Normalization approaches for the descent search direction in isogeometric shape optimization

Abstract

In isogeometric shape optimization, the use of the search direction directly predicted from the discrete shape gradient makes the optimization history strongly dependent on the discretization. This discretization-dependency can affect the convergence and may lead the optimization process into a sub-optimal solution. The source of this discretization-dependency is traced to the lack of consistency with the local steepest descent search direction in the continuous formulation. In the present contribution, this inconsistency is analyzed using the shape variation equations and subsequently illustrated with a volume minimization problem. It is found that the inconsistency originates from the NURBS discretization which induces a discrete quadratic norm to represent the continuous Euclidean norm. To fix this inconsistency, three normalization approaches are proposed to obtain a discretization-independent normalized descent search direction. The discretization-independence of the proposed approaches is verified with a benchmark problem. The superiority of the proposed search direction and its suitability for numerical implementation is illustrated with examples of shape optimization for mechanical and thermal problems. Although the present work focuses on a NURBS-based discretization usually used in conjunction with isogeometric analysis, the proposed methodology may also be applied to alleviate the “mesh-dependency” in (traditional) Finite Element-based shape optimization.