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Problems with numerical integration of discontinuous functions

Hi everybody,

 I am a very beginnerin doing research :-) and my topic is about "micro indentation analysis using continuum dislocation theory". I am applying high-order finite element method for this nonlinear problem.

My plan is first writing a subroutine for the element. However, when I intend to compute the internal force by using Gauss integration, I see a problem with the integrand function of some index of the internal force vector. This integrand is discontinuous function. It is therefore, I cannot get a good approximation with the standard Gauss integration. 

Does anybody have idea on how to deal with this problem? I attached two images: one of the integrand function before and one of the integrand function after using the standard Gauss integration in order to clarify my question. Thank you very much.

 Best Regards,


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