Stefan Bilbao | University of Edinburgh (original) (raw)

Papers by Stefan Bilbao

The accurate simulation of wave propagation is a problem of longstanding interest. In this articl... more The accurate simulation of wave propagation is a problem of longstanding interest. In this article, the focus is on higher-order accurate nite dierence schemes for the wave equation in any number of spatial dimensions. In particular, two step schemes (which operate over three time levels) are studied. A novel approach to the construction of schemes exhibiting both isotropy and accuracy is presented using modied equation techniques, and allowing for the specication of precise stencils of operation for the scheme, and thus direct control over stencil size and thus operation counts per time-step. Both implicit and explicit schemes are presented, as well as parameterised families of such schemes under conditions specifying the order of isotropy and accuracy. Such conditions are framed in terms of a set of coupled constraints which are nonlinear in general, but linear for a xed Courant number. Depending on the particular choice of stencils, it is often possible to develop schemes for which the traditional Courant-Friedrichs-Lewy condition is exceeded. A wide variety of families of such schemes is presented in one, two and three spatial dimensions, and accompanied by illustrations of numerical dispersion as well as convergence results conrming higher-order accuracy up to eighth order.

In room acoustics simulation and virtualization applications, accurate wall termination is a perc... more In room acoustics simulation and virtualization applications, accurate wall termination is a
perceptually crucial feature. It is particularly important in the setting of wave-based modeling
of 3D spaces, using methods such as the finite difference time domain method or finite volume
time domain method. In this article, general locally
reactive impedance boundary conditions are incorporated into a 3D finite volume time domain
formulation, which may be specialized to the various types of finite difference time domain
method under fitted boundary termination. Energy methods are used to determine stability
conditions for general room geometries, under a large family of nontrivial wall impedances, for
finite volume methods over unstructured grids. Simulation results are presented, highlighting in
particular the need for unstructured or fitted cells at the room boundary in the case of the
accurate simulation of frequency-dependent room mode decay times.

Modern computer hardware provides parallelism at various different levels - most obviously, multi... more Modern computer hardware provides parallelism at various different levels -
most obviously, multiple multicore processors allow many independent threads
to execute at once. At a finer-grained level, each core contains a vector unit
allowing multiple integer or floating point calculations to be performed with a
single instruction. Additionally, GPU hardware is highly parallel and performs
best when processing large numbers of independent threads. At the same time,
tools such as CUDA have become steadily more abundant and mature, allowing
more of this parallelism to be exploited.
In this paper we describe the process of optimising a physical modelling sound
synthesis code, the Multiplate 3D code, which models the acoustic response of
a number of metal plates embedded within a box of air. This code presented
a number of challenges and no single optimisation technique was applicable
to all of these. However, by exploiting parallelism at several different levels
(multithreading, GPU acceleration, and vectorisation), as well as applying other
optimisations, it was possible to speed up the simulation very significantly.

Recent bowed string sound synthesis has relied on physical modelling techniques; the achievable r... more Recent bowed string sound synthesis has relied on physical modelling techniques; the achievable realism and flexibility of gestural control are appealing, and the heavier computational cost becomes less significant as technology improves. A bowed string is
simulated in two polarisations by discretising the partial differential equations governing its behaviour, using the finite difference
method; a globally energy balanced scheme is used, as a guarantee
of numerical stability under highly nonlinear conditions. In one polarisation, a nonlinear contact model is used for the normal forces
exerted by the dynamic bow hair, left hand fingers, and fingerboard.
In the other polarisation, a force-velocity friction curve is used for
the resulting tangential forces. The scheme update requires the solution of two nonlinear vector equations.Sound examples and video
demonstrations are presented.

This paper presents a physical modelling sound synthesis environment for the production of valved... more This paper presents a physical modelling sound synthesis environment for the production of valved brass instrument sounds. The
governing equations of the system are solved using finite-difference
time-domain (FDTD) methods and the environment is implemented
in the C programming language. Users of the environment can
create their own custom instruments and are able to control player
parameters such as lip frequency, mouth pressure and valve openings through the use of instrument and score files.
The algorithm for sound synthesis is presented in detail along with
a discussion of optimisation methods used to reduce run time. Binaries for the environment are available for download online for
multiple platforms.

One goal of physical modelling synthesis is the creation of new virtual instruments. Modular appr... more One goal of physical modelling synthesis is the creation of new
virtual instruments. Modular approaches, whereby a set of basic
primitive elements can be connected to form a more complex instrument have a long history in audio synthesis. This paper examines such modular methods using finite difference schemes, within
the constraints of real-time audio systems. Focusing on consumer
hardware and the application of parallel programming techniques
for CPU processors, useable combinations of 1D and 2D objects
are demonstrated. These can form the basis for a modular synthesis environment that is implemented in a standard plug-in architecture such as an Audio Unit, and controllable via a MIDI keyboard.
Optimisation techniques such as vectorization and multi-threading
are examined in order to maximise the performance of these computationally demanding systems.

Stencil operations are often a key component when performing acoustics simulations, for which the... more Stencil operations are often a key component when performing
acoustics simulations, for which the specific choice of implementation
can have a significant effect on both accuracy and computational
performance. This paper presents a detailed investigation
of computational performance for GPU-based stencil operations
in two-step finite difference schemes, using stencils of varying
shape and size (ranging from seven to more than 450 points in
size). Using an Nvidia K20 GPU, it is found that as the stencil size
increases, compute times increase less than that naively expected by
considering only the number of computational operations involved,
because performance is instead determined by data transfer times
throughout the GPU memory architecture. With regards to the
effects of stencil shape, performance obtained with stencils that are
compact in space is mainly due to efficient use of the read-only data
(texture) cache on the K20, and performance obtained with standard
high-order stencils is due to increased memory bandwidth usage,
compensating for lower cache hit rates. Also in this study, a brief
comparison is made with performance results from a related, recent
study that used a shared memory approach on a GTX 670 GPU
device. It is found that by making efficient use of a GTX 660Ti
GPU—whose computational performance is generally lower than
that of a GTX 670—similar or better performance to those results
can be achieved without the use of shared memory

The Journal of the Acoustical Society of America, 2008

ABSTRACT

Journal of the Acoustical Society of America

A continued fraction expansion to the immittances defining viscothermal wave propagation in a cyl... more A continued fraction expansion to the immittances defining viscothermal wave propagation in a cylindrical tube has been presented recently in this journal, intended as a starting point for time domain numerical method design. Though the approximation has the great benefit of passivity, or positive realness under truncation, its convergence is slow leading to approximations of high order in practice. Other passive structures, when combined with optimisation methods, can lead to good accuracy over a wide frequency range, and for relatively low order.

journal of the audio engineering society, 2015

In the quest for synthetic sound of a natural acoustic character, fretted instruments such as the... more In the quest for synthetic sound of a natural acoustic character, fretted instruments such as the guitar present numerous challenges. For fully articulated synthesis, sample-based methods become unwieldy due to the large range of subtle variations in timbre and resultant storage requirements. A physical modeling approach thus becomes an attractive option. Here, a vibrating string is subject to intermittent contact/recontact phenomena along the length of the fretboard---and furthermore, the string is driven by a plucking interaction, and stopped by a finger, the position of which and force applied by are gestural parameters. The hypotheses underlying this model thus depart significantly from those which inform standard physical modeling methodologies, such as digital waveguides or modal synthesis, and an appeal to direct time space integration techniques is of interest. In this article, a finite difference time domain method is developed, with a penalty potential allowing for a convenient model of distributed collision. Implementation details are discussed, and simulation results and visualisations are presented illustrating a variety of typical playing gestures. Finally, given that such methods for highly nonlinear systems are prone to numerical instability, a brief description of an energy-balanced or Hamiltonian framework is provided, allowing for convenient numerical stability conditions.

Numerical Methods for Partial Differential Equations

This article is concerned with the numerical solution of the full dynamical von K{\'a}rm{\'a}n pl... more This article is concerned with the numerical solution of the full dynamical von K{\'a}rm{\'a}n plate equations for geometrically nonlinear (large-amplitude) vibration in the simple case of a rectangular plate under periodic boundary conditions. This system is composed of three equations describing the time evolution of the transverse displacement field, as well as the two longitudinal displacements. Particular emphasis is put on developing a family of numerical schemes which, when losses are absent, are exactly energy conserving. The methodology thus extends previous work on the simple von K{\'a}rm{\'a}n system, for which longitudinal inertia effects are neglected, resulting in a set of two equations for the transverse displacement and an Airy stress function. Both the semi-discrete (in time) and fully discrete schemes are developed. From the numerical energy conservation property, it is possible to arrive at sufficient conditions for numerical stability, under strongly nonlinear conditions. Simulation results are presented, illustrating various features of
plate vibration at high amplitudes, as well as the numerical energy conservation property, using both simple finite difference as well as Fourier spectral discretisations

Collisions play an important role in many aspects of the physics of musical instruments. The stri... more Collisions play an important role in many aspects of the physics of musical instruments. The striking action of a hammer or mallet in keyboard and percussion instruments is perhaps the most important example, but others include reed-beating effects in wind instruments, the string/neck interaction in fretted instruments such as the guitar as well as in the sitar and the wire/membrane interaction in the snare drum. From a simulation perspective, whether the eventual goal is the validation of musical instrument models or sound synthesis, such highly nonlinear problems pose various difficulties, not the least of which is the risk of numerical instability. In this article, a novel finite difference time domain simulation framework for such collision problems is developed, where numerical stability follows from strict numerical energy conservation or dissipation, and where a power law formulation for collisions is employed, as a potential function within a passive formulation. The power law serves both as a model of deformable collision, and as a mathematical penalty under perfectly rigid, non-deformable collision. Various numerical examples, illustrating the unifying features of such methods across a wide variety of systems in musical acoustics are presented, including numerical stability and energy conservation/dissipation, bounds on spurious penetration in the case of rigid collisions, as well as various aspects of musical instrument physics.

The computational requirements of finite difference schemes for the solution of the wave equation... more The computational requirements of finite difference schemes for the solution of the wave equation for physical modelling can be huge. Field programmable gate arrays (FPGAs) provide an ideal platform for performing highly parallel DSP computations, but the challenge is to be able to implement complex systems quickly and efficiently on FPGA platforms. The paper presents a system level design approach based on a dataflow model of computation using a particular finite difference scheme for the solution of a 2+1D wave equation. The results suggest that 84000 nodes could be accommodated on a single Virtex II FPGA.

The computational complexity of the finite difference (FD) schemes for the solution of the plate ... more The computational complexity of the finite difference (FD) schemes for the solution of the plate equation prevents them from being used in musical applications. The explicit FD schemes can be parallelized to run on multi-processor ar- rays for achieving real-time performance. Field Program- mable Gate Arrays (FPGAs) provide an ideal platform for implementing these architectures with the advantages of low- power and small form factor. The paper presents a design for implementing FD schemes for the plate equation on a multi-processor architecture on a FPGA device. The results show that 64 processing elements can be accommodated on a Xilinx X2VP50 device, achieving 487 kHz throughput for a square FD grid of 50x50 points. Among the methods for the digital synthesis of sound, phys- ical modelling approaches involve modelling of the sound production mechanism rather than the actual sound. This brings advantages of greater expressivity and wider ranges of sounds. Sound production mechanisms are described by partial differential equations (PDEs). Finite difference (FD) methods are the most obvious way to solve PDEs iteratively on a computer. The methods involve discretization of time and space to transform the PDEs to difference equations that can be implemented digitally. However, the major drawback of these methods is the massive computational requirements arising from the high space and time sampling rates due to the conditions on stability and convergence of the finite dif- ference schemes. The computational complexity exceeds the capabilities of a single computer implementation and as a re- sult parallel implementations should be sought. A particular class of FD schemes, named explicit schemes, naturally lend themselves to parallel execution. As a result explicit FD schemes have traditionally been imple- mented on parallel computer networks or massively parallel computers for various applications involving the solution of PDEs. In the sound synthesis context, using parallel com- puter networks can be impractical due to the trend towards smaller audio hardware devices. Field Programmable Gate Arrays (FPGAs) are programmable logic devices having a large number of logic gates and dedicated units for signal processing such as RAMsand multiplierson chip. Withthese properties these devices are suitable for implementing mas- sively parallel processing element (PE) networks for parallel implementation of FD schemes. With the added advantages of low-power and small form-factor, they can be used as hardware accelerators for computationally demanding phys- ical modelling sound synthesis applications. The sound production mechanism in plates can be de- scribed by the classical Kirchoff plate model which gov- erns the small transverse displacements of a thin plate. This model can be used for synthesizing plate-based instrument sounds such as gongs and guitar bodies (2) and for digitally implementing plate reverberation. The PDE can be solved numerically by a number of explicit FD schemes. In this paper we will describe the implementation of an FD scheme for the solution of the plate equation on an array multi-processor designed on a Xilinx Virtex II FPGA device. This implementation can be connected to a host device or a computer to provide the acceleration needed for real-time musical performance. In a previous paper (1), we have de- scribed FPGA implementation of wave equation. The FD schemes for the plate equation differ from those of the wave equation in terms of computation and communication pat- terns which will be explained in Sections 2 and 3. In addi- tion, in this paper we discuss the aspects of boundary condi- tions and excitation of the scheme and their implications for hardware implementation.

A new approach to spatialisation, based on the idea of projecting sounds through virtual sounding... more A new approach to spatialisation, based on the idea of projecting sounds through virtual sounding objects, is proposed, and initial experiments are carried out using a modal synthesis model of a rectangular membrane. The effects of varying the dimensions and damping parameters of the model on the perception of spatial attributes are discussed, as well as the relevance to sound diffusion.

The implementation of physical models of musical instruments by finite difference methods can be ... more The implementation of physical models of musical instruments by finite difference methods can be highly computationally complex. This paper investigates the use of FPGAs to accelerate these numerical methods to allow real time production of sounds for musical applications. The methodology to derive a circuit architecture that effectively exploits the types of concurrency in the algorithm for real time performance is described. An initial implementation on Xilinx XC2VP50 device allows computation to be performed for producing 1 second of plate sound sampled at 44.1 kHz on a finite difference grid of size 100times100 in 0.84 s compared to around half an hour on a P4 Centrino 1.6 GHz laptop using MATLAB

arge vibrating plates are used as thunder sheets in orches-tras. We have extended the use of flat... more arge vibrating plates are used as thunder sheets in orches-tras. We have extended the use of flat plates by cementing a flat panel electroacoustic transducer on a large brass sheet. Because of the thickness of the panel, the output is subject to nonlinear distortion. When combined with a real-time input and signal processing algorithm, the active brass plate can become an effective musical instrument for performance of new music.

Les instruments de type cymbale et gongs peuvent être représentées, en vue de la synthèse sonore ... more Les instruments de type cymbale et gongs peuvent être représentées, en vue de la synthèse sonore par modèle physique, par des plaques et coques minces. Le son typique qu'ils produisent, brillant, sans hauteur tonale précise et avec un très large contenu fréquentiel, s'explique par la non-linéarité géométrique que l'on ne peut plus négliger étant donné l'ordre de grandeur de l'amplitude des vibrations, qui peut atteindre jusqu'à dix fois l'épaisseur. Dans cet étude, l'analogue dynamique de von Karman pour les plaques minces est utilisé comme modèle continu. Un attention particulière est portée au cas d'une plaque forcée harmoniquement avec une amplitude croissante. De nombreuses expériences, menées sur différentes cymbales et gongs, ont permis de mettre en évidence un scénario de transition vers le chaos impliquant deux bifurcations successives et trois régimes distincts. Le premier régime est périodique, le second quasipériodique avec excitation de modes présentant des relations de résonance interne avec l'excitation, le troisième est le régime chaotique où perceptivement on retrouve le son de la cymbale en mode de jeu usuel. L'objectif de cette étude est de reproduire numériquement cette transition. Pour ce faire, une plaque rectangulaire à bord libre est considérée. Un schéma aux différences finies, du second ordre en temps et en espace, et utilisant un schéma d'intégration numérique conservatif est utilisée. Un tel schéma est nécessaire afin d'avoir la stabilité nécessaire à la simulation de tels régimes (régime chaotique, problème numériquement raide, grande dimension de l'espace des phases), ce qui est démontrée en comparant la méthode conservative avec un schéma symplectique de Störmer-Verlet. les résultats numériques permettent la reproduction du schéma de transition observé expérimentalement. Une fois le régime chaotique atteint, le formalisme de la turbulence d'ondes peut être utilisé afin de décrire les propriétés statistiques de la vibration. On montre que le schéma numérique reproduit exactement les prédictions théoriques du régime turbulent pour les spectres de puissance de la vitesse en un point.

The simulation of classic electromechanical musical instruments and audio effects has seen a grea... more The simulation of classic electromechanical musical instruments and audio effects has seen a great deal of activity in recent years, due in part to great recent increases in computing power. It is now possible to perform full emulations of relatively complex musical instruments in real time, or near real time. In this paper, time domain finite difference schemes are applied to the emulation of the Hohner Clavinet, an electromechanical stringed instrument exhibiting special features such as sustained hammer/string contact, pinning of the string to a metal stop, and a distributed damping mechanism. Various issues, including numerical stability, implementation details, and computational cost will be discussed. Simulation results and sound examples will be presented.

We know that the starting transient of a note is very important for the listener in determining t... more We know that the starting transient of a note is very important for the listener in determining the character of the note, and that this is also true of inter-note transients, or slurs. These transients, and the ease with which they can be executed, play an important role for the player in assessing the quality of a brass instrument. A skilled player may be able to make a slurred transient, for example, played on a poor instrument sound convincing to the listener, but is likely to prefer an instrument on which the same slur can be performed more easily. Recent studies using high speed video cameras, and mouthpieces designed to allow optical access, have revealed much about the mechanics of the brass player's lips and the initiation of the coupling between the lips and the air column, for both starting transients and slurs. In this paper, through the exploitation of a recently developed time domain model of brass instruments, we explore upward and downward slurs from one note to another. Of particular interest is the ease with which the player can slur over larger intervals which encompass one or more intermediate resonant modes.

The Journal of the Acoustical Society of America, 2008

ABSTRACT

Journal of the Acoustical Society of America

journal of the audio engineering society, 2015

Numerical Methods for Partial Differential Equations