Adaptively Weighted Numerical Integration in the Finite Cell Method

Abstract

With Adaptively Weighted (AW) numerical integration, for a given set of quadrature nodes, order and domain of integration, the quadrature weights are obtained by solving a system of suitable moment fitting equations in least square sense. The moments in the moment equations are approximated over a simplified domain that is homeomorphic to the original domain, and then are corrected for the deviation from the original domain using shape sensitivity analysis.In this paper, we demonstrate the application of AW integration scheme in the context of the Finite Cell Method which must perform numerical integration over arbitrary domains without meshing. The standard integration technique employed in FCM is the characteristic function method that converts the continuous integrand over a complex domain into a discontinuous integrand over a simple (box) domain. Then, well known integrand adaptivity techniques are employed to integrate the resulting discontinuous integrand over the box domain. Although this method is simple to implement, it becomes computationally very expensive for realistic complex 3D domains such as sculptures, bones and engines.In contrast, in AW scheme the quadrature weights directly adapt to the complex geometric domain without the need to making the integrand discontinuous leading to superior computational properties. In this paper, we demonstrate the computational efficiency of AW over the characteristic function method as it requires fewer subdivisions and less time to achieve a given accuracy in both two and three dimensions. In addition, AW offers a number of advantages including flexibility in the choice of quadrature points and basis functions.