Diagonal Hessian Proxy for Efficient Elastic Simulation Using Peridynamics

Meshless simulation of elasticity is important for deformable simulation in computer graphics. While shape matching is a popular meshless solution, it is limited to a subset of elastic constitutive models, challenging the simulation of generic elastic constitutive models using meshless integration. In contrast, peridynamics offers a more versatile capacity and can describe various material behavior through non-local interactions between vertices. However, the size of each stencil Hessian matrix varies with the number of nearby integration points, leading to inefficiency and accuracy loss. To address these challenges, we present an efficient and robust solver for generic elastic models based on peridynamics. We propose an efficient first-order Hessian proxy derived from the positive-negative decomposition of the stress tensor. The proposed symmetric positive definite proxies ensure convergence within a reasonable number of iterations while also being easy to parallelize on GPU. To further enhance stability, particularly for hyperelastic models, we propose enforcing strain limiting between peridynamics bonds to prevent tensile instability in meshless integration. Our algorithm includes two iteration loops of strain limiting and elastic Jacobis, and the pipeline is well-suited for GPU implementation. We evaluated the performance of our approach with a wide range of elastic constitutive models in diverse testing scenarios against the alternative numerical solvers. Our method features superior efficiency and faster convergence compared to existing numerical solvers. These compelling results underscore the practicality and effectiveness of our method for simulating elasticity via meshless integration.