Discretization methods based on meshes that do not conform to the geometry of the problem under consideration require special treatment when it comes to the integration of finite elements that are broken by the boundary or internal interfaces. To this end, we propose a numerical approach suitable for integrating broken elements with a low number of integration points. In this method, which is based on the moment fitting approach, an individual quadrature rule is set up for each cut element. The approach requires a B-rep representation of the broken element, which can be either achieved by processing a triangulated surface obtained from a CAD software or by taking advantage of a voxel model resulting from computed tomography. The numerical examples presented in this paper reveal that the proposed method delivers for a wide variety of geometrical situations very accurate results and requires a rather low number of integration points.
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