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Metal Forming and the Finite-Element Method$
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Shiro Kobayashi, Soo-Ik Oh, and Taylan Altan

Print publication date: 1989

Print ISBN-13: 9780195044027

Published to Oxford Scholarship Online: November 2020

DOI: 10.1093/oso/9780195044027.001.0001

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The Finite-Element Method—Part II

The Finite-Element Method—Part II

7 (p.111) The Finite-Element Method—Part II
Metal Forming and the Finite-Element Method

Shiro Kobayashi

Soo-Ik Oh

Taylan Altan

Oxford University Press

Numerical integration is an important part of the finite-element technique. As seen in Section 6.5 of Chap. 6, volume integrations as well as surface integrations should be carried out in order to represent the elemental stiffness equations in a simple matrix form. In deriving the variational principle, it is implicitly assumed that these integrations are exact. However, exact integrations of the terms included in the element matrices are not always possible. In the finite-element method, further approximations are made in the procedure for integration, which is summarized in this section. Numerical integration requires, in general, that the integrand be evaluated at a finite number of points, called Integration points, within the integration limits. The number of integration points can be reduced, while achieving the same degree of accuracy, by determining the locations of integration points selectively. In evaluating integration in the stiffness matrices, it is necessary to use an integration formula that requires the least number of integrand evaluations. Since the Gaussian quadrature is known to require the minimum number of integration points, we use the Gaussian quadrature formula almost exclusively to carry out the numerical integrations.

Keywords:   Area-weighted averaging, Direct iteration method, Element assemblage, Frictional stress, Gaussian elimination, Newton-Raphson method, Rezoning, Simpson's formula, Time increment

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