ABSTRACT. We consider a Landau-de Gennes model for a connected cubic lattice scaffold in a nematic host, in a dilute regime. We analyse the homogenised limit for both cases in which the lattice of embedded particles presents or not cubic symmetry and then we compute the free effective energy of the composite material.

In the cubic symmetry case, we impose different types of surface anchoring energy densities, such as quartic, Rapini-Papoular or more general versions, and, in this case, we show that we can tune any coefficient from the corresponding bulk potential, especially the phase transition temperature.

In the case with loss of cubic symmetry, we prove similar results in which the effective free energy functional has now an additional term, which describes a change in the preferred alignment of the liquid crystal particles inside the domain.

Moreover, we compute the rate of convergence for how fast the surface energies converge to the homogenised one and also for how fast the minimisers of the free energies tend to the minimiser of the homogenised free energy.

ABSTRACT. We present a numerical method that implements the Maier-Saupe bulk potential for uniaxial, nematic liquid crystals. A tensor order parameter is defined as the second moment of an orientational probability distribution and a free energy is defined so that it goes to infinity when the order parameter is non-physical. An elastic free energy is added which is an expansion of spatial gradients of the order parameter up to third order, allowing for parametric control over the anisotropy and elasticity of the liquid crystal. We show that this elastic free energy is unbounded in the case of the traditional double well bulk potential but remains bounded for the Maier-Saupe potential. Further, we show that the method yields stable configurations for topological defects in 2D and 3D and the elastic anisotropy leads to anisotropic defect configurations.

ABSTRACT. Existing phenomenological and micromechanical elastic models for rubbery polymers are unable to account for polymer molecular structure and inter-segment interaction. To address these limitations, we have developed a statistical mechanics-based field theoretic model for elastomers. We start with a continuous Gaussian chain model for a single polymer chain. In the many chain setting, we use statistical field theory to account for inter-segment interaction which is modelled using excluded volume effect. The model is solved numerically using a finite element approach to obtain the total free energy and equilibrium elastomer density as a function of applied deformation gradient. We find that in the absence of inter-segment interaction, the elastic response of the polymer chain matches with the classical rubber elasticity, whereas a sufficiently strong inter-segment interaction leads to unexpected instabilities in the structure and response of the polymer network. We extend our statistical mechanics-based field theoretic elastomer model for Liquid Crystal Elastomers (LCEs). Using our model, we probe the limitations of existing LCE models to predict LCE mesoscale structure and response, and further examine the effects of temperature, geometry and loading conditions on LCE response.

ABSTRACT. Ionic crystals such as solid electrolytes and complex oxides are central to modern technologies for energy storage, sensing, actuation, and other functional applications. An important fundamental issue in the atomic and quantum scale modeling of these materials is the question of defining the macroscopic polarization. In a periodic crystal, the usual definition of the polarization as the first moment of the charge density in a unit cell is found to depend qualitatively – allowing even a change in the sign! – and quantitatively on the choice of unit cell. We examine this issue using a rigorous approach based on the framework of two-scale convergence. By examining the continuum limit of when the lattice spacing is much smaller than the characteristic dimensions of the body, we prove accounting for the boundaries consistently provides a route to uniquely compute electric fields and potentials despite the non-uniqueness of the polarization. Specifically, different choices of unit cell in the interior of the body leads to correspondingly different partial unit cells at the boundary; while the interior unit cells satisfy charge neutrality, the partial cells on the boundary typically do not, and the net effect is for these changes to compensate for each other.

ABSTRACT. Optimizing the properties of the mosaic morphology of bulk heterojunction (BHJ) organic photovoltaics (OPV) is not only challenging technologically but also intriguing from the mechanistic point of view. Among the recent breakthroughs in studies of BHJ morphology, is the identification and utilization of a three-phase (donor/mixed/acceptor) BHJ, where the mixed-phase can inhibit morphological changes, such as phase separation. We develop and analyze a continuum model that undertakes the coupling between the spatiotemporal evolution of the material and charge dynamics along with charge transfer at the device electrodes. Starting from an ideal BHJ represented by stripes, we reveal and distinguish between generic mechanisms that alter the evolution of stripes: the bending (zigzag mode) and the pinching (cross-roll mode) of the donor/acceptor domains. We then emphasize that the latter instability mode, is notorious as it leads to the formation of disconnected domains and hence to loss of charge flux. Consequently, the analysis implies that donor-acceptor mixtures with higher mixing energy are more likely to develop pinching under charge-flux boundary conditions. We believe that these results provide a qualitative roadmap for morphological BHJ optimization, using mixed-phase composition, and moreover, open new vistas to application involving three-phase morphologies, such as ion-intercalated rechargeable batteries.

ABSTRACT. A mixture of confined block copolymers and/or homopolymers presents a variety of exotic morphologies including polyhedrons and a variety of patterns of micro-phase separation inside the particle. We discuss how we control and predict the shape of the particles and patterns inside based on a mathematical model consisting of a set of Cahn-Hilliard type of equations derived from appropriate free energy. One of the key things is the (in)compatibility between each polymer inside a particle and outer solvent, namely the pattern inside is strongly influenced by the environments. Our model allows us firstly to classify most of the experimentally observed patterns, secondly to describe the dynamical metamorphosis of block copolymers particle via annealing process from striped ellipsoids into an onion-like pattern, and thirdly to predict a variety of new morphologies including Ashura particle composed of three different homopolymers. If time allows, we discuss the landscape of our free energy. The main part of my talk is joint work with E. Avalos, T. Teramoto, and H. Yabu.

ABSTRACT. We look at phase transitions that occur when pure fluids or liquid mixtures are placed in electric forces. This is a model for systems under the influence of long-range forces with integral constraints. For an initially homogeneous system above the bulk binodal curve, when the external potential is sufficiently small, only smooth composition variations appear. When the voltage exceeds a critical value, however, electro prewetting transitions take place marked by the appearance of a sharp interface between two or more coexisting phases. The use of model B or model H dynamics allows us to track the domain growth and equilibrium location of the interface, and to calculate the surface tension associated with it. Due to the long-range nature of the forces, the surface tension depends on the location of the interface and the boundary conditions on remote surfaces. This leads to a new interfacial instability where the external field stabilizes the smooth interface and surface tension destabilizes it.

ABSTRACT. Active fluids consume fuel at the microscopic scale, converting this energy into forces that can drive macroscopic motions over scales far larger than their microscopic constituents. In some cases, the mechanisms that give rise to this phenomenon have been well characterized, and can ex- plain experimentally observed behaviors in both bulk fluids and those confined in simple stationary geometries. More recently, active fluids have been encapsulated in viscous drops or elastic shells so as to interact with an outer environment or a deformable boundary. Such systems are not as well un- derstood. In this work, we examine the behavior of droplets of an active nematic fluid. We study their linear stability about the isotropic equilibrium over a wide range of parameters, identifying regions in which different modes of instability dominate. Simulations of their full dynamics are used to iden- tify their nonlinear behavior within each region. When a single mode dominates, the droplets behave simply: as rotors, swimmers, or extensors. When parameters are tuned so that multiple modes have nearly the same growth rate, a pantheon of modes appears, including zigzaggers, washing machines, wanderers, and pulsators.

ABSTRACT. I will discuss some recent progress on a model system consisting of a flexible surface and a vector field defined on the surface in the case in which an interaction between the vector field and the surface is present. First the model will be introduced, the governing equations will be derived and some qualitative features of the model will be illustrated in the special situation of curves. Then I will state a short term existence and a stability result in the general case of surfaces.

ABSTRACT. The starting point is a system of interacting particles in 1D, e.g., charged particles with a nonlocal repelling interaction force, that are kept together by an externally applied force. The unknowns are the particle positions at equilibrium. The interest is in passing to the many-particle limit (i.e., number of particles to infinity), where the limiting problem is a nonlocal, nonlinear ODE for the particle density. The main challenge for proving such limits is dealing with the singularity in the particle interaction force.

Many-particle limits for various interacting particle systems have already been established in the literature, mainly by variational tools such as Gamma-convergence. However, such results do not quantify how close the particle positions are to the limiting density. In my talk I will present our first result on this quantification in terms of a convergence rate. This is joint work with M. Kimura (Kanazawa University, Japan).

ABSTRACT. Charged domain walls are a type of transition layers in thin ferromagnetic films which appear due to global topological constraints. The underlying micromagnetic energy is determined by a competition between a diffuse interface energy and the long-range magnetostatic interaction. The underlying model is non-convex and vectorial. In the macroscopic limit we show that the energy Gamma--converges to a limit model where jump discontinuities of the magnetization are penalized anisotropically. In particular, we identify a supercritical regime which allows for tangential variation of the domain walls.

ABSTRACT. We will discuss a variational model that describes the emergence of polycrystalline structures as a result of elastic energy minimisation. The setting is that of linearised planar elasticity. Starting from the variational semi-discrete model for edge dislocations introduced in [Garroni et al. 2010] within the so-called core radius approach, we derive by Gamma-convergence as the lattice spacing tends to zero, a limit energy given by the sum of a plastic term, acting on the dislocation density, and an elastic term, which depends on the symmetric strains. Minimisers under suitable boundary conditions are piece-wise constant antisymmetric strain fields, representing in our model a polycrystal whose grains are mutually rotated by infinitesimal angles. In this respect our result can be regarded as a linearised version of the Read-Shockley formula. We will also discuss a possible derivation from a non-linear energy.

ABSTRACT. The talk will describe how twinning and slip in Bravais lattices can be understood in terms of rank-one connections between energy wells in the Ericksen picture of elastic crystals, placing classical results of materials science in a more modern language.

ABSTRACT. We use the minimizing-movements approach to study an evolution of spin-systems defined on the two-dimensional square lattice of amplitude depending on a vanishing parameter ε. The driven energy is given by a nearest-neighbors anti-ferromagnetic potential. We analyze both the discrete and the continuous (limit, as ε goes to 0) flow from a geometric point of view identifying spin-systems with the union of lattice squares corresponding to the positive statuses of the system. Through this identification, the anti-ferromagnetic energy corresponds to a negative perimeter. We consider evolutions starting from a single point (nucleation).

We show that, the competition between short-range-repulsion (negative perimeter) and long-range-attraction (dissipation) produces the emergence of a checkerboard microstructure of the minimizers at the discrete level and a "backward" evolution at the continuous one. We prove that the scheme converges to a family of expanding sets with constant velocity. The "shape" of the limit motion depends on the choice of the scale between the time and space parameters and on the norm defining the dissipation term.

ABSTRACT. Symmetric quasiconvexity plays a key role for energy minimization in the setting of the geometrically linear theory of elasticity and in computational relaxation. A sufficient condition for this is symmetric polyconvexity, which can be nicely characterised. I will discuss this as well as an example that shows that symmetric rank-one convex quadratic forms need not be symmetric polyconvex in three dimensions. This is joint work with Omar Boussaid and Carolin Kreisbeck.

ABSTRACT. We first review the standard blow-up formula for the homogenized surface tension of the Gamma-limit of rapidly oscillating energies defined on partitions in stationary, ergodic media. We then present a simplified formula which allows to transfer quantitative information on the statistics to quantitative decay rates of the variance of the blow-up formula.

ABSTRACT. We derive geometrically linearized theories for incompressible materials from nonlinear elasticity theory in the small displacement regime. Our nonlinear stored energy densities may vary on the same (small) length scale as the typical displacements. This allows for applications to multiwell energies as, e.g., encountered in martensitic phases of shape memory alloys and models for nematic elastomers. Under natural assumptions on the asymptotic behavior of such densities we prove Gamma-convergence of the properly rescaled nonlinear energy functionals to the relaxation of an effective model. The resulting limiting theory is geometrically linearized in the sense that it acts on infinitesimal displacements rather than finite deformations, but will in general still have a limiting stored energy density that depends in a nonlinear way on the infinitesimal strains. Our results, in particular, establish a rigorous link of existing finite and infinitesimal theories for incompressible nematic elastomers.

ABSTRACT. We describe a debonding model for a thin film, where the wave equation on a time-dependent domain is coupled with a flow rule (Griffith's principle) for the evolution of the domain. The problem can be solved in closed form by assuming that the displacement depends on only one space variable. In order to attack the two-dimensional case, we propose a general definition of energy release rate, which is central in the formulation of Griffith's criterion. Next, by means of an existence result, we show that such definition is well posed in the special case of radial solutions, which allows us to employ representation formulas typical of one-dimensional models. From joint works with L. Nardini, R. Molinarolo, and F. Solombrino.

ABSTRACT. We prove discrete-to-continuum convergence of interaction energies defined on lattices in the Euclidean space (with interactions beyond nearest neighbours) to a crystalline perimeter, and we discuss the possible Wulff shapes obtainable in this way. Exploiting the "multigrid construction" of quasiperiodic tilings (which is an extension of De Bruijn's "pentagrid" construction of Penrose tilings) we adapt the same techniques to also find the macroscopical homogenized perimeter when we microscopically rescale a given quasiperiodic tiling. This is joint work with Giacomo Del Nin from University of Warwick.

ABSTRACT. Important qualitative features of two-phase systems related to phase separation processes can be described by Cahn-Hilliard equations. For these equations, many different boundary conditions are available. While the simplest boundary conditions dictate a static contact angle and no flux across the boundary, more sophisticated boundary conditions also allow for dynamic contact angles or mass transfer. In the recent years, different models were developed, which describe boundary effects using an additional Cahn-Hilliard equation on the boundary (cf. Goldstein, Miranville, Schimperna, Physica D, 2011; Liu, Wu, Arch. Ration. Mech. Anal, 2019). Both of these models satisfy similar physical properties but differ greatly in their mass conservation behavior.

In this talk, we analyze a new model interpolating between these previous models and discuss its numerical treatment. We present a unified numerical scheme, which is able to deal with the complete class of interpolation models despite the different structures of its limit models. The results presented in this talk are based on a joint work with Patrik Knopf (Universität Regensburg), Kei Fong Lam (Hong Kong Baptist University), and Chun Liu (Illinois Institute of Technology).

ABSTRACT. In this talk, I will give an overview of energetic variational approaches to multiphase flows for Navier-Stokes and thin-film type free boundary problems. The focus of the talk will be on deriving variational formulations of the corresponding models in terms of gradient flows or GENERIC structures and reducing them to thin film type evolution, cf. [1,2]. A special focus will be on free boundary problems and the treatment of moving interfaces and contact lines and the modeling of their dynamics in a thermodynamically consistent way. As a particular example, I will present experimental and theoretical results showing the flow of a fluid over another (immiscible) fluid, see [3].

[1] Huth, R., Jachalski, S., Kitavtsev, G., & Peschka, D. (2015). Gradient flow perspective on thin-film bilayer flows. Journal of Engineering Mathematics, 94(1), 43-61.

[2] Peschka, D. (2018). Variational approach to dynamic contact angles for thin films. Physics of Fluids, 30(8), 082115.

[3] Peschka, D., Bommer, S., Jachalski, S., Seemann, R., & Wagner, B. (2018). Impact of energy dissipation on interface shapes and on rates for dewetting from liquid substrates. Scientific Reports, 8(1), 1-11.

ABSTRACT. Mass transfer of gaseous components from rising bubbles to the ambient liquid is the basis for many chemical processes of industrial and technical importance. Besides experimental investigations, the necessary intensification requires numerical simulations based on detailed mathematical modeling. Our approach is based on continuum mechanical sharp-interface balances of mass, momentum and species mass. The occurrence of extremely thin liquid-sided concentration boundary layers at bubble interfaces for realistic, i.e., high Schmidt numbers is a severe obstacle for the detailed, accurate numerical simulation of mass transfer processes in gas-liquid systems. This contribution explains a subgrid-scale modeling approach to compute nonlinear flux corrections, both for physical and for reactive mass transfer. This is incorporated into the geometric Volume of Fluid (VOF) or the Interface Tracking method to obtain local transfer rates and reactive enhancement factors at rising bubbles. For mass transfer with complex chemical reactions, machine learning is successfully employed to estimate the diffuse and advection species fluxes. This allows for computing chemical selectivities under realistic physical parameters.

ABSTRACT. Consider a layer of viscous incompressible fluid bounded below by a flat rigid boundary and above by a moving boundary. The fluid is subject to gravity, surface tension, and an external stress that is stationary when viewed in coordinate system moving at a constant velocity parallel to the lower boundary. The latter can model, for instance, a tube blowing air on the fluid while translating across the surface. In this talk we will detail the construction of traveling wave solutions to this problem, which are themselves stationary in the same translating coordinate system. While such traveling wave solutions to the Euler equations are well-known, to the best of our knowledge this is the first construction of such solutions with viscosity. This is joint work with Giovanni Leoni.

ABSTRACT. Machine learning is finding applications to more and more tasks, in science as much as in everyday life. In this talk I will focus on how atomic and molecular simulations are being transformed by the use of statistical regression models, that make it possible to approximate accurately and efficiently atomistic properties computed from a few reference electronic-structure calculations. I will argue about the advantages that are brought about by an integrated framework, that makes it possible to access not only the interatomic potential, but all sort of properties from NMR chemical shieldings to the electronic charge density, and provide examples from molecular to condensed-phase systems.

ABSTRACT. The dream of machine learning in chemical and materials science is for a model to learn the underlying physics of an atomic system, allowing it to move beyond the interpolation of the training set to the prediction of properties that were not present in the original training data. In addition to advances in machine learning architectures and training techniques, achieving this ambitious goal requires a method to convert a 3D atomic system into a feature representation that preserves rotational and translational symmetries, smoothness under small perturbations, and invariance under re-ordering. The multiscale wavelet scattering transform, which is a specialized type of convolutional neural network, preserves these symmetries by construction. We will show that supervised machine learned models that use wavelet scattering representations of atomic systems are able to estimate the energies of unseen small molecules and materials, which are “similar” to those atomic systems that the model was trained on, at a fraction of the cost of density functional theory but with comparable accuracy. Furthermore, in certain use cases, such as the prediction of elastic constants, migration barriers, and energies of larger atomic systems, the wavelet scattering model still provides accurate predictions despite not being trained for such tasks.

ABSTRACT. Machine learning potentials (MLPs) for molecular dynamics simulations have been found to be capable of approaching ab initio accuracy with the computational efficiency closer to that of empirical potentials. The accuracy and performance of an MLP is highly dependent upon the choice of descriptors used to characterize the local atomic environment though, as well as on the training data that the algorithm uses to predict the energies and forces. The descriptors should respect the symmetries of the atomic environment, be differentiable with respect to the atomic coordinates, and contain sufficient information. If a Gaussian process is used for machine learning, the covariance function that quantifies the similarity or closeness between two points in descriptor space also has requirements to consider. Specifically, the choice of covariance function should impose minimal constraints on the potential to reduce the risk of systematic error. This work reports on recent progress using Gaussian process regression and a novel set of descriptors to construct accurate and efficient interatomic potentials.

ABSTRACT. Deep neural networks (DNNs) have recently emerged as a new wavefunction Ansatz method for variational Monte Carlo (VMC) approaches. First results already suggest that the superior approximation properties of DNNs with respect to high-dimensional functions can help breaking the curse of dimensionality in this setting and enable us to obtain highly accurate solutions for the electronic Schrödinger equation. In an attempt to further improve the optimization procedure of a DNN for VMC, we investigate if and under which circumstances already computed solutions for a specific DNN architecture can be utilized to quickly obtain good solutions for neighboring molecule configurations. In particular, we present empirical  evidence that the optimization of a DNN through VMC can indeed be significantly accelerated by sharing neural network weights when considering different nuclear configurations of the same molecule.

ABSTRACT. Contact angle hysteresis (CAH) is widely studied both experimentally and mathematically in recent years. Its qualitative mechanism has been studied in great details while there is still a lack of quantitative theory. In this work, we first propose a reduced model for the contact line and contact angle dynamics based on Onsager variational principle. Then we apply this reduced model to study the contact angle dynamic on chemically inhomogeneous surfaces. Multiscale expansion and averaging techniques are employed to approximate the model for asymptotically small chemical patterns. An effective dynamics is obtained in a two-stage structure: when the dynamic angle is within the range of the chemical pattern, the fast variable in small scale is slave to the slow dynamics in macroscopic scale; when the dynamic angle is out of the range of the chemical pattern, the fast dynamics generates an invariant measure so that the slow variable is averaged to an effective dynamics. Numerical simulations are presented to validate the analytical results. Moreover, we obtain a quantitative formula which is used to explain the asymmetric and speed dependent CAH in comparison with the experimental results both numerically and analytically.

ABSTRACT. We develop a mesoscopic dislocation dynamics model for vacancy-assisted dislocation climb by upscalings from a stochastic model on the atomistic scale. Our model incorporates various microscopic mechanisms associated with vacancy bulk/pipe diffusion and vacancy attachment-detachment kinetics on jogged dislocations. Our mesoscopic model consists of the vacancy bulk diffusion equation and a dislocation climb velocity formula. The effects of these microscopic mechanisms are incorporated by a Robin boundary condition near the dislocations for the bulk diffusion equation and a new contribution in the dislocation climb velocity due to vacancy pipe diffusion driven by the stress variation along the dislocation. Our climb formulation is able to quantitatively describe the self-climb of prismatic loops at low temperatures when the bulk diffusion is negligible. Simulations performed show evolution, translation, coalescence of prismatic loops by self-climb that agree with the experimental observations. The formation and evolution of these prismatic loops crucially influence the physical and mechanical properties of materials with irradiation.

ABSTRACT. We present a continuum model to compute the energy of low angle grain boundaries for any given degrees of freedom (arbitrary rotation axis, rotation angle and boundary plane orientation) based on a continuum dislocation structure. In our continuum model, we minimize the grain boundary energy associated with the dislocation structure subject to the constraint of Frank's formula for dislocations with all possible Burgers vectors. Uniqueness of the solution of the constrained minimization problem is obtained and numerical methods for solving the problem are discussed.

ABSTRACT. The talk will present a two scale elastic shape optimization approach based on optimal distribution of printable material pattern on the fine scale. To this end fine cell material geometries with fixed material bridges to the neighbouring cells are taken into account. For general classes of Dirichlet data on the cell boundary a shape optimization of the interior material distribution is performed resulting in a data base of pairs of effective elasticity tensors and invested volume of the cell pattern. Then, on the macro scale a free material optimization with volume cost is performed constraining the local elasticity tensor to realisable tensors on the fine scale and retrieving fine scale patterns from the pattern data base. This is joint work with Stefan Simon from Bonn University.

ABSTRACT. Sensitivity analysis of the eigenvectors corresponding to multiple eigenvalues is a complex and difficult problem. The main difficulty is that for given multiple eigenvalues, the eigenvector derivatives can only be computed for a specific eigenvector basis, the so-called adjacent eigenvector basis. These adjacent eigenvectors depend on individual variables, which makes the eigenvector derivative calculation very elaborate and expensive from a computational perspective for problems with a moderately large number of variables. In this talk, we present a method that avoids passing through adjacent eigenvectors in the calculation of the partial derivatives of any prescribed eigenvector basis. Furthermore, as our method fits into the adjoint sensitivity analysis framework, it is very efficient for computing the complete Jacobian matrix. In view of these two points, our method clarifies and unifies existing alternatives on eigenvector sensitivity analysis. Moreover, it provides a highly efficient computational method with a significant saving of the computational cost as compared to those of the existing methods. We will also present the extension of the method to the infinite dimensional situation.

ABSTRACT. Optimization of materials via the homogenization theory requires an approach at two levels. At the macroscopic level, a PDE is solved providing shape, topological and periodicity sensitivities. At the microscopic level these sensitivities are used to solve elliptic PDEs with periodicity conditions (the cellular problems). Besides solving these elliptic PDEs, one seeks for the optimal shape of the hole/inclusion of the inner structure in each point of the body. It is of great interest to allow the hole/inclusion o vary freely during the optimization process with no constraints due to the choice of the cell or its border. Theoretically, one solves the PDE on a quotient space, which is the flat torus. A tool for meshing, re-meshing and solving PDEs on manifolds, as flexible as it can be, is needed.

This is the motivation for implementing Manifem, a C++ library created from scratch which allows to mesh and re-mesh general manifolds, describe variational problems and solve them using directly C++ objects.

Examples of meshes on complicated manifolds will be given.

The process of meshing the flat torus will be discussed.

ABSTRACT. In this work, we present a fast one-step imaging algorithm based on the computation of topological derivatives for the detection of multiple 2D and 3D acoustic obstacles fully coated by a complex surface impedance. New closed-form formulae for the topological derivative of the misfit least squares functional are obtained by combining the use of shape derivatives and asymptotic expansions. Numerical experiments showing the performance of the method when either monochromatic or multi-frequency data are available will be shown both for full and limited aperture noisy measurement data.

ABSTRACT. Models of systems biology, climate change, ecology, complex instruments, and macroeconomics have parameters that are hard or impossible to measure directly. If we fit these unknown parameters, fiddling with them until they agree with past experiments, how much can we trust their predictions? We have found that predictions can be made despite huge uncertainties in the parameters – many parameter combinations are mostly unimportant to the collective behavior. We will use ideas and methods from differential geometry and approximation theory to explain sloppiness as a ‘hyper-ribbon’ structure of the manifold of possible model predictions. We show that physics theories are also sloppy – that sloppiness may be the underlying reason why the world is comprehensible. We will present new methods for visualizing this model manifold for probabilistic systems – such as the space of possible universes as measured by the cosmic microwave background radiation.

ABSTRACT. The fascinating convolutions of the human brain are believed to be caused by mechanical forces generated in the rapid expansion of the cortex with respect to the subcortical areas of the brain. These intricate folded shapes have fascinated generations of scientists and mathematicians and have, so far, defied a complete description. How do they emerge? How are they arranged? How is the brain shape related to its function? In this talk, I will review our current understanding of brain morphogenesis and how it can be modeled. In particular, I will discuss an ideal version of this problem that can be solved exactly, underlying the beautiful interplay between (differential) geometry and mechanics in the shaping of our most intricate organ.

ABSTRACT. By exerting mechanical forces, biological cells cause striking spatial patterns of localised deformation in the surrounding fibrous collagen matrix. Tether-like paths of high densification and fiber alignment form between cells, and radial hair-like bands emanate from cell clusters. While tethers may facilitate cell communication, the mechanism for their formation is unclear. In this study we show that tether formation is a densification phase transition of the fibrous extracellular matrix, caused by microbuckling instability of network fibers under compression. The mechanical behaviour of the extracellular matrix (ECM) caused by cell contraction is modelled and analysed from a macroscopic perspective employing the theory of nonlinear elasticity for phase transitions. From the one dimensional response of a single fiber a two dimensional strain energy function is constructed that fails to be rank one convex. This failure of rank-one convexity implies a loss of ellipticity of the Euler-Lagrange PDEs of the corresponding energy functional. In order to exclude mesh dependence in our approximations, a higher gradient term is added to the model. This term also introduces an internal length scale, which is an additional material parameter related to characteristic fiber length, bending stiffness and other parameters of the fiber network.

ABSTRACT. We consider a non-isothermal Cahn-Hilliard model based on a microforce balance. The model was derived by A. Miranville and G. Schimperna starting from the two fundamental laws of Thermodynamics, following M. Gurtin’s two-scale approach. The main working assumptions are on the choice of the Ginzburg-Landau free energy, and on the behaviour of the heat flux as the absolute temperature tends to zero and to infinity. By deriving suitable a priori estimates and by showing weak sequential stability of families of approximating solutions, we prove global-in-time existence for the initial-boundary value problem associated to the entropy formulation and, in a subcase, also to the weak formulation of the model. (Joint work with G. Schimperna)

ABSTRACT. For interface evolution problems not admitting a geometric comparison principle, as it is for instance the case in multi-phase mean curvature flow or fluid-fluid interface evolution, the derivation of a stability estimate or a weak-strong uniqueness principle often represents an open problem. In this talk, I will introduce a notion of gradient flow calibrations which adapts the classical notion for minimizers of the interface area functional from the static setting to the evolutionary setting. This construction in turn serves as the key building block for a multi-phase generalization of the notion of relative entropies for interface evolution problems. Both concepts taken together enable us to establish a weak-strong uniqueness principle for multi-phase mean curvature flow in the plane: We show that as long as a classical solution exists, any weak solution in the sense of the BV formulation of Laux and Otto (Calc. Var. Partial Differential Equations 2016) starting from the same initial data must coincide with it.

ABSTRACT. We consider the interaction of elastic stress and anisotropic surface energy on the shape of a void in an elastically-stressed solid. The void shape is a free-boundary problem determined by the minimization of elastic strain energy and surface energy. The equilibrium shape of the void has corners for sufficiently strong surface energy anisotropy, and the presence of corners will generate integrable singularities in the elastic stress. The contribution of these singularities to the total elastic strain energy can alter the entire void shape. We develop a numerical method for solving the free-boundary problem using a complex-variables boundary integral equation for the elasticity problem, coupled to a spectral method for determining the energy-minimizing shape. A novel feature of our work is the incorporation of an explicit corner term in the representation of the stress field that is derived from the asymptotic analysis of the corner region. Our results determine how the singularities associated with corners have a global effect on the entire shape of the void and affect the equilibrium corner angles on the void shape.

ABSTRACT. In 1914, a magnetic alloy with unusually high permeability and low hysteresis was discovered at the Bell Labs. This alloy resulted from a series of investigations on the Fe-Ni system, in which the nickel content was systematically varied. Under a very specific combination of the alloy composition (78.5% Ni), the magnetic alloy (now known as the permalloy) demonstrates a drastic increase in magnetic permeability and a decrease in the coercive force. In the past, this unusual behavior of the permalloy has been attributed to the anisotropy constant, however, there is still no theory that explains this drastic decrease in hysteresis in magnetic alloys. Our goal is to identify a mathematical relation between the magnetic material constants that is key in reducing hysteresis in magnetic materials. We hypothesize that a combination of a large local disturbance and material properties, such as anisotropy and magnetostriction coefficients, contribute to the drastic decrease in hysteresis. We formalize this idea using the micromagnetics theory in a phase-field framework and use energy minimization methods to investigate the links between microstructures and material constants in nickel-iron alloys. Our results demonstrate agreement with the permalloy experiments and provide theoretical insights into developing magnetic alloys with negligible hysteresis.

ABSTRACT. Developing simplified computational models of complex fundamental phenomena in physics, chemistry, astronomy, biology and materials science is an ongoing challenge. The purpose of such simplified models is to reduce computational cost at minimal loss of accuracy. At the same time, more importantly, these models can provide fundamental understanding of underlying phenomena. Recently, the following two concepts have gained significant importance in computational science: (i) machine learning (in particular neural networks) and (ii) structure-preserving (mimetic or invariant-conserving) computing for mathematical models in physics, chemistry, astronomy, biology, and materials science. While neural networks are very strong as high-dimensional ‘universal function approximators’, they require large datasets for training and tend to perform poorly outside the range of training data. On the other hand, structure-preserving methods are strong in providing accurate solutions to complex mathematical models from science, but are typically computationally expensive (this can be reduced by using model order reduction). The goal of our research is to better understand neural networks to enable the design of highly efficient, tailor-made neural networks built on top of and interwoven with fundamental properties of the underlying science problems. In this presentation, we will present the basic ideas, as well as a number of preliminary examples.

ABSTRACT. Energy storage technology (EST) has already proven to offer great opportunities for technological progress. A clear example of this is the Li-ion rechargeable battery. But far away from previous achievements, EST possess a significant potential to offer a number of new economic and environmental benefits by means of replacing systems powered by fossil fuels. Progress in EST demands new materials or finding novel properties of existing compounds. Yet trial-and-error exploratory research based on extending known compounds into new compositional spaces implies long-lasting work of synthesis and characterization. To mitigate this issue and accelerate the process of energy materials discovery, computational modelling is emerging as a powerful complementary tool. Based on theoretical chemistry grounds together with machine learning, computational predictions of the performance of materials are now sufficiently mature to be applied successfully in many cases. Here we discuss a selection of our most exciting examples of recent work in this area. In particular, we focus on general concepts that emerge about how such calculations can provide valuable understanding into key thermodynamic and kinetic properties for a selection of electrochemical and thermal energy storage systems, revealing insights into key phenomena such as phase stability and ion diffusion.

ABSTRACT. Microtubules are tubular protein complexes representing one of the key elements of the cytoskeleton of biological cells. They are the prime targets of some of the treatment strategies aimed at either triggering the apoptosis or interrupting uncontrolled cell division. Despite being such a vital organelle of biological cells, only few physics-based mathematical models relating to the dynamics of microtubules exist to date. In the present study, the influence of coupled thermo-electro-mechanical field on the properties of the microtubules has been evaluated within the cellular domain, taking into account different organelles. The main focus has been given to the arrangement of microtubules within the cellular domain, and the effects of the electric field strength and temperature on the mechanical stresses and strains, induced within the biological cells, have also been evaluated. Further, the synergy between the machine learning and multiscale modelling techniques will be exploited to provide a more comprehensive model for predicting the microtubules dynamics subjected to coupled fields. The proposed model is expected to play an important role in our better understanding of the influence mechanisms of coupled fields on the microtubules embedded within the biological cells, and it can be extended to other applications in biomedical engineering.

ABSTRACT. One of the recurring anomalies in the hydrogen induced fracture of high strength steels is the apparent disconnect between their toughness and uniaxial tensile strength in identical hydrogen environments. Here we propose, supported by detailed atomistic and continuum calculations, that unlike macroscopic toughness, hydrogen-mediated tensile failure is a re- sult of a fast-fracture mechanism. Specifically, we show that failure originates from the fast propagation of cleavage cracks that initiate from cavities that form around inclusions such as carbide particles. A combined atomistic/continuum model is used to explain a host of well-established experimental observations including (but not limited to): (i) insensitivity of the strength to the concentration of trapped hydrogen; (ii) the extensive microcracking in addition to the final cleavage fracture event and (iii) the higher susceptibility of high strength steels to hydrogen embrittlement.

ABSTRACT. Traditional simulations of complex mechanical deformation are technologically crucial and computationally expensive.Developing comparably accurate models with lower computational cost can enable more robust design and uncertainty quantification, as well as exhaustive structure-property exploration.Currently, high-throughput experimental techniques and microscale simulators can produce quantities of data that overwhelm traditional constitutive modeling methods and can provide sufficient data to train neural networks. As a modeling technique, neural networks are flexible in that their graph-like structure can be rearranged and functions of their nodes can be adapted to suit particular applications, such as image processing. On the other hand, Gaussian process models can outperform neural networks in the data limited regime. In this talk, I will discuss: how we construct input spaces and architectures to capture how microstructure affects mechanical outcomes given observations of the initial microstructure; and, how we draw upon classical constitutive theory to make predictions of the individual stress response of polycrystals that satisfy fundamental physical constraints. We compare the accuracy of these predictions to both traditional and recent homogenization theory.

ABSTRACT. Close to macrointerfaces in martensites one often observes microstructures with characteristic needle-shaped domains. The specific needle shape is mainly described by the bending and tapering of the needle towards the interface. In this talk, we shall outline that the tapering length can be understood within nonlinear elasticity but not within linearized elasticity. This is based on joint work with S. Conti, M. Lenz, N. Lüthen, and M. Rumpf.

ABSTRACT. Nonlocal energies are continuum models for large systems of particles with long-range interactions. Under the assumption that the interaction potential is radially symmetric, several authors have investigated qualitative properties of energy minimisers. But what can be said in the case of anisotropic kernels? Motivated by the example of dislocation interactions in materials science, we pushed the methods developed for nonlocal energies beyond the case of radially symmetric potentials, and discovered surprising connections with random matrices and fluid dynamics.

This is work in collaboration with J.A. Carrillo, J. Mateu. M.G. Mora, L. Rondi and J. Verdera.

ABSTRACT. We study the branching of twins appearing in shape memory alloys at the interface be- tween austenite and martensite. In the framework of three-dimensional non-linear elasticity theory, we propose an explicit, low-energy construction of the branched microstructure, generally applicable to any shape memory material without restrictions on the symmetry class of martensite or on the geometric parameters of the interface. We show that the suggested construction has optimal scaling under a two-well hypothesis and certain boundary conditions. In particular, the construction follows the classical "Kohn-Muller" type energy scaling, i.e., that for the surface energy of the twins being sufficiently small the branching leads to energy reduction. Furthermore, the construction can be modified to capture different features of experimentally observed microstructures without violating this scaling law. As a demonstration, we show that the construction is able to accurately predict the twin width in a Cu-Al-Ni single crystal and provide a realistic proxy (upper bound) to the number of the branching generations.

ABSTRACT. We present our results concerning the rigidity of a shape memory alloy undergoing cubic-to-tetragonal transformations. Starting from a geometrically linear elastic energy augmented by an interface penalization we derive a non-convex differential inclusion in the energy regime of branching microstructures. Without assuming additional regularity we classify all possible solutions and describe the qualitative rigidity properties of such microstructures. Furthermore, we give insight into quantitative aspects, such as the possibly fractal dimension of the set of macroscopic interfaces, by analyzing the H-measures generated by the microstructures.

ABSTRACT. I will present some conditions under which a trace for a function can be identified on a set with strictly positive codimension. The conditions are compatible with nonlocal problems where convolution-like operators with integrable kernels are employed. The functions are required to satisfy an oscillation constraint that does not require any differentiability away from the boundary. The assumptions allow domains with very rough boundaries that possess cusp-like features.

ABSTRACT. We present some recent results on consistency of higher order derivatives on random point clouds. Joint work with Dejan Slepcev.

ABSTRACT. Skin pigmentation, animal coats and block copolymers can be considered as multi-constituent inhibitory systems. Exquisitely structured patterns arise as orderly outcomes of the self-organization principle. The free energy functional combines an interface energy favoring micro-domain growth with a Coulomb-type long range interaction energy which prevents micro-domains from unlimited spreading. Analytically, via the weighted sharp interface model, we study the exact shape of global minimizers. Numerically, via the diffuse interface model, the core-shell assembly pattern and the transition from double bubble to core-shell were demonstrated.

ABSTRACT. WegiveamultiplicityresultforsolutionsoftheVanderWaals-Cahn-Hilliard two-phase transition equation with volume constraints on a closed Riemannian manifold. Our proof employs some results from the classical Lusternik–Schnirelman and Morse theory, together with a technique, the so-called photography method, which allows us to obtain lower bounds on the number of solutions in terms of topological invariants of the underlying manifold. The setup for the photography method employs recent results from Riemannian isoperimetry for small volumes.

ABSTRACT. Some problems related to curvature flow of networks in presence of an anisotropy will be discussed. Besides existence and uniqueness, some qualitative properties of the flow for long times will be addressed

ABSTRACT. We introduce a notion of uniform convergence for local and nonlocal curvatures and we propose an abstract method to prove the convergence of the corresponding geometric flows, within the level set formulation. We apply such a general theory to characterize the limits of several nonlocal geometric evolutions. We study the limit of the s-fractional mean curvature flows as s→0^+ and s →1^-. Moreover, in analogy with s-fractional mean curvature flows, we introduce the notion of s-Riesz curvature flows and characterize its limit as s → 0^-. Furthermore, using a suitable core-radius regularization, we define s-fractional perimeters and s-fractional curvatures also for s≥1 and we show that - as the core-radius tends to 0 - the corresponding geometric flows converge to the classical mean curvature flow. Eventually, we discuss the limit behavior as r → 0^+ of the flow generated by a regularization of the r-Minkowski content. The results discussed here are obtained in collaboration with A. Cesaroni (Padova), A. Kubin (Roma “La Sapienza”), M. Novaga (Pisa), M. Ponsiglione (Roma “La Sapienza”).

ABSTRACT. It has been experimentally observed (see for instance Mullins, Two-Dimensional Motion of Idealized Grain Boundaries, Journal of Applied Physics 27, 900 (1956)) that the grain boundaries of a recrystallized metal, 
when annealed, moves with a velocity proportional to their mean curvature. In order to analyse from the mathematical point of view this phenomenon, in the 70’s Brakke proposed a weak notion of Mean Curvature Flow defined on a broad class of singular surfaces which can undergo topological changes during the evolution. Differently from Brakke, our aim is to describe this motion as a strong solution of a system of PDEs that describes the evolution of a regular network in the plane. Because of topological changes during the flow, the occurrence of singularities is unavoidable. We will present the state-of-the-art of the problem of the motion by curvature of a regular network mainly focusing on singularity formation and on the need of a “restarting” theorem to define the flow past singularities. This is a project in collaboration with Jorge Lira (UFC Fortaleza), Rafe Mazzeo (Stanford), Mariel Saez (PUC Santiago de Chile).

ABSTRACT. We call a network the union of sufficiently smooth curves that meet at junctions. The simplest configuration is a three-network: this is the union of three curves that meet at one or two junctions. In the first case we speak of a triod, in the second of a theta-network. Networks and flow of networks arise naturally in the study of multiphase systems and when considering the dynamic of their interfaces. In this talk I will present recent results for the elastic flow of three- networks.

ABSTRACT. We propose the use of a liquid crystal polymer network for soft robotics where the various molecular accessories are assembled in the two dimensions of a coating. For instance, the LCN surface deforms dynamically into a variety of pre-designed topographic patterns by means of various triggers, such as temperature, light and the input of electric field. These microscopic deformations exhibit macroscopic impact on, for instance, tribology, haptics, laminar mixing of fluids in microchannels and directed cell growth. Another robotic-relevant function we brought into the LCN coating is its capability to secret liquids under UV irradiation or by an AC field. This controlled release is associated with many potential applications, including lubrication, controlled adhesion, drug delivery, and agriculture, antifouling in marine and biomedical devices, personal care and cosmetics. With this we bring together a tool box to form two dimensional soft robots designed to operate in area where man and machine come together.

ABSTRACT. Liquid crystal elastomers and other photoresponsive materials are attracting considerable attention due to their ability to perform mechanical work when excited by light. Although aspects of the performance of such materials are reported in the literature, there appears to be no unified approach to their comprehensive characterization. In this talk, I will describe one scheme to accomplish such a characterization, and report on the success of its implementation to date.

ABSTRACT. Progress in density-functional theory (DFT) is often the outcome of efforts involving physically sound approximate models, performant numerical schemes and optimal use of available computational hardware. In such an interdisciplinary field the lack of flexibility of many state-of-the-art codes poses an obstacle for the different fields to join forces in one code. To close this gap, we have recently developed the density-functional toolkit, DFTK (https://dftk.org). In about 6000 lines DFTK is capable of performing ground-state DFT simulations at a level of performance comparable to well-established packages. At the same time one may conduct calculations using toy Hamiltonians with potentials ranging from a 1D harmonic oscillator to standard GGA functionals.

This flexibility to use one code to prototype and perform calculations on a few hundred electrons is a key ingredient in our research on reliable black-box methods for high-throughput DFT calculations. Such applications require a high degree of automation and especially reliable numerical methods. In the past we used DFTK to design a robust, inexpensive and parameter-free preconditioner for simulating inhomogeneous materials such as metallic surfaces. Other research investigated computable a posteriori error estimates for simple Kohn-Sham models allowing to tackle adaptive automatic error balancing strategies in the future.

ABSTRACT. Kohn-Sham density functional theory (DFT) calculations have been instrumental in providing many crucial insights into materials behavior and occupy a sizable fraction of world’s computational resources today. However, the stringent accuracy requirements in DFT needed to compute meaningful material properties, in conjunction with the asymptotic cubic-scaling computational complexity with number of electrons, demand huge computational resources. Thus, these calculations are routinely limited to material systems with at most few thousands of electrons. In this talk, we present a massively parallel real-space DFT framework (DFT-FE), which is based on a local real-space variational formulation of the Kohn-Sham DFT energy functional discretized with higher-order adaptive spectral finite-element, and handles pseudopotential and all-electron calculations in the same framework. We will present the efficient and scalable numerical algorithms in conjunction with mixed precision strategies for the solution of Kohn-Sham equations, that has enabled computationally efficient, fast and accurate DFT calculations on generic material systems reaching ~100,000 electrons, and demonstrate an order of magnitude performance advantage over widely used plane-wave codes both in CPU-times and wall-times. Finally, we demonstrate the successful use of DFT-FE for a range of large-scale scientific applications ranging from DNA molecules to dislocations in crystalline materials.

ABSTRACT. We present an approach to accelerate real-space density functional theory (DFT) several fold, without loss of accuracy, by reducing the dimension of the discrete eigenproblem that must be solved. To accomplish this, we construct an efficient, systematically improvable, discontinuous basis spanning the occupied subspace, on to which the real-space Hamiltonian is projected. Through selected examples, we demonstrate that accurate energies and forces are obtained with 8–25 basis functions per atom, reducing the dimension of the associated real-space eigenproblems by 1–3 orders of magnitude.

ABSTRACT. Electronic structure theory and its computational implementation have become ubiquitous in the fields of materials science and molecular design. However, the steep computational scaling of these methods (cubic to septic) with system size poses a limiting factor in the simulation of physical systems of practical size. Due to its prevalence in physical simulation and its computational expense, much effort has been afforded to high-performance computational implementations of these methods over the years. Through these endeavors, these implementations have been able to leverage the latest advances in high-performance computing architectures to enable the simulation of larger and more realistic physical systems. As the field of high-performance computing is ever-evolving, development of high-performance electronic structure methods must be a central focus in the field of physical simulation. In this talk, several recent advancements and current trends in high-performance electronic structure methods development will be discussed.

ABSTRACT. We revisit the problem of curvature-induced delamination of a thin solid film from an adhesive, spherically-shaped substrate. Comparison of energies and stresses in fully laminated and partially delaminated states suggests two distinct instabilities. The first is triggered solely by the compressive component of the film stress, induced by conforming to the spherical topography. This “Euler-like” instability is akin to the delamination of a uniaxially-compressed film from a flat adhesive substrate, which occurs when the compressive stress in the fully laminated state exceeds a thickness-dependent threshold. The second type of instability occurs when the total strain energy, associated with both tensile and compressive components of the stress throughout the film, induced by the change in its Gaussian curvature, reaches a threshold value that is independent of the bending modulus. We refer to this mode of instability as “Gauss-like". We argue that the Gauss-like instability, which has been considered previously as the building block of curvature-induced blister networks, dominates only for moderately bendable films, while the Euler-like instability governs delamination of highly bendable films and gives rise to patterns populated by elongated, radially-oriented folds or rucks. Our experimental results support the existence of distinct instability mechanisms and corresponding routes for pattern formation.

ABSTRACT. A traditional and powerful method for studying the behavior of thin liquid sheets is to harness the analogy to elastic media. However, while in elastic sheets Gaussian curvature G is the source of elastic stress, in liquids the source of viscous stress is curvature dynamics dG/dt. I will discuss why this leads to qualitative differences from the “elastic” picture, and gives rise to nonlinear phenomena where dynamics and geometry are intertwined. I will show that rapidly depressurizing a bubble causes formation of a moving front, eerily similar to the front in stable Laplacian growth systems. Furthermore, I will argue that the pronounced corona of radial wrinkles observed in a rapidly depressurized bubble of (Oratis et al., Science 368 685, 2020) reflects the build-up of hoop compression at very early stages of the front formation, before any significant deformation occurs in the shape of the liquid film.

ABSTRACT. Thin elastic sheets supported on compliant media form wrinkles under lateral compression. Since the pressure is coupled to the sheet's deformation, varying it periodically in time represents a parametric excitation. We study the resulting parametric resonance of wrinkling modes in sheets supported on semi-infinite elastic or viscoelastic media, at pressures smaller than the critical pressure of static wrinkling. We find distinctive behaviors as a function of excitation amplitude and frequency. These include, in particular: (a) a different dependence of the dynamic wrinkle wavelength on sheet thickness compared to the static wavelength; (b) a discontinuous decrease of the wrinkle wavelength upon increasing excitation frequency at sufficiently large pressures. In the case of a viscoelastic substrate, resonant wrinkling requires crossing a threshold of excitation amplitude. We discuss various experimental implications of the results.

ABSTRACT. Size, in terms of a length scale, is often the dominant feature taken into account in diverse clinical problems from cancer treatment to aneurysm management. Aortic aneurysms, dilations of the largest blood vessel in the human body, are treated through surgery at a given size to prevent rupture, a catastrophic event of wall fracture that often leads to death. For any pressurized shell, the change in size over time provides information about strain and correlates to instabilities such as fracture. However aortas, normal or diseased, are not simple geometries. The established extreme linearization of aortic geometry to a single scalar measure is why aneurysm management has failed personalization, despite the immense amount of rich geometric data available from modern cross-sectional imaging. Utilizing a large data set of CTA images in longitudinally followed aortic aneurysm patients, we define their geometry by calculating local aortic wall curvatures. To allow optimal application of machine learning methodologies, the aortic geometry is mapped into a projected two-dimensional plane. An aortic feature space, the eigenaorta, is engineered using the dominant modes from a singular value decomposition analysis. This aortic eigenspace defines an optimal clustering of aneurysm patients allowing identification of high-risk geometric shape features.

ABSTRACT. We consider a problem of energy and sensor-efficient flocking. It is assumed that ambient forces act to decrease collision energies and dissipate kinetic energies of agents. Then we formulate the problem of finding self-propulsion (control) forces that enable flocking behavior while requiring the least amount of on-board energy. We present theoretical results and numerical simulations showing that abandoning formation control is energy-efficient.

ABSTRACT. Quasiperiodic and aperiodic materials present a novel class of metamaterials that possess very unusual, extraordinary properties such as superconductivity, unusual mechanical properties and diffraction patterns, extremely low thermal conductivity, etc. As all these properties critically depend on the microgeometry of the media, the methods that allow characterizing the effective properties of such materials are of paramount importance. We derive homogenized equations for the effective behavior of the composite, discuss an inverse problem of reconstructing microstructural parameters, and discover a variety of new effects that could have interesting applications in the control of wave and diffusion phenomena.

ABSTRACT. We consider the two scale asymptotic expansion for a transmission problem modeling scattering by a bounded inhomogeneity with a periodic coefficient in the lower order term of the Helmholtz equation. The squared index of refraction is assumed to be a periodic function of the fast variable, specified over the unit cell with characteristic size epsilon. Since the convergence of the boundary correctors to their limits is in general slow, we explore in detail their use in a second order approximation and show a new convergence estimate for the second order boundary corrector on a square. We show numerical examples of the higher order forward approximation in one and two dimensions. We then use the first order boundary correction as an asymptotic model for inversion and show numerical examples of inversion in the two dimensional case.

ABSTRACT. Flows of several immiscible viscous liquids can be found in a variety of applications including coating and cooling. The spatiotemporal dynamics and resulting integrity of the interfaces separating different fluid layers is central to practical applications and is particularly relevant in problems with underlying shear (e.g. due to driving pressure gradients in channels, or gravity in falling films). This talk will begin with general mathematical models that describe such viscous flows, and will introduce an asymptotic analysis framework that enables the derivation of reduced-dimension nonlinear evolution equations to describe the dynamics in different setups (such as surfactant effects, multidimensional effects, viscoelastic effects), leading to a canonical class of equations. These are of the active-dissipative type and include pseudo-differential operators that faithfully represent the physics. Analytical and numerical results will be shown. Some novel models that allow for interfacial slip will also be described. These are viscous analogues of the well-known Kelvin-Helmholtz instability and they can induce instabilities and nonlinear waves even at zero Reynolds numbers.