Optimization of piecewise smooth shapes under uncertainty using the example of Navier-Stokes flow

TL;DR

This paper develops a novel framework combining differential geometry and multi-shape calculus to optimize multiple piecewise smooth shapes under uncertainty in Navier-Stokes flow, with applications demonstrated through numerical experiments.

Contribution

It introduces a new multi-shape calculus framework for optimizing piecewise smooth shapes with geometric constraints and uncertainty in fluid dynamics.

Findings

Successful application of the framework to Navier-Stokes obstacle problems.

Effective use of stochastic augmented Lagrangian method for shape optimization.

Insights into algorithmic parameter choices for uncertain fluid flow problems.

Abstract

We investigate a complex system involving multiple shapes to be optimized in a domain, taking into account geometric constraints on the shapes and uncertainty appearing in the physics. We connect the differential geometry of product shape manifolds with multi-shape calculus, which provides a novel framework for the handling of piecewise smooth shapes. This multi-shape calculus is applied to a shape optimization problem where shapes serve as obstacles in a system governed by steady state incompressible Navier-Stokes flow. Numerical experiments use our recently developed stochastic augmented Lagrangian method and we investigate the choice of algorithmic parameters using the example of this application.