Download Fluids

67535: Computer Games Programming
Week 5:
Physics II:
• rigid body collision
• fluids
Tutorial: Geometry shader, transform
feedback
Rigid body description
Position p and orientation o (together known as pose)
Velocity v and spin (angular velocity) ω
Acceleration a and angular acceleration dω
Force f and torque τ ( F = ma; τ = I x dω )
Mass m and inertia tensor I
Momentum p and angular momentum L
Inertia Tensor
http://techhouse.brown.edu/~dmorris/projects/tutorials/inertia.tensor.summary.pdf
Inertia tensor defines distribution of mass in the volume
Volume integral of mass
Precomputed numerically
3x3 symmetric matrix (ok, tensor)
Diagonal elements: Ixx = m ∫v (y2 + z2) dV
Non-diagonal elements: Ixy = -m ∫v xy dV
– zero for symmetrical objects
Non-constrained motion
For every force acting on body
–
calculate the force and the torque
Calculate acceleration and angular acceleration
–
–
a = Σ F/m
dω = Σ τ x I-1
Integrate motion to get new position and orientation
Use an ODE solver
Fixed objects
Setting 1/m = 0 and I-1 = zero 3x3 matrix ensures that
the body moves nowhere and supports any weight
Collision resolution
During collision of two bodies, strong forces act in
a short time
Can be calculated by dynamics, with noninterpenetration approximated by forcefields
Good for soft, deformable bodies (e.g. cloth of jelly)
Rigid body collision
No gradual deformation
Shortcut: infinite force applied in zero time
v(t) is not continuous, dynamics do not work
Rigid body collisions
Rigid body Dynamics with Collisions
http://www.pixar.com/companyinfo/research/pbm2001/pdf/notesg.pdf
For each time step tn
For each object
Compute sum of all forces
Calculate acceleration an
Integrate motion to get vn and pn
If: collisions detected at tc
Stop ODE solver
Resolve collisions
Update velocities (keep position & orientation)
Restart ODE solver with new system state
Collision detection
Fact of collision
Time of collision
Points of contact
NB: use the fact that things do not change a lot
between t and t+h
– e.g. Find a separating plane once and run collision
detection only for elements of a separating plane
and their neighbours
Time of collision
No intersection at tn; intersection at tn+h
Find time of collision tc
– e.g. by bisecting h
– Rerun collision detection only for elements that are
intersecting at tn+h
Types of points of contact
Face-to-vertex (F2V)
Edge-to-edge (E2E)
Vertex-to-vertex and vertex-to-edge
– Degenerate cases
– ignore and wait
Everything else is a combination of F2V and E2E
– e.g. Face-to-face is a 3x F2V or F2V + 2x E2E or
2x F2V + 2x E2E
Managing points of contact
Maintain a list of active points of contact
– Pointers to bodies in contact
• p: Point of contact in world coordinates
– for vertex/face
• n: face normal
– for edge/edge
• ea and eb: edge directions (n = ea x eb)
Calculating relative velocities
Calculate Vpa and Vpb: velocity of point of contact
for both bodies:
Vpa = Va + ωa x (pa-xa)
Calculate velocity of pa relative to pb:
Vrel = n . (Vpa -Vpb)
Relative velocity
Vrel>ε: bodies moving away from each other;
remove contact point
Vrel=0±ε: resting contact
Vrel<-ε: collision contact
Collision contact resolution
Impulse: like force, but acts immediately, changing v
and ω
No friction: impulse acts along n only
Calculated so that
Vrel_after = -b*Vrel_before
where b is a bounciness factor, [0 1]
(derivation and code at
http://www.pixar.com/companyinfo/research/pbm2001/pdf/notesg.pdf)
Resting contact resolution
Calculate and update a contact force at point of
contact that:
* prevents interpenetration
* does not prevent separation
* becomes zero upon separation
A set of linear constraints on quadratic polynomials
Requires a Quadratic Programming (QP) solver
Collision resolution summary
Update points of contact
Update velocities at colliding contacts
Calculate forces at resting contacts
Update ODE solver with new velocities and resting
forces
NB: still no friction
Rigid body Dynamics with Collisions
http://www.pixar.com/companyinfo/research/pbm2001/pdf/notesg.pdf
For each time step tn
For each object
Compute sum of all forces
Calculate acceleration an
Integrate motion to get vn and pn
If: collisions detected at tc
Stop ODE solver
Resolve collisions
Update velocities (keep position & orientation)
Update forces
Restart ODE solver with new system state
Friction
http://www.cs.cmu.edu/~baraff/papers/sig94.pdf
Friction is a force ft at a contact point that is:
•
•
•
•
Tangential to the contact surface
Opposite to tangential acceleration
Depends on normal force and contact face area
At most μfn
Accounting for friction adds constraints to QS; not solvable
in extreme cases but usually O(n)
Static and dynamic friction
Static Friction
•Tangential relative velocity is 0
•Added constraints:
• ft must keep angular velocity at 0
• But not be higher than μfn
Dynamic friction
•ft = μfn
•Direction is opposite to velocity at contact point
Fluids
Fluids
http://software.intel.com/en-us/articles/fluid-simulation-for-video-games-part-1
Liquid, gas or plasma. Smoke is almost fluid: an
aerosol, a combination of gas and particles
Precise fluid models
• Navier-Stokes (for viscous fluid)
• Euler (for non-viscous fluid)
https://www.youtube.com/watch?v=vOFcHqImXJ8
https://www.youtube.com/watch?v=MlNxgmPVF6U
Simpler fluid models
As particles:
•
•
Properties averaged over nearby particles
Or weighted by distance
https://www.youtube.com/watch?v=Qve54Z71VYU
https://www.youtube.com/watch?v=7ZEpYxqcqbc
• As a volume: model flow at grid points (aka lattice)
•
lattice Bolzmann models
• Or as both (Hybrid)
https://www.youtube.com/watch?v=p67-Qiad5zc
https://www.youtube.com/watch?v=pxDeVrJO5yY
Volumetrics
The unspoken assumption
The world is mostly transparent with some non-transparent
objects
– Human vision is fine-tuned to perceive (most) solid objects as nontransparent, and (most) non-solid objects as transparent
Can be modelled using outer surfaces only
– That is why we get polygon meshes
Complications
The world is actually a 3D space full of matter with various
visual and, generally, physical properties
Some simple exceptions
– Fog
– Particle systems: fire, smoke, water mist
– Transparent and semi-transparent objects
Sometimes it is important to know what’s inside an object
– Breaking, bending, exploding, cutting through…
Modelling volumes
Voxel – a volume element
– A cube of space or a point of space in a grid
Each voxel holds information about its content:
–
–
–
–
Simplest case: boolean empty/full
Transparency: float 0..1 or short uint 0...255
Material
And other options, depending on task
3D object can be modelled as a 3D array of voxels
MyVoxel volume[100][100][100];
Simple, but computationally heavy: 1million voxels for a not-too-good
resolution
3D textures on GPU are useful here
Use case: Realistic terrain
– Caves, overhangs, arches...
– Height map is too simple
Use case: modeling physics
Realistic destruction
– Worms 4: Mayhem
Solid objects, e.g. vehicles
– Masters of Orion III
– Red Alert
Soft body dynamics
– Deformation, tearing, bounce
– http://www.alecrivers.com/fastlsm/
Use case: medicine
Imaging: Volumetric data is Xray/Ultrasound
transparency values
Illustration: http://grahamj.com/
Images: Wikimedia commons
Use case: fluid dynamics
http://http.developer.nvidia.com/GPUGems3/gpugems3_ch30.html
gas/liquid/plasma flow
Tutorial
Geometry shaders
Particle system on
GPU using
transform
feedback
Geometry Shader
Primitives
Input Primitives:
Points (1)
Lines (2)
Lines_adjacency (4)
Triangles (3)
Triangles_adjacency (6)
Output Privitives:
points
line_strip
triangle_strip
Format
layout (triangles) in;
layout (line_strip, max_vertices = 4) out;
* Input primitive is discarded
* 0 or more primitives can be output
* no more than max_vertices!
#version 330
layout (points) in;
Pos.y += 1.0;
layout (triangle_strip) out;
gl_Position = gVP * vec4(Pos, 1.0);
layout (max_vertices = 4) out;
TexCoord = vec2(0.0, 1.0);
EmitVertex();
uniform mat4 gVP;
uniform vec3 gCameraPos;
Pos.y -= 1.0;
Pos += right;
out vec2 TexCoord;
gl_Position = gVP * vec4(Pos, 1.0);
TexCoord = vec2(1.0, 0.0);
void main() {
EmitVertex();
vec3 Pos = gl_in[0].gl_Position.xyz;
vec3 toCamera =
Pos.y += 1.0;
normalize(gCameraPos - Pos);
vec3 up = vec3(0.0, 1.0, 0.0);
gl_Position = gVP * vec4(Pos, 1.0);
vec3 right = cross(toCamera, up);
TexCoord = vec2(1.0, 1.0);
EmitVertex();
Pos -= (right * 0.5);
gl_Position = gVP * vec4(Pos, 1.0);
TexCoord = vec2(0.0, 0.0);
EmitVertex();
EndPrimitive();
}
Instancing using GS
layout(invocations = 8) in;
...
gl_Position = gl_InvocationID *
gl_in[0].gl_Position.xyz;
Setting up GS
GLuint shader =
glCreateShader(GL_GEOMETRY_SHADER);
glShaderSource(shader, 1, “test.gs.glsl”, NULL);
glCompileShader(shader);
NOTE: the first one to extend ShaderIO class to GS and publish code on
moodle gets +1 to next exercise