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

Abstract

CAD geometries are most often exchanged between analysis tools using NURBS patches to represent the boundary (BRep). We present a method where the control points of the BRep are used to automatically derive parametrisations suitable for shape optimisation with gradient-based methods. Particular focus is on ensuring geometric continuity between the NURBS patches, which is achieved through formulation of discrete constraints. Design variables then arise from formulating an orthogonal basis to the remaining design space using a singular value decomposition (SVD). The manuscript presents the extension of earlier work on B-spline surfaces to full NURBS surfaces and investigates the effect of the cutoff threshold of the SVD on the optimisation results. To enable routine automatic use, an estimation of the effective rank is proposed which allows to automatically determine the suitable cutoff threshold. The effectiveness of the algorithm is demonstrated for the minimisation of total pressure loss over a section of an automotive climate duct, and a U-bend cooling channel.

Cad-Based Adaptive Shape Parametrisation for Aerodynamic Shape Optimisation

“…where X p,L and X p,R are the positional coordinates, n L and n R are the unit normal tangent plane, suffix L and R corresponds to Patch-L and Patch-R respectively. Then we need to assemble Jacobian for each of the constraint equation and after linearization we obtain the following linear system of equations [21],…”

“…An additional constraint recovery step is also required for non-linear constraints. Details can be found in [21]. This module makes the required modification to the initial NURBS patches and compute displacements to surface grid points δX s using the parametric coordinates computed from step 2.…”

“…NURBS have become the de-facto industry standard for data exchange between CAD systems and offers local shape modification property. The NSPCC method derives parametrisation directly from the boundary representation (BRep) of a CAD-model [21]. The BRep, in the typical standardised STEP format, represents the shape in a number of NURBS patches.…”

“…The important contribution of NSPCC to CAD-based parametrisation based on BRep is the formulation of geometric constraints, e.g. G 0 − G 2 continuity at NURBS patch interfaces [21] or box, radius and thickness constraints [22]. In the NSPCC approach, surface node sensitivities are projected onto the control points of the design surface.…”