Real-Time Subspace Integration for St. Venant

Real-Time Subspace Integration for St. Venant
Real-Time Subspace Integration for St.Venant-Kirchhoff Deformable Models
Jernej Barbič
Doug James ∗
Carnegie Mellon University
Abstract
In this paper, we present an approach for fast subspace integration
of reduced-coordinate nonlinear deformable models that is suitable
for interactive applications in computer graphics and haptics. Our
approach exploits dimensional model reduction to build reducedcoordinate deformable models for objects with complex geometry. We exploit the fact that model reduction on large deformation models with linear materials (as commonly used in graphics)
result in internal force models that are simply cubic polynomials
in reduced coordinates. Coefficients of these polynomials can be
precomputed, for efficient runtime evaluation. This allows simulation of nonlinear dynamics using fast implicit Newmark subspace
integrators, with subspace integration costs independent of geometric complexity. We present two useful approaches for generating
low-dimensional subspace bases: modal derivatives and an interactive sketching technique. Mass-scaled principal component analysis (mass-PCA) is suggested for dimensionality reduction. Finally,
several examples are given from computer animation to illustrate
high performance, including force-feedback haptic rendering of a
complicated object undergoing large deformations.
CR Categories:
I.6.8 [Simulation and Modeling]: Types of
Simulation—Animation, I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Physically based modeling, I.3.7 [Computer Graphics]: Three-Dimensional Graphics and
Realism—Animation
Keywords: simulation, animation, deformation, precomputation,
model reduction, finite element method, interactive, haptics
1
Introduction
Objects undergoing physically based large deformations play an
important part of computer graphics and animation where shape
changes must be visible, and their simulation is notorious for being
computationally demanding. For example, the high update rates
of force-feedback haptic rendering make it difficult to accurately
simulate large deformations, especially with complex geometry and
distributed contact interactions (see Figure 1). Many interactive and
offline simulations, such as those used in the computer animation
industry, would also benefit from having highly interactive large
deformation models.
In this paper, we show that dimensional model reduction on deformable models with geometric nonlinearities but linear materials,
as commonly used in graphics (the so-called St. Venant-Kirchhoff
model, or StVK), can lead to extremely fast and precomputable approximations for real-time applications. We exploit the fact that
∗ e-mail:
barbic | djames @ cs.cmu.edu
Figure 1: Large-deformation dynamics at kilohertz rates: Forcefeedback haptic rendering of distributed contact interactions between a
user-controlled ball and a flexible bridge model (59630 triangles, r = 15).
Subspace dynamics and contact handling are simulated at a hard real-time
(1000 Hz) update rate. Reduced coordinates are exploited for real-time
Bounded Deformation Tree collision processing [James and Pai 2004].
dimensional model reduction in this case results in internal force
models that are simply cubic polynomials in reduced coordinates.
Coefficients of these reduced force polynomials can be precomputed for efficient runtime evaluation of exact internal forces and
stiffness matrices. All the integration costs are independent of geometric complexity. Consequently, large deformation physics can be
integrated at extremely fast rates using trusted subspace integrators,
e.g., implicit Newmark, while graphical rendering is done at slower
rates. For example, the large bridge example shown in Figure 1
can only be dynamically rendered at about 40 Hz, but its dynamics can be integrated at more than a kilohertz, thus enabling haptic
simulations of complex large-deformation models. In general, the
integration speed is proportional to the number of subspace dimensions employed, e.g., with 4 dimensions the bridge dynamics can
be integrated at over 200 kHz.
Our proposed approach is most closely related to linear modal
vibration models, first introduced to graphics by Pentland and
Williams [July 1989]. These linear dynamics models are simple and
fast, and have seen extensive use [Shinya and Fournier 1992; Stam
1997; Basdogan 2001; James and Pai 2002; Hauser et al. 2003;
Choi and Ko 2005]. Unfortunately, geometric linearity leads to distortions for large deformations, which is a significant limitation for
computer graphics. Our proposed approach preserves several nice
properties of linear modal analysis, but overcomes the serious limitation of linear Cauchy strain by employing full quadratic Green
strain in all computations. Namely, we can still capture the largescale motion of a model with very few modes. We also preserve
the property that progressively more modes can be used to increase
simulation accuracy.
Given the deformation basis, our approach can automatically
generate a fast reduced-coordinate model. A key challenge there-
fore is to construct a good reduced deformation basis for describing general large deformation problems. To this end, we present
two approaches to good quality basis motion generation: modal
derivatives and a sketch interface. Modal derivatives provide a
fully-automatic approach where the standard linear modal analysis basis is augmented by the derivatives of the linear modal basis
vectors. In the sketch-based interface, the user is presented a linear
modal analysis model and interacts with it. The imposed forces are
recorded, and then an offline FEM solver generates the deformation samples. Finally, we use a variant of the PCA data-reduction
method to process the obtained samples, extracting the nonlinear
modal shape basis (the empirical eigenvectors) of the characteristic deformation space. We demonstrate our method on a variety of
examples, including force-feedback haptic rendering.
2
Related Work
Real-time deformable objects: Simulation of largedeformation models is a well-understood area in interactive
computer graphics and offline solid mechanics. Physics-based
large-deformation models have been used successfully in graphics
for almost two decades [Terzopoulos et al. 1987; Baraff and Witkin
1992; Metaxas and Terzopoulos 1992], and enjoy widespread application in mature graphics areas, such as cloth simulation [Baraff
and Witkin 1998; Bridson et al. 2002].
StVK models are often sufficient for the purposes of computer animation, and their use in many recent papers attests to
that [Zhuang and Canny 2000; O’Brien and Hodgins 1999; Picinbono et al. 2001; Debunne et al. 2001; Capell et al. 2002a]. For example, interactive simulations using direct integration include geometric nonlinearities, however the runtime assembly of all the cubic
force terms for every element limits the interactivity to only a few
hundred elements [Zhuang and Canny 2000; Picinbono et al. 2001].
Multi-resolution approaches use hierarchical deformation bases
to adaptively refine the analysis based on deformation activity of
the model [Debunne et al. 2001; Capell et al. 2002a; Grinspun
et al. 2002]. Domain embedding approaches are commonly used in
graphics for interactive applications, since high resolution meshes
can be deformed using coarse deformable models [Pentland and
Williams July 1989; Faloutsos et al. 1997; Müller and Gross 2004].
For linear material models, nonlinear kinematics can be simplified by exploiting local frames of reference. Multibody dynamics
approaches exploit local frames of reference when time-stepping
small deformations, and are widely used in graphics [Terzopoulos
and Witkin 1988; Metaxas and Terzopoulos 1992; Shabana 1990].
Closely related to this are so-called “stiffness warping” methods
(c.f. corotational formulations) [Müller et al. 2002; Müller and
Gross 2004; Irving et al. 2004], wherein an element undergoing
large deformations, with linear materials, simply reuses the undeformed linear element by rotating it to the current frame of reference. Linear materials have also been exploited for fast largedeformation kinematics of Cosserat rods [Pai 2002].
Modal warping [Choi and Ko 2005] is an approximation of StVK
models that is based on extrapolating per-element rotations during modal dynamics to produce a fast parametric nonlinear shape
model. This approach is easy to implement and is useful for
eliminating gross distortions associated with linear modal analysis. However, by virtue of linear modal analysis, the dynamics of
warped modes are driven by independent simple harmonic oscillators. Consequently, an initial condition exciting only one of the
modes will generate single-mode motion (regardless of amplitude),
and hence the well-known nonlinear coupling of modes cannot be
captured. On the other hand, our nonlinear modes are accurately
coupled via an analytic reduction of the StVK model. Also, there is
no guarantee that “warped modes” are sufficient for large deformation dynamics. In contrast, our approach uses a reduced displace-
ment basis produced from actual nonlinear shape statistics. Another
difference is illustrated by deformations in which no element rotations occur, such as a beam’s axial extension mode. With modal
warping, forces and volume grow linearly as the beam extends,
whereas in our model, forces are cubic polynomials and structure
becomes stiffer with extension. Modes also couple to counteract
volume growth. Finally, one benefit of our linear shape model is
that it can accelerate collision detection [James and Pai 2004].
StVK models are perhaps the simplest kind of physical largedeformation model, and one well-known deficiency is that forces
are inaccurate under larger compressions (see [Irving et al. 2004]
for a discussion). In the worst case, elements may actually invert
without proper restoring forces, and suitable steps must be taken to
address element inversion [Irving et al. 2004]. Although our approach is not suitable for simulating the general and complex deformations found in Irving et al. [2004], it is designed to be substantially faster for interactive applications. Finally, we note that
concerns about element inversion are constrained to our precomputation phase, and are not a major concern for runtime subspace
integration, since the shape subspace greatly restricts the likelihood
of element inversion.
To this date, most precomputation-based approaches for realtime simulation have considered geometrically and materially linear models. For fast elastostatics, condensation approaches have
been used to obtain boundary responses [Bro-Nielsen and Cotin
1996], as well as precomputation of boundary Green’s function responses [Cotin et al. 1999; James and Pai 1999].
James and Fatahalian [2003] precompute nonlinear deformation responses to a finite set of user impulses, and apply dimensional reduction using PCA. Although their approach handles selfcollisions, it greatly restricts the range of possible runtime interactions to a small discrete set of pre-selected impulses. On the
other hand, our approach allows general runtime forcing within the
reduced-dimensional subspace.
Subspace integration is closely related to discretizations using
global displacement bases that are commonly used in graphics to
avoid solving large systems (e.g., during semi-implicit integration),
and reducing numerical stiffness (for explicit timestepping), e.g.,
global polynomial shape functions [Baraff and Witkin 1992], deformable super-quadrics [Metaxas and Terzopoulos 1992], freeform deformation basis functions [Faloutsos et al. 1997], and multiresolution discretizations also project dynamical equations using
multiresolution scaling functions [Grinspun et al. 2002]. However,
one drawback with these approaches for interactive applications is
that they all suffer from evaluating unreduced internal forces (and
any Jacobians) at each time step, with cost typically proportional to
geometric complexity.
Model reduction in solid mechanics: Dimensional model reduction is a technique to simplify simulation of dynamical systems
described by differential equations. Complex systems can be simulated by reducing the dimensionality of the problem, yielding systems of differential equations involving fewer equations and fewer
unknown variables. These equations can be solved much more
quickly than the original problem, with some accuracy cost to the
solution. This method also appears in literature under the names
of Principal Orthogonal Directions Method, and Subspace Integration Method, and it has a long history in the engineering and applied
mathematics literature [Lumley 1967].
In nonlinear solid mechanics, early methods extended the principle of mode superposition for linear vibration analysis by using local tangent mode superposition [Nickell 1976], and later the derivatives of tangent eigenmode vectors were also included [Idelsohn
and Cardona 1985b]. Explicit computation of the coefficients of
reduced force polynomials for a time-varying basis of motion is
suggested in [Almroth et al. 1978]. These techniques are not suit-
able for interactive applications because they periodically involve
timesteps with a large amount of computation, such as when the
local basis is updated, and the number of derivative modes required
for accuracy can grow too quickly to be efficient. Recently, a statistical approach to basis generation for finite element models was
presented by Krysl et al. [2001], wherein a full-degree of freedom
system is first simulated, and then standard PCA is applied to the
resulting deformations to obtain a typical deformation basis. This is
a non-interactive technique with external forces known and fixed in
advance, and the simulated nonlinear deformations were relatively
small compared to deformations in our method. Also, reduced internal forces and reduced stiffness matrices were assembled by first
constructing unreduced quantities (followed by subspace projection), which is prohibitively expensive for interactive simulation of
complex models.
3
Background: Subspace Integration
3.1
Basic Deformation Concepts
Continuum mechanics provides the physical background to modeling deformable objects, and we refer the reader to [Fung 1977]
for an introduction. Background on nonlinear solid mechanics can
be found in [Belytschko 2001; Bonet and Wood 1997; Holzapfel
2000]. StVK material is defined by a linear stress-strain relationship of the form
S = λ (tr E)I3 + 2µE,
(1)
where S is the second Piola stress tensor, E is the Green-Lagrange
strain tensor, I3 is the 3 × 3 identity matrix, and λ and µ are (possibly spatially varying) Lamé constants. It is an example of a hyperelastic isotropic material: elastic strain energy is a unique function
of body deformation only (and not of deformation history), and at
any location, material is equally stretchable in all directions.
Without loss of generality, we use the Finite Element Method
(FEM) to discretize partial differential equations of solid continuum mechanics. The deformable body is represented as a volumetric mesh consisting of 3D polyhedra called elements. A particular
body deformation is specified by the displacements of mesh vertices. For a volumetric mesh consisting of n vertices, the displacement vector u ∈ R3n contains the x, y, z world-coordinate displacements of model vertices. A small set of vertices are constrained to
have zero displacements 1 .
In computer graphics, it is often useful to simulate models which
are essentially polygon soups. We follow a common approach in
graphics, wherein a 3D volumetric simulation mesh drives the deformations of a triangle mesh. The volumetric mesh is obtained by
voxelizing the triangle mesh into tiny elastic cubes (8-node first order brick elements) [James et al. 2004; Müller et al. 2004]. Inhomogeneous material parameters can be assigned to the cubes. External
forces acting on the triangle mesh vertices are transfered to simulation mesh vertices via simple trilinear interpolation. Likewise,
resulting displacements are transfered back to the triangle mesh.
While we found this discretization convenient during precomputation, we remind the reader that this paper’s contribution is general,
and can be applied to arbitrary elements.
3.2
Here, u ∈ R3n is the displacement vector (the unknown), M ∈ R3n,3n
is the mass matrix, D(u, u̇) are damping forces, and R(u) are internal deformation forces. The mass matrix depends only on the object’s mesh and mass density distribution in the rest configuration.
In general, it is a sparse non-diagonal matrix, however for algorithmic convenience, it is often simplified into a diagonal matrix by
accumulating all the row entries onto the diagonal element (mass
lumping). Our approach can handle both lumped and non-lumped
versions of the mass matrix. Internal forces corresponding to the
displacement u are given by the vector R(u) ∈ R3n . The mapping
R is nonlinear due to the nonlinearity of the Green-Lagrange strain
tensor, and (in general) due to any material nonlinearities. Note that
the matrix M, and the mappings D and R are independent of time.
Apart from u, the only time-dependent term in the equation is the
vector of external forces f , used to model, e.g., user interactions or
collision response. Let K(u) ∈ R3n,3n denote the Jacobian matrix of
the internal forces R, evaluated at u, i.e., the tangent stiffness matrix. Also, let K = K(03n ) denote the stiffness matrix at the origin
(here 03n denotes the 3n−dimensional zero vector). We use a local
Rayleigh damping model of the form
D(u, u̇) = αM + β K(u) u̇,
(3)
This damping model is controlled by two positive real-valued parameters, α and β , which, roughly, have the effect of damping low
and high time-frequency components of deformations, respectively.
This damping model is a generalization of the more familiar linear
Rayleigh damping model, which would be obtained if K(u) were
replaced by K. In practice, the presence of high frequency damping
significantly improves the stability of the simulation.
3.3
Reduced Equations of Motion
In model reduction for solid mechanics, the displacement vector is
expressed as u = Uq, where U ∈ R3n,r is some displacement basis
matrix, and q ∈ Rr is the vector of reduced coordinates. Here, U is a
time-independent matrix specifying a basis of some r-dimensional
(r 3n) linear subspace of R3n . There is an infinite number of
possible choices for this linear subspace and for its basis. Good
subspaces are low-dimensional spaces which well-approximate the
space of typical nonlinear deformations. The choice of subspace
Equations of Motion
After the FEM discretization, the motion of a deformable solid
can be described by the Euler-Lagrange equation [Shabana 1990],
which is a second order system of ordinary differential equations
M ü + D(u, u̇) + R(u) = f .
1 An
Figure 2: Simulation Meshes: Blue vertices are constrained.
(2)
extension to unconstrained meshes is possible, see Appendix D.
spoon
bridge
tower
heart
rendering
vertices triangles
3321
6638
41361
59630
45882
105788
12186
23616
voxel
resolution
100
128
140
80
simulation
vertices elements
3698
2005
11829
5854
20713
11304
28041
14444
Figure 3: The characteristics of models used in our paper.
depends on geometry, boundary conditions and material properties.
Selection of a good subspace is a non-trivial problem and we will
return to it in the next sections. For now, simply assume that a good
subspace basis U is available. Also, for a given r−dimensional subspace of the full deformation space R3n , there are many choices for
a specific basis for this subspace, and this choice can impact numerical stability. One choice would be to pick an orthogonal basis,
however, it is more natural to make columns of U mass-orthonormal
(see Appendix A), i.e., impose U T MU = Ir , where Ir is the r × r
identity matrix. By inserting u = Uq into Equation 2, and premultiplying by U T , one obtains the reduced equations of motion.
These equations determine the dynamics of the reduced coordinates
q = q(t) ∈ Rr , and therefore also the dynamics of u(t) = Uq(t) :
q̈ + D̃(q, q̇) + R̃(q) = f˜
(4)
energy of a given deformation u ∈ R3n is a fourth order multivariate polynomial function in the components of u. The terms of this
polynomial are localized, in the sense that the displacements of two
vertices can only appear together in a term if the two vertices share
an element. Full internal force on a mesh vertex equals the gradient
of the energy with respect to the x, y, z coordinates of the deformation of the vertex. Consequently, each component of the unreduced
force is a third-order multivariate polynomial function in the displacements of the vertex and all its immediate mesh neighbors.
4.1
Reduced Internal Forces are Cubic Polynomials
Consequently, for deformations of the form u = Uq, each component of the reduced internal force R̃(q) ∈ Rr is a multivariate cubic
polynomial in components of reduced coordinates q:
where D̃, R̃ and f˜ are r-dimensional reduced forces,
T
R̃(q)
D̃
=
U D(Uq,U q̇),
(5)
R̃(q)
f˜
=
U T R(Uq),
(6)
=
UT f .
(7)
Similarly, one can form the reduced tangent stiffness matrix,
K̃(q) = U T K(Uq)U ∈ Rr,r .
(8)
The existence theorem for systems of ordinary differential equations assures that the system in Equation 4 has a well-defined
unique solution, given a specific instance of initial conditions and
time-dependent external forces. Since r 3n, the integration of
(4) is much faster than the integration of the unreduced system (2),
albeit with some accuracy loss.
=
=
U T R(Uq) =
i
ij
(9)
i jk
P qi + Q qi q j + S qi q j qk ,
(10)
where Pi , Qi j , Si jk ∈ Rr are some constant vector coefficients. Furthermore, the reduced tangent stiffness matrix K̃(q) ∈ Rr,r is just
the Jacobian of R̃(q), and therefore, each component of K̃(q) is a
multivariate quadratic polynomial in q. Specifically, column ` of K̃
equals
∂ R̃(q)
= P` + (Q`i + Qi` )qi + (S`i j + Si` j + Si j` )qi q j .
∂ q`
(11)
In general all polynomial coefficients Pi , Qi j , Si jk are non-zero.
4.2
Precomputing Polynomial Coefficients
The coefficients of all the cubic and quadratic polynomials from
the previous subsection can be efficiently precomputed. Note that
there is one cubic polynomial per reduced force dimension (r cubic polynomials total), and one quadratic polynomial per entry of
the reduced tangent stiffness matrix (r(r + 1)/2 quadratic polynomials total due to symmetry of the stiffness matrix). Precomputation proceeds by first computing all the coefficients of the reduced
force polynomials. This can be done in O(r4 ) time per element,
and the algorithm is given in Appendix B. After these coefficients
are known, the coefficients of the reduced tangent stiffness matrix
polynomials can be obtained easily (Equation 11).
Model
r
spoon
bridge
tower
heart
12
15
30
30
num
elements
2005
5854
11304
14444
precomputation
time
60 sec
186 sec
79.2 min
97.4 min
size of
coefficients
98 Kb
223 Kb
3.0 Mb
3.0 Mb
Figure 5: Precomputing polynomial coefficients: Reported numbers are
totals for both reduced force and reduced stiffness matrix.
5
Figure 4: Subspace integration of Eiffel tower and heart models
4
Polynomial Reduced Forces
Following equations of continuum mechanics, it can be shown that
for StVK material with nonlinear Green-Lagrange strain, the strain
Deformation Basis Generation
Deformation basis generation is a hard open problem in solid mechanics, and there exist no algorithms for automatic proven-quality
global deformation basis generation under general forcing. Existing approaches use PCA on example motion to generate a lowdimensional basis for a specific context, i.e., “empirical eigenvectors” [Krysl et al. 2001]. However two problems with this approach
are that (a) for interactive applications, it is unclear what example motion would best describe the essential deformation behavior
of future uses, and (b) it is not automatic, since we can not simply press a button and build a general purpose model. In this section, we present two techniques, one providing an automatic, and
one providing an interactive way to building nonlinear deformation
bases. Both basis generation techniques apply to general nonlinear
materials and aren’t limited to StVK.
5.1
Modal Derivatives
Linear modal analysis [Shabana 1990] (LMA) provides the best deformation basis for small deformations away from the rest pose.
Intuitively, modal basis vectors are directions into which the model
can be pushed with the smallest possible increase in elastic strain
energy. A generalization is possible: for any deformation u0 ∈ R3n ,
tangent linear vibration modes give the best basis for small deformations away from the deformation pose u0 . The first k ≥ 1 tangent linear vibration modes at u0 (denoted by Ψi (u0 ), i = 1, . . . , k)
are the mass-normalized eigenvectors corresponding to the k smallest eigenvalues 0 < λ1 ≤ λ2 ≤ . . . ≤ λk of the symmetric generalized eigenproblem (K(u0 ))x = λ Mx. Tangent linear modes coincide with LMA modes at the origin (define Ψi := Ψi (03n )). Standard LMA simulation uses linear modes with linear forces and suffers from very visible errors for large deformations. A small improvement can be achieved by using U = {Ψi | i = 1, . . . , k} as a
deformation basis in a reduced subspace integrator (i.e. with nonlinear internal forces). In our experiments, we clearly detected a
modest improvement.
Alternatively, one can investigate how tangent linear vibration
modes change with u0 . We combine this approach with mass-PCA
to generate the deformation basis U. We evaluate the directional
derivative of Ψi (u0 ), at the origin, for the LMA directions u0 (p) =
Ψ` p` (note the summation convention), as shown in [Idelsohn and
Cardona 1985b]. Here, parameter p = p` e` ∈ Rk is the vector of
modal participation factors. The unnormalized modal derivatives
can be defined as
∂ i ` Ψ (Ψ p` )
.
(12)
Φi j =
∂ pj
|p=0k
Figure 6: Dominant linear modes and modal derivatives We exploit the
statistical redundancy of these modes using mass-PCA of suitably scaled
modes. All vectors are shown mass-normalized.
Scaling is necessary to put greater weight on the more important
low-frequency modes and their derivatives, which could otherwise
be masked by high-frequency modes and derivatives. Note that K
is a sparse symmetric matrix, and that different modal derivatives
can be computed in parallel. Preconditioning K by the incomplete
Cholesky factorization speeds up the computation.
It can be shown that a quadratic term now extends the LMA linear
deformation space into a parabola:
1
(13)
u(p) = Ψi pi + Φi j pi p j + O(p3 ).
2
If the effects of inertia terms are neglected, derivatives are symmetric (Φi j = Φ ji ), and can be precomputed by solving linear systems
KΦi j = −(H : Ψi )Ψ j ,
∂ Hi j` =
Ki j (u)
,
∂ u`
|u=03n
where
i, j, ` = 1, . . . , 3n
(14)
Figure 7: Extreme shapes captured by modal derivatives: Although
modal derivative are computed about the rest pose, their deformation subspace contains substantial nonlinear content to describe large deformations. (Left) Spoon (k = 6, r = 15) is constrained at far end. (Right) Beam
(r = 5, twist angle=270◦ ) is simulated in a subspace spanned by “twist”
linear modes and their derivatives Ψ4 , Ψ9 , Φ44 , Φ49 , Φ99 .
(15)
denotes the Hessian stiffness tensor. This third rank tensor is the
derivative of the stiffness matrix at the origin (see Appendix B).
Contraction H : a (for a vector a = a` e` ) denotes the matrix where
element (i, j) equals Hi j` a` , for i, j = 1, . . . , 3n. Normalized modal
Model
k
spoon
tower
heart
6
20
20
ij
derivatives Φ are obtained by mass-normalizing Φi j .
Equation 13 suggests that the linear space spanned by all vectors Ψi and Φi j is a natural candidate for a motion subspace. It
could be processed with mass-Gramm-Schmidt to obtain a massorthonormal basis [Idelsohn and Cardona 1985b]. However, its dimension k + k(k + 1)/2 quickly becomes prohibitive. Instead, we
scale the derivatives according to the eigenvalues of the corresponding linear modes. Namely, we obtain the low-dimensional deformation basis by applying mass-PCA on
o n λ2
o
nλ
ij
1 j
1
Ψ | j = 1, . . . , k ∪
Φ | i ≤ j; i, j = 1, . . . , k (16)
λj
λi λ j
Compute
linear modes
24 sec
65 sec
111 sec
Build right-hand
sides of Eq. 14
6.5 sec
226 sec
291 sec
Solve
Eq. 14
33 sec
26 min
28 min
Figure 8: Computation of Modal Derivatives: All performance data is
given for a single 3.0 Ghz Pentium workstation with 2Gb of memory. Massnormalization and mass-PCA times are small.
5.2
Interactive Sketching
Fast interactive linear models are available, and they can be used as
a bootstrapping mechanism to obtain a basis of nonlinear deformations. The user first interacts with a linear vibration model [James
and Pai 2002]. We use a static model to avoid the dynamic effects
which could confuse the user. Due to linearity, the model distorts
badly for large deformations, but still provides a clue to the deformation involved. The forces imposed by the user are recorded
to disk. A subset of these forces is automatically selected so that a
certain separation distance is maintained among consecutive forces.
These forces are then sent as input to a full unreduced offline static
solver which for every imposed load f computes the static rest configuration u. Again, a subset of all deformations is automatically
selected to maintain a certain separation mass-distance. Mass-PCA
is then applied on the resulting shapes to extract the basis of motion
U. When this basis is later used for an interactive nonlinear simulation, the model will be able to simulate nonlinear deformations
similar to those sketched. Additional sketches can be used to refine
the motion basis as desired.
ear system. However, in line with previous research in graphics, we
found it sufficient to perform a single Newton-Raphson iteration per
timestep. This is a speed-accuracy tradeof, and if necessary, multiple Newton-Raphson iterations can be performed per timestep. The
linear system to be solved is a dense r × r symmetric linear system, and we solve it using a direct symmetric matrix solver. Note
that iterative solvers are not as attractive in this case due to relatively small r and dense matrices. The implicit Newmark integrator is given in Appendix C. At any timestep, with the system in
state q ∈ Rr , it is necessary to evaluate reduced internal forces R̃(q)
and the reduced tangent stiffness matrix K̃(q). We note that for a
general nonlinear material, R̃(q) is a complicated function. For a
general isotropic hyperelastic material, it is a large sum of rational functions involving logarithmic terms. In general, it has several
poles, and doesn’t possess an immediate compact and simple analytical expression. Hence, direct evaluation of such functions is
non-trivial. One could proceed by evaluating full unreduced forces
R(Uq) ∈ R3n and forming explicit projection R̃(q) = U T R(Uq)
(and similarly for the reduced tangent stiffness matrix), however
such approach would currently not be real-time for large models.
6.2
Figure 9: Basis from Sketch: (Left) User interacts with a linear model.
Resulting shape is distorted. (Center) Applied force is recorded and sent to
an unreduced offline static solver to solve for the corresponding nonlinear
shape. Several such shapes are then processed by mass-PCA to obtain a
basis of motion. (Right) If same force is re-applied during the reduced runtime simulation, a shape which is visually almost indistinguishable from the
center image emerges.
Model
spoon
bridge
num selected
force loads
353
326
num selected
deformations
45
142
static
solve
45 min
2.4 hours
Figure 10: Precomputation Timings for the Basis from Sketch.
6
6.1
Runtime Computation
Implicit Integration
To timestep the simulation at runtime, we numerically integrate the
system from Equation 4. This is a nonlinear system of r coupled
second order differential equations. Nonlinearity is due to the forcing and damping terms. We use the implicit Newmark integrator
(see [Wriggers 2002]), which is second-order accurate and commonly used in structural dynamics. An alternative choice would be
the central differences explicit Newmark integrator, which doesn’t
require the assembly of the reduced tangent stiffness matrix nor a
linear system solve at every time step. However, we found it hard
to control the explicit timestep as numerical stiffness can cause the
explicit integrator to be unstable. A necessary condition for the explicit integrator to be stable is that the timestep be able to represent
the oscillations of the highest eigenfrequency of the linearized reduced system around the origin. When r is increased, more high
frequency content tends to enters the solution, and explicit timestep
is progressively limited. Moreover, stability of the model at the
origin doesn’t guarantee global stability, since stiffness typically
increases as the model moves away from the origin. Because guaranteed stability is very important for interactive applications, and
because local Rayleigh damping model requires the assembly of
the reduced tangent stiffness matrix anyway, we decided to use the
implicit integrator.
In general, one implicit Newmark step involves several NewtonRaphson sub-steps, each requiring the solution of a dense r × r lin-
Runtime Polynomial Evaluation
For the special case of the StVK material, there is a simple exact
polynomial formula for reduced internal forces, as shown in the
previous section. At runtime, given a state q, we directly evaluate
the precomputed polynomials. Evaluation of each component of
R̃(q) involves Θ(r3 ) operations, and evaluation of each component
of the reduced tangent stiffness matrix involves Θ(r2 ) operations,
so both evaluations can be performed in Θ(r4 ) time. Note that evaluation time is independent of the number of vertices and elements
in the model. About half of the computation time can be saved with
the tangent stiffness matrix by exploiting that it is symmetric. Even
though polynomials are low-degree and involve all possible terms,
evaluation order does matter. During pre-process, we organize all
the precomputed coefficients of the quadratic terms of the reduced
stiffness matrix K̃(q) into a constant matrix S. Each row of this matrix corresponds to one entry of K̃(q) : it contains all the quadratic
coefficients of the entry. Then, to evaluate the quadratic terms of
K̃(q) at runtime, we first assemble qi q j for all i ≤ j into a vector
q, and multiply S by q. A similar scheme was used to quickly evaluate the cubic terms of R̃(q). The number of lower-order terms is
smaller and their evaluation is faster.
6.3
External Forces
Before each rendering step, we reconstruct the full 3n-dimensional
displacement vector u by performing matrix-vector multiply u =
Uq. A collision detection routine can then use vector u to determine
the external forces f for the next timestep. External forces also
occur as a result of user interaction, e.g. a user pulling a certain
vertex or set of vertices in certain directions. Subsequently, the
external forces are projected into the basis U by equation f˜ = U T f .
Implementation can make use of the fact the user interaction vector
f is typically sparse.
6.4
GPU-accelerated Implementation
Matrix-vector multiply u = Uq can be easily performed on CPU.
We have also implemented it in graphics hardware. Matrix U is
stored in texture memory (16-bit floating point format is sufficient).
In pass 1, a fragment shader multiplies u = Uq and renders the
resulting deformation vector u to texture. In pass 2, a vertex shader
fetches u from texture memory, and a standard rendering pipeline
follows. Such an implementation leaves more room on CPU for
other computations. Also, model geometry is now effectively static
Model
spoon
bridge
tower
heart
r
12
15
30
30
force
8.2
22.0
550
550
evaluate [µs]
stiffness matrix
9.5
25.0
770
770
solve linear
system [µs]
12.5
18.4
75
75
integration
total [µs]
30.2
65.4
1395
1395
N
25
10
15
15
time for
u = Uq [µs]
565
14500
25500
6500
graphics frame rate
standard impl. GPU-accelerated
275 Hz
470 Hz
38 Hz
84 Hz
17 Hz
40 Hz
31 Hz
45 Hz
Figure 11: Runtime Computation Performance. Integration times refer to one integration step. The number of integration steps per graphics frame is N.
and can be efficiently cached in a display list, which avoids busbandwidth bottlenecks of rendering dynamic deformable geometry.
6.5
Runtime Modification of Material Parameters
If necessary, our method allows for interactively changing material parameters of the mesh at runtime. Exact polynomials for the
new values of material parameters can be generated interactively,
since Lamé coefficients λ , µ and mass density appear linearly in
the formulas for internal forces and the mass matrix. Mesh needs to
be divided into separate groups, with constant material parameters
over each group. Two polynomials are precomputed for each group,
one collecting only the λ -terms (and setting λ = 1), and one involving only the µ-terms (and setting µ = 1, see Appendix B). To edit
parameters, polynomials for each group are weighted by current
group values of λ , µ, and all the group polynomials are summed
together to produce the exact global polynomials. Changing mass
density for different parts of the mesh can be done in a similar fashion. Note that the precomputed basis will become less optimal if
material parameters deviate too far from those used for precomputation. It can however be shown that the modal derivative basis is
invariant under uniform global scaling of Young’s modulus and/or
mass density. Also, it is possible to omit any subset of basis vectors
from the basis before each individual runtime invocation: the terms
corresponding to omitted dimensions simply need to be dropped
from the polynomials. In particular, any first r0 ≤ r basis vectors
can be used for a particular runtime invocation.
7
Evaluation
We compared our methods to an unreduced implicit Newmark simulation with full internal force and stiffness matrix computation.
Same simulation parameters were used in all cases. Using a reduced
interactive model, we recorded a short user-exerted vertical external
force impulse, applied at the end of the spoon. This impulse was
used to generate all the simulations, and was strong enough to push
the spoon deeply into the nonlinear region. If mass-PCA is applied
on unreduced motion, and the resulting basis is used to re-simulate
the motion, the resulting trajectory lies very close to the original
motion. At around the first maximum, a short transient wave motion occurs in the full solution and such traveling localized deformations are difficult to capture by subspace dynamics. The modal
derivatives and sketching bases produce almost correct amplitudes
and 4.6%, 10.1% smaller nonlinear frequencies, respectively.
8
Discussion
Deformation modes in our paper have global support. Typically, the
number of modes is too small to represent deformations involving
high spatial frequencies, so such deformations can’t be simulated.
Of course, one solution is to add the corresponding localized basis functions into the basis. However, doing so for all localities
on the model would quickly result in a basis whose size prohibits
interactive applications. In the future, we plan to incorporate our
simulation into an adaptive multi-resolution FEM framework.
Figure 12: Vertical displacement of a spoon simulation mesh vertex, located
centrally at the end of the spoon. Length of spoon is about 2.5 units. Triangle mesh poses are shown for reference.
Deformations in our paper are large and self-collisions can occur
in extreme poses. Self-collisions were not a focus of our paper, but
could be addressed in the future, for example by augmenting the
Bounded Deformation Tree [James and Pai 2004] method to detect self-collisions efficiently. During self-contact, basis refinement
may be required due to the changed boundary conditions.
For certain isolated extreme deformation poses, and for extremely low values of r (e.g. r = 2 for the bridge), the reduced
internal force field can contain spurious stable equilibriums. This is
a manifestation of the fact that the chosen value of r is simply too
small to represent the problem. In our experience, this problem can
always be solved by increasing r.
Strain-rate damping (see [Debunne et al. 2001]) could be used
instead of local Rayleigh damping. The damping forces are again
cubic polynomials in q and q̇, and the coefficients could be precomputed. However, we found local Rayleigh damping model sufficient
for our applications.
Acknowledgements: We would like to thank NSF (CAREER0430528), The Link Foundation, Pixar, The Boeing Company, and
NVIDIA for generous support. We also thank Guido Dhondt for
help with his solid mechanics package CalculiX.
Appendix
A
Mass-scaled PCA
PCA is usually performed with respect to the standard Euclidean
metric, however a generalization is possible to any inner-productoriginating distance metric between pairs of deformation vectors u
and v. Standard Euclidean metric is suboptimal: for non-uniform
meshes it over-emphasizes deformations in parts of the mesh where
vertices are dense. It also ignores the mass distribution of the object.
Alternatively, mass-scaled metric (M > 0 is the mass matrix)
p
(17)
||u − v||M := < M(u − v), u − v >
weights the vertices according to the local amount of mass. Given
a set of deformations u(1) , u(2) , . . . , u(N) , and dimensionality r, the
objective of mass-scaled PCA is to find the r−dimensional hyperplane for which the sum of squared mass-projection errors in the
mass metric is minimized. Only the hyperplanes passing through
the origin are of interest, since we want the zero deformation to
be representable by the model. Using Cholesky decomposition
M = LLT , it can be shown that substitution z(i) = LT u(i) translates the problem to a standard Euclidean PCA problem for the
dataset Z = {z(i) | i = 1, . . . , N}. Also, the resulting best Euclideanorthonormal basis V for Z satisfies V = LT U, where U is the optimal mass-scaled basis. To perform mass-scaled PCA, we explicitly
form the z(i) , and perform standard PCA. Mass-orthonormal basis
U is then obtained by solving linear systems LT U = V. Note that
for models of constant mass density mass-scaled PCA reduces to
volume-scaled PCA.
B
Reduced Force Polynomials
Let uia ∈ R3 denote the deformation of vertex a under deformation mode i, for i = 1, . . . , r. Denote the contribution of element
e to the global reduced internal force polynomial coefficients by
i j i jk
Pei , Qe , Se . These contributions can be obtained by inserting standard FEM formulas for StVK unreduced internal forces [Capell
et al. 2002b] into Equation 9 (summation is over all vertices of e):
Pei
=
Qiej
=
Sei jk
=
Aab
i
ac i
ac i
Aca
(18)
1 ua + B1 ua + A2 ua
1
UcT ( C1cab +C2abc )(uia · ubj ) + (uib ⊗ uaj )(C1abc +C2cab +C2bac ) (19)
2
1
+ Dacbd
)(uia · ubj )ukd
(20)
UcT ( Dabcd
2
2 1
UcT
Z
=
e
ab
B
Z
=
e
Cabc
Z
=
e
abcd
D
Z
=
e
∇φa ⊗ ∇φb dV
∇φa · ∇φb dV
(21)
∈R
(22)
∇φa (∇φb · ∇φc )dV
∈ R3
(23)
(∇φa · ∇φb )(∇φc · ∇φd )dV
ab
Aab
1 = λA ,
C2abc
∈ R3,3
ab
Aab
2 = µA ,
= µC
abc
,
Dabcd
1
∈R
ab
Bab
1 = µB ,
abcd
= λD
,
Dabcd
2
(24)
C1abc = λCabc ,
= µDabcd .
(25)
Here, φa denotes the shape function corresponding to vertex a, i.e.
φa (a) = 1 and φa (b) = 0 for a 6= b. Lamé constants λ and µ relate
to Young’s modulus E and Poisson ratio ν as follows:
λ=
νE
,
(1 + ν)(1 − 2ν)
µ=
E
.
2(1 + ν)
(26)
To obtain the global coefficients Pi , Qi j , Si jk , sum the contributions
of all the elements. Efficient parallel implementations are possible. Contribution of element e to blocks corresponding to vertices
a, b, c of the full unreduced stiffness matrix and Hessian tensor at
the origin, and the mass matrix (ρ is mass density) are:
Meab =
Z
e
ρφa · φb dV I3 ∈ R3,3 ,
Heabc = (C1abc +C2bca +C2cba ) ⊗ I3
+
ba
ba
3,3
Keab = Aab
1 + B1 + A2 ∈ R ,
(27)
I3 ⊗ (C1cba+C2acb +C2bca ) +
(28)
+ ∑3i=1 ei ⊗ (C1bca +C2abc +C2cba ) ⊗ ei
∈ R3,3,3 .
Note that all the coefficients A, B,C, D and parameters λ , µ, ρ are
in general element-specific.
C
Implicit Newmark Subspace Integration
Algorithm One step of implicit Newmark subspace integration
Input: values of q, q̇, q̈ at timestep i, reduced external force f˜i+1 at
timestep i + 1; max number of Newton-Raphson iterations per
step jmax (semi-implicit solver: jmax = 1); tolerance TOL to
avoid unnecessary Newton-Raphson steps; timestep size ∆t.
Output: values of q, q̇, q̈ at timestep i + 1
1. qi+1 ← qi ;
2. for j = 1 to jmax // perform a Newton-Raphson iteration:
3.
Evaluate reduced internal forces R̃(qi+1 );
4.
Evaluate reduced stiffness matrix K̃(qi+1 );
5.
Form the local damping matrix
6.
C̃ = α M̃ + β K̃(qi+1 ); // in our work M̃ = Ir
7.
Form the system matrix A = α1 M̃ + α4C̃ + K̃(qi+1 );
8.
residual ← M̃(α1 (qi+1 − qi ) − α2 q̇i − α3 q̈i )+
9.
+C̃(α4 (qi+1 − qi ) + α5 q̇i + α6 q̈i ) + R̃(qi+1 ) − f˜i+1 ;
10.
if (||residual||2 < T OL)
11.
break out of for loop;
12.
Solve the
r × rdense symmetric linear system:
13.
A ∆qi+1 = −residual
14.
qi+1 ← qi+1 + ∆qi+1 ;
15. q̇i+1 ← α4 (qi+1 − qi ) + α5 q̇i + α6 q̈i ; // update velocities
16. q̈i+1 ← α1 (qi+1 − qi ) − α2 q̇i − α3 q̈i ; // update accelerations
17. Return qi+1 , q̇i+1 , q̈i+1 ;
Integrator uses parameters 0 ≤ β̃ ≤ 0.5, 0 ≤ γ̃ ≤ 1, and constants
α1 =
1
1
1 − 2β̃
γ̃
γ̃ γ̃
, α2 =
, α3 =
, α5 = 1 − , α6 = 1 −
, α4 =
∆t.
2
β̃ (∆t)
β̃ ∆t
β̃ ∆t
β̃
2β̃
2β̃
We chose β̃ = 0.25, γ̃ = 0.5, which is a common setting for many
applications. Explicit central differences integrator is defined by
β̃ = 0, γ̃ = 0.5. Constants α, β are Rayleigh damping constants.
D
Modal Derivatives for Unconstrained
Deformable Models
Section 5.1 demonstrated how to determine modal derivatives for
anchored meshes. For models with no constrained vertices, the first
six eigenvalues λ1 , . . . , λ6 are zero with the eigenvectors spanning
the space of infinitesimal rigid body motions. Derivatives are however still defined via Equation 12. To form a motion basis, we comij
bine linear modes Ψi , i ≥ 7 with derivatives Φ , i, j ≥ 7 (appropriately scaled, followed by mass-PCA). Rigid body motion can then
be coupled with deformations [Terzopoulos and Witkin 1988].
To compute the derivatives, first note that the approach from
Equation 14 is not directly applicable: stiffness matrix K is now
singular (nullspace dimension is six), and there is no guarantee that
Equation 14 has a solution. One approach to determine Φi j , i, j ≥ 7
is to use the full formulation from [Idelsohn and Cardona 1985a]:
K − λi M Φi j = MΨi (Ψi )T − I3n (H : Ψi )Ψ j .
(29)
Note that Equation 14 follows by neglecting mass terms and
that modal derivatives are now no longer symmetric. The matrix K − λi M is singular and its nullspace consists of multiples
of Ψi . However, it can be shown that Equation 29 always has
a solution, and that to find a solution, one can solve the regularized version of the system, obtained by replacing K − λi M with
K := K − λi M + Ψi (Ψi )T . Any multiple of Ψi can be added to any
solution of Equation 29. A particular solution Φi j can be chosen
by imposing (Ψi )T MΦi j = 0. Even though K is not sparse and
will often have negative eigenvalues, the “black-box” multiplication x 7→ Kx can be efficiently performed and can be used in a fast
T
sparse symmetric (since K = K) solver, such as MINRES. This
gives one approach to generating a motion basis for unconstrained
models, however the topic is a subject of ongoing research.
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