Adjoint-Based Aerodynamic Optimisation of Wing Shape Using Non-uniform Rational B-Splines

Abstract

Numerical shape optimisation with adjoint CFD is applied using the NURBS based Parametrisation method with Continuity Constraints (NSPCC) for aerodynamically optimising three dimensional surfaces. The ONERA M6 wing is re-parametrised with NURBS surfaces including weight adjustments to represent the three dimensional wing accurately, resulting in fewer control points and smoother variation of curvature. The NSPCC CAD kernel is coupled with the inhouse flow and adjoint solver STAMPS and a gradient-based optimiser to minimise the drag of the ONERA M6 wing in transonic Euler flow conditions. Optimisation results are presented for the B-Spline and NURBS parametrisations.

Imposing C0 and C1 continuity constraints during CAD‐based adjoint optimization

“…Using this information, the process shown in Sections 2.1-2.3 is followed which leads to the computation of matrices B and C. Both matrices are sparse due to the locality property of NURBS. Because of this sparsity, rank revealing QR decomposition 38 is preferred to compute the singular values and vectors of matrix C. Kernel(C) is then computed through Equation (15). The nodes of the boundary mesh are projected onto the BRep by using point inversion.…”

“…The differentiation of the NURBS was done through algorithmic differentiation techniques. [12][13][14] Zhang et al 15,16 performed NURBS-based shape optimization with continuity constraints imposed between adjacent patches. Both control points and weights were used for parameterizing the shape and the AD tool TAPENADE 17 was used for computing the derivatives.…”

Geometric continuity constraints of automatically derived parametrisations in CAD-based shape optimisation

“…The interfaces between surfaces require at least G 0 continuity (watertightness), but for immersed surfaces typically G 1 (tangency) or in case of wing and blade profiles G 2 (curvature) to avoid pressure spikes. NSPCC is a novel approach to discretely impose geometric constraints on Boundary representations (BRep) which naturally provides a parametrisation with orthogonal modes (Xu, Jahn, and Müller 2013;Xu et al 2015;Zhang, Jesudasan, and Müller 2019). NSPCC evaluates discrete constraint equations in sets of test points that are distributed along patch interfaces as illustrated in Fig.…”

Deep learning frameworks for point cloud reconstruction

“…We model our module on prior work that incorporates structured priors as modules in the deep learning framework, similar to Sheriffdeen et al [72], Joshi et al [36], and Djolonga and Krause [22]. Beyond deep learning-based approaches, automatic differentiation for NURBS parametric coordinates for obtaining the surface derivatives for adjoint-based sensitivity analysis has been performed by Zhang [87]. Ugolotti et al…”