ABSTRACT. We consider a curve shortening equation of graphs with non-local mobility, related to the evolution of grain boundaries. First, I explain the derivation of a problem by the maximal dissipation principle for the grain boundary energy. To understand the relationship between misorientations and the effect of curvature, we impose the periodic boundary condition on the curve shortening equation. Next, we show the unique existence and long time asymptotics of a solution. I focus on constructing the so-called monotonicity formula for time-dependent mobility, a crucial ingredient of the proof. Finally, I discuss how to extend the results to the non-graphical case.

ABSTRACT. We investigate the existence and weak stability of global weak solutions for certain complex fluid models. These comprise a suitably adapted version of the Navier-Stokes equations and a harmonic map heat flow-like equation. The Ericksen-Leslie model for liquid crystals is the simpliest non-trivial system of such a type. A second, more general example represents the flow of ferromagnetic thin films and involves a variant of the Landau-Lifshitz-Gilbert equation. The construction of weak solutions relies on variants of the Ginzburg-Landau approximation where the main problem consists of the limit passage in the Navier-Stokes equation. This issue is handled by invoking partial regularity techniques and the method of concentration-cancellation originally introduced for the incompressible Euler equations.

ABSTRACT. In this talk, we are concerned with a class of singular perturbation problems for systems of PDEs related to the dynamics of geophysical fluids. More precisely, we consider viscous weakly compressible flows, which undergo the action of a strong Coriolis force. We are interested in performing the incompressible and fast rotation limits simultaneously, allowing the Mach and Rossby numbers to have different orders of magnitude, and in understanding the asymptotic dynamics.

We will show how weak compactness methods allow to handle the multiple scales appearing in the system, and to prove convergence for a wide range of parameters. The key point is to exploit the algebraic structure of the system, and especially a compactness property hidden in the system of acoustic-Poincaré waves. The drawback of this technique is that it does not yield any rate of convergence of the solutions to the primitive system towards the solutions of the target problem, which in fact is (in some cases) underdetermined.

ABSTRACT. Nematic colloids are composite materials, obtained as mixtures of microparticles in a nematic liquid crystal host. The presence of the inclusions induces, by surface anchoring effects, a deformation in the arrangement of the nematic molecules. As a consequence, the dopant microparticles have a measurable effect on the macroscopic properties, even in the dilute regime. In this talk, we will discuss a homogenisation limit for a variational model of nematic colloids, based on the Landau-de Gennes theory. Using variational methods, we will identify an effective free energy for the composite material and we will prove convergence results for local minimisers in the dilute regime. The talk is based on a joint work with Arghir D. Zarnescu (BCAM, Bilbao, and Simion Stoilow Institute of the Romanian Academy, Bucharest).

ABSTRACT. Atom probe tomography (APT) provides a snapshot of the local atomic environment of highly disordered metallic alloys, such as high entropy alloys (HEAs). Classifying crystal lattice structures of an APT dataset is an important first step in understanding patterns and properties of crystalline materials. However, APT has two main drawbacks: an abundance of missing data and experimental noise. In this work, we specifically classify unit cells that are either body-centered cubic (BCC) or face-centered cubic (FCC). These two crystal structures are distinct when viewed through the lens of topology. More precisely, topological data analysis (TDA) extracts and summarizes essential topological properties of these lattice structures such as the empty space and connectedness. We build a Bayesian framework to compute posterior distributions of these topological summaries. Relying on these posterior distributions, we apply the Bayes factor based learning algorithm to classify the crystal lattice structures of a noisy and sparse materials dataset.

ABSTRACT. Rich functionalities of quantum and strongly correlated materials emerge from the interplay between the electronic, orbital, lattice, and spin degrees of freedom that often lead to complex structural and electronic phenomena spanning atomic to mesoscopic scales. Over the last decade, Scanning Transmission Electron Microscopy has emerged as a powerful quantitative probe of materials structure and functionality on the atomic level, providing high veracity information on local chemical bonding, composition, and symmetry breaking distortions. We aim to harness the power of machine learning methods to build a comprehensive picture of the chemistry and physics of quantum materials from these observations. In this presentation, I will illustrate the application of rotationally-invariant variational autoencoders (rVAE) towards the effective exploration of the chemical evolution of the system based on local structural changes, effectively discovering molecular building blocks and chemical reactions pathways in unsupervised manner. This approach is further extended to the engineering the latent space of the system by defining relevant discrete and continuous dimensions. I will further illustrate the extension of this approach in encoder-decoder architectures to establish the parsimonious structure-property relationships in complex materials on an example of plasmonic nanostructures.

ABSTRACT. We view a bond network's structure as a collection of local atomic environments that are sampled from some underlying probability distribution of such environments. In some sense this probability distribution generalizes the unit cell of a crystalline material to disordered materials. This framework provides a practical, computable method to compare local structures appearing in different materials.

In my talk, I describe this methodology, as well as applications to molecular dynamics simulations of silica glasses. This is joint work with Jeremy Mason and David Rodney.

ABSTRACT. In recent decades, a huge amount of effort has been invested in the development of tools for the consistent description of the microstructure of materials (micro-structure fingerprinting), because microstructure is a key determinant of properties. Having a mathematical descriptor of microstructure allows a rigorous approach for comparing/classifying the characteristics of different materials or it can be used for automation of various processes (e.g. microstructure screening or quality assurance) in industry and science.

Due to recent advances of Machine-Learning (ML) and Artificial Intelligence (AI) a variety of novel techniques for compressed representation of material microstructure have emerged. In the current work we demonstrate that model stacking together with statistical analysis provide a unifying framework for micro-structure fingerprinting, which encapsulate and generalize existing methods. Additionally, we propose a novel technique for the description of steel alloy microstructure which exploits scale-specific information. This can be important for characterizing materials and is missing from many existing methods which have been derived from generic scale invariant object recognition technology. Performance of the methods are examined on several publicly available datasets.

ABSTRACT. A variety of microstructured materials have been designed extensively in recent years to achieve enhanced mechanical properties, such as specific stiffness and strength, and to explore the distinct mechanical behaviors with unique functionalities as termed mechanical metamaterials. Periodic polyhedral units connected with each other belong to a class of mechanical metamaterials, whose mechanisms underpinning flexible frameworks provide remarkable mechanical capabilities including auxeticity given negative Poisson ratios. In this study, we present the design of orthotropic polyhedral structures with 3D connectivity and discuss their nonlinear auxetic deformation induced by the 3D rotation of internal polygonal components. We propose the periodic framework made up of stellated octahedral units, which are connected with eight tetrahedra per unit cell; the two adjacent polyhedra share their one edge and the other vertices are linked to those of the different edge-shared polyhedral units in the periodic fashion. Modeling the structure, the effective on-axis Poisson ratios, stress-strain curves and self-induced vibrations are analyzed. The bimode model with the two types of local rotational parameters of the polyhedral unit exhibits a rich range of elastic behaviors, e.g., the auxetic out-of-plane deformation with the in-plane negative Poisson ratio and the auxetic vibration behavior excited by a nearly zero-frequency mode.

ABSTRACT. Periodic structures allow a strictly geometric approach to auxetic deformations. We present a concise account of the basic principles and results of this theory. The structural understanding of periodic bar-and-joint frameworks with auxetic deformations leads to a systematic design methodology for meta-materials with auxetic behavior and we illustrate the procedure with several framework blueprints with one or few degrees of freedom.

ABSTRACT. The design of auxetic structures is often a theoretical exercise. One determines the degrees of freedom of a periodic framework or mechanism and uses this to calculate the deformation of the lattice. However, the goal of this research is to create physical structures with the prescribed property: lateral widening under tension and narrowing under compression. This means that the theoretical designs need to be translated into the physical world, a step that can often be more difficult than the initial design.

In this talk, I will show examples of this translation step. We will see how periodic frameworks and mechanisms can be converted into compliant structures and how we can measure their effective properties.

ABSTRACT. We show some new results about the variational linearization of finite elasticity, including the incompressible case and pure traction problems.

ABSTRACT. We address the study of the asymptotic behavior of variational problems for quasicrystalline composites. A key feature of these materials is the lack of translational symmetry, but instead exhibit self-similarity on large scales. Our main result concerns the characterization of the homogenized problem associated with highly oscillating integral energies whose integrand has a quasicrystalline structure in the fast variable and whose admissible vector fields are constrained to satisfy a system of linear partial differential equations. These differential constraints are a stand-in for physical constraints such as mechanical compatibility in linear elasticity or a div/curl constraint such as in Maxwell's equations of magnetostatics. We stress that this study lies in the framework of non-periodic homogenization. This talk is based on a work in collaboration with Irene Fonseca (CMU) and Raghavendra Venkatraman (CMU).

ABSTRACT. We present recent results on optimal energy scaling laws in micromagnetics where the effect of magnetostriction is severe. Joint work with Dick James and Vivekanand Dabade.

ABSTRACT. Some representation results for relaxed energies emerging in the modeling of polycrystals and composite materials will be presented.

ABSTRACT. Heart disease is the leading killer worldwide, responsible for about 30% of all deaths each year. The incidence of heart failure has remained persistently high due to maladaptive growth and remodelling (G&R). Mathematical modelling of cardiac growth has potential for providing effective, reliable, and consistent risk-stratification to patients. Volumetric growth has been widely used in modelling cardiac G&R in the past, while it has limitations to link with the pathological adaptation of different constituents. To overcome this, we adopt the constrained mixture theory framework for modelling cardiac G&R by treating myocyte and collagen separately in a beating human heart under various physiological and pathological conditions. In the developed approach, all constituents share the same total deformation gradient tensor from one configuration to another grown configuration, and a remodelling tensor is introduced to ensure the geometrical compatibility after G&R. Two scenarios have been studied with a human left ventricle, one is strain driven and the other one is stress driven. Our results show that the interplays of these individual G&R processes in myocytes and collagen have the potential to shed light on the biomechanical mechanism which is responsible for the transition from the compensation stage to the decompensation stage.

ABSTRACT. To sustain different biological functions, it is important for living cells to develop specific attachments with the extracellular matrix (ECM), such as focal adhesion.

Among the various contributions focusing on the mechanical aspects inherent in focal adhesion, only few attempts have been done to devise simplified one-dimensional models taking into account the mechanical stresses governing the cell-ECM interaction.

In the present study, we extend the one-dimensional model presented in [1] by including elasto-plastic remodelling. In particular, we employ a Perzyna-like model of plasticity [2], in which the evolution of the structural changes of focal adhesion is triggered by the stress state [2, 3, 4].

The numerical results show that remodelling influences, both qualitatively and quantitatively, the time and space distribution of the main mechanical quantities characterising focal adhesions.

This work is supported by the research project Integrated mechanobiology approaches for a precise medicine in cancer treatment (PRIN-20177TTP3S).

References

1. Cao, X., et al., Biophysical Journal, 109.9, 1807-1817. (2015)

2. Mićunović, M., Thermomechanics of viscoplasticity: Fundamentals and applications. Springer New York (2009).

3. Simo J.C., Hughes T. J. H., Computational Inelasticity. Springer Science & Business Media (1988).

4. Grillo, A., et al., Continuum Mechanics and Thermodynamics, 28, 579-601 (2016).

ABSTRACT. Natural and synthetic matrices constitute the skeleton of biological tissues. Transport in this kind of porous materials is generally described directly at the macroscale, by means of advection-diffusion-reaction equations. However, such level of description might neglect coupling effects that arise from the interactions between the solute and the solid skeleton. In this work, we derive the governing equations for the transport of an osmolyte in a poroelastic material characterized by pores exhibiting two different characteristic scales. The solute is sterically excluded by the smallest pores, and exerts an osmotic pressure on the solid scaffold. We start from the pore-scale and describe the transient solute transport in a Newtonian fluid coupled to a saturated poroelastic material. Upscaling to the macroscale is performed via asymptotic homogenization and provides a system of governing equations for the pore pressure, osmolyte concentration and solid displacement. These equations are strongly coupled and highlight the presence of a mechanically-induced solute transport mechanism.

ABSTRACT. The modelling of poroelastic media under finite deformation is usually carried out via phenomenological models neglecting complex micro-macro scales interdependency. One reason is that the mathematical two-scale analysis is only straightforward assuming infinitesimal strain theory. Exploiting the potential of ANNs for fast and reliable upscaling and localisation procedures, we propose an incremental numerical approach for the analysis of poroelastic media under finite deformation considering infinitesimal strain in each time increment. The brain tissue mechanical response under uniaxial cyclic test is simulated and studied.

ABSTRACT. Technological interest in nematic liquid crystal (nematics) thin films and nematic droplets includes their applications for liquid crystal display manufacturing (Stewart 2004 and Cousins et al. 2019, 2020) and new emerging technologies, such as microfluidics (Sengupta et al. 2013), adaptive lenses (Algorri et al. 2019) and microelectronics (Zou 2018). These situations often involve a solid-nematic interface between a solid substrate and the nematic, a nematic-gas interface (i.e., a free surface) between the nematic and a surrounding atmosphere, and a three-phase contact line. Using constrained energy minimisation, we formulate the full governing equations for a sessile two-dimensional ridge of nematic. This system allows us to further the understanding of nematic molecular orientation at the three-phase contact line and the interaction of the free surface, nematic elasticity, and interfacial molecular alignment forces within similar systems. Our findings allow us to construct parameter planes in terms of the nematic material parameters, which describe the wetting behaviour, the molecular orientation at the three-phase contact line and the possibility of multiple solutions for the free surface height.

ABSTRACT. Thin films of nematic liquid crystal (NLC) find widespread industrial use, in applications ranging from liquid crystal display devices to liquid lenses and optical shutters. Understanding how such films spread and flow is therefore important from an industrial perspective as well as being of fundamental scientific interest. We will describe how asymptotic methods (lubrication theory) can be applied to derive a simplified model for the free surface evolution of NLC films in a number of different settings. Of particular importance for film behavior is the orientation of the NLC molecules, both within the bulk film and at interfaces. The former is dictated by elastic effects and by the presence of applied external fields such as an electric field; while the latter depends primarily on interactions of the NLC with the adjacent material (a phenomenon known as anchoring). We will present simulations of our model that illustrate film behavior both without (dewetting) and with (dielectrowetting) an applied electric field, showing good qualitative agreement with available experimental data.

ABSTRACT. We propose a phase-field model to study interfacial flows of nematic liquid crystals that couple the surface tension with the elastic stresses in the nematic phase. The theoretical model has two key ingredients, a tensor order parameter that provides a consistent description of the molecular and distortional elasticity, and a phase-field formalism that represents the interfacial tension and the nematic anchoring stress by approximating a sharp-interface limit. Using this model, we carry out finite-element simulations of drop retraction in a surrounding fluid, with either component being nematic. The results are summarized by eight representative steady-state solutions in planar and axisymmetric geometries, each featuring a distinct configuration for the drop and the defects. The dynamics is dominated by the competition between the interfacial tension and the distortional elasticity in the nematic phase, mediated by the anchoring condition on the drop surface. As consequences of this competition, the steady-state drop deformation depends linearly on the elasto-capillary number.

ABSTRACT. We present a new mechanism whereby spatially varying order in a thin liquid crystal film, induced by chemical or topographic patterning on a substrate, can lead to spontaneous topography on an opposing free interface. Analytical and numerical results will be shown to elucidate this mechanism, together with recent experimental evidence and connections to other free interface problems.

ABSTRACT. Phase field modeling provides an extremely general framework for predicting microstructural evolution in complex systems. However, its numerical implementation requires a discretization grid with a small enough grid spacing to preserve the diffuse character of the theory. Recently, a new formulation, called S-PFM, in which interfaces are resolved with essentially only one grid point, has been proposed [A. Finel, et al. , Phys.Rev.Lett. 121 (2018) 025501]. This formulation significantly improves the numerical performance of this type of approaches. The work proposed here is a first extension of the S-PFM approach to grain growth. Specifically, a multi-field S-PFM model was developed to study normal grain growth in isotropic systems. In addition, a dynamic reallocation algorithm has been developed to limit the number of fields necessary to describe the microstructure of a polycrystalline material. In a first step, we will compare the predictions and possibilities of this model with those resulting from the phase field model with diffuse interfaces proposed in [L.Q. Chen, et al. Phys. Rev. B, 50 (1994) 15752]. Then, we will use our model to perform an accurate statistical study of normal grain growth, particularly on size and topology distributions [A. Dimokrati et al. Acta Materialia 201 (2020) 147].

ABSTRACT. Simulations can be used to measure the properties of interfaces in materials. The central role of quantitative phase field simulations in this effort is illustrated by a rapid throughput method to determine grain boundary properties. By comparing the evolution of experimentally determined three-dimensional grain structures to those derived from simulation, we measure the reduced mobilities of thousands of grain boundaries. Using a time step from the experiment as an initial condition in a phase-field simulation, the computed structure is compared to that measured experimentally at a later time. An optimization technique is then used to find the reduced grain boundary mobilities of over 1300 grain boundaries in iron that yield the best match to the simulated microstructure. We find that the grain boundary mobilities are largely independent of the five macroscopic degrees of freedom given by the misorientation of the grains and the inclination of the grain boundary. The challenge of developing quantitatively accurate phase field simulations of grain growth will be highlighted, with an emphasis on the methods that can account for the five degrees of freedom of the grain boundary energy

ABSTRACT. Orientation-field models describe a polycrystalline solid in terms of two continuous fields: a phase field, which distinguishes between liquid and solid, and an orientation field, which indicates the local orientation of the crystallographic axes with respect to a reference frame. In two dimensions, the orientation field is scalar and represents a single angle. The corresponding order parameter space is the unit circle, which is not simply connected. This topological property has important consequences. For each pair of grains there exist two different grain boundary solutions that cannot continuously transform into each other; if both solutions appear along a grain boundary, a topologically stable, singular point defect forms between them. These singularities cause lattice pinning in numerical simulations. To overcome these problems, we have developed two extensions of the model. They use a three-component unit vector field and a two-component vector field with an additional potential, respectively. In both cases, the additional degree of freedom makes the order parameter space simply connected, which removes the topological stability of these defects. We test our models in simulations of grain coarsening and find that all topological defects are rapidly eliminated. We also determine the limiting grain size distribution.

ABSTRACT. Shape-memory alloys display a very rich, non-quasiconvex energy landscape leading to the formation of a variety of microstructures. In the mathematical analysis of stress-free microstructures as differential inclusions a striking dichotomy arises: While surface energy constrained microstructures can often be shown to be ``rigid'' obeying (non-linear) hyperbolic compatibility equations, the unconstrained problem admits a plethora of ``wild'', fractal and highly non-unique convex integration solutions. In this talk I discuss recent results on this ``phase transition'' between the rigid and flexible regimes presenting both analytical and numerical observations. I will also connect the presence of wild solutions to ``wild'', irregular nucleation mechanisms.

ABSTRACT. Plasticity is the property of a material of undergoing permanent deformations in response to an external load. Plastic behavior is observed in most materials; however, the physical mechanisms that cause plastic deformation can vary widely. In metals, plasticity is the macroscopic consequence of the presence of defects at the microscopic scale, called dislocations. In soils, plasticity is caused by their granular structure: soils are made of many small solid particles, whose irreversible rearrangement produces permanent deformation. The understanding of the plastic mechanism at the microscopic scale and of its implications on the macroscopic response is a necessary step towards the formulation of predictive models of plasticity at the continuum level.In this talk I will discuss the mathematics of plasticity at different scales, ranging from its mechanisms at the microscopic scale to its macroscopic description.

ABSTRACT. We will present a derivation and generalization of the mass action kinetics of chemical reactions using an energetic variational approach. The dynamics of the system is determined through the choice of the free energy, the dissipation (the entropy production), as well as the kimenatics (conservation of species). The method enables us to capture the coupling and competition of various mechanisms, including mechanical effects such as diffusion, viscoelasticity in polymerical fluids and muscle contraction, as well as the thermal effects. We will also discuss several applications such as the modeling of wormlike micellar solutions. This is a joint work with Bob Eisenberg, Pei Liu, Yiwei Wang and Tengfei Zhang.

ABSTRACT. Dyadic models for the magnetohydrodynamics are derived in the way intermittency dimension enters the modeling naturally as a parameter. For models representing different energy cascade scenarios, finite time blow-up can occur for positive solutions when the intermittency dimension is below certain threshold.

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

ABSTRACT. In this talk, we present a systematic framework of deriving variational numerical schemes for generalized diffusions and gradient flows. The proposed numerical framework is based on the energy-dissipation law, which describe all the physics and the assumptions in a given system, and can combine different types of spatial discretizations including both Eulerian and Lagrangian approaches. The resulting semi-discrete equation inherits the variational structures from the continuous energy-dissipation law. As examples, we apply such an approach to construct variational Lagrangian schemes to porous medium type generalized diffusions and Allen-Cahn type phase-field models. Numerical examples show the advantages of variational Lagrangian schemes in capturing singularities, thin diffuse interfaces, and free boundaries.

ABSTRACT. I will present techniques to find reaction coordinates to be used in conjunction with free energy biasing techniques such as the adaptive biasing force method. This allows for instance to improve the sampling of configurations of complex proteins. However, reaction coordinates are often based on an intuitive understanding of the system, and one would like to complement this intuition or even replace it with automated tools. One appealing tool is autoencoders, for which the bottleneck layer provides a low dimensional representation of high dimensional atomistic systems. I will discuss some mathematical foundations of this method, and present illustrative applications including alanine dipeptide. Some on-going extensions to more demanding systems, namely HSP90, will also be mentioned.

ABSTRACT. Machine learning models are proving to be extremely effective in predicting the properties of atomistic configurations of matter, circumventing the need for time consuming electronic structure calculations. The most successful schemes achieve transferability by means of a local representation of structures, in which the problem of predicting a property is broken down into the prediction of local, atom-centred contributions. This approach is however not efficient in describing long-range interatomic forces, such as those arising due to electrostatics. I will present a possible solution to this conundrum based on the long-distance equivariant (LODE) framework, that combines a local description of matter with the appropriate, long-range asymptotic behaviour of interactions.

ABSTRACT. The cubic form of garnet-type electrolyte Li7La3Zr2O12 (LLZO) is of great interest for all-solid-state Li-ion batteries due to improved energy density and safety over traditional liquid electrolytes. Its performance is stabilized at room temperature by aliovalent substitution of Li+ by Al+3 or Ga+3. The high computational time required in the selection of adequate structures prior to large-scale atomistic simulations can be reduced by using efficient optimization techniques for screening energetically stable structures from a pool of LLZO garnets with Li vacancies and Ga substitutions. With this purpose, we have proposed an approach that converts a high-dimensional optimization problem into a reduced discrete optimization model with the pre-computed component of a score function for fast calculations. Two meta-heuristic algorithms, Simulated Annealing (trajectory-based) and Harmonic Search (population-based) have been implemented and demonstrated that our method can achieve the ground state with a high probability of success. Several features are introduced to enhance convergence and reduce computational time. This includes various cooling schemes, temperature length adaptation, modified Metropolis-based criteria, adaptive termination, and the neighbour algorithms. Our approach demonstrates a significant improvement in performance (~1000 times faster) and accuracy (at least twice more accurate) over the random search algorithm previously proposed.

ABSTRACT. Atomic probe tomography (APT) is an experimental technique to interrogate the atomic structure of materials by resolving the elemental identity and spatial distribution of millions of atoms in a sample. APT data sets are subject to well-known data sparsity and noise issues. Thus the current attempt to extract the maximum amount of information from an APT data set has become a data analysis issue. Particularly, establishing clear indications of atomic ordering on the Ångstrom scale remains a challenge. In this work a computational approach to evaluate atomic ordering by generating radial distribution functions (RDF) from APT data sets is developed. Atomic ordering is rendered as the Fractional Cumulative Radial Distribution Function (FCRDF) which allows for greater visibility of local compositions from short to medium range in the structure. An atomic ordering metric is proposed. A parametric sensitivity analysis of the ability to identify atomic ordering with respect to data sparsity and noise is performed. We investigate APT experiment of two complex alloys, Ni3Al, which contains known atomic ordering, and the high entropy alloy, Al1.3CoCrCuFeNi, for which a complete understanding of atomic ordering remains unknown.

ABSTRACT. Encoding the complex features of an energy landscape is a challenging task, and often chemists pursue the most salient features (minima and barriers) along a highly reduced space, i.e. 2- or 3-dimensions. Even though disconnectivity graphs or merge trees summarize the connectivity of the local minima of an energy landscape via the lowest-barrier pathways, there is more information to be gained by also considering the topology of each connected component at different energy thresholds (or sublevelsets). We propose sublevelset persistent homology as an appropriate tool for this purpose. Our computations on the configuration phase space of n-alkanes from butane to octane allow us to conjecture, and then prove, a complete characterization of the sublevelset persistent homology of the alkane C_m H_{2m+2} potential energy landscapes, for all m, and in all homological dimensions. We further compare both the analytical configurational potential energy landscapes and sampled data from molecular dynamics simulation, using the united and all-atom descriptions of the intramolecular interactions. Joint work with Joshua Mirth, Yanqin Zhai, Johnathan Bush, Enrique Alvarado, Howie Jordan, Mark Heim, Bala Krishnamoorthy, Markus Pflaum, Aurora Clark, Yang Zang for DELTA, an NSF Harnessing the Data Revolution project.

ABSTRACT. Traditionally, neural networks are parameterized using optimization procedures such as stochastic gradient descent, RMSProp and ADAM which drive the parameters of the network toward a local minimum. In this article, we employ alternative ``sampling'' algorithms (referred to here as ``thermodynamic parameterization methods'') which rely on discretized stochastic differential equations for a defined target distribution on parameter space. We show that the thermodynamic perspective improves neural network training. Moreover, by partitioning the parameters based on natural layer structure we obtain schemes with rapid convergence for data sets with complicated loss landscapes. This is joint work with Tiffany Vlaar and Charlie Matthews.

ABSTRACT. Materials science researchers are able to investigate the local structure of any material through a process called atom probe tomography (APT). This process yields the elemental type and spatial coordinates of each successfully detected atom, but does not yield small-scale elemental ordering nor lattice structure. Moreover, each successfully detected point has its coordinates perturbed by some experimental noise. I will present work incorporating a novel Bayesian framework that yields a snapshot of the atomic- level structure present in these novel alloys. By formulating a posterior density as a Gaussian Mixture Model, we are able to recover distributions of the lattice spacing and chemical ordering present in a material. Motivated by an inquiry into the structure of High-Entropy Alloys, we focus on noisy and sparse observations of affine transformations of a known lattice structure. We will present examples on both synthetic and real APT data sets. This is joint work with Vasileios Maroulas, David Keffer, and Kody J. H. Law.

ABSTRACT. Critical mechanical structures are structures that are on the verge of mechanical instability, and they offer fascinating properties such as auxetic mechanical response and topologically protected zero modes and states of self stress, all stemming from their proximity to instability. In this talk, we will discuss our recent studies on these critical mechanical structures, unveiling interesting relations between auxeticity and topological mechanics, and discuss their applications as transformable topological mechanical metamaterials.

ABSTRACT. Auxetics are a type of metamaterial — a conventional material that has been patterned at an intermediate length scale to generate bulk material properties — that expand in all directions when exposed to uniaxial strain. Two dimensional auxetics have been studied for several decades in systems, from origami to 3D prints to linkages. Here, we propose two design schemes for realizable 3D printed three-dimensional auxetics based upon hinges. The first is a set of Lego-like interchangeable modules that combine counterrotating regular polygons with scissor mechanisms. This is guaranteed to have a one dimensional configuration space because it can be reduced to series of one degree of freedom Sarrus linkages. The second mechanism is based upon a branched cover over the simple two arm scissor mechanism which adds stability and enables force to be transferred around corners.

ABSTRACT. Periodic pseudo-triangulations are a remarkable class of planar flexible frameworks with one degree of freedom. They provide the quintessential model for expansive behavior in 2D. Since expansive implies auxetic, we obtain an infinite catalog of auxetic frameworks. We present several examples which illustrate the versatility of this approach to auxetic design.

ABSTRACT. Peridynamic models have successfully predicted fractures and deformations for a variety of materials. In this talk, I will present some results on the convergence of solutions to a nonlocal state-based linear elastic model to their local counterparts as the interaction horizon vanishes. Our results provide explicit rates of convergence that are sensitive to the compatibility of the nonlocal boundary data and the extension of the solution for the local model.

ABSTRACT. Nonlocal continuum models are in general integro-differential equations in place of the conventional partial differential equations. While nonlocal models show their effectiveness in modeling a number of anomalous and singular processes in physics and material sciences, for example, the peridynamics model of fracture mechanics, they also come with increased difficulty in computation with nonlocality involved. Aiming at both rigorous numerical analysis and computational efficiency, we propose the reproducing kernel (RK) collocation methods, a class of meshfree methods, for approximating nonlocal models characterized by a length parameter that may change with the models. We present asymptotically compatible RK collocation method for nonlocal diffusion and nonlocal mechanics models with Dirichlet boundary condition. The scheme is shown to be convergent to both nonlocal models and their corresponding local limits as nonlocal interaction vanishes. The key idea for the stability of the RK collocation scheme is to compare the collocation scheme with the standard Galerkin scheme which is stable. This is a joint work with Yu Leng, Nat Trask and John Foster.

ABSTRACT. We study the dispersion relations of two consistent nonlocal-to-local coupling methods in 1D: (1) the quasi-nonlocal coupling and (2) the blending-based coupling. We analyze the dispersion relations and the reflection coefficients on the transitional regions for both methods, respectively. Both coupling methods are rigorously proved that the imaginary artifacts of their dispersion relations due to the coupling will disappear with first order speed as the horizon size goes to zero. In addition, we employ a finite difference numerical discretizations and find that the artificial imaginary parts of the dispersion relation and the reflection coefficients of the blending-based coupling are all cases smaller than those of the quasi-nonlocal coupling on the transitional region, hence, the waves are preserved better by the blending-based method. Several numerical tests are performed to confirm these theoretical findings. This is a joint work with Hayden Pecoraro, Kelsey Wells and Dr. Pablo Seleson.

ABSTRACT. We analyze regularity estimates for solutions to nonlocal space time equations driven by fractional powers of parabolic operators in divergence form. These equations are fundamental in semi-permeable membrane problems, biological invasion models and they also appear as generalized Master equations. We develop a parabolic method of semigroups that allows us to prove a local extension problem characterization for these nonlocal problems. As a consequence, we obtain interior and boundary Harnack inequalities and sharp interior and global parabolic Schauder estimates for solutions. For the latter, we also prove a characterization of the correct intermediate parabolic Holder spaces in the spirit of Sergio Campanato.

ABSTRACT. We study relaxation to a flat interface for the Mullins-Sekerka evolution in the plane. The method combines algebraic and differential information linking the energy, dissipation, and distance to the flat interface. An algebraic rate of convergence is obtained, which is optimal under our assumption on the initial data.

ABSTRACT. The theory of first order structured deformations introduced in [2] enriches the purely macroscopic field theory of non-linear elasticity by taking into account the effects of dissarangements that occur at a single submacroscopic level. Nonetheless, many natural and man-made physical systems have a rich enough geometrical structure to permit the identification of hierarchies consisting of more than one physically meaningful submacroscopic level. In [3], the field theory of hierarchies of structured deformations was developed. In this work we extend the variational model of structured deformations treated in [1] to hierachies of structured deformations. References [1] R. Choksi and I. Fonseca: Bulk and interfacial energy densities for structured deformations of continua. Arch. Rational Mech. Anal. 138 (1997), 37-103. [2] G. Del Piero and D. -R. Owen: Structured deformations of continua: Arch. Ration. Mech. Analysis -124 (1993), 99?155. 18 L. C. Evans and R. F. Gariepy. [3] L. Deseri and D. R. Owen: Elasticity with Hierarchical Disarrangements: A Field Theory That Admits Slips and Separations at Multiple Submacroscopic Levels: J.of Elasticity Vol- 135 (2019) Issue 1?2, 149?18.

ABSTRACT. We consider different phase field approximations of the elastica or Willmore energy that modify the well-known approach based on the Cahn-Hilliard phase separation energy. For a `gradient free' Willmore approximation we present an asymptotic analysis of the respective energies and gradient flows. For another higher-order approximation we prove the Gamma convergence of energies.

ABSTRACT. Microstructures in martensites are often modeled variationally by singularly perturbed multiwell elastic energies. In this talk, I shall disusss recent analytical results on the associated energy minimisation problems, including in particular geometrically linearized elasticity models for the two-well case with small volume fraction of one martensitic variant. This is partly based on joint work with S. Cont, J. Diermeier, and D. Melching.

ABSTRACT. Viscoelastic homogenization in the time-domain play a vital role in the multiscale simulations. Effective viscoelastic behavior obtained from incremental variational based mean-field homogenization (MFH) is compared with the behavior of regular periodic microstructures evaluated using Finite Element. Two different elastic bounds-based homogenization methods are coupled with MFH scheme. Relaxation in the effective modulus of microstructures of unidirectional (UD) fibre-reinforced polymer (FRP) (i.e., short, and long fiber) composite and particulate composites are plotted as a function of direction and time for comparisons. Face centered cubic (FCC), body centered cubic (BCC) and simple cubic (SC) microstructures are considered for particulate composites. MFH compared well with the full-field solution of FCC arrangement. Additionally, FCC and BCC arrangement indicated nearly the same response with different local field statistics. In the case of UD FRP composites, MFH solution compared well with the hexagonal arrangement of long fiber and staggered arrangement of ellipsoidal short fibers. Double inclusion MFH scheme coupled with the incremental variational approach indicated better comparisons with the full-field solution of UD FRP composites.

ABSTRACT. In nature, we typically found that biological materials are characterized by a viscoelastic behavior. These structures have combined a variety of their constituents to form reinforced structures with hierarchical arrangement. Recent contributions have addressed the multi-scale modeling of viscoelastic composites. In this regard, micromechanical models are particularly used when the aim is to determine the effective constitutive response at the macroscale based on the information available at the smaller scales. In the present work, we focus on the calculation of the effective properties of non-aging linear viscoelastic and hierarchical composite materials. We consider the elastic-viscoelastic correspondence principle and the Laplace-Carson transform, and we apply the three-scale asymptotic homogenization method (AHM). We exploit the potential of the approach and study the overall properties of biological tissues with hierarchical structure and viscoelastic mechanical response.

ABSTRACT. Periodic egg carton surfaces are universal patterns that can be found in nature such as in exoskeletons and flower petals. The multiscale nanowrinkling patterns play an important role in material multifunctionalities including structural colour and superhydrophobicity. We propose a novel linear method derived from the anisotropic surface anchoring model, which shows that a linear model is sufficient for an anisotropic patch to generate an egg carton surface by using the principle of superposition. In the novel linear method, the liquid crystal director effect is compensated by local surface dilation, which causes an egg carton pattern in geometry. The amplitude of the egg carton surface depends on the wrinkling direction which determines the nature of the shape equation found to be an elliptic PDE. The egg-carton amplitude reaches the maximum when the elliptic PDE reduces to a Helmholtz equation with the eigenvalue only dependent on the helicity of the chiral liquid crystal. The relation between mean and Gaussian curvature distributions and chirality is established. In the end, previous experimental data validate our theoretical computation and they contribute to the design and application of bioinspired materials.

ABSTRACT. From weak nematic phases to highly ordered columnar phases, liquid crystals (LCs) teem biological systems. Spanning medical, environmental and bio-inspired systems, LC phases underpin transport properties across length and time scales, ultimately regulating the dynamics and functions in living matter. Outside living systems, synthetic LC phases offer experimentally tractable systems whereby the microscale transport laws can be precisely understood. Liquid crystal microfluidics [1] has been instrumental in driving recent advancements in this field, yet the process of translating the fundamental transport laws to possible biological functions is still at its infancy. In this talk, I will discuss our efforts to bridge this gap using two specific examples: flow of lyotropic phases and transport in bacterial populations, highlighting the similarities and differences therein [2,3]. I will conclude with a perspective on liquid crystal microfluidics, specifically in biological parlance, touching upon the key challenges (and opportunities!) in exploring open questions in biology and translational medicine.

References: [1] Liquid crystal microfluidics: Sengupta et al. Liquid Crystals Reviews 2, 73, 2014 [2] Microbial Active Matter: A Topological Framework: Sengupta, Frontiers in Physics 8, 184, 2020 [3] Time dependent lyotropic chromonic textures in microfluidic confinements: Sharma, Ong, Sengupta, Crystals 11, 35, 2021

ABSTRACT. We study the dynamics of a model microswimmer, a squirmer, in a rheologically complex fluid, namely a nematic liquid crystal. We focus on the importance of anisotropy and elasticity of the suspending fluid in geometrical confinement towards the dynamics of an individual microswimmer. We employ our recently developed mesoscopic simulation method for nematic liquid crystals to simulate the squirmer dynamics which captures both hydrodynamics and thermal fluctuations. A squirmer in a homeotropically aligned nematic liquid crystal cell shows remarkable differences as compared to squirmer dynamics in Newtonian fluids. The squirmer trajectories depend strongly on the self-propulsion mechanism, self-propulsion strength, and degree of confinement. We have obtained three distinct types of behaviour: (i) steady swimming along the channel centreline for pullers, (ii) steady hovering near a wall for strong pushers, and (iii) oscillating motion for weak pushers. The steady hovering state of strong pushers near a wall has been found in recent experiments. We identify that anisotropic viscosity-induced hydrodynamic torques and wall-induced elastic repulsive forces are the key effects which govern the squirmer dynamics. We propose an analytical model that qualitatively reproduces all the dynamical regimes found in simulations.

ABSTRACT. Chiral liquid crystals exhibit various ordered structures arising from twisted or helical orientational ordering. Here we numerically investigate the structures of a chiral liquid crystal sandwiched by two parallel substrates. Our study is based on the Landau-de Gennes continuum theory describing the orientational order by a second-rank tensor. We show that diverse ordered structures are observed, depending on temperature, thickness of the liquid crystal, and type of surface anchoring. These structures include a hexagonal lattice of Skyrmions, a swirl-like structure found in diverse condensed matter systems. We also discuss their optical properties, paying particular attention to Kossel diagrams that visualize the directions of strongly reflected monochromatic light when focused light is incident onto the sample. Experimental Kossel diagrams are accounted for by numerical and analytical arguments.

ABSTRACT. Novel 2D materials have unusual properties, many of which are coupled to their large scale mechanical and structural properties. Modeling is a formidable challenge due to a wide span of length and time scales. I will review recent progress in structural multiscale modeling of 2D materials, including graphene [1], h-BN [2], and their heterostructures [3] based on the Phase Field Crystal (PFC) model combined with Molecular Dynamics and Quantum Density Functional Theory. The PFC model allows one to reach diffusive time scales at the atomic scale, which facilitates quantitative characterization of domain walls, dislocations, grain boundaries, and strain-driven self-organization up to micron length scales. This allows one to study e.g. thermal conduction and electrical transport in realistic multi-grain systems [2,4]. 1. P. Hirvonen et al., PRB 94, 035414 (2016). 2. H. Dong et al., CPPC 20, 4263 (2018). 3. P. Hirvonen et al., PRB 100, 165412 (2019). 4. Z. Fan et al., PRB 95, 144309 (2017); Nano Lett. 7b172 (2017); K. Azizi et al., Carbon 125, 384 (2017); K. Xu et al., PRB 99, 054303 (2019); K. Xu et al., PRB 99, 054303 (2019); Z. Fan et al., PRB 99, 064308 (2019); Z. Li et al., JCP (2019).

ABSTRACT. In this paper a two dimensional phase field crystal model of graphene and hexagonal boron nitride (hBN) is extended to include out of plane deformations in stacked multi-layer systems. As proof of principle the model is shown analytically to reduce to standard models of flexible sheets in the small deformation limit. Applications to strained sheets, dislocation dipoles and grain boundaries are used to validate the behavior of a single flexible graphene layer. For the multi-layer systems, parameters are obtained to match existing theoretical density functional theory calculations for graphene/graphene, hBN/hBN and graphene/hBN bilayers. More precisely it is shown that the parameters can be chosen to closely match the stacking energies and layer spacing calculated by Zhou et. al. (Phys. Rev. B, 92, 155438 (2015)). Further validation of the model is presented in a study of rotated graphene bilayers and stacking boundaries. The flexibility of the model is illustrated by simulations that highlight the impact of a complex microstructures in one layer on the other layer in a graphene/graphene bilayer.

ABSTRACT. We present a mesoscopic description of the dynamics of dislocations in binary alloys. The velocity of dislocations is derived analytically to incorporate and predict the effects induced by the preferential solute segregation and Cottrell atmospheres in two-dimensional and three-dimensional binary systems of various crystalline symmetries. The work is based on the description of defect dynamics obtained through the amplitude formulation of the phase-field crystal model (APFC). Modifications of the Peach-Koehler force resulting from solute concentration variations and compositional stresses are presented, leading to interesting new predictions of defect motion due to the effects of Cottrell atmospheres. These include the deflection of dislocation glide paths, the variation of climb speed and direction, and the change or prevention of defect annihilation. Numerical APFC simulations verify the analytic results and tackle the investigation of defect motion in binary systems for complex dislocation networks.

ABSTRACT. The purpose of the current work is a comparative modeling and experimental investigation of the interaction between defects and chemistry in metallic alloy systems at the nanoscopic scale. On the modeling side, three approaches are employed and compared, i.e., hybrid Monte-Carlo molecular dynamics, diffuse molecular dynamics [1], and atomistic phase-field chemomechanics [2]. All three are energy-based approaches. In the first two, the energy model is based on an interatomic potential. For comparability, the same potential is used to calibrate the phase-field energy model. On this basis, the three methods are applied in the current work to the modeling of solute segregation to dislocations in the binary system Pt-Au and compared with analogous experimental results from atom probe tomography and TEM. In particular, results for the effect of solute segregation on the dislocation, as well as maximum segregation to the dislocation at different bulk compositions, are investigated.

[1] J. Mendez, M. Ponga. MXE: A package for simulating long-term diffusive mass transport phenomena in nanoscale systems. arXiv:1910.01235v1 (2019). [2] J. R. Mianroodi, P. Shanthraj, P. Kontis, J. Cormier, B. Gault, B. Svendsen, D. Raabe. Atomistic phase field chemomechanical modeling of solute segregation and dislocation-precipitate interaction in Ni-Al-Co. Acta Materialia, 175, 1–30 (2019).

ABSTRACT. Self-folding structures, especially origami, are promising as means of creating 3D shapes from a flat, patterned structures, yet real systems suffer from the propensity to mis-fold. This mis-folding arises because flat, folding structures are at a critical point in the energy leading to proliferation of competing energy minima. These critical points exist generically in soft mechanisms and lead to a number of surprising phenomena. In this talk, I will discuss the rigidity of soft mechanisms at critical points and how to use them control folding.

ABSTRACT. We study periodic arrays of impurities that create localized regions of expansion, embedded in two-dimensional crystalline membranes. These arrays provide a simple elastic model of shape memory. As the size of each impurity increases (or the relative cost of bending to stretching decreases), it becomes energetically favorable for the impurities to buckle up or down into the third dimension, allowing for a vast number of metastable states. Guided by an analogy to the Ising antiferromagnet, we characterize the buckling transition and conjecture the lowest energy states of the membrane with discrete simulations and the nonlinear continuum theory of elastic plates. We then consider the influence of thermal fluctuations, and find that the system undergoes a structural phase transition characterized by an anomalous thermal expansion.

ABSTRACT. The fundamental topology of cellular structures - the location, number, and connectivity of nodes and compartments - can profoundly impact their acoustic, electrical, chemical, mechanical, and optical properties, as well as heat, fluid and particle transport. Here we introduce a two-tiered dynamic strategy to achieve systematic reversible transformations of the fundamental topology of cellular microstructures that can be applied to a wide range of material compositions and geometries. Our approach only requires exposing the structure to a liquid whose composition is selected to have the ability to first infiltrate and soften the material at the molecular scale, and then, upon evaporation, to form a network of localized capillary forces at the architectural scale that zip the edges of the softened lattice into a new topological structure, which subsequently re-stiffens and remains kinetically trapped. We then harness dynamic topologies for developing active surfaces with information encryption, selective particle trapping, and tunable mechanical, chemical and acoustic properties, as well as multi-stimuli actuation.

ABSTRACT. Classical rod theories account for the stretching, bending and twisting strains of slender structures in a linear way and do not represent finite-strains. Understanding finite-thickness effects arising during the compression of wide columns, predicting the emergence of shape due to heterogeneous pre-stress generated by growth or thermal effects or designing structures made of complex nonlinear materials therefore remain challenging tasks. The example of localization highlights these limitations: from necks in polymer bars to bulges in cylindrical balloons and folds in tape-springs, classical one-dimensional (1D) models fail to describe interfaces arising between distinct states of deformation.

I will introduce a systematic method to establish 1D models for non-linear, slender elastic structures starting from the three-dimensional description of a hyper-elastic prismatic solid. Using a formal asymptotic expansion performed near a finitely pre-strained state, I will derive 1D models that account for stretching, bending and twisting modes in a non-linear way. The resulting models retain sources of nonlinearity coming from the geometry and the constitutive law, include higher-order terms depending on strain gradient and thus accurately capture interfaces during localisation. I will illustrate the method on elasto-capillary necking and discuss its application to a wider range of structures.

ABSTRACT. Material properties at nanoscale exhibit size-dependent behavior which can be attributed to the influence of surfaces and interfaces on the properties of materials. Due to this, contact problems at nanoscale have to take into account surface energy. In this talk, we consider an isotropic half-space subjected to nanoscale contact with a rigid punch. The surface energy in the Steigmann-Ogden form is used to model the surface of the half-space while linear elasticity is used to model the bulk of the material. The nanoindentation problem is solved using Boussinesq’s displacement potentials and Hankel integral transforms. The problem is reduced to a single integral equation, the character of which is studied, and a numerical method of solution to the corresponding integral equation using Gauss-Chebyshev quadrature is presented.

ABSTRACT. Thin layers are key elements of virtually all composite structures; they are typically used to prevent damage and increase the durability of structures and products. There exists tremendous interest in better understanding the processes that occur in composite structures with thin layers. In this talk, we will discuss the novel asymptotic complex-variables-based approach for modeling thin layers in the context of 2D potential problems (e.g., conductivity or antiplane elasticity). This talk includes theoretical developments, numerical examples, and a discussion of interface regimes.

ABSTRACT. This talk presents the rigorous homogenization of a particulate flow consisting of a non-dilute suspension of a viscous Newtonian fluid with magnetizable particles. The fluid is assumed to be described by the Stokes flow, while the particles are either paramagnetic or diamagnetic, for which the magnetization field is a linear function of the magnetic field. Coefficients of the corresponding PDEs are locally periodic. A one-way coupling between the fluid domain and the particles is also assumed. The homogenized or effective response of such a suspension is derived, and the mathematical justification of the obtained asymptotics is carried out. The two-scale convergence method is adopted for the latter. As a consequence, the presented result provides a justification for asymptotic analysis of L\'{e}vy and Sanchez-Palencia for particulate steady-state Stokes flows.