Abstract While semi‐analytical boundary handling techniques have proven effective for modeling particle‐based fluid‐solid interactions, they can become unstable when applied to mesh boundaries undergoing dynamic motion or featuring complex, sharp geometries. We propose a novel semi‐analytical energy model for boundary handling that unifies fluid simulation and boundary interactions within a variational framework. The model comprises two key components: a semi‐analytical bulk energy formulation that mitigates particle deficiency issues in the evaluation of bulk energy, and a nonlocal contact potential that effectively prevents particle penetration into boundaries. Both energy terms are naturally compatible with the Semi‐Implicit SPH (SISPH), and a unified Hessian‐free solver combined with reduced‐order collision detection enables an efficient and stable GPU‐based implementation for both fluid dynamics and nonlinear fluid‐solid interactions. Furthermore, the unified treatment of fluid bulk energy and boundary energy via the semi‐analytical formulation robustly corrects penetrations in practice, even under severe compression scenarios involving complex moving boundaries. Compared with existing semi‐analytical boundary treatments, our method is more robust under fast boundary motion and strong compression. Across challenging benchmarks with sharp features, narrow gaps, and moving meshes, it remains stable and penetration‐free where prior methods often fail.

Abstract In simulating fluids using position‐based dynamics, the accuracy and robustness depend on numerous numerical parameters, including the time step size, iteration count, and particle size, among others. This complexity can lead to unpredictable control of simulation behaviors. In this paper, we first reformulate the problem of enforcing fluid compressibility/incompressibility into an nonlinear optimization problem, and then introduce a semi‐implicit successive substitution method (SISSM) to solve the nonlinear optimization problem by adjusting particle positions in parallel. In contrast to calculating an intermediate variable, such as pressure, to enforce fluid incompressibility within the position‐based dynamics (PBD) framework, the proposed semi‐implicit approach eliminates the necessity of such calculations. Instead, it directly employs successive substitution of particle positions to correct density errors. This method exhibits reduced dependency to numerical parameters, such as particle size and time step variations, and improves consistency and stability in simulating fluids that range from highly compressible to nearly incompressible. We validates the effectiveness of applying a variety of different techniques in accelerating the convergence rate.

Prior research has demonstrated the efficacy of balanced trees as spatially adaptive grids for large-scale simulations. However, state-of-the-art methods for balanced tree construction are restricted by the iterative nature of the ripple effect, thus failing to fully leverage the massive parallelism offered by modern GPU architectures. We propose to reframe the construction of balanced trees as a process to merge N -balanced Minimum Spanning Trees ( N -balanced MSTs) generated from a collection of seed points. To ensure optimal performance, we propose a stack-free parallel strategy for constructing all internal nodes of a specified N -balanced MST. This approach leverages two 32-bit integer registers as buffers rather than relying on an integer array as a stack during construction, which helps maintain balanced workloads across different GPU threads. We then propose a dynamic update algorithm utilizing refinement counters for all internal nodes to enable parallel insertion and deletion operations of N -balanced MSTs. This design achieves significant efficiency improvements compared to full reconstruction from scratch, thereby facilitating fluid simulations in handling dynamic moving boundaries. Our approach is fully compatible with GPU implementation and demonstrates up to an order-of-magnitude speedup compared to the state-of-the-art method [Wang et al. 2024]. The source code for the paper is publicly available at https://github.com/peridyno/peridyno.

Tensile instability is one of the major obstacles to particle methods in fluid simulation, which would cause particles to clump in pairs under tension and prevent fluid simulation to generate small-scale thin features. To address this issue, previous particle methods either use a background pressure or a finite difference scheme to alleviate the particle clustering artifacts, yet still fail to produce small-scale thin features in free-surface flows. In this article, we propose a dual-particle approach for simulating incompressible fluids. Our approach involves incorporating supplementary virtual particles designed to capture and store particle pressures. These pressure samples undergo systematic redistribution at each time step, grounded in the initial positions of the fluid particles. By doing so, we effectively reduce tensile instability in standard SPH by narrowing down the unstable regions for particles experiencing tensile stress. As a result, we can accurately simulate free-surface flows with rich small-scale thin features, such as droplets, streamlines, and sheets, as demonstrated by experimental results.

Real-time simulation of hyperelastic membranes like cloth still faces a lot of challenges, such as hyperplasticity modeling and contact handling. In this study, we propose projective peridynamics that uses a local-global strategy to enable fast and robust simulation of hyperelastic membranes with contact. In the global step, we propose a semi-implicit strategy to linearize the governing equation for hyperelastic materials that are modeled with peridynamics. By decomposing the first Piola-Kirchhoff stress tensor into a positive and a negative part, successive substitutions can be taken to solve the nonlinear problems. Convergence is guaranteed by further addressing the overshooting problem. Since our global step solve requires no energy summation and dot product operations over the entire problem, it fits into GPU implementation perfectly. In the local step, we further present a GPU-friendly gradient descent method to prevent interpenetration by solving an optimization problem independently. Putting the global and local solves together, experiments show that our method is robust and efficient in simulating complex models of membranes involving hyperelastic materials and contact.