A Level Set Topology Optimization Theory Based on Hamilton's Principle

In this article, we propose a unified variational 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. Notably, the proposed formulation inherently supports the nucleation of new holes and maintains a well-defined level set function without requiring explicit re-initialization procedures, both of which emerge naturally from the physically motivated variational framework. 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.