Constrained particle dynamics

Abstract

This paper presents a constrained particle dynamics (CPD) framework with explicit and implicit algorithms for simulating differential–algebraic equations of motion for systems of particles under constraints. Addressing limitations in existing techniques such as position-based dynamics (PBD), commonly used in computer graphics but prone to inaccuracies, the CPD approach utilizes Hamilton’s variational principle and Lagrange multipliers to ensure accurate constraint enforcement. The explicit CPD (xCPD) algorithm employs a central difference scheme, enhancing efficiency by advancing the system under external forces and applying a correction term for constraints. The implicit CPD (iCPD) algorithm uses the Trapezoidal rule, solving a saddle point problem that integrates dynamic and constraint equations, offering robustness for larger time steps. The effectiveness of the CPD algorithms is demonstrated through mathematical analysis and numerical comparisons of benchmark problems. Results indicate that CPD algorithms achieve higher accuracy and superior energy conservation properties compared to PBD, exhibiting second-order convergence rates; whereas, PBD shows only first-order convergence.