Jan Oellerich | Searcharxiv

arXiv2026

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

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.

math.OC↗

arXiv2025

A level set topology optimization theory based on Hamilton's principle

In this paper, we present a novel framework for deriving the evolution equation of the level set function in topology optimization, departing from conventional Hamilton-Jacobi based formulations. The key idea is the introduction of an auxiliary domain, geometrically identical to the physical design domain, occupied by fictitious matter which is dynamically excited by the conditions prevailing in the design domain. By assigning kinetic and potential energy to this matter and interpreting the level set function as the generalized coordinate to describe its deformation, the governing equation of motion is determined via Hamilton's principle, yielding a modified wave equation. Appropriate combinations of model parameters enable the recovery of classical physical behaviors, including the standard and biharmonic wave equations. The evolution problem is formulated in weak form using variational methods and implemented in the software environment FreeFEM++. The influence of the numerical parameters is analyzed on the example of minimum mean compliance. The results demonstrate that topological complexity and strut design can be effectively controlled by the respective parameters. In addition, the method allows for the nucleation of new holes and eliminates the need for re-initializing the level set function. The inclusion of a damping term further enhances numerical stability. To showcase the versatility and robustness of our method, we also apply it to compliant mechanism design and a bi-objective optimization problem involving self-weight and compliance minimization under local stress constraints.

math.OC↗