This paper deals with the optimisation of the geometry of screw compressors. The goal of the European Union funded project 'MOTOR' (Multi-ObjecTive design Optimization of fluid eneRgy machines) is the optimisation of fluid energy machines by using isogeometric analysis. Unfortunately, all of the practically viable PDE-based approaches impose certain requirements on the employed spline-basis, in particular global C≥1-continuity.

PDE-based EGG is well-established in classical meshing and first generalization attempts to spline-based descriptions (as is mandatory in IgA) have been made. The main challenge lies in finding a formulation of the problem that is suitable for a computational approach and a common strategy is to approximate the mapping operator by means of solving a PDE-problem. A subclass of the multitude of the available parameterization algorithms are those based on the principles of Elliptic Grid Generation (EGG) which, in their most basic form, attempt to approximate a mapping operator whose inverse is composed of harmonic functions. The first step towards applying isogeometric analysis techniques to solve PDE problems on a given domain consists in generating an analysis-suitable mapping operator between parametric and physical domains with one or several patches from no more than a description of the boundary contours of the physical domain. We provide an open-source implementation of the algorithm in MATLAB. We compare the quadrature schemes produced using our method to other methods in the literature and show that they are much more efficient both in terms of number of quadrature points and computational time. The quadrature schemes have applications to computation of geometric moments, immersogeometric analysis, conservative field transfer between high-order meshes, and initialization of multi-material simulations with rational geometry.

The second guarantees exactness for polynomial integrands up to a pre-specified degree k with an a priori-known number of quadrature points and retains the convergence properties of the first, but is slightly more complicated. The first is spectrally accurate (it converges faster than any algebraic order with respect to number of quadrature points) and is relatively easy to implement, but has no guarantee of polynomial exactness. We present two methods to compute the intermediate quadrature rule.

(2) We then use a high-order quadrature rule to compute the numerical antiderivative of the integrand along a coordinate axis, which is used to evaluate the Green’s theorem line integral.

Our algorithm proceeds in two steps: (1) We first find intermediate quadrature rules for line integrals along the region’s boundary curves corresponding to Green’s theorem. This work presents spectral, mesh-free, Green’s theorem-based numerical quadrature schemes for integrating functions over planar regions bounded by rational parametric curves.

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