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Contact and Friction Simulation for Computer Graphics(Siggraph course 2022)

相关的course：SIGGRAPH’20 Course: An Introduction to Physics-Based Animation

SIGGRAPH’22 Course: Contact and Friction Simulation for Computer Graphics (siggraphcontact.github.io)

Syllabus

Basics：

• contact generation using discrete collision detection
• numerical techniques for solving the associated LCPs and NCPs

advanced topics:

• soft body contact approaches
• penalty and barrier functions
• anisotropic friction modeling

Chapter 1. Introduction to Contact Simulation

there are three main paradigms for contact simulation

• constraint based methods(很精确): constrained optimization

• penalty based approaches: small abstract springs(也可以不使用弹簧) working between objects to keep them from overlapping. Hence, the springs “penalize” overlap.

  • similarities to penalty methods in numerical optimization: barrier methods

• impulse-based methods: contact between objects is conceptualized as sequences of micro-collisions

1.1 The Equation of Motion

/[M(t)/dot u(t)=f(q(t),u(t),t)
/]

Notations in this note:

• /(M/): mass matrix
• /(q/): position
• /(u/): velocity
• /(h=/Delta t/): timestep

1.2 Time Integration

Notations:

• /(u^-=u^n/): velocity of the last time step
• /(u^+=u^{n+1}/): velocity of the next time step

1.3 Constraints

流形优化方法求解约束问题

two types of constraint equations:

• bilateral: /(/phi(q)=0/), hinges, ball-and-socket, and prismatic joints
• unilateral: /(/phi(q)/ge0/).

/(m/) constraint functions /(/phi(q)/in R^m/) implicitly define a manifold that is embedded in the /(n/)-dimensional space of the simulation degrees of freedom. We can instead formulate the constraint equations in terms of the velocities by computing the gradient of /(