A consistent formulation for imposing packaging constraints in shape optimization using Vertex Morphing parametrization

Abstract

This paper aims at imposing no-penetration condition over arbitrary surfaces which act as bounding surfaces, also known as packaging constraints, on the design surface of shape optimization problem. We use Vertex Morphing technique for the shape parametrization. Vertex Morphing is a consistent surface control approach for node-based shape optimization. The suitability of this technique has been assessed and demonstrated for a wide range of engineering applications without geometric shape constraints. In this contribution, a consistent formulation is presented for the implementation of numerous point-wise geometric constraints in four main steps. First, a potential contact between optimization surface points and the bounding surface is identified via the so-called gap function. Second, the shape gradients of objective functions and active constraints are mapped onto the Vertex Morphing’s control space, where the optimization problem is formulated. Third, the linear least squares method is used to project the steepest-descent search direction onto the subspace tangent to the mapped active constraints. Finally, the feasible design update is mapped onto the geometry space. To verify the perfect consistency between the geometry space (where the constraints are formulated) and the control space (where the optimization problem is solved) two applications of CFD shape optimization in the automotive industry are presented.