Latin hypercube sampling with inequality constraints

In some studies requiring predictive and CPU-time consuming numerical models, the sampling design of the model input variables has to be chosen with caution. For this purpose, Latin hypercube sampling has a long history and has shown its robustness capabilities. In this paper we propose and discuss a new algorithm to build a Latin hypercube sample (LHS) taking into account inequality constraints between the sampled variables. This technique, called constrained Latin hypercube sampling (cLHS), consists in doing permutations on an initial LHS to honor the desired monotonic constraints. The relevance of this approach is shown on a real example concerning the numerical welding simulation, where the inequality constraints are caused by the physical decreasing of some material properties in function of the temperature

Global sensitivity analysis in welding simulations -- what are the material data you really need ?

International audienceIn this paper, the Sensitivity Analysis methodology is applied to numerical welding simulation in order to rank the importance of input variables on the outputs of the code like distorsions or residual stresses. The numerical welding simulation uses the Finite Element Method, with a thermal computation followed by a mechanical one. Classically, a Local Sensitivity Analysis is performed, hence the validity of the results is limited to the neighborhood of a nominal point, and cross effects cannot be detected. This study implements a Global Sensitivity Analysis which allows to screen the whole material space of the steel family mechanical properties. A set of inputs of the mechanical model --material properties that are temperature-dependent-- is generated with the help of Latin Hypercube Sampling. The same welding simulation is performed with each sampling element as input data. Then, output statistical processing allows us to classify the relative input influences by means of different sensitivity indices estimates. Two different welding configurations are studied. Considering their major differences, they give a different ranking of inputs, but both of them show that only a few parameters are responsible of the variability of the outputs. To prove it a posteriori for the first configuration, two series of computations are performed for a complete sample and for its reduced copy --where all the secondary parameters are set to mean values. They match perfectly, showing a substantial economy can be done by giving to the rest of the inputs mean values. Sensitivity analysis has then provided answers to what we consider one of the probable frequently asked questions regarding welding simulation: for a given welding configuration, which properties must be measured with a good accuracy and which ones can be simply extrapolated or taken from a similar material? That leads us to propose a comprehensive methodology for welding simulations including four sequential steps: a problem characterization, a sensitivity analysis, an experimental campaign, simulations

Analyse de sensibilité globale de modèles thermomécaniques de simulation numérique du soudage

La démarche actuelle de simulation numérique d'une opération de soudage (SNS) concernant la définition du modèle mécanique consiste à utiliser au mieux des connaissances, les paramètres matériau et à utiliser le modèle qui semble le plus adapté à la situation. Cette démarche comporte un impondérable de taille : les données du matériau sont toujours incomplètes ! Ainsi, cette manière de procéder admet, omet ou néglige l'influence de la méconnaissance des paramètres du modèle sur le résultat de la simulation. Dans ce cas quelle pertinence, quelle fiabilité peut on accorder au modèle ? Cette thèse propose d'apporter les éléments permettant de lever ce verrou et de les appliquer à des cas concrets de simulation du soudage. La détermination des paramètres influents sera réalisée par une analyse de sensibilité globale. Pour la mettre en oeuvre, des expériences numériques de soudage sont planifiées, couvrant l'espace matériau des aciers où toutes les propriétés du matériau varient à la fois. La méthodologie de l'analyse de sensibilité globale permet alors d'identifier les propriétés du matériau les plus influentes sur la variabilité des sorties du code de calcul (contraintes et distorsions). La méthodologie d'analyse de sensibilité globale, totalement innovante en SNS, a démontré la possibilité de réduire la phase d'identification aux seuls paramètres importants et de quantifier les effets des variables d'entrée sur le résultat du calcul. Ainsi, l'analyse de sensibilité apporte des réponses à quelques questions fréquemment posées en SNS : Quelles propriétés du matériau doivent être mesurées avec précision et lesquelles peut-on se contenter de prendre dans la littérature ?Current approach of most welding modellers is to content themselves with available material data, and to chose a mechanical model that seems to be appropriate. Among inputs, those controlling the material properties are one of the key problems of welding simulation: material data are never characterized over a sufficiently wide temperature range! This way to proceed neglect the influence of the uncertainty of input data on the result given by the computer code. In this case, how to assess the credibility of prediction? This thesis represents a step in the direction of implementing an innovative approach in welding simulation in order to bring answers to this question, with an illustration on some concretes welding cases. The global sensitivity analysis is chosen to determine which material properties are the most sensitive in a numerical welding simulation and in which range of temperature. Using this methodology require some developments to sample and explore the input space covering welding of different steel materials. Finally, input data have been divided in two groups according to their influence on the output of the model (residual stress or distortion). In this work, complete methodology of the global sensitivity analysis has been successfully applied to welding simulation and lead to reduce the input space to the only important variables. Sensitivity analysis has provided answers to what can be considered as one of the probable frequently asked questions regarding welding simulation: for a given material which properties must be measured with a good accuracy and which ones can be simply extrapolated or taken from a similar material?DIJON-BU Sciences Economie (212312102) / SudocSudocFranceF

Latin hypercube sampling with inequality constraints

Computer experiment, Latin hypercube sampling, Design of experiments, Uncertainty analysis, Dependence,