Stochastic Boundary Value Problem Research Articles

Побудова критерію довготривалого руйнування втоми для тонкостінних шаруватих оболонок

A model and criterion of long-term fatigue failure for thin-walled layered shells is built, taking into account the influence of the type of stress state. The problem of calculating the number of cycles to failure under combined loading is considered. Solutions are built on the basis of the concept of equivalent stresses. The problem of determining local stresses in composites of random structure is formulated within the framework of the second-order nonlinear theory. The solution of the stochastic boundary value problem on determining the stress concentration in a unidirectional composite with a metal matrix (MMC) was obtained. To build a complete system of equations of the second order, the method of successive approximations is used. The parameters of the stress concentration at the boundary of the components are determined. The given examples show the importance of the influence of nonlinear properties on the redistribution of stresses near the fibers. The possibility of predicting the long-term strength of the material is shown. The necessary information about the material for the formulation of failure criteria is the S-N curves for individual components of the combined stresses.

Probabilistic learning inference of boundary value problem with uncertainties based on Kullback–Leibler divergence under implicit constraints

This paper deals with a probabilistic learning inference that allows for integrating data (target set) into predictive models for which the target set is constituted of statistical moments of the quantity of interest (QoI) and for which the training set is constituted of a small number of points. In a first part, we present a mathematical analysis of a general methodology that allows for estimating a posterior probability model for a stochastic boundary value problem from a prior probability model of a control parameter. The given targets are statistical moments of the QoI for which the underlying realizations are not available. Under these conditions, the Kullback–Leibler minimum principle is used for estimating the posterior probability measure. The constraints are represented by an implicit nonlinear mapping for which a statistical surrogate model is introduced. The MCMC generator and the necessary numerical elements are given to facilitate the implementation of the methodology. In a second part, an application is presented to illustrate the proposed theory and is also, as such, a contribution to the three-dimensional stochastic homogenization of heterogeneous linear elastic media in the case of a non-separation of the microscale and macroscale. Consequently, the macroscale is another mesoscale at larger scale with random effective/apparent elastic properties. For the construction of the posterior probability measure by using the probabilistic learning inference, in addition to the constraints defined by given statistical moments of the random effective/apparent elasticity tensor, the second-order moment of the random normalized residue of the stochastic partial differential equation has been added as a constraint. This constraint guarantees that the algorithm seeks to bring the statistical moments closer to their targets while preserving a small residue.

Stochastic isogeometric analysis on arbitrary multipatch domains by spline dimensional decomposition

This paper presents a new stochastic method by integrating spline dimensional decomposition (SDD) of a high-dimensional random function and isogeometric analysis (IGA) on arbitrary multipatch geometries to solve stochastic boundary-value problems from linear elasticity. The method, referred to as SDD-mIGA, involves (1) analysis-suitable T-splines with significant approximating power for geometrical modeling, random field discretization, and stress analysis; (2) Bézier extraction operator for isogeometric mesh refinement; and (3) a novel Fourier-like expansion of a high-dimensional output function in terms of measure-consistent orthonormalized splines. The proposed method can handle arbitrary multipatch domains in IGA and uses standard least-squares regression to efficiently estimate the SDD expansion coefficients for uncertainty quantification applications. Analytical formulae have been derived to calculate the second-moment properties of an SDD-mIGA approximation for a general output random variable of interest. Numerical results, including those obtained for a 54-dimensional, industrial-scale problem, demonstrate that a low-order SDD-mIGA is capable of efficiently delivering accurate probabilistic solutions when compared with the benchmark results from crude Monte Carlo simulation.

ЗАДАЧИ ШОУОЛТЕРА–СИДОРОВА И КОШИ ДЛЯ ЛИНЕЙНОГО УРАВНЕНИЯ ДЗЕКЦЕРА С КРАЕВЫМИ УСЛОВИЯМИ ВЕНТЦЕЛЯ И РОБЕНА В ОГРАНИЧЕННОЙ ОБЛАСТИ

Deterministic and stochastic initial boundary value problems for the Dzekzer equation describing the evolution of the free surface of a filtering fluid in a bounded region with a smooth boundary are considered. Wentzell and Robin conditions are set on the boundary of the domain, and either the Showalter-Sidorov condition or the Cauchy condition is taken as the initial condition. Note that for the filtration model under study, the Wentzell condition is considered, which is not a classical condition. In recent years, the boundary condition has been considered in the mathematical literature from two points of view (classical and neoclassical). Since Cauchy and Showalter–Sidorov initial conditions have been studied earlier in various situations, in this work, in the particular case of classical Wentzell and Robin conditions, by methods of the theory of degenerate holomorphic semigroups, exact solutions have been constructed, which allow to determine quantitative predictions of changes in geochemical regime of groundwater under unpressurized filtration. Nelson–Glicklich derivative theory was used in stochastic case. In particular, the investigation of the set problems in the context of Wentzell boundary conditions allowed to determine the processes occurring at the boundary of two media (in the region and at its boundary).

Stochastic elliptic operators defined by non-Gaussian random fields with uncertain spectrum

This paper presents a construction and the analysis of a class of non-Gaussian positive-definite matrix-valued homogeneous random fields with uncertain spectral measure for stochastic elliptic operators. Then the stochastic elliptic boundary value problem in a bounded domain of the 3D-space is introduced and analyzed for stochastic homogenization.

Scaling in Anti-Plane Elasticity on Random Shear Modulus Fields with Fractal and Hurst Effects

The scale dependence of the effective anti-plane shear modulus response in microstructures with statistical ergodicity and spatial wide-sense stationarity is investigated. In particular, Cauchy and Dagum autocorrelation functions which can decouple the fractal and the Hurst effects are used to describe the random shear modulus fields. The resulting stochastic boundary value problems (BVPs) are set up in line with the Hill–Mandel condition of elastostatics for different sizes of statistical volume elements (SVEs). These BVPs are solved using a physics-based cellular automaton (CA) method that is applicable for anti-plane elasticity to study the scaling of SVEs towards a representative volume element (RVE). This progression from SVE to RVE is described through a scaling function, which is best approximated by the same form as the Cauchy and Dagum autocorrelation functions. The scaling function is obtained by fitting the scaling data from simulations conducted over a large number of random field realizations. The numerical simulation results show that the scaling function is strongly dependent on the fractal dimension D, the Hurst parameter H, and the mesoscale δ, and is weakly dependent on the autocorrelation function. Specifically, it is found that a larger D and a smaller H results in a higher rate of convergence towards an RVE with respect to δ.

Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets

A novel extension of the Probabilistic Learning on Manifolds (PLoM) is presented. It makes it possible to synthesize solutions to a wide range of nonlinear stochastic boundary value problems described by partial differential equations (PDEs) for which a stochastic computational model (SCM) is available and which depend on a vector-valued random control parameter. The cost of a single numerical evaluation of this SCM is assumed to be such that only a limited number of points can be computed for constructing the training dataset (small data). Each point of the training dataset is made up of realizations from a vector-valued stochastic process (the stochastic solution) and the associated random control parameter on which it depends. The presented PLoM constrained by PDE allows for generating a large number of learned realizations of the stochastic process and its corresponding random control parameter. These learned realizations are generated so as to minimize the vector-valued random residual of the PDE in the mean-square sense. Appropriate novel methods are developed to solve this challenging problem. Three applications are presented. The first one is a simple uncertain nonlinear dynamical system with a nonstationary stochastic excitation. The second one concerns the 2D nonlinear unsteady Navier–Stokes equations for incompressible flows in which the Reynolds number is the random control parameter. The last one deals with the nonlinear dynamics of a 3D elastic structure with uncertainties. The results obtained make it possible to validate the PLoM constrained by stochastic PDE but also provide further validation of the PLoM without constraint.

Multipoint stochastic approach to localization of microscale elastic behavior of random heterogeneous media

The paper is devoted to the method of evaluation of local mechanical state in microscale components of random heterogeneous media taking into account their morphological features. According to the proposed approach, a representative volume element (RVE) of heterogeneous medium can be considered as a system of random interacting components with distinguishable properties. In this case, internal mechanical response to the load applied to an RVE can be expressed via continuous distributions of microscale stress and strain fields. The novelty of this work is connected with the methodology of restoration of parameters of these distributions using the solutions of the stochastic boundary value problems (SBVPs) that consider multipoint statistical correlations between different points within RVE, in contrast to the traditionally widely used two-point correlations. This allow to obtain more accurate results for the problems of localization of mechanical behavior for materials with complex random microstructure. The computational techniques, which are essential for numerical implementation of the developed approach, are described. The case studies connected with analysis of local stress and strain fields distributions in random bicontinuous media with interpenetrating phases are investigated using the developed approach and verified with the finite element analysis.

Numerical solution of an inverse random source problem for the time fractional diffusion equation via PhaseLift

This paper is concerned with the inverse random source problem for a stochastic time fractional diffusion equation, where the source is assumed to be driven by a Gaussian random field. The direct problem is shown to be well-posed by examining the well-posedness and regularity of the solution for the equivalent stochastic two-point boundary value problem in the frequency domain. For the inverse problem, the Fourier modulus of the diffusion coefficient of the random source is proved to be uniquely determined by the variance of the Fourier transform of the boundary data. As a phase retrieval for the inverse problem, the phaselift method with random masks is applied to recover the diffusion coefficient from its Fourier modulus. Numerical experiments are reported to demonstrate the effectiveness of the proposed method.

Спадкова повзучість ізотропних композитів випадкової структури при складному напруженому стані

Nonlinear hereditary creep problem of the mechanics of composites is solved within the framework of a second-order theory. The hereditary functionals are used to construct general constitutive relations. A stochastic boundary value problem for determining the stress concentration and its relaxation in metal matrix composite (PMC) is solved in Laplace-Carson image space. Shapery's correspondence principle for quasi-linear viscoelastic media is generalised on the hereditary creep problem and the method of successive approximation is used. The reduced creep functions 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 with possibility to predicting the long-term strength.

Stochastic isogeometric analysis in linear elasticity

A new stochastic method, integrating spline dimensional decomposition (SDD) and isogeometric analysis (IGA), is proposed for solving stochastic boundary-value problems from linear elasticity. The method, referred to as SDD–IGA, involves Galerkin isogeometric analysis as a deterministic solver for governing partial differential equations and a novel Fourier-like orthogonal spline expansion generated from the analysis-of-variance decomposition of a high-dimensional function. For the stochastic part of the SDD–IGA method, an innovative dimension-reduction integration technique is presented for efficiently calculating the expansion coefficients. Analytical formulae have been derived to calculate the second-moment properties of an SDD–IGA approximation for a general output random variable of interest. Numerical examples demonstrate the capability of a low-order SDD–IGA in efficiently delivering probabilistic solutions with an approximation quality as good as, if not better than, that obtained from a high-order polynomial dimensional decomposition. The proposed SDD–IGA method is most suitable in the presence of locally nonlinear or nonsmooth behavior commonly found in applications.

Physics‐constrained non‐Gaussian probabilistic learning on manifolds

SummaryAn extension of the probabilistic learning on manifolds (PLoM), recently introduced by the authors, has been presented: In addition to the initial data set given for performing the probabilistic learning, constraints are given, which correspond to statistics of experiments or of physical models. We consider a non‐Gaussian random vector whose unknown probability distribution has to satisfy constraints. The method consists in constructing a generator using the PLoM and the classical Kullback‐Leibler minimum cross‐entropy principle. The resulting optimization problem is reformulated using Lagrange multipliers associated with the constraints. The optimal solution of the Lagrange multipliers is computed using an efficient iterative algorithm. At each iteration, the Markov chain Monte Carlo algorithm developed for the PLoM is used, consisting in solving an Itô stochastic differential equation that is projected on a diffusion‐maps basis. The method and the algorithm are efficient and allow the construction of probabilistic models for high‐dimensional problems from small initial data sets and for which an arbitrary number of constraints are specified. The first application is sufficiently simple in order to be easily reproduced. The second one is relative to a stochastic elliptic boundary value problem in high dimension.

Consistency and stability difference scheme study for random diffusion second type problem under mean square

The present article is devoted to the construction and investigation of the difference scheme for approximating the solutions of the stochastic boundary-value problem for the heat model in one dimension. This approach permits us to discuss the effect of the random inputs in the stochastic heat model from through the consistency and stability theorems for the random difference scheme under mean square

Uncertainty quantification under dependent random variables by a generalized polynomial dimensional decomposition

This paper is concerned with uncertainty quantification analysis of complex systems subject to dependent input random variables. The analysis focuses on a new, generalized version of polynomial dimensional decomposition (PDD), referred to as GPDD, entailing hierarchically ordered measure-consistent multivariate orthogonal polynomials in dependent variables. Under a few prescribed assumptions, GPDD exists for any square-integrable output random variable and converges in mean-square to the correct limit. New analytical formulae are proposed to calculate the mean and variance of a GPDD approximation of a general output variable in terms of the expansion coefficients and second-moment properties of multivariate orthogonal polynomials. However, unlike in PDD, calculating the coefficients of GPDD requires solving a coupled system of linear equations. Besides, the variance formula of GPDD contains extra terms due to statistical dependence among input variables. The extra terms disappear when the input variables are statistically independent, reverting GPDD to PDD. Two numerical examples, the one derived from a stochastic boundary-value problem and the other entailing a random eigenvalue problem, illustrate second-moment error analysis and estimation of the probabilistic characteristics of eigensolutions.

A polynomial chaos expansion in dependent random variables

This paper introduces a new generalized polynomial chaos expansion (PCE) comprising measure-consistent multivariate orthonormal polynomials in dependent random variables. Unlike existing PCEs, whether classical or generalized, no tensor-product structure is assumed or required. Important mathematical properties of the generalized PCE are studied by constructing orthogonal decomposition of polynomial spaces, explaining completeness of orthogonal polynomials for prescribed assumptions, exploiting whitening transformation for generating orthonormal polynomial bases, and demonstrating mean-square convergence to the correct limit. Analytical formulae are proposed to calculate the mean and variance of a truncated generalized PCE for a general output variable in terms of the expansion coefficients. An example derived from a stochastic boundary-value problem illustrates the generalized PCE approximation in estimating the statistical properties of an output variable for 12 distinct non-product-type probability measures of input variables.

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.

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.

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.

Uncertainty quantification and stochastic polynomial chaos expansion for recovering random data in Darcy and Diffusion equations

Abstract A probabilistic collocation based polynomial chaos expansion method is developed to solve stochastic boundary value problems with random coefficients and randomly distributed initial data. In this paper we deal with two different boundary value problems with random data: the Darcy equation with random lognormally distributed hydraulic conductivity, and a diffusion equation with absorption, with random distribution of the initial concentration under periodic boundary conditions. Special attention is paid to the extension of the probabilistic collocation method to input data with arbitrary correlation functions defined both analytically and through measurements. We construct the relevant Karhunen–Loève expansion from a special randomized singular value decomposition of the correlation matrix, which makes possible to treat problems of high dimension. We show that the unknown statistical characteristics of the random input data can be recovered from the correlation analysis of the solution field.

A Stochastic Optimal Control Model for BCG Immunotherapy in Superficial Bladder Cancer

Urologic studies have reported in several papers that the optimal dose of Bacillus Calmette-Guerin (BCG) immunotherapy in superficial bladder cancer, is still a subject of research. Our main goal from this paper, is to find treatment regimens that minimize the total number of tumors in the presence of a diffusion process. For this, we devise a stochastic model in the form of a nonlinear system of four stochastic differential equations (SDEs) that describe tumor-immune dynamics after BCG instillations. Therefore, we study the existence and the stability results. Then, we introduce a control function in the mathematical model, to represent the dose of BCG intravesical therapy, and we seek its optimal values through the application of a stochastic version of Pontryagin's maximum principle. Finally, we present some numerical simulations using iterative stochastic Runge-Kutta progressive-regressive schemes which we propose for solving the optimality system of the obtained stochastic two-point boundary value problem.