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Concept# Numerical integration

Summary

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for numerical integration, especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take quadrature to include higher-dimensional integration.
The basic problem in numerical integration is to compute an approximate solution to a definite integral
:\int_a^b f(x) , dx
to a given degree of accuracy. If f(x) is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision.

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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.

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