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Equation (1.4) generally defines the XFEM. For a particular realization of the

XFEM, the choice of the nodal subset I?, global enrichment function r(x), and the

partition of unity functions N?i (x) has to be defined.

The displacement approximation given by Equation (1.4) is called ‘extrinsic’ global enrichment

(i.e., FE approximation basis is augmented with additional functions and all

the nodes in the FE mesh are enriched with r(x)). This does not satisfy the Kronecker-

d property (i.e., Ni(xj) =di j) which renders the imposition of essential boundary conditions

and the interpretation of results difficult, expect for the phantom node method

60, 54. In most cases, the region of interest is localized, for example, cracks ormaterial

interfaces and hence the enrichment could be restricted closer to the region

of interest. This type of enrichment is called ‘local enrichment’. Moreover, a global

enrichment is computationally demanding because the number of degrees of freedom

is proportional to the number of nodes and the number of enrichment functions and

the resulting system matrix is not banded. A ‘shifted enrichment’ is used to retain the

Kronecker-d property, given by:

uh(x) = uh

f em(x)+uh enr(x)

=å

i ?I

Ni(x)ui+ å

j?I?

Nj(x)r(x)?r(xj)aj. (1.5)

where r(xj) is the value of the enrichment function evaluated at node j. An example

will be considered of this shifting for 1D when r(x)= |f (x)|=|x?xb|, where xb is the

location of the interface from the left end. Different enrichment functions are proposed

in the literature to capture strong and weak discontinuities arising in different problems.

For weak and strong discontinuities, the nodal subset I? is built from all nodes of

elements that are cut by the discontinuity. Whether or not an element is cut by the

discontinuity can conveniently be determined on element-level by help of the level-set

function f (x)

??

?

cut elements min

i?Iel

fi.max

i?Iel

fi < 0
uncut elements min
i?Iel
fi.max
i?Iel
fi < 0.
For weak discontinuities, where a solution shows a kink, or in other words, a jump
in the gradient, the global enrichment function is typically chosen as the abs-function
of the level-set function,
r(x) = |f (x)|
Along strong discontinuities, a jump is present in the solution. A typical choice for the
global enrichment function is the sign-function (or Heaviside-function) of the level-set
function,
r(x) = signf (x) =
??
?
?1 f (x) < 0,
0 f (x) = 0,
1 f (x) > 0