Position Based Dynamics

Matthias Müller et al.VRIPHYS 2006

Copyright of figures and other materials in the paper belongs original authors.

Presented by Eunki Hong

2014. 10. 21

Computer Graphics @ Korea University

Eunki Hong | 2014. 10. 21 | # 2Computer Graphics @ Korea University

• Introduction

• Related Work

• Position Based Simulation

Algorithm Overview

Constraint Projection

Collision Response

• Cloth Simulation

Representation of Cloth

Self Collision

Cloth Balloons

• Result

Index

Eunki Hong | 2014. 10. 21 | # 3Computer Graphics @ Korea University

Introduction

• Traditional approach to simulationdynamic objects had been workwith forces

• F = ma Calculate accelerations Error

• This paper use only constraint and position

• Main Feature of PBD

Control over explicit integration.

Removes the typical instability problems.

Positions of vertices can directly be manipulated during the simulation.

Easy to understand and implement.

Eunki Hong | 2014. 10. 21 | # 4Computer Graphics @ Korea University

• Large Steps in Cloth Simulation

[Baraff D. and Witkin A. / Proceedings of ACM SIGGRAPH 1998]

• Using constraint function

• Instead of computing constraint function energy, this paper solve for the equilibrium configuration and project positions

Related Work

Eunki Hong | 2014. 10. 21 | # 5Computer Graphics @ Korea University

• Meshless Deformations Based on Shape Matching

[Matthias Müller et al / Proceedings of ACM SIGGRAPH 2005]

• Closest method with this paper

• Only treat one specialized global constraint. Therefore, do not need a position solver.

Related Work

Position Based Simulation

Eunki Hong | 2014. 10. 21 | # 7Computer Graphics @ Korea University

Algorithm Overview

Initialize

Move Position(external forces)

Solver(internal forces)

Eunki Hong | 2014. 10. 21 | # 8Computer Graphics @ Korea University

Algorithm Overview - Solver

projectConstraints(C1,...,CM+Mcoll,p1,...,pN)

for(C = C1 to CM)

q C.get_using_vertices_set();

ConstraintProjection(C,q);

endfor

for(C = CM+1 to CM+Mcoll)

q C.get_using_vertices_set();

CollisionResponse(C,q);

endfor

end

Eunki Hong | 2014. 10. 21 | # 9Computer Graphics @ Korea University

• Constraint is a function

C : R3nj R

nj : the number of vertices that constraint use

• Constraints expression internal energy

• C(p) = 0 mean System is stable

Define Constraint

Eunki Hong | 2014. 10. 21 | # 10Computer Graphics @ Korea University

• C(p1,p2) = ||p1 – p2|| - d

C(p1,p2) = ||Δp1|| + ||Δp2||

C(p1,p2) = 0

Constraint Example

Eunki Hong | 2014. 10. 21 | # 11Computer Graphics @ Korea University

• Let define terms like

C : Current constraint

n : The number of vertices that constraint use

pi : i‘s vertex position vector (size is 3)

wi : Inverse of i‘s vertex mass ( = 1/mi)

p : (p1,p2,…,pn) Connect of vertices position vector (size is 3n)

Δp : next timestep vertices position (size is 3n)

• We want to find Δp s.t. C(p + Δp) = 0

• and ∑i miΔpi = 0 (momentum conserved)

Constraint Projection

Eunki Hong | 2014. 10. 21 | # 12Computer Graphics @ Korea University

• Taylor expansion

C(p + Δp) ≒ C(p) + ∇pC(p)∙Δp = 0

• Direction of Δp

Δp = λ ∇pC(p)

• Putting together

Δp = –C(p)

||∇pC(p)||2∇pC(p)

• For individual position

Δpi = –s ∇piC(p) ,

s = C(p)

∑j ||∇pjC(p)||2(is scaling factor)

Constraint Project Equation (1/2)

(only if all vertices mass are equal)

Eunki Hong | 2014. 10. 21 | # 13Computer Graphics @ Korea University

• Consider mass

Δpi = –s wi ∇piC(p) ,

s = C(p)

∑jwj||∇pjC(p)||2

• Why?

∑i miΔpi = 0 (linear momentum conserved)

= –s ∑i∇piC(p) = 0 (translation invariance)

Constraint Project Equation (2/2)

Eunki Hong | 2014. 10. 21 | # 14Computer Graphics @ Korea University

• C(p1,p2) = ||p1 – p2|| - d

∇p1C(p) = p1 – p2

||p1 – p2||

∇p2C(p) = –p1 – p2

||p1 – p2||

• ∴ s = C(p)

∑jwj||∇pjC(p)||2=

||p1 – p2|| – dw1 + w2

. ∴ Δp1 = –||p1 – p2|| – d

w1 + w2w1

p1 – p2

||p1 – p2||

Δp2 = + ||p1 – p2|| – d

w1 + w2w2

p1 – p2

||p1 – p2||

Constraint Project Example

Eunki Hong | 2014. 10. 21 | # 15Computer Graphics @ Korea University

• pi pi + k Δpi (k ∈ [0…1])

• Distance between prev point and result point is Δpi(k)

• After ns iteration, distance become Δpi(1 – (1 – k)ns)

• So we need to change k’ dependent on iteration time

• k‘ = 1 – (1 – k)1/solverIterations

pi pi + k’ Δpi

Stiffness parameter of Constraint

Eunki Hong | 2014. 10. 21 | # 16Computer Graphics @ Korea University

• If xi pi ray enters an object,

qc : entry point

nc : suface normal

• Make ConstraintC(pi) = (pi – qc)∙nc

Stiffness k=1

• This constraint goalis C(pi + Δpi) ≥ 0,not C(pi + Δpi) = 0

Collision Constraint

xi

pi

qc

nc

Eunki Hong | 2014. 10. 21 | # 17Computer Graphics @ Korea University

• Collision constraint will work only C(pi) < 0

Δpi = 0 C(pi) ≥ 0

–s wi ∇piC(pi) C(pi) < 0

Collision Response

Cloth Simulation

Eunki Hong | 2014. 10. 21 | # 19Computer Graphics @ Korea University

• There are two type of internal force

• Stretch

Cstretch(p1,p2) = ||p1 – p2|| – l0 Stiffness k = kstretch

• Bending

Cbend(p1,p2,p3,p4) =

acos((p2 – p1) ×(p3 – p1)

||(p2 – p1) ×(p3 – p1)||∙

(p2 – p1) ×(p4 – p1)||(p2 – p1) ×(p4 – p1)||

) – φ0

Stiffness k = kbend

Representation of Cloth

Eunki Hong | 2014. 10. 21 | # 20Computer Graphics @ Korea University

With Bending Without Bending

Bending Constraint

Eunki Hong | 2014. 10. 21 | # 21Computer Graphics @ Korea University

• If vertex q move though a triangle p1, p2, p3

C(q,p1,p2,p3) = . (q – p1)∙n – h and

C(q,p1,p2,p3) = –(q – p1)∙n – h

• When

n : normal vector of triangle p1, p2, p3

h : cloth thickness

Self Collision

Eunki Hong | 2014. 10. 21 | # 22Computer Graphics @ Korea University

• C(p1,…,pN) = ∑i (pt𝑖1× pt

𝑖2) ∙ pt

𝑖3

– kpressureV0

└ (Volume of mesh)

• If kpressure > 1 mesh will get overpressure

• ∇piC(p) = ∑j:t𝑗1=i (pt𝑗

2× pt

𝑗3) + ∑j:t

𝑗2=i (pt𝑗

3× pt

𝑗1) + ∑j:t

𝑗3=i (pt𝑗

1× pt

𝑗2)

= ∑(normal vector who contain pi) * (area of flat)

Cloth Balloons

Eunki Hong | 2014. 10. 21 | # 23Computer Graphics @ Korea University

Result