bouaziz14projective

bouaziz14projective
Projective Dynamics: Fusing Constraint Projections for Fast Simulation
Sofien Bouaziz∗
EPFL
Sebastian Martin†
VM Research
Tiantian Liu‡
University of Pennsylvania
Ladislav Kavan§
University of Pennsylvania
Mark Pauly¶
EPFL
Figure 1: We propose a new “projection-based” implicit Euler integrator that supports a large variety of geometric constraints in a single
physical simulation framework. In this example, all the elements including building, grass, tree, and clothes (49k DoFs, 43k constraints), are
simulated at 3.1ms/iteration using 10 iterations per frame (see also accompanying video).
Abstract
1
We present a new method for implicit time integration of physical
systems. Our approach builds a bridge between nodal Finite Element
methods and Position Based Dynamics, leading to a simple, efficient,
robust, yet accurate solver that supports many different types of
constraints. We propose specially designed energy potentials that
can be solved efficiently using an alternating optimization approach.
Inspired by continuum mechanics, we derive a set of continuumbased potentials that can be efficiently incorporated within our solver.
We demonstrate the generality and robustness of our approach in
many different applications ranging from the simulation of solids,
cloths, and shells, to example-based simulation. Comparisons to
Newton-based and Position Based Dynamics solvers highlight the
benefits of our formulation.
Physics-based simulation of deformable material has become an indispensable tool in many areas of computer graphics. Virtual worlds,
and more recently character animations, incorporate sophisticated
simulations to greatly enhance visual experience, e.g., by simulating
muscles, fat, hair, clothing, or vegetation. These models are often based on finite element discretizations of continuum-mechanics
formulations, allowing highly accurate simulation of complex nonlinear materials.
CR Categories: I.3.7 [Computer Graphics]: Three-Dimensional
Graphics—Animation; I.6.8 [Simulation and Modeling]: Types of
Simulation—Animation
Keywords: physics-based animation, implicit Euler method, position based dynamics, continuum mechanics.
Links:
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∗ [email protected][email protected][email protected]
§ [email protected][email protected]
Introduction
Besides realism and accuracy, a number of other criteria are also
important in computer graphics applications. By generality we mean
the ability to simulate a large spectrum of behaviors, such as different types of geometries (solids, shells, rods), different material
properties, or even art-directable extensions to classic physics-based
simulation. Robustness refers to the capability to adequately handle
difficult configurations, including large deformations, degenerate
geometries, and large time steps. Robustness is especially important
in real-time applications where there is no “second chance” to re-run
a simulation, such as in computer games or medical training simulators. The simplicity of a solver is often important for its practical
relevance. Building on simple, easily understandable concepts –
and the resulting lightweight codebases – eases the maintenance of
simulators and makes them adaptable to specific application needs.
Performance is a critical enabling criterion for realtime applications.
However, performance is no less important in offline simulations,
where the turnaround time for testing new scenes and simulation
parameters should be minimized.
Current continuum mechanics approaches often have unfavorable
trade-offs between these criteria for certain computer graphics applications, which led to the development of alternative methods, such
as Position Based Dynamics (PBD). Due to its generality, simplicity,
robustness, and efficiency, PBD is now implemented in a wide range
of high-end products including PhysX, Havok Cloth, Maya nCloth,
and Bullet. While predominantly used in realtime applications, PBD
is also often used in offline simulation. However, the desirable qualities of PBD come at the cost of limited accuracy, because PBD is
not rigorously derived from continuum mechanical principles.
We propose a new implicit integration solver that bridges the gap
between continuum mechanics and PBD. The key idea is to introduce energy potentials with a specific structure. More precisely, our
potentials consist of a convex quadratic distance measure from a constraint. The constraints are general nonlinear functions that express
the desired state of an element, for example, that the volume of a
tetrahedron must remain within given bounds. The distance measure
quantifies how much individual constraints are violated in a given
deformed configuration. While our solver can handle arbitrary geometric constraints, we propose a specific set of constraints derived
from continuous deformation energies. These continuum-based
constraints are very practical because they considerably simplify
parameter tuning especially when dealing with meshes of different
resolutions and non-uniform tessellation.
The main advantage of our constraint-based potentials is that their
structure enables an efficient local/global optimization (block coordinate descent). Specifically, the local step consists of projecting
every element onto the constraint manifold, i.e., solving a small
nonlinear problem per element. The global step combines the results
of individuals projections, finding a compromise between all of the
individual constraints, while also taking into account global effects
such as inertia and external forces.
The local/global approach allows us to formulate an implicit integration solver that is guaranteed to weakly decrease the energy in every
iteration without requiring any specific precautions. This contrasts
with classical Newton’s method which requires line search strategies
and safeguards against singular or indefinite Hessians to guarantee
robustness. Furthermore, with a fixed set of constraints, we can
pre-factor the linear system of the global step, which greatly reduces
computation time. The local steps consists of small independent
optimization problems, which can be all executed in parallel.
To our knowledge, our method is the first to apply local/global optimization to simulate general dynamical systems. We demonstrate
that this solution provides a robust and efficient approach to implicit
integration, often significantly outperforming the classical Newton
method. The connection between PBD and our solver reveals new
insights on how PBD relates to traditional approaches based on finite
element methods and Newtonian mechanics.
2
Related Work
Since the pioneering work of Terzopulous and colleagues [1987],
models derived from continuum mechanics play an important role in
physics-based animation. The basic principle is that the resistance of
an elastic object to deformations is quantified using an elastic potential energy – a scalar function whose variational derivative leads to
the elastic force [Sifakis and Barbic 2012]. Unfortunately, the elastic
forces are usually non-linear even for basic material models, which
complicates time integration of the resulting equations of motion.
The simplest time integration schemes used in computer graphics
are explicit and very fragile to large time steps [Press et al. 2007].
Implicit Euler methods significantly improve robustness [Baraff
and Witkin 1998], but at the cost of solving a system of non-linear
equations at every step. As shown in [Martin et al. 2011], this can
be equivalently formulated as a non-convex optimization problem
that operates directly on elastic potentials instead of forces. One
of the main shortcomings of implicit Euler integration is artificial
numerical damping. This motivated the development of symplectic
integrators [Hairer et al. 2002; Kharevych et al. 2006] and mixed
implicit-explicit methods (IMEX) [Bridson et al. 2003; Stern and
Grinspun 2009], featuring better energy conservation properties. Another approach is energy budgeting [Su et al. 2013] which enforces
energy conservation explicitly. However, implicit Euler integration continues to be one of the popular choices in applications of
physics-based animation where robustness is an important criterion
and numerical damping is not a major concern. Our solver is derived
from the variational form of implicit Euler integration [Martin et al.
2011] as it gives an intuitive way of thinking about time integration in our framework – simply by adding another constraint to the
system. This further allows us to draw connections between PBD
and the implicit Euler integration scheme and results in a robust and
efficient approach that is stable under large time steps.
Regardless of the particular flavor and formulation of implicit integration, Newton’s method remains the computational workhorse
for solving the system of non-linear equations. However, its robust
implementation requires precautions such as conservative line search
procedures and safeguards against indefinite Hessians [Boyd and
Vandenberghe 2004]. From a performance standpoint, a serious
drawback of Newton’s method is the fact that the Hessian matrix
and the gradient change at every iteration. Quasi-Newton methods
therefore employ approximate Hessians, trading faster linear system
solves for suboptimal descent directions (and therefore slower convergence) as demonstrated by [Desbrun et al. 1999; Hahn et al. 2012].
A similar strategy, explored in the context of co-rotated elasticity,
is to use carefully scheduled updates of sparse Cholesky factorization [Hecht et al. 2012]. Recently, Liu and colleagues [2013] presented a method for efficient implicit time integration of mass-spring
systems by introducing auxiliary variables that enable alternating
local/global optimization. This approach, also known as block coordinate descent, has been previously used with great success in
geometry processing [Sorkine and Alexa 2007; Bouaziz et al. 2012].
We also employ local/global alternation in our approach, but contrary to [Liu et al. 2013], which is limited to mass-spring systems
and assumes only linear springs (Hooke’s law), we show how to
generalize this concept employing projection onto constraint sets to
simulate general nodal dynamical systems.
Our constraint-based formulation bears some similarity with recent
non-traditional approaches based on constraint projection. The idea
of constraint projection is central to the Nucleus system [Stam 2009]
and Position Based Dynamics [Müller et al. 2007; Bender et al.
2013]. In contrast to our solution, these methods do not treat the
constraints in a global manner, but iteratively project onto them in a
(non-linear) Gauss-Seidel-like fashion [Müller et al. 2007]. While
the resulting algorithm is very easy to implement, this approach has
a number of shortcomings: the Gauss-Seidel optimization does not
converge very rapidly, the material stiffness depends on the number
of iterations, and the result depends on the traversal order. In contrast, our method uses constraints to formulate elastic potentials that
are rigorously combined with inertial terms as dictated by Newton’s
laws of motion. Our solver first computes all constraint projections
separately and then finds the best compromise between them, which
makes the solution independent of the order of constraints. To obtain faster convergence, constraints are expressed using differential
coordinates, which often yields satisfactory results after just a few
iterations. Furthermore, our solver converges to a true implicit Euler solution with our elastic energy, in contrast to Position Based
Dynamics which converges to completely inelastic behavior.
Another closely related concept is shape matching [Müller et al.
2005; Rivers and James 2007] where, in contrast to our method,
constraint projections are used to directly build elastic forces instead
of potentials to simulate deformable objects. Constraint projections
were also used in strain limiting [Provot 1995; Goldenthal et al.
2007; Thomaszewski et al. 2009; Wang et al. 2010; Narain et al.
2012] not as a standalone simulation technique but rather as a way
to improve handling of stiff systems with standard time integration
methods. In our approach we can also perform strain limiting but it
is directly included in the implicit solver.
3
Continuum Mechanics View
Ψ(E(·))
In this section we introduce the special structure of our potentials
that form the basis of our method. We start with the implicit time
integration of FEM-discretized elastic models.
q
p
d(q, p)
E(p) = 0
3.1
Implicit Euler Solver
Let us briefly review the variational form of implicit Euler integration [Martin et al. 2011]. We assume a mesh consisting of m vertices
with positions q ∈ Rm×3 and velocities v ∈ Rm×3 . The system
evolves in time according to Newton’s laws of motion through a discrete set of time samples t1 , t2 , . . .. At time tn , the system is defined
as {qn , vn }. The sum of the external forces is defined as fext and
the sum of internal forces as fint . WeP
consider position dependent
internal forces such that fint (q) = − i ∇Wi (q), where Wi (q) is
a scalar potential energy function. Implicit Euler time integration
results in the following update rule:
qn+1 = qn + hvn+1
(1)
−1
(2)
vn+1 = vn + hM
(fint (qn+1 ) + fext )
where M is the mass-matrix and h represents the simulation step
size. Note that fext and M are held constant for any given time step.
Using these equations we can derive
M(qn+1 − qn − hvn ) = h2 (fint (qn+1 ) + fext ).
min
X
1
1
kM 2 (qn+1 − sn )k2F +
Wi (qn+1 ),
2h2
i
We
introduce potential functions W that make use of an auxiliary variable p as
Decoupling distance measure and constraint manifold.
(4)
where sn = qn + hvn + h2 M−1 fext and k.kF denotes the Frobenius norm. Intuitively, this minimization problem describes the
compromise between the momentum potential
1
1
kM 2 (qn+1 − sn )k2F ,
2h2
(5)
which states that the solution should follow its momentum (plus
external forces), and the elastic potential, that requires the solution
to minimize the elastic deformation. The corresponding weighting
terms, i.e., the mass distribution in M, the time step h and the material stiffness of W , determine which potential has more importance
in this balance. Furthermore, according to Noether’s theorem, linear and angular momenta are always conserved when the elastic
potential is rigid motion invariant.
The minimization of Equation 4 is commonly performed using careful implementations of Newton’s method [Martin et al. 2011]. However, this is quite costly because at each iteration a different linear
system needs to be solved, as the Hessian changes from one iteration
to the next. To simplify notation, we will drop below the subscript
in qn+1 and just use q.
3.2
Nonlinear Elasticity
We analyze the classical form of FEM-based nonlinear elastic energies to reveal how we can restrict the elastic potentials in Equation 4
to a structure that will allow deriving our novel solver.
In nonlinear continuum mechanics
the deformation from a rest state is measured using a discrete, elemental strain E(q), e.g., the quadratic Green’s strain [Irving et al.
2004]. Numerous elastic potentials used in practice are formulated
Nonlinear elastic potentials.
as a function of the strain using a (often nonlinear) material model
Ψ(·), resulting in elastic potentials W (q) = Ψ(E(q)). From a
geometric point of view, we can observe that E(q) = 0 defines a
constraint manifold of all possible undeformed configurations, while
Ψ(E(q)) measures how far the deformed configuration is from this
manifold (level sets in Figure 2). Our key observation is that these
two concepts can be decoupled; the distance metric does not have to
be a complicated nonlinear function because the nonlinearities are
already captured by the constraint manifold.
(3)
This system can be converted to an optimization problem
qn+1
Figure 2: The function Ψ(E(·)) defines both the constraint manifold
E(·) = 0 as its zero level set and the elastic potential given by its
isolines. By introducing a projection variable p in the manifold,
we can decouple the manifold definition from the elastic potential,
modeled as the distance function d(q, p).
W (q, p) = d(q, p) + δE (p).
(6)
Here, δE (p) is an indicator function that evaluates to zero if E(p) =
0 and to +∞ otherwise, and formalizes the requirement that p
should lie on the constraint manifold. The function d(q, p) then
measures a distance between q and p. Minimizing Equation 6 over
p corresponds to a projection of q onto the constraint manifold, as
illustrated in Figure 2. An elastic potential analogous to Ψ(E(q))
can therefore be defined as W̃ (q) = minp W (q, p).
Quadratic distance measures. With this separation in mind, we
can build a solver that alternates between distance minimization and
projection. An important advantage of this formulation is that the
distance measure can be freely chosen. The constraint nonlinearity
(also known as geometric nonlinearity) is already taken care of by
the projection on the constraint set, so the distance metric can be
kept simple, trading general material nonlinearity against efficiency
and robustness. Specifically, we consider distance metrics leading
to the following potentials:
w
W (q, p) = kAq − Bpk2F + δC (p),
(7)
2
where A and B are constant matrices and w is a nonnegative weight.
The distance to the constraint set is thus modeled by a quadratic
function in q and p, which allows us to deploy an efficient solver.
Moreover, we are not restricted to Green’s strains but can use any
constraint definition C(q) = 0 for the set of desired configurations [Baraff and Witkin 1998], e.g., describing desired bending
angles between triangles, goal volumes for tetrahedrons, or boundary conditions, as discussed below.
3.3
Projective Implicit Euler Solver
Using simplified potentials as given in Equation 7, we can reformulate the implicit integration defined in Equation 4 as the minimization
Algorithm 1: Projective Implicit Euler Solver
sn = qn + hvn + h2 M−1 fext
qn+1 = sn
loop solverIteration times
forall the constraints i do
pi = P rojectOnConstraintSet(Ci , qn+1 )
end
qn+1 = SolveLinearSystem(sn , p1 , p2 , p3 , . . .)
end
vn+1 = (qn+1 − qn )/h
1
2
3
4
5
6
7
8
9
X wi
1
1
kAi Si q − Bi pi k2F + δCi (pi )
kM 2 (q − sn )k2F +
2
2h
2
i
(8)
over q and the auxiliary variables pi , where Si is a constant selection
matrix that selects the vertices involved in the ith constraint. We
minimize Equation 8 using a local/global alternating minimization
technique.
Local solve. First, we minimize Equation 8 over the auxiliary
variables keeping the positions fixed. Since each constraint has its
own set of auxiliary variables pi , the minimization can be performed
independently for each constraint as
pi
wi
kAi Si q − Bi pi k2F + δCi (pi ),
2
(9)
which allows massive parallelization of the local step. We will
discuss specific constraint types in Section 5.
Second, we minimize Equation 8 over the positions,
keeping the auxiliary variables fixed. Since Equation 8 is quadratic
in the unknowns q, we can minimize it with a single linear solve.
Requiring that the gradient vanishes at the critical point leads to the
linear system
Global solve.
(
Ci
Gauss-Seidel
of
min
Cj
Cj
Ci
X
M X
M
+
wi STi ATi Ai Si )q = 2 sn +
wi STi ATi Bi pi .
2
h
h
i
i
(10)
The system matrix is constant as long as the constraints are not
changing and therefore can be prefactored at initialization, allowing
for very efficient global solves. The right hand side requires recomputation in each iteration after the projection variables have been
updated in the local step. Note that the objective is bounded below
and that both local and global steps are guaranteed to weakly decrease it, even for non-convex sets. Consequently, the optimization
converges, making safeguards unnecessary.
We summarize our optimization procedure in Algorithm 1. On line 2 we warm start the optimization using the momentum estimate sn . We observe that this is favorable when using only
few solver iterations, leading to less damped systems than when
using the last time step’s solution as starting point. After solving
multiple local/global iterations the velocities are updated in line 9.
Jacobi
Figure 3: Gauss-Seidel vs. Jacobi. The Gauss-Seidel algorithm
used in PBD consecutively projects the current estimate on each
constraint set (Ci and Cj in this case). If there is no feasible
solution, i.e., the constraint sets do not overlap, the Gauss-Seidel
algorithm will oscillate between the different constraints (between
the two red points). On the contrary, the Jacobi algorithm projects
the current estimate on each constraint set in parallel (green points)
and reaches a consensus in a second step. This allows the Jacobi
algorithm to converge (red point).
linear system to solve. However, this choice corresponds to working
directly with absolute positions, which results in a poor convergence
rate because changes propagate slowly through the (usually locally)
coupled points [Bouaziz et al. 2012].
The convergence can be greatly improved if we make use of the fact
that internal physical constraints are translation invariant (i.e., applying a common translation to all involved points in the constraints
does not change the values of the constraints). In this case, we can
choose Ai = Bi as differential coordinate matrices (global translation in their null space). Various such matrices can be used, for
example one can subtract the mean [Bouaziz et al. 2012] or simply
one of the vertices involved in the constraint [Liu et al. 2013]. Note
that the choice of Ai and Bi only impacts the numerical solution
procedure and does not affect the conservation of momentum.
Using such differential coordinates greatly improves the convergence speed of the resulting local/global solver [Bouaziz et al. 2012].
However, without further precautions, the resulting behavior is tessellation and resolution dependent. We show in Section 5 that in
certain cases the Ai and Bi matrices can be derived from continuum
formulations in order to avoid these shortcomings.
4
Position Based Dynamics View
While [Liu et al. 2013] hint at the similarity between general variational implicit Euler and PBD, in this section, we derive the exact
relationship not just between implicit Euler and PBD, but also between the local/global formulation and PBD, for general constraints.
This analysis highlights the close connections of PBD to our solver,
but also identifies fundamental differences that explain the higher
accuracy of results obtained with our approach.
Algorithm.
Choice of A and B.
If we choose Ai = Bi = I, Equation 7
measures the squared Euclidean distance from Si q to its closest
point on the constraint set. With diagonal matrices, the Hessian of
the global solve ends up being diagonal as well, leading to a trivial
4.1
Gauss-Seidel Solver
A classical PBD solver [Müller et al. 2007] performs three steps.
In the first step, the positions are initialized by an explicit Euler
step, ignoring internal forces. In the second step, the positions are
updated by projecting the current configuration consecutively on
each constraint set respecting the mass weighting. In the last step,
the velocities are updated as vn+1 = (qn+1 − qn )/h.
We can show that the constraint resolution strategy of PBD actually
12ms/frame
61ms/frame
121ms/frame
240ms/frame
Figure 4: For a piece of cloth with 19683 DoFs and 19360 edge
constraints, PBD exhibits different material stiffness depending on
the allowed time budget for a time step (top). Due to the additional
momentum term and the differential coordinate formulation, our simulation behaves consistently even for different number of iterations
(bottom).
implements a Gauss-Seidel type minimization on the energy
1
1X
kMi2 (Si q − pi )k2F + δCi (pi ),
2 i
(11)
using a lumped mass matrix Mi only involving the constraint’s
points. A Gauss-Seidel approach minimizes this energy by optimizing each summand sequentially, i.e., minimizing potentials of the
1
form 12 kM 2 ∆qk2F + δC (q + ∆q) where we introduce corrections
∆q = p − q to simplify the derivation. Using Lagrange multipliers
for the linearized constraint C(q) + tr(∇C(q)T ∆q) = 0, we can
define the Lagrangian
1
1
kM 2 ∆qk2F + λ C(q) + tr(∇C(q)T ∆q) .
(12)
2
Using the critical point condition w.r.t. ∆q , we find the optimal
direction ∆q = −λM−1 ∇C(q). The Lagrange multiplier λ can
then be found by requiring that the linearized constraint vanishes in
1
this direction, i.e., C(q) − λkM− 2 ∇C(q)k2F = 0, leading to the
final update
∆q = −M−1 ∇C(q)
C(q)
1
−2
kM
∇C(q)k2F
,
Figure 5: For a given continuous surface, discretizing our continuum based constraints on piecewise simplicial approximations of
different resolutions results in very similar qualitative behaviors.
momentum of the entire object, however the individual momenta
of the points are washed away the longer the optimization iterates –
contrary to finding a compromise between momentum and internal
elasticity as suggested by the Implicit Euler solver that we propose
(see Figure 4).
4.2
Jacobi Solver
In the view of Equation 11, we can solve these issues in a straightforward manner by performing two steps. First, we replace the
Gauss-Seidel by a Jacobi solver (see Figure 3) that is able to deal
with incompatible constraints. Jacobi solvers have in general slower
convergence than Gauss-Seidel solvers [Thomaszewski et al. 2009].
However, they allow the use of differential coordinate representations for faster convergence and efficient parallelization of the
constraint projections that resolve this shortcoming. Second, we
introduce the momentum constraint into the optimization to take
into account the inertia of each point. As seen in the continuum
mechanics view, to achieve a correct behavior we need to add back
the inertia of each point by integrating the momentum constraint
term defined in Equation 5
(13)
corresponding exactly to the mass-weighted update rule of
PBD [Bender et al. 2013].
Theoretically, Gauss-Seidel has good convergence,
however only for feasible constraint sets. For non-feasible sets,
lacking a global view on the optimization problem, Gauss-Seidel
will oscillate between the incompatible sets (see Figure 3). As an
example, when simulating the compression of an elastic material
with stretch constraints and boundary conditions or collisions, the
constraints can become unfeasible and thus the solution will oscillate
and not converge.
Discussion.
More severely, the same is true for the momentum estimation performed in the first step, which consists of first solving the constraint
given in Equation 5. If added as a true constraint to the optimization,
it could lead to completely incompatible constraint sets and make
convergence even worse. By solving the momentum constraint first
with the initial explicit Euler step, it is possible to maintain the linear
X wi
1
1
1
kM 2 (q − sn )k2F +
kMi2 (Si q − pi )k2F + δCi (pi ).
2
2h
2
i
(14)
The Jacobi solver then becomes a two-step optimization: In the local
step, the current solution q is first projected onto the constraints
independently by solving Equation 11 for all pi . Then, a consensus
can be reached between the different solutions by solving the global
step over q.
At this stage, we can
see how close this Jacobi solver is to our projective implicit solver
procedure presented in the last section – we recover this solver
Connection to Projective Implicit Euler.
1
by choosing Ai = Bi = Mi2 . By deriving constraints from a
continuum principle in the next section we furthermore achieve
better independence on mesh tessellation and convergence than with
the simpler mass-based weighting used in PBD (see Figure 5).
Figure 6: Starting from the same mesh, strain limiting allows simulating material that can undergo small to moderate amount of
stretching. From left to right, we use strain limits of [-10%, +10%],
[-20%, +20%] and [-30%, +30%]. Notice how the cloth stretches
and how the folds get absorbed when the limit increases.
5
Continuum-Based Constraints
Differential representations are important for our local/global solver
to improve convergence. In geometry processing the gradient and
the Laplace-Beltrami operators play an essential role in the design
of efficient and robust models. In this section we will present a set of
continuous energies based on these operators that allow the control
of the differential properties of the material under deformation. We
will show that their discretizations will have a form similar to Equation 7 that allow for correct behavior under mesh refinement and
non-uniform discretizations. The local optimization of the discrete
potentials will be discussed in Appendix A.
5.1
Strain
Strain energies are important for simulating
materials that can stretch. We first discuss 2-manifold surfaces and
then extend the results to volumes and curves. Let the undeformed
surface be a differentiable 2-manifold surface S embedded in R3 .
We define the piecewise linear coordinate function of the undeformed
surface by g : S → R3 and its deformed counterpart by f : S → R3 .
Introducing a set M of desired point-wise transformations T, we
formulate an energy measuring the change of local variation between
the deformed and the undeformed surface as
Z
w
E(f , T) =
k∇S f − T∇S gk2F + δM (T) dA,
(15)
2 S
Figure 7: Varying weight combinations of volume preservation and
strain constraints allow the simulation of different types of materials
for volumetric objects.
embedded in 2D, and similarly Xg contains the triangle edges of
the rest configuration. Note that this discrete potential has the same
form as the one in Equation 7 where A is a function of the rest
state edges and the area and B only depends on the rest state area.
Figure 6 shows the strain limiting constraint applied to a curtain
example.
Volumes and curves. This potential can be defined in a similar
way for volumes: If S is a 3-manifold simplicial complex the energy can be discretized over tetrahedrons replacing the areas of the
triangles by the volumes of the tetrahedrons and having 3 × 3 edge
matrices. Note that if we perform a 1D discretization of this energy
over a set of edges, we arrive at a model similar to the fast simulation
of mass spring models of Liu et al. [2013] where, in addition, the
edge potentials are now properly weighted by the edge length.
Continuous energy.
where ∇S is the gradient operator defined on the manifold surface S.
The choice of M determines all allowed rest configurations T∇S g.
If M is the set of rotation matrices SO(3), we are simply measuring
the local deviation from a rigid motion. In this case this energy is
identical to the deformation model presented by Chao et al. [2010].
If M is the set of matrices with bounded singular values σmin < σ <
σmax , we can also achieve isotropic strain limiting similar to Wang
et al. [2010]. This could be further extended to anisotropic material
by using reference frames following Hernandez et al. [2013].
If S is a 2-manifold simplicial complex this
energy can be discretized over triangles using a piecewise linear hat
basis [Botsch et al. 2010]. The integral is then transformed to a sum
of per triangle potentials of the form
Discrete potential.
W (q, T) =
w
2
AkXf X−1
g − TkF + δM (T),
2
(16)
where A is the triangle area, Xf = [qj − qi , qk − qi ] ∈ R2×2
contains the triangle edges of the current configuration isometrically
5.2
Area and Volume Preservation
Area and volume preservation is important for simulating incompressible materials. Using the continuous energy of Equation 15
we can define M as the set of matrices with bounded determinants σmin < det(T) < σmax , effectively enabling us to control
the amount of volume change. If σmin < 1 the modeled material
allows for compression and similarly if σmax > 1 then the material
allows for expansion. Figure 7 shows the combination of volume
preservation and strain constraints.
5.3
Example-Based
Example-based simulation allows modeling artistic elastic material
behavior by supplying a few deformation examples that the material
should follow [Martin et al. 2011; Koyama et al. 2012; Jones et al.
2013]. We use an energy comparable to Equation 15 defined on
3-manifold surfaces as
Z
w
E(f , R, w) =
k∇S f − R∇S h(w)k2F + δSO(3) (R) dV.
2 S
(17)
where h(w) is a parametrized rest shape defined
Pby the examples.
We formulate the rest shape as h(w) = g + i wi (Ri gi − g),
where the gi define the piecewise linear coordinate functions of the
examples and Ri are precomputed rotation matrices defined pointwise such that it rotates gi locally to best align with the undeformed
configuration g, similar in spirit to Koyama et al. [2012].
We can discretize this continuous energy using a piecewise linear
Figure 9: Simulation of a thin shell cylinder using increasing bending weights from left to right. When the cylinder is compressed,
buckling patterns of different frequencies appear.
Figure 8: Adding the deformation examples (top) to the simulation
using the example-based constraint allows the simulation of complex
artistic materials. In this scene, three cars collide and react in a
cartoonish manner following the prescribed examples (bottom).
If S is a 2-manifold simplicial complex Equation 20 can be discretized using a piecewise linear hat basis leading
to per vertex potentials of the form
Discrete potential.
W (q, R) =
hat basis leading to a sum of per tetrahedron potentials
w
−1 2
V kXf X−1
g − RXh (w)Xg kF + δSO(3) (R),
2
(18)
W (q, R, w) =
P
where Xh (w) = Xg + i wi (Ri Xgi − Xg ). Note that the
example weights w can either be defined locally per element or
globally, resulting in local or global coupling of the deformation,
respectively. An example of three colliding cars using this constraint
can be found in Figure 8.
5.4
Bending
Thin shells and thin plates are commonly
simulated using a bending energy based on dihedral angles across
edges [Grinspun et al. 2003]. More recently, efficient models for
bending of inextensible surfaces relating the Laplace-Beltrami operator to the mean curvature normal have been presented [Bergou et al.
2006; Garg et al. 2007]. We introduce a bending energy measuring
the squared difference of absolute mean curvatures
Continuous energy.
E(f ) =
w
2
Z
(|Hf | − |Hg |)2 dA,
(19)
S
where Hf and Hg are the mean curvature functions of the deformed
and undeformed surface, respectively. For an isometric deformation
(inextensible surface) we can then rewrite the energy using auxiliary
rotation matrices as
Z
w
E(f , R) =
k∆S f − R∆S gk22 + δSO(3) (R) dA, (20)
2 S
where ∆S is the Laplace-Beltrami operator defined on the manifold
surface S. This is because the mean curvature vector is equal to the
surface’s Laplace-Beltrami operator applied to the coordinate function. For an isometric deformation the Laplace-Beltrami operator
does not change and therefore can be defined on the undeformed
surface. Please notice how similar Equation 20 is to Equation 15
replacing the gradient by the Laplace-Beltrami. It could therefore be
interesting to apply the strain limiting and example-based concepts
to the bending energy as well.
w
AkXf c − RXg ck22 + δSO(3) (R),
2
(21)
where A is the Voronoi area of the vertex, and Xf and Xg contain
the one-ring edges of the vertex for the current configuration and
for the rest configuration, respectively. The vector c stores the
common cotangent weights divided by the Voronoi area [Botsch
et al. 2010]. An example of the bending constraint can be found in
Figure 9. As can be seen in the appendix this bending constraint
allows for a very efficient local solve as it can be implemented just as
a simple normalization of the mean curvature vector of the deformed
configuration.
6
Discrete Constraints
The constraints derived from continuous energies presented in the
previous section allow modeling a large variety of elastic bodies.
For practical animation systems additional constraints are equally
important. We model these directly as discrete constraints.
As seen earlier, individual DoFs can be
directly constrained by simply choosing Ai = Bi = I in Equation 7.
Dirichlet boundary conditions can then be realized by defining the
constraint set as the desired goal positions, in order to fix objects or
create interactive handles.
Positional constraints.
Handling collisions in an implicit manner fits naturally into our general solver and allows respecting the equilibrium
of momentum and internal constraints during the collision resolution. When detecting a collision, we dynamically add new unilateral
plane constraints. As for positional constraints, we again choose
Ai = Bi = I in Equation 7. For a colliding point qc we first
find the closest surface point b with normal n, defining a collision
plane, such that the constraint set C is defined by the half space
nT (q − b) ≥ 0. The projection into this half space in the local
step is trivial as it is either a plane projection or the identity map.
Note that defining the collision constraint unilaterally allows us to
overcome the commonly known sticking problems in implicit collisions handling. Similar to PBD, we handle friction and restitution
by changing the velocities of colliding vertices when updating velocities. A simple damping model can also be implemented by filtering
velocities [Müller et al. 2007].
Collisions.
General types of geometric constraints, as for
example bending constraints using hinge angles [Bender et al. 2013],
More constraints.
Figure 10: Even under extreme wind forces our projective implicit
solver remains stable. The solver weakly decreases the energy at
each iteration making any safeguards unnecessary (top). The pirate
flag is torn by the wind in real-time using dynamic updates of the
constraints (bottom).
can be easily incorporated into our solver. The local solve can be
performed in a general manner by minimizing Equation 7 over the
auxiliary variables. For many geometric constraints closed-form
solutions for this minimization can be found [Bouaziz et al. 2012].
If no closed-form solutions exist, the optimization can be solved
using sequential quadratic programming (SQP) [Nocedal and Wright
1
2006]. As shown in Section 4.1, for the case of A = B = M 2 one
step of SQP is similar to the PBD update [Bender et al. 2013].
7
7.1
Results
Generality
Our solver does not rely on any particular type of constraint and
is able to deal with any variety of geometric constraints within the
same setup, making it possible to simulate complex sceneries using
a single solver and to also handle object interactions robustly in
an implicit manner. In Figure 1 we show such a complex scene
with different constraint types, where the objects are also coupled
together. For example, the tree and the house are modeled with
volumetric strain constraints whereas the washing line, the cloth, the
grass and the leaves use edge strain and bending constraints.
In Figure 10 we simply use edge strain constraints to model the behavior of a pirate flag. Wind forces are added
as a function of wind direction and triangle normal. When the wind
forces are too strong, the pirate flag is torn. This is realized by removing edge constraints when the strain exceeds a certain threshold.
More complex cloths that can undergo small to moderate amounts of
stretching can also be modeled using a limit on the triangle strain in
combination with bending constraints (see Figure 6). By varying the
weights of the strain and bending constraints other types of materials
such as thin plates and thin shells can be simulated. In Figure 9 we
can see an example of a thin cylinder compressed from the top and
showing different buckling patterns due to different ratios of strain
and bending constraint stiffnesses.
Cloths and shells.
We simulate solids by
using a combination of strain and volume constraints applied on
tetrahedral meshes. As shown in Figure 7, different type of materials
Solids and exampled-based simulation.
1 iteration
10 iterations
3.3ms/frame
31.7ms/frame
20 iterations
Newton converged solution
62.6ms/frame
420ms/frame
Figure 11: This volumetric hippopotamus with 7161 DoFs and 8406
strain constraints is simulated with 1, 10, and 20 iterations of our
local/global solver. It is interesting to notice that already after 10
iterations our approach looks very similar to the converged solution
computed using Newton’s method for a fraction of the computational
cost.
can be modeled by varying the weights combining these constraints.
While we cannot model arbitrary non-linear materials, we are able to
approximate some non-linear behaviour by combining weak strain
constraints with stronger strain limiting constraints. Then, the material is soft for small deformations while becoming stiffer when
the deformation reaches the strain limit and the second constraint
becomes active (see the accompanying video) – a behavior commonly modeled by nonlinear material models. Combining different
quadratic potentials has been used earlier for collision handling
in [Harmon et al. 2009] but also suits very well our framework to
model non-linear material behavior.
Example-based simulation of volumetric meshes is also possible in
our formulation. This allows an artistic control over the physical
simulation. In Figure 8 three cars deform in a cartoonish manner
following the input examples after colliding. Similar to [Martin et al.
2011] the car surface is embedded into a volumetric mesh, which is
then deformed using our solver.
7.2
Robustness and Simplicity
One important advantage of our approach is numerical stability.
In Figure 10 we show that even under extreme forces our solver
stays robust. Similarly, our method remains reliable in situations
where the mesh elements degenerate. This can be seen during the
simulation of the balls compressed between two planes (see the
accompanying video). The only requirement of our approach is that
the mesh elements of the input model are well behaved in order
to compute the discretization of the gradient and Laplace-Beltrami
operators of the original manifold.
We illustrate the simplicity of our approach by laying out our optimization procedure in Algorithm 1. By removing line 7 and changing pi to qn+1 in line 5, we are able to completely recover the
structure of the original PBD algorithm [Müller et al. 2007]. Moreover, notice that introducing a new constraint only requires the
definition of the constraint projection used in the local solve (either
exact if known or the general approximate projection scheme given
in [Müller et al. 2007] if not) and the definition of suitable quadratic
distance metrics (matrices Ai and Bi ).
0
10
−5
10
−10
Local Global
Newton
relative error
relative error
10
0
10
10
1
number of iterations
10
2
10
0
10
−5
10
−10
3
10
Local Global
Newton with Cholesky
Newton with PCG
0
0.1
0.2
0.3
0.4
0.5
0.6
time (second)
0.7
0.8
0.9
1
Figure 12: By comparing the decrease of the relative error with respect to the iteration count, we observe that Newton’s method converges
faster than our local/global approach. However, this does not reflect the cost of each iteration as for each Newton iteration a changing linear
system needs to be solved. Looking at the decrease of the relative error with respect to the computation time, we notice that our local/global
approach exhibits a better performance up to a relative error of 10−10 making our approach particularly attractive for interactive applications.
In these curves, the relative error is defined as the normalized error relative to the optimal solution ((qi ) − (q∗ ))/((q0 ) − (q∗ )) and measured
for a twisting bar example (left) with 4290 DoFs and 4099 tetrahedral strain constraints.
7.3
Accuracy and Performance
In Figure 12 we compare the performance of our local/global solver to Newton’s method when solving
the discretization of Equation 15 for M = SO(3) similar to [Chao
et al. 2010]. As shown in Figure 12 the local/global approach converges slower in number of iterations. This is perfectly logical as
Newton’s method exhibits quadratic convergence while local/global
solvers (block coordinate descent methods) have linear convergence.
However, when looking at the convergence in terms of computational
time, we notice that our approach is faster than Newton’s method
for interactive applications. For 1 Newton iteration approximatively
30 local/global iterations can be performed. This is due to the fact
that at each Newton iteration the Hessian needs to be recomputed
and therefore a new linear system needs to be solved.
Comparison with Newton.
Moreover, in Figure 11 and the accompanying video we observe
that with approximatively 10 iterations the simulation looks visually similar to the converged one using Newton’s method making
our scheme a better choice for realtime applications where high
accuracy is not the main focus. This type of behavior has already
been observed in some of the previous local/global solvers used in
geometry processing and simulation [Myles and Zorin 2012; Liu
et al. 2013]. Note that implementing Newton’s method for the continuous energies presented in Section 5 is nontrivial as one needs to
differentiate SVD [McAdams et al. 2011] and new Hessian matrices
have to be computed in each time step. Moreover, some safeguards
need to be integrated in the optimization as the Hessian matrix may
become indefinite and a line search procedure is also needed to avoid
overshooting.
We also compared our approach to PBD using edge strain constraints. As explained in Section 4, PBD does not include the momentum constraints making the material stiffness dependent of the number of
iterations. This can be seen in the accompanying video and in Figure 4, where for different number of iterations the stiffness of the
material simulated by PBD drastically changes. This is not the case
in our approach where the material stiffness is much less dependent
of the number of iterations.
behavior under different piecewise simplicial approximations of
the same underlying surface. This is an important property for
computer graphics applications and interactive environments where
mesh resolutions can frequently change during development and
where geometric levels of detail are widely employed to increase
performance. The lack of convergence of PBD approaches makes
it difficult to handle geometric level of detail properly due to the
dependence of the material behavior on the underlying meshing and
of the number of iterations [Häggström 2009].
8
Implementation
The complete framework presented in this paper is implemented in
C++. We use OpenMP to parallelize the local step and we solve
the global step in parallel for the x, y and z coordinates by prefactorizing the linear system using sparse Cholesky factorization and
performing three times back-substitution in parallel. Dynamic constraints are handled by rank updates and downdates of the linear
system. The Eigen library (eigen.tuxfamily.org) is used for dense
and sparse linear algebra. We use either the standard simplicial mass
discretization [Hughes 2000] or its lumped version to compute the
mass matrix without any noticeable difference.
For simulation of medium sized models (< 30K constraints and < 30K DoFs), 5-10 iterations are usually sufficient. At
1-6ms per iteration, this enables realtime simulation on a MacBook
Pro 2.7 GHz Intel Quad-core i7 with 16GB of memory. Statistics on
timings and meshes can be found in the accompanying video. Moreover, the accompanying application demonstrates the performance
on multiple examples.
Timing.
Comparison with Position Based Dynamics.
In Section 5 we presented a set of new
constraints derived from continuous energies. As shown in Figure 5,
these new constraints allow our solver to maintain the deformation
Meshing independence.
9
Limitations and Future Work
While our implicit Euler solver is efficient and robust, it exhibits
implicit damping. In the near future we plan to extend our approach
to symplectic integrators [Kharevych et al. 2006] which provide
better energy behavior. Damping can also be observed when the
optimization is terminated early. This is due to the fact that external
forces may not be able to propagate fully through the mesh if the optimization is not run for enough iterations. This effect is accentuated
in large meshes as more iterations are needed until convergence. As
a future work, we would like to improve the speed of our solver by
implementing a GPU version of our code and focus on topological
changes (cutting, fracturing) that result in dynamically changing constraints. While the local steps remain simple to solve on the GPU,
the global system is changing, making it even more involved to solve
efficiently. This problem becomes even more accentuated if we want
to extend our approach to fluid simulation similar to [Macklin and
Müller 2013] where neighborhood relations always change.
B ENDER , J., M ÜLLER , M., OTADUY, M. A., AND T ESCHNER , M.
2013. Position-based methods for the simulation of solid objects
in computer graphics. In EG State of the Art Reports.
We are trading hard constraints for simplicity and efficiency. Treating all constraints in a soft manner allows us to handle them in a
unified and effective manner. However, in certain situations, being
able to enforce hard constraints, such as for collision handling or
boundary conditions, would be advantageous. Hard constraints can
still be approximated by increasing the weight of the constraints.
However, this can degrade the conditioning of the linear system and
can result in locking artifacts.
B OTSCH , M., KOBBELT, L., PAULY, M., A LLIEZ , P.,
B. 2010. Polygon Mesh Processing. AK Peters.
Another interesting area of further research is enlarging our set
of constraints. One direction we want to explore is modeling more
complex deformation behaviors such as in anisotropic and non-linear
materials. Furthermore, it would also be attractive to integrate rigid
bodies into the same simulation framework.
10
Conclusion
We introduce a new implicit constraint-based solver for real-time
simulation. Our approach is based on an abstract, constraint-based
description of the physical system making our method general in its
use to simulate a large variety of different geometries and materials.
To solve the constraint problem, we apply a local/global solver that
is guaranteed to weakly decrease the energy making any safeguards
unnecessary and giving us robustness. Our simple constraint-based
formulation only requires the definition of a projection operator for
a given constraints (local solve), making it very easy to implement
and to introduce new models into the solver. Furthermore, the global
solve only requires solving a linear system, where the system matrix
is constant if the number of constraints is kept fixed, leading to
efficient computation. Due to the independence of the local solves,
the approach is also very well suited for parallelism, further boosting
performance. We derive a broad set of constraints directly from
continuous energies using proper discretization that make the solver
robust to non-uniform meshing with different resolutions. With
these qualities in mind we believe that our approach strikes the right
balance between the simplicity, generality, robustness and performance of position-based simulations with the rigor and accuracy of
continuum mechanics. We believe this makes our method suitable
for many applications in both realtime and offline simulation in
computer graphics.
We thank James O’Brien, Adam Bargteil,
Basil Fierz and Bernhard Thomaszewski for the insightful discussions and the reviewers for their valuable comments. We are grateful to luismigabril for providing the cartoon house model and to
Daniel Grauer for modeling the teaser scene. We also thank Yuliy
Schwartzburg for his narration of the accompanying video. This
research is supported by the Swiss National Science Foundation
grant 20PA21L 129607, by the European Research Council under
the European Unions Seventh Framework Programme (FP/20072013)/ERC Grant Agreement n. 257453: COSYM and by the NSF
Career Award IIS-1350330.
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A
Local Solves
Strain.
When minimizing over T while keeping q fixed in the
local step
2
min kXf X−1
g − TkF + δM (T),
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T
the optimization can be reformulated as
min
kΣ − Σ∗ k2F s.t. σmin ≤ Σ∗ii ≤ σmax ,
∗
Σ
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Similar to the strain constraint the local minimization of the volume constraint can be reformulated as
Y ∗
min
kΣ − Σ∗ k2F s.t. σmin ≤
Σii ≤ σmax .
(24)
∗
Area and Volume.
Σ
i
This problem can be further transformed in
Y
min kDk22 s.t.
(Σii + Di ) = σ,
D
Q
with Σ∗ii = Σii + DiQ
and where σ = σmin when i Σ∗ii < σmin
∗
and σ = σmax when i Σii > σmax . This constrained minimization can be solved by iteratively solving a quadratic programming
problem by linearizing the constraint leading to a simple update rule
Dk+1 =
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(25)
i
where C(D) =
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modeling. In Proc. EG Symp. Geometry Processing.
(23)
T
∗ T
where Xf X−1
g = UΣV and T = UΣ V . The optimal solution can be computed as Σ∗ being the singular values Σ clamped
between σmin and σmax . For tetrahedrons, if det(Xf X−1
g ) < 0,
the last singular value is negated to avoid reflections.
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(22)
Q
∇C(Dk )T Dk − C(Dk )
∇C(Dk ),
k∇C(Dk )k22
i (Σii
(26)
+ Di ) − σ.
We solve the optimization
−1 2
min kXf X−1
g − RXh (w)Xg kF + δSO(3) (R),
R,w
(27)
using a local/global approach by minimizing over R and w iteratively. The minimization over R is solved using SVD following [Sorkine and Alexa 2007] and solving over w corresponds to
solve a simple linear system.
Bending.
The local solve of the bending constraint can be formu-
lated as
min kvf − Rvg k22 + δSO(3) (R),
R
(28)
where vf = Xf c and vg = Xg c. This corresponds in finding a
rotation R such that the rotated vector vg matches best the vector
vf . While R could be found using SVD [Sorkine and Alexa 2007]
this problem has an easier closed form solution where Rvg can be
vf kvg k2
replaced by kv
.
f k2
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