Gaussian quadrature rules for arbitrary cut-volumes in embedded interface methods

Quadrature schemes, for arbitrary volumes, are constructed by solving the moment fitting equations. The integration of base functions over the volumes is carried out by using the divergence theorem. The integration process involves three key steps: convertion of volume to surface integral using the divergence theorem, projection of the integral from arbitrary plane to the coordinate plane, and reduction of surface integral to contour integrals using the divergence theorem. The present scheme is capable of constructing quadrature rules for arbitrary convex and concave volumes, without decomposing the volume into a number of tetrahedra. The position of the quadrature points are predefined, and moment fitting equations are solved for the quadrature weights. The present procedure is applied to generate accurate quadrature rules over many complex volumes. The accuracy of the method is demonstrated by integrating polynomials over complex volumes, and comparing the results with the exact value. The method is used to integrate the weak forms in embedded interface simulations, and it is shown that the present method is much more efficient than the volume decomposition approach.