9 Combined Extended Finite Element and Level Set Method (XFE-LSM) for Free Boundary Problems
Ravindra Duddu
Department of Civil and Environmental Engineering, Vanderbilt University, USA
As described in Chapter 3, it is convenient to represent embedded interfaces using a scalar level set function 
; however, if the interface is moving with some velocity 
one needs to solve a first-order nonlinear partial differential equation (PDE) given by the level set evolution to determine its new location. The interface velocity 
can be obtained by numerically solving the governing elliptic or parabolic PDEs associated with the physics of the problem; by using the extended finite element method (XFEM) to solve these PDEs, one can eliminate the need for the finite element mesh to coincide with the embedded interface. The level set method proposed in Osher and Sethian (1988) and Sethian (1999b) enables us to capture the motion of interface in a consistent and efficient way through fast marching techniques. An important point is that the interface velocity 
is only defined for points that lie on the ...
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