Is there classical XFEM examples available

The current implementation of the PUM library only supports strong discontinuities on which solutions shows jumps. Inside PUM class, we have used the Heaviside function as enrichment functions and we have assumed that discontinuities end up in the cell facets (no tip enrichment).

To support weak discontinuities, you need to derive a class from GenericPUM to support other enrichment functions (like level set). You may also need to support discontinuous Quadrature elements to consider different material properties in each side of discontinuities (As a temporary solution, you can manually modify the generated code to support different material properties on quadrature points on the elements crossed by discontinuities).

If I derive another class, like level set, to support weak discontinuity and different material properties, It seems to me that I have to modify ffcpum scripts to enable the numerical quadratures in the enriched element. am I right? I think the current pum library also doesn't implement numerical integration, instead it is done in the ffc-pum scripts.

In order to define discontinuous QuadratureElement, ffcpum needs to be modified. I have created a blueprint for it in the PUM form compiler page.

https://blueprints.launchpad.net/ffc-pum/+spec/discontinuous-quadrature-element

However, i can't promise that this will be implemented in near future. You are welcome to try to implement it.

As a temporary solution (hack), you can define a constant function for the material property and compile your form. Then you can manually modify the generated code to have different values based on the position of the quadrature point w.r.t. discontinuity surface.