Stochastic Boundary Conditions Research Articles

A Boundary Control Problem for Stochastic 2D-Navier–Stokes Equations

AbstractWe study a stochastic velocity tracking problem for the 2D-Navier–Stokes equations perturbed by a multiplicative Gaussian noise. From a physical point of view, the control acts through a boundary injection/suction device with uncertainty, modeled by stochastic non-homogeneous Navier-slip boundary conditions. We show the existence and uniqueness of the solution to the state equation, and prove the existence of an optimal solution to the control problem.

Global well-posedness and interior regularity of 2D Navier–Stokes equations with stochastic boundary conditions

AbstractThe paper is devoted to the analysis of the global well-posedness and the interior regularity of the 2D Navier–Stokes equations with inhomogeneous stochastic boundary conditions. The noise, white in time and coloured in space, can be interpreted as the physical law describing the driving mechanism on the atmosphere–ocean interface, i.e. as a balance of the shear stress of the ocean and the horizontal wind force.

Combination of Karhunen-Loève and intrusive polynomial chaos for uncertainty quantification of thermomagnetic convection problem with stochastic boundary condition

Uncertainty propagation analysis plays a crucial role in understanding the impact of variations in initial condition, boundary condition, and fluid physical properties on simulation results for thermomagnetic convection heat transfer problems. To effectively analyze the uncertainty problem of thermomagnetic convection caused by random temperature fluctuations, a novel and comprehensive computational framework based on intrusive polynomial chaos approach was proposed in this paper. Our proposed approach aims to represent random temperature boundaries using Karhunen-Loève expansion (KLE) and stochastic output response using polynomial chaos expansion (PCE). In addition, we use spectral projection method to transform the stochastic control equation into a group of deterministic control equations. We obtained the uncertainty characteristics of the stochastic output response by solving each polynomial chaos basis. Compared with Monte Carlo simulation (MCS), our approach accurately and efficiently predicts the uncertainty characteristics of thermomagnetic convection problems under stochastic temperature boundary conditions while significantly reducing computational resources.

Reduced-Order Modeling with Time-Dependent Bases for PDEs with Stochastic Boundary Conditions

Reduced-Order Modeling with Time-Dependent Bases for PDEs with Stochastic Boundary Conditions

On equilibrating non-periodic molecular dynamics samples for coupled particle-continuum simulations of amorphous polymers

In the context of fracture simulations of polymers, the molecular mechanisms in the vicinity of the crack tip are of particular interest. Nevertheless, to keep the computational cost to a minimum, a coarser resolution must be used in the remaining regions of the numerical sample. For the specific case of amorphous polymers, the Capriccio method bridges the gap between the length and time scales involved at the different levels of resolution by concurrently coupling molecular dynamics (MD) with the finite element method (FEM). Within the scope of the Capriccio approach, the coupling to the molecular MD region introduces non-periodic, so-called stochastic boundary conditions (SBC). In similarity to typical simulations under periodic boundary conditions (PBC), the SBC MD simulations must reach an equilibrium state before mechanical loads are exerted on the coupled systems. In this contribution, we hence extensively study the equilibration properties of non-periodic MD samples using the Capriccio method. We demonstrate that the relaxation behavior of an MD-FE coupled MD domain utilizing non-periodic boundary conditions is rather insensitive to the specific coupling parameters of the method chosen to implement the boundary conditions. The behavior of an exemplary system equilibrated with the parameter set considered as optimal is further studied under uniaxial tension and we observe some peculiarities in view of creep and relaxation phenomena. This raises important questions to be addressed in the further development of the Capriccio method.

Uncertainty quantification of two-phase flow in porous media via the Coupled-TgNN surrogate model

The uncertainty quantification (UQ) of subsurface two-phase flow usually requires numerous executions of forward simulations under varying conditions. In this work, a novel coupled theory-guided neural network (TgNN) based surrogate model is built to facilitate computation efficiency under the premise of satisfactory accuracy. The core notion of this proposed method is to bridge two separate blocks on top of an overall network. They underlie the TgNN model in a coupled form, which reflects the coupling nature of pressure and water saturation in the two-phase flow equation. The TgNN model not only relies on labeled data, but also incorporates underlying scientific theory and experiential rules (e.g., governing equations, stochastic parameter fields, boundary and initial conditions, well conditions, and expert knowledge) as additional components into the loss function. The performance of the TgNN-based surrogate model for two-phase flow problems is tested by different numbers of labeled data and collocation points, as well as the existence of data noise. The proposed TgNN-based surrogate model offers an effective way to solve the coupled nonlinear two-phase flow problem, and shows good accuracy and strong robustness when compared with the purely data-driven surrogate model. By combining the accurate TgNN-based surrogate model with the Monte Carlo method, UQ tasks can be performed at a minimum cost to evaluate statistical quantities. Since the heterogeneity of the random fields strongly impacts the results of the surrogate model, corresponding variance and correlation length are added to the input of the neural network to maintain its predictive capacity. In addition, several more complicated scenarios are also considered, including dynamically changing well conditions and dynamically changing variance of random fields. The results show that the TgNN-based surrogate model exhibits satisfactory accuracy, stability, and efficiency in the UQ problem of subsurface two-phase flow.

Stochastic boundary condition effects on supersonic leading edge blowing

The present manuscript studies the robustness of leading edge blowing for active control of the upstream bow shock and overall aerothermal characteristics of a supersonic airfoil subject to variability in the injection conditions. The injection boundary conditions were considered stochastic and their randomness was propagated to the flow topology and aerothermal quantities of interest by means of non-intrusive Polynomial Chaos Expansions. The evaluations needed to build these expansions were carried out via steady two-dimensional Reynolds-Averaged Navier Stokes simulations. Two different injection port sizes were considered, with the smallest promoting a Coanda effect that gives rise to non-symmetric flow topologies. The uncertain boundary conditions lead to significant variations in the flow topology, particularly at the injection exit and the bow shock position, with variations of up to 75% of the nominal value. This study identifies the key operating conditions to ensure the optimal application of leading-edge injection for active control of supersonic airfoils.

Uncertainty quantification in a mechanical submodel driven by a Wasserstein-GAN

The analysis of parametric and non-parametric uncertainties of very large dynamical systems requires the construction of a stochastic model of said system. Linear approaches relying on random matrix theory Soize (2000) and principal component analysis can be used when systems undergo low-frequency vibrations. In the case of fast dynamics and wave propagation, we investigate a random generator of boundary conditions for fast submodels by using machine learning. We show that the use of non-linear techniques in machine learning and data-driven methods is highly relevant.Physics-informed neural networks Raissi et al. (2017) are a possible choice for a data-driven method to replace linear modal analysis. An architecture that supports a random component is necessary for the construction of the stochastic model of the physical system for non-parametric uncertainties, since the goal is to learn the underlying probabilistic distribution of uncertainty in the data. Generative Adversarial Networks (GANs) are suited for such applications, where the Wasserstein-GAN with gradient penalty variant Gulrajani et al. (2017) offers improved convergence results for our problem.The objective of our approach is to train a GAN on data from a finite element method code (Fenics) so as to extract stochastic boundary conditions for faster finite element predictions on a submodel. The submodel and the training data have both the same geometrical support. It is a zone of interest for uncertainty quantification and relevant to engineering purposes. In the exploitation phase, the framework can be viewed as a randomized and parametrized simulation generator on the submodel, which can be used as a Monte Carlo estimator.

Tailoring the Structural and Electronic Properties of Graphene through Ion Implantation.

Ion implantation is a superior post-synthesis doping technique to tailor the structural properties of materials. Via density functional theory (DFT) calculation and ab-initio molecular dynamics simulations (AIMD) based on stochastic boundary conditions, we systematically investigate the implantation of low energy elements Ga/Ge/As into graphene as well as the electronic, optoelectronic and transport properties. It is found that a single incident Ga, Ge or As atom can substitute a carbon atom of graphene lattice due to the head-on collision as their initial kinetic energies lie in the ranges of 25–26 eV/atom, 22–33 eV/atom and 19–42 eV/atom, respectively. Owing to the different chemical interactions between incident atom and graphene lattice, Ge and As atoms have a wide kinetic energy window for implantation, while Ga is not. Moreover, implantation of Ga/Ge/As into graphene opens up a concentration-dependent bandgap from ~0.1 to ~0.6 eV, enhancing the green and blue light adsorption through optical analysis. Furthermore, the carrier mobility of ion-implanted graphene is lower than pristine graphene; however, it is still almost one order of magnitude higher than silicon semiconductors. These results provide useful guidance for the fabrication of electronic and optoelectronic devices of single-atom-thick two-dimensional materials through the ion implantation technique.

A domain decomposition method for the Navier\u2013Stokes equations with stochastic input

The paper is committed to studying the domain decomposition method for the incompressible Navier–Stokes equations(NSEs) with stochastic input. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as trial basis to represent the random space, and the stochastic NSEs system are transformed into deterministic ones via the polynomial chaos expansion. The corresponding deterministic equations are transformed into the constrained optimization problem by minimizing the cost function on the common interface after the whole domain decomposed into two sub-domains. The constrained optimization problems are transformed into unconstrained problems by the Lagrange multiplier rule. A gradient method-based approach to the solutions of domain decomposition problem is proposed to solve the unconstrained optimality system. Finally, one numerical simulation experiment for square cavity flow problem with the stochastic boundary conditions are performed to demonstrate the feasibility and applicability of the gradient method.

An approach to numerical estimating the stability of multilevel constitutive models

Multilevel constitutive models of materials give the possibility to explicitly describe the mechanisms of inelastic deformation, evolution of a material structure and changes in the physical and mechanical properties of materials determined by their chemical composition and their internal structure. Therefore, these models seem to be very effective for improving metal processing and forming techniques. A study of the solutions (response change histories) obtained using constitutive models under perturbation of input data (influence history and initial conditions) and model operator is actual due to the fact that the material mechanical characteristics (including the lower scale level properties), physical processes occurring during deformation (for example, acts of interaction of defect structures at the microscale level) and the resulting influences (produced by stochastic boundary conditions) are stochastic in nature. Finding the solution to this problem is particularly important when researchers need to justify the use of new constitutive models for describing modern technological processes of thermomechanical treatment, in particular, those focused on creation of functional materials. The disadvantages of traditional analytical approaches (Lyapunov methods) taken to analyze the stability of multilevel material models have been briefly discussed. The definition of the solution stability is introduced; in contrast to the traditional definition, it takes into account the parametric perturbation of operators and the perturbation of the history of influences, which determine the right-hand side of the system of equations. A procedure for the model stability numerical assessment includes consideration of solutions stability for various values of the parameters that determine the operator and input data. The description of the program of computational experiments for the implementation of the proposed approach is presented. This program can be used to study various perturbations of initial conditions, the influence history, the operator, as well as to analyze the norms of their deviations and the integral norm of deviation of perturbed solutions from the base ones obtained in calculations with unperturbed parameters.

基于嵌入式多项式混沌展开法的随机边界下流动与传热问题不确定性量化

An intrusive polynomial chaos expansion method and a finite element program framework were proposed to quantify the uncertainty of flow and heat transfer problems under stochastic boundary conditions. In this method, the Karhunen-Loeve expansion was used to express the stochastic boundary condition, and the polynomial chaos expansion method was used to express the output random field. At the same time, the control equation was transformed into a set of deterministic equations with the spectral decomposition technology, and each polynomial chaos was solved to obtain the statistical characteristics of the numerical solution. Compared with the Monte Carlo method, this method can accurately and efficiently predict the uncertainty characteristics of flow and heat transfer problems under stochastic boundary conditions, and can save a lot of computing resources.

Numerical simulation of a supersonic turbulent compressible jet in a co-flow with stochastic spectral inflow boundary conditions

A compressible supersonic turbulent jet of a perfect gas in a co-flow with the formulation of stochastic spectral inflow boundary conditions is numerically modeled. The base equations are the LES averaged Navier – Stokes equations closed by the Smagorinsky model, the solution of which is carried out by the ENO scheme of the third order of accuracy. The stochastic boundary conditions at the inlet are constructed on the basis of the spectral method of generating fluctuations of gas-dynamic variables to obtain an inhomogeneous anisotropic turbulent flow. The numerical results of turbulent characteristics are compared with experimental data for the shear layer problem. The thickness of the shear layer is obtained, in which the growth of the shear layer between the jet and the co-flow for three types of grid (coarse, medium and fine) is demonstrated. Coherent vortex structures appearing in the jet are constructed in dynamics, which made it possible to analyze in detail the growth and development of vortices over time.

Nonequilibrium ab initio molecular-dynamics simulations of lattice thermal conductivity in irradiated glassy Ge2Sb2Te5

An analysis of thermal transients from nonequilibrium ab initio molecular-dynamics simulations can be used to calculate the thermal conductivity of materials with a short phonon mean-free path. We adapt the approach-to-equilibrium methodology to the three-dimensional case of a simulation that consists of a cubic core region at higher temperature approaching thermal equilibrium with a thermostatted boundary. This leads to estimates of the lattice thermal conductivity for the glassy state of the phase-change memory material, Ge2Sb2Te5, which are close to previously reported experimental measurements. Self-atom irradiation of the material, modeled using thermal spikes and stochastic-boundary conditions, results in glassy models with a significant reduction of diffusive thermal transport compared to the pristine glassy structure. This approach may prove to be useful in technological applications, e.g., for the suppression of thermal cross talk in phase-change memory and data-storage devices.

Uncertainty quantification based optimization of centrifugal compressor impeller for aerodynamic robustness under stochastic operational conditions

Centrifugal compressor impeller operates at stochastic boundary conditions. The operational uncertainties cause performance deviation from design value and consequently affect the reliability of the compressor. Considering the stochastic operational uncertainties in the early design stage, this paper presents an uncertainty quantification based optimization of centrifugal compressor impeller with splitter blades for aerodynamic robustness. The nonlinear response relation between the design variables, the aerodynamic boundary uncertainties and the impeller performance is modelled by a combination of the Latin Hypercube Sampling, the three dimensional CFD, the Kriging surrogate model and the Non-intrusive Probability Collocation method. A sensitivity analysis with single and multiple random geometry variations is carried out to identify the most sensitive parameters. The effects of rotor speed uncertainty on pressure ratio and efficiency are quantified. A case study is conducted to search for optimal impellers with higher performance and lower sensitivity to boundary uncertainty using NSGA-II. The optimization is verified by the Monte Carlo method. The results demonstrate that the aerodynamic robustness of the compress impeller with splitter blades is enhanced by the proposed approach, which provides references for compressors, turbines and other turbo machinery.

Stochastic field representation using bi-fidelity combination of proper orthogonal decomposition and Kriging

In the current paper efficient uncertainty quantification (UQ) of high dimensional stochastic fields is performed via a bi-fidelity surrogate model. The method is based on combination of proper orthogonal decomposition (POD) and Kriging method. In the developed method, the trend function of the Kriging model is estimated via a low cost POD, whilst the stochastic Gaussian contribution is tuned using a limited number of high-fidelity computations. The proposed method is applied to three test cases, including a highly nonlinear test function, thermally driven flow in a cavity with stochastic wall temperatures and turbulent heat transfer in a ribbed channel with stochastic heat flux boundary condition. Implementing both POD and POD–Kriging methods leads to significant reduction of computational cost in comparison to the classical full polynomial chaos expansion (PCE) with least cost saving of more than 60%. Furthermore, in all cases combined POD–Kriging method performs either superior or at least similar to the POD method. More specifically, in surrogate models with lower number of low- and high-fidelity samples greater gains in accuracy are obtained. Consequently, for a specific level of accuracy, POD–Kriging method requires lower computational cost.

On periodic boundary conditions and ergodicity in computational homogenization of heterogeneous materials with random microstructure

Due to high computational costs associated with stochastic computational homogenization, a highly complex random material microstructure is often replaced by simplified, parametric, ergodic, and sometimes periodic models. This replacement is often criticized in the literature due to unclear error resulting from the periodicity and ergodicity assumptions. In the current contribution we perform a validation of both assumptions through various numerical examples. To this end we compare large-scale non-simplified and non-ergodic models with simplified, ergodic, and periodic solutions. In addition we analyze the Hill–Mandel condition for stochastic homogenization problems and demonstrate that for a stochastic problem there are more than three classical types of boundary conditions. As an example, we propose two novel stochastic periodic boundary conditions which possess a clear physical meaning. The effect of these novel periodic boundary conditions is also analyzed by comparing with non-ergodic simulation results.

Efficient uncertainty quantification of stochastic heat transfer problems by combination of proper orthogonal decomposition and sparse polynomial chaos expansion

To increase the contribution and reliability of computational fluid dynamics efforts in design process of industrial equipments, it is necessary to quantify the effects of uncertainties on the system performance. Due to exponentially increment of the computational cost with number of uncertain variables for uncertainty quantification using classical polynomial chaos expansion methodology, reducing the required number of samples for uncertainty quantification is a real engineering challenge. In this paper, the proper orthogonal decomposition method based on the multifidelity approach is combined with the full and sparse polynomial chaos expansions for efficient uncertainty quantification of complex heat transfer problems with large number of random variables. The conjugate conduction heat transfer in NASA C3X cooled gas turbine blade with geometrical uncertainties and the convective heat transfer in ribbed passage with the stochastic wall heat flux boundary condition are considered as the test cases. Results of uncertainty quantification analysis in both test cases showed that proposed multi-fidelity approaches are able to produce the statistical quantities with much lower computational cost compare to the classical regression-based polynomial chaos method. It is shown that the combination of the proper orthogonal decomposition with the sparse polynomial chaos gives a computational gain at least 2 times greater than combination of the proper orthogonal decomposition with the full polynomial chaos expansion.

Two extensions of exact nonequilibrium steady states of a boundary-driven cellular automaton

Recently Prosen and Mejía-Monasterio (2016 J. Phys. A: Math. Theor. 49 185003) obtained exact nonequilibrium steady states of an integrable and reversible cellular automaton driven by some stochastic boundary conditions. In this paper, we explore the possible extensions of their method by generalizing the boundary conditions. As the result, we find two cases where such an extension is possible. One is obtained by generalizing probabilities for values of virtual cells while imposing a special condition on emission and absorption rates. Although this actually coincides with the conditional boundary driving discussed in Prosen and Buča (2017 J. Phys. A: Math. Theor. 50 395002), we present a solution in a different form. The other is obtained by considering a conserved quantity as energy, and by considering boundaries as heat reservoirs. The latter contains the original solution and the former one as the special cases. Properties of both solutions are discussed.

A Copula-Based Bayesian Network for Modeling Compound Flood Hazard from Riverine and Coastal Interactions at the Catchment Scale: An Application to the Houston Ship Channel, Texas

Traditional flood hazard analyses often rely on univariate probability distributions; however, in many coastal catchments, flooding is the result of complex hydrodynamic interactions between multiple drivers. For example, synoptic meteorological conditions can produce considerable rainfall-runoff, while also generating wind-driven elevated sea-levels. When these drivers interact in space and time, they can exacerbate flood impacts, a phenomenon known as compound flooding. In this paper, we build a Bayesian Network based on Gaussian copulas to generate the equivalent of 500 years of daily stochastic boundary conditions for a coastal watershed in Southeast Texas. In doing so, we overcome many of the limitations of conventional univariate approaches and are able to probabilistically represent compound floods caused by riverine and coastal interactions. We model the resulting water levels using a one-dimensional (1D) steady-state hydraulic model and find that flood stages in the catchment are strongly affected by backwater effects from tributary inflows and downstream water levels. By comparing our results against a bathtub modeling approach, we show that simplifying the multivariate dependence between flood drivers can lead to an underestimation of flood impacts, highlighting that accounting for multivariate dependence is critical for the accurate representation of flood risk in coastal catchments prone to compound events.