Annual Examination Papers

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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.

x

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