We introduce optimal quadrature rules for spline spaces that are frequently used in Galerkin discretizations to build mass and stiffness matrices. Using the homotopy continuation concept (Bartoň and Calo, 2016) that transforms optimal quadrature rules from source spaces to target spaces, we derive optimal rules for splines defined on finite domains. Starting with the classical Gaussian quadrature for polynomials, which is an optimal rule for a discontinuous odd-degree space, we derive rules for target spaces of higher continuity. We further show how the homotopy methodology handles cases where the source and target rules require different numbers of optimal quadrature points. We demonstrate it by deriving optimal rules for various odd-degree spline spaces, particularly with non-uniform knot sequences and non-uniform multiplicities. We also discuss convergence of our rules to their asymptotic counterparts, that is, the analogues of the midpoint rule of Hughes et al. (2010), that are exact and optimal for infinite domains. For spaces of low continuities, we numerically show that the derived rules quickly converge to their asymptotic counterparts as the weights and nodes of a few boundary elements differ from the asymptotic values

Manufacturing techniques that are applied to turbomachinery components represent a challenge in the aeronautic sector. These components require high resistant super-alloys in order to satisfy the extreme working conditions they have to support during their useful life. Besides, in the particular case of Integrally Bladed Rotors (IBR), usually present complex geometries that need to be roughed and finished by milling and grinding processes, respectively. In order to improve their manufacturing processes, Super Abrasive Machining (SAM) is presented as a solution because it combines the advantages of the use of grinding tools with milling feed rates. However, this innovative technique usually needed high tool rotary speed and pure cutting oils cooling. These issues implied that SAM technique was not feasible in conventional machining centers. In this work, these matters were tackled and the possibility of using SAM in these five-axis centers with emulsion coolants was achieved. To verify this approach, Inconel 718 single blades with non-ruled surfaces were manufactured with Flank-SAM technique and conventional milling process, analyzing cutting forces, surface roughness, and dimension accuracy in both cases. The results show that SAM implies a suitable, controllable, and predictable process to improve the manufacture of aeronautical critical components, such as IBR.

We introduce Gaussian quadrature rules for spline spaces that are frequently used in Galerkin discretizations to build mass and stiffness matrices. By definition, these spaces are of even degrees. The optimal quadrature rules we recently derived (Bartoň and Calo, 2016) act on spaces of the smallest odd degrees and, therefore, are still slightly sub-optimal. In this work, we derive optimal rules directly for even-degree spaces and therefore further improve our recent result. We use optimal quadrature rules for spaces over two elements as elementary building blocks and use recursively the homotopy continuation concept described in Bartoň and Calo (2016) to derive optimal rules for arbitrary admissible numbers of elements.We demonstrate the proposed methodology on relevant examples, where we derive optimal rules for various even-degree spline spaces. We also discuss convergence of our rules to their asymptotic counterparts, these are the analogues of the midpoint rule of Hughes et al. (2010), that are exact and optimal for infinite domains