A Hybrid Lagrangian-Eulerian Method for Topology Optimization

We propose LETO, a new hybrid Lagrangian-Eulerian method for topology optimization. At the heart of LETO lies in a hybrid particle-grid Material Point Method (MPM) to solve for elastic force equilibrium. LETO transfers density information from freely movable Lagrangian carrier particles to a fixed set of Eulerian quadrature points. The quadrature points act as MPM particles embedded in a lower-resolution grid and enable sub-cell resolution of intricate structures with a reduced computational cost. By treating both densities and positions of the carrier particles as optimization variables, LETO reparameterizes the Eulerian solution space of topology optimization in a Lagrangian view. LETO also unifies the treatment for both linear and non-linear elastic materials. In the non-linear deformation regime, the resulting scheme naturally permits large deformation and buckling behaviors. Additionally, LETO explores contact-awareness during optimization by incorporating a fictitious domain-based contact model into the static equilibrium solver, resulting in the discovery of novel structures. We conduct an extensive set of experiments. By comparing against a representative Eulerian scheme, LETO's objective achieves an average quantitative improvement of 20% (up to 40%) in 3D and 2% in 2D (up to 12%). Qualitatively, LETO also discovers novel non-linear functional structures and conducts self-contact-aware structural explorations.