ABSTRACT. In adsorption separations, mixtures flow through a column packed with solid particles. The stronger adsorbing component moves more slowly, causing the exiting mixture to separate relative to the inlet. By exploiting differences in affinity for a solid material, rather than heating and cooling (e.g., conventional distillation), adsorption separations can be very energy efficient. Understanding the so-called “break-through curve” measurement, the outlet fluid concentrations as a function of time, is central to scale-up. Unfortunately, interpretation relies on the assumption of (i) infinitely-fast mass transfer and (ii) infinitely-long columns, which are seemingly unrelated. To aid in understanding, boundary layer theory is used to analyze isothermal single-solute adsorption with plug flow in the limit of fast adsorption. The leading order “outer” form of the problem yields shock waves, and the associated boundary layers can be resolved by rescaling in a moving coordinate system. Previous theories are found to be related as the form of the problem outside (i) and inside (ii) the boundary layer. Each boundary layer, whose existence depends on the convexity of the isotherm, has one associated inflection point in the break-through curve. A comparison to numerical simulations is presented to support the validity of the approach.

ABSTRACT. In this work we introduce a generalized machine-learning approach for stochastic multi-scale material modeling that utilizes coarse-grained (CG) atomistic simulation data to inform upper-scale models in the form of nonlinear partial differential equations (PDEs) for material applications. As a first step, atomistic data generated by a lower-scale discrete model are coarse-grained using a Gaussian filtering of relevant atomic quantities (in our case, potential energies). To calibrate the upper-scale model, we perform a dimensionality reduction by projecting both the CG atomistic data and the continuum model solutions onto the Grassmann manifold and exploit its structural information to calculate the solution error, defined as the distance on the manifold between the atomistic and continuum model deformation fields. Gaussian Process (GP) regression is employed to construct a mapping between model parameters and solution error. Next, we perform surrogate-based optimization to obtain a set of parameters that minimizes discrepancies between the models. For this purpose, a local surrogate updating algorithm is employed that reduces significantly the computational cost. We demonstrate the applicability of the proposed methodology on a simulated metallic glass system and demonstrate that one can recover continuum mechanics model parameters that result in material response consistent with the atomistic simulations.

ABSTRACT. Optimal control and estimation of spatially-distributed dynamical systems is a mathematically and computationally challenging problem. For large-scale and spatially-distributed systems, localized control and estimation schemes – i.e., architectures in which measurements of the state of the system are only shared locally – are desirable in order to reduce the computational burden. In this work, we focus on spatially-invariant dynamics containing inertial, elastic and damping terms. We show that: 1) the optimal controller and state-estimator are inherently spatially-localized and 2) that elasticity together with environmental and measurement disturbances can further enhance this localization, decreasing the communication requirements to achieve optimal performance. Implications for the control of the continuum will be discussed.

ABSTRACT. In this talk, I will present some recent theoretical results on the approximation of electronic wave functions by (deep) ReLU Neural Networks. Based on the regularity of the solutions of the Hartree-Fock problem and on bounds on the approximation error by suitable piecewise polynomials, we obtain that neural networks provide exponentially convergent approximations. Specifically, we show that, given a target accuracy $\epsilon>0$ and a wave function, there exists a neural network that emulates the wave function with error at most $\epsilon$, and whose depth and width are bounded polylogarithmically in $\epsilon$. Results of this kind help in giving a first mathematical justification to electronic structure calculations that use neural networks for the representation of the wavefunctions. In addition, they can contribute to the design of efficient algorithms for the solution of problems in quantum mechanics.

ABSTRACT. Numerous algorithms exist to solve the Kohn-Sham equations of electronic structure. They are either based on the direct minimization of the energy under constraints or based on fixed point iterations to solve a self-consistent formulation of the problem. It is not clearly understood which class of algorithms is more efficient and robust in which situation. We propose in this talk a first approach to the understanding of the intrinsic differences between two simple algorithms of each class: a damped self-consistent field algorithm and a projected gradient descent. We perform a local analysis and derive explicit convergence rates, confirmed by numerical experiments.

ABSTRACT. Machine learning configuration interaction takes a different approach to machine learning in quantum chemistry by embedding an artificial neural network within a variational method to accelerate the calculation of an ab initio wavefunction. Here the neural network learns on the fly to predict important configurations in an iterative scheme that efficiently constructs a compact yet sufficiently accurate wavefunction.

I will discuss the recent developments of this concept including a more scalable, faster approach to the generation of trial configurations where the neural network predictions are also used as a hash function for the efficient removal of duplicates. Investigations into the transferability of the neural network and applications to ab initio potential energy curves will be presented. I will also present work on the parallelization of the method and ways to calculate excited states within the approach.

ABSTRACT. The dimension of the vector space in the full configuration interaction (FCI) method renders its application to molecules with even modest numbers of electrons intractable. Fast randomized iteration (FRI) addresses this challenge by stochastically sparsifying vectors and leveraging this sparsity to reduce the storage and computational costs of iterative linear algebra methods. Averaging is applied to reduce the resulting statistical errors. I will discuss the fundamentals of FRI and illustrate its application to the calculation of ground- and excited-state FCI energies, i.e., the low-lying eigenvalues of the FCI Hamiltonian matrix, for various molecules. These methods are sufficiently general to be applicable beyond the domains of FCI and electronic structure.

ABSTRACT. We examine the possibility of using a reinforcement learning algorithm to solve large-scale eigenvalue problems in which the desired eigenvector can be approximated by a sparse vector with at most k nonzero elements, where k is relatively small compare to the dimension of the matrix to be partially diagonalized. This type of problem arises in applications in which the desired eigenvector exhibits localization properties and in large-scale eigenvalue computations in which the amount of computational resource is limited. When the positions of these nonzero elements can be determined, we can obtain the k-sparse approximation to the original problem by computing eigenvalues of a k*k submatrix extracted from k rows and columns of the original matrix. We review a previously developed greedy algorithm for incrementally probing the positions of the nonzero elements in a k-sparse approximate eigenvector and show that the greedy algorithm can be improved by using an RL method to refine the selection of k rows and columns of the original matrix. We describe how to represent states, actions, rewards and policies in an RL algorithm designed to solve the k-sparse eigenvalue problem and demonstrate the effectiveness of the RL algorithm on two examples originating from quantum many-body physics.

ABSTRACT. Full configuration interaction (FCI) solvers are limited to small basis sets due to their expensive computational costs. An optimal orbital selection for FCI (OptOrbFCI) is proposed to boost the power of existing FCI solvers to pursue the basis set limit under a computational budget. The optimization problem coincides with that of the complete active space SCF method (CASSCF), while OptOrbFCI is algorithmically quite different. OptOrbFCI effectively finds an optimal rotation matrix via solving a constrained optimization problem directly to compress the orbitals of large basis sets to one with a manageable size, conducts FCI calculations only on rotated orbital sets, and produces a variational ground-state energy and its wave function. Coupled with coordinate descent full configuration interaction (CDFCI), we demonstrate the efficiency and accuracy of the method on the carbon dimer and nitrogen dimer under basis sets up to cc-pV5Z. We also benchmark the binding curve of the nitrogen dimer under the cc-pVQZ basis set with 28 selected orbitals, which provide consistently lower ground-state energies than the FCI results under the cc-pVDZ basis set. The dissociation energy in this case is found to be of higher accuracy.

ABSTRACT. Surface diffusion is one of the most important mechanisms driving crystal growth. Albeit usually neglected, adatoms (atoms freely diffusing on the surface of the crystal) seem to play a fundamental role in the description of the behaviour of a solid-vapor interfaces. For this reason, some years ago Fried and Gurtin introduced a system of evolution equations to describe such a situation, where adatoms are treated as a separate variable of the problem.

In this talk a first step in the programme of studying the above mentioned evolution equations from a variational point of view is presented. In particular, the focus is on the static problem in the small mass regime, where the elastic energy is negligible. Ground states, effective energy, and phase filed approximation suitable for numerical purposes are discussed. This latter is embedded in a general framework that allows to treat similar problems for a large class of functionals.

The talk is based on works in collaboration with Marco Caroccia (University of Rome 'Tor Vergata'), and Laurent Dietrich (Lycée Fabert).

ABSTRACT. In this seminar, I will present some recent results about upscaling of dislocations.

In the one-dimensional case, a discrete-to-continuum limit passage including an annihilation rule is studied. Considering both positive and negative particles, the empirical measures of the discrete dynamics satisfy a continuum evolution equation, whose solution can be identified with the limiting particle density under a mild separation assumption.

In the two-dimensional case, a system of screw dislocations confined in a domain by an external strain is considered. The energies associated with the empirical measures of the discrete system are shown to Gamma-converge to an energy associated with a measure which is absolutely continuous with respect to the H^1 measure restricted to the boundary of the domain.

These results are from joint works with Ilaria Lucardesi, Patrick van Meurs, Riccardo Scala, and Davide Zucco.

ABSTRACT. In this talk I will speak about a variational model from [Kholmatov, Piovano: ARMA 2020] to simultaneously treat Stress-Driven Rearrangement Instabilities (SDRI) such as boundary discontinuities, internal cracks, external filaments, delaminations, wetting, and brittle fractures. The model is characterized by an energy displaying both elastic and surface terms, and allowing a unified treatment from thin films to crystal cavities, and from capillarity to fracture models.

Existence of minimizing configurations is established in 2d: Compactness and energy lower semicontinuity are first shown w.r.t a proper selected topology in a class of (constrained) admissible configurations under the assumption that the free crystalline interface is the boundary consisting of an at most m connected components.

Next we show that, as m converges to infinity, the energy of constrained minimal admissible configurations tends to the minimum energy in the general class of configurations without the bound on the number of connected components for the free interface. Also using constraint m-minimizers as well as uniform density estimates for the local decay of the energy at the m-minimizers' boundaries, we will directly construct a global (unconstraint) minimizer of the SDRI model. Finally, we study some regularity properties for the morphology of any minimizer.

ABSTRACT. I present the study of the discrete-to-continuum variational limit of the antiferromagnetic XY model on the two-dimensional triangular lattice in the vortex regime. Within this regime, the spin system cannot overcome the energetic barrier of chirality transitions, hence one of the two chirality phases is prevalent. I discuss the order parameter that describes the vortex structure of the spin field in the majority chirality phase. The Γ-limit of the scaled energy can be calculated, showing that it concentrates on finitely many vortex-like singularities of the spin field.

ABSTRACT. We discuss the singular limit of diffuse to sharp interface models for the two-phase flow of viscous incompressible fluids. More precisely, we consider the sharp interface limit of a coupled Stokes/Cahn-Hilliard system, when a parameter $\varepsilon>0$ that is proportional to the thickness of the diffuse interface tends to zero. For sufficiently small times we prove in a two dimensional domain convergence of the solutions of the Stokes/Cahn-Hilliard system to solutions of a sharp interface model, where the Stokes system together with the Young-Laplace law for the jump of the stress tensor is coupled to a Mullins-Sekerka equation with additional convection term, which describes the evolution of the interface. To this end we construct a suitable approximation of the solution of the Stokes/Cahn-Hilliard system and estimate the difference with the aid of a suitable refinement of a spectral estimate of the linearized Cahn-Hilliard operator. This is a joint-work with Andreas Marquardt. If time permits we will also discuss recent advances in the corresponding analysis for a Stokes/Allen-Cahn system.

ABSTRACT. I will present well-posedness and regularity results for a free boundary problem for the two-dimensional Stokes equations, modelling a droplet on a substrate close to its moving front. The free boundary is the liquid-gas interface whose evolution is driven by surface tension. The considered model is subject to a linear Navier-slip condition and to a fixed microscopic contact angle (but all angles strictly between 0° and 90° are allowed).

It is well-known that this relatively parsimonious model leads to a singularity in the pressure close to the triple junction which excludes the existence of smooth solutions. This has sparked new proposed physical models seeking to remove this singularity.

In my talk, I want to advocate for keeping the simple model, accepting that solutions will develop singularities and building a suitable mathematical framework that can capture this behaviour. In particular, the presented (L^2-based) analysis gives precise information on the singular expansion of the velocity, the pressure and the profile in the vicinity of the triple junction, and yields global well-posedness of solutions for initial data close to a linear equilibrium profile.

The talk is based on ongoing work with Manuel Gnann, Hans Knüpfer and Nader Masmoudi.

ABSTRACT. Since the Seventies, various formal arguments predicted that, in the complete wetting regime, the "macroscopic" profile of a spreading droplet depends only logarithmically on the slippage model at intermediate time-scales. These arguments rely on the uniform validity of lubrication theory, a fact which does not hold in different situations, such as that of an "advancing meniscus": a ideal plate is plunged with constant, "small" velocity into a liquid reservoir, leading to the formation of a meniscus. In such geometry, the full curvature term must be accounted for.

In this talk, I will present a (formal and rigorous) asymptotic analysis of this model, which characterizes advancing menisci in terms of four regions: an inner foot region, dominated by slippage; a Tanner region, balancing slippage and viscous friction; a meniscus region, balancing friction and gravity; an outer "geometry" region, where gravity dominates and the full curvature term becomes relevant. The analysis suggest a natural notion of macroscopic contact angle as the leading-order value of the slope, attained at the boundary between Tanner and meniscus region. I will conclude with possible generalizations and open questions. This is a joint work with Felix Otto.

ABSTRACT. We study self-similar solutions of the thin-film equation that describe the lifting of an isolated touch-down point given by an initial profile of the form |x|. This provides a mechanism for non uniqueness of the thin-film equation with mobility exponent between 2 and 4 , since solutions with a persistent touch-down point also exist in this case. In order to prove the existence of the self-similar solutions, we study a four-dimensional continuous dynamical system. The proof consists of a shooting argument based on the identification of invariant regions and on suitable energy formulas.

ABSTRACT. Lattice structures are periodic porous bodies which are becoming popular since they are a good compromise between rigidity and weight and can be additively manufactured. Their optimization has been widely studied. We propose here to optimize structures built with particular lattice material: parametrized cells, previously optimized for stress.
 First, homogenized structures are optimized for the L^p-norm of stress. Second, using the de-homogenization method, we compute genuine structures. The efficiency of these structures is then discussed, they are also compared to optimized structures with non-optimized lattice structures.

ABSTRACT. We present a novel method for the solution of a particular class of structural optimzation problems: the continuous stochastic gradient method (CSG). In the simplest case, we assume that the objective function is given as an integral of a desired property over a continuous parameter set. The application of a quadrature rule for the approximation of the integral can give rise to artificial and undesired local minima. However, the CSG method does not rely on an approximation of the integral, instead utilizing gradient approximations from previous iterations in an optimal way. Although the CSG method does not require more than the solution of one state problem (of infinitely many) per optimization iteration, it is possible to prove in a mathematically rigorous way that the function value as well as the full gradient of the objective can be approximated with arbitrary precision in the course of the optimization process. Moreover, numerical experiments for a linear elastic problem with infinitely many load cases are described. For the chosen example, the CSG method proves to be clearly superior compared to the classic stochastic gradient (SG) and the stochastic average gradient (SAG) method.

ABSTRACT. In shape optimization it is desirable to obtain deformations of a given mesh without negative impact on the mesh quality. We propose a new algorithm using least square formulations of the Cauchy--Riemann equations. Our method allows us to deform two-dimensional meshes in a nearly conformal way and thus approximately preserves the angles of triangles during the optimization process. It outperforms similar approaches based on elasticity through the symmetric gradient on relevant test cases, but its good behavior is also observed to depend on the existence of conformal transformations between the initial and final shapes.

ABSTRACT. We consider the problem of minimizing the first eigenvalue of the p-Laplacian operator where the coefficients take two possible positive values. The control variable is the set where we place each of these coefficients. The measure of the set where we pose the best one is limited. We obtain a relaxed formulation and some regularity results for the solutions. As a consequence, we get some non-existence results for the unrelaxed formulation. The work extends previous results for the Laplacian operator.

ABSTRACT. In recent years, nonlocality has been given increasing attention in the modeling of materials properties and other complex systems, especially in the presence of anomalies and singularities. The effective modeling and simulation of nonlocal interactions bring on new challenges to mathematicians and many questions remain open. In particular, it is interesting to ask what are the signs for one to consider nonlocal modeling as a potentially helpful approach. We will present examples to address this question. We will further discuss some related mathematical and computational issues, as well as recent works.

ABSTRACT. The ability to remain cohesive is a crucial function of cell aggregates, and is required during tissue development and homeostasis. Interestingly, cell-cell separation or fracture is also biologically essential, a simple yet fundamental example being the formation of a new cavity or lumen. Tissue fracture can be the result of externally applied forces, actively generated endogenous forces, or downregulation of molecules involved in cell-cell adhesion. Over the last few years, it has been shown that hydraulic forces can also cause cell-cell separation, and that major morphogenetic events across species and including mammals rely on the formation and evolution of patterns of hydraulic cracks. In this talk I will discuss our recent work to understand the mechanisms governing the formation and evolution of patterns of hydraulic cracks between semi-permeable membranes bridged by adhesion molecules. To this end, we have developed mathematical and computational models to track the dynamics of osmolytes, of water flow, of adhesion molecules and of membrane mechanics during the hydraulic fracture of a biomimetic experimental system. Besides providing the physical rules governing hydraulically driven morphogenesis, our work may provide a background to control the formation of cavities in artificial systems formed by epithelial living materials.

ABSTRACT. In order to take advantage of biodegradable metallic materials (magnesium, zinc, and iron) in tissue engineering applications, their degradation parameters should be tuned to the rate of regeneration of new tissue. One approach for investigating biodegradation behavior is to construct computational models to assess the biodegradation properties prior to conducting experiments. Our developed model captures the release of metallic ions, changes in pH, the formation of a protective film, the dissolution of this film in presence of different ions, and the effect of perfusion of the surrounding fluid. This has been accomplished by deriving a system of time-dependent reaction-diffusion-convection partial differential equations from the underlying oxidation-reduction reactions. The level set formalism was employed to track the biodegradation interface between the biomaterial and its surroundings. The equations were solved implicitly using the finite element method for spatial terms (with a 1st order Lagrange polynomial as the shape function) and the backward-Euler finite difference method for temporal terms on an Eulerian mesh. A Bayesian optimization routine was used to calibrate the model and estimate the unknown parameters. The model was validated by comparing the predicted and experimentally obtained values of global pH changes, for which a good agreement was observed.

ABSTRACT. Self-Healing Ceramic Matrix Composites (SH-CMC) are at the frontline of technological developments in many application domains, and in particular in the aerospace sector. These composites are characterized by matrix layers which react with oxygen at the crack onset. The resulting oxide seals the fissure locally, delaying the oxidation and breaking of the fibres, thus considerably increasing the material lifetime. The complexity of the material, as well as its durability, makes numerical simulations that account for all the chemical, physical and structural aspects a very useful complement of experimental investigations. The work discussed in this contribution focuses on the self-healing process in transverse cracks inside a bundle under tensile load. We present a crack-averaged model coupling the oxygen diffusion through the crack, the evolution of the chemical species produced and consumed during oxidation, and a static fatigue model for fibres. A PDE model is established, and its dimensionless form studied to fully understand the contribution of each term, and perform the most appropriate discretization choices in terms of boundary conditions, and time integration. The present model can be used for different material configurations and environmental conditions, showing lifetime predictions in agreement with literature.

ABSTRACT. The synthesis of composite (multi-phase) polymer particles has a strong practical interest due to the performance superiority of multi-phase particles over particles with uniform compositions. The properties of resultant materials are strictly related to their morphology, defined by a pattern of phase-separated domains forming produced particles. Thus, an accurate prediction of the morphology formation is vital for the synthesis of new materials, but still not feasible due to its complexity. Intending to assist in the rational design of composite particles, we develop a Population Balance Equations (PBE) model for predicting a size distribution of polymer agglomerates forming the multi-phase morphology. Here, we propose a novel approach called Optimal Scaling with Constraints (OSC) which, due to its capability to identify and set up a scale-separated regime, allows us to significantly improve the performance of our PBE model. In particular, using OSC and taking advantage of the multi-scale nature of the involved processes, we derive an approximate PBE model for multi-phase particles morphology formation. In comparison with the original PBE model, the new model demonstrates the orders of magnitude higher computational performance and a better numerical stability, while still maintaining the same level of accuracy.

ABSTRACT. The Semistochastic Heatbath Configuration Interaction (SHCI) method is a fast and memory-efficicient Selected Configuration Interaction plus Perturbation Theory (SCI+PT) method. Two key innovations are: 1) It takes advantage of the fact that the Hamiltonian matrix elements for double excitations depend only on the four orbitals whose occupations change to make the determinant selection in both the variational and the perturbative steps much faster. A similar idea is used to speed up the selection of 1-body excitations as well. This enables a procedure in which only the important determinants are ever looked at, resulting in orders of magnitude saving in computer time. 2) It overcomes the memory bottleneck of the perturbative step by evaluating the perturbative energy correction using a 3-step semistochastic approach. The method has been applied to challenging molecular systems with more than 2 billion variational determinants and trillions of perturbative determinants, and to the homogeneous electron gas using more than 30,000 orbitals. Correlating 28 valence and semicore electrons, an accurate potential energy surface is obtained for the chromium dimer. Applications to the G1 set of molecules are also described. The efficiency of the method depends on the choice of orbitals. An accelerated orbital optimization method will be described.

ABSTRACT. Excited state computations are of great importance in understanding and predicting many physical phenomena in photochemistry, spectroscopy, and others. Comparing to the ground state computation, excited state computations are more difficult to mean-field methods Under the full configuration interaction (FCI) framework, it is also considered more difficult but is not as severe as that under mean-field framework. In general, the difficulties of excited state computations come from two parts. First, excited states are naturally of multi-reference feature. Second, the energy gaps within excited states are in general smaller than that between ground state and the first excited state. In this talk, we extend the recently developed efficient FCI solver, Coordinate Descent Full Configuration Interaction (CDFCI) to excited state computations.

ABSTRACT. Selected configuration interaction (sCI) methods exploit the sparsity of the full configuration interaction (FCI) wave function, yielding significant computational savings and wave function compression without sacrificing the accuracy. Despite recent advances in sCI methods, the selection of important determinants remains an open problem. We explore the possibility of utilizing reinforcement learning approaches to solve the sCI problem. By mapping the configuration interaction problem onto a sequential decision-making process, the agent learns on-the-fly which determinants to include and which to ignore, yielding a compressed wave function at near-FCI accuracy. This method, which we call reinforcement learned configuration interaction (RLCI), adds another weapon to the sCI arsenal and highlights how reinforcement learning approaches can potentially help solve challenging problems in electronic structure theory.

ABSTRACT. A molecular dynamics (MD) model for single-sheet graphene is coarse-grained into the peridynamic nonlocal continuum theory using homogenized displacements as the continuum degrees of freedom. The resulting momentum balance for the continuum displacements includes nonlocal bond forces that are obtained from the MD simulation. These bond forces replace the stress tensor that is used in the local theory. Fitting an algebraic expression to the bond forces as a function of the deformation provides a peridynamic material model. Machine learning techniques can be applied to help obtain the form of the material model and its parameters. The technique can include heterogeneities in the MD description and the effect of temperature. The application of the nonlocal model to boundary value problems demonstrates the potential of the coarse-graining method to achieve a multiscale description of graphene.

ABSTRACT. A hallmark of fracture modeling using peridynamics is the numerical emergence of cracks though the use of nonlocal modeling. Here, local interactions between neighboring points result in global consequences like the emergence of fracture surfaces. Emergent phenomena can be modeled non-locally and examples include motion of flocks of birds modeled through the Cuker Smale model. We provide a peridynamics model for calculating dynamic fracture. The force interaction is derived from a double well strain energy density function. The fracture set emerges from the model and is part of the dynamics. The material properties change in response to evolving internal forces eliminating the need for a separate phase field to model the fracture set. In the limit of zero nonlocal interaction, it is seen that the model reduces to a sharp crack evolution characterized by the classic Griffith free energy of brittle fracture with elastic deformation satisfying the linear elastic wave equation off the crack. The non-local model is seen to encode the well known kinetic relation between crack driving force and crack tip velocity. A rigorous connection between the nonlocal fracture theory and the wave equation posed on cracking domains given in Dal Maso and Toader is found. 


ABSTRACT. In this presentation, we discuss the restrictions imposed by bond-based peridynamics, particularly with respect to plane strain and plane stress models. We begin with a review of the derivations in [1] wherein for isotropic materials a Poisson's ratio restriction of 1/4 for plane strain and 1/3 for plane stress is deduced. Next, we show that Cauchy's relations are an intrinsic limitation of bond-based peridynamics and specialize this result to plane strain and plane stress models, generalizing the results of [1] and demonstrating that the Poisson's ratio restrictions in [1] are simply a consequence of Cauchy's relations. We conclude with a discussion of the validity of peridynamic plane strain and plane stress models formulated from two-dimensional bond-based peridynamic models.

Reference:

[1] Gerstle, W., Sau, N., Silling, S., Peridynamic modeling of plain and reinforced concrete structures, in: 18th International Conference on Structural Mechanics in Reactor Technology (SMiRT 18) (2005): 54-68.

[2] Trageser, J., Seleson, P., Bond-based peridynamics: A tale of two Poisson’s ratios, Journal of Peridynamics and Nonlocal Modeling 2 (2020): 278-288.

ABSTRACT. Glass-Ceramics (GCs), obtained by controlled crystallization of glass are an interesting class of materials. Controlled crystallization enables the simultaneous presence of multiple phases at the microscale – with at least one type of crystalline phase and a glassy phase. The properties of GCs are highly dependent on their microstructure, controlled by their composition and processing parameters, which enables a wide array of desirable properties. Of particular interest is the increase in fracture toughness of glass-ceramics compared to traditional glass. While experimental data shows this effect in several systems there are multiple mechanisms that can be operative in the multiphase materials. Experimental data suggests a wide variety of possible toughening mechanisms at the microscale. Computational modeling of GC microstructures is a promising tool than can further probe the microstructure but such studies in literature are scarce. Peridynamics, the recently developed non-local theory of continuum mechanics is uniquely suited to study GCs at the microscale. In this work, peridynamics has been applied to study crack propagation at the microscale by explicitly considering multiple phases of the microstructure. Toughening mechanisms are revealed which are functions of crystallinity and morphology. Suggestions are provided to improve the microstructure to increase toughening due to the microstructure.

ABSTRACT. A two species interacting system motivated by the density functional theory for triblock copolymers contains long range interaction that affects the two species differently. In a two species periodic assembly of discs, the two species appear alternately on a lattice. A minimal two species periodic assembly is one with the least energy per lattice cell area. There is a parameter b in [0,1] and the type of the lattice associated with a minimal assembly varies depending on b. There are several thresholds defined by a number B=0.1867... If b is in [0, B), a minimal assembly is associated with a rectangular lattice; if b is in [B, 1-B], a minimal assembly is associated with a square lattice; if b is in (1-B, 1], a minimal assembly is associated with a rhombic lattice. Only when b=1, this rhombic lattice is a hexagonal lattice. None of the other values of b yields a hexagonal lattice, a sharp contrast to the situation for one species interacting systems, where hexagonal lattices are ubiquitously observed.

ABSTRACT. We consider the minimization of an energy functional given by the sum of a crystalline perimeter and a nonlocal interaction of Riesz type, under volume constraint. We show that, in the small mass regime, if the Wulff shape of the anisotropic perimeter has certain symmetry properties, then it is the unique global minimizer of the total energy. In dimension two this applies to convex polygons which are reflection symmetric with respect to the bisectors of the angles. We further prove a rigidity result for the structure of (local) minimizers in two dimensions. This is a joint work with M. Bonacini and I. Topaloglu

ABSTRACT. We consider scalar transport equations involving nonlocal interaction terms and different kinds of mobility and we present how to obtain weak solutions (in some regimes even entropy solutions) as many particle limit of a suitable nonlocal version of the deterministic follow-the-leader scheme, which can be interpreted as the discrete Lagrangian approximation of the target pde. We discuss both the cases of linear and nonlinear mobilities as well as how the evolution is affected when a diffusive term is taken into account. The content of this talk is based on several works obtained in collaborations with S. Daneri, M. Di Francesco, S. Fagioli and E. Runa.

ABSTRACT. Nonlocal models have seen a rapid resurgence in recent years, motivated by successes and promising advances in the theory of peridynamics, applications in image processing, mixing alloys, biology and many more fields. In this talk I will discuss recent results for nonlinear nonlocal models that appear as conservation laws, diffusion models, where the integral operators converge to classical counterparts in the limit of the vanishing horizon.

ABSTRACT. In this talk I will discuss the minimization problem for a family of nonlocal interaction energies defined on probability measures. The energy functionals consist of a purely nonlocal term of convolution type, whose interaction kernel is given by a perturbation of the Coulomb kernel, and a quadratic confinement. I will show that, for even 0-homogeneous perturbations that are small in a suitable norm, the energy has a unique minimizer, which is the normalized characteristic function of an ellipse.

ABSTRACT. The Merriman-Bence-Osher thresholding scheme is an efficient numerical algorithm for multiphase mean curvature flow. In this talk, I want to present a new convergence result which applies to the generalized scheme for arbitrary surface tensions and mobilities by Salvador and Esedoglu.

The basis of the proof is the observation by Esedoglu and Otto that thresholding respects the gradient-flow structure of multiphase mean curvature flow: it can be interpreted as a minimizing movements scheme for an energy that approximates the interfacial area. De Giorgi's framework for general gradient flows provides an optimal energy dissipation relation for the scheme in which we pass to the limit to derive a dissipation-based weak formulation of multiphase mean curvature flow.

This is based on joint work with Felix Otto and with Jona Lelmi.

ABSTRACT. The system for magnetic fluids under investigation comprises the incompressible Navier-Stokes equations coupled with an evolutionary equation for the magnetization vector and the Cahn-Hilliard equations. In joint work with Martin Kalousek and Sourav Mitra we show global in time existence of weak solutions to the system using the time discretization method.

ABSTRACT. In this talk, I will describe recent results on the relation between the two-dimensional Euler equations and Kirchhoff's point vortex system in the case where the vorticity is an unbounded function. At the heart of the result is a stability estimate in terms of the Wasserstein distance between vorticity fields and singular point vortex solutions. The stability estimate is simple but remarkable, as it applies to a setting in which uniqueness to the Euler equations is not known.

ABSTRACT. One of the stimulating mathematical challenges behind the modelling of nematic elastomers is to prove the lower semicontinuity of energy functionals combining terms defined in the reference configuration with terms defined using Eulerian coordinates. The first existence result for a fully geometrically nonlinear model was obtained by Barchiesi & De Simone (ESAIM:COCV, 2015), assuming that the growth of the energy density with respect to the elastic part of the deformation gradient is at least of the form |F|^p for some p strictly larger than the space dimension n. In previous work (joint with M. Barchiesi and C. Mora-Corral) it was shown how to obtain the existence result assuming only that p > n-1. In this talk we present how to relax the coercivity to an energy density growing only as |F|^{n-1}(log |F|)^{\alpha} (joint with B. Stroffolini).

We will also discuss the related borderline problem of proving the existence of minimizers for the neoHookean energy (joint work with M. Barchiesi, C. Mora-Corral & R. Rodiac).

ABSTRACT. Morphing strategies of elastic surfaces based on exploiting the connection between surface metric and Gaussian curvature (Gauss’ Theorema Egregium) are becoming the focus of increasing attention, starting from seminal work by E. Sharon, M. Warner, and others.

We will discuss our own contributions to this emerging research field, starting form our studies of shape-shifting unicellular organisms and 3d-printed bio-inspired structures, and continuing with Liquid Crystal Elastomer sheets and pneumatic systems.

ABSTRACT. Thin solids often develop elastic instabilities and subsequently complex, multiscale deformation patterns. Motivated by experiments and simulations of ultrathin sheets subjected to radial stretching or thickness gradients, we employ a coarse-graining approach, and draw parallels between the meso-scale motifs in geometrically frustrated wrinkle patterns, and Landau-deGennes theory of chiral nematogens. As was pointed recently by Aharoni et al. the presence of an elastically-favored uniform wavelength of wrinkles imparts a local smectic order to the pattern. Nevertheles, a global smectic order is in conflict with a non-uniform director field, dictated by confinement forces, which imparts macroscale bend and splay rigidities to the pattern. We show that the conflict between these elements is akin to the frustrated smectic phase of chiral nematogen, and discover a defect- proliferated, amplitude-modulated state of wrinkles, analogous to the celebrated Renn-Lubensky twist-grain- boundary (TBG) phase of chiral liquid crystals.

ABSTRACT. Consider a localized defect (point defect of straight dislocation) embedded in a homogeneous host crystal. It is generally assumed / understood that the defect core is highly localised, while the far-field bulk behaviour is more or less generic. In this talk I will present a multi-pole inspired expansion of the elastic far-field, which makes this intuition precise, and discuss some applications, including e.g. high-accuracy boundary conditions for defect simulations, rigorous definition of the defect dipole moment, sharp estimates on the decay of the elastic field and the order at which defects interact.

ABSTRACT. Plasticity of submicron crystals defies conventional continuum theories because of the wild fluctuations in the plastic response, unpredictable yield stress and strong finite-size effects. These features are likely emergent from the cooperative effects of dislocations and their collective dynamics. We propose a mesoscopic theory based on the phase field crystal (PFC) modelling approach in which dislocations, defect kinetics, core structures and mutual interactions are emergent from a more fundamental description based on symmetry-broken states and their topological constraints.

We show that the dislocation velocity is determined by the evolution of the order parameter of the crystal symmetry, and becomes proportional to the Peach-Kohler force, in certain limits. We propose a versatile method to separate the slow timescale of the overdamped defect motion and the instantaneous relaxation of elastic disturbances by deriving an expression of the stress field from the order parameter and constraining it to be in mechanical equilibrium. We implement this method both in 2D and 3D for various crystalline symmetries. Coupling the model to an external stress, we shows that nucleation of edge dislocation dipoles in 2D is foreshadowed by the monotonic increase in the lattice incompatibility field from continuum dynamics, the latter derived from the phase-field model.

ABSTRACT. Amplitude expansion of the phase field crystal model (APFC) is an extension of the phase-field crystal (PFC) model that expresses the atomic density function of the PFC model in terms of complex amplitudes, describing amplitudes of waves aligned with a chosen set of base vectors.

As the APFC model enables modeling of phenomena with atomic resolution on adaptively refined meshes (AMR), it is ideally suited for the study of phenomena and industrial processes where both, large simulation volume and atomic resolution are required.

Its wider use has so far been hindered by the phenomena of beats in the complex amplitudes which occur in all grains rotated with regard to the initial choice of base vectors, and an unphysical grain boundary that appears in the model between grains, rotated by the crystal's symmetry rotation.

Simulations of dynamically rotating grains on an adaptive computational mesh using an improved APFC model will be presented. No unphysical grain boundary is observed in the entire rotation period and the grain boundary structure repeats in accordance with the rotational symmetry of the crystal lattice. The observed phenomena match PFC simulations. Improved model successfully overcomes the previous limitations of the APFC models.

ABSTRACT. From stadium covers to solar sails, we rely on deployability for the design of large-scale structures that can quickly compress to a fraction of their size. Historically, two main strategies have been pursued to design deployable systems. The first and most common approach involves mechanisms comprising interconnected bar elements, which can synchronously expand and retract, occasionally locking in place through bistable elements. The second strategy instead, makes use of inflatable membranes that morph into target shapes by means of a single pressure input. Neither strategy however, can be readily used to provide an enclosed domain able to lock in place after deployment. Here, we draw inspiration from origami, the Japanese art of paper folding, to design rigid-walled deployable structures that are multistable and inflatable. Guided by geometric analyses and experiments, we create a library of bistable origami shapes that can be deployed through a single fluidic pressure input. We then combine these units to build functional structures at the meter-scale, such as arches and emergency shelters, providing a direct pathway for a new generation of large-scale inflatable systems that lock in place after deployment and offer a robust enclosure through their stiff faces.

ABSTRACT. Lack of stiffness often limits thin shape-shifting structures to small scales. The large in-plane transformations required to program the reference metrics are indeed commonly achieved by using soft hydrogels or elastomers. We present here a versatile single-step method to shape-program stiff inflated structures, opening the door for numerous large scale applications. This technique relies on channel patterns obtained by heat-sealing superimposed flat quasi-inextensible fabric sheets. Inflating channels induces a coarse-grained anisotropic in-plane contraction through out-of-plane bending and thus a possible change of Gaussian curvature. Seam lines, which act as a director field for the in-plane deformation, encode the shape of the deployed structure. We present three patterning methods to quantitatively and analytically program shells with non-Euclidean metrics. In addition to shapes, we discuss the mechanical properties of the inflated structures.

ABSTRACT. Shape-morphing finds widespread utility, from the deployment of small stents and large solar sails to actuation and propulsion in soft robotics. Origami and Kirigami structures provide a template for shape-morphing, but the geometric rules governing their design are challenging to integrate into broad and versatile design tools, and the physics governing their mechanical behavior under loads and stimuli is challenging to model. This talk will describe our efforts to attack the design and modeling challenges inherent to origami and kirigami structures.

ABSTRACT. In this work, we study the most basic and fundamental geometric building block of Kirigami: a thin sheet with a single cut. We consider the deformation of a circular thin plate with a radial slit following its opening by a given excess angle. In the isometric limit—as the thickness of the disk approaches zero—the elastic energy has no stretching contribution and the shape of the disk is governed by the bending energy as it approaches that of an e-cone: a conical solution where all the generators remain straight and intersect at a singularity at its apex. We solve the post-buckling problem for the e-cone in the geometrically nonlinear setting assuming a Saint Venant-Kirchhoff constitutive plate model; we find closed-form expressions for the stress fields and show the geometry of the e-cone to be governed by the spherical elastica equation. This allows us to fully map out the space of solutions and investigate the stability of the post-buckled e-cone problem assuming mirror symmetric boundary conditions on the rotation of the lips on the open slit.

ABSTRACT. In this talk, we show that the permeability of porous media and that of a bubbly fluid are limiting cases of the complexified version of the two-fluid models posed in in the 1990 paper by Lipton and Avellaneda We assume the viscosity of the inclusion fluid is $z\mu_1$ and the viscosity of the hosting fluid is $\mu_1$, $z\in\field{C}$. The proof is carried out by construction of solutions for large $|z|$ and small $|z|$ by an iteration process inspired by the one used in the 1993 paper by Bruno and Leo and the analytic continuation. Moreover, we show that for a fixed microstructure, the permeabilities of these three cases share the same integral representation formula (IRF) \eqref{IRF_Ks_prime} with different values of $s'$, as long as the 'contrast parameter' $s':=\frac{1}{z-1}$ is not in the interval $[-\frac{2E_2^2}{1+2E_2^2},-\frac{1}{1+2E_1^2}]$, where the constants $E_1$ and $E_2$ are the extension constants that depend on the geometry of the microstructure.

ABSTRACT. We consider the evolution of the solid-liquid interface during melting and solidification of a material with constant internal heat generation and prescribed temperature and flux at the boundary of a cylinder and a sphere. The approach involves an assumption that the transition between solid and liquid phases occurs at a point. Using the method of separation of variables we find transient solutions. The steady-state part of the solution is obtained by direct integration. Finally, employing the energy balance equation at the interface, we derive ordinary differential equations for the interface that involve Fourier type infinite series terms. The obtained initial boundary value problems are solved numerically and the results show that during melting the solution in the solid phase exhibits the effect of overheating. We compare these solutions with the numerical solutions obtained using the enthalpy method which allows the presence of a transition or a mushy zone. We investigate the extent to which sharp interface models are able to capture the mushy zone.

ABSTRACT. The paper studies dense lattices in which the number of links exceeds the number of degrees of freedom of the nodes. In such lattices, the lengths of the links always satisfy additional compatibility conditions that are analogous to the compatibility conditions in continuum mechanics. We derive and analyze these conditions for infinitesimal and finite elongations and prove limiting theorems. The theory is applied to a quantitive description of a state of a partially damaged lattice, providing a measure of the degree of damage.