Stochastic Reduced Basis Methods Research Articles

Structural Reliability Analysis of a Beam with Different Distributed Variables Via Stochastic Reduced Basis Method

In this paper, the reliability of a random beam structure with different distributed variables is calculated, and the scope of application of the stochastic reduced basis methods (SRBMs) method is expanded. With the help of stochastic Krylov subspace, the undetermined coefficients of the response expression under static force can be obtained. Then using Galerkin projection technology to obtain the final expression of the random response expression. Furthermore, an example is investigated in order to demonstrate the precision of the method, the results show that this method has high calculation accuracy comparing with direct Monte Carlo simulation.

Deterministic and Stochastic Model Order Reduction for Vibration Analyses of Structures With Uncertainties

To reduce the computational effort using polynomial chaos expansion to predict the dynamic characteristics of structures with several uncertain parameters, hybrid techniques combining stochastic finite element analysis with either deterministic or stochastic model order reduction (MOR) are developed. For the deterministic MOR, the Arnoldi-based Krylov subspace technique is implemented to reduce the system matrices of the finite element model. For the stochastic MOR, a stochastic reduced basis method is implemented in which the structural modal and frequency responses are approximated by a small number of basis vectors using stochastic Krylov subspace. To demonstrate the computational efficiency of each reduced stochastic finite element model, variability in the natural frequencies and frequency responses of a simply supported flexible plate randomized by uncertain geometrical and material parameters is examined. Results are compared with both Monte Carlo (MC) simulations and nonreduced stochastic models. Using the reduced models, the effects of the individual uncertain parameters as well as the combined uncertainties on the dynamic characteristics of the plate are examined.

Joint Uncertainty Propagation in Linear Structural Dynamics Using Stochastic Reduced Basis Methods

Uncertainties in the properties of joints produce uncertainties in the dynamic response of built-up structures. Line joints, such as glued or continuously welded joints, have spatially distributed uncertainty and can be modeled by a discretized random field. Techniques such as Monte Carlo simulation can be applied to estimate the output statistics, but computational cost can be prohibitive. This paper addresses how uncertainties in joints might be included straightforwardly in a finite element model, with particular reference to approaches based on fixed-interface (Craig– Bampton) component mode synthesis and a stochastic reduced basis method with two variants. These methods are used to determine the output statistics of a structure. Unlike perturbation-based methods, good accuracy can be achieved even when the coefficients of variation of the input random variables are not small. Undamped as well as proportionally damped components are considered. Efficient implementations are proposed based on an exact matrix identity that leads to a significantly lower computational cost if the number of joint degrees of freedom is sufficiently small compared with the structure’s overall number of degrees of freedom. A numerical example is presented. The proposed formulation is an efficient and effective implementation of a stochastic reduced basis projection scheme. It is seen that the method can be up to orders of magnitude faster than direct Monte Carlo simulation, while providing results of comparable accuracy. Furthermore, the proposed implementation is more efficient when fewer joints are affected by uncertainty.

Recursive approach for random response analysis using non-orthogonal polynomial expansion

Using non-orthogonal polynomial expansions, a recursive approach is proposed for the random response analysis of structures under static loads involving random properties of materials, external loads, and structural geometries. In the present formulation, non-orthogonal polynomial expansions are utilized to express the unknown responses of random structural systems. Combining the high-order perturbation techniques and finite element method, a series of deterministic recursive equations is set up. The solutions of the recursive equations can be explicitly expressed through the adoption of special mathematical operators. Furthermore, the Galerkin method is utilized to modify the obtained coefficients for enhancing the convergence rate of computational outputs. In the post-processing of results, the first- and second-order statistical moments can be quickly obtained using the relationship matrix between the orthogonal and the non-orthogonal polynomials. Two linear static problems and a geometrical nonlinear problem are investigated as numerical examples in order to illustrate the performance of the proposed method. Computational results show that the proposed method speeds up the convergence rate and has the same accuracy as the spectral finite element method at a much lower computational cost, also, a comparison with the stochastic reduced basis method shows that the new method is effective for dealing with complex random problems.

Multi-element stochastic reduced basis methods

This paper presents multi-element stochastic reduced basis methods (ME-SRBMs) for solving linear stochastic partial differential equations. In ME-SRBMs, the domain of definition of the random inputs is decomposed into smaller subdomains or random elements. Stochastic reduced basis methods (SRBMs) are employed in each random element to evaluate the response statistics. These elemental statistics are assimilated to compute the overall statistics. The effectiveness of the method is demonstrated by solving the stochastic steady-state heat transfer equation on two geometries involving different types of boundary conditions. Numerical studies are conducted to investigate the h-convergence rates of global and local preconditioning strategies.

Stochastic Krylov subspace methods for flow in random porous media

AbstractThe present work establishes a comparative analysis of two different Krylov‐based methods for assessing uncertainty in porous media flow applications. They are: (1) the stochastic reduced basis method (SRBM) and the (2) Krylov‐Karhunen‐Loeve moment equation (KKLME). The former relies on the construction of an orthogonal basis from which projectors may be realized to perform the uncertainty analysis on a lower‐dimensional space. The second approach relies on the idea of expressing input and corresponding outputs in terms of stochastic and perturbative polynomial expansions. By grouping terms and moments of the same order, the original stochastic equation is replaced by successive (and parallel) solutions of a set of deterministic equations. We provide a set of numerical experiments illustrating the capabilities of SRBM and KKLME against Monte Carlo simulations (MCS) on stationary permeability cases. We show that KKLME is superior to SRBM. Furthermore, KKLME appears to be also a more efficient alternative than MCS for non‐stationary permeability field distributions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Hybridization of stochastic reduced basis methods with polynomial chaos expansions

We propose a hybrid formulation combining stochastic reduced basis methods with polynomial chaos expansions for solving linear random algebraic equations arising from discretization of stochastic partial differential equations. Our objective is to generalize stochastic reduced basis projection schemes to non-Gaussian uncertainty models and simplify the implementation of higher-order approximations. We employ basis vectors spanning the preconditioned stochastic Krylov subspace to represent the solution process. In the present formulation, the polynomial chaos decomposition technique is used to represent the stochastic basis vectors in terms of multidimensional Hermite polynomials. The Galerkin projection scheme is then employed to compute the undetermined coefficients in the reduced basis approximation. We present numerical studies on a linear structural problem where the Youngs modulus is represented using Gaussian as well as lognormal models to illustrate the performance of the hybrid stochastic reduced basis projection scheme. Comparison studies with the spectral stochastic finite element method suggest that the proposed hybrid formulation gives results of comparable accuracy at a lower computational cost.

Comparative study of projection schemes for stochastic finite element analysis

We present a comparison of subspace projection schemes for stochastic finite element analysis in terms of accuracy and computational efficiency. More specifically, we compare the polynomial chaos projection scheme with reduced basis projection schemes based on the preconditioned stochastic Krylov subspace. Numerical studies are presented for two problems: (1) static analysis of a plate with random Young’s modulus and (2) settlement of a foundation supported on a randomly heterogeneous soil. Monte Carlo simulation results based on exact structural analysis are used to generate benchmark results against which the projection schemes are compared. We show that stochastic reduced basis methods require significantly less computer memory and execution time compared to the polynomial chaos approach, particularly for large-scale problems with many random variables. For the class of problems considered, we find that stochastic reduced basis methods can be up to orders of magnitude faster, while providing results of comparable or better accuracy.

Forced response statistics of mistuned bladed disks: a stochastic reduced basis approach

This paper presents a stochastic reduced basis approach for predicting the forced response statistics of mistuned bladed-disk assemblies. In this approach, the system response in the frequency domain is represented using a linear combination of complex stochastic basis vectors with undermined coefficients. The terms of the preconditioned stochastic Krylov subspace are used here as basis vectors. Two variants of the stochastic Bubnov–Galerkin scheme are employed for computing the undetermined terms in the reduced basis representation, which arise from how the condition for orthogonality between two random vectors is interpreted. Explicit expressions for the response quantities can then be derived in terms of the random system parameters, which allow for the possibility of efficiently computing the response statistics in the post-processing stage. Numerical studies are presented for mistuned cyclic assemblies of mono-coupled single-mode components. It is demonstrated that the accuracy of the response statistical moments computed using stochastic reduced basis methods can be orders of magnitude better than classical perturbation methods.

Stochastic Reduced Basis Methods

Stochastic reduced basis methods for solving large-scale linear random algebraic systems of equations, such as those obtained by discretizing linear stochastic partial differential equations in space, time, and the random dimension, are introduced. The fundamental idea employed is to represent the system response using a linear combination of stochastic basis vectors with undetermined deterministic coefficients (or random functions). We present a theoretical justification for employing basis vectors spanning the preconditioned stochastic Krylov subspace to approximate the response process. Subsequently, variants of the Bubnov–Galerkin scheme are employed to compute the undetermined coefficients, which allow explicit expressions for the response quantities to be derived. We also examine some theoretical properties of the projection scheme and procedures for computing the response statistics. Numerical studies are presented for static and dynamic analysis of stochastic structural systems. We demonstrate that significant improvements over the Neumann expansion scheme, as well as other relevant techniques in the literature, can be achieved.

Stochastic reduced basis methods

Stochastic reduced basis methods