[Paper Review] Towards Real-time Simulation of Hyperelastic Materials

Tiantian Liu, Sofien Bouaziz+1 more|arXiv (Cornell University)|Apr 25, 2016

Advanced Numerical Analysis Techniques48 references17 citations

TL;DR

This paper presents a novel quasi-Newton formulation of Projective Dynamics that enables fast, accurate, and robust real-time simulation of hyperelastic materials—including Neo-Hookean and spline-based materials—by reinterpreting the method as a constant, positive-definite Hessian approximation. The approach achieves over 10× speedup over one Newton iteration while maintaining or improving accuracy and simplifying implementation.

ABSTRACT

We present a new method for real-time physics-based simulation supporting many different types of hyperelastic materials. Previous methods such as Position Based or Projective Dynamics are fast, but support only limited selection of materials; even classical materials such as the Neo-Hookean elasticity are not supported. Recently, Xu et al. [2015] introduced new "spline-based materials" which can be easily controlled by artists to achieve desired animation effects. Simulation of these types of materials currently relies on Newton's method, which is slow, even with only one iteration per timestep. In this paper, we show that Projective Dynamics can be interpreted as a quasi-Newton method. This insight enables very efficient simulation of a large class of hyperelastic materials, including the Neo-Hookean, spline-based materials, and others. The quasi-Newton interpretation also allows us to leverage ideas from numerical optimization. In particular, we show that our solver can be further accelerated using L-BFGS updates (Limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm). Our final method is typically more than 10 times faster than one iteration of Newton's method without compromising quality. In fact, our result is often more accurate than the result obtained with one iteration of Newton's method. Our method is also easier to implement, implying reduced software development costs.

Motivation & Objective

• Address the lack of support for general hyperelastic materials like Neo-Hookean and spline-based models in existing real-time methods such as Projective Dynamics.
• Overcome the performance bottleneck of Newton’s method, which is slow due to repeated Hessian computation and solution of linear systems per iteration.
• Enable artist-friendly material modeling via spline-based materials while maintaining real-time performance.
• Improve convergence and robustness of real-time physics simulation through integration of L-BFGS updates within a quasi-Newton framework.
• Bridge the gap between real-time and off-line physics simulation by achieving high accuracy at interactive speeds.

Proposed method

• Reformulate Projective Dynamics as a quasi-Newton method with a constant, positive-definite Hessian approximation matrix A, avoiding costly Hessian updates.
• Use a line search to ensure convergence stability and prevent energy increase during optimization, compensating for the fixed Hessian approximation.
• Integrate L-BFGS updates to improve the Hessian approximation using curvature information from previous iterates, enhancing convergence speed.
• Leverage matrix-free linear system solves to maintain efficiency, especially when combined with iterative solvers like CG.
• Apply the method to hyperelastic energy functions satisfying the Valanis-Landel assumption, including spline-based materials and classical models.
• Use implicit Euler time integration with the new solver, ensuring stability and enabling simulation of large deformations and inverted elements.

Experimental results

Research questions

• RQ1 Can Projective Dynamics be reinterpreted as a quasi-Newton method to support a broader class of hyperelastic materials?
• RQ2 Can a constant Hessian approximation in a quasi-Newton framework maintain accuracy and stability while enabling faster simulation than Newton’s method?
• RQ3 How does the integration of L-BFGS updates affect convergence speed and robustness in real-time physics simulation?
• RQ4 To what extent can the proposed method outperform one-iteration Newton’s method in terms of speed, accuracy, and implementation complexity?
• RQ5 Can the method handle extreme deformations and inverted elements without loss of stability or accuracy?

Key findings

• The proposed method achieves over 10× speedup compared to one iteration of Newton’s method while maintaining or improving solution accuracy.
• The method supports a wide range of hyperelastic materials, including Neo-Hookean and spline-based materials, which were previously incompatible with Projective Dynamics.
• Robustness is preserved even under extreme initial conditions, as demonstrated by a deformed hippo model recovering to its rest pose with inverted elements.
• L-BFGS integration significantly accelerates convergence, especially when combined with a well-chosen initial Hessian approximation.
• The method outperforms CG-based solvers in terms of both speed and convergence quality, even with 15 CG iterations, due to the superior Hessian approximation.
• The solver is easier to implement than Newton’s method, reducing software development costs while maintaining high performance and accuracy.