We precisely and rigorously characterise the decay of elastic fields generated by dislocations in crystalline materials, focusing specifically on the role of multilattices. Concretely, we establish that the elastic field generated by a dislocation in a multilattice can be decomposed into a continuum field predicted by a linearised Cauchy-Born elasticity theory, and a discrete and nonlinear core corrector representing the defect core. We demonstrate both analytically and numerically the consequences of this result for cell size effects in numerical simulations.

Motivated by the emergence of dispersive, bianisotropic materials occurring as an effective medium model of metamaterials, we develop a time-domain algorithm capable of simulating general dispersive, bianisotropic media. This method utilizes auxiliary differential equations for the polarization and magnetization, which are solved in conjunction with Maxwell's equations using a method-of-lines approach with high-order finite-differences in space and integrated using the fourth order Runga-Kutta method.

We formulate the blended force-based quasicontinuum method for multilattices and develop rigorous error estimates in terms of the approximation parameters: choice of atomistic region, blending region, and continuum finite element mesh. Balancing the approximation parameters yields a convergent atomistic/continuum multiscale method for multilattices with point defects, including a rigorous convergence rate in terms of the computational cost. The analysis is illustrated with numerical results for a Stone–Wales defect in graphene.

The multiscale optimization of metafilm-based photonic systems may still remain computationally expensive, especially if the individual unit cell evaluations are not efficient. To circumvent this, we propose the use of effective bianisotropic tiles. In the simplest case, the homogenized metafilm can be obtained using a standard technique that approximates each cell by a tile with effective permittivity and permeability producing the same far-field response. This approach fails in the cases of symmetry breaking inside the cell - the cases common in metafilms with anisotropic unit cells fabricated on substrates. To overcome this problem, we are using bianisotropic homogenization (BAH), where a BA tensor accounts for spatial asymmetries. Here, we introduce new approaches to BAH of metafilms capable to interpolate their angular-dependent and frequency dispersive behavior. Our proposed approach is most general - by taking advantage of the fundamental theorems of linear algebra, we compare the retrieved eigenvalues with known theoretical cases. We verified our approaches with the conventional BAH techniques using normal incidence or tangential surface polarizations. The use of the optical dispersion, based on the generalized dispersive material model imposed by a set of Padé approximants allows for a flexible time-domain operation. Furthermore, accounting for the illumination intensity implies an innovative development of the BAH for non-linear metafilms. We demonstrate that, even with existing limitations, the proposed BAH technique offers extraordinary means for multiscale design of functional metadevices in frequency and time domains with useful expansions to the most general problems of non-linear optics, quantum computing, and imaging.

We formulate a model for a point defect embedded in a homogeneous multilattice crystal with an empirical interatomic potential interaction. Under a natural phonon stability assumption, we quantify the decay of the long-range elastic fields with increasing distance from the defect. These decay estimates are an essential ingredient in quantifying approximation errors in coarse-grained models and in the construction of optimal numerical methods for approximating crystalline defects.

We examine the Petviashvilli method for solving the equation $ \phi - \Delta \phi = |\phi|^{p-1} \phi$ on a bounded domain $\Omega \subset \mathbb{R}^d$ with Dirichlet boundary conditions. We prove a local convergence result, using spectral analysis, akin to the result for the problem on $\mathbb{R}$ by Pelinovsky & Stepanyants, 2004. We also prove a global convergence result by generating a suite of nonlinear inequalities for the iteration sequence, and we show that the sequence has a natural energy that decreases along the sequence.

We formulate and analyze an optimization-based Atomistic-to-Continuum (AtC) coupling method for problems with point defects. Application of a potential-based atomistic model near the defect core enables accurate simulation of the defect. Away from the core, where site energies become nearly independent of the lattice position, the method switches to a more efficient continuum model. The two models are merged by minimizing the mismatch of their states on an overlap region, subject to the atomistic and continuum force balance equations acting independently in their domains. We prove that the optimization problem is well-posed and establish error estimates.

Many important materials processes take place on time scales that vastly exceed the nanoseconds accessible to molecular dynamics simulation. Typically, this long-time dynamical evolution is characterized by a succession of thermally activated infrequent events involving defects in the material. In the accelerated molecular dynamics (AMD) methodology, known characteristics of infrequent-event systems are exploited to make reactive events take place more frequently, in a dynamically correct way. For certain processes, this approach has been remarkably successful, offering a view of complex dynamical evolution on time scales of microseconds, milliseconds, and sometimes beyond. Examples include metallic surface diffusion and growth, radiation damage annealing processes, and carbon nanotube dynamics. In this talk, I will discuss some recent advances that are extending the range of applicability of the AMD methods to larger and more complex systems. For example, we now understand that the parallel replica dynamics method can give exact state to state dynamics even when the state definitions are not necessarily Markovian, provided the dephasing stage is continued long enough [1]. This provides a clean framework for treating more complex systems where it is difficult to define simple, deep states, or where it is desirable to lump many basins together into one superstate. We have also formulated a new, local version of the hyperdynamics method that gives constant boost as the system size is increased, in contrast to standard hyperdynamics, for which the boost decays towards unity as the system size is increased. We have also been exploring the limiting boost that is possible with hyperdynamics when the bias potential can be designed in an ideal way.

We present a new optimization-based method for atomistic-to-continuum (AtC) coupling. The main idea is to cast the latter as a constrained optimization problem with virtual Dirichlet controls on the interfaces between the atomistic and continuum subdomains. The optimization objective is to minimize the error between the atomistic and continuum solutions on the overlap between the two subdomains, while the atomistic and continuum force balance equations provide the constraints. Separation, rather then blending of the atomistic and continuum problems, and their subsequent use as constraints in the optimization problem distinguishes our approach from the existing AtC formulations. We present and analyze the method in the context of a one-dimensional chain of atoms modeled using a linearized two-body potential with next-nearest neighbor interactions.

Several measurements are used to describe the behavior of a diabetic patient’s blood glucose. We describe a new, wavelet-based algorithm that indicates a new measurement called a PLA index could be used to quantify the variability or predictability of blood glucose. This wavelet-based approach emphasizes the shape of a blood glucose graph. Using continuous glucose monitors (CGMs), this measurement could become a new tool to classify patients based on their blood glucose behavior and may become a new method in the management of diabetes.