This article proposes a novel simulation approach that combines implicit integration with the Bonded Discrete Element Method (BDEM) to achieve faster, more stable, and more accurate fracture simulation. The new method leverages the efficiency of implicit schemes in dynamic simulation and the versatility of BDEM in fracture modeling. Specifically, an optimization-based integrator for BDEM is introduced and combined with a manifold optimization approach to accelerate the solution process of the quaternion-constrained system. Our comparative experiments indicate that our method offers better scale consistency and more realistic collision effects than finite element method and material point method fragmentation approaches. Additionally, our method achieves a computational speedup of 2.1 to 9.8 times over explicit BDEM methods.

Simulating stiff materials has long posed formidable challenges for traditional physics-based solvers. Explicit time integration schemes demand prohibitively small time steps, while implicit methods necessitate an excessive number of iterations to converge, often yielding visually objectionable transient configurations in the early iterations, severely limiting their real-time applicability. Position-based dynamics techniques can efficiently simulate stiff constraints but are inherently restricted to constraint-based formulations, curtailing their versatility. We present “Reliable Iterative Dynamics” (RID), a novel iterative solver that introduces a dual descent framework with theoretical guarantees for visual reliability at each iteration, while maintaining fast and stable convergence even for extremely stiff systems. Our core innovation is an iterative method that circumvents the need for numerous iterations or small time steps to handle stiff materials robustly. Experimental evaluations demonstrate our method’s ability to handle a wide range of materials, from soft to infinitely rigid, while producing visually reliable results even with large time steps and minimal iterations. The versatile formulation allows seamless integration with diverse simulation paradigms like the finite element method, material point method, smoothed particle hydrodynamics, and incremental potential contact for applications ranging from elastic body simulations to fluids and collision handling.

We present a novel multigrid solver framework that significantly advances the efficiency of physical simulation for unstructured meshes. While multi-grid methods theoretically offer linear scaling, their practical implementation for deformable body simulations faces substantial challenges, particularly on GPUs. Our framework achieves up to 6.9× speedup over traditional methods through an innovative combination of matrix-free vertex block Jacobi smoothing with a Full Approximation Scheme (FAS), enabling both piecewise constant and linear Galerkin formulations without the computational burden of dense coarse matrices. Our approach demonstrates superior performance across varying mesh resolutions and material stiffness values, maintaining consistent convergence even under extreme deformations and challenging initial configurations. Comprehensive evaluations against state-of-the-art methods confirm our approach achieves lower simulation error with reduced computational cost, enabling simulation of tetrahedral meshes with over one million vertices at approximately one frame per second on modern GPUs.

The Bonded Discrete Element Method (BDEM) has raised interests in the graphics community in recent years because of its good performance in fracture simulations. However, current explicit BDEM usually needs to work under very small time steps to avoid numerical instability. We propose a new BDEM, namely Rod-BDEM (RBDEM), which uses Cosserat energy and yields integrable forces and torques. We further derive a novel Cosserat rod discretization method to effectively represent the three-dimensional topological connections between discrete elements. Then, a complete implicit BDEM system integrating the appropriate fracture model and contact model is constructed using the implicit Euler integration scheme. Our method allows high Young’s modulus and larger time steps in elastic deformation, breaking, cracking, and impacting, achieving up to 8 times speed up of the total simulation.

Along with motion and deformation, fracture is a fundamental behaviour for solid materials, playing a critical role in physically-based animation. Many simulation methods including both continuum and discrete approaches have been used by the graphics community to animate fractures for various materials. However, compared with motion and deformation, fracture remains a challenging task for simulation, because the material's geometry, topology and mechanical states all undergo continuous (and sometimes chaotic) changes as fragmentation develops. Recognizing the discontinuous nature of fragmentation, we propose a discrete approach, namely the Bonded Discrete Element Method (BDEM), for fracture simulation. The research of BDEM in engineering has been growing rapidly in recent years, while its potential in graphics has not been explored. We also introduce several novel changes to BDEM to make it more suitable for animation design. Compared with other fracture simulation methods, the BDEM has some attractive benefits, e.g., efficient handling of multiple fractures, simple formulation and implementation, and good scaling consistency. But it also has some critical weaknesses, e.g., high computational cost, which demand further research. A number of examples are presented to demonstrate the pros and cons, which are then highlighted in the conclusion and discussion.