dimensional rigid structures. Depending on the organization of folds, structures can be flat-packed for ease of transport. By beginning with a flat plane, there is the potential to reduce production costs associated with manufacturing parts with curvature. Additionally, there are numerous variations possible with one systematic method.

This research seeks to move beyond origami — the art of folding paper — by embracing material and structural constraints. Computation-based simulation and analysis is used to design stronger lightweight structures. Additionally, more efficient methods to manufacture parts are being developed in direct dialogue with industry sponsors and collaborators. Lastly, new methods and materials are being invented to expand the possibilities of translating paper folding into materials which have the potential to scale-up. Folding is embraced as a means to tackle industry related problems. Applications include: lightweight deployable structures, ultra-thin formwork for concrete casting, and stay-in-place formwork for shell structures and concrete slabs.

ABSTRACT. We will discuss the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised functional, whose volume and surface integrands are characterised by asymptotic formulas involving minimisation problems on larger and larger cubes with special boundary conditions. In the proof we combine a recent deterministic Gamma-convergence result for free-discontinuity functionals with the Subadditive Ergodic Theorem by Akcoglu and Krengel. This is a joint work in collaboration with Gianni Dal Maso (SISSA), Lucia Scardia (Heriot-Watt University), and Caterina Zeppieri (University of Münster).

ABSTRACT. In this talk I will present a regularity result and a Korn inequality in the space $GSBD^p$ of generalised functions with bounded deformation, for any $p> 1$ and any dimension $n \geq 2$. Then I will present some consequences and applications, including an extension result.

This is work in collaboration with F. Cagnetti, A. Chambolle and M. Perugini.

ABSTRACT. In this work, we provide a characterization result for lower semicontinuity of surface energies defined on piecewise rigid functions, i.e., functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is a skew symmetric matrix. This characterization is achieved by means of an integral condition, called BD-ellipticity, which is in the spirit of BV -ellipticity defined by Ambrosio and Braides. By specific examples we show that this novel concept is in fact stronger compared to its BV analog. We further provide a sufficient condition implying BD-ellipticity which we call symmetric joint convexity. This notion can be checked explicitly for certain classes of surface energies which are relevant for applications, e.g., for variational fracture models. Finally, we give a direct proof that surface energies with symmetric jointly convex integrands are lower semicontinuous also on the larger space of GSBD^p functions.

ABSTRACT. The talk is based on a joint work with Annika Bach and Caterina Ida Zeppieri. There we study the $\Gamma$-convergence of singularly perturbed elliptic functionals that generalizes the functionals of Ambrosio-Tortorelli type. It is well known that the latter approximate, in the sense of $\Gamma$-convergence, the free discontinuity functional of Mumford-Shah which models for example image segmentation. Accordingly to this, the asymptotic analysis of these functionals gives rise to a decoupling volume-surface effect also in the more general setting that we consider. In particular, if $u$ is a function describing the image segmentation effect, we derive that the surface term of the limit functional does not depend on the jump opening $[u]$. Our general result applies, for example, to the case of stochastic homogenization.

ABSTRACT. The Bilbao Crystallographic Server (www.cryst.ehu.es) is a free web site with an access to crystallographic data of space and point groups, magnetic and subperiodic groups, their representations and group-subgroup relations. A database on incommensurate structures incorporating modulated structures and composites, a magnetic-structure database and a k-vector database with Brillouin-zone figures are also available. Wide range of complex solid-state physics and structure-chemistry aspects of materials studies are facilitated by the specialized software provided by the server. The server offers a set of structure-utility programs including tools for crystal-structure transformations, for a quantitative analysis of structure-model similarities and for studies of crystal-structure relationships. There are tools for a systematic research of phase-transition mechanisms including the evaluation of the structure pseudosymmetry which proves to be a powerful method for the prediction of new ferroic materials. Recently implemented computational tools and databases allow the systematic application of symmetry arguments in the study of magnetic structures. The presentation of the databases and programs offered by the Bilbao Crystallographic Server will be accompanied by case studies illustrating the capacity and efficiency of the online tools in material studies.

ABSTRACT. Operating since 1997, the Bilbao Crystallographic Server (www.cryst.ehu.es) is a free web server that grants access to specialized databases and tools for the resolution of different types of crystallographic, structural chemistry and solid-state physics problems. Recently, new tools dedicated to the study of materials with layer and multilayer symmetry have been developed due to the arising interest in these type of materials. The aim of this contribution is to report the current state of the programs in the server related to the study of materials with layer group symmetry.

The section dedicate to Subperiodic groups gives access to the layer groups databases which contains the basic crystallographic information (generators, general positions, Wyckoff positions and maximal subgroups) and the Brillouin zone and k-vectors tables that form the background and classification of the irreducible representations of layer groups. Moreover, programs to identify the layer symmetry of periodic sections and to describe the electronic structure and surface states of crystals are also available. The last tool to join this section is able to calculate the irreducible representations of layer groups. The utility of the available applications will be demonstrated by illustrative examples.

ABSTRACT. Incommensurately modulated structures (IMS) are aperiodic, but long-range ordered, in physical space. IMS can be described as periodic structures in higher-dimensional superspace by defining a 3-dimensional basic structure and a set of modulation parameters accounting for displacive and/or occupation modulations. Experimentally determined IMS cannot be validated by means of density functional theory (DFT) or similar computational methods, which require an atomic model described in 3D space. Likewise, structures that show static disorder, e.g. atomic sites that are partially occupied in the average structure, are another class of materials for which a clear approach for structure model validation from first principles is missing. The industrially important ceramic material mullite, an aluminium silicate, is both incommensurately modulated and disordered with respect to Al/Si ordering and vacancy ordering. The underlying ordering patterns were identified based on DFT calculations on fully ordered, commensurate approximations that were generated taking the full superspace symmetry into account. A consistent model for a broad range of compositions was derived, capable of explaining the vacancy and Al/Si ordering in the IMS of mullite.

ABSTRACT. Alkaline sulfates of transition elements attract interest because of their electrochemical properties and application as electrode materials. The crystal structures of Na2Mn3(SO4)4 polymorphs are based upon heteropolyhedral frameworks formed by Mnφn polyhedra (n = 5,6) and SO4 tetrahedra and display order-disorder (OD) character and can be described using the same OD groupoid family, more precisely a family of OD structures built up by two kinds of non-polar layers, with layer symmetry Pmc21 (L2n+1 type) and Pbcm (L2n type) (category IV). All ordered polytypes as well as disordered structures can be obtained by means of the following symmetry operators that may be active active in the L2n type layers with symmetry Pbcm: the 21 screw axis parallel to c [- - 21] or inversion centres and the 21 screw axis parallel to b [- 21 -]. The relationships between different polytypes is represented by the presence in their structures of common tilings, in particular, 3T2M-[43], 4T3M-[4.62], and 2T3M-[32.42]. The geometry optimization of the hypothetical structure of the MDO2-polytype predicted using the OD approach demonstrates its reasonability as another possible stable polymorph of Na2Mn3(SO4)4. The calculated enthalpy factors of the polytypes decreases in the following order: non MDO-4O > MDO1 > MDO2.

ABSTRACT. This work (IFB 2020) forms part of the variational analysis of cavitation (Gent & Lindley '59, Ball '82, Muller & Spector '95, Sivaloganathan & Spector '00). It is known (Sivaloganathan & Spector '10; H. & Serfaty '13) that the energetically most favourable cavities are spherical; nonetheless, for large external loads incompressiblity makes it impossible to create only spherical cavities. For a simplified 2D setting, we give a geometric sufficient condition for a given load to be compatible with the existence of deformations opening only round cavities: that the radius of each cavity, after the deformation, is smaller than a certain value comparable to the distance, in the reference configuration, between cavitation points. When this condition is satisfied, we prove that the minimizers of the Dirichlet energy among incompressible deformations of the domain with epsilon-initial holes tend to produce circular cavities as epsilon tends to zero. This gives a lower bound for the coalescence load: for coalescence the displacement load on the boundary must be large enough for our geometric sufficient condition to fail.

ABSTRACT. Rate-independent systems (RIS) are characterized by the lack of any internal time length scale: rescaling the input of the system in time leads to the very same rescaling of its solution. In continuum mechanics, rate-independent models represent a reasonable approximation whenever the external conditions change slowly enough so that the system can always reach its equilibrium. This applies if inertial, viscous, and thermal effects are neglected. RIS have proven to be useful in modeling of hysteresis, phase transitions in solids, elastoplasticity, damage, or fracture in small and large strain regimes.

The talk introduces the notion of separately global solutions for RIS with non-convex functionals and describes an existence result for a model of bulk damage at large strains. The analysis covers non-convex energies blowing up for extreme compression, yields solutions excluding interpenetration of matter, and allows for handling nonlinear couplings of the deformation and the internal variable, which emerges e.g. from the interplay between Eulerian and Lagrangian description. It extends the theory developed so far in the small strain setting.

ABSTRACT. We present a nonlocal variational model in nonlinear elasticity, based on the concept of Riesz fractional gradient. We explain how the theory of polyconvexity applies in this context. We also prove the Gamma-convergence to the local (classical) model as the nonlocality vanishes.

ABSTRACT. It has been long known in geophysics that materials such as silica glass exhibit anomalous pressure-dependent strength characterized by a strongly non-convex elastic domain. This lack of convexity violates the standard Drucker principles of classical plasticity, including the principle of maximum dissipation, and may result in fine-scale patterning of the stress field. In order to understand these phenomena, we consider problems of static equilibrium in which the primary unknown is the stress field and the solution maximizes a complementary energy subject to equilibrium constraints. A necessary and sufficient condition for the sequential lower-semicontinuity of such functionals is symmetric div-quasiconvexity in the sense of Fonseca and Muller's A-quasiconvexity. We specifically consider the example of the static problem of plastic limit analysis and seek to characterize its relaxation in the nonstandard case of a non-convex elastic domain. We show that the symmetric div-quasiconvex envelope of the elastic domain, which describes the effective or macroscopic plastic behavior of the material, can be characterized explicitly by means of a rank-2 hull construction for isotropic materials whose elastic domain depends on pressure and Mises effective shear stress. The resulting relaxed elastic domain closely matches that obtained by means of molecular dynamics calculations for silica glass.

ABSTRACT. In the context of stochastic homogenization, the Bourgain-Spencer conjecture states that the ensemble-averaged solution of a divergence-form linear elliptic equation with random coefficients allows for an intrinsic description in terms of higher-order homogenized equations with an accuracy four times better than the almost sure solution itself. While previous rigorous results are restricted to a perturbative regime with small ellipticity ratio, we shall explain how half of the conjecture can be established in a non-perturbative regime. Our approach involves the construction of a new corrector theory in stochastic homogenization: while only a bounded number of correctors can be constructed as L^2 stationary random fields, we show that twice as many can be usefully defined in a Schwartz-like distributional sense on the probability space. We shall also discuss corresponding results for linear waves in random media.

ABSTRACT. I will discuss regularity properties for solutions of linear second order non-uniformly elliptic and parabolic equations in divergence form. Assuming certain integrability conditions on the coefficient field, we obtain local boundedness and validity of Harnack inequality. The assumed integrability assumptions are essentially sharp and improve upon classical results in the literature. These determistic regularity results are key ingredients in homogenization results, namely quenched invariance principle and local limit theorem, for random walks in a random degenerate environment.

ABSTRACT. In 1997, A. Naddaf and T. Spencer established that the scaling limit of the Ginzburg-Landau random surface model is a Gaussian free field. Their proof relies on the observation that the behavior of the random surface is characterized by the large-scale properties of the solutions of an infinite dimensional elliptic PDE, called the Helffer-Sjöstrand equation, which can be understood through techniques of stochastic homogenization. In this talk, we will describe the arguments developed by Naddaf and Spencer, explain how they can be reworked using the tools of the quantitative theory of stochastic homogenization, and present some of the (quantitative) results on the model which can be obtained by these techniques. We will then explain how these arguments can be used to understand the large-scale behavior of another model of statistical physics: the Villain rotator model. This is based joint work with W. Wu.

ABSTRACT. We are interested in the endpoint distribution a directed polymer in a Gaussian random environment. A PDE hierarchy governing the evolution of the n-point density functions is derived, and we will discuss its applications.

ABSTRACT. Colloidal particles in liquid crystals exhibit a flow behaviour that is strikingly different from that in isotropic media. On one hand, the emerging topological defects give rise to highly complex, anisotropic elastic forces that the liquid crystal exerts on the colloidal particles. On the other hand, non-linear, flow-dependent forces are also involved, which are impossible to predict analytically. Studies of particles in nematics were therefore generally limited to how the molecules reorder around the solute particles.

We present new results of lattice-Boltzmann simulations of colloid-liquid crystal composite materials in Poiseuille flow. At the single-particle level we observe a new, hydrodynamic and very fast separation effect that is distinct from the well-known Segre-Silberberg effect in isotropic media at low volume fractions. Contrary to the latter, there is no single attractors: where and how particles accumulate depends sensitively on the imposed flow rate and particle size. At higher volume fractions we observe a range of other intriguing features, like the reorientation of the local defect network around the colloidal particles, states where a significant fraction of the particles has temporarily negative mobilities, as well as percolating networks of topological defects that appear to act as low-viscosity conduits for the flow.

ABSTRACT. Flow of liquid crystals is inherently complex due to the coupling between their orientational order and the imposed shear velocity. Experiments carried out with channel-confined nematic fluids reveal strong influence of boundary conditions, elasticity, viscosity, and external fields on observed nonequilibrium states. I will present how different orientational states in pressure-driven nematics can coexist and get stabilized by precisely engineered microflows. The mechanisms that connect the behavior of the nematic liquid crystal with its material parameters will be discussed in the framework of Ericksen-Leslie-Parodi formulation of nematodynamics, and supported with the results of numerical modelling. The symmetry-breaking transition to chiral states will be explained through geometric and topological assumptions of a phenomenological Landau model. I will also demonstrate how geometric variations of the confinement can contribute to dynamic reshaping of the steady-state flow structures. Our findings could find use in out-of-equilibrium lyotropic liquid crystals and active nematics where additional dynamic response is expected due to density- and activity-dependence of their elastic properties.

The research was conducted with support from the Slovenian Research Agency (ARRS), and with support from the National Science Foundation (NSF) of the United States of America.

ABSTRACT. Cavitation, the nucleation of vapor in liquids, is ubiquitous in fluid dynamics, and is often implicated in a myriad of industrial and biomedical applications. Although extensively studied in isotropic liquids, corresponding investigations in anisotropic liquids are largely lacking. We reveal flow-induced cavitation in liquid crystals at low Reynolds numbers [1]. We combine liquid crystal microfluidic experiments, nonequilibrium molecular dynamics simulations and theoretical arguments to identify the mechanism underpinning LC cavitation. The cavitation domain nucleates due to sudden pressure drop upon flow past a cylindrical obstacle within a microchannel. The inception and growth of the cavitation domain ensued in the Stokes regime, while no cavitation was observed in isotropic liquids flowing under similar hydrodynamic parameters. Using simulations we identify a critical value of the Reynolds number for cavitation inception that scales inversely with the order parameter of the fluid. Strikingly, the critical Reynolds number for anisotropic fluids can be 50% lower than that of isotropic fluids.

[1] T. Stieger, Hakam Agha, Martin Schoen, Marco G. Mazza, Anupam Sengupta, Nature Communications 8, 15550 (2017).

ABSTRACT. We report results and simulations for Poiseuille flow, parallel to the zero-pretilt surface alignment direction, of a thin (d = 25 micrometer) layer of nematic liquid crystal with positive dielectric anisotropy and an electric field orthogonal to the layer and to the flow direction. We demonstrate how flow significantly extends the low director distortion state to higher voltages and removes the tilt re-orientation direction degeneracy of the classical Freedericksz switching. We show switching between the flow stabilized, low director distortion state and the flow modified high director distortion state via abrupt removal and re-application of externally driven flow. Our numerical simulations using the Ericksen-Leslie dynamic equations reveal how the asymmetry of the director distortion (about z = d/2) in the flow modified state becomes more pronounced as the flow rate is increased. At very high flow rates there is predicted to be a narrow re-orientation region in which there is transition between director alignment close to the positive Leslie angle for z < d/2 to director alignment close to the pi radians minus the Leslie angle for z > d/2.

ABSTRACT. Nuclear magnetic resonance (NMR) spectroscopy is an effective tool for identifying molecules in a sample. Although many previously observed NMR spectra are accumulated in public databases, they cover only a tiny fraction of the chemical space, and molecule identification is typically accomplished manually based on expert knowledge. Herein, we propose NMR-TS, a machine-learning-based python library, to automatically identify a molecule from its NMR spectrum. NMR-TS discovers candidate molecules whose NMR spectra match the target spectrum by using deep learning and density functional theory (DFT)-computed spectra. As a proof-of-concept, we identify prototypical metabolites from their computed spectra. After an average 5451 DFT runs for each spectrum, six of the nine molecules are identified correctly, and proximal molecules are obtained in the other cases. This encouraging result implies that de novo molecule generation can contribute to the fully automated identification of chemical structures.

ABSTRACT. With an incredible volume of chemical and crystal database accumulated so far well as the hardware to process such large information, data-driven chemical science is expected play an important role, particularly in discovering novel functional molecules and crystals. In this talk, I will present several machine learning models that can allow efficient exploration of chemical space towards a new discovery. Examples could include crystal structure prediction, synthesizability prediction, and chemical pathway planning.

ABSTRACT. Machine learning (ML) has proven invaluable in accelerating materials discovery, both in forward design, where we use ML models to rapidly screen large chemical spaces, and in inverse design where we build generative models to design molecules with targetted properties. Large datasets are indispensable for training reasonably accurate ML models that drive high-throughput virtual screening. High-throughput computation with density functional theory (DFT) enables the cost-efficient generation of relatively accurate datasets. A naive workflow for DFT-based high-throughput computation, however, may suffer from both wasting computational resources on failed calculations and domain of application errors from DFT, especially for transition metal containing molecules. We apply ML to aid this high-throughput computation in two ways: i) we build classification models to both pre-screen a large chemical space and monitor a calculation on-the-fly to avoid unproductive calculations that would waste computational resources, and ii) we build a semi-supervised classification model to predict when molecules contain strong static correlation and thus require more accurate but more expensive wavefunction theory calculations. We anticipate these ML models to be useful in making the data generation process more efficient and accurate with currently available electronic structure methods.

ABSTRACT. It is well known that kinetic models satisfying the so-called detailed balance condition have an entropy functional which can be used to derive convergence to equilibrium results. On the other hand, there are many physical situations (typically open systems) where it is natural to use kinetic equations for which a detailed balance condition does not hold. In these cases, more complicated dynamical behavior can arise, for instance, oscillatory behaviors. A class of kinetic equations where it is not a priori evident if temporal oscillations can occur are the coagulation-fragmentation equations.

In the talk, we concentrate on Becker-Döring type dynamics, in which only a single monomer can attach or detach from a cluster. These equations have been extensively used to model chemical-physical systems and especially bubbleator dynamics. In this talk, I will describe such models for which the onset of periodic oscillations can be proven by formal asymptotics. One of the models represents the formation of large clusters in a Becker-Döring equation having a source of monomers and removal of large clusters.

ABSTRACT. The Lifshitz-Slyozov system describes the temporal evolution of a mixture of monomers and aggregates, where individual monomers can attach to or detach from already existing clusters. The aggregate distribution follows a transport equation with respect to a size variable, whose transport rates are coupled to the monomer dynamics through a mass conservation relation. Being a model traditionally designed to describe phase transitions, the attachment and detachment rates proposed by Lifshitz and Slyozov do not require a boundary condition at zero size. However, this model is now used in a number of different contexts (e.g. protein polymerization or tentative applications to Oceanography). These situations impose attachment and detachment rates that requiere a boundary condition at zero size, which is intepreted as the synthesis of new clusters from monomers by a nucleation process. Up to date, the mathematical results on this new setting are scarce. In this contribution we study existence and uniquenes of local-in-time solutions for nonlinear boundary conditions, together with continuation criteria and results on long-time behavior. We are able to deal with attachment and detachment rates that may lack Lipschitz regularity, like power law rates. This requires a careful analysis of the characteristic curves associated to the transport process.

ABSTRACT. In this lecture, we discuss the global in time solvability of the continuous growth-fragmentation-coagulation equation with unbounded coagulation kernels, in spaces of functions having finite moments of a sufficiently high order. The main tool is the recently established result on moment regularization of the linear growth-fragmentation semigroup that allows us to consider coagulation kernels whose growth for large clusters is controlled by how good the regularization is, in a similar manner to the case when the semigroup is analytic.

ABSTRACT. In many growing filamentary structures such as neurons, roots, and stems, the intrinsic shapes and material response are produced by differential growth of the tissue. Therefore, a key problem is to link the growth field at the microscopic level to the macroscopic shape and properties of the filaments. In this talk, starting with a morphoelastic tubular structure and assuming an arbitrary local growth law on the growth tensor, I will give a multiscale method to obtain the overall curvature, torsion, and material parameters of a growing filament. Various examples of curvature and torsion generation are given and the impact of residual stress on the generation of curvature is demonstrated. Applications to plants and axons will be given

ABSTRACT. Thin nematic elastomer sheets can be programmed, via the nematic director field embedded into them, to take different shapes in different environments. Recent experiments from various groups demonstrate excellent control over the director field, thus opening a door for achieving accurate and versatile designs of shape-shifting surfaces. At the crux of any effort to implement this design mechanism lies the inverse design problem -- given an arbitrary surface geometry, constructing the director field that would induce it upon actuation. In this talk I describe several aspects of this inverse problem. I present a numerical algorithm for finding approximate global solutions to the inverse problem for any 2D geometry. I show analytically the existence of many local solutions for any smooth geometry, provide an algorithm for their integration, and show a convenient classification useful for finding optimized director fields per given surface geometry. I further discuss non-smooth surface geometries, and how these can be realized via topological defects and domain walls in the nematic director field.

ABSTRACT. Consider a prototypical “stretching plus bending” functional of an elastic shell, E(f), where the shell is modeled as a d-dimensional Riemannian manifold endowed with a reference second fundamental form, which is being immersed by f into a (d + 1)-dimensional ambient space. It is well known that when the ambient space is Euclidean, there exists a zero-energy configuration, f, if and only if the reference metric and second fundamental form satisfy the Gauss-Codazzi-Mainardi compatibility conditions. It is therefore natural to ask whether inf E = 0 implies that the reference metric and second fundamental form are compatible, and in this case, whether any minimizing sequence converges to a zero-energy reference configuration. In this lecture I will show that this is indeed true. In particular, under the assumption that the ambient space is of constant sectional curvature, any sequence of immersions of asymptotically vanishing energy converges to an isometric immersion of the shell into ambient space, having the reference second fundamental form. I will present a sketch of the proof of this theorem, emphasizing its main ideas.

ABSTRACT. Chirality in materials, such as liquid crystals, often entails a high degree of geometric frustration, leading to a rich array of morphological phenomena. The mathematics of such systems is similarly complex, with chirality often manifesting non-linearly. Here I will discuss how methods from contact geometry and topology can be used to give a qualitative understanding of the behaviour of chiral materials. Using experiments on cylindrically confined lyotropic chromonic liquid crystals as a model system, I will show how contact topology-theoretic methods lead to a topological description of the various metastable minima in the energy landscape, as well as explaining the existence of novel chirally-protected solitons. I will further show how the same perspective can be used to understand the structural stability of highly-twisted Skyrmions in chiral ferromagnets, with applications to racetrack memory devices. Finally, I will present the results of transition path sampling calculations describing generic features of chiral morphological transitions in liquid crystalline systems.

ABSTRACT. The aim of this talk is to give a survey of recent results on quantitative aspects of stochastic homogenization. I shall start with the model problem of conductivity in a random checkerboard. The solution of this conductivity problem displays oscillations (at the scale of the checkerboard boxes) and random fluctuations (the medium has itself random fluctuations). When the boxes are small with respect to the scale of observation, it is standard to replace this heterogeneous medium by an effective deterministic medium. Quantitative homogenization aims at answering three main questions: how accurately does the effective medium describe the heterogeneous medium, how accurately can we approximate this effective medium, what do the fluctuations of the solution look like? In the first part of the talk I will answer these questions on the model problem of conductivity in a random checkerboard, which is by now well-understood, and introduce important tools and quantities (sensitivity calculus, flux corrector, and homogenization commutator). In the second part of the talk, I will give significant extensions of these results of interest to material sciences: linear elasticity with long-range correlations, log-normal coefficients, nonlinear elliptic systems, elasto-dielectric coupling, heat equation, wave equation, sedimentation of rigid particles in a Stokes flow.

ABSTRACT. For more than 120 years, experiments on a range of systems have shown an unexpected correlation between diffusion pre-exponential factors and energy barriers. This compensation effect, or the Meyer-Neldel rule, implies a universal linear relation between the activation energy and the entropy. While a number of phenomenological explanations have been proposed, these could not be directly tested against experimental data. Building on extensive catalogs of activated events generated with the activation-relaxation technique (ART nouveau) in a number of disordered systems, we lift this restriction and demonstrate how the deformation of the energy landscape affects the vibrational density of state that dominates the entropic contributions. Showing that the harmonic approximation captures the essential part of the compensation effect, we find that the deformation associated with diffusion both softens low and stiffens high frequency phonons. Softening generally dominates as barrier increase, creating a compensation effect. This correlation, however, is held only on average and huge fluctuations in prefactors are observed on an event by event basis. These can affect the evolution of materials and cannot be neglected without being assessed.

Part of the work is done in collaboration with Alecsandre Sauvé-Lacoursière (U. Montréal), Gilles Adjanor and Christophe Domain (EDF, France)

ABSTRACT. Atomistic simulations of solid-solid phase transformations provide insight into the microscopic processes that govern the kinetics and mechanisms of the transition. Understanding these processes is of considerable technological importance in a wide range of materials such as metals and alloys, minerals, or molecular crystals. To achieve an efficient sampling of the underlying high dimensional phase space with often complex features typically requires the application of enhanced simulation approaches.

Here, we combine machine learning with enhanced sampling algorithms to explore structural phase transformations induced by interface migration. Specifically, we employ a neural network based identification of local structural environments that are then used to construct global structural classifiers. Within the space of global classifiers a one-dimensional path collective variable is defined that is subsequently used in the enhanced sampling.

We illustrate our approach by exploring the complex migration of a phase boundary during the solid-solid transformation between the bcc and A15 phase in molybdenum. We were able to observe the growth of both phases, and estimate the free energy profile along the transformation. The approach presented here is generally applicable, and the efficient sampling of the phase space can facilitate the study of structural transformations in various complex condensed matter systems.

ABSTRACT. Exascale computing presents a challenge for the scientific community as new algorithms must be developed to take full advantage of the new computing paradigm. Atomistic simulation methods that offer full fidelity to the underlying potential, i.e., molecular dynamics (MD) approaches, fail to use the whole machine speedup, leaving a region in time and sample size space that is unattainable with current algorithms. We present here an extension of the Parallel Trajectory Splicing algorithm that relies on a synchronous sublattice decomposition to further parallelize the process in space. This algorithm is based on a domain decomposition in which events happen independently in different regions in the sample as Parallel Trajectory Splicing is run in each subdomain. Extra care must be taken to ensure that the dynamics remain accurate including in the local subdomain ghost atoms from neighboring subdomains. We validate the algorithm comparing the rates for adatom hopping on a {110} surface in W obtained from direct MD, traditional Parallel Replica Dynamics, Parallel Trajectory Splicing and the Sub-Lattice Parallel Trajectory Splicing. Once validated we applied the algorithm to study clustering formation of adatoms in large samples relevant for fusion applications.

ABSTRACT. In molecular dynamics, several algorithms have been designed over the past few years to accelerate the sampling of the exit event from a metastable domain, i.e. the time spent and the exit point from the domain. Some of them are based on the fact that the exit event from a metastable region is well approximated by a Markov jump process. In this talk, I will present recent results on the exit event from a metastable region for the overdamped Langevin dynamics. These results aim in particular at justifying the use of a Markov jump process parametrized by the Eyring-Kramers law to model the exit event from a metastable region.

This is a joint work with G. Di Gesù, T. Lelièvre, and D. Le Peutrec.

ABSTRACT. Most origami structures and much of recreational origami is designed as piecewise rigid origami. In some cases it would be desirable to allow smooth, nonaffine isometric deformations of the tiles, for example, when one wishes to build-in some elastic energy that would bias the structure toward some particular configuration so that the structure would tend to return to that configuration upon release. Additional possibilities would arise for functional isometrically deforming tiles, such as shape memory tiles. Using a Lagrangian description of isometric deformations and formalizing the compatibility conditions at the folds, we present a theory for piecewise isometric origami. For related ideas, particularly in the case of functional materials, see #442 of this session by Feng, Duffy, Biggins and Warner.

ABSTRACT. Liquid crystal elastomers (LCEs) can undergo large reversible contractions along their nematic director upon heating or illumination. A spatially patterned director within a flat LCE sheet thus encodes a pattern of contraction on heating, which can morph the sheet into a curved shell, akin to a pattern of growth sculpting a developing organism. Here we consider, theoretically, numerically and experimentally, patterns constructed from regions of radial and circular director, which, in isolation, would form cones and anticones. The resultant surfaces contain curved ridges with sharp V-shaped cross-sections, associated with the interfaces between regions in the patterns. Such ridges may be created in positively and negatively curved variants and, since they bear Gauss curvature, they cannot be flattened without energetically prohibitive stretch. Our experiments and numerics highlight that, although such ridges cannot be flattened isometrically, they can deform isometrically by trading the (singular) curvature of the “V” angle against the (finite) curvature of the ridge line. In finite thickness sheets, the sharp ridges are non-isometrically blunted to relieve bend, resulting in a “smearing out” of the encoded singular Gauss curvature. We discuss possible applications as actuators, as well as similarities to shape changes found during the morphogenesis of several biological systems.

ABSTRACT. Curved fold, non-isometric origami, actuated by light or heat, can be realised in flat sheets of nematic glasses and elastomers with a programmed in-plane director field. Large ($\sim 30$\%), reversible contractions arise along the director in elastomers.

An in-plane log-spiral/circular director pattern actuates to a cone with Gaussian curvature (GC) localised at its tip and Gaussian-flat flanks. Joining such cones requires an interface between two such spiral/circular director patterns in the reference domain that gives a target space crease where lengths from actuation under each pattern are identical. Such interfaces obey a metric compatibility condition in the reference space and are characterized as hyperbolic-type or elliptic-type. Fitting the two cones together forms curved creases and requires isometries that we compute analytically for the circular cases and numerically for the spiral cases. Unlike isometric curved-fold origami, the creases carry non-trivial concentrated GC that can be calculated analytically. The computational results illustrate that creases are slightly diffuse and so is the GC, whereby the divergent bend is relieved at the cost of some stretch.

A rich variety of shapes with points and lines of GC will be presented.

ABSTRACT. The morphology of spherically confined flexoelectric fluid membrane vesicles in an external uniform electric field is studied numerically. Due to the deformations induced by the confinement, the membrane gets polarized resulting in an interaction with the external field. The equilibrium shapes of the vesicle without electric field can be classified in a geometrical phase diagram as a function of scaled area and reduced volume. When the area of the membrane is only slightly larger than the area of the confining sphere, a single axisymmetric invagination appears. A non-vanishing electric field induces an additional elongation of the confined vesicle which is either perpendicular or parallel depending on the sign of the electric field parameter. Higher values of the surface area or the electric field parameter can reduce the symmetry of the system leading to more complex folding. We present the resulting shapes and show that transition lines are shifted in the presence of an electric field. The obtained folding patterns could be of interest for biophysical and technological applications alike.

ABSTRACT. I will talk about some geometric questions that arise in the study of soft/thin objects with negative curvature. I will motivate the need for new "geometric" methods, for studying the mechanics of leaves, flowers, and sea-slugs. I will also discuss how this view naturally leads to a formulation of thin-sheet mechanics in terms of additional "natural" frames, in contrast with the usual formulation in terms of the Lagrangian (material) and Eulerian (Lab) frames.

ABSTRACT. From soap films and liquid crystals to biological membranes, minimal surfaces arise as the ground state of a variety of manmade and naturally occurring membranes and lamellar structures. Recently, helical motifs were discovered in the lamellar structure of the endoplasmic reticulum and thylakoids [1]. In both systems right- and left-handed motifs were observed in stoichiometry that suggested global pitch balance [2]. Theoretical treatment of such helical motifs in minimal surfaces has been so far limited to the small-slope approximation, which breaks in the biological settings. We provide a recipe for constructing exact and analytically tractable minimal surfaces in which the appropriate distribution of helical motifs are embedded [3]. We analyze the fundamental structure and interaction between two helical motifs of opposite and similar handedness. Finally, we exploit the newly devised method to construct a minimal surface with an arbitrary array of $N$ helical motifs and show that area minimization leads to global pitch balance.

[1] Mustárdy, et al. Plant Cell, 20 (2008) 2552; Liberton, et al. Plant Physiol., 155 (2011) 1656; Terasaki, et al. Cell, 154 (2013) 285.

[2] Bussi, et al. PNAS, 116 (2019) 22366.

[3] da Silva and Efrati, Proc. R. Soc. A, 477 (2021) 20200891.

ABSTRACT. Anthocyanins are cell vacuolar pigments responsible for the orange, pink, red, purple, and blue colors of flowers, fruits, vegetables, and leaves. Chromatic patterns form in vitro and in vivo.

ABSTRACT. We study the stochastic motion of a droplet in a stochastic Cahn–Hilliard equation in the sharp interface limit for sufficiently small noise. The key ingredient in the proof is a deterministic slow manifold parametrized by the position of the droplet. We show its stability for long times under small stochastic perturbations, and also reduce the dynamics to a stochastic differential equation for the stochastic motion of the center of the droplet.

ABSTRACT. We study the minimizers for the Cahn-Hilliard energy functional with a symmetric quartic double-well potential and under a strong anchoring condition on the domain boundary. We show a bifurcation phenomenon determined by the boundary value and a parameter $\kappa$ that describes the thickness of a transition layer separating the two phases. When the boundary value is exactly the average of the two pure phases, if $\kappa$ is larger than or equal to a critical value, the minimizer is unique and is exactly the homogeneous state. Otherwise, there are exactly two symmetric minimizers. The critical bifurcation value is inversely proportional to the first eigenvalue of the negative Laplace operator with the zero Dirichlet boundary condition. When the boundary value is larger (or smaller) than the average of the two pure phases, the symmetry is broken and there is only one minimizer. We also obtain the bounds and morphological properties of the minimizers under additional assumptions on the domain. Our analysis utilizes the notion of the Nehari manifold and connects it to the eigenvalue problem for the negative Laplacian with the homogeneous boundary condition. We will also present our numerical simulations and discuss them in the context of our analytical results.

ABSTRACT. The large-scale behavior of phase transitions has a long history. In this talk, I want to present two recent projects which establish convergence results based on a new relative entropy for phase-field models.

With Julian Fischer and Theresa M. Simon, we prove optimal convergence rates for the Allen-Cahn equation to mean curvature flow before the onset of singularities. The proof does not rely on the maximum principle and does not require to understand the spectral properties of the linearized Allen-Cahn operator.

With Yuning Liu, we consider the dynamics in the Landau-de Gennes theory of liquid crystals. We show that at the critical temperature, a scaling limit can be derived: The interface between the isotropic and nematic phases moves by mean curvature flow. Furthermore, in the nematic phase, the director field is a harmonic map heat flow with homogeneous Neumann boundary conditions. To derive the equations, we combine the relative entropy method with weak convergence methods.

ABSTRACT. Can atomistic simulations tell us their corresponding evolution equations in the continuum limit? Can a non-equilibrium process be interpreted as an equilibrium one? In this talk, various coarse-graining strategies will be discussed to elucidate these questions and provide important connections between atomistic and continuum models.

ABSTRACT. There are multiple tools in computational topology that can be used in material science. Starting from persistent homology (joint with vectorization methods and learning procedures), Euler characteristic curves, Mapper algorithm and beyond. In this talk I will give a general overview of those techniques and show how they can be applied to solve concrete problems.

ABSTRACT. Statistics, machine learning, and signal processing are dominant paradigms used to analyze data; unfortunately, such techniques provide limited capabilities to capture topological features of data. This limitation often requires models that are over-parameterized and difficult to interpret. Random field theory is an area of mathematics that integrates topology, geometry, and statistics and introduces descriptors to characterize data topology, such as the Euler characteristic (EC). The EC does not require strict statistical assumptions for data (e.g., stationarity, isotropy), generalizes to high dimensions, and provides dimensionality reduction capabilities.

In this talk, I discuss concepts of the EC, and demonstrate its application to data from liquid crystal experiments and molecular dynamics (MD) simulations. I demonstrate how the EC provides concise summaries of topological defect textures in liquid crystal films. I also show how the EC can be used to quantify 2-dimensional MD simulations for the analysis of self-assembled monolayers (SAMs). The EC can be used as a pre-processing step for these datasets, which simplifies the models needed to perform classification and regression tasks (i.e., complex CNNs simplified to linear regression models). I will also explore some future research areas, such as applications to model fitting and parameterization.

ABSTRACT. Many natural process, including various types of avalanches and earthquakes involve stick-slip type of dynamics. While there has been many attempts to understand in precise terms the mechanisms that lead to granular failure, important aspects of stick-slip dynamics are still not understood. One important question is whether there are clear precursor events to slip, or in other words, is there some information in the system that could be used to predict slip events. In this presentation, we will discuss the insights that can be reached by analyzing the force networks obtained in discrete element simulations of sheared granular systems using the tools of computational topology. The topological tools have been already successfully used to discuss evolution of force networks through jamming, as well as influence of particle geometry on force networks properties. One significant advantage of this approach is significant data reduction, allowing to describe crucial properties of complicated force networks by considering only a few relatively simple measures. We will use this approach to discuss whether the static and dynamic measures describing the force networks before slip events can be used to provide more information about the conditions that lead to stick-slip type of dynamics.

ABSTRACT. Large-scale molecular dynamic simulations routinely involve millions of atoms, and identifying a process of interest can usually be reduced to a comparison of local atomic environments. These consist of the intersection of a point cloud with a ball of fixed radius, though any local atomic environments related by the action of various symmetry groups are physically indistinguishable. Specifically, the space of local atomic environments is equivalent to that of a set of points in the unit ball quotiented by the orthogonal group acting on the point coordinates and the symmetric group acting on the index set of the points. This work proposes to metrize this space by viewing a local atomic environment as a finite metric measure space, and constructing a distance function on the space of local atomic environments using a variant of the Gromov-Wasserstein distance. This approach differs from those already in the literature in that the distance is continuous rather than discrete, and that local atomic environments of arbitrary spatial extent and composition can be compared. Numerical examples illustrate the efficiency and versatility of the algorithm which is expected to be useful in analyzing the points clouds that arise in other contexts.

ABSTRACT. Establishing structure-property relationship for material design still faces grand challenge due to enormous chemical space and structural reconfiguration. To probe this challenge, in this talk, we will present recently developed deep learning networks. The first class of networks is for predicting material properties with any given structure (forward design) while the other one named regression and conditional generative adversarial network (RCGAN) will autonomously generate new material structures upon demanded properties (inverse design). The employed material system for this study was the two-dimensional graphene/boron nitride hybrids described by numerical matrixes. The prediction performance of several convolutional neural networks (VGGNet, ResNet, ConcateNet and InceptionNet) was evaluated by benchmark ab initio calculation. They can achieve impressive prediction accuracy with fractional mean absolutely error (MAEF) of < 10%. The transfer learning was also proved effective to apparently enhance the prediction accuracy based on a small dataset. In the inverse design task, key features of the RCGAN, including the implementation of Regressor and the concatenation of latent space, enable the successful training and generation of new distinguished structures upon any given property values within 10% error of targeted ones. Finally, a potential application to inverse molecular design will be discussed.

ABSTRACT. Physically transparent and predictive models that quantify structure-property relationships of materials are of key importance in numerous fields, including chemistry, physics, and materials science. In this talk, I will discuss two machine learning applications with the goal of interpretability in mind (i.e., "glass box"). I will first present on the use of the Sure Independence Screening and Sparsifying Operator (SISSO) algorithm to find descriptors of perovskite stability. SISSO is shown to find a physically meaningful descriptor that predicts the stability of perovskite oxide and halide materials with superior performance compared to the well-known Goldschmidt tolerance factor. Second, I will present a collaborative work on the use of generalized additive models (GAMs) to clarify geometric structure-property relationships for chemisorption on alloys (e.g., O, OH, S, and Cl on Rh-, Pd-, Ag-, Ir-, Pt- and Au alloy surfaces). By comparing the GAM-derived chemisorption models to previously established electronic-structure models, we clarify the critical physical parameters that control the chemisorption process on metal surfaces.

ABSTRACT. We consider a fully discrete model for interface propagation in a random environment. The interface, given as the graph of a discrete function, may jump to the next site with a rate depending on the random force at the current site as well as the states of its neighbors and is zero if the sum of the local force and the discrete Laplacian at a site is non-positive. Pinning then occurs when the jump rate is zero for all points on the interface. We first consider the case where the pinning force's expectation vanishes (or is even positive). Our proof for pinning consists of the construction a supersolution employing a local path optimization procedure. A similar construction also leads to the phenomenon of infinite pinning if the pinning force has infinite second moment. This stands in contrast to the results for related continuum models, which require much fatter tails to obtain infinite pinning. This is joint work with Martin Jesenko (Ljubljana) and Michael Scheutzow (Berlin).

ABSTRACT. In this talk we discuss some results concerning the homogenization of solutions to a Stokes system in a domain having many small random holes. The main application for these models arises from the study of the flow of a viscous incompressible fluid through a porous medium or in the quasi-static regime of sedimenting particles. Our main focus is the rigorous derivation of the homogenization limit and the quantification of the convergence to the effective problem. In particular, we are interested in the case in which the holes in the domain under consideration are randomly distributed and may give rise to complicated geometries by overlapping and concentrating in certain regions of the domain.

ABSTRACT. We study the asymptotic behavior of free-discontinuity functionals defined on randomly perforated domains, as the size of the perforations goes to zero. To gain compactness for minimizing sequences we need to overcome the lack of equi-coercivity of the functionals by means of an extension result, under the assumption that the random perforations cannot come too close to one another. The effective limiting energy is then obtained by first applying a general stochastic homogenization result for free-discontinuity functionals to a perturbed coercive version of the energies and eventually by identifying the homogenized volume and surface integrands by means of a careful limit procedure.

ABSTRACT. We prove existence of uniformly bounded extension and trace operators on stationary randomly perforated domains. The extension operators are uniformly preserving the norm of the gradient and the symmetric gradient of vector valued functions. This leads to suitable Poincare and Korn inequalities on the perforated domains. In contrast to earlier work, we can deal with non-uniformly Lipschitz domains with infinitely connected holes.

ABSTRACT. The plasticity transition at the yield strength of a crystal typically signifies the tendency of dislocation defects towards relatively unrestricted motion. For an isolated dislocation the motion is in the slip plane with velocity proportional to the shear stress, while due to the long range interaction dislocation ensembles move towards satisfying emergent collective elastoplastic modes. Such collective motions have been discussed in terms of the elusively defined back-stress. In this talk, a two-dimensional stochastic continuum dislocation dynamics theory will be presented that clarifies the role of back-stress and demonstrates a precise agreement with the collective behavior of its discrete counterpart, as a function of applied load and with only three essential free parameters. The main ingredients of the continuum theory are the evolution equations of statistically stored and geometrically necessary dislocation densities, which are driven by the long-range internal stress, a stochastic flow stress term and, finally, two local diffusion-like terms. The agreement is shown primarily in terms of the patterning characteristics that include the formation of dipolar dislocation walls.

ABSTRACT. I will present an overview of our recent efforts to apply machine learning to study the problem of predicting the fluctuating plastic deformation process of small crystals containing dislocations. First, I will discuss the problem of predicting the stress-strain curves of individual samples from 2D discrete dislocation dynamics (DDD) simulations [1]. There, the key findings include non-monotonic strain dependence of predictability, as well as a size effect "larger is more predictable". Then, I will proceed to consider dislocation pileups interacting with frozen disorder, exhibiting a depinning phase transition [2]. Also in that case we find a non-monotonic dependence of predictability on strain. I will also discuss our attempts to predict the individual strain bursts in that case. Finally, I will describe our very recent results using data from strain-rate-controlled 2D DDD simulations. Specifically, we find that predictability of the deformation process exhibits rate dependence.

[1] H. Salmenjoki, M. J. Alava, and L. Laurson, Machine learning plastic deformation of crystals, Nat. Commun. 9, 5307 (2018).

[2] M. Sarvilahti, A. Skaugen, and L. Laurson, Machine learning depinning of dislocation pileups, APL Mater. 8, 101109 (2020).

ABSTRACT. Predicting features of mechanical deformation represents a fantastic challenge as it depends on a myriad of factors related to material microstructure, specimen geometry and loading conditions. Micromechanical approaches try to bridge microstructural features of solids with their mechanical response, but they are confronted to the high level of complexity of microstructures and microscopic damage processes involved in materials, especially in multicomponent crystalline materials, such as high entropy alloys. Recently, new approaches inspired from data science have emerged: the key microscale parameters that control mechanical deformation at large scale can be identified, and even learned, from the statistical analysis of a large amount of data. Statistical predictions range in a wide gamut of applications, from indentation of composites, to neural networks for understanding of elastic and plastic features of deformation in crystals. I will present a set of tools that utilize the dynamical stability of elasticity as an asset towards microstructural characterization and parameter-free multiscale modeling. Through this approach, I will demonstrate applications for synthetic data and will discuss possible experimental validation routes.

ABSTRACT. Atmospheric aerosol particles undergo complex phenonena that influence their movement, size and chemical composition. These particles grow due to coalescence, however, how they grow and how their composition changes is not well understood. We consider a simplified setting where particles are homogeneously distributed in space. The distribution of particle composition over time may be described by the multicomponent Smoluchowski's coagulation equation. We study existence and properties of mass-conserving solutions for a large class of kernels. Curiously, the mass tends to localize along a direction in the size space, forming a pattern for large times. This property is specific to the multicomponent setting and it occurs on all solutions for all kernels in the class considered.

ABSTRACT. Smoluchowski’s coagulation equation is a classical model for mass aggregation phenomena extensively used in the analysis of problems of polymerization, particle aggregation in aerosols, drop formation in rain and several other situations. In this talk I will present some recent results on the problem of existence or non-existence of stationary solutions to coagulation equations, for single and multi-component systems, under non-equilibrium conditions which are induced by the addition of a source term for small cluster sizes. The most striking feature of these stationary solutions is that, whenever they exist, the solutions to multi-component systems exhibit an unusual “spontaneous localization” phenomena. More precisely, the stationary solutions to the multi-component coagulation equation asymptotically localize into a direction determined by the source term. (Joint work with M.A. Ferreira, J. Lukkarinen and J.J.L. Velázquez)

ABSTRACT. We will show that solutions of Smoluchowski's classical coagulation model exhibit a universal self-similar long-time behaviour if the rate kernel is sufficiently close to the constant one. In particular, we will also provide explicit rates of convergence towards the scaling profile.

ABSTRACT. Exploiting connections between coagulation and random graphs I will present a large deviations result that explicitly includes microscopic and macroscopic (gel) particles as well as giving some information about intermediate scales. One sees that multiple gel particles are possible for finite large deviations cost, but that having only one gel particle is optimal. The initial results assume an initial particle population of identical monomers. However, I will also discuss the extension of the work to include heterogeneous initial particle distribution.

This is joint work with Luisa Andreis, Wolfgang Koenig and Heida Langhammer.

ABSTRACT. Starting from the theory of elastic plates, we derive a non-linear one-dimensional model for elastic ribbons with thickness t, width a and length L, assuming t<<a<<L. It takes the form of a rod model with a special non-linear constitutive law accounting for both the stretching and the bending of the ribbon mid-surface. The model is asymptotically correct and can handle finite rotations. Both Kirchhoff's linear beam theory and Sadowsky's inextensible ribbon model can be recovered as limit cases, for small (respectively, large) values of the bending and twisting strain. The lateral-torsional buckling of an elastic ribbon is solved as an illustration: an excellent agreement with both table-top experiments and finite-element shell simulations is obtained.

ABSTRACT. The bending behavior of plates is often described using dimensionally reduced models. We propose a practical method for the numerical simulation of bilayer plate bending that is based on a nonlinear two-dimensional plate model. Our method employs a discretization of the resulting energy using DKT (discrete Kirchhoff triangle) elements in space and a discrete gradient flow restricted to appropriate tangent spaces for the minimization of the energy functional. Particularly, we discuss the simulation of (self)-contact and the applicability of the method for simulating plates of nematic liquid crystal elastomers.

ABSTRACT. Imagine a 1D curve. Use it to fill a 2D manifold that covers an arbitrary 3D object. This computationally intensive materials challenge has been realized in the ancient technology known as knitting. This process for making functional materials 2D materials from 1D filaments dates back to prehistory. Knitted textiles are ubiquitous as they are easy and cheap to create, lightweight, portable, flexible and stretchy. As with many functional materials, the key to knitting’s extraordinary properties lies in its microstructure.

Knits are composed of an interlocking series of slip knots. There is only one manipulation that creates a knitted stitch – pulling a loop of yarn through another loop. However, there exist hundreds of books with thousands of patterns of stitches with seemingly unbounded complexity.

The topology of knitted stitches has a profound impact on the geometry and elasticity of the resulting fabric. This puts a new spin on additive manufacturing – not only can stitch pattern control the local and global geometry of a textile, but the creation process encodes mechanical properties within the material itself. Unlike additive manufacturing techniques, the innate properties of the stitch microstructure has a direct effect on the global geometry and mechanics of knitted fabrics.

ABSTRACT. The current pandemic has sparked a great deal of debate over the efficacy of cloth masks as an alternative to medical masks for the general public. In this talk we will discuss the mechanics of masks. Despite an abundance of experimental studies on the filtration of aerosols by cloth masks, we currently lack the capability to predict mask performance a priori. In this talk I will establish quantitative models for pressure drops across cloth masks and filtration efficiencies which include the mesoscale heterogeneities that naturally arise in woven textiles. This trade space between filtration efficacy and breathability is presented in a decision map to offer practical guidance for designing and/or purchasing cloth masks. Finally, the impact of the predicted mask efficiencies on the spread of the pandemic is quantified as a correction factor to the basic reproduction number, R0.

ABSTRACT. A key challenge to nonlocal models is the analytical complexity of deriving them from first principles, and frequently their use is justified a posteriori. In this work we extract nonlocal models from data, circumventing these challenges and providing data-driven justification for the resulting model form. Extracting data-driven surrogates is a major challenge for machine learning (ML) approaches, due to nonlinearities and lack of convexity - it is particularly challenging to extract surrogates which are provably well-posed and numerically stable. Our scheme not only yields a convex optimization problem, but also allows extraction of nonlocal models whose kernels may be partially negative while maintaining well-posedness even in small-data regimes. To achieve this, based on established nonlocal theory, we embed in our algorithm sufficient conditions on the non-positive part of the kernel that guarantee well-posedness of the learnt operator. These conditions are imposed as inequality constraints to meet the requisite conditions of the nonlocal theory. We demonstrate this workflow for a range of applications, including reproduction of manufactured nonlocal kernels; numerical homogenization of Darcy flow associated with a heterogeneous periodic microstructure; nonlocal approximation to high-order local transport phenomena; and approximation of globally supported fractional diffusion operators by truncated kernels.

ABSTRACT. We show that machine learning can improve the accuracy of simulations of stress waves in one-dimensional composite materials. We propose a data-driven technique to learn non-local constitutive laws for stress wave propagation models. The method is an optimization-based technique in which the nonlocal kernel function is approximated via Bernstein polynomials. The kernel, including both its functional form and parameters, is derived so that when used in a nonlocal solver, it generates solutions that closely match high-fidelity data. The optimal kernel, therefore, acts as a homogenized nonlocal continuum model that accurately reproduces wave motion ina smaller-scale, more detailed model that can include multiple materials. We apply this technique to wave propagation within a heterogeneous bar with a periodic microstructure. Several one-dimensional numerical tests illustrate the accuracy of our algorithm. The optimal kernel is demonstrated to reproduce high-fidelity data for composite material in applications that are substantially different from the problems used as training data.

ABSTRACT. As a nonlocal extension of continuum mechanics, peridynamics has been widely and effectively applied in different fields where discontinuities in the field variables arise from an initially continuous body. An important component of the constitutive model in peridynamics is the influence function which weights the contribution of all the interactions over a nonlocal region surrounding a point of interest. Recent work has shown that in solid mechanics the influence function has a strong relationship with the heterogeneity of a material's micro-structure. However, determining an accurate influence function analytically from a given micro-structure typically requires lengthy derivations and complex mathematical models. To avoid these complexities, the goal of this paper is to develop a data-driven regression algorithm to find the optimal bond-based peridynamic model to describe the macro-scale deformation of linear elastic medium with periodic heterogeneity. We generate macro-scale deformation training data by averaging over periodic micro-structure unit cells and add a physical energy constraint representing the homogenized elastic modulus of the micro-structure to the regression algorithm. We demonstrate this scheme for examples of one- and two-dimensional linear elastodynamics and show that the energy constraint improves the accuracy of the resulting peridynamic model.

ABSTRACT. We develop a physics-informed machine learning algorithm to infer the fractional order of derivatives in fractional differential equations. In particular, we use the variational physics-informed neural networks (VPINNs). The VPINN formulation constructs a general framework based on the nonlinear approximation of deep neural networks and projection onto the space of high-order polynomials. It provides the flexibility of reducing the order of differential operators via integration-by-parts. Here, we extend this formulation to fractional differential equations with distributed/variable order derivatives. Given the sparse space-time observations of dynamics, we discover the corresponding kernel of fractional derivatives in an inverse problem setting.

ABSTRACT. We study global minimizers of an energy functional arising as a thin sample limit in the theory of light-matter interaction in nematic liquid crystals. We show that depending on the parameters various defects are predicted by the model. In particular we show existence of a new type of topological defect which we call the shadow vortex. Finally, we discovered that at the boundary of the illuminated region, the profile of the minimizers is given by the universal equation of Painlevé.

ABSTRACT. The talk will discuss critical points of the Onsager functional for liquid crystals, giving in particular a simple proof of their axisymmetry in the case of the Maier-Saupe molecular interaction, a classical result of Fatkullin & Slastikov (2005) and Liu, Zhang & Zhang (2005). The proof avoids spherical polar coordinates, instead using an integral identity on the sphere. The connection with the singular potential of Majumdar and the author for the Maier-Saupe interaction is addressed. For general molecular interactions the smoothness of critical points is proved, and for a wide class of interactions the existence of non-axisymmetric critical points established.

ABSTRACT. Hardy and Poincaré-type inequalities are among the most frequently used inequalities in the analysis of PDE. The aim of this talk is to present a unified strategy to derive Hardy-Poincaré inequalities on bounded and unbounded domains. The approach allows proving a general Hary-Poincaré inequality from which the classical Poincaré and Hardy inequalities immediately follow. The argument also applies to the more general context of variable exponent Sobolev spaces. The argument, concise and constructive, does not require a priori knowledge of compactness results and returns geometric information on the optimal constants.

ABSTRACT. In my talk I will describe a geometrically nonlinear model for microstructure formation under bending in shape-memory alloys, restricted to a two-dimensional situation with a single rank-one connection. We show that the optimal scaling on the energy with Dirichlet boundary conditions is attained by a complex pattern with microstructure around both boundaries. Furthermore, if one prohibits the microstructure around the boundaries, then the optimal scaling law changes significantly. Both scaling laws are different from the one, previously obtained in the literature for multiple variants of the classic Kohn-Muller energy, which mimics the situation with two rank-one connections. This is a joint project with S.Conti and R.Kohn.