The peridynamics theory of solid mechanics is a nonlocal extension of classical continuum mechanics proposed by Stewart Silling. Peridynamics attempts to unify the mechanics of particle systems, cracks, and continuous media. It describes the behavior of systems by integral equations, in contrast to classical models based on partial differential equations. Dependence on differentiability of the displacement field limits the applicability of classical mechanics models, whereas discontinuous displacements represent no mathematical or computational difficulty for peridynamics. As a consequence, peridynamics has been applied to the study of material failure and damage.

Peridynamics is similar to molecular dynamics as it involves long-range interactions between material points. I investigated the connection between these models by studying peridynamics as an upscaling or continualization of molecular dynamics. I found that peridynamics recovers the same dispersion behavior as in molecular dynamics, when appropriate length scales are chosen; those effects are lost in classical elasticity. This work was performed in collaboration with Max Gunzburger, Michael L. Parks, and Richard B. Lehoucq.

Further investigations of peridynamic equations, in collaboration with Michael L. Parks, suggest that the nonlocality in the models can be controlled either by the interaction range or by the singularity of the integrand or kernel in the model. This study was applied to problems in wave propagation and fracture dynamics. Peridynamics converges to classical elasticity in the limit of vanishing nonlocality.

I investigated multiscale methods to concurrently couple peridynamics and classical elasticity. In collaboration with Samir Beneddine and Serge Prudhomme I developed and implemented a force-based blended approach. The domain of interest is decomposed into subdomains described by different models and a bridge region where the models are blended. A blending function is introduced to characterized each domain as well as to weigh the contribution of each model on the bridge region. The model is based on equilibration of forces at each point using blended equations of motion. In contrast to common approaches, this method derives a blended model from a single framework, preventing the appearance of spurious efects along bridging regions.

In atomistic-to-continuum (AtC) coupling techniques, an atomistic model is used in regions where microscale resolution is necessary, but elsewhere a (discretized) continuum model is applied. The main issues in AtC coupling methods are how to couple local continuum and nonlocal atomistic models and where to locate the interface region.

I investigated an energy-based blending approach for AtC coupling and studied its convergence behavior. I derived analytical relations to connect the atomistic and continuum models, which led to a consistent implementation of both models on the same system. Furthermore, different approaches for the implementation of Dirichlet-type boundary conditions in the atomistic region were proposed, which allowed for appropriate comparison between the performance of the discrete and continuum models. Numerical results for singular loads demonstrate that the AtC coupling approach performs better than the corresponding continuum model, compared to a reference atomistic solution. This motivates the implementation of AtC coupling models for cases where classical continuum models fail. This research was performed in collaboration with Max Gunzburger.

Model adaptivity techniques are focused on refining the mathematical models to control and determine the accuracy of surrogate models with respect to their reference models, as opposed to mesh refinement methods. In the context of concurrent multiscale modeling, where different models are coupled together, model adaptivity attempts to answer the basic question of where each model should be used.

I investigated a novel methodology for adaptive modeling applied to the Arlequin framework, based on phase-field methods. In contrast to more traditional approaches, where model adaptivity is driven by the geometry of the domain, the proposed method uses a diffuse interface approach to optimize the shape of the blending function to control errors with respect to a goal quantity. This method allows to handle highly complex geometries, avoids the introduction of a priori analytical functional forms for blending functions, and eliminates the need to track model interfaces. This work was done in collaboration with Timo van Opstal, Kris van der Zee, Serge Prudhomme, and Qiang Du.

Many complex systems nowadays involve processes occurring across several length and time scales. Unfortunately, the use of accurate fine-scale models is often too computationally expensive for many applications of practical interest. The need to simulate larger time and length scales than the ones characterizing the fine scale models, has motivated the design of modeling reduction techniques, as an attempt to achieve feasible and accurate descriptions of such systems. A particular class of methods commonly used for molecular dynamics systems is referred to as coarse-graining; these methods cluster groups of degrees of freedom in the fine-scale model and map each group to a single degree of freedom in a coarser description of the system of interest. A major challenge in coarse-graining is how to achieve a simpler description of the effective interactions in a system while preserving given properties of the system or quantities of interest.

In collaboration with Max Gunzburger, Michael L. Parks, and Richard B. Lehoucq, I developed a coarse-graining approach for crystal structures through a continuous upscaling of molecular dynamics, which preserves quantities of interest, such as the energy of the system. Having fewer degrees of freedom, simulations based on a coarse discretization of the continuum model require substantially less computer resources than their molecular dynamics counterparts.

Complex systems, such as polymeric materials, contain uncertainties which should be accounted for in the estimation of the coarse-grained model parameters. The Bayesian inference method is a common technique used in the field of uncertainty quantification to quantify and propagate sources of uncertainty. I collaborated with J. Tinsley Oden, Peter J. Rossky, Serge Prudhomme, Eric Wright and Kathryn Farrell, in the derivation of consistent coarse-grained models for polymeric materials. The proposed methodology aims at deriving the effective parameters of coarse-grained models based on virtual experiments using all-atom simulations. Calibration and validation of the coarse-grained models is based on Bayesian inference.

One dimensional hydrodynamical simulations reveal a critical mass scale for combined gas and dark matter systems, above which we have an expanding stable shock and below which we observe cold infall of gas that builds a disc. However, submillimeter observations reveal massive disc galaxies above the critical mass scale at high redshifts, which appear to be inconsistent with the theoretical picture.

In my Physics M.S. thesis, at the Hebrew University of Jerusalem, I investigated the filamentary structure appearing in N-body simulations of dark matter and its relation to galaxy formation processes simulated using hydrodynamical simulations. In particular, I focused on the study of cold streams penetrating through shocked gas. Important differences in the filamentary structure of big and small halos at redshift z = 0 were found that explain the cold flows effect, as well as its dependence on redshift. In addition, a strong correlation between the gas temperature and dark matter density profiles was found. Furthermore, I calculated linear correlations between different parameters of dark matter halos and found that dark matter properties are consistent with buildup by mergers. These results provided numerical support to a model postulated by Avishai Dekel and Yuval Birmboim, which explains the above observations. This research was performed in collaboration with Avishai Dekel.

As part of my undergraduate studies, I have been involved in the following research projects:

• Ellipsometric measurements of thin layer structures in porous silicon
  During the third year of my undergraduate physics program, I was involved in a research project in the group of Amir Sa’ar at the Department of Applied Physics of the Hebrew University of Jerusalem. During the project, I performed ellipsometric measurements of thin layer structures in porous silicon media, in order to determine their width and dielectric coefficients. The project involved the use of the ellipsometer machine in the visual and near infrared ranges, and the development of numerical algorithms to compute dielectric properties of materials.

• Shot noise measurements and development of techniques for the spectrum analyzer
  During my undergraduate studies, I participated in a Summer Internship at the Weizmann Institute of Science in Rehovot, Israel. I was involved in a research project in the group of Mordehai Heiblum, at the Department of Condensed Matter Physics. The project involved measurements of shot noise, which results were used to check Miliken’s experimental results for the value of the electron’s charge. In addition, I developed alternative techniques for the spectrum analyzer.