Abstract: We propose a novel variance reduction strategy based on control variables for simulating the averaged equation of a stochastic slow-fast system. In this system, we assume that the fast equation is ergodic, implying the existence of an invariant measure, for every fixed value of the slow variable. The right hand side of the averaged equation contains an integral with respect to this unknown invariant measure, which is approximated by the heterogeneous multiscale method (HMM). The HMM method corresponds to a Markov chain Monte Carlo method in which samples are generated by simulating the fast equation. As a consequence, the variance of the HMM estimator decays slowly. Therefore, we introduce a variance-reduced HMM estimator based on control variables: from the current time HMM estimation, we subtract a second HMM estimator at the previous time step using the exact same seed as the current time HMM estimator. To avoid introducing a bias, we add the previously calculated variance-reduced estimator. We analyze convergence of the proposed estimator and apply it to a linear and nonlinear model problem.

Abstract: Porous materials are instrumental in a wide range of micro- and nanoscale engineering applications, and porous media research continues to spawn innovations, e.g., in the biomedical and energy domains. For instance, oil recovery from reservoir rocks relies on multiphase flows that are governed by a complicated interplay of capillary pressures, permeability, and wettability of the porous formation. When a porous medium is filled with an electrolyte, dissociation or adsorption of ionic groups lead to a net surface charge which is compensated by an excess distribution of counterions in the bulk fluid. Under an applied electric field the charged ions accelerate the fluid resulting in electro-osmotic flow. Simple analytical theories can explain the basic effects but fail to predict quantitatively more complex electrokinetic transport phenomena that are affected by multiscale effects due to coupling of hydrodynamic, electrokinetic and diffusive transport. Mesoscopic simulation techniques have proven successful in solving numerically the coupled partial differential equations describing such systems, in particular when complex boundary conditions have to be taken into account. We present pore-scale simulations of electro-osmotic flow through a charged porous geometry using coupled lattice Boltzmann and link-flux methods implemented in our LB3D code. We investigate the dependence of the macroscopic fluxes on bulk ionic concentration, salt concentration, and applied potential gradient. Moreover, we apply the moment propagation method to calculate diffusion coefficients for neutral, cationic and anionic tracer particles in a simple model pore. The results reveal a crossover of the effective diffusion coefficient for charged tracers, and a non-monotonic dispersion coefficient depending on the salt concentration and Debye length. These findings are relevant for potential upscaling strategies between microscopic and macroscopic transport coefficients. Future extensions of our simulation approach include multiphase flows with charged amphiphilic surfactants, and more generally charged fluids for novel functional materials.

Abstract: A multicale modeling is developed to study rolling contact fatigue and predict fatigue lives. At the nanoscale, molecular dynamics simulations of confined n-alkanes are performed to calculate the friction coefficient of contact surface in the presence of lubrication. Then, the finite element method is used to conduct fatigue analysis of roller contact elements at the macroscale. The fatigue crack initiation life and the position of the initial crack can be estimated. This work can be viewed as a frame work for studying mechanical systems subject to cyclic loads with the consideration of lubrication effects.

Abstract: We compute the interaction between the light field and the non-linear polarization of an optically-active, semiconductor medium. This coupling strongly influences laser light behavior. For the macroscale calculations we have adapted the streamline-upwind/Petrov-Galerkin finite element method (FEM) to compute the laser propagation within the paraxial approximation. On the microscale, we compute the medium polarization local to the FEM Gauss points, using both a simple model (optical Kerr) and a sophisticated model (Semiconductor-Maxwell-Bloch equations within Monte Carlo simulation). The coupled medium-lightfield response is handled using the Hierarchical Multiscale Method. This approach enables large-scale, high-fidelity calculations on high-performance computers of both the laser propagation and the material response.

Abstract: The strength of crystalline materials are controlled, to a large extent, by the motion of dislocations. In materials with a high density of microstructural features, the motion of dislocations is restricted, resulting in increased strength. The small scale plasticity occurring in the vicinity of microstructure is a fundamentally multiscale phenomena. Bridging the characterization of individual dislocations at the atomistic scale with the macroscopic plastic response from cooperative dislocation motion at the continuum scale remains an open challenge. A method well suited for small scale plasticity is discrete dislocation dynamics (DDD) where plasticity is explicitly captured by the motion of dislocations. However, the computational expense of DDD grows immensely with the introduction of microstructure. Furthermore, parallel scalability is limited by the inherent differences in domain decomposition and load balancing when modeling both dislocations and microstructure. To address these issues, we have developed a multiple program multiple data (MPMD) approach to incorporating the effects of microstructure on plasticity. In this method, we couple DDD with a finite element (FE) solver to account for microstructural effects. Each application is executed separately to provide optimal domain decomposition, load balancing, and concurrency. Communication between applications is performed in parallel using distributed shared memory (DSM). In the present work, we analyze the performance of this algorithm and demonstrate the ability to model small scale plasticity in previously intractable systems.