Stability of Karhunen–Loève expansion for the simulation of Gaussian stochastic fields using Galerkin scheme

The paper investigates the problem of numerical stability of the Karhunen–Loève expansion for the simulation of Gaussian stochastic fields using Galerkin scheme. The instability is expressed as loss of positive definiteness of covariance matrix and is the result of modifications of standard exponential covariance functions that are commonly applied to increase the sparsity of the covariance matrix. The loss of positive definiteness of covariance matrix limits the use of efficient eigenvalue solvers that are needed for the solution of the resulting generalized eigenvalue problem. Two modifications of the shape of covariance function to avoid instability problems and at the same time to raise the numerical efficiency of Karhunen–Loève expansion by increasing the sparsity of the covariance matrix are proposed. The effects of the proposed modifications are demonstrated on numerical examples.     

An efficient method of solving the Navier–Stokes equations for flow control

A new method of solving the Navier–Stokes equations efficiently by reducing their number of modes is proposed in the present paper. It is based on the Karhunen–Loève decomposition which is a technique of obtaining empirical eigenfunctions from the experimental or numerical data of a system. Employing these empirical eigenfunctions as basis functions of a Galerkin procedure, one can a priori limit the function space considered to the smallest linear subspace that is sufficient to describe the observed phenomena, and consequently reduce the Navier–Stokes equation defined on a complicated geometry to a set of ordinary differential equations with a minimum degree of freedom. The present algorithm is well suited for the problems of flow control or optimization, where one has to compute the flow field repeatedly using the Navier–Stokes equation but one can also estimate the approximate solution space of the flow field based on the range of control variables. The low-dimensional dynamic model of viscous fluid flow derived by the present method is shown to produce accurate flow fields at a drastically reduced computational cost when compared with the finite difference solution of the Navier–Stokes equation. © 1998 John Wiley & Sons, Ltd.     

A Fourier–Karhunen–Loève discretization scheme for stationary random material properties in SFEM

In order to overcome the computational difficulties in Karhunen–Loève (K–L) expansions of stationary random material properties in stochastic finite element method (SFEM) analysis, a Fourier–Karhunen–Loève (F–K–L) discretization scheme is developed in this paper, by following the harmonic essence of stationary random material properties and solving a series of specific technical challenges encountered in its development. Three numerical examples are employed to investigate the overall performance of the new discretization scheme and to demonstrate its use in practical SFEM simulations. The proposed F–K–L discretization scheme exhibits a number of advantages over the widely used K–L expansion scheme based on FE meshes, including better computational efficiency in terms of memory and CPU time, convenient a priori error‐control mechanism, better approximation accuracy of random material properties, explicit methods for predicting the associated eigenvalue decay speed and geometrical compatibility for random medium bodies of different shapes. Copyright © 2007 John Wiley & Sons, Ltd.     

Karhunen–Loève decomposition of random fields based on a hierarchical matrix approach

The simulation of the behavior of structures with uncertain properties is a challenging issue, because it requires suitable probabilistic models and adequate numerical tools. Nowadays, it is possible to perform probabilistic investigations of the structural performance, which take into account a space‐variant uncertainty characterization of the structures. Given a structural solver and the probabilistic models, the reliability analysis of the structural response depends on the continuous random fields approximation, which is carried out by means of a finite set of random variables. The paper analyzes the main aspects of discretization in the case of 2D problems. The combination of the well‐known Karhunen–Loève series expansion, the finite element method and the hierarchical matrices approach is proposed in the paper. Copyright © 2013 John Wiley & Sons, Ltd.     

On the investigation of shell buckling due to random geometrical imperfections implemented using Karhunen–Loève expansions

For the accurate prediction of the collapse behaviour of thin cylindrical shells, it is accepted that geometrical and other imperfections in material properties and loading have to be accounted for in the simulation. There are different methods of incorporating imperfections, depending on the availability of accurate imperfection data. The current paper uses a spectral decomposition of geometrical uncertainty (Karhunen–Loève expansions). To specify the covariance of the required random field, two methods are used. First, available experimentally measured imperfection fields are used as input for a principal component analysis based on pattern recognition literature, thereby reducing the cost of the eigenanalysis. Second, the covariance function is specified analytically and the resulting Friedholm integral equation of the second kind is solved using a wavelet‐Galerkin approach. Experimentally determined correlation lengths are used as input for the analytical covariance functions. The above procedure enables the generation of imperfection fields for applications where the geometry is slightly modified from the original measured geometry. For example, 100 shells are perturbed with the resulting random fields obtained from both methods, and the results in the form of temporal normal forces during buckling, as simulated using LS‐DYNA^®, as well as the statistics of a Monte Carlo analysis of the 100 shells in each case are presented. Although numerically determined mean values of the limit load of the current and another numerical study differ from the experimental results due to the omission of imperfections other than geometrical, the coefficients of variation are shown to be in close agreement. Copyright © 2007 John Wiley & Sons, Ltd.     

Gaussian process emulators for the stochastic finite element method

This paper explores a method to reduce the computational cost of stochastic finite element codes. The method, known as Gaussian process emulation, consists of building a statistical approximation to the output of such codes based on few training runs. The incorporation of emulation is explored for two aspects of the stochastic finite element problem. First, it is applied to approximating realizations of random fields discretized via the Karhunen–Loève expansion. Numerical results of emulating realizations of Gaussian and lognormal homogeneous two‐dimensional random fields are presented. Second, it is coupled with the polynomial chaos expansion and the partitioned Cholesky decomposition in order to compute the response of the typical sparse linear system that arises due to the discretization of the partial differential equations that govern the response of a stochastic finite element problem. The advantages and challenges of adopting the proposed coupling are discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

Hybrid stress quadrilateral finite element approximation for stochastic plane elasticity equations

This paper considers stochastic hybrid stress quadrilateral finite element analysis of plane elasticity equations with stochastic Young's modulus and stochastic loads. Firstly, we apply Karhunen–Loève expansion to stochastic Young's modulus and stochastic loads so as to turn the original problem into a system containing a finite number of deterministic parameters. Then we deal with the stochastic field and the space field by k ?version/p ?version finite element methods and a hybrid stress quadrilateral finite element method, respectively. We derive a priori error estimates, which are uniform with respect to the Lamè constant λ ∈(0,+∞ ). Finally, we provide some numerical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Recursive solution of inverse natural convection problems by means of mode reduction

 A recursive method based on the Kalman filtering is developed to solve inverse natural convection problems of estimating the unsteady nonuniform wall heat flux from temperature measurements in the flow. By employing the Karhunen–Loève Galerkin procedure that reduces the Boussinesq equation to a small set of ordinary differential equations, the computational difficulties associated with the Kalman filtering for the partial differential equations are overcome. The present method is assessed through several numerical experiments, and is found to yield satisfactory results. Received 20 January 2001 / Accepted 31 May 2001     

Elastic wave propagation in a 3‐D unbounded random heterogeneous medium coupled with a bounded medium. Application to seismic soil–structure interaction (SSSI)

We present a sub‐structuring method for the coupling between a large elastic structure, and a stratified soil half‐space exhibiting random heterogeneities over a bounded domain and impinged by incident waves. Both media are also weakly dissipative. The concept of interfaces classically used in sub‐structuring methods is extended to ‘volume interfaces’ in the proposed approach. The random dimension of the stochastic fields modelling the heterogeneities in the soil is reduced by introducing a Karhunen–Loéve expansion of these stochastic fields. The coupled overall problem is solved by Monte‐Carlo simulation techniques. A realistic example of a large industrial structure interacting with an uncertain stratified soil medium under earthquake is finally presented. This case study and others validate the presented methodology and its ability to handle complex mechanical systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Stochastic model construction of observed random phenomena

A method for constructing probabilistic models of non-stationary time dependent natural hazards is proposed. It is based on the use of Karhunen–Loève expansion and of a kernel estimator for the distribution of the multivariate random variables appearing in the expansion. The terms of the expansion and the distribution are identified from available measures. The approach is assessed through an academic example and is then applied to seismic ground motion modelling based on recorded data.     

An adaptive spectral Galerkin stochastic finite element method using variability response functions

A methodology is proposed in this paper to construct an adaptive sparse polynomial chaos (PC) expansion of the response of stochastic systems whose input parameters are independent random variables modeled as random fields. The proposed methodology utilizes the concept of variability response function in order to compute an a priori low‐cost estimate of the spatial distribution of the second‐order error of the response, as a function of the number of terms used in the truncated Karhunen–Loève (KL) expansion. This way the influence of the response variance to the spectral content (correlation structure) of the random input is taken into account through a spatial variation of the truncated KL terms. The criterion for selecting the number of KL terms at different parts of the structure is the uniformity of the spatial distribution of the second‐order error. This way a significantly reduced number of PC coefficients, with respect to classical PC expansion, is required in order to reach a uniformly distributed target second‐order error. This results in an increase of sparsity of the coefficient matrix of the corresponding linear system of equations leading to an enhancement of the computational efficiency of the spectral stochastic finite element method. Copyright © 2015 John Wiley & Sons, Ltd.     

Simulation of random fields on random domains

This paper focuses on the simulation of random fields on random domains. This is an important class of problems in fields such as topology optimization and multiphase material analysis. However, there is still a lack of effective methods to simulate this kind of random fields. To this end, we extend the classical Karhunen–Loève expansion (KLE) to this class of problems, and we denote this extension as stochastic Karhunen–Loève expansion (SKLE). We present three numerical algorithms for solving the stochastic integral equations arising in the SKLE. The first algorithm is an extension of the classical Monte Carlo simulation (MCS), which is used to solve the stochastic integral equation on each sampled domain. However, such approach demands remeshing each sampled domain and solving the corresponding integral equation, which can become computationally very demanding. In the second algorithm, a domain transformation is used to map the random domain into a reference domain, and only one mesh for the reference domain is required. In this way, remeshing different sample realizations of the random domain is avoided and much computational effort is thus saved. MCS is then adopted to solve the corresponding stochastic integral equation. Further, to avoid the computational effort of MCS, the third algorithm proposed in this contribution involves a reduced-order method to solve the stochastic integral equation efficiently. In this third algorithm, stochastic eigenvectors are represented as a sum of products of unknown random variables and deterministic vectors, where the deterministic vectors are efficiently computed by solving deterministic eigenvalue problems. The random variables and stochastic eigenvalues that appear in this third algorithm are calculated by a reduced-order stochastic eigenvalue problem constructed by the obtained deterministic vectors. Based on the obtained stochastic eigenvectors, the target random field is then simulated and reformulated as a classical KLE-like representation. Finally, three numerical examples are presented to demonstrate the performance of the proposed methods.     

A stochastic approach to model material variation determining tow impregnation in out-of-autoclave prepreg consolidation

Material variations are always present even though out-of-autoclave prepregs are machine-made. They strongly determine the consolidation and may eventually lead to voids within the final part, depending on applied process conditions. To capture any contingencies, stochastic differential equations are derived to describe various interacting phenomena in OoA consolidation. In a second step the probabilistic space is discretized using the Karhunen–Loève truncation and the Probabilistic Collocation method is applied in order to use deterministic solvers for flow and compaction problems. The initial degree of impregnation is represented by an Ornstein–Uhlenbeck process and calibrated with CT-images.     

Dimension reduction in stochastic modeling of coupled problems

Coupled problems with various combinations of multiple physics, scales, and domains are found in numerous areas of science and engineering. A key challenge in the formulation and implementation of corresponding coupled numerical models is to facilitate the communication of information across physics, scale, and domain interfaces, as well as between the iterations of solvers used for response computations. In a probabilistic context, any information that is to be communicated between subproblems or iterations should be characterized by an appropriate probabilistic representation. Although the number of sources of uncertainty can be expected to be large in most coupled problems, our contention is that exchanged probabilistic information often resides in a considerably lower dimensional space than the sources themselves. This work thus presents an investigation into the characterization of the exchanged information by a reduced‐dimensional representation and in particular by an adaptation of the Karhunen‐Loève decomposition. The effectiveness of the proposed dimension–reduction methodology is analyzed and demonstrated through a multiphysics problem relevant to nuclear engineering. Copyright © 2012 John Wiley & Sons, Ltd.     

Transient analysis of electro‐osmotic transport by a reduced‐order modelling approach

Transient behaviour of electro‐osmotic transport in typical electrokinetic channels is studied in this paper. The time needed for the electro‐osmotic flow to reach steady‐state exhibits multiple time scales depending on whether the flow is governed by either a viscous force, electrokinetic force or by a combination of both. When an intersection is present in the electrokinetic channel, such as in a cross or a T‐channel, the flow in the main channel and in the intersection gets to steady‐state at different times. A weighted Karhunen–Loève (KL) decomposition method is proposed in this paper to generate the global basis function for reduced‐order simulation. The key idea in a weighted KL approach is that, instead of minimizing a least‐squares measure of ‘error’ between the linear subspace spanned by the basis functions and the observation space, we minimize the weighted ‘error’ between the two spaces. The global basis functions in a weighted KL approach can be generated by computing the singular value decomposition (SVD) of the matrix containing the weighted snapshots. We show that the weighted KL decomposition based reduced‐order model is computationally more efficient and can capture the multiple time scales encountered in electro‐osmotic transport much more effectively compared to the classical KL decomposition based reduced‐order model. Copyright © 2003 John Wiley & Sons, Ltd.     

A meshfree-Galerkin method in modelling and synthesizing spatially varying soil properties

Physical properties of soil vary from point to point in space and exhibit great uncertainty, suggesting random field as a natural approach in modelling and synthesizing these properties. The significance of considering spatial variability and uncertainty of soil properties is greatly manifested in the probabilistic seismic risk analysis of soil–structural system (nonlinear dynamic analysis under earthquake loading), where modelling and synthesis of the spatial variability and uncertainty of soil properties are necessary. This paper introduces a meshfree-Galerkin approach within the Karhunen–Loève (K–L) expansion scheme for representation of spatial soil properties modelled as the random fields. The meshfree shape functions are introduced and employed as a set of basis functions in the Galerkin scheme to obtain the eigen-solutions of integral equation of K–L expansion. An optimization scheme is proposed for the resulting eigenvectors in treating the compatibility between the target and analytical covariance models. Assessments of the meshfree-Galerkin method are conducted for the resulting eigen-solutions and the representation of covariance models for various homogeneous and nonhomogeneous random fields. The accuracy and validity of the proposed approach are demonstrated through the modelling and synthesis of the spatial field models inferred from the field measurements.     

Statistical reconstruction and Karhunen–Loève expansion for multiphase random media

Termed as random media, rocks, composites, alloys and many other heterogeneous materials consist of multiple material phases that are randomly distributed through the medium. This paper presents a robust and efficient algorithm for reconstructing random media, which can then be fed into stochastic finite element solvers for statistical response analysis. The new method is based on nonlinear transformation of Gaussian random fields, and the reconstructed media can meet the discrete‐valued marginal probability distribution function and the two‐point correlation function of the reference medium. The new method, which avoids iterative root‐finding computation, is highly efficient and particularly suitable for reconstructing large‐size random media or a large number of samples. Also, benefiting from the high efficiency of the proposed reconstruction scheme, a Karhunen–Loève (KL) representation of the target random medium can be efficiently estimated by projecting the reconstructed samples onto the KL basis. The resulting uncorrelated KL coefficients can be further expressed as functions of independent Gaussian random variables to obtain an approximate Gaussian representation, which is often required in stochastic finite element analysis. Copyright © 2015 John Wiley & Sons, Ltd.     

A unified stochastic framework for robust topology optimization of continuum and truss-like structures

In this article, a unified framework is introduced for robust structural topology optimization for 2D and 3D continuum and truss problems. The uncertain material parameters are modelled using a spatially correlated random field which is discretized using the Karhunen–Loève expansion. The spectral stochastic finite element method is used, with a polynomial chaos expansion to propagate uncertainties in the material characteristics to the response quantities. In continuum structures, either 2D or 3D random fields are modelled across the structural domain, while representation of the material uncertainties in linear truss elements is achieved by expanding 1D random fields along the length of the elements. Several examples demonstrate the method on both 2D and 3D continuum and truss structures, showing that this common framework provides an interesting insight into robustness versus optimality for the test problems considered.     

Spatiotemporal regularization for digital image correlation: Application to infrared camera frames

The spatiotemporal response of a stainless steel plate undergoing cyclic laser shock is recorded with an infrared camera, and digital image correlation is used to analyze both displacement and temperature fields. Two very challenging difficulties are addressed: (i) large gray‐level variations (due to temperature changes) and (ii) convection effects affecting images. To this aim, a spatiotemporal regularization is designed exploiting a numerical model of the test. The thermomechanical space‐time predictions are first processed through Karhunen‐Loève decomposition to extract dominant temporal and spatial modes. The temporal modes are then inserted in a spatiotemporal digital image correlation framework to estimate the experimental spatial modes that account for both gray‐level variations (and hence temperature) and displacement fields. It is shown that with only 3 modes, the full thermomechanical response of the material is captured. The temporal regularization issued from the model also allows the spurious effect of convection to be filtered out. Due to the drastic decrease in the number of degrees of freedom because of data reduction, the number of analyzed frames can be reduced from 50 down to 6 to capture the thermomechanical response, thereby leading to enhanced efficiency.     

Towards an improved critical wave groups method for the probabilistic assessment of large ship motions in irregular seas

A novel approach for the systematic construction of wind-generated, high probability, wave groups, is presented. The derived waveforms originate from a Markov chain model allowing for the incorporation of cross-correlations between successive wave heights and periods. Analytical expressions of the transition probability distributions are provided in terms of copulas. Rank correlations are estimated from an envelope-process-based approach. The Karhunen–Loève theorem is employed in order to construct the continuous analogs of discrete height and period successions. The method seems to predict well the expected wave heights. The period predictions are conservative, yet they follow the trends of simulated wave trains. Comparisons with predictions of the “Quasi-Determinism” theory for very high runs indicate good coincidence. The derived wave groups are intended to be used for the assessment of ship stability in irregular seas.