Level-set shape optimization via polytopic discontinuous Galerkin methods

TL;DR

This paper presents a novel level-set shape optimization method utilizing polytopic discontinuous Galerkin techniques, enabling accurate and efficient geometric resolution and shape updates in complex domains.

Contribution

It introduces a new shape optimization approach combining polytopic discontinuous Galerkin methods with Runge-Kutta schemes for improved geometric accuracy and computational efficiency.

Findings

Accurate zero-level set resolution demonstrated

Efficient shape updates via Runge-Kutta DG methods

Numerical experiments confirm method's robustness

Abstract

We introduce a new level-set shape optimization approach based on polytopic (i.e., polygonal in two and polyhedral in three spatial dimensions) discontinuous Galerkin methods. The approach benefits from the geometric mesh flexibility of polytopic discontinuous Galerkin methods to resolve the zero-level set accurately and efficiently. Additionally, we employ suitable Runge-Kutta discontinuous Galerkin methods to update the level-set function on a fine underlying simplicial mesh. We discuss the construction and implementation of the approach, explaining how to modify shape derivate formulas to compute consistent shape gradient approximations using discontinuous Galerkin methods, and how to recover dG functions into smoother ones. Numerical experiments on unconstrained and PDE-constrained test cases evidence the good properties of the proposed methodology.