Stochastic Boundary Research Articles

Simulation and Data Analytics in the Conceptual Design of Industrial Networks in Stochastic Environments

AbstractThe cross‐linking of industrial processes and plants is gaining great importance. The project Carbon2Chem® deals with industrial networks in a stochastic environment with volatilely available resources. Thus, processes in the networks must be designed for discontinuous operation, and design challenges arise due to complex dynamic interdependencies within the system. Here, a V‐model based framework for the conceptual design of industrial networks is presented, which integrates simulation and data analytics methods for a hierarchical derivation of stochastic boundary conditions on different levels of process integration. This framework is applied to study the utilization of steel mill gases for methanol synthesis in a stochastic dynamic environment.

A stochastic boundary element method for piezoelectric problems

In this paper, the stochastic boundary element method of piezoelectric problems with randomness of material parameters and randomness of applied loads is proposed. By using the method of first-order Taylor expansion, each random quantity is written into the sum of the mean and deviation. The boundary integral equations corresponding to the means and deviations of the displacements and electric potential are derived, respectively. It is demonstrated that the randomness of material parameters can be transformed into the equivalent random body forces and equivalent random charge density, so that the fundamental solutions of deterministic piezoelectric problems can be used in boundary integral equations of the means or deviations due to the similarity between the governing equations with randomness and those of deterministic piezoelectric problems. Finally, several numerical examples are performed to verify the validity of the proposed stochastic boundary element method.

Rates of convergence to equilibrium for collisionless kinetic equations in slab geometry

This work deals with free transport equations with partly diffuse stochastic boundary operators in slab geometry. Such equations are governed by stochastic semigroups in L1 spaces. We prove convergence to equilibrium at the rate O(t−k2(k+1)+1)(t→+∞) for L1 initial data g in a suitable subspace of the domain of the generator T where k∈N depends on the properties of the boundary operators near the tangential velocities to the slab. This result is derived from a quantified version of Ingham's tauberian theorem by showing that Fg(s):=limε→0+⁡(is+ε−T)−1g exists as a Ck function on R\\{0} such that ‖djdsjFg(s)‖≤C|s|2(j+1) near s=0 and bounded as |s|→∞(0≤j≤k). Various preliminary results of independent interest are given and some related open problems are pointed out.

Driving Solar Giant Cells through the Self-organization of Near-surface Plumes

Abstract Global 3D simulations of solar giant-cell convection have provided significant insight into the processes which yield the Sun’s observed differential rotation and cyclic dynamo action. However, as we move to higher-resolution simulations a variety of codes have encountered what has been termed the convection conundrum. As these simulations increase in resolution and hence the level of turbulence achieved, they tend to produce weak or even anti-solar differential rotation patterns associated with a weak rotational influence (high Rossby number) due to large convective velocities. One potential culprit for this convection conundrum is the upper boundary condition applied in most simulations, which is generally impenetrable. Here we present an alternative stochastic plume boundary condition which imposes small-scale convective plumes designed to mimic near-surface convective downflows, thus allowing convection to carry the majority of the outward solar energy flux up to and through our simulated upper boundary. The use of a plume boundary condition leads to significant changes in the convective driving realized in the simulated domain and thus to the convective energy transport, the dominant scale of the convective enthalpy flux, and the relative strength of the strongest downflows, the downflow network, and the convective upflows. These changes are present even far from the upper boundary layer. Additionally, we demonstrate that, in spite of significant changes, giant cell morphology in the convective patterns is still achieved with self-organization of the imposed boundary plumes into downflow lanes, cellular patterns, and even rotationally aligned banana cells in equatorial regions. This plume boundary presents an alternative pathway for 3D global convection simulations where driving is non-local and may provide a new approach toward addressing the convection conundrum.

Robust boundary conditions for stochastic incompletely parabolic systems of equations

We study an incompletely parabolic system in three space dimensions with stochastic boundary and initial data. We show how the variance of the solution can be manipulated by the boundary conditions, while keeping the mean value of the solution unaffected. Estimates of the variance of the solution is presented both analytically and numerically. We exemplify the technique by applying it to an incompletely parabolic model problem, as well as the one-dimensional compressible Navier–Stokes equations.

Boundary controllability of nonlinear stochastic fractional systems in Hilbert spaces

Abstract Sufficient conditions for the controllability of nonlinear stochastic fractional boundary control systems are established. The equivalent integral equations are derived for both linear and nonlinear systems, and the control function is given in terms of the pseudoinverse operator. The Banach contraction mapping theorem is used to obtain the result. A controllability result for nonlinear stochastic fractional integrodifferential systems is also attained. Examples are included to illustrate the theory.

Variability and Clustering of Midlatitude Summertime Convection: Testing the Craig and Cohen Theory in a Convection-Permitting Ensemble with Stochastic Boundary Layer Perturbations

Abstract The statistical theory of convective variability developed by Craig and Cohen in 2006 has provided a promising foundation for the design of stochastic parameterizations. The simplifying assumptions of this theory, however, were made with tropical equilibrium convection in mind. This study investigates the predictions of the statistical theory in real-weather case studies of nonequilibrium summertime convection over land. For this purpose, a convection-permitting ensemble is used in which all members share the same large-scale weather conditions but the convection is displaced using stochastic boundary layer perturbations. The results show that the standard deviation of the domain-integrated mass flux is proportional to the square root of its mean over a wide range of scales. This confirms the general applicability and scale adaptivity of the Craig and Cohen theory for complex weather. However, clouds tend to cluster on scales of around 100 km, particularly in the morning and evening. This strongly impacts the theoretical predictions of the variability, which does not include clustering. Furthermore, the mass flux per cloud closely follows an exponential distribution if all clouds are considered together and if overlapping cloud objects are separated. The nonseparated cloud mass flux distribution resembles a power law. These findings support the use of the theory for stochastic parameterizations but also highlight areas for improvement.

Automatic Between-Pulse Analysis of DIII-D Experimental Data Performed Remotely on a Supercomputer at Argonne Leadership Computing Facility

For the first time, an automatically triggered, between-pulse fusion science analysis code was run on-demand at a remotely located supercomputer at Argonne Leadership Computing Facility (ALCF, Lemont, Illinois) in support of in-process experiments being performed at DIII-D (San Diego, California). This represents a new paradigm for combining geographically distant experimental and high-performance computing facilities to provide enhanced data analysis that is quickly available to researchers. Enhanced analysis improves the understanding of the current pulse, translating into a more efficient use of experimental resources and quality of the resultant science. The analysis code used here, called SURFMN, calculates the magnetic structure of the plasma using Fourier transform. Increasing the number of Fourier components provides a more accurate determination of the stochastic boundary layer near the plasma edge by better resolving magnetic islands, but requires 26 min to complete using local DIII-D resources, putting it well outside the useful time range for between-pulse analysis. These islands relate to confinement and edge-localized mode suppression, and may be controlled by adjusting coil currents for the next pulse. ALCF has ensured on-demand execution of SURFMN by providing a reserved queue, a specialized service that launches the code after receiving an automatic trigger, and network access from the worker nodes for data transfer. Runs are executed on 252 cores of ALCF’s Cooley cluster and the data are available locally at DIII-D within 3 min of triggering. The original SURFMN design limits additional improvements with more cores; however, our work shows a path forward where codes that benefit from thousands of processors can run between pulses.

Seed Production and Technical Efficiency of Sorghum Farmers in Mali

Seed Production and Technical Efficiency of Sorghum Farmers in Mali Abdoul Karim DIAMOUTENE, Lamissa DIAKITE, Adama COULIBALY Abstract This study aimed to determine the effect of self-seed production on the technical efficiency of certified sorghum seed users in Mali. We applied the estimated Cobb Douglas stochastic production boundary using Frontier 4.1 and Stata 14software. The data are made of 63 certified seed producers during the March-April 2013 seed value chain survey as part of the research project "Strengthening Sustainable Seed Systems for Food Crops and Oilseeds and Proteins" to ensure food security and increase farmers' incomes in West and Central Africa. ". The results found out that, the average technical efficiency score of sorghum producers is 0.67, implying that sorghum cultivation in the study area can be improved by 0.33, without additional cost. In addition, the technical efficiency determinants are age, education and motorized equipment. Otherwise, the data revealed that, the self-seed production by farmer does not affect its technical efficiency in the sorghum production, in Mali. Full Text: PDF DOI: 10.15640/jaes.v7n1a7

Multipoint Approximation of Statistical Descriptors of Local Strain and Stress Fields in Heterogeneous Media Using Integral Equation Method

This paper is devoted to derivation of analytic expressions for statistical descriptors of stress and strain fields in heterogeneous media. Multipoint approximations of solutions of stochastic elastic boundary value problems for representative volume elements are investigated. The stress and strain fields are represented in the form of random coordinate functions, for which analytical expressions for the first- and second-order statistical central moments are obtained. Such moments characterize distribution of fields under prescribed loading of a representative volume element and depend on the geometry features and location of components within a volume. The information of the internal geometrical structure is taken into account by means of multipoint correlation functions. Within the framework of the second approximation of the boundary value problem, the correlation functions up to the fifth order are required to calculate the statistical characteristics. Using the method of Green’s functions, analytical expressions for the moments in distinct phases of the microstructure are obtained explicitly in a form of integral equations. Their analysis and comparison with previously obtained results are performed.

Yields versus Expected Returns of Corporate Bonds: Some Unexpected Results

We derive expected bond return equations for various structural credit valuation models with alternative stochastic processes and boundary conditions for default given in Merton [1974], Merton [1976], Black and Cox [1976], Heston [1993], Longstaff and Schwartz [1995], and Collin-Dufresne and Goldstein [2001]. We investigate the relationship between bond yields and expected returns under each of these models. We show that for the case of low leverage, bond yields are inversely related to expected returns for a range of maturities that lie beyond the hump of the term structure of bond yields. We also show that for the bonds of same maturity, the option characteristics of the bonds often lead to an inverse relation between bond yields and expected returns. For example, callable bonds have higher yields, but lower expected returns, than the corresponding yields and expected returns of the underlying straight bonds. Convertible bonds have higher yields but lower expected returns for the case of high leverage, and lower yields but higher expected returns for the case of low leverage, than the corresponding yields and expected returns of the underlying straight bonds.

Design optimization under uncertainties of a mesoscale implant in biological tissues using a probabilistic learning algorithm

This paper deals with the optimal design of a titanium mesoscale implant in a cortical bone for which the apparent elasticity tensor is modeled by a non-Gaussian random field at mesoscale, which has been experimentally identified. The external applied forces are also random. The design parameters are geometrical dimensions related to the geometry of the implant. The stochastic elastostatic boundary value problem is discretized by the finite element method. The objective function and the constraints are related to normal , shear, and von Mises stresses inside the cortical bone. The constrained nonconvex optimization problem in presence of uncertainties is solved by using a probabilistic learning algorithm that allows for considerably reducing the numerical cost with respect to the classical approaches.

Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system.

A subcritical pattern-forming system with nonlinear advection in a bounded domain is recast as a slow-fast system in space and studied using a combination of geometric singular perturbation theory and numerical continuation. Two types of solutions describing the possible location of stationary fronts are identified, whose origin is traced to the onset of convective and absolute instability when the system is unbounded. The former are present only for non-zero upstream boundary conditions and provide a quantitative understanding of noise-sustained structures in systems of this type. The latter correspond to the onset of a global mode and are present even with zero upstream boundary conditions. The role of canard trajectories in the nonlinear transition between these states is clarified and the stability properties of the resulting spatial structures are determined. Front location in the convective regime is highly sensitive to the upstream boundary condition, and its dependence on this boundary condition is studied using a combination of numerical continuation and Monte Carlo simulations of the partial differential equation. Statistical properties of the system subjected to random or stochastic boundary conditions at the inlet are interpreted using the deterministic slow-fast spatial dynamical system.

Yields versus Expected Returns of Corporate Bonds: Some Unexpected Results

We derive expected bond return equations for various structural credit valuation models with alternative stochastic processes and boundary conditions for default given in Merton [1974], Merton [1976], Black and Cox [1976], Heston [1993], Longstaff and Schwartz [1995], and Collin-Dufresne and Goldstein [2001]. We investigate the relationship between bond yields and expected returns under each of these models. We show that for the case of low leverage, bond yields are inversely related to expected returns for a range of maturities that lie beyond the hump of the term structure of bond yields. We also show that for the bonds of same maturity, the option characteristics of the bonds often lead to an inverse relation between bond yields and expected returns. For example, callable bonds have higher yields, but lower expected returns than the corresponding yields and expected returns of the underlying straight bonds. Convertible bonds have higher yields but lower expected returns for the case of high leverage, and lower yields but higher expected returns for the case of low leverage, than the corresponding yields and expected returns of the underlying straight bonds.

Investigation on effects of stochastic loading at the boundary under thermoelasticity with two relaxation parameters

• A problem of half space with stochastic loading at the boundary is solved. • Stochastic mechanical distribution is applied at the boundary plane. • Integral transform technique and stochastic simulation are applied. • The effects of stochastic type boundary condition are highlighted. • Results are compared with the results for deterministic cases. The main objective of the present work is to study the responses of stochastic type mechanical distribution at the boundary of an elastic half space in the context of generalized thermoelasticity. In order to compare the results under the stochastic mechanical distribution, we have also considered the case of deterministic mechanical distribution prescribed at the boundary. The stochastic mechanical distribution is considered in such a way that it reduces to a deterministic type distribution as a special case. Laplace transform technique is used to solve the problem and its inversion is carried out by using asymptotic expansions valid for short times to obtain the solution of all the physical field variables like, stress and temperature distributions in the physical domain. Numerical results are found out to compare the effects of stochastic and deterministic load prescribed at the boundary of the elastic half space.

Exact matrix product decay modes of a boundary driven cellular automaton

We study integrability properties of a reversible deterministic cellular automaton (Rule 54 of (Bobenko et al 1993 Commun. Math. Phys. 158 127)) and present a bulk algebraic relation and its inhomogeneous extension which allow for an explicit construction of Liouvillian decay modes for two distinct families of stochastic boundary driving. The spectrum of the many-body stochastic matrix defining the time propagation is found to separate into sets, which we call orbitals, and the eigenvalues in each orbital are found to obey a distinct set of Bethe-like equations. We construct the decay modes in the first orbital (containing the leading decay mode) in terms of an exact inhomogeneous matrix product ansatz, study the thermodynamic properties of the spectrum and the scaling of its gap, and provide a conjecture for the Bethe-like equations for all the orbitals and their degeneracy.

Absolute continuity of the law for solutions of stochastic differential equations with boundary noise

We study the existence and regularity of the density for the solution [Formula: see text] (with fixed [Formula: see text] and [Formula: see text]) of the heat equation in a bounded domain [Formula: see text] driven by a stochastic inhomogeneous Neumann boundary condition with stochastic term. The stochastic perturbation is given by a fractional Brownian motion process. Under suitable regularity assumptions on the coefficients, by means of tools from the Malliavin calculus, we prove that the law of the solution has a smooth density with respect to the Lebesgue measure in [Formula: see text].

Stochastic spline fictitious boundary element method for random vibration analysis of plane elastic problems with structural uncertainties

A stochastic spline fictitious boundary element method (SFBEM) is proposed for random vibration analysis of plane elastic problems with both structural and loading uncertainties modeled as random fields and random processes, respectively. Two sets of governing differential equations with respect to the means and deviations of dynamic responses are derived by including the first order terms of deviations. As these equations are in similar forms to those of deterministic plane elastostatic problems, they can be solved by the traditional procedure of SFBEM with deterministic fundamental solutions, leading to the mean and covariance solutions to dynamic responses. To deal with the domain integrals of equivalent loads involved in the covariance solution, the random fields and processes considered are effectively represented by Karhunen–Loeve (KL) expansion in conjunction with the Galerkin projection. Numerical examples indicate that the results of the proposed method agree well with those of Monte-Carlo simulation (MCS) with small variations of structural parameters, and the present approach has higher efficiency than the perturbation stochastic finite element method (FEM) with KL expansion techniques.

Stress concentration in nonlinear viscoelastic composites

A nonlinear viscoelastic problem of the mechanics of composites is solved within the framework of a second-order nonlinear theory. A viscoelastic functional is used to construct general defining relations. A stochastic boundary value problem for determining the stress concentration and its relaxation in polymer composite materials (PCM) is solved. To derive the complete system of second-order viscoelastic equations, the method of successive approximation is used. A generalization of the correspondence principle to nonlinear viscoelastic media is obtained. The relaxation functions averaged over the viscoelastic matrix and elastic inclusions and the stress concentration parameters are determined. Examples are given showing the importance of the mutual influence of nonlinear elastic and viscous properties of the components on stress redistribution near inclusions in multicomponent PCMs. As a practical result, one can note the possibility of predicting the long-term strength of a material when a viscoelastic stress field is known near inclusions.

Thermal performance evaluation of a massive brick wall under real weather conditions via the Conduction Transfer function method

The reliable estimation of buildings energy needs for cooling and heating is essential for any eventual thermal improvement of the envelope or the HVAC equipment. This paper presents an interesting method to evaluate the thermal performance of a massive wall by using the frequency-domain regression (FDR) method to calculate CTF coefficients by means of the Fourier transform. The method is based on the EN ISO 13786 (2007) procedure by using the Taylor expansion for the elements of the heat matrix. Numerical results were validated through an experimental heating box with stochastic boundary conditions on one side of the wall representing real weather conditions and constant temperature profile on the other side representing the inside ambiance in real cases. Finally, a frequency analysis was performed in order to assess the validity and accuracy of the method used. The results show that development to the second order is sufficient to predict the thermal behavior of the studied massive wall in the range of frequencies encountered in the building applications (one hour time step). This method is useful for thermal transfer analysis in real weather conditions where the outside temperature is stochastic; it also allows the evaluation of the thermal performance of a wall through a frequency analysis.