An Approximate Newton Smoothing Method for Shape Optimization

TL;DR

This paper introduces a new approximate Newton smoothing method for shape optimization that adaptively identifies regions needing smoothing and demonstrates effectiveness in fluid flow drag minimization tasks.

Contribution

It develops a semi-automated approach to approximate Hessians using local Fourier analysis, including adaptive smoothing and symbol identification for diverse problems.

Findings

Effective in drag minimization for Stokes and Navier-Stokes flows

Automatically detects regions where smoothing is physically meaningful

Extends to a wide range of shape optimization problems

Abstract

A novel methodology to efficiently approximate the Hessian for numerical shape optimization is considered. The method enhances operator symbol approximations by including body fitted coordinates and spatially changing symbols in a semi automated framework based on local Fourier analysis. Contrary to classical operator symbol methods, the proposed strategy will identify areas in which a non-smooth design is physically meaningful and will automatically turn off smoothing in these regions. A new strategy to also numerically identify the analytic symbol is derived, extending the procedure to a wide variety of problems. The effectiveness is demonstrated by using drag minimization in Stokes and Navier-Stokes flows.