Printone: Interac:ve Resonance Simula:on for
Free-form Print-wind Instrument Design

Nobuyuki Umetani
Athina Panotopoulou
Ryan Schmidt
Emily Whi:ng

Free-form Wind-musical Instruments
Simula:on & op:miza:on to guide the instrument design
with correct tones
func:onal
instrument
Internal cavity
input shapemouth pieceﬁnger holeInterac:ve Design Interface

Interac:ve Hole-size Op:miza:onWhy Musical Instruments?
• It is fun to play with sounds!!

• There are many prac:cal applica:ons

speakerhousingmagnetron

resonator

antennamuﬄer

wave guideRelated Work: Sound Simula:onCompu:ng acous:c transfer for solid vibra:on[Zhang et al. 2009][Chadwick et al. 2009][Li et al. 2015][Raghuvanshi et al. 2010][O'Brien et al. 2002][James et al. 2006][Bonneel et al. 2008][Dobashi et al. 2003]

Related Works: Contact Sound Design
Our goal is interac:ve design of wind-instrument[Umetani et al. 2010][Bharaj et al. 2015][Musialski et al. 2016]

Related Works: Acous:c Sound Design
Our goal is 3D func:onal interac:ve design

our work[Allen et al. 2015][li et al. 2016]

Interac:vity

Interac:ve

Oﬄine op:miza:onInterac:ve

Simula:on DoF

2D

3D: Concatena:ons ofparameterized voxel

3D: free-form

Anatomy of Wind-Musical Instruments
Mouth pieceResonatorOutput Becomes Large at Resonance Freq.

Output from the resonatorfrequency: !

Input from mouth piece

resonanceLowest Resonance Frequency Determines the Tone
Fundamental resonancePower (dB)0-20
Overtones resonance frequency-40-60-80-100-120

0

5000

10000

15000

20000

Frequeny (Hz)25000

Design of Free-form Resonator is Diﬃcult
cavity-neck shape
Helmholtz resonatortube-like shape
air-column resonator

length : L

length : larea : avolume : V

0D mass-spring model

1D distributed mass-spring

free-form shapeMethods to Compute Acous:cs for 3D Shapes• Geometric – Path Tracing– Linaccurate for resonance[Funkhouser et. al 2002]• Time Domain – FTDT,FEM- Lineﬃcient for sta:onary response[Raghuvanshi et. al 2010]• Frequency Domain – FEM, BEM- Jeﬃcient for sta:onary response- Jaccurate[James et. al 2006]

What is the Acous:c Resonance?

How Boundary Element Simula:on Worksreﬂec:ngincident
self- pressurepressureinterac:on

How Boundary Element Simula:on Works

How Boundary Element Simula:on Works

Au = freﬂec:ngpressure

uselfinterac:onincident
pressure

A f

Solving Helmholtz Equa:on with BEM
Helmholtz equa:on(r2 + k)u(x) = (x)Solu:on is a combina:onof kernels

u(r) = e ikr /4⇡rTotal reﬂec:on boundary condi:on@u(x)/@x = 0
Dipole distribu:onon the surface

Solving Helmholtz Equa:on with BEM
We assemble matrix to capture interplay of reﬂec:ng pressure

u:reﬂec:ng
pressureassembling
rela:onships
at all points

AHow Boundary Element Simula:on Works
Eigenvalue ofsystem matrix

A(!)u = f
Frequency dependencyuA fSmall min. Eigenvalue gives a Large Ampliﬁca:on
Our goal is to ﬁnd a frequency ! that makes

Amplify ra:o ='uf'min (A) nearly zero

1min (A(!))

Eigen Analysis in Resonance Simula:on
Simple harmonic oscilla:on problem

Acous:c resonance simula:onKM
decomposable

A(!) = K(A) = 0!2 M

Solu:on with General
Eigenvalue Problem

A(!)

No decomposi:onavailable(A) = 0No easy solu:on!

Tradi:onal Frequency Sweep Method is Very Slow
Find smallest min. eigenvalue by compu:ng matrices andtheir eigenvalue for many sampled frequencies
Smallest minimum eigenvalueA(!)

Sample frequencies

frequency !

We Propose “Sparse Matrix Sampling”
We incrementally solve from sparsely sampled frequency
Smallest minimum eigenvalue

Ai 1

Ai+1

AiA(!)
!
i!

!

1i!

!
i+1

frequency !

Compu:ng Resonance Incrementally
Newton-Raphson

Our schememin [A(!)] = 0f (x) = 0
x=f (x)✓dfdx◆ 1f!==ff (x) = 0x

minhmax

Ai+1!i+1dAd!

⇥ dAd! A

Ai 1!i 11A⇤1

We use the power method to ﬁnd
the maximum eigenvalueiAutoTune: Automa:c Hole-size Op:miza:on
Analy:cally compu:ng the sensi:vity of the tone withrespect to the hole size
sensi:vity@@

Itera:ve minimiza:on ofdiﬀerence between simulated
and goal frequencies
1st order real-:me es:ma:on[Umetani et al 2011]

Live Demo!

“Puﬀ the Magic Dragon”with the Stanford Dragon

Beethoven Symphony #9

Ocarina with Genus 1 topology

FluteBig Ocarina

Saxophone

Chroma:c Tuner Test

Accuracy: Instrument can Produce Right Tones
The target tone and the lower & higher bound of sound
Only 4 cases in 104 target tones are out of tune

Passive Resonance Measurement: (<10Hz error)
loud speakermicrophone

Simulated
Resonance
frequencies

Key Contribu:onsu• Eigenvalue modeling of acous:c resonancef• Fast small minimum eigenvalue searchf (x)f• Interface to design wind instrumentf (x) = 0x

Limita:on: Simula:ng Timbre (Sound Quality)
Simula:ng overtones
Power (dB)0mouthpiece modelingturbulence simula:on-20-40-60-80-100-120

0

5000

10000

15000

20000

Frequeny (Hz)25000

[N Giordano 2013]

Thank You for Your Auen:on!
Acknowledgements• Anonymous reviewers• Alexander Barneu at Dartmouth College• NSF grant, Award #1464267