SUMMARYAn eXtended Stochastic Finite Element Method has been recently proposed for the numerical solution of partial dierential equations dened on random domains. This method is based on a mariage between the eXtended Finite Element Method and spectral stochastic methods. In this paper, we propose an extension of this method for the numerical simulation of random multi-phased materials. The random geometry of material interfaces is described implicitly by using random level-set functions. A xed deterministic nite element mesh, which is not conforming the random interfaces, is then introduced in order to approximate the geometry and the solution. Classical spectral stochastic nite element approximation spaces are not able to capture the irregularities of the solution eld with respect to spatial and stochastic variables, which leads to a deterioration of the accuracy and convergence properties of the approximate solution. In order to recover optimal convergence properties of the approximation, we propose an extension of the partition of unity method to the spectral stochastic framework. This technique allows the enrichment of approximation spaces with suitable functions based on an a priori knowledge of the irregularities in the solution. Numerical examples illustrate the eciency of the proposed method and demonstrate the relevance of the enrichment procedure.

Inspection by non-destructive testing (NDT) techniques of existing structures is not perfect and it has become a common practice to model their reliability in terms of probability of detection (PoD), probability of false alarms (PFA) and receiver operating characteristic (ROC) curves. These results are generally the main inputs needed by owners of structures in order to achieve inspection, maintenance and repair plans (IMR). The assessment of PoD and PFA is even deduced from intercalibration of NDT tools or from the modelling of the noise and the signal. In this last case when the noise and the signal depend on the location on the structure PoD and PFA are spatially dependent. This paper presents how to define PoD and PFA when damage and detection are stochastic fields or spatially dependent. Corrosion of coastal structures in harbours is considered for illustration and ROC curves are deduced. Identification of probability density functions on polynomial chaos is shown to be more suitable than predefined probability distribution functions (pdf) in view of fitting noise and signal plus noise distributions.

This paper is devoted to the computational nonlinear stochastic homogenization of a hyperelastic heterogeneous microstructure using a nonconcurrent multiscale approach. The geometry of the microstructure is random. The nonconcurrent multiscale approach for micro-macro nonlinear mechanics is extended to the stochastic case. Because the nonconcurrent multiscale approach is based on the use of a tensorial decomposition, which is then submitted to the curse of dimensionality, we perform an analysis with respect to the stochastic dimension. The technique uses a database describing the strain energy density function (potential) in both the macroscopic Cauchy green strain space and the geometrical random parameters domain. Each value of the potential is numerically computed by means of the FEM on an elementary cell whose geometry is given by the random parameters and the corresponding macroscopic strains being prescribed as boundary conditions. An interpolation scheme is finally introduced to obtain a continuous explicit form of the potential, which, by derivation, allows to evaluate the macroscopic stress and elastic tangent tensors during the macroscopic structural computations. Two numerical examples are presented.A. CLÉMENT, C. SOIZE AND J. YVONNET concurrent multiscale methods were developed and aim at solving simultaneously the nonlinear problems at both scales (see for instance [6][7][8][9][10][11][12][13]). The macroscopic problem provides the strain states in all integration points and gives the boundary conditions used to solve each microscopic problem. This kind of techniques allows taking into account different types of nonlinearities but may lead to significant computational costs. Alternatively, the nonconcurrent multiscale methods aim at decoupling the numerical simulations at both scales. They consist in identifying parameters characterizing the macroscopic constitutive law from experimental or numerical test [14] or using a database formed of microscopic calculations to build a numerical constitutive law without any assumption on the constitutive equations [15][16][17][18][19]. In this work, we focus on the methodology proposed in [17][18][19].On the other hand, it may also be necessary to take into account the different sources of uncertainty at the microscopic level if one seeks to obtain a reliable model of the macroscopic behavior of the material. Thus, the general area of stochastic multiscale modeling of complex random heterogeneous microstructures has recently attracted a continuously growing attention in the scientific community. A simple way to solve this kind of problem consists in coupling numerical simulations of random microstructures (based on experimental identification or some geometrical characteristics such that the volume fraction or others parameters such that the spatial and size distributions [20,21]) with classical finite element computations. Then, one can perform a statistical study of average quantities of interest characterizing the macroscopic behavior of the material (e.g. s...

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. AbstractThis paper is devoted to a computational stochastic multiscale analysis of nonlinear structures made up of heterogeneous hyperelastic materials. At the microscale level, the nonlinear constitutive equation of the material is characterized by a stochastic potential for which a polynomial chaos representation is used. The geometry of the microstructure is random and characterized by a high number of random parameters. The method is based on a deterministic non-concurrent multiscale approach devoted to micro-macro nonlinear mechanics which leads us to characterize the nonlinear constitutive equation with an explicit continuous form of the strain energy density function with respect to the large scale Cauchy Green strain states. To overcome the curse of dimensionality, due to the high number of involved random variables, the problem is transformed into another one consisting in identifying the potential on a polynomial chaos expansion. Several strategies, based on novel algorithms dedicated to high stochastic dimension, are used and adapted for the class of multi-modal random variables which may characterize the potential. Numerical examples, at both small and large scales, allow analyzing the efficiency of the approach through comparisons with classical methods.

The article contains sections titled: 1. Polyols, General 1.1. Definition 1.2. Physical, Chemical, and Organoleptic Properties 1.3. Metabolism and Nutrition 1.3.1. Uptake, Digestion, and Tolerance 1.3.2. Nutritional Aspects 1.3.3. Oral Health and Hygiene 1.3.4. Oxidative Stress 1.3.5. Conclusion 1.4. Regulatory Aspects 2. Xylitol 2.1. Physical, Chemical, and Organoleptic Properties 2.2. Production 2.3. Specifications, Analysis, and Legal Aspects 2.4. Uses 2.5. Metabolism, Tolerance, and Safety 3. Sorbitol 3.1. Physical, Chemical, and Organoleptic Properties 3.2. Production 3.3. Regulatory and Quality Aspects 3.3.1. Purity Requirements 3.3.2. Analysis 3.4. Uses 3.5. Physiology, Tolerance, Toxicology 3.6. Economic Aspects 4. Mannitol 4.1. Physical, Chemical, and Organoleptic Properties 4.2. Production 4.3. Quality Aspects 4.4. Uses 4.5. Physiology, Tolerance, Toxicology 5. Isomaltulose and Trehalulose, Isomalt 5.1. Isomaltulose and Trehalulose 5.1.2. Physical and Chemical Properties 5.1.3. Production 5.1.4. Uses 5.1.5. Economic Aspects 5.2. Isomalt 5.2.1. Physical and Chemical Properties 5.2.2. Production 5.2.3. Uses 5.2.4. Economic Aspects 6. Lactitol 6.1. Physical, Chemical, and Physiological Properties 6.2. Production 6.3. Analysis and Regulatory Status 6.4. Uses 6.5. Economic Aspects 7. Maltitol and Maltitol‐Containing Syrups 7.1. Physical, Chemical, and Organoleptic Properties 7.2. Uses 7.3. Economic Aspects 8. Erythritol 8.1. Physical, Chemical, and Physiological Properties 8.2. Production 8.3. Analysis and Regulatory Status 8.4. Uses 8.5. Economic Aspects

No abstract

SUMMARYAn eXtended Stochastic Finite Element Method has been recently proposed for the numerical solution of partial differential equations defined on random domains. This method is based on a marriage between the eXtended Finite Element Method and spectral stochastic methods. In this article, we propose an extension of this method for the numerical simulation of random multi-phased materials. The random geometry of material interfaces is described implicitly by using random level set functions. A fixed deterministic finite element mesh, which is not conforming to the random interfaces, is then introduced in order to approximate the geometry and the solution. Classical spectral stochastic finite element approximation spaces are not able to capture the irregularities of the solution field with respect to spatial and stochastic variables, which leads to a deterioration of the accuracy and convergence properties of the approximate solution. In order to recover optimal convergence properties of the approximation, we propose an extension of the partition of unity method to the spectral stochastic framework. This technique allows the enrichment of approximation spaces with suitable functions based on an a priori knowledge of the irregularities in the solution. Numerical examples illustrate the efficiency of the proposed method and demonstrate the relevance of the enrichment procedure.

To cite this version:G. Stefanou, Anthony Nouy, Alexandre Clement. Identification of random shapes from images through polynomial chaos expansion of random level-set functions. International Journal for Numerical Methods in Engineering, Wiley, 2009, 79 (2) SUMMARYIn this paper, an e cient method is proposed for the identi cation of random shapes in a form suitable for numerical simulation within the eXtended Stochastic Finite Element Method (X-SFEM).The method starts from a collection of images representing di erent outcomes of the random shape to identify. The key-point of the method is to represent the random geometry in an implicit manner using the level-set technique. In this context, the problem of random geometry identi cation is equivalent to the identi cation of a random level-set function, which is a random eld. This random eld is represented on a polynomial chaos basis and various e cient numerical strategies are proposed in order to identify the coe cients of its polynomial chaos decomposition. The performance of these strategies is evaluated through some "manufactured" problems and useful conclusions are provided. The propagation of geometrical uncertainties in structural analysis using the X-SFEM is nally examined.key words:Random geometry; Level-set method; Probabilistic identi cation; Polynomial chaos; Maximum likelihood; Extended stochastic nite element method.