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

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.

Figures (25)

• Figure 1: Different topologies $\Omega(\phi_i(\mathbf{c}))$ obtained from $\Phi(\mathbf{c})$.
• Figure 2: Evolution of the topology $\Omega(\phi(\mathbf{c}))$ in the interval $[\mathbf{c}_1,\mathbf{c}_2]$.
• Figure 3: Construction of the auxiliary domain $\mathscr{D}$ containing fictitious material.
• Figure 4: Transition of the level set function from state $\mathbf{c}(s_1)$ to $\mathbf{c}(s_2)$. ➀ shows the acceleration of the surface element $\mathrm{d}\mathscr{D}$, ➁ shows the compression and tension of a surface element.
• Figure 5: (a) Different solutions obtained for varying weights $\mathbf{w}$, (b) qualitative illustration of the mapping $\varphi$.