Contents 1. Abstract 2. Introduction 3. Related Work 4. Cloth
Model 5. Constrained Dynamics 1. Implicit Constraint Direction(ICD)
2. Step and Project(SAP) 3. Fast Projection Method 4.
Implementation 6. Results 7. Discussion
Slide 4
Abstract A Simulation of Woven Fabrics with LOW Strain
Constrained Lagrangian Mechanics Fast Projection Method Resulting
algorithms acts as a Velocity Filter Weft Warp Woven Fabric
Slide 5
Algorithm I Simulation based-on Elastic Materials Constraints
Lagrangian Mechanics Oscillation occurs Rest length Direction :
Length : Eq(1)
Slide 6
Algorithm II The problem : Direction : Length : The Solution:
to use Constraint at n+1 Direction : Length :
Slide 7
Algorithm III However, we dont know, Compute Implicitly Step
and Project Find the closest point Fast Projection Find a close
point until it converges or less than the tolerance
Slide 8
Introduction I The Problem Most fabrics do not stretch under
their own weight Unfortunately, Popular cloth solvers allow to
Stretch because of Performance Unrealistic 10% Strain Inaccuracy +
Instability of Discrete Time Integration Time Step, h Force, F For
stability or The most powerful!! Integrated velocity
Slide 9
Introduction II The Solution Strain and Strain Rate Limiting
Algorithms are required [Provot 1995] Our Method to solve Prohibits
Stretching by enforcing Constraint Equations NOT by Integrating
internal spring forces Constraint based method is well-suit to
operate in Quasi-Inextensible Regime. Motivation : [Bridson et al.
2002] Velocity filtering passes
Slide 10
Contributions (2) We summarize the relevant Literature (3) We
describe the basic discrete cloth model. (4.1) We propose a novel
CLM formulation that is implicit on the constraint gradient. (4.2)
We prove that the implicit methods nonlinear equations correspond
to a minimization problem (4.3) This result motivates a fast
projection method for enforcing inextensibility (4.4) We describe
an implementation of fast projection as a simple and efficient
velocity filter, as part of a framework that decouples time
stepping, inextensibility, and collision passes (5) Consequently,
the fast projection method easily incorporates with a codes
existing bending, damping, and collision models, to yield
accelerated performance
Slide 11
Figure 1 Importance of capturing inextensibility. Actual Denim
PatchSimulation results of progressively smaller permissible
strain
Slide 12
Related Work I Broad Surveys on Cloth Simulation [House and
Breen 2000; Choi and Ko 2005] Treating Cloth as an Elastic Material
[Terzopoulos et al. 1987] [Breen et al. 1994], [Eberhardt et al.
1996] [Baraff and Witkin 1998], [Choi and Ko 2002] To reduce
visible stretching, Large elastic moduli or stiff springs Degrading
numerical stability Hookes Law
Slide 13
Related Work II Implicit Integration [Baraff and Witkin 1998]
Large, stable time steps Adaptive time stepping Implicit-Explicit
(IMEX) formulation [Eberhardt 2000], [Boxerman et al. 2003] Only a
subset of forces implicitly Stretching forces are singled out for
special treatment (Related to this papers method) Problem Improved
performance by allowing stretch of fabrics Implicit Systems perform
many iterations with elastic material stiffness Timestepping
algorithms introduce undesirable numerical damping Backward Euler
& BDF2
Slide 14
Related Work III An alternative method To reduce the stiff
component and reformulate it as a constraint [Hairer et al. 2002]
Discrete Setting: various iterative or global Algorithms Iterative
Enforcement Correcting Edge lengths by displacing incident vertices
[Provot 1995] Bounding the rate of change of spring length per a
time step less than 10% of the current length [Desbrun et al. 1999;
Meyer et al. 2001;Fuhrmann et al. 2003;Bridson et al. 2002; 2003]
Non-linear Gauss-Seidel: enforce inextensibility on each
constraints [Muller et al. 2006] Iterative strain Limiting
algorithms behave as Jacobi or Gauss-Seidel [Bridson et al.]
Iterative constraint enforcement requires a number of
iterations
Slide 15
Related Work IV Global Enforcement [House et al. 1996]
Lagrangian Multiplier with CLM to treat stretching Alleviates the
difficulties associated with poor numerical conditioning and
artificial damping Difficulties in handling collision response in
[2000] Our Method : Overcome by velocity-filter paradigm to handle
inextensibility and complex collision Numerical Drift Exacerbated
by the discontinuities introduced during collision responses :
overcome by constraint-restoring springs But hard to adjust the
spring coefficients Caused by linearization of constraint equation
Our method : use nonlinear equation
Slide 16
Related Work V [Tsiknis 2006] Triangle strain limiting with a
global stitching step for stability [Hong et al. 2005] Linearized
implicit formulation to improve stability Summary
Spring-based/Strain-limiting approach is costly and evenly
intractable The alterative : Constraint-based Methods
Slide 17
Cloth Model Woven Fabrics A complex mechanical network of
interleaving yarn Materials warp and weft directions do not stretch
perceptively Quadrangulation Less edges (2n VS 3n), given n
vertices Subtracting constraints from positional DOFS : O(n) DOFs
remain, corresponding to the normal direction at each vertex (VS
zero DOFs for a triangulation) Constraints are linearly independent
Full-rank Jacobian treatable by direct solver
Slide 18
Cloth Model Stretch Warp- or weft aligned quad edge maintains
its undeformed length l, by enforcing Other : compatible with
existing systems Shear : fabrics actually perceptibly shear Bending
/ collisions / and so on.. (1) (2) Its gradient: Constraint:
Slide 19
Constrained Dynamics Augmented Lagrangian Equation [Marsden
1999] : the time-varying 3n-vector of vertex position : time
derivative : 3n x 3n mass matrix : the stored energy (e.g. bending,
shear, and gravity) : the m-vector of constraints (i-th edge) : the
m-vector of Lagrange multipliers 1 x 3n 3n x 3n 3nx13nx1 3nx13nx1 1
1 1 x m Mx1Mx1 Mx1Mx1 xx--x
Discretization Discretization of Continuous functions SHAKE and
RATTLE Widely-used family of discretization Extend the Verlet
scheme [Hairer et al. 2002] Consider a constraint force direction
at the beginning of the time step where and are the position and
velocity of the mesh at time replacewith
Slide 22
Discretization - SHAKE
Slide 23
Discretization SHAKE (Failure) : : Derivative of C There might
be solutions even in the case that the matrix cannot be calculated
The gradient directions The magnitudes of the moves are
proportional to the entries of
Slide 24
Discretization SHAKE (Failure) (Q1) No Solution, R(0)
nonsingular (Q3) Solutions Exist C rank deficient (Q2) Solutions
Exist is singular (Q4) Redundant constraints G rank deficient
Slide 25
Implicit Constraint Direction I Evaluating the constraint
direction, at the end of the time step Explicit : nImplicit : n+1
ICD resolves (Q1), (Q2) and (Q4). (Q3) remedied by decreasing the
time step
Slide 26
Implicit Constraint Direction II ICD time steps
ExplicitImplicit Define To obtain, we need to calculate using
Slide 27
Step and Project Problems of Solving using Newton method Many
unknowns (5n) Requires the solution of an indefinite linear system
Constraint integration in two steps (a) step forward only the
potential forces (b) enforce the constraints by projecting onto the
constraint manifold
Slide 28
Step and Project II The projected point extrimizes the
objective function The point on the constraint manifold closest to
Closest : minimum S discretization The L2 norm of the mass-weighted
displacement of the mesh as it moves from to
Slide 29
Theorem 1 : ICD SAP
Slide 30
Newtons Method An iterative method to solve non-linear function
Nonlinear system of equations
Slide 31
Fast Projection Method Solving SAP using Newtons method Each
iteration would improve upon a guess for the shortest step Fast
Projection Also uses a sequence of iterations But, it relax the
requirement of SAP The closest a close point by taking a sequence
of smallest steps
Slide 32
A Step of Fast Projection I Progressively closer approximations
to the constrained position The (j+1)th step of fast projection
extremizes the objective function Expanding the constraint to first
order =start goal tolerance
Slide 33
A Step of Fast Projection II A quadratic objective function,
whose stationary equations with respect to and are Substituting (5)
into (6), we eliminate and solve a linear system in Finally, As
with ICD/SAP, a projection step require a linear solver Fast
projections system, (7), is smaller(2 5), positive define and
sparser. (5) (6) (7)
Slide 34
Algorithm 1 Fast Projection Implementation : Step 3 requires
solving a sparse symmetric positive definite linear system PARDISO
[Schenk and Gartner 2006]
Slide 35
Fast Projection Algorithm Convergence condition Until the
maximal strain is below a threshold The constraint may be satisfied
up to a given tolerance FP finds a manifold point,, that is close
(not the closest) Referring Corollary, FP solves C=0 while it
approximates F=0 The FPs error is acceptable? First iteration of PF
and ICD/SAP is identical At the end of this first iteration :
Additional PF iteration seek C 0 Since, increments in x
are,therefore F remains in is considered acceptable in many
contexts [Baraff and Witkin 1998; Choi and Ko 2002] halt the Newton
process after a single iteration
Slide 36
Figure 3 The effect of fast projection on the residual F After
the first and last iterations of fast projection
Slide 37
Figure 4 The performance of FP against an implicit treatment of
stiff springs Ex) 1D chain resists stretching, but not bending (a)
permissible strain (b) Mesh resolution Using 80 vertices and 1%
strain, FP achieves a 25X speed up
Slide 38
Draping Cloth To compare fast projection with (1) ICD Is able
to use larger timesteps than SHAKE and still converge Each time
step is substantially more expensive than SHAKE (2) SHAKE Uses the
accelerating suggested in [Barth et al. 1994] to rebuild the matrix
once per step or when it fails to converge Extremely small
timesteps to converge, but each step is relatively small (3) The
strain limiting algorithms (Both Jacobi and Gauss-Siedel) Iterates
until strain is in the permissible range
Slide 39
Figure 5 Performance (a) Permissible strain (5041 vertices) (b)
Discretization Resolution (1% permissible strain) As the stiffness
is increased for a cloth mesh All CLM methods scale equally well,
FP is the fastest
Slide 40
Figure 6 The simulation results using FP and ICD methods
Qualitatively similar results
Slide 41
Figure 7 FP is capable of producing complex, realistic
simulation of cloth
Slide 42
Figure 8
Slide 43
Discussion Experiment focus The performance of enforcing
inextensibility using CLM Inextensibility : [Bergou et al. 2006]
assumes inextensibility to accelerate bending computation Adapting
the velocity-filtering speed, simplicity, and software modularity
No way to enforce both ideal inextensibility and ideal collision
handling Since one filter must execute before the other Both ideals
corresponds to sharp constraints combining them in a single pass
but, introducing considerable complexity and convergence challenges
Practically, the drawback does not cause artifacts in our
simulation First, we execute collision handling last Second, the
constraint is maintained at the end of the time step errors are not
accumulated
Slide 44
Conclusion The most common fabrics do not stretch visibly The
trend is to use stretching formulations based on penalty-springs
(a) relaxing realism by allowing 10% strain (b) adapting simple
iterative strain and strain-rate algorithms that have poor
convergence behavior Constrained Lagrangian Mechanics
Straightforward filter Good convergence behavior Enforcing
inextensibility One immediate and pragmatic approach to fast and
realistic fabric simulation