## p.p1 span.s1 {font: 9.0px Helvetica} Polynomial ‘finite elements’ by

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Polynomial ‘finite elements’ by appropriate functions chosen to accurately represent the local
behaviour.
A discontinuity may be defined as a rapid change of a field quantity over a length
which is negligable compared to the dimensions of the observed domain. In the real
world, discontinuities are frequently found. In solids, stresses and strains are discontinuous
across material interfaces, and displacements are discontinuous at cracks. Tangential
displacements are discontinuous across shear bands. In fluids, velocity and
pressure fields may involve discontinuities at the interface of two fluids. Consider
W ? Rn to be the computational domain containing two different immiscible incompressible
phases. Let ¶W be the boundary of W. The sub-domains containing the two
phases are denoted by W1and W2 with W = W1 ?W2 W1 ?W2 = /0. We assume that W1
and W2 are connected. The separation of W into two fluid phases is defined by means
of the level-set method. The scalar level-set function f has a positive sign in one phase
and a negative sign in the other. By definition the interface between the two phases is
given by
G = {x ? W| f (x) = 0}. (1.1)
while f is usually defined as a signed-distance function:
f (x) = min
¯ x?G kx? ¯ xk, ?x ? W. (1.2)
and
?(f ) =
??
?
?1, f (x) 0.
(1.3)
is the corresponding dynamic viscosity.
The necessary steps involved in the implementation of the XFEM are:
1. Representation of the interface: The interface or discontinuity can be represented
explicitly by line segments or implicitly by using the level set method (LSM)
47, 58.
2. Selection of enriched nodes: In case of the local enrichment, only a subset of the
nodes closer to the region of interest is enriched. The nodes to be enriched can
be selected by using an area criterion or from the nodal values of the level set
function.
3. Choice of enrichment functions: Depending on the physics of the problem, different
enrichment functions can be used.

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