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4. Integration: A consequence of adding custom tailored enrichment functions to

the FE approximation basis, which are not necessarily smooth polynomial functions

(for example, ?r in case of the LEFM) is that special care has to be taken

in numerically integrating over the elements that are intersected by the discontinuity

surface. The standard Gauss quadrature cannot be applied in elements

enriched by discontinuous terms, because Gauss quadrature implicitly assumes

a polynomial approximation.

Suppose that the domain W is discretized by nel elements, numbered from 1 to nel.

I is the set of all nodes in the domain, and I? is the nodal subset of the enrichment

(I? ? I). A standard extended finite element approximation of a function u(x) is of the

form

uh(x) = uh

f em(x)+uh enr(x)

=å

i ?I

Ni(x)ui+ å

j?I?

N?j (x)r(x)aj. (1.4)

For simplicity only one enrichment term is considered. The approximation consists

of a standard finite element (FE) part and the enrichment. The individual variables

stand for

• uh(x): approximated function,

• Ni(x): Standard FE function of node i,

• ui: unknown of the Standard FE part at node i,

• N?j (x): standard FE shape functions which are not necessarily the same than

those of the standard part of the approximation (Ni(x)), These functions build a

partition of unity, å

j?I?

N?j (x) = 1. in elements whose nodes are all in the nodal

subset I?. In these elements, the global enrichment function can be reproduced

exactly; we call these elements reproducing elements. In elements with only

some of their nodes in I?, does not build a partition of unity, å

j?I?

N?j (x) 6= 1.

As a consequence, the global enrichment function r(x) cannot be represented

exactly in these elements. Elements with only some of their nodes in I? are

called blending elements. Several publications discuss problems arising from

blending elements.

• r(x): global enrichment function. The enrichment function r(x) carries with it

the nature of the solution or the information about the underlying physics of the

problem, for example, r(x)= H, is used to capture strong discontinuities, where

H is the Heaviside function.

• aj: unknown of the enrichment at node j