Neural-network-assisted Newton-Raphson Method for Elastoplastic Problems in Implicit MPM

  • Suzuki, Misato (Keio University)
  • Sugai, Riichi (Tohoku University)
  • Terada, Kenjiro (Tohoku University)
  • Muramatsu, Mayu (Keio University)

Please login to view abstract download link

In this study, we propose a method that reduces the number of Newton–Raphson (NR) iterations in the implicit Material Point Method (MPM) [1] by using neural networks. MPM combines particles carrying physical quantities with a background grid where governing equations are solved to simulate large deformations. While implicit MPM allows for larger time steps than explicit method, it requires solving nonlinear equilibrium equations at each time step, typically via the NR method. To find the solution, the linearized equation is solved repeatedly to obtain a corrector vector, and the solution is progressively updated by adding this corrector to the predictor until convergence. When solving finite strain elastoplastic problems with implicit MPM, the solution is updated by accumulating many small correctors, thereby substantially increasing the number of required NR iterations. To address this using machine learning techniques, some approaches predict the solution itself to accelerate the solver [2, 3], but we take an approach to enhancing the conventional NR corrector by introducing a neural corrector that guides the solution closer to the converged state. To predict this neural corrector, a U-Net is adopted because NR correctors are computed on the background grid and the neural corrector likewise needs to be represented as a grid-based field. The inputs to the U-Net are the conventional NR corrector computed in the first iteration, the accumulated plastic strain, the von Mises stress, and the material distribution, and the output is the neural corrector. As a numerical example, we demonstrate tensile analysis of plates with a hole. The trained network accurately predicts neural correctors even for geometries not included in the training data, with varying hole radius and hole position. When the neural corrector is used in the iterative solution process, an average reduction of approximately 40% in the number of iterations is observed. REFERENCES [1] Yamaguchi Y., Moriguchi S., Terada K., Extended B-spline-based implicit material point method, Int. J. Numer. Methods Eng., Vol.122, pp.1746-1769, 2021. [2] Odot A., Haferssas R., Cotin S., DeepPhysics: A physics aware deep learning framework for real-time simulation, Int. J. Numer. Methods Eng., Vol.123(10), pp.2381-2398, 2022. [3] Jin, T., Maierhofer, G., Schratz, K., Xiang, Y., A fast neural hybrid Newton solver adapted to implicit methods for nonlinear dynamic, J. Comput. Phys., Vol.529, 113869, 2025.