Comparative Visualization for Wave-based and Geometric Acoustics Eduard Deines 1, Martin Bertram 3, Jan Mohring 4, Jevgenij Jegorovs 4, Frank Michel 1, Hans Hagen 2, and Gregory M. Nielson 5 1 IRTG Kaiserslautern 2 University of Kaiserslautern 3 DFKI Kaiserslautern 4 ITWM Kaiserslautern 5 Arizona State University

2 Comparative Visualization for Wave-based and Geometric Acoustics Overview Motivation Acoustic Simulation Modified Phonon Tracing FEM-based solver Comparison approach Interference pattern visualization Gain visualization

3 Comparative Visualization for Wave-based and Geometric Acoustics Motivation Shortcomings of geometric and wave-based methods Combination of both approaches Numerical and visual comparison

4 Comparative Visualization for Wave-based and Geometric Acoustics Previous Work Phonon tracing, based on sound particles Computation of energy decomposition for each phonon (phonon emission) Construction of response filters for each listener position (phonon collection)

5 Comparative Visualization for Wave-based and Geometric Acoustics Previous Work Scattered data Listener based Surfaces Single particle

6 Comparative Visualization for Wave-based and Geometric Acoustics Overview Motivation Acoustic Simulation Modified Phonon Tracing FEM-based solver Comparison approach Interference pattern visualization Gain visualization

7 Comparative Visualization for Wave-based and Geometric Acoustics Modified Phonon Tracing Modeling linear pressure attenuation by Gaussian basis functions dilated proportional with traversed distance => approximates partition of unity for the basis functions Tracing pressure rather then energy => better comparison with FEM

8 Comparative Visualization for Wave-based and Geometric Acoustics Phonon Emission For each phonon: Pressure spectrum Virtual source Phonons current position One emmission, multiple collection passes for different listeners

9 Comparative Visualization for Wave-based and Geometric Acoustics Phonon Emission Reflection at scene elements Update virtual source Multiply pressure by Re-use path by next phonons Record phonons on scene elements Emission according to source distribution

10 Comparative Visualization for Wave-based and Geometric Acoustics Phonon Collection For each visible phonon: Add unit pulse translated by distance Scaling decreases with: Accumulated length Wall absorption Scaling depends on: Phonon outgoing direction Direction from virtual source to listener position

11 Comparative Visualization for Wave-based and Geometric Acoustics Phonon Collection Scaling depends on: Reference pressure at 1m from source Gaussian weighting function split unit pulse into frequency bands

12 Comparative Visualization for Wave-based and Geometric Acoustics Overview Motivation Acoustic Simulation Modified Phonon Tracing FEM-based solver Comparison approach Interference pattern visualization Gain visualization

13 Comparative Visualization for Wave-based and Geometric Acoustics Finite Element Method (FEM) Solving wave equation by FEM FEM approximates the wave equation by a large system of ODEs To many unknowns to solve the ODEs => reduction needed

14 Comparative Visualization for Wave-based and Geometric Acoustics Finite Element Method (FEM) Representation of the system dynamics by a superposition of eigenmodes The coefficients of the eigenmodes are the unknowns of the reduced system Samplewise constant input => transformation of the continuous state-space model into a discrete one

15 Comparative Visualization for Wave-based and Geometric Acoustics Finite Element Method (FEM) Typically few hundred unknowns rather than 10,000 degrees of freedom Required number of unknowns increases as the third power of the frequency => only for low- frequency bands

16 Comparative Visualization for Wave-based and Geometric Acoustics Overview Motivation Acoustic Simulation Modified Phonon Tracing FEM-based solver Comparison approach Interference pattern visualization Gain visualization

17 Comparative Visualization for Wave-based and Geometric Acoustics Test Scenario Geometry Simulation scenario

18 Comparative Visualization for Wave-based and Geometric Acoustics Simulation regular grid wave number k = {3, 6, 9, 12, 15} f = {164.25, 328.5, 492.7, 657, 821.2} Hz Phonon Tracing: point source, phonons FEM: small membrane

19 Comparative Visualization for Wave-based and Geometric Acoustics Overview Motivation Acoustic Simulation Modified Phonon Tracing FEM-based solver Comparison approach Interference pattern visualization Gain visualization

20 Comparative Visualization for Wave-based and Geometric Acoustics Interference Pattern Visualization Color mapping: Positive pressure to red Negative pressure to blue Reduction of saturation, depending on the absolute pressure value Mapping of pressure values below the hearing threshold to gray Bilinear interpolation of color values for additional grid points

21 Comparative Visualization for Wave-based and Geometric Acoustics Interference Pattern (k=6 / f=328.5 Hz) FE M Phonon Tracing

22 Comparative Visualization for Wave-based and Geometric Acoustics Interference Pattern (k=12 / f=657 Hz) FE M Phonon Tracing

23 Comparative Visualization for Wave-based and Geometric Acoustics Interference Pattern k=6 k=12 FE M Phonon Tracing

24 Comparative Visualization for Wave-based and Geometric Acoustics Overview Motivation Acoustic Simulation Modified Phonon Tracing FEM-based solver Comparison approach Interference pattern visualization Gain visualization

25 Comparative Visualization for Wave-based and Geometric Acoustics Gain Visualization Acoustic metric: influence of the room Error:

26 Comparative Visualization for Wave-based and Geometric Acoustics Gain Visualization Color coding: Use of HSV color space Red for maximum value Blue for minimum value Same range for different simulations Same range for all error plots

27 Comparative Visualization for Wave-based and Geometric Acoustics Gain - Error

28 Comparative Visualization for Wave-based and Geometric Acoustics Gain (k=3 / f= Hz) FEM Phonon Tracing Error

29 Comparative Visualization for Wave-based and Geometric Acoustics Gain (k=6 / f=328.5 Hz) FEM Phonon Tracing Error

30 Comparative Visualization for Wave-based and Geometric Acoustics Gain (k=12 / f=657 Hz) FEM Phonon Tracing Error

31 Comparative Visualization for Wave-based and Geometric Acoustics Acknowledgments Stiftung Rheinland-Pfalz fuer Innovation Kaiserslautern Excellence Cluster Dependable Adaptive Systems and Mathematical Modeling US Army Research Office

32 Comparative Visualization for Wave-based and Geometric Acoustics Thank you for attention!