IEEE TRANSACTIONS ON MAGNETICS, VOL. 48, NO. 2, FEBRUARY 2012 759 Adaptive Method for Non-Intrusive Spectral Projection—Application on a Stochastic Eddy Current NDT Problem K. Beddek , S. Clenet , O. Moreau , V. Costan , Y. Le Menach ,and A. Benabou Universite Lille1, L2EP, Villeneuve d’Ascq 59650, France Electricite De France R&D, dept. THEMIS, Clamart 92141, France Arts et Metiers ParisTech, L2EP, Lille 59046, France The Non-Intrusive Spectral Projection (NISP) method is widely used for uncertainty quantification in stochastic models. The deter- mination of the expansion of the solution on the polynomial chaos requires the computation of multidimensional integrals. An automatic adaptive algorithm based on nested sparse grids has been developed to evaluate those integrals. The adapted algorithm takes into ac- count the weight of each random variable with respect to the output of the model. To achieve that it constructs anisotropic sparse grid of the mean, leading to a reduction of the number of numerical simulations. Furthermore, the spectral form of the solution is explicitly identified from the constructed quadrature scheme. Numerical results obtained on an industrial application in NDT demonstrate the efficiency of the proposed method. Index Terms—Adaptive sparse grids, non-destructive testing, polynomial chaos, uncertainty quantification. I. INTRODUCTION F OR many years, deterministic modeling approaches as- suming material properties, sources and geometric dimen- sions to be known, have made it possible to deal with a great number of applications in electromagnetism. However, design, reliability or risk management requires more and more to es- timate the influence of the uncertainties of the input data on the output data. One way to take into account the variability of the input data is to consider them as random fields or vari- ables. In computational electromagnetics, a stochastic partial differential equations system has then to be solved. The poly- nomial chaos (PC) methods consist in expanding the stochastic solution in an orthogonal polynomial basis. The Non-Intrusive Spectral Projection (NISP) makes it possible to compute the PC expansion coefficients from an appropriate set of deterministic solutions. This approach has the advantage to not require any modifications of the original deterministic software contrary to the spectral stochastic finite element method (SSFEM) [2]–[4]. The expansion coefficients involve multidimensional integrals for which each evaluation point is a deterministic computation for a set of input random variables. The number of these deter- ministic computation greatly depends on the chosen method for the integral calculation: sampling, full tensor-product quadra- ture (Gauss, Clenshaw-Curtis) or Smolyak sparse grids These schemes are isotropic formulas in the sense that the different in- tegral directions are equally discretized. Even with a sparse grid, the number of quadrature points can be very important when the number of input variables is high. Moreover, integrating in the same way along each stochastic dimension may turn out to be not adapted when the numerical model is sensitive only to a small number of input variables. An adaptive procedure is a way to reduce the number of quadrature points (deterministic Manuscript received July 06, 2011; revised October 07, 2011; accepted Oc- tober 22, 2011. Date of current version January 25, 2012. Corresponding author: O. Moreau (e-mail: olivier.moreau@edf.fr). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2011.2175204 simulations) by taking advantage of the difference of sensitivity along the stochastic dimensions. In this work, we propose an adaptive algorithm based on the non-isotropic nested Gauss-Pat- terson formulas which takes into account the global sensitivity of the model with respect to the input random variables. This method has been applied to the Eddy Currents Non Destruc- tive Testing inspection of steam generator tubes with regard to clogging of the quatrefoil support plate in steam generators of nuclear power plants [7]. II. STOCHASTIC PROBLEM Let us consider a spatial domain divided into con- ducting disjoint subdomains and non-conducting disjoint subdomains. The permeability and the conductivity of are assumed to be random fields, written as and . is the spatial variable and denotes the elementary event be- longing to the random space . Since the behavior laws are random, and are unknown random fields defined on . By introducing the magnetic vector potential and the electric scalar potential , the stochastic magneto-harmonic formulation is given by: (1) where is the angular speed. To solve the stochastic problem, the non-intrusive spectral projection is used. The method will be thereafter described. III. NON-INTRUSIVE SPECTRAL PROJECTION In the following, we will assume that and can be expressed as finite sums: (2) (3) where and are functions of space. The random variables and are assumed to be independent and considered 0018-9464/$31.00 © 2012 IEEE