Topology optimization with material point method: investigation into the design sensitivity and the effect of shape functions

The Material Point Method (MPM) is considered promising for analyzing structures that experience large displacements and extreme events, areas where typical mesh-based analysis methods falter due to mesh distortion. By leveraging both Eulerian and Lagrangian descriptions, MPM facilitates the easy application of boundary conditions, overcoming the inherent limitations of mesh-based approaches. However, integrating MPM into topology optimization has been hindered by challenges in deriving analytical sensitivities and inherent numerical inaccuracies. This study introduces a novel topology optimization approach that employs MPM instead of the finite element method. The design variable is parameterized by material point volumes that utilize point-wise properties to represent design layouts. This approach addresses the calculation and validation of analytical design sensitivities and significantly enhances design flexibility, as the density of the design variable is not confined to an existing grid and can be user-defined. Furthermore, the research explores the effects of cell crossing errors on the stress field within topology optimization and proposes modifications to the shape function to mitigate these errors, thereby improving the applicability of MPM.