Abstract This work introduces a novel nonlinear Topology Optimization (NLTO) approach that effectively addresses both geometric nonlinearity and internal contact within the design domain. We propose an NLTO method-based on the implicit Material Point Method (MPM) to overcome two critical limitations of traditional approaches: element distortion in void regions and the inability to handle internal contact between surfaces formed during the design process. MPM is a hybrid Lagrangian–Eulerian method that inherently manages contact without requiring boundary identification or collision detection, while maintaining numerical stability under large deformations. This hybrid discretization is leveraged for the design variable discretization, sensitivity computation, filter application, and linear algebra solution. This results in ease of implementation and enhanced computational efficiency. Our approach discretizes design variables on Lagrangian particles rather than cells, enabling sub-resolution designs without increasing the degrees of freedom in the nonlinear structural solver. Unlike previous works, our approach comprehensively explores MPM’s inherent ability to address internal contact in nonlinear topology optimization, providing a solid foundation for future developments. Small cut stability, inversion-free line search, and re-evaluation with convection are some of the key improvements presented in this work that are specific to nonlinear MPM in the NLTO context. We investigate the behavior of void regions during internal contact and under severe distortion; and the effect of Lagrangian particle convection and Eulerian grid resetting is analyzed to maintain cell distortion within manageable limits. Alongside these methodological contributions, we present the computational effort required to solve challenging nonlinear 3D structural problems in topology optimization with internal contact, with up to 2.4 million design variables, and highlight implementation strategies that enable this efficient and robust performance.
Hassan et al. (Sat,) studied this question.