This document summarizes two technical paper sessions from SIGGRAPH 2013: Fluid Grids & Meshes and Sounds & Solids. One paper presented a method for reducing degrees of freedom in fluid simulations through subspace integration. This allows re-simulating variations of an existing high-resolution fluid simulation more efficiently. Another paper discussed using subspace methods to efficiently simulate structures by projecting displacement vectors onto a reduced basis.

1. SIGGRAPH 2013
@yamo_o
Technical Papers Preview Seminar @UT
Fluid Grid & Mesh Session
+
Sound & Solid Session
Monday, July 1, 13

2. このスライドは2013/7/1に東大で行われたSIGGRAPH 2013 Preview Seminarの捕捉付き
資料です.
SIGGRAPH 2013 review Seminarでは半日かけて2013年の全SIGGRAPH Technical Papers
を持ち回りで紹介し合いました.
この資料では中でもFluid Grids & MeshesセッションとSounds & Solidsセッションにつ
いて紹介しています.
Monday, July 1, 13

3. Subspace Fluid Re-Simulation
Theodore Kim, John Delaney (University of California, Santa Barbara)
前処理によって中規模以上の流体シミュレーションを高速にやるための自由度削減
についての論文.
Subspace Fluid Re-Simulation
Theodore Kim⇤
Media Arts and Technology Program
University of California, Santa Barbara
John Delaney†
Media Arts and Technology Program
University of California, Santa Barbara
Figure 1: An efficient subspace re-simulation of novel fluid dynamics. This scene was generated an order of magnitude faster than the
original. The solver itself, without velocity reconstruction (§5), runs three orders of magnitude faster.
Abstract
We present a new subspace integration method that is capable of
efficiently adding and subtracting dynamics from an existing high-
resolution fluid simulation. We show how to analyze the results of
an existing high-resolution simulation, discover an efficient reduced
approximation, and use it to quickly “re-simulate” novel variations
of the original dynamics. Prior subspace methods have had diffi-
1 Introduction
Fluid simulation methods have made great recent progress, but
working with high-resolution fluids can still be a time-consuming
process. Once a large-scale simulation has completed, the results
are usually considered static; obtaining new results involves launch-
ing another long-running simulation. However, having already paid
the cost of simulating a sequence, can we somehow analyze itsMonday, July 1, 13

4. 高解像度の流体シミュレーションをやったけど結果が気に食わなかったのでパラ
メータを変えてもう一度シミュレーションを走らせたい、がもう一度やるのは時間
がかかり過ぎる、といった状況で既にある結果からシーンの特徴をうまく表現でき
るような基底を抽出してやって新しい状況に使いまわして高速化してやりましょ
う, という技術.
Figure 6: Left to right: 1. Last frame of the original semi-Lagrangian fluid simulation. 2. Last frame of our subspace simulat
discard threshold 10 7
and rank r = 150. Note that our subspace integrator successfully reproduces the dynamics of the origin
3. Discard 10 6
, r = 130. 4. Discard 10 5
, r = 105. 5. Discard 10 4
, r = 61. 6. Discard 10 3
, r = 28. 7. Discard 10 2
dynamics are surprisingly resilient to low-accuracy bases. Even when the basis fails (e.g. 10 2
), it stably generates complex, no
ficiently re-simulated (Fig. 7, lower and upper right). When the
constant was set much higher, stable motion was still produced, but
the dynamics became more abstract. These motions can be seen in
Dirichlet Test Example: We tested our Dirichlet IOP
adding static obstacles to a plume simulation (Figure
tion induced by MacCormack advection was already q
Original Re-simulation
部分空間を使って流体シミュの自由度削減をするような研究は今までもあったけど,
この使いまわせる, っていうところが新しい.
Monday, July 1, 13

5. あと過去の研究だとこんな風に低い周波数領域しか再現できな
かったけどそれも解決しましたとのこと.
the diffusion
ected to eV =
be performed
riable. In lieu
perform a full
eV
eu, where e
003].
)u, must also
e written as a
ed repeatedly
ed, the (u ·
, and the full
er, u changes
and frustrates
r tensor, A 2
u = Auu =
2 U ⇥3 U =
eAeueu = eu⇤
.
e
eAeu eu. The
Figure 2: Left: Frame from a standard [Stam 1999] fluid simula-
tion. Right: Results of trying to reproduce the left simulation using
the projected tensor [Treuille et al. 2006] approach. Different spa-
Original Reduced
Monday, July 1, 13

6. 自由度削減は構造体でも一般的なように
s
s
y
e
.
c
n
u
-
t
d
h
-
times be written as a tensor, and a projecte
to efficiently compute ef. What if F cannot
tensor? A slow but viable method [Krysl et a
and project the full 3N-dimensional f vector
ef = UT
f = UT
F(x) = UT
F
Unfortunately, this N-dependent computatio
speedups that would be obtained by a “su
However, Eqn. 5 is in fact a multi-dimensi
entire simulation domain ⌦, with respect to
coded by the columns of U:
ef = UT
F(U˜x) =
Z
UT
F
というようにもとの自由度のベクトルfに適当な行列Uをかけてずっ
と少ない次元のベクトルに変換することで行います. このベクトル
は変位であったり速度であったりします.
Monday, July 1, 13

7. 一般的な流体シミュレーションの各ステップ, 拡散, 非圧縮拘束の式に対しても適当
な行列で挟み込んでやるとずっと次元の低い簡単な問題になります.
この適当な行列には前もって行っておいた高解像度での問題における速度発散, 圧
力-速度変換の各行列を特異値分解して得たものをここでは使います.
Subspace diffusion and pressure: Putting aside the force and
advection stages for the moment, we can show that the last four
stages of the integration can be performed very efficiently in re-
duced coordinates. Assume we have an orthonormal velocity basis,
U 2 R3N⇥r
, and a pressure-divergence basis, P 2 RN⇥r
, both
obtained by taking the SVD of a pre-existing simulation. The last
four stages can then be projected as follows:
UT
u2
=
“
UT
VU
”
UT
u1
PT
d =
“
PT
WU
”
UT
u2
PT
p =
“
PT
XP
” 1
PT
d
UT
ut+1
= UT
u2
+
“
UT
YP
”
PT
p
)
eu2
= eVeu1
ed = fWeu2
ep = eX 1ed
eut+1
= eu2
+ eYep
.
Note that the matrix inverse and the projection have been inter-
changed the row of equations above. This swap is discussed in
detail in [Stanton et al 2013]. Further details on the implementation
and computation of the necessary SVDs are available in §3.4.
All of these operations are linear, and the projections have made
the matrices very small, so it becomes practical to combine the
diffusion and pressure stages into a single matrix-vector multiply:
eut+1
=
h
I eY eX 1 fW
i »
I
I
–
eVeu1
= eZeu1
, (9)
Diffusion:
Velocity to Divergence:
Poisson Equation:
Pressure to Velocity:
u: Velocity, p: Pressure
Monday, July 1, 13

8. さらにこれら一つにまとめて一回の行列とベクトルの積にできるのでさらに高速化で
きる. この行列は前計算しておくことができ, かつ中に逆行列が含まれているが密行列か
つ条件数がよいので精度の低下も問題にならない.
UT
ut+1
= UT
u2
+
“
UT
YP
”
PT
p
eut+1
= eu2
+ eYep
Note that the matrix inverse and the projection have been inter-
changed the row of equations above. This swap is discussed in
detail in [Stanton et al 2013]. Further details on the implementation
and computation of the necessary SVDs are available in §3.4.
All of these operations are linear, and the projections have made
the matrices very small, so it becomes practical to combine the
diffusion and pressure stages into a single matrix-vector multiply:
eut+1
=
h
I eY eX 1 fW
i »
I
I
–
eVeu1
= eZeu1
, (9)
where I denotes an r ⇥ r identity matrix. Directly computing the
inverse of a Poisson matrix is often discouraged because the sparse
ti
in
p
o
p
b
b
is
w
O
te
in
u: Velocity
障害物が動く場合にはこの行列が変化してしまいますが, Iterated Orthogonal Projectionとい
う手法を使うと対角行列をさらに一つZに右から掛けるだけで対応ができます.
Monday, July 1, 13

9. 面倒なのが移流と外力のステップでこれは全体を積分しなければいけないのでそ
のまま変換行列を適用することができません. なのでここでは立体求積法を使いま
す.
立体求積法というのはある関数を積分するのが難しいときにいくつかの値をサン
プリングしてきてそれに重みを掛けて足し合わせることでそれを近似する方法で
す.
これを使って代表点pでの移流行列Aと重みwを足し合わせてこのように移流ス
テップの自由度を削減してやります.
oints
§4.
o the
milar
ant.
e ap-
t in-
gian
d in-
ature
racy.
with
(N)
[An
ound
ncies
lver.
split
matrix becomes dense, and a large condition number can introduce
numerical error. In our case, both the matrix and its inverse are
dense, so space is not a consideration, and the projection by U
clusters the eigenvalues sufficiently that we found the results of the
direct inverse and a matrix solve to match to within working preci-
sion. Thus, as with other subspace methods [Treuille et al. 2006],
no matrix inversion is necessary at runtime.
Subspace Advection: Once a cubature scheme has been precom-
puted, the advection stage can be efficiently computed in reduced
coordinates. We will describe the runtime here, and the precom-
putation in §4. Assume we have a function A(up) that computes
advection at a grid cell p, containing velocity up. We can compute
the reduced quantity eu1
as:
eu1
=
PX
p=1
wpUT
p A(Upeu0
). (10)
As A is generic, it can correspond to any scheme that supports point
sampling, and we have successfully used it to compute both semi-
Lagrangian and MacCormack advection (Figs. 6 and 7). These
schemes are already essentially point-based, so rewriting existing
code to support cubature took minimal effort.
u: Velocity, U: Reduction Matrix
ここでこのときの代表点と重みの決定が問題となります.
Monday, July 1, 13

10. 既にある高解像度のシミュレーション結果から時間的にTフレームを教師データとし
て持ってきて, 各フレームからランダムにPポイント代表点を選んで速度fをサンプリ
ングすると, 重みwを変数とする最小二乗問題になります.
この論文ではこれに非負値制約(負の固有値を避けるため)をつけて解いています. 代
表点は残差に与える影響を計算してImportance Samplingで決定します.
e greedy search that took nearly six
omputed in less than half an hour
er each iteration of the outer while
n Table 1. As a constant number of
o converge to a desired error bound,
call, so we characterize the average
2 as O(rTP3
).
r approach: The complexity of our
ably with the projected tensor ap-
he main bottleneck of that approach
3N⇥3N
. This tensor is very sparse,
entries. This sparsity can only ex-
. F ⇥1 U ⇥3 U = eF13 can be
the third projection, i.e. eF13 ⇥2 U,
parison, the O(rTP3
) complexity
ethod does not depend on N. The
need to be projected once to form
t the very beginning of training.
d Results
were generated using a Precondi-
G) solver with a Modified Incom-
[Fedkiw et al. 2001]. Faster algo-
maker et al. 2008] or highly tuned
son 2012] are also available, but
n unoptimized prototype and PCG
ur comparisons. Our code, includ-
mented in C++, and our tests were
ac Pro with 96 GB of RAM. We
1$ 11$ 21$ 31$ 41$
Simula%on'Timestep'
Figure 3: Error of subspace simulations over the first 50 timesteps
of each simulation. Interestingly, integration error does not appear
to accumulate. Instead, it stays proportional to the 1% error of the
advection cubature. Note the logarithmic scale.
least-squares problem,
2
6
6
6
6
6
6
6
4
ef1
1 · · · ef1
p · · · ef1
P
...
...
...
...
eft
1 · · · eft
p · · · eft
P
...
...
...
...
efT
1 · · · efT
p · · · efT
P
3
7
7
7
7
7
7
7
5
2
6
4
w1
...
wP
3
7
5 =
2
6
6
6
6
6
6
6
4
ef1
...
eft
...
efT
3
7
7
7
7
7
7
7
5
. (14)
The superscipt indexes the t 2 [1 . . . T] snapshots, and the sub-
script p 2 [1 . . . P] the cubature points. On the right, each tth
row is a projected full-rank example computed using Eqn. 5. These
are the projections of the example velocity fields immediately post-
advection. The left is a matrix of point samples computed using
Eqn. 7, corresponding to the advection of a handful of cells.
This least squares problem yields a set of cubature weights that
closely approximate the examples. We abbreviate the above system
as Aw = b, where A 2 RrT ⇥P
, w 2 RP
and b 2 RrT
. If the
fit given by Eqn. 14 does not meet a user-specified error bound ✏
(i.e. if krk2 = kb Awk2 > ✏) then a new cubature point must
a significant constant speedup. While it
ity problem, we used it to accelerate all
Discarded Alternatives: The existing
rithm, so it is tempting to investigate oth
such as dynamic programming and divi
and Tardos 2006]. In the case of dyna
clear that the necessary optimal sub-prob
our problem. Even if it is, dynamic pro
solution incrementally, so it is likely to
gorithm. Divide-and-conquer is more
points discovered by smaller, independe
rithm can be combined into a higher qu
nately, this approach was still too slow
4.2 An Importance Sampling Ap
We instead design an asymptotically fa
O(rTP3
) time. This algorithm is moti
tions. First, much of the work of the gre
as many A matrices are solved that on
suggests that many candidates should b
c at once to amortize the cost of fitting
Red: Cubature Point
Monday, July 1, 13

11. Synthesizing Waves from Animated Height Fields
Michael Nielsen (Aarhus University),Andreas Soderstrom (Weta Digital), Robert Bridson (University of British Columbia)
今までの映画などでの海面の動きは, アーティストがシーンに合わせて大まかな波の形
を作る→シミュレーションを走らせて望ましい結果が出るまで繰り替えす, というよう
に非常に面倒だった.
2 • M. B. Nielsen et al.
Fig. 1. (Left) Input previs of waves from a visual effects production environment. Only geometry is provided. Angular frequencies and amplitudes vary
in time. (Middle) Output synthesized waves. Horizontal displacements and high frequencies have been added based on an estimation of wave parameters
(amplitude, phase shift and angular frequency) for each wave vector present in the input. This enhances the previs with flat troughs and sharp peaks that
characterize true deep water waves while retaining a correspondence to and the timing of the previs. (Right) Analytical surface velocities projected onto a
plane.
(2) Resolving mesh self-intersections caused by adding
physically-based horizontal displacement using a new
optimization method which is significantly better at tracking
the desired result than previous methods;
(3) Estimating physical depth required for the animation to drive
a consistent dispersion relation for synthesizing higher fre-
quency detail with standard wave spectrum models;
(4) Optionally providing a physically consistent velocity (or time-
varying displacement) field at and below the surface for addi-
tic oceans which have since become standard across the industry.
Tessendorf also discusses adding “chop” with horizontal displace-
ments, and a simple approach to detect where this causes a self-
intersection. Angelidis et al. [2011] mention a procedural method
for removing self-intersections (but do not provide any detail) and
develop a design tool for layout artists.
Many authors have worked in the general area of fluid control,
matching input simulations/animations but not tackled ocean waves
[Treuille et al. 2003; McNamara et al. 2004; Shi and Yu 2005b;
→ なのでアーティストが手付けで付けた大まかなHeight Fieldに最も形状が近いような
物理的に正しい波を自動で生成する, というのがこの論文.
Monday, July 1, 13

12. h
e
mation and syn-
dic.
and sea state
ng a deep wa-
between prop-
estimated by
orf 1999] (the
) to the video
a state can be
timate disper-
vectors.
S
for estimating
field. For the
s= w/2 p=0 q=0
|h(xp, zq, tr+s) ˜h(xp, zq, tr, s)|2
(1)
where xp = pLx/nx, zq = qLz/nz, tr = rLt/nt, Lx ⇥ Lz is
the size of the domain in world space, nx ⇥ nz is the number of
sample points, w is the window of samples in time contributing to
the estimation at time tr and ˜h is the sum of cosine waves given by
˜h(xp, zq, tr, s) =
nx/2 1
X
i= nx/2+1
nz/2 1
X
j= nz/2+1
Aij(tr) · (2)
cos(~k · (xp, zq) ⌦rs
ij + ✓ij(tr))
where ✓ij(tr) is the phase shift, ⌦rs
ij =
R tr
t=0
!ij(t)dt+s t!ij(tr)
integrates the angular frequency, !ij(tr) is the angular frequency,
t is the time-step used in the estimation, ~k = 2⇡(i/Lx, j/Lz)
is the wave vector, k = ||~k||2 is the wave number and Aij is the
amplitude. The unknowns at each point in time tr are the phase
shifts ✓ij(tr), the angular frequencies !ij(tr) and the amplitudes
Aij(tr). The known quantity
R tr
t=0
!ij(t)dt contained in ⌦rs
ij rep-
resents the phase change of the cosine wave solved for sequentially
ACM Transactions on Graphics, Vol. VV, No. N, Article XXX, Publication date: Month YYYY.
海面は単純な複数の調和的な波の周波数成分や大きさ, 位相などを決めて足し合わせる
ことによって作るのが一般的です.
なのでこの論文ではそれらを変数としてアーティストの作ったアニメーションする
Height Fieldと各時間での差分ノルムの最小化を行っています.
on and syn-
applications considered in this paper, the estimation of the wave
parameters is important mainly for synthesizing the velocity field
and adding higher frequency detail that moves consistently with
the coarse waves; knowledge of the exact wave parameters is not
strictly required to add horizontal displacements for flatter troughs
and sharper peaks.
In the approach to synthesizing waves summarized by
Tessendorf [1999], the wave parameters are all fixed in time. For
matching previs input, however, it is necessary to allow these to
vary with time and we perform the estimation as follows. Given a
non-physically based animated input height field sampled in space
and time, h(xp, zq, tr), we wish to fit to h a sum of constant-
frequency cosine waves. This cosine representation of waves is
used in both graphics [Bridson 2008; Tessendorf 1999] and contin-
uum mechanics [Lautrup 2005]. At each instant the cosine waves
are estimated by keeping their phase-shift, amplitude and angular
frequency parameters constant inside a small temporally symmet-
ric window. In particular the following expression is minimized at
each point in time, tr:
w/2
X
s= w/2
nx 1X
p=0
nz 1X
q=0
|h(xp, zq, tr+s) ˜h(xp, zq, tr, s)|2
(1)
where xp = pLx/nx, zq = qLz/nz, tr = rLt/nt, Lx ⇥ Lz is
the size of the domain in world space, nx ⇥ nz is the number of
sample points, w is the window of samples in time contributing to
the estimation at time tr and ˜h is the sum of cosine waves given by
n /2 1 n /2 1
A: Amplitude, k: Wave number, Ω: Angular velocity, θ: Phase
Artist’s input Simulated wave
Monday, July 1, 13

13. ただこのままだと波が位置に対して独立ではなくて解くのが難しくなってしまってい
るのでフーリエ変換して周波数領域で最小二乗法を使って係数を求めます.
again.
In the above formulation of the minimization problem, each
point sampled in space depends on all cosine waves, hence all
unknowns are coupled. However, it turns out that by transform-
ing the minimization problem into Fourier space, only the two co-
sine waves with spatial frequencies (i, j) and ( i, j) are coupled.
This simplifies the minimization problem and allows us to solve for
each such pair of spatial frequencies independently.
In particular, the discrete form of Parseval’s Theorem [Wong
2011] implies that ||F[g]||2
2 = nxnz||g||2
2, where F[g] is the dis-
crete Fourier transform of g. Hence minimizing Eq.(1) is identical
to minimizing
w/2
X
s= w/2
nx/2 1
X
i= nx/2+1
nz/2 1
X
j= nz/2+1
|F[h]ij(tr) F[˜h]ij(tr, s)|2
. (3)
Since both h and ˜h are real-valued, their Fourier transform is com-
plex conjugate even. The Fourier coefficients of a sum of cosine
waves similar to Eq.(2) is given in [Bridson 2008]. However, in our
case the wave parameters depend on time and the angular frequency
depends on direction of travel as well. The Fourier coefficients of
the wave, ˜h, in Eq.(2) are given by:
F[˜h]ij(tr) =
1
2
e
p
1(✓ij (tr) ⌦rs
ij )
Aij(tr) + (4)
1
2
e
p
1(✓ i, j (tr) ⌦rs
i, j )
A i, j(tr).
6
4
be the line
Ignoring d
the time-w
tion syste
three sam
over-deter
sis of the
mated par
if ⌦rs
i, j
numbered
⌦rs1
ij = ⌦
and ⌦rs
i,
system ha
rank-2 sys
traveling
zero.
The est
of the inp
Artist’s input Simulated wave
F[]: Fourier Transform
Monday, July 1, 13

14. ここまでで高さ方向の波が出来上がったのでこれにさらに水平方向の変動と高周波数
の細かい波を足し合わせます. これらの成分は推定された低い周波数の波の角速度から
計算されます[Tessendorf 1999][Bridson 2008].
しかし水平方向にも動かすとメッシュが交差する場所がでてくるのでそれを解消しま
す. これは各頂点の交差しない最初の位置からの最小の移動量を求めるQuadratic
Programmingとなります.
Synthesizing Waves from Animated Height Fields • 3
(a) Input height field.
(b) Estimated waves.
(c) Estimated waves with x-displacements and
self-intersections.
Algorithm 1 (Aij(tr), ✓ij(tr), !ij(tr)) = estimateWaveParameters
Input: h {input height field}
Input: nx, nz, nt {dimensions in space and time}
Input: ⌧ {convergence threshold}
Input: t {time-step between samples in time}
Input: w {window of samples in time}
Require: w 3
for r = 0 ! nt do
{solve for wave parameters at time tr:}
for (i, j) = ( nx/2 + 1, 0) ! (nx/2 1, nz/2 1) do
{solve for Fourier modes (i, j) and ( i, j):}
{outer loop of non-linear least squares:}
while residual ⌧ do
non-linear search for (!ij, ! i, j)
{linear least squares solve:}
(residual, Bij, B i, j) = LSQ(Eq.(7),!ij,! i, j)
end while
end for
end for
(a) Input height field.
(b) Estimated waves.
(c) Estimated waves with x-displacements and
self-intersections.
(d) Self-intersection handling based on Jaco-
bian (approach is conservative and subject to
temporal instability).
(e) Our proposed self-intersection handling
based on explicit optimization.
(f) Addition of synthesized high-frequency
detail.
No self-intersection
Monday, July 1, 13

15. 交差が解消されると最後に水底までの深さが有限であることの影響を波の速度場に取
り入れてやってできあがりです.
(b) Estimated waves.
(c) Estimated waves with x-displacements and
self-intersections.
(d) Self-intersection handling based on Jaco-
bian (approach is conservative and subject to
temporal instability).
(e) Our proposed self-intersection handling
based on explicit optimization.
(f) Addition of synthesized high-frequency
detail.
(g) Shows an additional layer at finite depth
with attenuated movement compared to the
Require: w 3
for r = 0 ! nt do
{solve for wave parameters at time tr:}
for (i, j) = ( nx/2 + 1, 0) ! (nx/2 1, nz/2 1) do
{solve for Fourier modes (i, j) and ( i, j):}
{outer loop of non-linear least squares:}
while residual ⌧ do
non-linear search for (!ij, ! i, j)
{linear least squares solve:}
(residual, Bij, B i, j) = LSQ(Eq.(7),!ij,! i, j)
end while
end for
end for
applications considered in this paper, the estimation of the wav
parameters is important mainly for synthesizing the velocity fiel
and adding higher frequency detail that moves consistently wit
the coarse waves; knowledge of the exact wave parameters is no
strictly required to add horizontal displacements for flatter trough
and sharper peaks.
In the approach to synthesizing waves summarized b
Tessendorf [1999], the wave parameters are all fixed in time. Fo
matching previs input, however, it is necessary to allow these t
vary with time and we perform the estimation as follows. Given
non-physically based animated input height field sampled in spac
and time, h(xp, zq, tr), we wish to fit to h a sum of constan
frequency cosine waves. This cosine representation of waves
used in both graphics [Bridson 2008; Tessendorf 1999] and contin
uum mechanics [Lautrup 2005]. At each instant the cosine wave
are estimated by keeping their phase-shift, amplitude and angula
frequency parameters constant inside a small temporally symme
ric window. In particular the following expression is minimized
each point in time, tr:
w/2
X
s= w/2
nx 1X
p=0
nz 1X
q=0
|h(xp, zq, tr+s) ˜h(xp, zq, tr, s)|2
(1
No self-intersection
temporal instability).
(e) Our proposed self-intersection handling
based on explicit optimization.
(f) Addition of synthesized high-frequency
detail.
(g) Shows an additional layer at finite depth
with attenuated movement compared to the
surface layer.
Fig. 2. Illustration of the various steps involved in the estimation
thesis of waves. Note that the signal in question is not periodic.
Spencer et al. [2006] determine the scale of waves and s
from video. In particular the scale is found by assuming a d
ter dispersion relation and measuring the difference betwee
agation speeds for different wavelengths. Sea state is estim
The final wave
Monday, July 1, 13

16. Simulating Liquids and Solid-Liquid Interactions
with Lagrangian Meshes
Pascal Clausen, Martin Wicke, Jonathan Shewchuk, James O'Brien (University of California at Berkeley)
Lagrangianメッシュを使った流体-構造体の連成シミュレーションに関する論文
huk, and James F. O’Brien
s the
ove-
en as
tum,
ume
ions.
a de-
opo-
tting
rmu-
ume
in a
iffu-
nges.
from
ernal
Images copyright Clausen, Wicke, Shewchuk, and O’Brien.
Fig. 1: A simulation of liquid dripping onto a hydrophobic surface. The
top row shows rendered images; the bottom row visualizes the dynamic
tetrahedral simulation mesh.
Monday, July 1, 13

17. 2008年のSIGGRAPHで"Two-way Coupling of Fluids to Rigid and Deformable Solids and
Shells"[Avi Robinson-Mosher el al. 2008] というすごく画期的な論文が発表されました.
その論文ではデュアルセル, という概念を導入して構造体と流体の移流以外の項を単一
の式で表現して複雑なシーンでも非常に効率的に解けることができるようになってい
ました.
Figure 11: An elastic cloth bag is submerged in and then quickly pulled from a pool of water, carrying fluid with it (14
grid, 1k triangles). The bag bounces as it is raised, expanding and contracting, which causes water to splash out. The b
outer iterations where a conjugate gradient method is applied re-
peatedly to invert A, etc. are quite common, and thus casting our
solid/fluid coupling problem into similar form allows us to benefit
from previous insights. We consider this a promising avenue for
future investigation.
We use an Incomplete Cholesky preconditioner as usual for the
pressure rows of our system, and a block diagonal mass-inverse
7 Time Integration
Our two-way coupled time integration scheme
lution with a Newmark method for solid integ
2003]. Typically Newmark iteration require
ear system for the solid velocities twice per
solves are replaced with Equation (16). The
done with the positions frozen at time tn
and
are used except convection (i.e. except u · ru
is used to update the momentum, and there cMonday, July 1, 13

18. しかし, その論文にも問題があって, Eulerianで離散化を行っていたために結果として流
体の構造体への影響が表面への負荷質量という形になってしまい, 系が非正定にになり
ます. また, Eulerianである限り移流拡散誤差は抑えきれません.
そこでこの論文では似たような定式化をLagrangianで離散化することを行っています. 離
散化は四面体メッシュで行っています.
Monday, July 1, 13

19. これで移流拡散も起きないし系も正定なので高速に解けるよ！
de i, its
on both
rent in-
gas, and
ear sys-
. [2005]
balance
gravity.
ines the
without
termine
e appro-
a node
% of the
quation,
(9)
he ther-
an implicit Euler time integration scheme and piecewise linear ba-
sis functions over a tetrahedral finite element mesh [Cook et al.
2001]. We combine both the equations of motion and the continu-
ity equation in a single linear system where the unknowns are the
node velocities vn+1
and pressures pn+1
at time tn+1
= tn
+ t:
✓ 1
t
M + t K + D 1
t
BT
1
t
B 0
◆ ✓
vn+1
t pn+1
◆
=
✓
t Kvn
f
0
◆ (12)
where M is the (lumped) mass matrix, K is the stiffness matrix, D
is the damping matrix (containing viscous terms), B and BT
are the
discretized divergence and gradient operators, and f is the external
forces. The vectors with superscripts denote system-sized state vec-
tors containing the values for all nodes in the system at the indicated
time. The pressure vector pn+1
functions as Lagrange multipliers
for the incompressibility constraints. We scale the constraints by
1/ t to make the linear system’s conditioning less dependent on
the time step. After vn+1
is computed, the node positions in world
space are updated as xn+1
= xn
+ t vn+1
.
The heat equation is modeled by a separate linear system whose
solution is the new vector of nodal temperatures tn+1
.
✓
1
◆
n+1 1 n
v: Velocity, p: Pressure
M: Mass Matrix
K: StiffnessMatrix
D: Damping Matrix
構造体の速度と流体の移流項以外の更新式を統合
B: Interaction Matrix
Only for solids
Monday, July 1, 13

20. しかし, Lagrangianで離散化した副作用としてメッシュの再構築がすごく面倒になってし
まった(Eulerianでレベルセット使えば何も考える必要なし)ので工夫というかもうかな
り根性で対応しています.
結局ここで超時間かかるんじゃん...orz
Monday, July 1, 13

21. 例えば図のように両端が拘束されてる方向に運動量が生じると直感的にも動きが堅く
なってしまうことが分かる. なので中間に新しいノードを一時的に配置して動きやすく
してやったり...
ting Liquids and Solid-Liquid Interactions with Lagrangian Meshes • 17:5
that are axis-
the linear sys-
s we rotate the
Chentanez et al.
o enforce con-
stress and the
or large defor-
he corotational
ble the stiffness
ard for simula-
n into rotation
esses from the
sses back into
¨uller and Gross
(a) (b)
(c) (d)
Fig. 3: (a) An example of a locking configuration in a two-dimensional tri-
angular mesh. The red nodes are constrained not to move, and the volume
preservation constraints keep the unconstrained nodes from moving verti-
cally. No mass can be transferred between the two elements. (b), (c) After aMonday, July 1, 13

22. メッシュが移動するたびに潰れたメッシュ付近だけメッシュを組み直すのですが, ただ
潰れたメッシュを削除するだけだと体積と運動量が失われてしまうので周りのノード
に物理量を分散させたり...
Fig. 4: New operations to remove a bad tetrahedron. Left: A face contrac-
tion. A new vertex is inserted on an edge, splitting the tetrahedron in two,
and a newly created edge is immediately contracted, eliminating the tetra-
hedron. Right: A tetrahedron contraction. A new vertex is inserted on a face,
splitting the tetrahedron in three, and the new interior edge is contracted.
(a) (b) (c)
Fig. 5: Resampling can cause loss of momentum. (a) If only the red node
has nonzero velocity, its momentum is lost when (b) the node is deleted
from the mesh. (c) A compensating force is applied to the nodes for one
time step, preserving the total momentum.
would like to change the surface of the object as little as possible, a
much smaller set of operations is available to the remesher near the
6.3
Afte
resa
Wic
with
store
resa
weig
C
mov
tal m
forc
recti
C
men
Vij b
Give
we r
Monday, July 1, 13

23. 応力が強くかかった場所は分離しますがその分離面の決定(3パターン)を行ったりや逆
にくっつくときに一時的に重複するノードを作って後から削除したり...
Solid-Liquid Interactions with Lagrangian Meshes • 17:7
(a) (b) (c)
Fig. 6: Creating split faces (red) in the splitting plane. (a) Two split faces
propagate through an element having no snap edge. (b) This element has
one snap edge, from which a split face propagates. (c) Two snap edges. A
preexisting face becomes a split face.
Simulating Liquids and Solid-Liquid Interactions with Lagrangian Meshes • 17:7
shes require explicit treatment of these operations, but the ef-
is compensated by reduced numerical viscosity, better vol-
e preservation, and greater accuracy of surface evolution and
lision detection. We use a local tetrahedron splitting operation
deled after fracture algorithms for solids [O’Brien and Hodgins
99] and merging operations based on tetrahedron subdivision.
r method supports all the operations necessary to represent a
wing liquid undergoing topological changes.
1 Mesh Splitting and Material Fracture
itting has been extensively studied in the context of fracture and
ting of elastic materials [Bielser et al. 1999; Smith et al. 2001;
ller et al. 2001; O’Brien et al. 2002; Molino et al. 2004; Steine-
nn et al. 2006]. In fracture simulations, a material is typically
t where the stress exceeds a specified threshold.
We use a combination of physical and geometric criteria to deter-
ne whether a material can split. For liquid elements, we initiate
ological separation only at surface elements and only in concave
ions—where at least one of the two surface curvatures is nega-
. Moreover, we require that there be a tensile principal stress
ose magnitude is higher than a threshold lg · ⇣ where ⇣ is the
illary threshold and lg the surface tension. Where both criteria
met, we fracture the material with a splitting plane perpendicu-
to the direction of highest principal stress.
Fracture can be triggered in inappropriate locations by field fluc-
tions caused by discretization, especially when it is driven by
cous stresses in turbulent liquids; these stresses can vary wildly
ween elements. To obtain physically plausible splitting, we have
nd it necessary to compute viscous stresses from a velocity field
t has been smoothed over the 2-ring neighborhood of each vertex
a moving least squares approximation.
f the largest principal stress in an element e exceeds a speci-
(a) (b) (c)
Fig. 6: Creating split faces (red) in the splitting plane. (a) Two split face
propagate through an element having no snap edge. (b) This element ha
one snap edge, from which a split face propagates. (c) Two snap edges. A
preexisting face becomes a split face.
(a) (b)
Fig. 7: Merging operations. (a) A tetrahedron split inserts a node of on
mesh (red) into an element of the other. (b) An intersection between a
edge of one mesh and a face of the other is treated by inserting a new nod
(red) into both.
of e which together contain all the snap edges, and whose normal
deviate least from the original principal stress direction.
Creating surfaces. To propagate the fracture, the split face
must become surface faces. For each node, we check whether th
split faces separate its adjoining elements into two or more face
Red: Split faces
Merge
Red:Temporary nodes
Monday, July 1, 13

24. といった涙ぐましい努力の跡が...
論文にもこのメッシュ関係のプログラミングは苦行だったと書かれていましたw
Monday, July 1, 13

25. その甲斐もあって結果は素晴らしくて水滴が細管から滴るときのプルプルした感じな
んかがとてもリアルに再現されています.
Simulating Liquids and Solid-Liquid Interactions with Lagrangian Meshes • 17:13
Images copyright Clausen, Wicke, Shewchuk, and O’Brien.
Fig. 15: The Stanford Bunny melts on a hot plate.
ちなみに説明を飛ばしましたが熱量の影響も考慮されています.
今年も虐待されるウサギ
Monday, July 1, 13

26. Wave-Based Sound Propagation in Large Open
Scenes using an Equivalent Source Formulation
Ravish Mehra (University of North Carolina at Chapel Hill), Nikunj Raghuvanshi (Microsoft Research), Lakulish Antani,Anish Chandak,
Sean Curtis, Dinesh Manocha (University of North Carolina at Chapel Hill)
この論文ではオープンスペースを含む広いフィールドでの残響を前計算結果を使うこと
によってリアルタイムに計算する手法を提案しています. なお, 音源とリスナーの位置は
フィールド上を自由に移動することができますが, 障害物は固定となります.
4 •
Fig. 3: Overview of our wave-based sound propagation technique based on equivalent sources on a simple scene composed of two objects
and a sound source (shown with a red dot). The magnitudes of pressure fields are visualized using the color scheme shown.
4. SOUND PROPAGATION USING ESM
We give a brief overview of the precomputation and runtime stages
of our technique (see Figure 3). Our formulation is in the frequency
domain. We construct a complex frequency response (containing
magnitudes and phases), at regularly sampled frequencies, to model
the delay information in the propagated sound. Thus, the steps out-
lined in this section, except the offset surface calculation, need to
be performed for a regularly sampled set of frequencies in the range
Symbols Meaning
qin
i , qout
j ith
& jth
eq. src for incoming, outgoing field resp.
'in
ik , 'out
jh kth
& hth
multipole term of eq. src. qin
i & qout
j resp.
Q, P number of incoming, outgoing eq. srcs resp.
M, N order of incoming, outgoing field multipoles resp.
Table I. : Table of commonly used symbols.
4.3 Per-object Transfer FunctionMonday, July 1, 13

27. 背景となってるアイディアは光におけるPrecomputed Radiance Transferに近いものです.
2006年に既に物体からの音の放射を前計算するPrecomputed Acoustic Transferという手法
が発表されていますがこれはその延長線上にあるものです.
これは空間の音場を調和振動子, マルチポールの集合で近似するというもので, この論文
では物体内部からの音の放射ではなく, 物体に外から入射してくる音の散乱場を物体表
面に配置したマルチポールの集合で近似しています.
boundary of the domain, @A, the pressure is specified using a
Dirichlet boundary condition:
p = f(x) on @A. (2)
To complete the problem specification, the behavior of p at infin-
ity must be specified, usually by the Sommerfeld radiation condi-
tion [Pierce 1989]:
lim
r!1

@p
@r
+ ˆj
w
c
p = 0, (3)
where r = kxk is the distance of point x from the origin and ˆj =p
1. The equivalent source method [Ochmann 1995; 1999; Pavic
2006] relies on the existence of fundamental solutions also called
Green’s functions or equivalent sources q(x, y), of the Helmholtz
equation (1) subject to the Sommerfeld radiation condition (3) for
all x 6= y. An equivalent source q(x, yi) is the solution field in-
duced at any point x due to a point source located at yi, and can be
expressed as the sum:
q(x, yi) =
L 1X
l=0
lX
m= l
dilm'ilm(x) =
L2
X
k=1
dik'ik(x), (4)
where k is a generalized index for (l, m). The fundamental solution
'ilm(x) is the field due to a multipole source located at yi, dilm
is its strength, and L is the order of the multipole (L = 1 is just a
monopole, L = 2 includes dipole terms as well, and so on). The
field due to a multipole located at point yi is defined as
'ilm(x) = lmh
(2)
l (wri/c) lm(✓i, i) (5)
the Helmholtz equa
to an object, and th
on @A. Consider a
contained in the in
sources at any x 2 A
p(x) =
X
i
where cik = cidik a
sources. The main i
strengths cik and p
boundary condition
p(x) =
RX
i=
then p(x) is the cor
This process can al
object, the only diffe
placed in the exterio
condition (7), we ge
rior region A .
In practice, the bou
proximately for a fin
can be controlled by
of each source must
time, R is the main
formance and mem
highly suitable for i
ACM Transactions on Graphics, Vo
lim
r!1 @r
+ j
c
p = 0, (3)
where r = kxk is the distance of point x from the origin and ˆj =p
1. The equivalent source method [Ochmann 1995; 1999; Pavic
2006] relies on the existence of fundamental solutions also called
Green’s functions or equivalent sources q(x, y), of the Helmholtz
equation (1) subject to the Sommerfeld radiation condition (3) for
all x 6= y. An equivalent source q(x, yi) is the solution field in-
duced at any point x due to a point source located at yi, and can be
expressed as the sum:
q(x, yi) =
L 1X
l=0
lX
m= l
dilm'ilm(x) =
L2
X
k=1
dik'ik(x), (4)
where k is a generalized index for (l, m). The fundamental solution
'ilm(x) is the field due to a multipole source located at yi, dilm
is its strength, and L is the order of the multipole (L = 1 is just a
monopole, L = 2 includes dipole terms as well, and so on). The
field due to a multipole located at point yi is defined as
'ilm(x) = lmh
(2)
l (wri/c) lm(✓i, i) (5)
where c
sources.
strength
boundar
then p(x
This pro
object, t
placed i
conditio
rior regi
In pract
proxima
can be c
of each
time, R
formanc
highly s
ACM Transactions o
A point sound source:
A multipole:
d: strength
Γ: Spherical harmonics orthonomal
h: Spherical Hankel function(2)
ψ: Spherical harmonics function
Monday, July 1, 13

28. 例えば空間の中にAとBという２つの障害物があるとすると, 音源からリスナーに届く音
は直接音の他には音源からAにだけ当たって反射してくる音, Bだけ, Aに当たった後Bに
当たってから届く音, その逆, の４パターンに分類されます.
ちなみに例えばA→B→Aの場合は音源の位置を任意とすればB→Aと等価です. これの4
パターンを近似するマルチポール群をそれぞれ前計算します
10 •
Monday, July 1, 13

29. まずAとBそれぞれ単独で置いてで空間上で十分な数の位置を選んでそれら全ての組み
合わせだけ音源(マルチポール)とリスナーを配置して境界要素法で残響を計算します.
A
Sound Source 1
Listener 1
Sound Source 2
Listener 2
Monday, July 1, 13

30. これによって入射マルチポール-散乱場(音圧)の関係が算出できます. 音源の場所は物体
表面から法線方向に適当なオフセットをとった位置だと好ましいようです.
n uniformly-sampled locations {x1, x2, ..., xn} on @A. We sub-
tract the incident field from the total pressure field to compute the
outgoing scattered field at these sampled locations (see Figure 4),
denoted by ¯pik = {p(x1), p(x2), ..., p(xn)}.
We fit the outgoing field multipole expansion to the sampled scat-
tered field, in a least-squares sense, by solving an over-determined
linear system (n > PN2
) subject to a pre-specified error threshold
for all incoming field multipoles:
(P,N2)
X
(j,h)=(1,1)
'out
jh (xt) ↵ik
jh = p (xt) , for t = 1, ..., n; (10)
V↵ik
= ¯pik. (11)
The least-squares solution yields the coefficients ↵ik
correspond-
ing to the ikth
row of the scattering matrix T. This process is
repeated for all incoming field multipoles to compute the scatter-
ing matrix. The solution can be computed efficiently using a single
combined linear system
V Ttr
A =
⇥
¯p11 . . . ¯pQM2
⇤
, (12)
where Ttr
A is the transpose of TA. The per-object transfer func-
tion is computed for all objects at sampled frequencies. The error
threshold is used while deciding the number and placement of
equivalent sources (Section 4.5) such that the above linear system
gives error less than .
6
4
gB
A
where
ing mul
tipole '
thus enc
ally, the
the outg
GA
A and
(comple
Compu
GB
A ca
lar to th
the pre
{p(x1),
nous m
the free
tion 5).
the outg
offset su
a separa
(
次に物体表面上においてこの散乱場pを
のようにマルチポールφの集合として最小二乗法を使って求めてやるとこの係数行列α
は入射マルチポールから散乱マルチポールへの変換行列となります.
これを単体行列と呼ぶことにします.
us, the steps out-
culation, need to
ncies in the range
requency. We as-
Table I provides
tage, we classify
e for each object.
e compute a per-
field incident on
pair, we precom-
s how the outgo-
ming field for the
r-object transfer
odel acoustic in-
ve for the global
all the outgoing
add the pressure
field equivalent
ast computation,
-time.
o well-separated
d, we use the no-
Table I. : Table of commonly used symbols.
4.3 Per-object Transfer Function
In order to capture an object’s scattering behavior, we define the
per-object transfer function f, a function which maps an arbitrary
incoming field reaching the object to the corresponding outgoing
scattered field after reflection, scattering and diffraction due to the
object itself. This function is linear owing to the linearity of the
wave equation and depends only on the shape and material prop-
erties of the object.
The incoming and outgoing fields for an object A are both ex-
pressed using equivalent sources. The outgoing field is represented
by placing equivalent sources {qout
1 , qout
2 , qout
3 , ...} in the interior
region A of the object. Similarly, the incoming field is repre-
sented by placing equivalent sources {qin
1 , qin
2 , qin
3 , ...} in the ex-
terior region A+
. The transfer function f maps the basis of the in-
coming field (multipoles 'in
ik ) to the corresponding outgoing field
expressed as a linear combination of its basis functions (multi-
poles 'out
jh ):
f('in
ik ) =
(P,N2)
X
(j,h)=(1,1)
↵ik
jh'out
jh ; (8)
2
6
6
6
6
4
f('in
11)
f('in
12)
.
.
f('in
2 )
3
7
7
7
7
5
=
2
6
6
6
6
6
6
4
↵11
11 ↵11
12 ... ↵11
P N2
↵12
11 ↵12
12 ... ↵12
P N2
. . ... .
. . ... .
↵QM2
11 ↵QM2
12 ... ↵QM2
P N2
3
7
7
7
7
7
7
5
2
6
6
6
6
4
'out
11
'out
12
.
.
'out
2
3
7
7
7
7
5
p: Sound pressure
Monday, July 1, 13

31. 次に今度はAB両方を空間に配置してそれぞれ単独で配置したときの散乱マルチポール
を入射マルチポールとしたときのもう片方の散乱マルチポールへの変換を前と同様に
計算します.
A
Listener 1
B
Listener 2
object transfer function. For two objects A and B, the inter-object
transfer function gB
A expresses the outgoing field of A in terms of
the basis of the incoming field of B. Like the per-object transfer
function, the inter-object transfer function is also a linear function.
The inter-object transfer function gB
A maps each basis function of
the outgoing field of A (multipoles 'out
jh ) to the corresponding in-
coming field of B expressed as a linear combination of its basis
functions (multipoles 'in
ik ):
gB
A ('out
jh ) =
(Q,M2)
X
(i,k)=(1,1)
jh
ik 'in
ik ; (13)
Choosing Incoming
the incoming field is n
the number and posit
solve this problem by
> from the objec
tance, and placing inco
(see Table II for the v
alent sources Q depen
pressure field. As bef
sources are uniformly
coming field on the in
ACM Transactions on Graphics, Vol. V
Monday, July 1, 13

32. 最後に以上のパターンを一つにまとめます. 既に鳴っている音場をC, 先程求めた全ての
障害物間の干渉行列をG, 単体行列をTとし, 定常状態を仮定すると
と表すことができます.
つまりこの連立方程式をCについて解いてこれのみを保存しておき, ランタイム時には
リスナーの位置でマルチポールとCの一度の積を行うだけで最終的な残響を計算するこ
とができます.
in the scene is C. This field when propagated through the scene,
transferred via all possible object pairs using interaction matrix G,
generates an incoming field GC that, in addition to the source field
S, generates the total incoming field (GC+S) on the objects. This
incoming field is then scattered by the object, via scattering matrix
T, to produce an outgoing field T(GC + S). Under steady state,
this outgoing field must equal C. Mathematically, this can be writ-
ten as
C = T(GC + S) (17)
lay
sol
ob
ple
an
de
Th
an
GH
ACM Transactions on Graphics, Vol. VV, No. N, Article XXX, Publication date: Month YYYY.
Monday, July 1, 13

33. 個人的な感想としてはちょっと総当たりすぎて前計算の計算コストが
高すぎな感も否めない
Monday, July 1, 13

34. Example-Guided Physically Based
Modal Sound Synthesis
Ravish Mehra (University of North Carolina at Chapel Hill), Nikunj Raghuvanshi (Microsoft Research), Lakulish Antani,Anish Chandak,
Sean Curtis, Dinesh Manocha (University of North Carolina at Chapel Hill)
Example-Guided Physically Based Modal Sound Synthesis • 1:11
g. 10. Parameter estimation for different materials. For each material, the material parameters are estimated using an example recorded audio (top row)
pplying the estimated parameters to a virtual object with the same geometry as the real object used in recording the audio will produce a similar sound
ottom row).
Monday, July 1, 13

35. サウンドレンダリングをゲームなどの実際のコンテンツに利用しようと考えたときの
要望として「こんな音が鳴るこんな形状の物体を作りたい」というのは当然上がって
くるわけです.
しかし, 例えば現実にあるドラムと見た目そっくりなドラムの3Dモデルを作ったとして
もそれと同じ思い通りの音はまず生成されません. 本当に厳密にモデルを作りこめば原
理的には可能ですがそれを制作サイドでやるのは非常に困難です.
Monday, July 1, 13

36. ならばそこまで作りこみされていない３Dモデルでもその形状とほぼ同じ形状の現実の
物体を叩いた音の録音からパラメータを抽出してそれを3Dモデルにマッピングできれ
ば便利でしょう, というのがこの研究.
Monday, July 1, 13

37. こういった研究は古くからありましたが回転代の上に物体を置いて回しながら様々な
点を専用の機械を使って叩いたときの音をこれまた物体周囲あらゆる方向からマイク
で収録しなければいけない, というとても大変な作業を要したので現実的ではありませ
んでした.
3.1 Estimation of th
The estimation of the mo
from the measurements was ach
three phases.
First, we estimated the moda
each sound sample separately. T
ing the windowed discrete Fou
transform) of the signal—extra
dampings, and amplitudes by fi
with a sum of a small number (
complex exponentials. The para
capable of very accurate frequen
is able to resolve very close mod
common in artificial objects tha
metries resulting in mode dege
impurities break this symmetry,
by a small amount. These nearb
tinctly audible as beating, or “s
Figure 3. An automated measuring system was used to acquire the
data. A robot arm moves the microphone to a preprogrammed set of
locations. An impulse force was then automatically applied to the
object with the solenoid and sounds were recorded for subsequent
analysis.
648 PRESENCE: VOLUME 16, NUMBER 6
TimbreFields: 3D Interactive Sound Models for Real-Time Audio
Richard Corbett, Kees van den Doel, John E. Lloyd,Wolfgang Heidrich, 2007
Monday, July 1, 13

38. この研究ではこれを解決するために適当に手で叩いた単一の録音素材からだけでマッ
ピングができるような工夫をしています.
Monday, July 1, 13

39. まず作った3Dモデルのの振動音は, 質量行列と合成行列を作成して一般化固有値問題の
解から
¨q + (αI + β )˙q + q = UT
f, (3)
where is a diagonal matrix, containing the eigenvalues of Eq. (2),
U is the eigenvector matrix, and transforms x to the decoupled
deformation bases q with x = Uq.
The solution to this decoupled system, (Eq. (3)) consists of a
bank of modes, that is, damped sinusoidal waves. The i’th mode
looks like
qi = aie−di t
sin(2πfit + θi), (4)
where fi is the frequency of the mode, di is the damping coefficient,
ai is the excited amplitude, and θi is the initial phase.
ACM Transactions on Graphics, Vol. 32, No. 1, Article 1, Publication date: January 2013
という減衰振動(モード)の和で表すことができます. ここで, 周波数が求まった固有値と
結びつきます.
このモードの周波数と減衰係数と録音素材とのマッチングを行います.
a: Amplitude, f: Frequency, d: Damping coefficient
Monday, July 1, 13

40. 録音素材のモードはスペクトログラム上でピークを探すことによって抽出できます. こ
の抽出されたモードは3Dモデルから計算されたモードとは値がずれてしまっています.
Monday, July 1, 13

41. ここで, この２つのモード間の距離を評価する関数を2つ定義します. ここに人間の聴覚
の特性を考慮するところもこの論文のコントリビューションの一つです. 最終的な評価
はこの２つの評価関数を掛け合わせて求めます.
Monday, July 1, 13

42. 一つめは画像距離で２つのスペクトログラムを画像と見立ててそれぞれのピクセルの
差分のノルムを評価関数とします. ただし, スペクトログラムをそのまま比べてしまうと
人間の聴覚にとって重要な情報が過小評価されてしまうので周波数を臨界帯域[Bark], 音
量をラウドネス[sone]として座標軸の変換を行います.
臨界帯域というのは音の大きさを弁別できる耳の周波数解像度で, ラウドネスは知覚的
な音の大きさです.
d Physically Based Modal Sound Synthesis • 1:7
time (s)
z(Bark)
0.2 0.4 0.6 0.8 1 1.2
0
10
20
0
1
2
f(kHz)
5
10
15
20
0
20
40
dB-SPLsone
A
A
B
B
. Different representation of a sound clip. Top: time domain signal
Middle: original image, power spectrogram P [m, ω] with intensity
ured in dB. Bottom: image transformed based on psychoacoustic prin-
. The frequency f is transformed to critical-band rate z, and the
ity is transformed to loudness. Two pairs of corresponding modes are
ed as A and B. It can be seen that the frequency resolution decreases
Example-G
Fig. 4. Psychoacoustics-related values: (a) the relationship between
critical-band rate (in Bark) and frequency (in Hz); (b) the relationship be-
tween loudness level LN (in phon), loudness L (in sone), and sound pressure
level Lp (in dB). Each curve is an equal-loudness contour, where a constant
loudness is perceived for pure steady tones with various frequencies.
it is constant when measured in Barks. Therefore, by transforming
the frequency dimension of a power spectrogram from f to z, we
obtain an image that is weighted according to human’s perceptual
Monday, July 1, 13

43. 評価関数２つ目はモード周波数と減衰係数の２次元分布間のノルムです. ただし, 3Dモ
デルから計算されたモードと録音素材から抽出されたモードが一対一に対応している
保障はないので近くの位置にあるモードに対して重み関数を掛けて正規化したのちに
ノルムを計算します.
1:8 • Z. Ren et al.
0 2 4 6 8 10 12 14
0
50
100
150
200
1
2
3
frequency (kHz)
damping(1/sec)
(a)
0 20 40 60 80 100 120 140
0
20
40
60
80
100
1
2 3
X(f)
Y(d)
(b)
Fig. 6. Point set matching problem in the feature domain: (a) in the original
frequency and damping, (f, d)-space; (b) in the transformed, (x, y)-space,
where x = X(f ) and y = Y(d). The blue crosses and red circles are
the reference and estimated feature points respectively. The three features
having the largest energies are labeled 1, 2, and 3.
For damping, although human can roughly sense that one mode
damps faster than another, directly taking the difference in damping
value d is not feasible. This is due to the fact that humans cannot dis-
tinguish between extremely short bursts [Zwicker and Fastl 1999].
For a damped sinusoid, the inverse of the damping value, 1/di, is
is a weighted averag
turn defined as
R
a weighted average o
point-to-point match
distance of feature p
values in the continu
points coincide, and
R(φi, ˜ ) = 1 when
and R( , ˜ ) = 1 w
matched. The weigh
adjust the influence o
feature points, ˜R, is
˜R
The match ratios f
sets are then combin
which measures how
Example-Guided Physically Based Modal Sound Synthesis • 1:9
by Eq. (8) in Section 3, the estimated modes
circle in the (ω, d)-space. Secondly, although
ence modes, they are not evenly excited by a
erve that usually the energy is mostly concen-
ant ones. Therefore, a good estimate of α and
that passes through the neighborhood of these
eature points. We also observe that in order to
lue, there must be at least one dominant esti-
quency of the most dominant reference mode.
ate our starting points by first drawing two
eature points from a total of Ndominant of them,
ssing through these two points. This circle is
circle, from which we can deduce a starting
ing Eq. (8). We then collect a set of eigenvalues
ned in Section 5.1) {(λ0
j , a0
j )}, such that there
λ0
k, a0
k ) that simultaneously satisfies λ0
k < λ0
j
Fig. 7. Residual computation. From a recorded sound (a), the reference
features are extracted (b), with frequencies, dampings, and energies depicted
as the blue circles in (f). After parameter estimation, the synthesized soundMonday, July 1, 13

44. 以上２つの画像距離評価関数と特徴量距離評価関数を最小化するようにモード周波数
と減衰係数を決定します.
Monday, July 1, 13

45. ここで, 求まったモードのみを鳴らしてしまうと倍音成分が非常に少なく, 非常に安っぽ
い音になってしまいます. これはより小さな雑音成分が抜けていることが原因です.
そこで録音素材からこの雑音成分を取り出してこれも3Dモデルにマッピングします.
Monday, July 1, 13

46. この雑音成分は録音素材のスペクトログラムと求まったモードによるスペクトログラ
ムの差で求まりますが, これをそのままランタイム時に鳴らしてしまうと物理的なシ
ミュレーションによるモードの励起状態の変化と分離してしまい不自然になってしま
います.
そこで実際にランタイム時に起こる録音素材のモードPsからの状態変化を関数としてこ
の雑音成分にも適用してやります.1:10 • Z. Ren et al.
Fig. 8. Single mode residual transform: The power spectrogram of a source
mode (f1, d1, a1) (the blue wireframe) is transformed to a target mode
ALGORITH
Input: source
source
Output: targe
← Determ
foreach mode
Ps
← Sh
Ps
← St
A ← Find
Ps
residual ←
sMonday, July 1, 13

47. 具体的にはランタイム時にPsからPtにモードの励起状態が変化したとき,
1: モード周波数がシフトする
2: 減衰係数(時間スケール)がスケーリングされる
3: 音量がスケーリングされる
が起こります. このシフトとスケーリングをそのモード周波数周辺の雑音成分にも適用
するとリアリスティックな効果が得られます.
Monday, July 1, 13

48. 課題は単一素材でできた物ならば上手くいくが, 接合部分を持ったようなもの(例えば多
くの楽器)には対応できないということ
Example-Guided Physically Based Modal Sound Synthesis • 1:13
Fig. 14. The estimated parameters are applied to virtual objects of various sizes and shapes, generating sounds corresponding to all kinds of interactions such
as colliding, rolling, and sliding.
Table II. Offline Computation for Material Parameter Estimation
Material #starting points average #iteration average time (s)
Wood 60 1011 46.5
Plastic 210 904 49.4
Metal 50 1679 393.5
Porcelain 80 1451 131.3
Table III. Material Recognition Rate Matrix: Recorded
Sounds
Recognized Material
Recorded Wood Plastic Metal Porcelain Glass
Material (%) (%) (%) (%) (%)
Wood 50.7 47.9 0.0 0.0 1.4
Monday, July 1, 13

49. Eulerian-on-Lagrangian Simulation
Ye Fan, Joshua Litven (University of British Columbia), David I.W. Levin (MIT CSAIL), Dinesh K. Pai (University of British Columbia)
2 •
(a) Eulerian-on-Lagrangian Grids (b) Large Deformation (c) Plasticity (d) Multiple Objects
Fig. 1: (a) The Eulerian-on-Lagrangian method embeds an Eulerian solid simulator in a Lagrangian grid. (b) Elastic spheres undergo a
collision with large deformations, and return to precise rest shapes. (c) Two plastic cylinders after colliding with a rigid rod. (d) Complex
contact between several Eulerian-on-Lagrangian objects.
tact. The end result is a simulator that can adaptively function as a
Lagrangian simulator, a fully Eulerian simulator or a combination
of the two.
3. METHODS
qi. 2 maps reference objects into their deformed state due to Eu-
lerian motion and 1 maps deformed objects into space via La-
grangian motion. The use of configuration variables is key to our
approach since it allows us to build in any Lagrangian motion us-
ing a reduced set of coordinates. In practice we compute 1
2 by
advecting material coordinates on a uniform grid located in the in-Monday, July 1, 13

50. 弾性体のシミュレーションはラグランジアンで離散化を行うのが一般的です. オイラリ
アンでやるのも可能ですが, 物体が広く動くと非効率な上に移流拡散誤差も生じるので
あまり一般的ではありません.
しかしメリットもあって明示的にメッシュを作る必要が無いのでスキャンしたボ
リュームデータをそのまま利用できたりメッシュが絡まる心配がありません.
そこでこの研究では両者のいいとこ取りをするために, 単一の物体の剛体運動を含む任
意の数の低次モードの運動だけをラグランジアンで計算して残りの細かい動きをオイ
ラリアン格子で計算します.
Monday, July 1, 13

51. まずEulerianとLagrangianを結ぶ中間座標系を定義します.
この中間座標系はLagrangianにとっては物体のローカル座標系であり, 一方Eulerianに
とっては変位の空間ドメイン表現となります.
中間座標系での変位は6面体有限要素法で離散化されているとし, そのときの形状関数
をNとします.
f n ⇥ 1 vector resulting from stacking all f
•
f df
dt in spatial domain
˙f df
dt in intermediate domain
[f] cross product matrix: f⇥
Material(z) Intermediate(y) Spatial(x )
Fig. 2: The continuous domains used for the Eulerian-on-Lagrangian sim-
ulator as well as the discrete grid structure stores in the Eulerian domain.
The mapping between the domains is described using a set of generalized
configuration variables.
Our exposition begins by describing three separate continu-
ous domains: The material domain, the intermediate domain (also
known as the reference domain) and the spatial domain (Figure 2).
We define invertible mappings i between these spaces and param-
eterize these mappings using generalized configuration variables
space) that map y
at z to its deform
(Lagrangian), wh
taking the time d
arrive at the spat
where [y] is the c
⌫ is a linear velo
from Equation 1
and so ˙u correspo
to the Eulerian-L
We take the L
axis-angle of rota
y) of the rigid fra
The displacem
main, y; this is
grangian approac
We use a hexahe
tions; the nodal d
q2. For a single e
ACM Transactions on Graphics, Vol. , No. , Article , Publication date: .
Monday, July 1, 13

52. 中間座標系において物体の低次から剛体モードを含む固有ベクトルをLagrangianで表現
したい数だけ並べた行列Uを使うと,
となる. よって速度は,
where N is a matrix of shape functions, U is the matrix of linear
modes for the element (comprised of the appropriate rows from the
global linear deformation basis), and q1 are the degrees of freedom
for the modal model. The spatial position of a given reference point
is then
x = N (z + u (z, t))) Uq1 (8)
and the velocity can be computed as
•
x = F 1v + NU ˙q1 (9)
where F 1 is the Lagrangian deformation gradient, given by
ry qT
1 UT
NT
and v is (as before) the intermediate space Eu-
lerian velocity. In this case we choose our generalized coordinates
to be q1 and the intermediate space, Eulerian displacements u. We
form the Lagrangian of the system and write the per-element mass
matrix, which in this case becomes
Mi
=
Z
⌦i
⇢ ¯J
✓
UT
NT
NU UT
NT
F 1N
NT
F T
1 NU NT
F T
1 F T
1 N
◆
d⌦i
. (10)
3.3 The Momentum Equation
We note that both the rigid body and linear modal element mass ma-
2 : y = z + u (z, t) ,
where N is a matrix of shape functions, U is the matrix of linear
modes for the element (comprised of the appropriate rows from the
global linear deformation basis), and q1 are the degrees of freedom
for the modal model. The spatial position of a given reference point
is then
x = N (z + u (z, t))) Uq1 (8)
and the velocity can be computed as
•
x = F 1v + NU ˙q1 (9)
where F 1 is the Lagrangian deformation gradient, given by
ry qT
1 UT
NT
and v is (as before) the intermediate space Eu-
lerian velocity. In this case we choose our generalized coordinates
to be q1 and the intermediate space, Eulerian displacements u. We
form the Lagrangian of the system and write the per-element mass
matrix, which in this case becomes
Mi
=
Z
i
⇢ ¯J
✓
UT
NT
NU UT
NT
F 1N
NT
F T
NU NT
F T
F T
N
◆
d⌦i
. (10)
となります. さらに微分すると, この系の運動方程式は
ocks are
diagonal
eference
e used to
ect’s rest
n which
ulation
n be uti-
eral case
ar modal
arbitrary
(7)
of linear
from the
freedom
nce point
(8)
where the diagonal blocks are the mass matrices for our individ-
ual Lagrangian and Eulerian dynamical systems (respectively) and
the off-diagonal blocks are the coupling matrices. Applying the
Lagrange-d’Alembert principle [Lanczos 1986] gives rise to the
equations of motion for our element. After assembling the per-
element mass matrices and force vectors we arrive at
M ¨q = fqvv + fe + fb + fc = f (12)
where M is the global mass matrix, fqvv are the centrifugal and
Coriolis forces acting on the Eulerian domain, fe are the forces of
elasticity, fb are body forces, and fc are forces due to contact.
The force fqvv results from the quadratic velocity terms in the
Euler-Lagrange equations [Murray et al. 1994]. We observe that
this global mass matrix retains the block structure of Equation 11
which is illustrated in Figure 3. From here we can proceed in a com-
pletely general fashion, assuming only that the mass matrix has the
described structure.
The crucial part of this system is the acceleration term ¨q, which
can be thought of as the solution to the set of differential equations
✓
¨q1
@v
@t
+ v · rv
◆
= M 1
f (13)
where we have expanded the material derivative for the sake of
clarity. One oft-used method of solving such a differential equation
is splitting, wherein one first solves
The displacement in Lagrangian:
z: Grid position, u: Displacement in Eulerian, q: Generalized coordinate
F: Lagrangian deformation gradient, v: The derivative of u
M: Mass Matrix
Monday, July 1, 13

53. この質量行列Mは
となり, 対角成分にLagrangian/Eulerianそれぞれ単独の質量行列, 対角に相互作用のブロッ
ク行列が並んだ形となっています.
ry q1 U N and v is (as before) the intermediate space Eu-
lerian velocity. In this case we choose our generalized coordinates
to be q1 and the intermediate space, Eulerian displacements u. We
form the Lagrangian of the system and write the per-element mass
matrix, which in this case becomes
Mi
=
Z
⌦i
⇢ ¯J
✓
UT
NT
NU UT
NT
F 1N
NT
F T
1 NU NT
F T
1 F T
1 N
◆
d⌦i
. (10)
3.3 The Momentum Equation
We note that both the rigid body and linear modal element mass ma-
trices share a similar block structure. In fact this structure is com-
mon to all Eulerian-on-Lagrangian (EoL) derivations. The mass
and the
which i
algorith
lerian g
Thus
grangia
at the v
describ
• 3
ivative of the Eulerian veloc-
rial derivative. In our method
t advection step (as is usually
n [2008] for a good descrip-
rial derivative into the equa-
11] for a depiction of such an
he Lagrangian function of our
V (q) (5)
[y] [y] Ni
I Ni
NiT
NiT
Ni
1
A d⌦i
, (6)
Fig. 3: The block structure of the Eulerian-on-Lagrangian mass matrix using
rigid motion. The diagonal blocks are Lagrangian and Eulerian respectively
while the off-diagonal blocks represent the coupling terms of the system.
ρ: Density, J: Deformation gradient determinant
Monday, July 1, 13

54. 以上の運動方程式に従ってEulerianでの変位, 速度, 中間座標系での一般化変位, 速度を更
新すると物体の運動が計算できます.
4 •
Algorithm 1 A high level overview of the Eulerian-on-Lagrangian
solution procedure. Here we explicitly denote the partial time
derivatives of Eulerian qualities for clarity.
1: Solve M
✓
¨q1
@v
@t
◆
= f for ˙qt+1
1 and v⇤
2: vt+1
= advect(v⇤
)
3: Solve
✓
q1
@u
@t
◆
=
✓
˙q1
t+1
vt+1
◆
for q1 and u⇤
4: u = advect(u⇤
)
To solve step 1 of Algorithm 1, we discretize the acceleration ¨q
and solve at the velocity-impulse level, which gives
M ˙qt+1
= tf + M ˙qt def
= pt+1
. (16)
Using the block structure of M and suppressing the time step t + 1
for clarity,
M1 ˙q1 + M12v⇤
= p1 (17a)
MT
12 ˙q1 + M2v⇤
= p2, (17b)
w
to
re
te
g
lu
a
˙q
Eulerian上での変位, 速度は流体シミュレーションの場合と同様に移流させます. また, 物
体の衝突はEulerian上でガウスの最小束縛の原理による最適化問題をQuadratic
Programmingを解いて解消します.
Monday, July 1, 13

55. Full Eulerianの場合と比べた結果がこちらで, 両者に一定の角初速度を与えた場合の角加
速度の時間変化です. 他に一切の外力を与えていないに関わらずFull Eulerianでは振動し
て収束していってしまう一方で提案手法では一定の値をキープできています.Fig. 6: Top Row: Purely Eulerian translation. Bottom Row: Eulerian-on-
Lagrangian Translation
0 2 4 6 8 10 12
0
1
2
3
Time (s)
AngularVelocity(m/s)
E−on−L
Eulerian
Fig. 7: The angular velocity of an Eulerian-on-Lagrangian object (E-on-L)
and a purely Eulerian object undergoing rotation. Both were given an initial
Fig. 9
Lagra
the Eo
Euleri
defor
Euler
Ackn
The a
anony
by gr
ter W
REFEMonday, July 1, 13

56. RealTime Dynamic Fracture withVolumetric
Approximate Convex Decompositions
Matthias Muller, Nuttapong Chentanez,Tae-Yong Kim (NVIDIA)
Figure 6: Impact-based fracture triggering. Approximately 600 compounds and 800 convexes are created during the simulation.
Figure 7: The arena after serious destruction resulting in 20k compounds and 32k convexes. Throughout the simulation, the fracture time
stays below 50ms.
NVIDIAのGPUデモでも使われてたリアルタイムに物体の破壊を再現するための研究.
Monday, July 1, 13

57. 始めに断っておくとこの研究は物理的に正しい破壊をシミュレーションするものでは
なくてゲームのフレームレート内でできるもっともらしい破壊を目指している.
従来ゲームなんかでは予め物体のどこが壊れるのかというのを階層的に決めておいて
ランタイム時にはそれを分離するということが多かった.
しかしこれだと例えばFPSでプレイヤーが銃で撃った場所と位置がずれた場所が崩壊し
てしまって非常に不自然だった.
この論文ではこれを解決するために物体の任意の場所を中心にリアルタイムに割れ方
を決定する方法について述べている.
Monday, July 1, 13

58. まず物体を凸抱の集合に分割する. これにはメッシュを囲む空間ごとボロノイ分割した
後に分割された面をメッシュにアラインすることで行う.
(a) (b)
(c) (d)
Figure 8: Separate fracture and rendering meshes. (a): A com-
pound mesh is created from a visual mesh (gray) via VACD. Inter-
active placement of Voronoi nodes provides intuitive control. (b):
The mesh is split along the Voronoi cell boundaries and an approx-
imate convex hull created for each sub-mesh. (c): At runtime a
Figure 9: The importance of working with polygons instead of tri-
angles. The figure shows a hexagon split twice using general polyg-
onal faces (top) and triangular faces only (bottom). In the latter
case, many ill shaped triangles are created even after only two cuts.
4 Mesh Preparation
As described in the previous sections, our fracture algorithm works
on compound meshes, not on visual meshes directly. Therefore we
have to create the compound mesh for a given visual mesh as a pre-
processing step. We would like to emphasize again that this mesh
preparation step is different from pre-fracturing models because it
is independent of the fracture pattern used at runtime.
Figure 2: Overview of the fracture algorithm. Lef
convex pieces are intersected with all cells. The
cell. Pieces within one cell become a new compou
Partial fracture. Pieces completely outside the imp
sphere (yellow) form one new compound.
Monday, July 1, 13

59. さらに任意のひび割れ模様の３次元テクスチャを用意する.
Red: Fracture Pattern by Artist,
Monday, July 1, 13

60. このテクスチャの中心を破壊が起こる中心にセットしてひび割れ模様と凸抱の交差し
た面を分割して破壊形状としています.
w of the fracture algorithm. Left: The fracture pattern (red) is aligned with the impact location (black
intersected with all cells. The green convex pieces can be welded to form a single piece because they
one cell become a new compound (coloring). Island detection finds that the dark red compound needs
ieces completely outside the impact sphere (orange) are not cut. All orange pieces plus the split pieces o
rm one new compound.
Red: Fracture Pattern by Artist,
Monday, July 1, 13

61. ただこのままだと破壊したい場所から離れた場所まで一緒に細かく破壊されてしまう
ので破壊中心から適当な半径外はテクスチャとの交差面でなく凸包で破壊します(図の
オレンジ).
ture pattern (red) is aligned with the impact location (black dot). Middle: All
ex pieces can be welded to form a single piece because they cover the entire
g). Island detection finds that the dark red compound needs to be split. Right:
(orange) are not cut. All orange pieces plus the split pieces outside the impact
Monday, July 1, 13

62. 課題は物体へかかる力を一切計算していないため, 自重で倒れる等が表
現できないこと
うさぎさんフルボッコ☝( ◠‿◠ )☝
Monday, July 1, 13

63. Thanks
Monday, July 1, 13