The flyer of the summer school initially distributed is available here ( French / German ).

The detailed program of the courses is given in the following table, with abstracts of all presentations.

Macroscopic properties (e.g., permeability, thermal conductivity or acoustic absorption) of cellular materials are highly influenced by the microstructure. Models from stochastic geometry are powerful tools for studying these relations. For cellular materials such as foams or polycrystalline materials, random tessellations are the models of choice.

In this talk, we will introduce various tessellation models and discuss their application as models for cellular materials. We will start by introducing the notion of a random tessellation, the most well-known model types (Voronoi and Laguerre tessellations, hyperplane tessellations), and their basic geometric characteristics.

In practice, a tessellation model can be fit to a real microstructure using geometric characteristics which are estimated from image data. In particular, three dimensional CT images are an important source of information on the microstructure of materials. We will introduce methods for the estimation of geometric characteristics of cellular materials from image data and describe how these characteristics can be used to fit tessellation models to the observed structure.

Finally, we will present some applications where tessellation models are used to investigate relations between the microstructure and macroscopic properties of foams.

[1] D. Stoyan, W.S. Kendall, J. Mecke, Stochastic geometry and its applications, 2nd edition, Wiley, Chichester, 1995
[2] C. Redenbach, Modelling foam structures using random tessellations, Stereology and Image Analysis, Proceedings of the 10th European Conference of ISS, 2009, www.mathematik.uni-kl.de/~redenbach/Ecs10_Redenbach.pdf

Complex microstructures in materials often involve multi-scale heterogeneous textures, modelled by random sets derived from Mathematical Morphology.
Starting from 2D or 3D images, a complete morphological characterization by image analysis is performed, and used for the identification of a model of random structure.
Morphological models enter into the prediction of effective properties by estimation, bounds, or from numerical simulations.
3D microtomographic images or simulations of realistic microstructures are introduced in a numerical solver to compute appropriate fields (in electrostatics, elasticity, …) and to estimate the effective properties by numerical homogenization, accounting for scale dependent statistical fluctuations of the fields. In addition to the estimation of effective properties, our approach provides essential indications on the size of the Representative Volume Element (RVE) for each property.
It will be illustrated by examples of multi-scale hierarchical models: Boolean random sets based on Cox point processes and various random grains (spheres, cylinders), showing a very low percolation threshold, and therefore a high conductivity or high elastic moduli for a low volume fraction of a second phase.

A powerful tool for uncertainty propagation in computational mechanics is the Stochastic Finite Element Method (SFEM). The considerable attention that SFEM received over the last two decades can be mainly attributed to the understanding of the significant influence of the inherent uncertainties on systems behavior and to the dramatic increase of the computational power in recent years, permitting the efficient treatment of complex realistic problems.
A fundamental issue in SFEM is the modelling of the uncertainty characterizing the system parameters, which is usually quantified by using the theory of stochastic functions (processes/fields). This presentation is devoted to the topic of simulation of stochastic processes and fields. The most important simulation methods for Gaussian and non-Gaussian processes and fields are presented and their basic characteristics (accuracy, computational efficiency) are examined. Some recent advances and open issues are also pointed out. Several applications to structural mechanics problems are finally presented, such as the identification of random fields representing structural parameters, the computation of the response variability of structures with random material and geometric properties, the stability analysis of shells with stochastic imperfections as well as the nonlinear dynamic analysis of structures with random system properties under earthquake loading.

In the last two decades, a growing attention has been given to functional approaches for uncertainty quantification, where uncertain (random) quantities are seen as functionals of parameters characterizing the input uncertainties. This functional view, combined with approximation theory and numerical analysis, has led to the development of a family of numerical method, the so-called spectral stochastic methods, for the propagation of uncertainties through a model, yielding a complete characterization of uncertain model outputs. In a first part, we will recall the basis of these methods, and we will present some recent advances on solution methods for stochastic parameterized partial differential equations (adapted or enriched approximation spaces, random geometry, multiscale methods…).
Classical spectral stochastic methods suffer from the curse of dimensionality, which is associated with the dramatic increase in the dimensionality of approximation spaces of multi-parametric functionals. In a second part, we will present recent advances on model reduction and tensor-based methods that try to overcome this curse of dimensionality. In particular, we will focus on a family of methods (Proper Generalized Decompositions) based on the a priori construction of separated representations of the solution. They can be interpreted as generalizations of spectral decomposition (Karhunen-Loeve decomposition), leading to the automatic construction of reduced bases of functions which are optimal for the representation of the solution.

[1] A. Nouy. Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations. Archives of Computational Methods in Engineering, 16(3):251–285, 2009.
[2] A. Nouy. Proper Generalized Decompositions and separated representations for the numerical solution of high dimensional stochastic problems. Archives of Computational Methods in Engineering, 17(4):403–434, 2010. doi:10.1007/s11831-010-9054-1.

We present a fully deterministic approach to a probabilistic interpretation of inverse problems in which unknown quantities are represented by random fields or processes, described by a non-Gaussian prior distribution. The description of the introduced random fields is given in a ``white noise'' framework, which enables us to solve the stochastic forward problem through Galerkin projection onto polynomial chaos. With the help of such representation, the probabilistic identification problem is cast in a polynomial chaos expansion setting and the linear Bayesian form of updating.By introducing the Hermite algebra this becomes a direct, purely algebraic way of computing the posterior, which is inexpensive to evaluate. In addition, we show that the well-known Kalman filter method is the low order part of this update. The proposed method has been tested on a stationary diffusion equation with prescribed source terms, characterised by an uncertain conductivity parameter which is then identified from limited and noisy data obtained by a measurement of the diffusing quantity. To speedup the computational process all ingredients of Bayesian update formula are approximated in low-rank data format. The approximation error, memory requirement and computing time are demonstrated on the example from numerical aerodynamic.

[1] A. Litvinenko, H.G. Matthies, Sparse data formats and efficient numerical methods for uncertainties quantification in numerical aerodynamics, Braunschweig, (Informatik-Bericht 2010-01), http://www.digibib.tu-bs.de/?docid=00036490
[2] B.V. Rosic, A. Litvinenko, O. Pajonk and H.G. Matthies, Direct Bayesian update of polynomial chaos representations, Braunschweig, (Informatik-Bericht 2011-02), http://www.digibib.tu-bs.de/?docid=00039000

Spectral methods have emerged as a powerful tool for uncertainty analysis in the last ten years. As far as an industrial implementation of these methods is concerned, non-intrusive approaches are well adapted since they only rely upon evaluating the mechanical model for a selected set of input parameters (the so-called experimental design) without any modification of the associated computer code. State-of-the art non-intrusive methods including spectral projection, stochastic collocation and regression will be reviewed. Emphasis will be put on sparse decompositions and validation techniques which allow the analyst to address large dimensional problems while mastering the computational cost and the accuracy.
Polynomial chaos expansions may also be used in the context of model calibration and identification under uncertainty. In the former case some prior knowledge on the basic parameters uncertainty is updated by observations of the model response. In the latter case (which is also referred to as stochastic inverse problems) the uncertainty on model input parameters is looked for from available (scattered) experimental data. A Bayesian framework will be presented that addresses both types of issues and illustrated on application examples.

Safety-based design generally implies the assessment of probabilities of rare and undesirable events, also called failure event. Solving reliability problems characterized by such low probabilities (10^-3 or less) is known to be rather challenging, due to the complex nature of the stochastic problem and the computational cost of the underlying model which is used to define the failure.
After a brief presentation of the framework of reliability problems, the talk will give an overview of off-the-shelf methods for reliability assessment, most of them available in freely or commercially distributed software. A special emphasis will be put on a number of points which make some of these methods either fail or lack of efficiency for certain types of problems: probability level of the failure event, role of most probable failure points, geometry of the limit-state, dimension of the random space (curse of dimensionality), nature of the stochastic model (basic random variable vs. random fields and processes). In an effort to reduce the computational burden, the recourse to surrogate models such as response surfaces, support vector machines or Gaussian processes represent an interesting alternative. The presentation will briefly recall the basics for building such surrogates and highlight their main advantages and limitations.
This course will provide some guidelines for selecting the most appropriate strategy for analysing highly reliable systems and useful hints for avoiding most common pitfalls in such approaches. It will be illustrated with examples in the field of time-invariant structural mechanics.

This course will deal with dynamical systems under random excitation. At the outset, model classes will be identified and their practical value will be discussed. Analytical solution techniques for linear systems will be briefly introduced and stability analysis methods will be presented.
The major focus of the course will be on the discretization of time variant problems and on reliability analysis of time variant systems with respect to first excursion failure. Reliability analysis of time variant systems is very challenging due to the large number of random variables that result from the discretization procedure. Several simulation methods that are able to provide estimates of the excursion probability will be presented and compared.
Practical examples from various fields of engineering will guide the participant through the course and help to get an understanding for the major achievements in time variant reliability analysis as well as for future challenges.
A second aspect of this course will be put on sensitivity analysis techniques that help to focus on the most important parameters of a model. Local and global sensitivity analysis methods will be distinguished and guidelines concerning the interpretation of a sensitivity study will be given.

The lecture addresses the reliability analysis of ceramic components under complex transient loading conditions. Ceramics are promising materials for high temperature and corrosive envronmental conditions but are difficult in design due to their inherent brittleness and ensuing scatter in mechanics strength [1]. Reliability assessment is performed using a weakest link analysis based on a Weibull distribution for the material strength. The failure probability is calculated on the basis of a Finite Element stress analysis with the help of suitable postprocessing routines. At IAM, the postprocessor STAU available which was continuously developed during the last years to allow increasing complexity in loading conditions as well as in material behaviour (crack propagation, thermal shock, fatigue, etc.) [2].
Uncertainties in the underlying database are reflected in uncertainties in the resulting failure probabilities. A bootstrap simulation approach is one way of obtaining confidence bounds thus dealing with uncertainties in the calculated failure probabilities and getting information on how to improve the data base [3].
Efficient use of the data base may also be possible by pooling strategies, if data obtained under different experimental conditions can be transformed to a joint reference condition [4].

[1] D. Munz, T. Fett, Ceramics: mechanical properties, failure behaviour, materials selection, Springer Verlag, Berlin 1999.
[2] H. Riesch-Oppermann, M. Härtelt, O. Kraft, STAU - a review of the Karlsruhe weakest link finite element postprocessor with extensive capabilities, International Journal of Materials Research 99 (2008), 1055-1065. doi: 10.3139/146.101735H.
[3] H. Riesch-Oppermann, S. Scherrer-Rudiy, T. Erbacher, O. Kraft, Uncertainty analysis of reliability predictions for brittle fracture, Engng. Fract. Mech. 74 (2007), 2933-2942. (Special Issue on "Reliability – Statistical Methods in Fracture and Fatigue", doi: 10.1016/j.engfracmech.2006.05.026)
[4] M. Härtelt, H. Riesch-Oppermann, T. Schwind, O. Kraft, Statistical evaluation of fatigue crack propagation from natural flaws in silicon nitride, J. Amer. Ceram. Soc. 94 (2011). doi: 10.1111/j.1551-2916.2011.04635.x.

In mass production, tolerance analysis aims at proving that mechanical behavior of product is robust with respect to manufacturing uncertainties. In general, tolerance analysis consists in evaluating the probability (often called defect probability, expressed in parts per million, ppm) that the product falls outside its functional bounds due to material deviations, parts geometry deviations, manufacturing assembly deviations… A general trend of the industry is to manage defect probability in the design stage for economic and environmental reasons, reducing warranty returns and wastage in production. The presentation aims at showing the wide applicability of probabilistic approaches to the tolerance analysis of products. The following main topics will be addressed:

• tolerance analysis objectives in mass production;
• definition of process capability;
• usual methods for evaluating the defect probability of mechanical products;
• advanced methods for predicting the defect probability in the design stage;
• applications: linear/non linear requirements, non explicit CAD requirements, hyperstatic problems;
• outlook on tolerance synthesis by cost optimization under defect probability constraints.

Ultimately, all engineering models are developed and applied to support decision making; examples include the identification of an optimal design, the selection of an efficient construction method, the development of a strategy for natural hazard protection or the choice of optimal inspection/maintenance schedules. When the relevant phenomena are subject to significant uncertainty and/or the potential consequences of adverse events are large, the decision processes should be supported by risk assessments. These include quantitative assessments of uncertainties (the main topic of this course) and consequences.

This talk will present the principles of risk-based decision making in engineering. It will outline the theory, including the Bayesian pre-posterior decision analysis, and will then present a number of applications of risk-based decision support. These are in the areas of infrastructure systems, offshore and ship structures, natural hazards and aircrafts. In the presentation of the examples, a particular focus will be set (a) on identifying the right level of detailing needed in the probabilistic analysis and (b) on the use of Bayesian approaches to uncertainty quantification with the goal of combining information from different sources for a more effective decision support.

Phimeca Engineering provides consulting services to industrial companies and research institutes in the domain of uncertainty analysis and structural reliability for ten years. Research projects and applied studies have been carried out in various domains including nuclear engineering, automotive & aerospace engineering and offshore structures. Various case studies will be presented that illustrate the theoretical courses given during the week by the various speakers. Emphasis will be put on the original industrial issue and the solutions that have been proposed to address it.

Methods of uncertainty treatment and uncertainty analysis become more and more important in virtual product development. In the last decade especially robustness, reliability and sensitivity analysis as well as robust design optimization have received an increasing interest and acceptance e.g. in automotive industry. Since ten years Dynardo develops software which helps to introduce uncertainty analysis in CAE-based product development. In our experience the key of a successful propagation of such methods is that the application of generally complex algorithms is simplified as much as possible. This is realized in our products with robust highly automatized algorithmic frameworks together with wizard based algorithmic settings which enable the beginner and even the expert to successfully apply the methods for a broad array of applications.

In the talk this concept is explained in detail by means of sensitivity and robustness analyzes. Several industrial applications from automotive industry are discussed, where uncertainty analysis was successfully applied and where it was introduced meanwhile as state of the art in virtual product development.