En analyse numérique, il existe une vaste famille d’algorithmes dont le but principal est d’estimer la valeur numérique de l’intégrale définie sur un domaine particulier pour une fonction donnée (par exemple l’intégrale d’une fonction d’une variable sur un intervalle). Ces techniques procèdent en trois phases distinctes :

Décomposition du domaine en morceaux (un intervalle en sous-intervalles contigus) ;

Intégration approchée de la fonction sur chaque morceau ;

Sommation des résultats numériques ainsi obtenus.

Définitions Formule de quadrature On appelle formule de quadrature une expression linéaire dont l’évaluation fournit une valeur approchée de l’intégrale sur un morceau typique (l’intervalle [0 ; 1] par exemple). Une transformation affine permet de transposer la formule sur un morceau particulier. La formule de quadrature fait intervenir des valeurs pondérées de la fonction (et parfois également celles de sa dérivée) en certains nœuds : les coefficien

Evaluation of Weld-induced Residual Stresses in S690 Steel

Katharina Rohr

The objective of this project is to analyse welding processes in high strength low-carbon steel grade S690 in numerical simulation with MORFEO codes and in experiments with a Gleeble® thermo-mechanical simulator instrument. The influence of phase transformation on residual stresses is investigated with simulations on plate to plate and on tube to plate connections. Furthermore, this study investigates the effects of element activation on numerical results and the possibilities to simplify multi-pass welding by lumped models. In addition to literature based descriptions of temperature depending steel properties, experimental tests are conducted to survey the behavior of steel grad S690 at elevated temperatures.

2013

The Scrambled Net Filter

Thomas Gsponer

The standard Kalman filter is a powerful and widely used tool to perform prediction, filtering and smoothing in the fields of linear Gaussian state-space models. In its standard setting it has a simple recursive form which implies high computational efficiency. As the latter is essentially a least squares procedure optimality properties can be derived easily. These characteristics of the standard Kalman filter depend strongly on distributional and linearity assumptions of the model. If we consider nonlinear non-Gaussian state-space models all these properties and characteristics are no longer valid. Consequently there are different approaches on the robustification of the Kalman filter. One is based on the the ideas of minimax problems and influence curves. Others use numerical integration and Monte Carlo methods. Herein we propose a new filter by implementing a method of numerical integration, called scrambled net quadrature, which consists of a mixture of Monte Carlo methods and quasi-Monte Carlo methods, providing an integration error of order of magnitude

N^{-3/2}\log(N)^{(r-1)/2}

in probability, where

r

denotes dimension. We show that the point-wise bias of the posterior density estimate is of order of magnitude

N^{-3}\log(N)^{r-1}

but grows linearly with time.

2002

The Scrambled Net Filter

Thomas Gsponer

N^{-3/2}\log(N)^{(r-1)/2}

in probability, where

r

denotes dimension. We show that the point-wise bias of the posterior density estimate is of order of magnitude

N^{-3}\log(N)^{r-1}

but grows linearly with time.

2002

The Scrambled Net Filter

Thomas Gsponer

N^{-3/2}\log(N)^{(r-1)/2}

in probability, where

r

denotes dimension. We show that the point-wise bias of the posterior density estimate is of order of magnitude

N^{-3}\log(N)^{r-1}

but grows linearly with time.

2002

The Scrambled Net Filter

Thomas Gsponer

N^{-3/2}\log(N)^{(r-1)/2}

in probability, where

r

denotes dimension. We show that the point-wise bias of the posterior density estimate is of order of magnitude

N^{-3}\log(N)^{r-1}

but grows linearly with time.

2002

Calcul numérique d'une intégrale

Décomposition du domaine en morceaux (un intervalle en sous-intervalles contigus) ;

Intégration approchée de la fonction sur chaque morceau ;

Sommation des résultats numériques ainsi obtenus.

Décomposition du domaine en morceaux (un intervalle en sous-intervalles contigus) ;

Intégration approchée de la fonction sur chaque morceau ;

Sommation des résultats numériques ainsi obtenus.

Evaluation of Weld-induced Residual Stresses in S690 Steel

The Scrambled Net Filter

The Scrambled Net Filter

The Scrambled Net Filter

Evaluation of Weld-induced Residual Stresses in S690 Steel

The Scrambled Net Filter