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Abstract

Simulating (elastically) deformable models that can collide with each other and with the environment remains a challenging task. The resulting contact problems can be elegantly approached using Lagrange multipliers to represent the unknown magnitude of the response forces. Typical methods construct and solve a Linear Complementarity Problem (LCP) to obtain the response forces. This requires the inverse of the generalized mass matrix, which is generally hard to obtain for deformable-body problems. In this article, we tackle such contact problems by directly solving the Mixed Linear Complementarity Problem (MLCP) and omitting the construction of an LCP matrix. Since a convex quadratic program with linear constraints is equivalent to an MLCP, we propose to use a Conjugate Residual (CR) solver as the backbone of our collision response system. By dynamically updating the set of active constraints, the MLCP with inequality constraints can be solved efficiently. We also propose a simple yet efficient preconditioner that ensures faster convergence. Finally, our approach is faster than existing methods (at the same accuracy), and it allows accurate treatment of friction.
17
Eicient and Accurate Collision Response for Elastically
Deformable Models
MICKEAL VERSCHOOR, Eindhoven University of Technology and Universidad Rey Juan Carlos
ANDREI C. JALBA, Eindhoven University of Technology
Fig. 1. A hyper-elastic Armadillo pulled through a set of (transparent) rotating cylinders at timesteps 5,000,7,500,10,000,and12,500. The timing results
of this experiment for various methods are shown later in Figure 10.
Simulating (elastically) deformable models that can collide with each other
and with the environment remains a challenging task. The resulting con-
tact problems can be elegantly approached using Lagrange multipliers to
represent the unknown magnitude of the response forces. Typical methods
construct and solve a Linear Complementarity Problem (LCP) to obtain the
response forces. This requires the inverse of the generalized mass matrix,
which is generally hard to obtain for deformable-body problems. In this
article, we tackle such contact problems by directly solving the Mixed Lin-
ear Complementarity Problem (MLCP) and omitting the construction of
an LCP matrix. Since a convex quadratic program with linear constraints is
equivalent to an MLCP, we propose to use a Conjugate Residual (CR) solver
as the backbone of our collision response system. By dynamically updat-
ing the set of active constraints, the MLCP with inequality constraints can
be solved eciently. We also propose a simple yet ecient preconditioner
that ensures faster convergence. Finally, our approach is faster than exist-
ing methods (at the same accuracy), and it allows accurate treatment of
friction.
CCS Concepts: • Computing methodologies Physical simulation;
Additional Key Words and Phrases: Collision response, conjugate residual
Authors’ addresses: M. Verschoor, Department of Mathematics and Computer Sci-
ence, Eindoven University of Technology, Eindhoven, P.O. Box 513, 5600 MB, The
Netherlands, and Department of Computer Science, Universidad Rey Juan Car-
los, Móstoles, Calle Tulipán s/n, 28933, Spain, Spain; email: mickeal.verschoor@
gmail.com; A. C. Jalba, Department of Mathematics and Computer Science, Eindhoven
University of Technology, Eindhoven, P.O. Box513, 5600 MB, The Netherlands; email:
a.c.jalba@tue.nl.
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https://doi.org/10.1145/3209887
ACM Reference format:
Mickeal Verschoor and Andrei C. Jalba. 2019. Ecient and Accurate Col-
lision Response for Elastically Deformable Models. ACMTrans.Graph.38,
2, Article 17 (March 2019), 20 pages.
https://doi.org/10.1145/3209887
1 INTRODUCTION
Physically based models are now widely used in many computer
graphics applications, such as animated movies and video games.
Typical animations consist of many (complex) objects that can in-
teract with each other and the environment. The behavior of such
objects is based on their internal dynamics, modeled through some
material simulation (e.g., rigid-body, cloth, or uid simulation).
However, the very motion of each object can also inuence the mo-
tion of other objects in the scene. Therefore, accurate treatment of
collision events dramatically improves the realism of the simula-
tion, delivering rich and visually pleasing animations. In addition
to translational and rotational motion, the motion of a deformable
object is also aected by shape changes (deformations). Since the
deformation is generally unknown, no exact collision time and
location can be directly computed. Furthermore, the collision re-
sponse inuences the dynamics of the objects and thus their mo-
tion, and ideally, it should also account for accurate friction.
Typical approaches for handling contact problems reformulate
them as velocity constraints and use Lagrange multipliers to
represent the unknown magnitudes of the contact forces. Then,
the resulting Mixed Linear Complementarity Problem (MLCP) has
to be solved, which yields a solution that agrees with both the
collision response problem and the model dynamics. The con-
strained problem is usually tackled by reformulating it as a Linear
Complementarity Problem (LCP), which typically is solved using
ACM Transactions on Graphics, Vol. 38, No. 2, Article 17. Publication date: March 2019.
17:2 M. Verschoor and A. C. Jalba
(Projected) Gauss-Seidel methods. The construction of the LCP
matrix requires the inverse of the generalized mass matrix, which
can be directly computed for rigid-body simulations. However,
for deformable bodies, this inverse is not directly available and
has to be approximated (e.g., see Otaduy et al. (2009)). Therefore,
one often seeks an approximation or solves a system of coupled
problems. Alternatively, directly solving the original MLCP
constitutes a viable approach for deformable models. Ramage and
Wathen (1994) compared the performance of the Conjugate
Residual (CR) method for solving coupled indenite problems,
stemming from nite-element discretizations of the Stokes
equations, to that of a two-level approach in which two nested
Conjugate Gradient (CG) solvers were used. Their results show
that the CR method outperforms the coupled CG approach for
such equality-constrained problems. Inspired by their results, we
shall investigate here how the CR method can be extended for e-
ciently solving contact problems involving inequality constraints.
For stable simulations of coupled rigid and deformable bodies, it
is important that all collisions are resolved completely at the end
of each timestep. However, when a collision is resolved, the result-
ing deformation may lead to new collisions or changes in contact
forces elsewhere. This often results in a state in which contacts do
not agree with the actual geometry. If such errors are not corrected,
these contacts will eventually introduce energy to the system, re-
sulting in oscillations and instabilities. Additionally, the larger the
deformation is, the more likely that such mismatches appear. Such
mismatches can occur even for rigid bodies. To minimize these er-
rors, one needs to solve non-linear contact problems.
In this article, we present a method that focuses on solving the
contact problem for deformable bodies, without the need to com-
pute the Delassus operator. Due to this, contacts can be eciently
linearized, which allows us to solve the non-linear contact prob-
lem, resulting in a guaranteed collision-free state at convergence.
The method is based on the CR method, which simultaneously
provides estimates for both deformation and contact forces. By al-
lowing constraints to change status (switch among active or in-
active states), inequality (non-penetration) constraints can be han-
dled directly. Furthermore, friction is modeled similarly, such as
by switching between static and kinetic friction constraints. By
continuously updating the sliding directions of the kinetic friction
constraints, Coulomb’s friction model is approximated. Addition-
ally, we propose a simple yet eective preconditioner that signif-
icantly reduces the number of iterations and computation time.
The method allows for large deformations and contact forces, sta-
ble stacking of (almost) rigid objects, and an accurate approxima-
tion of Coulomb’s friction cone (see Figures 1and 2, and, later,
Figure 8). Finally, our approach is faster compared to other meth-
ods, with the speedup typically increasing with the complexity of
the simulation.
1.1 Previous and Related Work
A tremendous amount of work has been done addressing physical
simulations for computer animation, including rigid-body, cloth,
deformable-object, and uid simulations. Here, we shall only give
a very brief overview of highly related methods. For background
material on contact mechanics, we refer to the work of Wriggers
(2002) and references therein.
Rigid bodies. Although penalty methods for collision response
(Barbič and James 2007; Moore and Wilhelms 1988) are relatively
fast and easy to implement, the computed penalty forces have
to compete with all other forces acting on the simulated bodies.
Therefore, the forces may fail to avoid inter-penetration. Impulse-
based methods (Bara 1989) model the response of a collision
event through the application of contact impulses. These methods
apply impulses on objects to resolve collisions but can trigger
new ones. This problem can be solved by using shock propagation
techniques (Guendelman et al. 2003). Other approaches used
in rigid-body simulations often model collision response as a
constrained optimization problem (Bara 1994; Erleben 2007;
Müller et al. 2007; Redon et al. 2002). This class of methods also in-
cludes approaches based on LCP formulations (Cottle et al. 1992),
typically used in rigid-body simulations (Bara 1996), interactive
applications (Catto 2005; Tonge et al. 2012), and cloth simulations
(Bridson et al. 2002). LCPs are frequently used in combination with
implicit time-integration schemes such that a collision-free state
is guaranteed for the next timestep (Stewart and Trinkle 1996). It
is this last property that makes such approaches very appealing
for solving the contact problem for deformable models as well.
Anitescu and Hart (2004) develop xed-point iteration algorithms
that solve convex sub-problems that are guaranteed, for small
friction coecients, to obtain the unique velocity solution of the
non-convex friction LCP. Their method converges at a linear rate.
A non-smooth non-linear CG method is presented by Silcowitz-
Hansen et al. (2010b), which combines Projected Gauss-Seidel
(PGS) with a Fletcher-Reeves CG method. Bertails-Descoubes
et al. (2011) present a method for simulating contacts between
small rods or bers. The method models exact Coulomb friction
and uses a non-smooth Newton solver, which relies on an easy
construction of the Delassus operator. Xu et al. (2014) present a
method for rigid bodies, which solves a Quadratic-Programming
(QP) problem based on contact constraints. A Singular-Value-
Decomposition (SVD) solver is used to form the (singular) Schur
complement matrix, which is then used in combination with
Cholesky decomposition and back-substitution. Friction is mod-
eled using friction anchors. Although a penalty-based method,
due to the implicit-integration scheme used, this approach can
also be interpreted as a method that solves constraints. Mazhar
et al. (2015) introduce the so-called Accelerated Projected Gradient
Descent approach to accelerate the simulation of large systems of
rigid bodies interacting through normal and frictional contact. The
method can easily be parallelized, resulting in speedups of one to
two orders of magnitude. Silcowitz-Hansen et al. (2010a)propose
a PGS method working on a subspace of the problem. At each
iteration, the method rst performs an approximation using PGS,
followed by a more accurate solve concerning a subset of active
constraints having non-clamped multipliers. A detailed overview
of numerical methods for solving LCPs is given by Erleben (2013).
Finally, an extensive overview of rigid-body simulations is given
by Bender et al. (2014). Other related projected Krylov methods,
used in rigid many-body/granular-material simulations, are found
in Heyn et al. (2012) and Renouf and Alart (2005).
ACM Transactions on Graphics, Vol. 38, No. 2, Article 17. Publication date: March 2019.
Eicient and Accurate Collision Response for Elastically Deformable Models 17:3
Fig. 2. Example simulations by our method. (a) Diicult case with many deformable objects in a moving wedge. (b) Large deformations and elastic forces
appear when the shown elastic strings are tangled; by releasing external forces, the strings are untangled. (c) Stable stacking of rigid objects, illustrating
accurate friction treatment.
Deformable models. Early work on collision handling for de-
formable models includes the method of Bara and Witkin (1992),
which models the collision response through constraints yielding
contact forces. By solving a QP problem, an optimal solution is
obtained. Although not directly used for deformable objects, Con-
straint Anticipation (Bara 1996) is a technique that transforms an
MLCP into an LCP. After solving the LCP for the unknown La-
grange multipliers, collision response impulses are applied on the
colliding models. When applied to rigid bodies, this technique is
very ecient. Otaduy et al. (2009) extend this approach for de-
formable models as Iterative Constraint Anticipation (ICA). ICA
computes the multipliers and the collision response using two
nested (Projected) Gauss-Seidel solvers that approximate both the
multipliers and velocities. Raghupathi and Faure (2006)useanAc-
tive Set approach for solving the contact response problem: Given
a set of active and inactive constraints, a QP problem is solved
using the CG method. On convergence, constraints can change
state between active and inactive. The Staggered Projections (SP)
method (Kaufman et al. 2008) computes an unconstrained velocity
that is corrected using two coupled projection steps (each solved
using a QP solver). This iterative process continues until the resid-
ual is minimized. The method of Allard et al. (2010)constructs
volume-based constraints that ensure a zero intersection volume
of two colliding objects. The resulting MLCP is solved using the
Gauss-Seidel–like method of Duriez et al. (2006). The latter ap-
proximates exact Coulomb friction and explicitly constructs the
Delassus operator (an LCP matrix), for which some approxima-
tions were made. Daviet et al. (2011) present a scalable and robust
solver for capturing Coulomb friction in large assemblies of tightly
packed bers, such as hair. The method can handle a few thousand
bers subject to tens of thousands frictional contacts at a reason-
able computational cost. Li et al. (2015) present an ecient Gra-
dient Projection method for computing contact responses by de-
coupling constraints. Unfortunately, their method cannot handle
coupled frictional constraints eciently. Works on collision detec-
tion (Teschner et al. 2004), coupling of rigid and deformable bodies
(Shinar et al. 2008), deformable bodies (Galoppo et al. 2006; Irving
et al. 2007; Pauly et al. 2004; Spillmann et al. 2007), and penalty
forces (Tang et al. 2012) are all closely related to the contact prob-
lem for deformable models.
Comparison to our method. We build on methods that involve
velocity-level constraints and compute Lagrange multipliers in
combination with implicit time integration. As seen earlier, many
methods found in rigid-body applications require the construc-
tion of an LCP, Schur complement matrix, or Delassus operator,
which relies on the availability of the inverted generalized mass
matrix. For rigid-body problems, this inverse can be obtained rel-
atively easily, whereas for Finite Element Method (FEM)-based
deformable-body problems, this is generally not immediately pos-
sible. Therefore, many of these methods are not ecient for sim-
ulating deformable bodies with a large number of degrees of free-
dom (see Section 6.2.4). Indeed, for deformable-body problems,
often coupled solvers (Kaufman et al. 2008;Otaduyetal.2009)
and/or PGS methods are used (Allard et al. 2010; Duriez et al.
2006). Otaduy et al. (2009) iteratively approximate interleaved nor-
mal and friction responses, but their method does not satisfy the
Maximum Dissipation Principle (MDP). In Kaufman et al. (2008),
this principle is guaranteed. This method iteratively solves two
coupled QP problems and has been demonstrated for rigid and
reduced-order exible multibody systems. The method of Duriez
et al. (2006) models exact Coulomb friction (satisfying the MDP)
and solves the problem using a PGS approach.
All methods just mentioned rst solve an unconstrained prob-
lem, which is corrected later, or rst compute collision impulses,
which are applied as external forces to objects. Our approach dif-
fers from all of these works in that no coupled solvers are used,
neither is an LCP, Delassus, or Schur complement matrix con-
structed or used. We approximate the Coulomb friction model us-
ing an additional kinetic friction force, aligned with the sliding ve-
locity, thus satisfying the MDP. The corresponding QP problem
(containing the generalized mass matrix of the simulation and its
ACM Transactions on Graphics, Vol. 38, No. 2, Article 17. Publication date: March 2019.
17:4 M. Verschoor and A. C. Jalba
constraints) is solved using the CR method. By changing the state
of the constraints, projections are performed that directly apply to
the global problem such that friction and normal responses aect
the internal (elastic) energy of the objects. In addition, constraints
can be added or re-linearized at convergence to guarantee that a
global, collision-free state is obtained, satisfying all constraints. To
accelerate the convergence of the method, we derive a precondi-
tioner that signicantly reduces the amount of iterations. Finally,
in this work, we create constraints per contact point, although we
could also use constraints based on a volumetric description of a
collision, similar to Allard et al. (2010).
2 BACKGROUND
2.1 Linear and Hyper-Elasticity
The deformation of a body can be described by a mapping ϕfrom
material coordinates Xto world coordinates x(i.e., x=ϕ(X)). The
stress Pat a point Xdepends solely on the deformation gradi-
ent F=ϕ
Xand is obtained through some energy density function
Ψ(F)via P(F)=Ψ
F. Since the stress is invariant under rotations,
P(UF)=UP(F)holds for any rotation U. Furthermore, if the ma-
terial is isotropic, P(FVT)=P(F)VTholds for any rotation V.By
diagonalizing Fusing SVD, Fbecomes Uˆ
FVT, so the stress can be
obtained through
P(F)=UP(ˆ
F)VT,(1)
with ˆ
Fa diagonal matrix containing the singular values of the
deformation gradient. Given the (isotropic) constitutional model
used in Ψ, a wide range of materials can be simulated. If Ψis a
quadratic function, a linear elastic model is obtained, for which
the unrotated force gradient is constant; this coincides with co-
rotational FEM (Müller and Gross 2004). Once the stress Pis ob-
tained, the nodal force fiis obtained by
fi=P(F)bi,(2)
with bithe area-weighted normals of the faces connected to node
ifor a particular volume and stress P(see Irving et al. (2004)). By
assembling the nodal forces for all volumes, net force fis obtained.
To obtain the force gradient, f
x=if
Fi
Fi
x, the gradients of the
SVD in Equation (1)withrespecttoFare required (see Sin et al.
(2011) for its derivation). When dealing with hyper-elastic mate-
rials, the computed force responses can become very large (due
to the non-linear nature of the energy model) when the compres-
sion becomes large. To robustly simulate such materials, we use
the method of Stomakhin et al. (2012), which linearly extrapolates
the energy density function after a certain amount of compression
is reached.
2.2 Dynamics and Numerical Integration
Given Newton’s second law of motion Ma =f, a rst-order Taylor
expansion of the net force fis performed, yielding
Ma =f+Δtf
x
x
t+Δtf
v
v
t=f+Δtf
xv+Δtf
va.(3)
Using a rst-order forward dierence approximation of the accel-
eration a, the following semi-implicit system is obtained,
M+ΔtCd+Δt2Kvi+1=(M+ΔtCd)vi+Δtf,(4)
Fig. 3. Vertex-face contact pair. Vertex xi
m(green) at iteration ipenetrates
a face at i+1(red). xjis the collision point on the face, wa,wb,wcits
barycentric weights, nkthe contact normal, and t1,2the tangent vectors;
dk,nrepresents the normal distance at the beginning of the timestep and
dk,tthe tangential distance.
with Mthe mass matrix of the system, vthe velocity, damping
matrix Cd=f
v, and stiness matrix K=f
x(see Bara and
Witkin (1998)). Positions xare updated using
xi+1=xi+Δtvi+1.(5)
We discretize Equation (4) in space using the FEM. Accordingly,
for each element (tetrahedron), a per-element equivalent of Equa-
tion (4) is generated, given mapping x=ϕ(X). By assembling all
of these local instances into a global one, the global version of
Equation (4) is obtained, now with M,Cd,and Kthe global mass-,
damping-, and stiness matrix, and x,v,fnow vectors represent-
ing positions, velocities, and all internal (Equation (2)) and exter-
nal forces (e.g., gravitational forces) of all nodes in the discretized
model. This system is stored using a large yet sparse matrix in the
form Avi+1=b, with AR3Nv×3Nvthe generalized mass matrix
(M+ΔtCd+Δt2K), bthe right-hand side, and Nvthe total num-
ber of vertices. Such systems are typically solved using the CG
method for the unknown velocities vi+1. Given the new velocities,
the positions are updated using Equation (5).
2.3 Non-Penetration Constraints
Two deformable bodies have collided if the signed distance be-
tween any two points on their surfaces Γ1and Γ2is negative. Such
collision events are detected by considering vertex-face and edge-
edge contact pairs. For the vertex-face case, let xi
mΓ1be a (dis-
crete) vertex on the surface of the rst object, which penetrates Γ2
in the next timestep, resulting in a collision point xjon a triangu-
lar face of Γ2(see Figure 3). The (signed) distance dk,nbetween xi
j
and xi
mis computed using
Ck(xi)=xi
mxi
j·ni
k=dk,n,(6)
with ni
kthe contact normal,kthe constraint identier, and xi
a vector containing all vertex positions at timestep i.Toavoid
penetration, Ck(xi)0 must hold for any xjand xmpair.
Clearly, this should also hold after the semi-implicit update (i.e.,
Ck(xi+Δtvi+1)0). Since Equation (4) solves for vi+1,Equa-
tion (6) is transformed into a velocity constraint using a rst-order
approximation—that is,
ΔtCk
xjΔtCk
xm·
vi+1
j
vi+1
m
≥−dk,n=ck,n,(7)
ACM Transactions on Graphics, Vol. 38, No. 2, Article 17. Publication date: March 2019.
Eicient and Accurate Collision Response for Elastically Deformable Models 17:5
with vi+1
jand vi+1
mthe new vertex velocities, and ck,nthe con-
straint constant. After collecting all resulting non-penetration con-
straints and assembling all instances of Equation (7), one obtains
Jvi+1cn,(8)
with JRNc×3Nvthe Jacobian matrix containing the derivatives
of all positional constraints, Ncthe number of non-penetration
constraints, Nvthe number of vertices, va vector representing
all vertex velocities, and cthe right-hand side of Equation (7),
for all constraints. Since xj=waxa+wbxb+wcxcand we con-
sider vertex-face and edge-edge pairs, Jcontains per contact point
asparserow-vectorjkR1×3Nv,e.g.,jk=[waΔtnT
k,wbΔtnT
k,
wcΔtnT
k,ΔtnT
k] for a vertex-face constraint Ck, with nkthe con-
tact normal (see Figure 3). At this point, we consider nkto be
constant within one timestep. In Section 3.3, an extension is pro-
posed that takes the change of the constraints into account. Fur-
thermore, self-collisions in which Γ1=Γ2and degenerate colli-
sions (vertex-vertex/vertex-edge) are treated similarly. However,
the latter will produce duplicate contacts that need special care (see
Section 3.4.1).
2.4 Constrained Velocity as a Complementarity Problem
We model collision response using Lagrange multipliers so that
normal contact forces are given by fn=JTλ, with λthe vector
of Lagrange multipliers representing the unknown magnitudes
of all contact forces. According to Signorini’s contact model (see
also Duriez et al. (2006)), a complementarity condition is fur-
ther imposed—to prevent penetration, the contact forces should
push the bodies apart, and hence λ0. Furthermore, if the non-
penetration constraint is not violated (Ck(x)>0 for some con-
straint), its corresponding multiplier should be λk=0. Hence,
(Jv cn)Tλ=(vTJTcT
n)λ=vTfncT
nλ=0, which gives the
complementarity condition.
Contact forces are simply included in the motion equations of
the model (see Section 2.2), resulting in
0AJ
T
J0
v
λb
cn1
λ0(9)
(see Cottle et al. (1992)), which is an MLCP for which the Karush-
Kuhn-Tucker (KKT) optimality conditions (Kuhn and Tucker 1950)
apply.
3 COLLISION RESPONSE THROUGH
THE CR METHOD
MLCPs can be solved by transforming them into LCPs (Bara
1996), provided that matrix Acan be easily inverted. They can also
be solved by any algorithm for strictly convex QP problems with
linear constraints. Given the MLCP from Equation (9), the corre-
sponding QP problem is
min f(v)=1
2vTAv vTb(10)
subject to Jv cn.
A sucient condition to guarantee strict convexity is for matrix
Ato be positive denite. In this case, f(v)is a strictly convex
function, and if the constraints also are linear, it has a unique
global minimizer (Boyd and Vandenberghe 2004). Since according
to Equation (4), Ais given by the linear combination of the
(positive-denite) mass and (positive semi-denite) stiness and
damping matrices, Ais positive denite. If the material simula-
tion does not produce a positive semi-denite matrix, this property
must be enforced, for example, by removing the negative eigenval-
ues from the element stiness matrices.
The quadratic function in Equation (10), but subject to Jv =cn,
is equivalent to
AJ
T
J0

B
v
λ

y
=b
cn

d
,(11)
or By =din short. Since matrix BR(3Nv+Nc)×(3Nv+Nc)is not
positive denite, the system in Equation (11) cannot be solved by
the CG method. The CR method (Luenberger 1970)isaKrylov
subspace method, similar to the (more popular) CG method, and
minimizes By d2and thus solves Equation (11). In the follow-
ing sections, we describe how the CR method is used to solve the
contact problem, including friction. Additionally, we describe how
constraints are re-linearized to obtain a collision-free state.
3.1 CR Method With Non-Penetration Constraints
Luenberger (1973) proposed a method for solving constrained non-
linear programming problems by treating inequality constraints as
equality ones that can regularly change states. The method uses a
descent step to improve the current approximation. When the prob-
lem is a QP with linear constraints, the CR method can be used to
perform these descent steps. In the following, we describe how
the method in Luenberger (1973)isappliedincaseofcontactand
friction and how it is used in combination with the CR method in
Algorithm 1.
For now, let jdenote the loop index inside the CR method. Fur-
thermore, recall that each constraint Ckis represented by a sparse
row-vector jkin Jacobian J, its corresponding Lagrange multi-
plier λk,and some constant ck,n(see Section 2.3). The comple-
mentarity condition in Equation (9) states that if λk>0istrue,
then jkv=ck,nalso holds for some constraint Ck, with vthe
global velocity. Conversely, if λk=0istrue,thenjkvck,nand
(jkvck,n)Tλk=0also hold. Therefore, constraint Ckdoes not
contribute to the computation of vand can be deactivated in line 13
or line 23. Due to this mechanism, summarized in Algorithms 2
and 3, the CR method can solve the problem in Equation (10). In
other words, each constraint is active or inactive throughout the
solver iterations, and its state is regularly updated. The method
converges when rj+1<ϵ1and Urj+1<ϵ2, with ϵ1and ϵ2a
relative and absolute tolerance, respectively, and Ua matrix that
selects the entries from rthat involve active constraints. This en-
sures a maximum constraint error less than ϵ2. The second con-
vergence criterion is used to detect local minima (see Section 4.3).
Furthermore, when the CR method converges, new constraints
can be added and inactive constraints are removed. However, the
main dierence with active set methods is that constraints’ state
changes are allowed before the solver converges. Therefore, the
problem is to decide when to activate or deactivate constraints prior
to convergence. Note that when a constraint is inactive, multipli-
cations Bpjand Byj+1are performed such that the corresponding
entries in Jare neglected. This also applies for the correspond-
ing terms of the preconditioner C(see Section 3.4 for details about
ACM Transactions on Graphics, Vol. 38, No. 2, Article 17. Publication date: March 2019.
17:6 M. Verschoor and A. C. Jalba
ALGORITHM 1: Pseudo-code of our collision handling system.
1Discretize computational domain;
2Initialize BVH and additional data structures;
3while Simulating do
4Update matrix Ausing FEM (see Equation (4));
5Find all potential collision candidates (Section 5.1);
6Linearize and check all active constraints;
7r0=dBy0;p0=C1r0,j=0;
8while Not converged do
9α=
rT
jC1BC1rj
pT
jBC1Bpj
;
10 yj+1=yj+αpj;
11 rj+1=rjαBpj;
12 if α>0then
13 EvaluateConstraints(rj+1,j,ϵ1,ϵ2);
14 if No constraints changed then
15 β=rT
j+1C1BC1rj+1
rT
jC1BC1rj;
16 pj+1=C1rj+1βpj;
17 Bpj+1=BC1rj+1βBpj;
18 else
19 rj+1=dByj+1;
20 pj+1=C1rj+1;
21 Bpj+1=BC1rj+1;
22 if rj+1<ϵ1∧Urj+1<ϵ2∨BC1rj+1<ϵ1
then
23 EvaluateConstraints(rj+1,0,ϵ1,ϵ2);
24 if No constraint states have changed then
25 Check all candidates for new collisions
(Section 5.2);
26 if No new constraints are added then
27 Re-linearize constraints (Section 3.3 and
Equation (21));
28 if No constraints are updated then
29 Advance simulation;
30 break;
31 Recompute residual (see lines 19–21);
32 j=j+1;
preconditioning). Furthermore, when the residual is recomputed
for inactive constraints, the values in dcorresponding to inactive
constraints are also neglected.
Constraint activation. The activation of a constraint Ckdepends
on ck,n,jkand vat solver iteration j. Distance dk,nis computed
using Equation (6) and stored in ck,nwhen the constraint is added
to the system (when a collision is detected), or when the constraint
is re-linearized. At the same moment, jkis computed and included
in J(see Section 5for more details). To determine when a constraint
should be activated (assuming it is inactive), the penetration depth
is evaluated regularly. If
jkvck,n0 (12)
ALGORITHM 2: Pseudo-code for constraint evaluations.
1Function EvaluateConstraints(r,i,ϵ1,ϵ2):
2interval = max(1,log2(r/ϵ1)
3);
3if imod interval == 0i== 0then
4foreach contact kdo
5if Non-penetration constraint active then
6if jkvck,n0and λk0then
7Deactivate constraint, λk=0;
8else if jkvck,n0then
9Activate constraint, set friction to kinetic friction;
10 if Non-penetration constraint active then
11 if Friction constraint in static friction mode then
12 if γk>μλkand (jk,tvck,t)Tγk>0
then
13 Switch to kinetic friction;
14 ˆ
δk=ˆ
γk,λ
k=γk/μ,γk=0;
15 else if (jk,tvck,t)Tˆ
δk0then
16 Switch to static friction;
17 γk=γ
k,γ
k=0;
18 UpdateKineticFriction();
19 if Z1Sd[Δλk,ΔγT
k+ΔγT
k]T>ϵ2then
20 Use updated constraint kand report change to
solver;
21 else
22 Discard changes for constraint k;
23 d=[(bJtγ)T,cT
n,cT
t]T/*Update RHS*/;
holds, the constraint is activated and the corresponding Lagrange
multiplier will be computed, which would eventually resolve the
collision (see line 8 of Algorithm 2).
Constraint deactivation. Since our solver allows for activation
and deactivation of constraints at any time, a criterion is needed to
decide when to deactivate them. It may seem logical to deactivate
a constraint Ckif jkvck,n0. However, since vdoes not reect
the nal velocity, this criterion cannot solely be used. When both
jkvand λkare used instead, the criterion can be formulated as
follows: if
jkvck,n0andλk0 (13)
hold, then there is no collision and the constraint is attracting the
objects—that is, it tries to enforce the equality jkv=ck,n. There-
fore, the constraint is deactivated (see line 6 of Algorithm 2). This
is thus the only state that allows deactivation of a non-penetration
constraint. For all other combinations, the constraint is kept active
since there is still a collision (jkvck,n<0) or objects are repelled
(λk>0).
Unlike Luenberger (1973), our CR-based solver for contact prob-
lems updates only the active set at dynamic intervals determined
by the residual norm (see line 2 of Algorithm 2). When the residual
is small, constraints are evaluated more frequently. Conversely, for
a larger residual norm, constraints are evaluated less frequently,
which reduces the number of residual evaluations (each requiring
a multiplication By). This makes our method more ecient than
ACM Transactions on Graphics, Vol. 38, No. 2, Article 17. Publication date: March 2019.
Eicient and Accurate Collision Response for Elastically Deformable Models 17:7
Fig. 4. A representation of the friction cone in which sliding velocity
jk,tvck,tis misaligned by angle θ(blue segment) with the friction vec-
tor δk. To address this, we compute the new friction direction δ
kthat is
more aligned with the sliding velocity (see Algorithm 3). If the magnitude
of the sliding velocity is below the solver tolerance (red area), no update
of δkis performed.
the CR solver in Luenberger (1973). Additionally, our method sup-
ports a left preconditioner (matrix C1in Algorithm 1).
3.2 Incorporating Friction Constraints
Friction is also treated as a constraint on the velocity. According to
Coulomb’s friction model, friction can be described by a cone that
bounds the friction force ffin all directions, given the magnitude
of the normal force fnand a friction coecient μ(see Figure 4).
When ff<μfn, the friction force is not bounded and the
sliding velocity is zero. When ff=μfn, the friction force
is bounded and is not large enough to keep the sliding velocity
at zero. Equation (9) can therefore be extended with additional
complementarity conditions, resulting in the following problem:
0
AJ
TJT
t0
J00 0
Jt00eT
0μe0
v
λ
γ
β
b
cn
ct
0
1
λ
γ
β
0,(14)
with JT
tγ=ffthe applied friction force, JtRNpNc×3Nvamatrix
containing tangent vectors for all friction constraints, γavectorof
unknown multipliers representing the magnitude of these vectors,
and ctcontaining all tangential distances for all constraints.
Per constraint, the tangential distances ck,t(see Figure 3)are
computed similar to Equation (6), albeit the normal vector is
replaced by a tangent one. Please note that ck,trepresents the
distance from conguration xi
mto the rst moment of impact at
xj. For persistent contacts, ck,t=0since xi
mis already in contact.
Matrix Jtusually contains, per contact point, a number Npof
scaled instances of tangent vectors, each corresponding to one
facet of the discretized friction cone. Furthermore, eand μare
matrices used to couple all multipliers in λ,γand β. The latter
representing the unknown sliding velocities. If βkis positive for
some friction constraint Ck, a positive sliding velocity exists and
the friction force is bounded, resulting in dissipation of energy. If
βkis zero, the friction force is not bounded and no sliding occurs.
Unfortunately, Equation (14) is not symmetric and non-convex,
and is therefore not equivalent to a QP problem. In the next
section, a modication is proposed that restores this relation.
ALGORITHM 3: Pseudo-code for kinetic friction update.
1Function UpdateKineticFriction():
2if Constraint kin kinetic friction mode then
3if γ
kμλkthen
4Compute Simple Moving Average λk,a;
5λ
k=λk,a;
6if 0<δT
k(jk,tvck,t)<cos(ϵθ)/*If angle θ>ϵθ*/ then
7Δδk=(jk,tvck,t)δk/*Compute dierence*/;
8δk=δk+αθΔδk/*Decreases θ*/;
9γ
k=μλ
kδk/*Update friction force*/;
3.2.1 Modified Friction Model. Given the model described in
Equation (14), we now proceed by adapting this model. Instead of
using a discretized friction cone, a continuous and bounded kinetic
friction force is modeled. This approach is summarized in Algo-
rithms 2and 3, and discussed here. Per constraint Ck, the friction
force is decomposed using only two perpendicular tangent vectors
tk,1and tk,2, which appear in jk,tR2×3Nvas part of Jt(see Fig-
ure 4). Due to this, γcontains, per constraint Ck, two Lagrange
multipliers (γk,1,γk,2)T=γk, which can be both positive and neg-
ative. For clarity of exposure, we assume for now that all friction
constraints in the system are either in the static or kinetic state. To
also keep the presentation brief, we may not make a clear distinc-
tion between model and method.
Static friction. For static friction, the sliding velocity is zero (β=
0), so Equation (14) can be written as
0
AJ
TJT
t
J00
Jt00
v
λ
γ
b
cn
ct
1
λ
|γ|
0,(15)
with |γ|the componentwise absolute values of γ,andthead-
ditional condition that 0μλ, which for a constraint Ck
means that 0 ≤γk≤μλk. The second inequality should hold
for the static case, because the friction force is not bounded by the
friction cone, and the solver computes a friction force that satises
jk,tv=ck,t.Conversely,ifboth
γk>μλkand (jk,tvck,t)Tγk>0 (16)
hold, the applied friction force is too large and acts in the direc-
tion of the sliding velocity; thus, the conditions for static friction
are violated and the constraint is switched to kinetic friction (see
line 12 of Algorithm 2).
Kinetic friction. For kinetic friction, the sliding velocity is pos-
itive (β>0), which makes Equation (14) non-symmetric. Such a
system can be solved as described in Bender et al. (2014). Alterna-
tively, this system can be symmetrized by removing slack vector
βfrom the equation. This has the advantage that solutions always
exist and can be found by Lemke’s algorithm (Stewart and Trinkle
1996). Unfortunately, dropping βresults in a violation of the MDP,
as βselects a discrete vector in Jtthat ensures this principle. In
our method, the MDP is enforced dierently.
Our approach for handling kinetic friction could be interpreted
as a Bounded Linear Complementarity Problem (BLCP) (Judice and
Pires 1994) reformulation of Equation (14), where the alignment
ACM Transactions on Graphics, Vol. 38, No. 2, Article 17. Publication date: March 2019.
17:8 M. Verschoor and A. C. Jalba
of the discretized friction cone depends on the state of the sys-
tem. The relation in the last row of Equation (14) is enforced sepa-
rately. This means that the kinetic friction force will never be over-
estimated, as it could happen in the BLCP model.
For kinetic friction, β>0, thus μλ =eγ, which for a particular
constraintCkmeans that μλk=γk. The sliding velocity is given
by jk,tvck,t=eTβ>0. The basic idea is now to estimate both
the magnitude and direction of γk.
Let γ
k=(μλ
kδk,1,μλ
kδk,2)Tbe such an estimate, with λ
kan
approximation of λk,andδk=δk,1tk,1+δk,2tk,2a unit vector,
representing the direction of the kinetic friction force in the tan-
gent plane, for example, at xj(see Figure 4). Next, we eliminate
from Equation (14) all rows and columns corresponding to βand
move JT
tγto the right-hand side so that our approximation can
beexpressedinmatrixformas
0AJ
T
J0
v
λbJT
tγ
cn1
λ0,(17)
supplemented per constraint by the condition (jk,tvck,t)Tδk>
0. This condition is violated when the sliding velocity changes di-
rection due to a too large friction force—that is, when
(jk,tvck,t)Tδk0.(18)
By then, switching the constraint to static friction, the solver will
estimate γk, which eventually satises jk,tvck,t=0(see line 15
of Algorithm 2).
Kinetic friction update. Earlier, a correct approximation of γ
k
was assumed to be available; here, we show how this vector is
updated such that the friction force and the sliding velocity are
aligned, and that the magnitude of the friction force is bounded.
Given the preceding conditions, a violation occurs if
μλkγ
k=μλ
kδk,1,μλ
kδk,2T.(19)
If this happens, Simple Moving Average (SMA) λk,ais updated by
adding the current λkto its histor y of nvalues (n=2 in our imple-
mentation; see line 4 of Algorithm 3). Next, λ
kis set to the SMA
of λk,aat line 5. To impose the MDP, the kinetic friction direction
δkand the sliding velocity have to be kept aligned. When the an-
gle θ(see Figure 4) between δkand the current sliding velocity
is larger than a given threshold (ϵθ=1.2 degrees in our imple-
mentation), the dierence Δδkbetween both vectors is computed
(line 7). Next, δkis updated using δk+αθΔδkand then normal-
ized, with αθa small step size (0.01 in our implementation; see
line 8 of Algorithm 3). Thus, δkis gradually updated to realign
it with the current sliding velocity. Finally, γ
kis approximated
(see line 9 of Algorithm 3). Using this strategy, eventually all ki-
netic friction vectors approach their nal conguration and satisfy
the conditions specied in Equation (17). By using this strategy, a
stabilization mechanism is introduced that suppresses undesired
oscillations while solving the contact problem. Finally, in a real
simulation, the system contains both static and kinetic friction
constraints, so the problem can be described in matrix form as
0
AJ
TJT
t
J00
Jt00
v
λ
γ
bJT
tγ
cn
ct
1
λ
γ
0,(20)
where, for static constraints, entries in γare set to zero, whereas
for kinetic constraints, all entries in γ,ctand those of matrix B
corresponding to Jt,JT
tare similarly set to zero.
3.3 The Non-Linear Contact Problem
The problem solved in Equation (20) assumes that Jand Jtare
constant during the timestep, which generally is not true. Hence,
C(xi+1)λmay not hold at the end of the timestep. To minimize
such errors, the complementarity conditions must also hold after
updating the geometry. To solve this non-linear problem, C(x)is
re-linearized using an interpolated approximation of xbased on
its previous and current approximations, followed by a solve of
Equation (20). Instead of interpolating C(x)globally, one can re-
linearize each individual Ck(x)—that is,
Cl+1
k(xi+1)=wCl+1
k(xi+1)+(1w)Cl
k(xi),(21)
with 0 <w<1 a weight such that the change in the contact nor-
mal is less than, say, 15 degrees, andCl
kindicates the last, Cl+1
kthe
current, and Cl+1
kthe new interpolated conguration of the con-
straining geometry. The re-linearization of the constraints results
in an updated J,Jt,and c(line 27 of Algorithm 1). Then, the method
recomputes the preconditioner and the residual vector, and contin-
ues until it converges again for the linearized problem. This pro-
cedure continues until, for each active constraint, 0 <Ck(x)<ϵ2
holds. When w=0, this procedure is exactly Newton’s method in
which the constraints are re-linearized followed by a solve of the
updated Equation (20). Since line 25 of Algorithm 1updates the set
of potential collisions, a collision-free state is eventually obtained.
Additionally, when a contact point slides o a face or edge, the
corresponding constraint must be removed from the system. The
main advantage of Algorithm 1is that updating J,Jt,and cdoes
not require an update of the LCP matrix or Delassus operator. This
allows us to eciently solve the non-linear contact problem for
large systems.
3.4 Preconditioning
In this section, we derive a preconditioner that is easy to compute,
yet it signicantly reduces the amount of iterations compared to a
simple diagonal preconditioner. A common approach for improv-
ing the convergence rate of Krylov-subspace methods is to use a
preconditioner matrix C1. Many methods exist for approximat-
ing preconditioner matrices, such as LU and Cholesky factoriza-
tions (Golub and Van Loan 1996). Unfortunately, these methods
assume that the matrix is positive denite, and therefore these
preconditioners are not useful for our indenite problem from
Equation (20).
Algorithm 1shows the pseudo-code of our preconditioned CR
solver with inequality constraints. Matrix Bin the solver repre-
sents the matrix from Equation (20)andC1the preconditioner
matrix such that C1B1. The blockwise inverse of Bis
B1=A1A1LTS1LA1(S1LA1)T
S1LA1S1,(22)
with LT=(JTJT
t)R3Nv×3Ncthe combined constraint matrix
and S=LA1LT. Next, we approximate matrices Aand Sby
ACM Transactions on Graphics, Vol. 38, No. 2, Article 17. Publication date: March 2019.
Eicient and Accurate Collision Response for Elastically Deformable Models 17:9
Ad=diag(A)and Sd=diag(ZLA1
dLT),respectively, with diag(·)
extracting the diagonal part of its matrix argument and Zadiag-
onal scaling matrix (see the following). Thus, the computations of
A1
dand S1
dbecome trivial. Additionally, S1
dLA1
dis easy to eval-
uate and yields the same sparsity structure as that of L.Basedon
this, we construct the following preconditioners:
Full:C1is obtained from Equation (22) by replacing Aand Sby
Adand Sd, respectively;
Diag:C1=A1
d0
0S1
d.
When the preconditioner is applied, only those parts correspond-
ing to active non-penetration and static friction constraints are
considered. To construct the Full preconditioner, we rst compute
the quantities S1
d,S1
dLA1
d,A1
dLTfor each constraint and use
them when evaluating the blocks of the preconditioner (see Equa-
tion (22)). Furthermore, when constraints are added, removed, or
re-linearized, these precomputed quantities are also updated.
If a positive-denite preconditioner is used, like our Diag pre-
conditioner, then the norm of the preconditioned residual will de-
crease monotonically when the states of the constraints are not
updated. As we will discuss in Section 4.2, updating constraint
states does not always result in a decrease of r(see Algorithm 1),
but the approximation will always be closer to the new optimum.
When an indenite preconditioner is used, the evolution of ris
more chaotic. More importantly, αcan also be negative, which in-
dicates that the current optimum for the preconditioned problem
lies in the opposite direction of p. Such an update does improve the
preconditioned problem (rTC1r), although generally not the
unpreconditioned problem (rTr). Since the constraint updates
are based on the unpreconditioned values xand λ,anupdateis
only performed if a step toward the unpreconditioned optimum is
taken (i.e., when α>0).
3.4.1 Preconditioner Scaling. The Full preconditioner approx-
imated by Equation (22) signicantly improves the convergence
rate of the method. Unfortunately, this preconditioner will have
large deviations in the upper-left block when Bbecomes singular
or ill conditioned due to (nearly) duplicate constraints in L. When
Lis full rank, both the inverse and pseudo-inverse of Bare equal.
By adding duplicate rows to L,Band thus Sbecome singular. In
our approximation, Sddoes not become singular. Computing then
the approximated block inverse for our preconditioner using Sd
yields a change in the upper-left block of the preconditioner com-
pared to the non-singular case. If this error becomes too large, the
CR method converges signicantly slower. In contrast, the pseudo-
inverse of Bdoes not introduce such a change.
To keep the upper-left block invariant for duplicates in L,itis
sucient to see that duplicate rows and columns can be added
together. As a result, the diagonal values in Sdwould have been
scaled up by the number of duplicates for their corresponding con-
straint. Since we do not add duplicates together, the same scaling
must be introduced to Sdusing Z, which contains, per constraint
in L,the number of similar or duplicate instances. When comput-
ing the upper-left block, duplicate constraints are added implicitly
through the multiplication LTS1
dL. Since this multiplication also
involves S1
dand thus Z1, the dierence introduced by the dupli-
cates is compensated by Z1.
Similar or identical constraints occur, for example, when an edge
intersects other edges close to one of their shared vertices. The
barycentric coordinate for the shared vertex is relatively large for
all constraints. The other involved vertices have a small contribu-
tion, which makes the constraints almost identical to each other.
For constraints involving deformable bodies, it is sucient to com-
pute, for each pair of constraints aand b, a weight based on the
dierences between their barycentric coordinates (i.e., (1−|wa,i
wb,i|), with ia vertex id). The similarity of a pair is the product of
these factors for each vertex that aand bhave in common. Even-
tually, the total scaling factor for one constraint is the sum of all
of these products. For contacts between rigid bodies (Figure 2(c)),
many constraints can act on the same degrees of freedom of the
rigid objects. Here the amount of duplicates, and thus the error in
the preconditioner, can be potentially large. In this case, it is suf-
cient to scale all constraints working on the same two bodies by
the total number of constraints between those bodies. This is in
some way similar to the method of Tonge et al. (2012).
3.5 Optimizations
The convergence rate of the method described in Algorithm 1is
improved by allowing constraints to update their status at dynamic
intervals. In other words, the closer the method is to convergence,
the smaller the intervals are between successive constraint evalua-
tions (see line 2 of Algorithm 2). Eventually, constraints are evalu-
ated every iteration when the residual is small. If we always allow
constraints to update their status every iteration, in the worst case,
the solver would need to recompute the residual vector as in line 19
every iteration, instead of using the recurrence from line 15. The
performance would then be comparable to that of a Gradient De-
scent method, which can converge slowly due to zigzagging.Con-
versely, when constraints are only allowed to update their status at
convergence, one has an Active Set method. In this case, the num-
ber of iterations between successive constraint evaluations can be
relatively large. Due to this, the solver can enter a loop in which
constraints continuously change their status. This constraint cy-
cling drastically aects solver convergence and should be avoided.
By evaluating constraints at dynamic intervals, the solver re-
sponds fast enough to situations in which constraints should
change states, although also slow enough for the solver to exploit
the conjugacy of the residual vectors. A positive side eect of this
strategy is that the number of residual re-evaluations (see line 19) is
signicantly reduced. We have run a large number of experiments
while tuning the relation between the computed interval and the
residual norm. We have found that this strategy works very well
for eciently solving contact-response problems. This approach
gave good results in all test cases but are not optimal for all cases.
Within our method, the possibility still exists that constraints
change their status a large number of times, decreasing the con-
vergence rate. To prevent this, one approach is to keep (at some
point) constraints in a xed state, as mentioned in Raghupathi and
Faure (2006). Unfortunately this results in a violation of the condi-
tions specied in Sections 3.1 and 3.2, which could introduce ad-
ditional energy to the system. However, even in such cases, the
ACM Transactions on Graphics, Vol. 38, No. 2, Article 17. Publication date: March 2019.
17:10 M. Verschoor and A. C. Jalba
current approximation is able to move toward the optimum, mean-
ing that the method is still able to converge (see Luenberger
(1973)).
When a constraint is about to change (see line 19 of Algo-
rithm 2), the change is postponed if the change in distance is less
than ϵ2. In this case, the eect of performing the update is negli-
gible. This strategy also eliminates undesired oscillations of con-
straint states (constraint cycling) that have no eect on the nal
result. This usually occurs when the solution for a particular con-
straint is located close to a discontinuity in the derivatives (i.e.,
when a vertex is about to slide or when two entities are barely
touching).
Each time constraints change states, vectors r,p,and Bp must be
recomputed to restore the conjugacy of the residual vector. Since
the computation of the residual requires a full multiplication with
matrix B, this is not very ecient. However, since each residual
vector can be decomposed as r=ru+rc, with ru=bAv the un-
constrained residual and rcthe remaining constraint residual, a
multiplication with matrix Acan therefore be completely omitted
since ruis not aected by a state change. Indeed, only rcneeds to
be recomputed, which signicantly improves performance. Please
note that this decomposition requires that other intermediate vec-
tors and computations are decomposed similarly.
4 CONVERGENCE
In this section, we discuss the convergence behavior of our
method, given the criteria described in Section 3for switching
constraints. First, we show that a minimal residual, in combina-
tion with all constraints satised, implies that a global solution is
found. Next, we show that successive updates of constraints lead to
a state in which all constraints are satised. Finally, we show that
when all constraints are satised but the residual does not reach
the minimum, its gradient will reach the minimum instead.
4.1 Global Convergence
The Delassus operator (LA1LT), where LT=(JTJT
t),de-
scribes, for each pair of constraints that share the same object in
the inverted generalized mass matrix A1, their inuence on each
other. The inverted Delassus operator can therefore be seen as the
“solution” to the contact problem and describes the inuence of
each pair of constraints onto each other, no matter how many ob-
jects and constraints are present in the chain between the two con-
straints. Hence, this solution is a global one. In our method, the De-
lassus operator is not used. However, the CR method minimizes
By d, with y=B1dits minimum. Given the blockwise in-
verse of B(see Equation (22)), minimizing By dalso implicitly
applies the Delassus operator. This is equivalent to solving both λ
and γthrough the Delassus operator, and applying these forces to
the bodies to obtain the unknown velocity v.
Within our method, the states of the constraints are actively
switched when they do not agree with the currently computed
v,γ,and λ. When some constraints change states, the method must
reset vectors r,p,and thus Bp (i.e., a state change inuences the
residual energy of the system). This reset allows the method to
search in a direction that minimizes the energy according to the
new state. Since all constraints and the deformation problem in
Fig. 5. Activation and deactivation of a constraint.
sub-matrix Acontribute to the residual, a suciently small resid-
ual should imply that the problem is solved. However, the solution
can only be the global minimizer if all constraints are in their nal
state at convergence . If not, then constraint states are changed, the
residual will be inuenced, and thus the current solution cannot
be a global minimizer for the problem. The method then continues
until all constraints are satised.
Assuming that the constraint states are known and only active
constraints are considered, a global minimum exists if the con-
straints are convex and Aand the second derivatives of the con-
straints (which vanish for linear constraints) are symmetric and
positive denite (see Luenberger (1973)). At this global minimum,
all multipliers of all active constraints are strictly positive (or not
clamped in case of friction constraints) and their distances are zero.
When inactive constraints are considered as well, their multipliers
are zero and their distances are strictly positive. Hence, the com-
plementarity conditions are satised, and the complementarity er-
ror vanishes (see Section 6.3). Let us dene such a residual that,
once minimized, solves Equation (20)—that is,
rд=
Av b+JT
A(λ)++JT
t,S(γ)μλ +JT
t,K(γ)μλ
(JAvcn)+(JIvcn)
γT(Jt,Svct)+δT(Jt,Kvct)
,(23)
with JAand JIcorresponding to all active and inactive non-
penetration constraints; similarly, Jt,Sand Jt,Kcorrespond to
all static and kinetic friction constraints, respectively. Further-
more, the multipliers and distances are clamped by operators
(·),(·)+,(·)μλ between (−∞,0),(0,),and (μλ,μλ),respec-
tively. At the global minimum, rд=0⇒r=0. Conversely,
r=0⇒rд=0 if and only if no constraints are violated. Un-
fortunately, the nal constraint states for which this relation holds
are not known apriori.
4.2 Constraint Updates
In the following, we reason about the implications of updating con-
straint states, as illustrated by Figure 5. We consider only non-
penetration and kinetic friction constraints, as static friction can
be treated similar to non-penetration constraints.
Activation of non-penetration constraints. Figure 5(a) shows
the activation of an inactive constraint Ckat iteration j.Its
ACM Transactions on Graphics, Vol. 38, No. 2, Article 17. Publication date: March 2019.
Eicient and Accurate Collision Response for Elastically Deformable Models 17:11
distance jkvjck,nis positive, thus not contributing to rдnor
to r. By minimizing rin the direction of pj,anewapproxima-
tion vj+1is obtained, which is closer to the current minimum v
(Luenberger 1970). If distance jkvj+1ck,nbecomes negative, the
constraint is activated. Due to this active constraint, the new min-
imum v∗∗ along pjnow lies on the (green) plane dened by the
constraint, which intersects the path between vjand vj+1. Since
vj+1is now closer to the new optimum v∗∗ along pjthan vj
was to the previous optimum, this constraint switch results in
an improvement of the approximation. However, the distance be-
tween vj+1and v∗∗ appears as an additional residual term via
jkvj+1ck,n=jk(vj+1v∗∗), through the rst term in the sec-
ond row of Equation (23). Depending on the used preconditioner,
rmight or might not decrease. However, since approximation
vj+1will now be closer to the new optimum, this step including
the constraint activation is an improvement.
Deactivation of non-penetration constraints. Figure 5(b) shows
the deactivation of a constraint Ck. At iteration j,distance jkvj
ck,nis negative. Next, a new step in direction pjis performed,
with vj+1the new approximation. When both λk<0andjkvj+1
ck,n>0 hold, the constraint must be deactivated. The negative
multiplier in combination with a positive distance indicates that
the minimum lies on the positive side of the constraint and not on
its plane, and hence the constraint is working in the wrong direc-
tion and can be deactivated. Since jkvck,nis zero somewhere
between vjand vj+1, and the new minimum vlies on the posi-
tive side of the constraint, vj+1is closer to vthan vjwas. Hence,
the state is improved. Since the positive distance jkvj+1ck,nand
negative multiplier are clamped in Equation (23), rдdecreases.
The actual ragain might or might not decrease, because it is
based on active constraints and possible negative multipliers.
Delayed evaluation. When constraints are evaluated and
switched at xed intervals, noptimization steps are performed
that move the current approximation closer to the current opti-
mum. This can also be considered as a single optimization step.
If constraints do change their state after nsteps, a new opti-
mum is obtained and the method proceeds in that direction. This
only demonstrates that the method eventually converges to an
optimum. The choice of nlargely aects the performance of the
method. In general, the larger nis, the larger is the possibility
that the method overshoots, which results in a smaller overall im-
provement per noptimization steps. Therefore, we keep nlarge for
large residuals and decrease nwith the residual norm (see line 2 of
Algorithm 2).
Kinetic friction constraint update. The update of a kinetic fric-
tion constraint changes the magnitude λ
kand direction δkof the
kinetic friction force. Since the kinetic friction force is placed on
the right-hand side of Equation (20), any update of δkand λ
k
results in a change of r. When directions jk,tvck,tand δk
are misaligned, δkis updated as described in Section 3.2. Since
each updated δkis somewhere between the previous δkand
jk,tvck,tdue to moving average δk,a, the angle between these
vectors becomes smaller (see Figure 4). Due to this, also the change
in jT
k,tδkμλ(the kinetic friction force; see the right-hand side of
Fig. 6. Plots of several residual norms for preconditioners Diag and Full
of the simulation timestep shown in Figure 7. The non-linear updates of
the constraints are indicated by circles. The preconditioned residual norm
is computed as C1/2r, the constraint residual norm as Ur ,and
the preconditioned constraint residual norm as UC1/2r. Due to the
infinity norm, the shown (preconditioned) constraint residuals represent
an upper bound of the actual constraint residual.
Equation (20)) becomes smaller. Therefore, successive increases of
rdue to these updates (through the last term in the rst row of
Equation (23)) become smaller and eventually vanish when δkand
jk,tvck,tare parallel.
Convergence plots. Figure 6depicts the convergence behavior of
our method for both preconditioners for the simulation timestep
shown in Figure 7. Figure 6(a) shows the residual norms obtained
with the Diag preconditioner. The residual shows a similar behav-
ior compared to the preconditioned residual and (preconditioned)
constraint residuals. Although the residual does not provide much
information about the convergence, the preconditioned residual
C1/2rshows a decreasing behavior on the lower bound. A state
change of a constraint results in an increase of the residuals. Af-
ter continuing for a few iterations, the preconditioned residual is
(in most cases) smaller compared to its norm before the state up-
date. Additionally, the (preconditioned) constraint residual shows
an overall decreasing trend on the upper bound of the error, indi-
cating that, in general, the update of the constraint states eventu-
ally results in a better approximation. This behavior also explains
the eect of the optimization discussed in Section 3.5 in which the
ACM Transactions on Graphics, Vol. 38, No. 2, Article 17. Publication date: March 2019.
17:12 M. Verschoor and A. C. Jalba
Fig. 7. Test case in which nine deformable spheres roll between a (trans-
parent) rigid torus and a rotating spherical object. Figure 6depicts the
residual norms at a certain simulation iteration.
constraints are evaluated at a certain rate depending on the resid-
ual norm. This allows the method proceed toward the new opti-
mum for a few iterations until constraints are re-evaluated. The
large peaks in the plots correspond to the non-linear updates of the
constraints. The intervals between successive non-linear updates
become smaller as the approximation approaches the nal solution
(see Section 3.3). Figure 6(b) shows the (preconditioned) residu-
als obtained with the Full preconditioner, which shows less cor-
respondence with the (preconditioned) constraint residuals. How-
ever, the constraint residuals show the same behavior compared to
the Diag preconditioned case, albeit the Full preconditioned case
results in a much faster convergence. The constraint tolerance for
both experiments were ϵ2=106and a relative tolerance for the
residual of ϵ1=105.
4.3 Local Minima
Whenever the problem is over-constrained, no unique solution ex-
ists. Hence, the problem has local minima. At such a local min-
imum, the gradient vanishes. Since the CR method minimizes
C1/2r2=|rTC1r|, the gradient of the minimized function
(2BC1r) becomes zero at a local minimum. When this happens, α
becomes zero in line 9 of Algorithm 1, so the approximation can-
not be further improved. Next, the computation of βwill result
in a division by zero, causing the method to break down. To pre-
vent this, the norm of the gradient function (second test in line 22
of Algorithm 1) is also inspected, as proposed by Hayami (2002).
If this norm drops below ϵ1, the method has computed a least-
square approximation for the current local minimum. This allows
the method to perform the checks in lines 23 through 25 and to
update the problem.
5 SYSTEM OVERVIEW
Non-penetration and friction constraints act on subsets of vertices
of the colliding models. To constrain the movement globally, all
vertex-face and edge-edge collision pairs should be detected, col-
lected, and assembled in (global) matrices Jand Jt. Since the solu-
tion of Equation (20) is collision free, a standard collision detection
system would not nd any collisions in the beginning of each sim-
ulation timestep. This suggests that all collisions would have to be
detected inside the internal loop of the CR method. However, doing
so would be prohibitively expensive in terms of computations.
Algorithm 1gives the overall pseudo-code of our collision han-
dling system, which is further described in the following sections.
Within our approach, we search once for all possible candidate col-
lision pairs at the beginning of each simulation timestep (see line 5
of Algorithm 1). Then, all existing constraints are updated. When
the solver converges, a collision check is performed (see line 25). If
no new collisions are found, then all active constraints are checked
if their conguration and state agree with the actual geometry
and velocity; if necessary, constraints are re-linearized or removed
from the system (see line 27). Finally, if the system is not updated,
the state of the simulation is guaranteed to be collision free, and
the error made by each constraint is guaranteed to be less than the
desired tolerance.
5.1 Collision Detection
When the simulation is initialized, a Bounding Volume Hierar-
chy (BVH) is constructed, containing axis-aligned bounding vol-
umes enclosing all surface faces (triangles) of the simulated bodies
(which coincides with the simulation meshes; see line 2 of Algo-
rithm 1); our hierarchical data structure is a binary tree. The BVH
is used to quickly nd potential face-face collisions by testing for
intersection of their bounding volumes. Since deformable objects
can have self-collisions, all faces of an object must also be tested for
collisions with all other faces of the same object. Since bounding
volumes of adjacent faces will intersect, one approach to discard
false positives is to terminate the tree traversal when both faces
belong to the same (self-collision free) surface patch (see Volino
and Thalmann (1994)). When there is a possibility that both faces
can collide at the end of the timestep, both faces must be included
in further collision checks to guarantee a collision-free state at the
end of the timestep.
To detect nearby faces of a given face, we extend each bounding
box by 3Δtvin the directions of its vertex velocities. For each face
represented in the BVH, a set is created that stores all nearby and
potentially colliding faces. All faces whose extended bounding vol-
umes intersect the one of the current face are selected by travers-
ing the tree. These face-face pairs are then used to create so-called
candidate lists. The candidate lists contain, for each vertex/edge, a
list of all nearby faces/edges (the candidates) that can potentially
collide; duplicate edge-edge pairs are discarded.
5.2 Raw Collision Detection
At convergence, all candidate pairs (vertex-face and edge-edge) are
checked for collisions given the current velocity (see line 25). First,
each collision pair is updated (time integrated) using the current
velocity. If the pair has intersected, a root-nding method, such
as the Brent-Dekker method (Press et al. 1992), is used to nd
the collision point, contact normal, and barycentric coordinates,
given the current and initial geometry state (at the beginning of
the timestep). Then, the corresponding non-penetration constraint
is initialized, activated, and stored in matrix J(see Section 2.3).
Likewise, the corresponding friction constraints are initialized and
stored in Jt. Since this collision check is performed when the solver
converges, new constraints are added as long as the state is not
ACM Transactions on Graphics, Vol. 38, No. 2, Article 17. Publication date: March 2019.
Eicient and Accurate Collision Response for Elastically Deformable Models 17:13
Fig. 8. Simulations of multiple elastic objects. The collision response is eiciently computed through our method, which can handle complex situations
such as tens of thousands of constraints, large impact forces, and complex interactions (see the text and accompanying video).
collision free. This is similar to the Constraint Manifold Rene-
ment (CMR) method of Otaduy et al. (2009), to ensure that no col-
lisions/constraints are missed. Please note that the collision check
relies on the candidate pairs so that (multiple) tree traversals are
avoided. In some exceptional cases, a vertex can move out of its
extended bounding volume such that collisions might be missed.
If such a case is detected at convergence (line 25), using a sim-
ple vertex-in-box check, we need to search locally for additional
missed collisions. This search is done by rst computing the ex-
tended bounding box and rechecking the BVH for collisions with
the updated box. If any new box-box intersections are found, the
corresponding candidate lists are updated, and a collision check
is performed on the updated candidate lists. The overhead of this
additional check is relatively small—up to 5% of the total collision
detection time.
6RESULTS
In this section, we validate our approach by providing quantitative
and qualitative results, and comparing these to those of existing
methods. An overview of example results is shown in Figure 8.
6.1 Varying Parameters
The plot in Figure 9shows the average number of solver iterations
during the rst second of the experiment shown in Figure 8(e) for
dierent values of Δt. In general, a larger Δtrequires more iter-
ations per simulation step, but fewer simulation steps are needed
for advancing the simulation to a given timestamp. However, in
all cases, the total amount of solver iterations is comparable. We
Fig. 9. Number of solver iterations per simulation step for dierent values
of Δt(s)andtheexperimentshowninFigure8(e), with E=5×106Pa.
observed that simulations with smaller timesteps reveal subtle mo-
tions of objects but require more iterations; however, such details
are not visible when larger timesteps are used.
We have also changed the stiness parameter Ewhile keeping
Δtxed at 0.001s. A smaller stiness parameter generally results in
faster convergence for unconstrained problems. When the objects
are less sti, we observed that the method requires more New-
ton steps due to larger deformations, and this often requires more
iterations. For instance, the setup shown in Figure 8(e) is simu-
lated using varying stiness parameters starting from 5 ×106Pa
to 3.125 ×105Pa by dividing Eby two for every simulation. Only
for the case E=3.125 ×105Pa was the total amount of iterations
ACM Transactions on Graphics, Vol. 38, No. 2, Article 17. Publication date: March 2019.
17:14 M. Verschoor and A. C. Jalba
about 1.5 times larger than for E=5×106Pa. In all other cases,
these numbers were similar. Please note that it is dicult to quan-
tify these results because a dierent stiness parameter results in
a dierent behavior of the material (and simulation). In case of a
collision, the time that objects have contact is larger for exible
objects, which inuences the behavior of the method.
6.2 Comparative Results
In this section, we compare various versions of our method against
ICA (Otaduy et al. 2009) and SP (Kaufman et al. 2008). For this
comparison we have simulated three setups with varying di-
culty degrees (see Figures 8(h), (e), and (b)). All methods used
double-precision arithmetic and converged to the same absolute
error (ϵ1=5×105). Additionally, our method uses a threshold
for updating vectors δkof ϵθ=1.2 degrees and used a constraint
tolerance/safety distance ϵ2=5×106. All objects had the fol-
lowing material properties: ν=0.2 (Poisson’s ratio), E=5×105Pa
(Young’s modulus), and ρ=1,000kg/m3(density). Furthermore, a
gravitational acceleration of д=9.81m/s2was used. The FEM ma-
chinery is based on co-rotated nite elements (Müller and Gross
2004), and no additional damping was used. Our test machine was
equipped with a 4GHz AMD FX-8350 CPU (using one core) and
16GB of RAM memory. All methods use a warm start (reuse val-
ues from the previous simulation iteration), the same collision de-
tection system, the same optimized routines for vector-vector and
matrix-vector operations, and the same tolerances/precision in an
attempt to make a fair comparison.
The rst experiment from Figure 8(h) contains 120 randomly
placed elastic rings, simulated with a timestep Δt=0.001s. The to-
tal number of contact points grew to more than 1,500. The second
experiment (Figure 8(e)) contains 140 objects, which are aligned
in a pyramid. For Δt,we used 0.0005. The experiment shown in
Figure 8(b) uses complex objects in which many internal elements
exist. For this experiment, we randomly placed 120 bunnies with a
Young’s modulus Eof 5 ×106Pa and simulated with Δt=0.001s.
The friction coecient was set to 0.5 for all simulations. Table 1
shows, for each method, the average (and median) time spent on
collision detection, contact resolution, solving the whole contact
problem, and the average number of iterations performed by all
methods. Furthermore, additional details about the simulations
are provided. The application of the Full preconditioner required
about 15% of the total time, whereas its construction time was
negligible—less than 0.1%.
6.2.1 Hyper-Elastic Materials. In addition to linear FEM, we
also made a comparison using highly deformable, hyper-elastic
models. The stiness of hyper-elastic materials changes with the
amount of compression. For a neo-Hookean material, the stress
will approach innity when the volume of the object approaches
zero. Due to this, numerical methods have to deal with large con-
tact forces. Unfortunately, directly using these materials can cause
problems, as small changes in deformation could result in exces-
sively large forces. To deal with this, the underlying energy den-
sity function is linearly extrapolated, as described in Stomakhin
et al. (2012). Furthermore, since more deformation is involved, all
methods need to re-linearize constraints more often. For ICA, this
implies that the LCP matrix also needs to be re-computed. On
Fig. 10. Total runtime and solver iterations for a hyper-elastic object
(see Figure 1and the Armadillo case in Table 1). The total running-
time includes all tasks performed during the process, including collision
detection.
average, constraints are re-linearized 5 times per timestep, with
peaks approaching 10 times. In our test case (see Figure 1), a neo-
Hookean energy density function is used and extrapolated when
the volume of an element drops below 70% of its initial volume.
Furthermore, the Lamé parameters were set using Young’s mod-
ulus of elasticity E=5×105Pa and Poisson ratio ν=0.2. An Ar-
madillo consisting of roughly 200K degrees of freedom is pulled
through a set of rotating cylinders, resulting in large compressions,
deformation, and contact forces. The total number of iterations and
total computation time are presented in Figure 10. For all meth-
ods, the total number of iterations and computation time increase
with the amount of compression. This test also faces the under-
lying collision detection with numerous challenging cases, which
are all resolved correctly.
6.2.2 Comparison to ICA. We compared our approach to the
elegant and exible ICA method of Otaduy et al. (2009). We chose
this method for comparison because it constructs and solves com-
plete LCPs (instead of an approximation (Duriez et al. 2006)) and
takes the full generalized mass matrix into account when comput-
ing collision responses. Both of these aspects are essential when
dealing with FEM-based deformable bodies. Furthermore, the com-
putation of non-penetration and friction multipliers are combined,
making the method very ecient. Please note that we needed
to compute the unconstrained velocity with a slightly lower
tolerance to guarantee that the nal approximation is accurate
enough.
In Figure 11 and Table 1, a comparison is shown between our
method and ICA. We compare the average numbers of iterations
and average time per timestep. ICA performs, per iteration, up
to two Sparse Matrix-Vector (SpMV) multiplications: one for per-
forming a (block) Gauss-Seidel step in the outer loop and one
(block) Jacobi step for setting up the right-hand side of the LCPs.
Additionally, the iterations spent on computing the unconstrained
velocity are considered. We did not explicitly count the number of
iterations used for solving the LCPs, as these matrices generally
are very sparse, and the LCPs converge quite quickly.
For both methods, we have observed that computing the multi-
pliers converged quickly compared to the global problem. Within
ACM Transactions on Graphics, Vol. 38, No. 2, Article 17. Publication date: March 2019.
Eicient and Accurate Collision Response for Elastically Deformable Models 17:15
Fig. 11. Number of solver iterations per timestep for dierent methods, performing the simulation shown in Figure 8(h) (le) and Figure 8(e) (right). CR
(Full) represents our complete method including the Full preconditioner. CR (Diag) represents our method with the diagonal preconditioner, in whichthe
friction cone is replaced by a friction pyramid. SP (Active Set) is an implementation of the SP method, in which the QP problems are solved using two
CR-based Active Set methods. SP (2xCR) represents an implementation of the SP method, in which the QP problems are solved by a modified version of our
diagonally preconditioned CR method. All methods converged to the same absolute error ϵ1=5×105, and used the same constraint tolerance ϵ2=106
and double-precision arithmetic. The dashed plots represent test case “Rings*,” in which the stiness was increased.
ICA, a small LCP is solved per iteration, which converges very
rapidly. In our approach, the convergence of the multipliers is
slower because each iteration improves the multipliers by just a
bit, whereas ICA actually solves an LCP. Nevertheless, the conver-
gence rate of the contact problem is mostly bounded (especially
for deformable bodies) by the convergence of the velocity (the de-
formation part) rather than that of the multipliers. For more com-
plex cases, like the Pyramid setup, a signicant amount of time
is spent on the LCPs as well. In the hyper-elastic test case (Fig-
ures 1and 10), ICA tends to perform slower over time. First, per
timestep, more re-computations of the LCPs are required due to
the non-linear motion and deformation. Second, due to the com-
putation of the unconstrained velocity (which releases all stress),
more collisions are detected, and more constraints than needed are
generated. This results in a much larger overhead spent on con-
structing and solving the LCPs. Due to the increased stiness of
the problem, the PGS and GS methods converge slower as well.
Finally, since the test case in Figure 1also involves rigid bodies,
the constructed LCP contains dense regions, which increases the
time per PGS iteration. In Section 7, we shall further discuss the
dierences between both methods.
6.2.3 Comparison to SP. We also compared our approach to SP
(Kaufman et al. 2008). The SP method solves the contact problem
by rst computing the unconstrained velocity, which is then cor-
rected by separately computing impulses of the non-penetration
and friction constraints. This correction procedure rst solves a QP
problem for non-penetration constraints, provided that some esti-
mate of the friction impulses is available, followed by a QP solve
of the friction problem using non-penetration impulse estimates.
This process continues until an optimum is found that minimizes
both the friction and non-penetration sub-problems. Since a FEM-
based discretization results in a large yet sparse matrix A,wehave
used a diagonally preconditioned MINRES/CR-based Active Set
approach for solving the QP problems rather than a dense solver,
as in the original work (Kaufman et al. 2008). Each individual solve
uses a “warm start.” The relative error of the friction impulses is
computed by solving a linear problem using the CG method. Since
the SP method does not guarantee a collision-free state, we have re-
placed the CR solver within our framework from Algorithm 1with
the SP method such that at convergence, additional constraints can
be added and corrected.
In Figure 11 and Table 1, a comparison between our approach
and the SP method is shown. Since SP solves two QP problems per
iteration, the total number of QP solver iterations per simulation
step is counted. We have also used a modied version of our CR
approach as a QP solver for SP. In this solver, the kinetic friction
treatment, described in Section 3.2, is replaced by one in which a
friction pyramid is used. This is because our approximation of the
friction cone is not suitable for use in the SP method.
The main reason for the dierence in the number of solver it-
erations is that SP solves two QP problems, which both inuence
each other. In our approach, both non-penetration and friction im-
pulses are approximated, and constraints are allowed to update
their state more frequently. Due to this, our approach switches
the state of a constraint when the current approximation starts
to deviate from the (unknown) global minimum. Since SP keeps
a part of the constraints xed, while updating the other set, it
can, for instance, omit changing the state of a non-penetration
constraint while solving for the friction impulses. This can result
in a sequence of approximations that is deviating from the (un-
known) global minimum for a longer stretch of iterations. In some
exceptional cases, we have observed a slow convergence due to
cycling between non-penetration and friction constraints. The to-
tal speedup of our method compared to the Active Set–based SP
method is roughly between 7 and 12 times. Replacing the Active
Set solvers by modied versions of our method, an improvement
in the iteration count of about 2 times was obtained when a diag-
onal preconditioner was used. Additionally, we have also applied
a modied version of our Full preconditioner (in which either the
active non-penetration or static friction constraints were present),
but this approach was not successful.
To run the experiment from Figure 8(e) using SP, we needed
to reduce Δtto 0.0005s, because for larger timesteps, the method
ACM Transactions on Graphics, Vol. 38, No. 2, Article 17. Publication date: March 2019.
17:16 M. Verschoor and A. C. Jalba
did not converge. Even with the smaller Δt, solving both QP sub-
problems converged properly, but applying each outcome to the
other sub-problem resulted in an oscillating and diverging behav-
ior. This problem appeared, for example, when the rst two layers
of objects collided with a large impact; the eects are visible in
Figure 11(b), in which larger iteration counts (spikes) can be seen.
A similar observation was done for the hyper-elastic test (see Fig-
ures 1and 10). Here also the method takes a large number of it-
erations for solving the individual QPs, and then it requires many
steps for nding a global optimum. A tighter coupling between the
friction and non-penetration constraints seems to be necessary in
these kinds of situations. Using our method, we were able to in-
crease Δteven further and even for stier models (see Section 6.1).
When the stiness of the models is increased (setup “Rings*”), SP
seems to converge much slower compared to both ICA and CR.
When stier and more complex objects are used (case “Bunnies”),
SP did not properly converge, whereas the QP problems did con-
verge. Therefore, these results are not present in Table 1.
6.2.4 Comparison to Reduced-form Methods. Reduced-form
methods transform the complete contact problem into a reduced
one and solve dynamics and contact separately. These forms re-
quire the Delassus operator W=LA1LT,whichisoftenusedin
methods in which the number of degrees of freedom per object is
relatively small, such as granular materials, rigid bodies, reduced
deformable bodies, or ber simulations. In those cases, the gen-
eralized mass matrix Acan be inverted eciently. For deformable
bodies with a large amount of degrees of freedom, the computation
of A1is more demanding and may bring extra inaccuracies. As
suggested in Bertails-Descoubes et al. (2011), A1can be found
through a Cholesky decomposition followed by forward and back-
ward substitutions. Since A1contains large and dense blocks, and
not all columns of A1are required for constructing W,itisprefer-
able to compute Wthrough mforward and backward substitu-
tions involving A,LT,andL, with mthe number of constraints (see
Daviet et al. (2011)). This omits the explicit form of A1, which can
have a very large memory footprint.
The method in Tonge et al. (2012) does not explicitly compute
Wbut computes, per contact block, each individual LA1LTfor
each constraint and object associated with the contact block and
accumulates this in a small matrix. The computational complexity
is similar to the method in Daviet et al. (2011), in which only the
non-zero blocks of Ware computed. Their running-time analysis
for constructing Wassumes that the degrees of freedom per ob-
ject are bounded, and that the computation time is dominated by
the squared number of constraints. Hence, by exploiting the block-
diagonal structure of A,Wcan be assembled eciently.
The main problem with deformable bodies is to compute W
in reasonable time. Unfortunately, all mentioned methods assume
that A1is easy to compute, which is not the case for deformable
bodies with a large amount of degrees of freedom. Figure 12 shows
the time consumed for computing Wthrough a Cholesky decom-
position (by also taking the block structure of Ainto account
(Davietetal.2011)) and our method for the same number of con-
straints. This clearly shows that the computation time of Wgrows
with the number of constraints, where its slope is determined by
Fig. 12. Timing comparison of our method compared to the time required
for constructing the Delassus operator LA1LTusing the optimized ap-
proach of Daviet et al. (2011) for the Rings, Armadillo, and Bearing test
cases, respectively, in Table 1 (see also Figures 1,8(h), and 6). The numbers
in the legend refer to the le (1) or right (2) y-axis.
the complexity of computing A1(which mainly depends on its
structure and/or conditioning). Once all blocks of the Delassus op-
erator are computed, the methods can start solving the problem,
typically using a (P)GS method in which Wis used as the iter-
ation matrix (Daviet et al. 2011; Tonge et al. 2012) (both meth-
ods explicitly require the diagonal blocks of W), or by using a
Newton method in which at each iteration the search direction
and step size is found by solving linear systems involving W
(Bertails-Descoubes et al. 2011). Alternatively, in the latter method,
the Delassus operator can be applied implicitly using an opera-
tor that solves a linear system involving A. However, we experi-
enced convergence issues when solving a linear problem involv-
ing this operator for the examples shown in Figures 1and 8(h).
This is likely caused by the bad conditioning of Wdue to dupli-
cate constraints or multiple constraints between the same degrees
of freedom, which typically occur between rigid and/or deformable
objects, or in cases of large compression. In fact, they mention
that their method cannot solve these types of problems, including
stacking of rigid bodies as shown in Figure 2(c) for our method.
Similar issues are mentioned in Daviet et al. (2011) when Lbe-
comes rank decient. Please note that Figure 12 only considers the
time required for constructing the Delassus operator. On top of
that, the actual running time of the method must be added. Fur-
thermore, the plot does not take into account additional updates
of Wdue to re-linearized constraint Jacobians, which boils down
to re-computing Wa few times per timestep.
6.2.5 Comparative Timings. Comparing only the iteration
count, our approach requires between 10 and 60 times fewer it-
erations compared to SP. Similarly, our method requires between
2 and 8 times fewer iterations than ICA. When the actual compu-
tation time is taken into account, the dierences are small for rela-
tively simple models. When the complexity of the models increases
(e.g., due to arbitrary shaped tetrahedral elements and many in-
ternal vertices), the timing dierences become larger. This is due
to the increased density and conditioning of matrix A. The con-
ditioning aects the convergence rate, whereas the density aects
the time spent per iteration. Furthermore, when a model has many
ACM Transactions on Graphics, Vol. 38, No. 2, Article 17. Publication date: March 2019.
Eicient and Accurate Collision Response for Elastically Deformable Models 17:17
Table 1. Comparison of Our Method (CR (Full)) Against Various Instances of SP (With Active Set Solvers or a Modified Version of Our Approach) and ICA
Method Resolution (s) Collision (s) Time (s) Iterations Constraints Objects Elements Faces Setup Δt(s)E(Pa)
SP (Active Set) 5.6 (3.2) 0.68 (0.72) 6.2 (3.9) 1,546 (780) 2,385 (2,919) 120 12,960 17,280 Rings 0.001 5 ×105
SP (2xCR Diag) 5.0 (4.4) 0.68 (0.72) 5.7 (5.2) 814 (728) 2,340 (2,886) 120 12,960 17,280 Rings 0.001 5 ×105
CR (Diag) 2.8 (2.3) 1.1 (1.0) 3.9 (3.5) 451 (340) 2,435 (2,976) 120 12,960 17,280 Rings 0.001 5 ×105
ICA 0.8 (0.7) 0.85 (0.85) 1.6 (1.5) 180 (134) 2,846 (3,180) 120 12,960 17,280 Rings 0.001 5 ×105
CR (Full) 0.52 (0.44) 0.94 (0.96) 1.46 (1.43) 97 (76) 2,280 (2,466) 120 12,960 17,280 Rings 0.001 5 ×105
SP (Active Set) 20.8 (11.75) 2.9 (2.9) 23.7 (14.9) 1,088 (604) 8,647 (9,651) 140 60,480 73,920 Pyramid 0.0005 5 ×105
SP (2xCR Diag) 15.6 (11.5) 3.7 (3.8) 19.3 (15.4) 463 (332) 8,901 (9,939) 140 60,480 73,920 Pyramid 0.0005 5 ×105
CR (Diag) 7.6 (6.1) 4.4 (4.3) 12.0 (10.8) 229 (179) 8,871 (9,877) 140 60,480 73,920 Pyramid 0.0005 5 ×105
ICA 14.2 (3.8) 3.77 (3.61) 17.95 (7.77) 558 (75) 8,734 (9,765) 140 60,480 73,920 Pyramid 0.0005 5 ×105
CR (Full) 2.3 (1.9) 3.77 (3.70) 6.0 (5.6) 69 (59) 8,565 (9,546) 140 60,480 73,920 Pyramid 0.0005 5 ×105
SP (Active Set) 20.2 (15.65) 0.9 (1.06) 21.7 (16.7) 3,373 (2,423) 2,290 (2,892) 120 12,960 17,280 Rings* 0.001 5 ×106
ICA 2.1 (1.7) 0.78 (0.75) 2.7 (2.5) 655 (550) 2,439 (3,217) 120 12,960 17,280 Rings* 0.001