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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,

Minh

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