Acoustic Modeling of Reverberation using Smoothed Particle Hydrodynamics

1. Acoustic Modeling of Reverberation using Smoothed Particle Hydrodynamics Thomas Wolfe

2. Introduction

3. Presentation Itinerary • Goals • Motivation • Reverberation • Smoothed Particle Hydrodynamics • New Algorithms • Implementation • Results • Possible Future Work • Conclusion

4. Goals of the Research • Is SPH is a viable method for simulation of sound waves? • Can we accurately simulate complex environments for the sound to interact with? • Is SPH robust enough to allow materials with different acoustic response properties?

5. Motivation • Create reverberation effects for music. • Expose strengths and weaknesses of SPH as an acoustic modeling platform. • Acoustic modeling • Music industry • Architecture • Engineering • Auto and marine industries

6. Reverberation

7. What Is Reverberation? • Natural property of enclosed spaces. • Sound continues to echo after source has been removed. • Sound reflects off of walls, floors and other obstacles. • These reflections are called reverberation of the original sound.

8. Previous Reverberation Methods(physical) • Chamber • Uses real reverberation chamber (such as room or hall) to capture a sound with its echoes. • Plate • Metal plate that resonates according to a sound source and is picked up via an electromagnetic transducer on the plate. • Spring • Resonance in the spring induced by the transducer is recorded at the other end by a transducer.

9. Previous Reverberation Methods(other) • DSP • Copies original sound source many times and adds the result back into the original sound. • Sound Tracing • Adaptation of existing ray-tracing methods. • Traces a “beam of sound” from the listener back through reflections and then back to the sound source.

10. SPH as a Possible Solution • Limitations of previous methods • Physical methods are expensive and time-consuming to set up properly. • DSP methods are fast, but not accurate. • Sound tracing methods only deal well with static, simple environments. • SPH can be used as DSP methods are now. • SPH has the potential to be more accurate than DSP methods. • SPH can handle complex, dynamic environments.

11. Smoothed Particle Hydrodynamics

12. Smoothed Particle Hydrodynamics • Lagrangian CFD method • First used for astrophysics simulation in 1977 • General enough for astrophysics, fluid dynamics and deformable bodies • Uses probability distribution to determine field values • Derivatives only apply to kernel

13. Smoothing Kernels • Probability function for the SPH summation • Must satisfy the following constraints: and

14. Monaghan’s B-spline kernel

15. Monaghan’s B-spline kernel

16. Desbrun’s “Spiky” kernel

17. Desbrun’s “spiky” kernel

18. Müller’s Viscosity kernel

19. Müller’s Viscosity kernel

20. Determining Field Values in SPH(continuous to discrete) • A smoothed average is calculated by: • For discrete particles:

21. Determining Field Values in SPH(gradient and Laplacian) • The gradient of a field variable: • And similarly, the Laplacian:

22. SPH Versions of the Equations of Motion (Density) • Density must be calculated at every time step:

23. SPH Versions of the Equations of Motion (Pressure and Viscosity) • Force due to pressure gradient: • Force due to viscosity:

24. Equation of State • Used to calculate pressure • Ideal Gas Equation • Keeping temperature constant: where k is a constant depending on the gas (78120.9 J/kg at 273.15 K for air).

25. Boundary Treatment(force field) • At each timestep, for each particle/boundary pair, add a force to the particle using: where ks is the stiffness coefficient, kd is the dampening coefficient and x is the distance from the fluid particle to the boundary.

26. Boundary Treatment(virtual particles) • Boundaries are comprised of virtual fluid particles. • Virtual particles do not move with the fluid or evolve parameters such as density. • VPs exert a force on fluid particles according to Lennard-Jones potential (simple mathematical model that includes a van Der Waal’s attractive force at long ranges and a Pauli repulsion force at short ranges).

27. Boundary Treatment(Lennard-Jones potential) • In SPH, the Lennard-Jones potential is represented by: where n1 and n2 are 12 and 6, respectively, D is taken as the square of the largest velocity and r0 is the cutoff distance.

28. New Algorithms

29. Equilibrium • System runs with no acoustic input until kinetic energy is below a certain threshold (L is 0.00001 J): • After equilibrium is reached, the emitter and receiver are engaged.

30. Sound Emitter • Acts as a boundary to the fluid. • Allowed to move according to an input sound file. • The movement from DC is computed according to: where st is the input sample at time t and k is the deflection factor.

31. Sound Receiver • Captures density fluctuations caused by sound waves propagating through the fluid. • When simulation first reaches equilibrium, the receiver computes its reference density. • The density deflection from reference is computed and stored as an audio sample according to:

32. Implementation

33. Implementation • Original idea was realtime reverb effects processing. • C++ and OpenGL were chosen for speed and visualization. • Part of system code was taken from previous SPH water simulation created for the Computer Animation and Scientific Visualization course in Spring 2005.

34. Class Hierarchy

35. Class SPHSimulator • Main program class • Manages ALL SPH-related objects in the simulation. • Provides the following methods: • Load() – Load a new scene file • StepSimulation() – Progress the simulation one time step and render the results both to the screen and to the output sound file.

36. SPHSimulator::StepSimulation() • For each particle, assign neighboring particles to neighbor list. • For each particle, compute its density. • For each particle, compute its pressure. • For each particle, compute the force acting on the particle due the pressure gradient and the viscosity. • For each particle, compute the force acting on the particle due to the boundary forces (either using the Lennard-Jones potential or the boundary force equation). • Integrate the equations of motion to find the new velocity and position. • For each sound emitter, read the next sample in the input WAV file and move the emitter accordingly. • For each sound receiver, compute the deflection from the reference density and store it as a sample in an output WAV file.

37. Class SPHParticle • Stores a single particle’s state: • Density (kg/m3) • Mass (kg) • Pressure (Pa) • Fluid viscosity constant • Position vector (m) • Velocity vector (m/s) • Acceleration vector (m/s2) • List of neighboring particles

38. Class SPHKernel & Derivatives • SPHKernel is an abstract class that exposes the following methods: • W() – Kernel • dW() – Kernel derivative • lap() – Kernel Laplacian • 3 kernels are currently implemented: • Monaghan’s B-spline for density • Desbrun’s “spiky” kernel for pressure gradient • Müeller’s viscosity kernel

39. Classes SPHGrid and SPHGridCell • Used to find neighboring particles for a given particle. • SPH kernels have compact support, so any potential neighboring particles beyond a distance of h from the current particle can be ruled out as neighbors. • The grid is divided into cells of length h and a nearest neighbor search is performed for each particle.

40. SPHGrid::AssignNeighbors() • For each SPHGridCell in SPHGrid, clear the particle list. • For each particle i, • Clear the neighbor list. • Add i to the neighbor list (i is its own neighbor). • Search the cell adjacent to the cell containing i for neighbors. • A particle j in an adjacent cell is considered a neighbor of i only if the distance from i to j < h. If j is a neighbor of i, add a reference to j to i’s neighbor list • Add a reference to i to the current cell’s contents.

41. Class SPHBoundary & Derivatives • Encapsulates a boundary (wall, floor, obstacle). • Base class has position and orientation. • Subclassing SPHBoundary allows for different shapes and/or collision detection and response algorithms.

42. Class SPHSoundEmitter • Derived from one of the SPHBoundary concrete classes. • A configured maximum deflection allows for amplitude control of the output sound file. • At each time step, the emitter reads the next sample in the input sound file and moves itself along its local x-axis accordingly.

43. Class SPHSoundReceiver • Computes reference density at its local origin when equilibrium is first reached. • At each time step, the density is again computed and the deflection from its reference density is stored as an output sound sample. • The deflections are initially stored as double-precision floating point numbers from -1 to 1. • When stored in the WAV file, they are converted to 16-bit signed integers.

44. Class WAVFile • Stores the contents of a WAV audio file in memory. • Allows for random and sequential access to the audio sample data.

45. Class Camera • Used to visualize the particle system. • Allows the viewer to pan and rotate in all three dimensions.

46. Results

47. Simulation Test Run Setup • The runs are set up as scene files with user-definable parameters. • Using scene files allows the simulation results to be compared on a change-by-change basis to determine the best parameters for the system. • Most of the simulation parameters are exposed, including: • Timestep • Fluid parameters (number, initial volume, mass, viscosity, stiffness). • Boundary parameters (shape, size, orientation and collision response algorithm). • Emitter and receiver parameters (input and output sound filenames and emitter deflection constant).

48. Simulation Results Movie 1 (boundary force) Movie 2 (virtual particles) Audio Results

49. Future Work

50. Future Work • Improve the reverberation accuracy. • Equation of state • Thermal energy • Smoothing kernels • Acoustic Modeling • Building design • Auto, aircraft and marine • Music production