A variational geometric framework for multi-objective level set topology optimization

TL;DR

This paper introduces a novel variational geometric framework for multi-objective level set topology optimization, interpreting the level set as a generalized coordinate and employing a dynamic, Hamiltonian-based approach to effectively explore Pareto frontiers.

Contribution

It presents a new geometric and dynamical perspective on multi-objective topology optimization, enabling adaptive exploration of the Pareto frontier with a stable and scalable method.

Findings

Provides a stable approximation of the Pareto frontier

Enables adaptive exploration of objective space

Scales effectively to higher-dimensional objectives

Abstract

This paper proposes a variational framework for multi-objective level set topology optimization. The approach interprets the level set function as a generalized coordinate of a fictitious material and derives its equation of motion from Hamilton's principle, resulting in a damped wave equation governing the optimization process. The objective functionals are combined using a weighted sum formulation. An analysis of the underlying system structure reveals a geometric interpretation of the problem, shifting the perspective beyond conventional approaches based on purely discrete approximations of the Pareto frontier. Under suitable regularity assumptions, the set of stationary solutions forms a structured subset in objective space, in which the Pareto frontier is locally embedded and the weighting factors act as intrinsic coordinates. This perspective motivates the introduction of a dynamic evolution of the weights, leading to a coupled dynamical system for the level set function and the weighting parameters that enables adaptive exploration of the objective landscape. Numerical results demonstrate that the proposed framework provides a stable and uniform approximation of the Pareto frontier and scales to higher-dimensional objective spaces.