Random Boundary Conditions Research Articles

Stochastic Analysis of Groundwater Flow Subject to Random Boundary Conditions

A stochastic model was developed to simulate the flow in heterogeneous media subject to random boundary conditions. Approximate partial differential equations were derived based on the Karhunen-Loeve (KL) expansion and perturbation expansion. The effect of random boundary conditions on the two-dimensional flow was examined. It is shown that the proposed stochastic model is efficient to include the random boundary conditions. The random boundaries lead to the increase of head variance and velocity variance. The influence of the random boundary conditions on head uncertainty is exerted over the whole simulated region, while the randomness of the boundary conditions leads to the increase of the velocity variance in the vicinity of boundaries.

Correction to “Explicit analytical solutions for one‐dimensional steady state flow in layered, heterogeneous unsaturated soils under random boundary conditions”

[1] In this study, we directly derive first-order analytical solutions to the pressure head moments (mean and variance) for one-dimensional steady state unsaturated flow in randomly heterogeneous layered soil columns under various random boundary conditions. We assume that the constitutive relation between the unsaturated hydraulic conductivity and the pressure head follows an exponential model, and treat the saturated hydraulic conductivity Ks as a random function and the pore size distribution parameter α as a random constant. Unlike the solution given in Lu and Zhang (2004) in which Kirchhoff transformation was used and the solution to pressure head variance was presented as a function of (cross-)covariances related to the intermediate, Kirchhoff-transformed variable, the solution to the pressure head variance presented in this paper is an explicit function of the input variabilities. In addition, we also give analytical solutions to the statistics of the unsaturated hydraulic conductivity and the effective water content. These first-order analytical solutions are compared with those from Monte Carlo simulations. We also investigated the effect of uncertain boundary conditions, the relative contribution of input variabilities to the head variance, and the possible errors introduced by treating the correlated α field as a random constant in the analytical solutions. The results indicate that the uncertain constant head at the bottom of a deep soil column may not have a significant effect on predicting flow statistics in the upper portion of the column. Furthermore, it is found that treating α as a random constant is justified when the correlation length of α is relatively large as compared to the layer thickness.

Expansion of random boundary excitations for elliptic PDEs

In this paper we deal with elliptic boundary value problems with random boundary conditions. Solutions to these problems are inhomogeneous random fields which can be represented as series expansions involving a complete set of deterministic functions with corresponding random coefficients. We construct the Karhunen–Loeve series expansion which is based on the eigen-decomposition of the covariance operator. It can be applied to simulate both homogeneous and inhomogeneous random fields. We study the correlation structure of solutions to some classical elliptic equations in respond to random excitations of functions prescribed on the boundary. We analyze the stochastic solutions for Dirichlet and Neumann boundary conditions to Laplace equation, biharmonic equation, and to the Lame system of elasticity equations. Explicit formulae for the correlation tensors of the generalized solutions are obtained when the boundary function is a white noise, or a homogeneous random field on a circle, a sphere, and a half-space. These exact results may serve as an excellent benchmark for developing numerical methods, e.g., Monte Carlo simulations, stochastic volume and boundary element methods.

Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over [formula omitted

We consider ergodic random Schrödinger operators on the metric graph Z d with random potentials and random boundary conditions taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable. This limit, the integrated density of states, can be expressed by a closed Shubin–Pastur type trace formula. It supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. Among other examples we discuss random magnetic fields and percolation models.

Explicit analytical solutions for one‐dimensional steady state flow in layered, heterogeneous unsaturated soils under random boundary conditions

In this study, we directly derive first‐order analytical solutions to the pressure head moments (mean and variance) for one‐dimensional steady state unsaturated flow in randomly heterogeneous layered soil columns under various random boundary conditions. We assume that the constitutive relation between the unsaturated hydraulic conductivity and the pressure head follows an exponential model, and treat the saturated hydraulic conductivity Ks as a random function and the pore size distribution parameter α as a random constant. Unlike the solution given in Lu and Zhang (2004) in which Kirchhoff transformation was used and the solution to pressure head variance was presented as a function of (cross‐)covariances related to the intermediate, Kirchhoff‐transformed variable, the solution to the pressure head variance presented in this paper is an explicit function of the input variabilities. In addition, we also give analytical solutions to the statistics of the unsaturated hydraulic conductivity and the effective water content. These first‐order analytical solutions are compared with those from Monte Carlo simulations. We also investigated the effect of uncertain boundary conditions, the relative contribution of input variabilities to the head variance, and the possible errors introduced by treating the correlated α field as a random constant in the analytical solutions. The results indicate that the uncertain constant head at the bottom of a deep soil column may not have a significant effect on predicting flow statistics in the upper portion of the column. Furthermore, it is found that treating α as a random constant is justified when the correlation length of α is relatively large as compared to the layer thickness.

Estimating fatigue sensitivity to polycrystalline Ni‐base superalloy microstructures using a computational approach

ABSTRACTA computational study is conducted to determine the influence of microstructure attributes and properties on driving forces for fatigue crack formation and microstructurally small crack growth in a polycrystalline Ni‐base superalloy, IN100, a turbine disk alloy. A principal objective is to obtain quantitative estimates of the effect of variability of microstructure features on scatter in fatigue life or fatigue strength for a given life. Understanding is sought regarding sensitivity of driving forces to various microstructure attributes that may guide selection of the process route to tailor microstructure to achieve fatigue resistance. A microstructure‐sensitive crystal plasticity model is used to explicitly model individual grains and polycrystals, which is then used to explore effects of: (a) grain size distribution and (b) secondary and tertiary coherent γ′ precipitate size distributions and volume fractions on the cyclic inelastic strain distribution. Multiple statistical volume elements (SVEs) are subjected to random periodic boundary conditions to build up statistically significant measures of distributions of cyclic microplasticity. Multiaxial fatigue criteria with critical plane approaches are used to estimate the crack initiation life. Methods are developed for assessing crack formation and microstructurally small crack growth as a function of microstructure attributes.

Homogenized Dynamics of Stochastic Partial Differential Equations with Dynamical Boundary Conditions

A microscopic heterogeneous system under random influence is considered. The randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium. This system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes (obstacles or heterogeneities), together with random dynamical boundary conditions on the boundaries of these small holes.A homogenized macroscopic model for this microscopic heterogeneous stochastic system is derived. This homogenized effective model is a new stochastic partial differential equation defined on a unified domain without small holes, with a static boundary condition only. In fact, the random dynamical boundary conditions are homogenized out, but the impact of random forces on the small holes’ boundaries is quantified as an extra stochastic term in the homogenized stochastic partial differential equation. Moreover, the validity of the homogenized model is justified by showing that the solutions of the microscopic model converge to those of the effective macroscopic model in probability distribution, as the size of small holes diminishes to zero.

Burgers equation with random boundary conditions

We prove an existence and uniqueness theorem for stationary solutions of the inviscid Burgers equation on a segment with random boundary conditions. We also prove exponential convergence to the stationary distribution.

Numerical analysis for random temperature fields of embankment in cold regions

The stochastic finite element equations for random temperature are obtained using the first-order perturbation technique taking into account the random thermal properties and boundary condition, based on heat transfer variational principle. The local average method for 2-D is used to discretize random fields. Then, the random temperature fields of embankment in cold regions are investigated on condition that the thermal properties and boundary condition are taken as random fields, respectively, by using the program, which is written by the methods. The expected value of temperature field and the standard deviation of the temperature field of embankment in cold regions are obtained and analyzed.

Nonlinear plate aeroelastic response with uncertain stiffness and boundary conditions

The probabilistic response of a nonlinear panel in supersonic flow is investigated using a new computational methodology. The aeroelastic system is modelled by coupling the von Karman plate equations with piston theory aerodynamics. Baseline deterministic results are compared to published data to establish validity of the aeroelastic model. Uncertainties in modulus of elasticity and boundary conditions (BCs) are considered, with their impact on panel response being quantified in terms of limit-cycle oscillation (LCO) onset pressure and panel amplitude. The panel's uncertain BCs are modelled using rotational springs in place of ideally fixed boundary conditions. The panel's uncertain elastic modulus is modelled as a second-order, spatially homogeneous, isotropic, random field. The boundary spring stiffness values are spatially constant for each random realization assuming either a uniform or Weibull distribution. Probabilistic response distributions are obtained through Monte Carlo simulation (MCS) for selected values of flowfield dynamic pressure. MCS is run for the case of random modulus only, as well as random modulus combined with either random BC distribution. It is observed that uncertainties have the greatest nonlinear influence on LCO amplitude near the deterministic point of LCO onset. Furthermore, panel response is much more sensitive to location of the modulus fields' extreme values than the correlation length used to generate the field. Novel aspects of the computational methodology include (1) the use of random fields (as opposed to a random variable) to model uncertainty in the elastic modulus, and (2) the use of proper orthogonal decomposition (POD) to yield insight into the local factors that govern the global response of the panel, permitting enhanced response prediction based on the nature of the random field realizations.

Eigenvalues of structures with uncertain elastic boundary restraints

The eigenvalue problems of structures with random elastic boundary supports are studied in this paper. Using the perturbation method, the differential equations including stochastic distributed parameters and random boundary conditions that govern the eigenproblems are transformed to a series of deterministic differential equations and boundary conditions. Then the differential equations and boundary conditions are discretized utilizing the finite element method (FEM). This process is easy to be implemented since the resulting perturbation equations with different orders own the same FEM meshes. The first-order and second-order sensitivities of eigenvalues are derived through solving the deterministic algebraic equations. Upon determining these sensitivities of eigenvalues, the approximate statistic expressions of random eigenvalues considering uncertain elastic boundary supports are established. At the end, several numerical examples are given to illustrate the application and effectiveness of the present method. Comparison of the present numerical results with those from the Monte-Carlo simulation method verifies the accuracy of the developed method.

Random temperature fields of embankment in cold regions

The stochastic finite element equations for random temperature are derived from using first-order perturbation technique taking into account the random thermal properties of material and boundary conditions. Then, the random temperature fields of an embankment in a cold region are investigated by the methods. The results will improve our understanding of random temperature field of embankment in cold regions taking into account random boundary conditions. The formalism presented may be directly employed for other problems.

Computation of moment functions for parabolic PDEs with random parameters

AbstractThis paper deals with different approaches for the computation of stochastic characteristics (such as mean, variance, correlation function, …) of the solution to parabolic PDEs with random parameters, in particular with a random Neumann boundary conditions and a random initial condition. Thereby Finite‐Element discretisation of the spatial variables is used for the construction of pathwise solutions. The random influences are assumed to be ε ‐correlated (cp. [4]) that means the correlation functions vanish if the difference of the arguments exceeds a given correlation length ε . So an asymptotic expansion of higher order with respect to the correlation length for the correlation function of the approximative solution is given. A second possibility is the so called explicit calculation. Examples which compare the different methods can be found in [2]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Solutions of partial differential equations with random Dirichlet boundary conditions by multiquadric collocation method

Problems described by deterministic partial differential equations with random Dirichlet boundary conditions are considered. Formulation of the solution to such a problem by the global collocation method using multiquadrics is presented. The quality of the solution to a stochastic problem depends on both its expected value and its variance. It is proposed that the shape parameter of multiquadrics should be chosen to optimize both the accuracy and the variance of the solution. Test problems described by the Poisson, the Helmholtz, and the diffusion–convection equations with random Dirichlet boundary conditions are solved by the multiquadric collocation method. It is found that there is a trade-off between solution accuracy and solution variance for each problem.

On the Ising Model with Random Boundary Condition

The infinite-volume limit behavior of the 2d Ising model under possibly strong random boundary conditions is studied. The model exhibits chaotic size-dependence at low temperatures and we prove that the `+' and `-' phases are the only almost sure limit Gibbs measures, assuming that the limit is taken along a sparse enough sequence of squares. In particular, we provide an argument to show that in a sufficiently large volume a typical spin configuration under a typical boundary condition contains no interfaces. In order to exclude mixtures as possible limit points, a detailed multi-scale contour analysis is performed.

Analytical solutions to steady state unsaturated flow in layered, randomly heterogeneous soils via Kirchhoff transformation

In this study, we derive analytical solutions of the first two moments (mean and variance) of pressure head for one-dimensional steady state unsaturated flow in a randomly heterogeneous layered soil column under random boundary conditions. We first linearize the steady state unsaturated flow equations by Kirchhoff transformation and solve the moments of the transformed variable up to second order in terms of σ Y and σ β , the standard deviations of log hydraulic conductivity Y=ln( K s) and of the log pore size distribution parameter β=ln( α). In addition, we also give solutions for the mean and variance of the unsaturated hydraulic conductivity. The analytical solutions of moment equations are validated via Monte Carlo simulations.

Nonlocal and localized analyses of conditional mean transient flow in bounded, randomly heterogeneous porous media

We consider the numerical prediction of transient flow in bounded, randomly heterogeneous porous media driven by random sources, initial heads, and boundary conditions without resorting to Monte Carlo simulation. After applying the Laplace transform to the governing stochastic flow equations, we derive exact nonlocal (integro‐differential) equations for the mean and variance‐covariance of transformed head and flux, conditioned on measured values of log conductivity Y = ln K. Approximating these conditional moment equations recursively to second order in the standard deviation σY of Y, we solve them by finite elements for superimposed mean‐uniform and convergent flows in a two‐dimensional domain. An alternative conditional mean solution is obtained through localization of the exact moment expressions. The nonlocal and localized solutions are obtained using a highly efficient parallel algorithm and inverted numerically back into the time domain. A comparison with Monte Carlo simulations demonstrates that the moment solutions are remarkably accurate for strongly heterogeneous media with σY2 as large as 4. The nonlocal solution is only slightly more accurate than the much simpler localized solution, but the latter does not yield information about predictive uncertainty. The accuracy of each solution improves markedly with conditioning. A preliminary comparison of computational efficiency suggests that both the nonlocal and localized solutions for mean head and its variance require significantly less computer time than is required for Monte Carlo statistics to stabilize when the same direct matrix solver is used for all three (we do not presently know how using iterative solvers would have affected this conclusion). This is true whether the Laplace inversion and Monte Carlo simulations are conducted sequentially or in parallel on multiple processors and regardless of problem size. The underlying exact and recursive moment equations, as well as the proposed computational algorithm, are valid in both two and three dimensions; only the numerical implementation of our algorithm is two‐dimensional.

Compaction of confined mono-sized spherical particle systems under symmetric vibration: a suspension model

This paper presents a simulation study on the compaction of confined mono-sized spherical particle systems under symmetric vibration. In the model used, a random loose packing is confined between two fixed random granular boundary conditions and is submitted to symmetric vibrations of constant amplitude until complete transformation into a “suspension”. The granular system is then compacted by letting particles settle down until an equilibrium of minimum potential energy is reached. The influence of the vibration amplitude on the final packing density is studied. The increase in the vibration amplitude leads to an increase in the final packing density. Packing densities ranging from 0.57 up to 0.67 are obtained. For small vibration amplitude a linear relation is found between the final packing density and the vibration amplitude. For large vibration amplitude high packing densities are obtained and an order appears amongst the system. A suspension model is then presented which considers that, in the “suspension”, all the particles can move symmetrically upwards and downwards independently of the displacements of other particles in the system. It is shown that in these conditions, and for random systems, the vibration amplitude can be considered as a homogeneous contraction of the system during sedimentation. A relation is found between the final packing density, initial packing density and vibration amplitude. A very good agreement is obtained between the simulation results and this suspension model as long as the packing is random.

Simulation of the compaction of confined mono-sized spherical particles systems under symmetric vibration

This paper presents a simulation study on the compaction of mono-sized spherical particles systems under symmetric vibration. The influence of the vibration amplitude on the density after 500 000 vibration cycles and on the dynamics of vibration is first studied. It is shown that vibration brings the initial packing to a “suspension” where the distance between particles depends on the vibration amplitude. A large vibration amplitude ( D/10) leads to a crystallization of the system whereas a small vibration amplitude induces an efficient compaction of the packing. The vibration procedure is then modified by decreasing the vibration amplitude during compaction in order to speed up the densification of the system and to reach high packing densities. Packing densities ranging from 0.64 up to 0.69 are easily obtained. In the latter case (obtained with an initial vibration amplitude of D/10), a crystallization of the packing is initiated along the flat walls of the container and progresses towards the bulk packing. Finally, the influence of the boundary conditions is studied. It is shown that a random granular boundary condition also leads to a crystallization of the system for a large initial vibration amplitude ( D/10). The order that appears amongst the system is due, in that case, to the vibration procedure itself and not to the boundary conditions.

Modeling uncertainty in flow simulations via generalized polynomial chaos

We present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as trial basis to represent the random space. A standard Galerkin projection is applied in the random dimension to obtain the equations in the weak form. The resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode. This approach can be considered as a generalization of the original polynomial chaos expansion, first introduced by Wiener [Am. J. Math. 60 (1938) 897]. The original method employs the Hermite polynomials (one of the 13 members of the Askey scheme) as the basis in random space. The algorithm is applied to micro-channel flows with random wall boundary conditions, and to external flows with random freestream. Efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by Monte Carlo simulations. It is shown that the generalized polynomial chaos method promises a substantial speed-up compared with the Monte Carlo method. The utilization of different type orthogonal polynomials from the Askey scheme also provides a more efficient way to represent general non-Gaussian processes compared with the original Wiener–Hermite expansions.