An improved MPM formulation for free surface flow problems based on finite volume method - He, Kai-Yuan; Jin, Yin-Fu; Zhou, Xi-Wen; Yin, Zhen-Yu; Chen, Xiangsheng (Computer methods in applied mechanics and engineering, 2025)
•Novel FVMPM framework: Introduces a semi-implicit finite-volume MPM formulation that accurately imposes Neumann conditions on irregular geometries within a Cartesian grid.•Adaptive cut-cell discretization: Combines marching squares with second-order geometric integration for precise governing equations discretization.•Enhanced volume conservation: Uses a density projection strategy to mitigate volume loss, with a smoothed Heaviside function employed to improve the accuracy of integration weight calculations.•Particle distribution: Utilizes an artificial weak-spring model to alleviate particle clustering, combined with a prefix-sum algorithm that reduces computational complexity. The Material Point Method (MPM) has made significant progress in modeling incompressible free-surface flows; however, handling arbitrary irregular solid boundaries within structured Cartesian background grids remains a thorny challenge. This study proposes a novel semi-implicit Finite Volume MPM (FVMPM) framework that effectively accommodates both fixed and moving solid boundaries on Cartesian background grids. In this framework, the Poisson Pressure Equation (PPE) is reformulated into a finite-volume integral form based on control volume discretization, with solid boundaries implicitly represented by a signed distance function to accurately capture complex geometric features. An adaptive cut-cell algorithm is developed by synergistically integrating the marching squares technique with a second-order geometric integration scheme for efficient discretization of the governing equations. To dynamically compensate for the inherent volume loss/expansion of conventional operator splitting methods, a density projection strategy is innovatively incorporated into the FVMPM. This strategy introduces an absolute density error term into the integral-form PPE, effectively suppressing density oscillations and ensuring rigorous volume conservation throughout simulations. Furthermore, particle shifting technology based on an artificial weak-spring force model is implemented to effectively mitigate the issue particle clustering. The framework also employs an Affine Particle-In-Cell (APIC) mapping scheme that enhances momentum conservation and numerical stability. Additionally, GPU-based parallel computing programs are developed using the Taichi language to ensure high computational efficiency. Numerical experiments demonstrate that the proposed FVMPM framework accurately enforces solid boundary conditions on complex irregular geometries, such as inclined walls and curved obstacles, and exhibits superior performance in free-surface flow simulations involving moving solid boundaries.