Problem and background: Elastodynamic systems are routinely used in visual computing (e.g. in computer graphics) to model the motion of objects such as cloth, fibers, textiles, flexible solids, the dynamic behavior of fluids, and deformable bodies. These systems can be described by using Newton’s second law of motion, leading to a system of second-order DEs:
$M x’~'(t) + Dx'(t) + Kx(t) =f(x(t)), \quad x(t_0)=x_0, \quad x'(t_0)=v_0$
where $x_0$ and $v_0$ are initial position and velocity, respectively. Here $x(t) \in \mathbb{R}^{3N}$ is the vector of positions, $M \in \mathbb{R}^{3N \times 3N}$ is the mass (nonsingular) matrix, $D \in \mathbb{R}^{3N \times 3N}$ is the damping matrix, $K \in \mathbb{R}^{3N \times 3N}$ is spring matrix (stiff), and $f(x) \in \mathbb{R}^{3N}$ is the total external forces ($N$ is the number of particles) acting on particles which can result from an external potential, collisions, etc., and can be dependent of all particle positions, velocities, or external forces set by user interaction.
Numerical solution to such elastodynamics systems is difficult since the linear spring forces possess very high frequencies ($\| K\| >>1$), leading to time scales differing by several orders of magnitude. Current methods have the shortcoming that their performance is highly dependent on the numerical stiffness of the underlying system that often leads to unrealistic behavior or a significant loss of efficiency.
Contributions:
1. Development of a stiffly accurate integrator for elastodynamics systems
I have developed a new integration method which is based on a mathematical reformulation of the underlying differential equations, an exponential treatment of the full nonlinear forcing operator as opposed to more standard partially implicit approaches, and the utilization of the concept of stiff accuracy which ensures that the efficiency of the simulations is significantly less sensitive to increased stiffness. Using this approach, in collaboration with Dominik L. Michels (Stanford and KAUST) and Mayya Tokman (UCM), we are able to tremendously accelerate the simulation of stiff systems compared to the state-of-the-art methods used in visual computing and significantly increase the overall accuracy. The advantage of this new approach is demonstrated on a broad spectrum of complex models including human hair, deformable bodies, textiles, and bristles. For more details, see our results in [1] below.
[1] D.L. Michels, V.T. Luan, M. Tokman, A stiffly accurate integrator for elastodynamic problems
ACM Transactions on Graphics, Vol. 36, No. 4, Article 116 (2017)
This work was highlighted as one of the amazing works at “Technical Papers Preview Trailer SIGGRAPH 2017“, see:
Later, the University of California and Southern Methodist University also announced this their research news:
- April 2018, SMU news: Animations that are true to life
- October 2017, UC news: “Animations that are true to life”: https://www.universityofcalifornia.edu/news/animations-are-true-life
- September 2017, UCM news: “Mathematicians Making Strides Toward Lifelike Animations”
http://www.ucmerced.edu/news/2017/mathematicians-making-strides-toward-lifelike-animations
Related pictures:
Simulation of tooth brushing
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2. Building efficient exponential Rosenbrock methods for simulating systems of nonlinear coupled oscillators
In this work, I proposed an advanced time integration technique associated with explicit exponential Rosenbrock-based methods for the simulation of large stiff systems of nonlinear coupled oscillators. In particular, a family of efficient exponential Rosenbrock schemes for simulating the reformulated system is derived and applied to an equivalent reformulation of these systems. Moreover, the required regularity conditions and the convergence of these schemes for the system of coupled oscillators are established. In collaboration with Dominik L. Michels (KAUST), we present an efficient implementation of this new approach and discuss several applications in scientific and visual computing. The accuracy and efficiency of our approach are demonstrated through a broad spectrum of numerical examples, including a nonlinear Fermi–Pasta–Ulam–Tsingou model, elastic and nonelastic deformations as well as collision scenarios focusing on relevant aspects such as stability and energy conservation, large numerical stiffness, high fidelity, and visual accuracy. For more details, see our results in [2] below.
[2] V.T. Luan and D.L. Michels, Efficient exponential time integration for simulating nonlinear coupled oscillators,
to appear in Journal of Computational and Applied Mathematics (2021). DOI: https://doi.org/10.1016/j.cam.2021.113429
Related pictures:
Simulation of a frontal crash of a car into a wall
Simulation of an oscillating coil spring
Elastic deformation of a dragon model (including 150000 vertices corresponding to N = 450000 equations of motion)