Spatially Adaptive Stochastic Methods for Fluid-Structure Interactions Subject to Thermal Fluctuations in Domains with Complex Geometries

We develop stochastic mixed finite element methods for spatially adaptive simulations of fluid-structure interactions when subject to thermal fluctuations. To account for thermal fluctuations, we introduce a discrete fluctuation-dissipation balance condition to develop compatible stochastic driving fields for our discretization. We perform analysis that shows our condition is sufficient to ensure results consistent with statistical mechanics. We show the Gibbs-Boltzmann distribution is invariant under the stochastic dynamics of the semi-discretization. To generate efficiently the required stochastic driving fields, we develop a Gibbs sampler based on iterative methods and multigrid to generate fields with $O(N)$ computational complexity. Our stochastic methods provide an alternative to uniform discretizations on periodic domains that rely on Fast Fourier Transforms. To demonstrate in practice our stochastic computational methods, we investigate within channel geometries having internal obstacles and no-slip walls how the mobility/diffusivity of particles depends on location. Our methods extend the applicability of fluctuating hydrodynamic approaches by allowing for spatially adaptive resolution of the mechanics and for domains that have complex geometries relevant in many applications

Stochastic turbulence modeling in RANS simulations via Multilevel Monte Carlo

A multilevel Monte Carlo (MLMC) method for quantifying model-form uncertainties associated with the Reynolds-Averaged Navier-Stokes (RANS) simulations is presented. Two, high-dimensional, stochastic extensions of the RANS equations are considered to demonstrate the applicability of the MLMC method. The first approach is based on global perturbation of the baseline eddy viscosity field using a lognormal random field. A more general second extension is considered based on the work of [Xiao et al.(2017)], where the entire Reynolds Stress Tensor (RST) is perturbed while maintaining realizability. For two fundamental flows, we show that the MLMC method based on a hierarchy of meshes is asymptotically faster than plain Monte Carlo. Additionally, we demonstrate that for some flows an optimal multilevel estimator can be obtained for which the cost scales with the same order as a single CFD solve on the finest grid level.Comment: 40 page

Holistic mesoscale modelling of concrete – recent developments

Modelling of concrete at the mesoscale is needed in many applications, but developing a realistic mesoscale model for the analysis of concrete behaviour under general loading conditions is challenging. This paper presents an overview of the development of mesoscale modelling of concrete within a finite element framework for both quasi-static and high strain rate applications. A 2D mesoscale model incorporating random aggregates and equivalent interfacial transition zones enables examination into the effects of random aggregate structure and the sub-scale non-homogeneity within the mortar matrix on the macroscopic behaviour of concrete. In applications where multi-axial stresses and confinement effects are significant, such as under high-strain rate loading where the inertial confinement plays an important role, a realistic representation of the multi-axial stress condition becomes necessary, and this requires 3D mesoscale model. Two types of 3D mesoscale concrete model have been developed, namely a pseudo-3D mesoscale model and a full 3D mesoscale model. For the explicit representation of the fracture process, a cohesivecontact approach has been implemented, at present in a 2D mesoscale framework. Illustrative examples are given to demonstrate the performance of the mesoscale models and the results are discussed

Efficient Bayesian estimation of Markov model transition matrices with given stationary distribution

Direct simulation of biomolecular dynamics in thermal equilibrium is challenging due to the metastable nature of conformation dynamics and the computational cost of molecular dynamics. Biased or enhanced sampling methods may improve the convergence of expectation values of equilibrium probabilities and expectation values of stationary quantities significantly. Unfortunately the convergence of dynamic observables such as correlation functions or timescales of conformational transitions relies on direct equilibrium simulations. Markov state models are well suited to describe both, stationary properties and properties of slow dynamical processes of a molecular system, in terms of a transition matrix for a jump process on a suitable discretiza- tion of continuous conformation space. Here, we introduce statistical estimation methods that allow a priori knowledge of equilibrium probabilities to be incorporated into the estimation of dynamical observables. Both, maximum likelihood methods and an improved Monte Carlo sampling method for reversible transition ma- trices with fixed stationary distribution are given. The sampling approach is applied to a toy example as well as to simulations of the MR121-GSGS-W peptide, and is demonstrated to converge much more rapidly than a previous approach in [F. Noe, J. Chem. Phys. 128, 244103 (2008)]Comment: 15 pages, 8 figure

Random field sampling for a simplified model of melt-blowing considering turbulent velocity fluctuations

In melt-blowing very thin liquid fiber jets are spun due to high-velocity air streams. In literature there is a clear, unsolved discrepancy between the measured and computed jet attenuation. In this paper we will verify numerically that the turbulent velocity fluctuations causing a random aerodynamic drag on the fiber jets -- that has been neglected so far -- are the crucial effect to close this gap. For this purpose, we model the velocity fluctuations as vector Gaussian random fields on top of a k-epsilon turbulence description and develop an efficient sampling procedure. Taking advantage of the special covariance structure the effort of the sampling is linear in the discretization and makes the realization possible

Efficient Monte Carlo for high excursions of Gaussian random fields

Our focus is on the design and analysis of efficient Monte Carlo methods for computing tail probabilities for the suprema of Gaussian random fields, along with conditional expectations of functionals of the fields given the existence of excursions above high levels, b. Na\"{i}ve Monte Carlo takes an exponential, in b, computational cost to estimate these probabilities and conditional expectations for a prescribed relative accuracy. In contrast, our Monte Carlo procedures achieve, at worst, polynomial complexity in b, assuming only that the mean and covariance functions are H\"{o}lder continuous. We also explain how to fine tune the construction of our procedures in the presence of additional regularity, such as homogeneity and smoothness, in order to further improve the efficiency.Comment: Published in at http://dx.doi.org/10.1214/11-AAP792 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Uncertainty Quantification of geochemical and mechanical compaction in layered sedimentary basins

In this work we propose an Uncertainty Quantification methodology for sedimentary basins evolution under mechanical and geochemical compaction processes, which we model as a coupled, time-dependent, non-linear, monodimensional (depth-only) system of PDEs with uncertain parameters. While in previous works (Formaggia et al. 2013, Porta et al., 2014) we assumed a simplified depositional history with only one material, in this work we consider multi-layered basins, in which each layer is characterized by a different material, and hence by different properties. This setting requires several improvements with respect to our earlier works, both concerning the deterministic solver and the stochastic discretization. On the deterministic side, we replace the previous fixed-point iterative solver with a more efficient Newton solver at each step of the time-discretization. On the stochastic side, the multi-layered structure gives rise to discontinuities in the dependence of the state variables on the uncertain parameters, that need an appropriate treatment for surrogate modeling techniques, such as sparse grids, to be effective. We propose an innovative methodology to this end which relies on a change of coordinate system to align the discontinuities of the target function within the random parameter space. The reference coordinate system is built upon exploiting physical features of the problem at hand. We employ the locations of material interfaces, which display a smooth dependence on the random parameters and are therefore amenable to sparse grid polynomial approximations. We showcase the capabilities of our numerical methodologies through two synthetic test cases. In particular, we show that our methodology reproduces with high accuracy multi-modal probability density functions displayed by target state variables (e.g., porosity).Comment: 25 pages, 30 figure