Physically Based Modeling

1. Physically Based Modeling Let physics take over!

2. Physically Based Modeling • Account for forces in system • Account for object interaction, e.g. friction, collision

3. Spring-Mass-Damper • Model: jello, cloth, muscle • Have gravity: add mass to vertices • Have stability (add stiffness to flag pole) • Put springs on each vertex, allowing to stretch a finite amount • Another use: angular springs on polygon corners to prevent self-penetration • Numerical integration for animation

4. Governing Equations F • Hooke’s Law (for graphics):Fs = ks(dist-len) where • len = rest length • dist = current length • ks = spring constant • Fij = -Fji = ks(distij(t) – lenij)dij • Where dij = unit vector along i-j • t = time • Fs = Σ Fij (sum of all edges coming out of a vertex) V1 E12 E31 V3 V2 E23 Look at neighbors when calculating spring force! V

5. Damping Force & Angular Springs • FD = -kd v(t) • Damper force is proportional to velocity • and acts in direction opposite to velocity • NET FORCE:F = Fs + Fd = ks(dist-len) -kd v(t) M2 M1 • Angular spring: τ (torque) = ks[Ө(t) -Ө(rest)] - kdӨ(t)

6. Object Representation • Vertices: mass • Edges: • Spring, damping constant • Resting length • Vertex IDs • Each vertex has • Current position • Current velocity • Current acceleration • Mass • Number of edges X, Y, Z components

7. How it all comes together • Use Newton’s Law (F = ma) to calculateacceleration for every vertex(big system of linear equations) • Basic strategy: accumulate acceleration from different sources (Gravity, Spring, Damper) and integrate 2 times to get velocity and position • Can use Runge-Kutta, for example • For project: you can use xspringies (2D) & ODE library or other library you can find or roll your own • Make sure to cite your sources

8. Details (source Paul Bourke) Create the particles Create the springs between the particles Initialize the particle and spring parameters loop in time { Update the particle positions (solve ODEs) Display the results somehow }

9. Update Particle Positions • Calculate force at each point • Add Positive force: Force*mass for each X,Y,Z • Subtract Drag: drag*velocity for each X,Y,Z • Handle spring interaction • For each spring • For each of X,Y,Z of attached points • Force = Hooke’s law • Force += damping constant(Δvelocity of points)(Δlen of points in direction)/lenx,y,z • Force *= -(Δlen of points)/lenx,y,z • Add or subtract force to/from point (if the point is not fixed)

10. Calculate derivatives for points • We already have velocities: • dpx/dt = velocityx • Velocity derivative: • dvx/dt = forcex/mass • And solve with your favorite ODE solver (Runge-Kutta..) • Update positions • Update velocity