The Use of Equivalent Source Models for Reduced Order Simulation in Room Acoustics

Transcription

1 Purdue University Purdue e-pubs Publications of the Ray W. Herrick Laboratories School of Mechanical Engineering The Use of Equivalent Source Models for Reduced Order Simulation in Room Acoustics Yangfan Liu Purdue University, J Stuart Bolton Purdue University, Follow this and additional works at: Liu, Yangfan and Bolton, J Stuart, "The Use of Equivalent Source Models for Reduced Order Simulation in Room Acoustics" (213). Publications of the Ray W. Herrick Laboratories. Paper 8. This document has been made available through Purdue e-pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information.

2 Yangfan Liu Advisor: J. Stuart Bolton

3 Room Acoustics with Source of Finite Size Sound prediction of a flat screen TV: Not point sources Arbitrary room shapes Fast Traditional Methods: Ray Tracing Image Source Mdl Model (hybrid) bid) /FEM Available techniques are not suitable for a fast and accurate room acoustics simulations of finite-size sources The euseo of Equivalent ae tsouce Source Models odes( (ESM) The actual sound field Governing Sound field Estimate ESM Equation from sources parameters by Boundary Boundary matching the Conditions Conditions BCs 2

4 Sound Field Components in Room Acoustics Room surface Source surface Free-space component Incoming component Scattering component Total sound field: p p p ut uf ur t f r Free-space ESM vs. Room Acoustics ESM Free-space ESM Outgoing waves only Pressure of the total sound field (sampled at a measurement surface) Source type Boundary condition Room Acoustics ESM Both incoming and outgoing waves Impedances of room component (sampled at both source and room surfaces) 3

5 BC of the Room Components in Room Acoustics Room surface Free-space component In-vacuo driving velocity on the source surface. (source characteristics) Source surface Room component Total sound field: p p p u u u t f r t f r Boundary conditions used in the room acoustics ESM: xpr() xunr() x x xpr() xunr() xunf() x xpf() x x

6 General Procedure of Constructing Room Acoustics ESM Location in the sound field N px ( ) gi( x, yi) Qi, i1 N 1 ux ( ) g( xyq, ), j i 1 i i i Location of each source Matrix form used to estimate the ESM parameters: (1) (1) Boundary condition BA 1 p A u on source surfaces Q u B p n BA A (2) (2) 2 p u nf n 2 f B1 diag( 1( x1), 1( x2),..., 1( xm )), 1 B diag( ( x ), ( x ),..., ( x )) 2 2 M 1 2 M 2 2 M 1 1 (1) (1) 1 ( Ap ) ij gj ( xi, yj ), ( Au ) (, ), n ij ngj xi yj j (2) (2) ( Ap ) ij gj( xm, ), ( ) (, ), 1i yj Aun ij ngj xm1i yj j 1 Boundary condition on room surfaces Construct a specific ESM (find the sound field expression) 5

7 Room Acoustics ESM Using Monopole Distributions Monopoles are distributed inside the source surface (outgoing wave) and outside the room surface (incoming wave) Sound field expression (2D): j (1) g( x, y) H ( kr) 4 Incoming waves Room Acoustics ESM Using Monopole Distributions Outgoing waves Multipoles of monopole, dipole, quadrupole, n th order, (both incoming and outgoing) Multipoles (incoming & Sound field expression (2D): outgoing) out j (1) in j (2) n = : P ( xy, ) H ( kr), P ( xy, ) H ( kr) 4 4 n th order: out out out P Sn S np n S nr n ( P ) v1v2... vn n, R in in in n PSn SnPn SnRn ( P ) v1v2... vn Details on next slide 6

8 Sound field expression for multipole sources (2D): Dipole: P X X d P X X v Sn1(, ) Sn (, ), n1 P Sd P v T P Sn( X X, ) PSn( X X, ) d,, vn1 x y S1 1, 1, Dipole strength: Sd 1 T Quadrupole: P Sd d v2 R v1 S , Quadrupole strength: Sd1d2 P P [, ] T x y P x P y x P x y P 2 y Order n: P Q g Sn n n Source strength: Q 1... n Sd d2 d g R ( P) v v... v, n n 1 2 n n R n - tensor outer product - tensor inner product n 7

9 Which source orientations should be included in the model? Even more nonlinear if the orientations are unknown. A source of arbitrary orientation can be decomposed into several standard source configurations. Derivative does not depend on the sequence of differentiation How many independent elements? Nnr (, ) C 1, n n nr 1, n Can be numerically enumerated for an arbitrary n. 8

10 Simulation in a 2 dimensional room (circular geometry) Geometry of the room (Rayls) Nomral Impedance ral Impedance (Rayls) Nomr Non-uniform Impedance in Case 1 Room Surface Source Surface Angle (radians) Impedance in Case Room Surface Source Surface Angle (radians) Norm mal driving veloc city Normal impedance on different surfaces Angle (radians) In-vacuo Driving velocity on the source surface 9

11 Singular Values ular Values Sing Different models used in the simulation Type of ESM Number of Parameters Multipole ESM (order up to 3) 1 Multipole ESM (order up to 6) 28 Monopole ESM (outside: 1; inside: 3) 13 Monopole ESM (outside: 15; inside: 45) 195 Result from a Boundary Monopole ESM (outside: 2; inside: 6) 26 Element Model was used as the true sound field. Monopole ESM (outside: 1; inside: 3) 13 Choice of regularization techniques Singular Value Distribution for Monople ESM (1,3) Index of Singular Values Singular Value Distribution for Monople ESM (2,6) Index of Singular Values Singular value distribution in monopole ESMs Use TSVD for multipole ESM Use GCV for monopole ESM Singular Values Sing gular Values Singular Value Distribution for Multipole ESM (3rd order) Index of Singular Values Singular Value Distribution for Multipole ESM (6th order) Index of Singular Values Singular value distribution in multipole ESMs 1

12 Results from monopole distribution ESMs Pressure Amplitude (Pa) Overall Prediction Comparison for Case 1 5 Monopole ESM (1,3 no regu) Monopole ESM (1,3 regu) 4 Pressure Amplitude (Pa) Overall Prediction Comparison for Case 1 Monopole ESM (1,3 no regu) Monopole ESM (1,3 regu) Frequency (Hz) Overall Prediction Comparison for Case Frequency (Hz) Overall Prediction Comparison for Case 2 Pressure Amplitu ude (Pa) Monopole ESM (1,3 no regu) Monopole ESM (1,3 regu) Pressure Amplitu ude (Pa) Monopole ESM (1,3 no regu) Monopole ESM (1,3 regu) Frequency (Hz) Spatially averaged prediction from monopole ESMs Frequency (Hz) Spatially averaged prediction from monopole ESMs Accurate (up to 5 Hz) if there are a large number of monopoles and with regularization. Regularization may cause instabilities. 11

13 Results from monopole distribution ESMs s sure A m plitude of Sound Pres Predicted Sound Pressure at All Receiver Locations (at 2Hz Case 1) Monopole ESM (1,3 regu) 25 Monopole ESM (15,45 regu) Monopole ESM (2,6 regu) Index of the Receivers s sure A m plitude of Sound Pres Predicted Sound Pressure at All Receiver Locations (at 4Hz Case 1) Monopole ESM (15,45 regu) Monopole ESM (2,6 regu) Index of the Receivers s ure A m plitude of Sound Press Predicted Sound Pressure at All Receiver Locations (at 2Hz Case 2) Monopole ESM (1,3 regu) Monopole ESM (15,45 regu) Monopole ESM (2,6 regu) Index of the Receivers Prediction from monopole ESMs at 2 Hz s ure p litude of Sound Press Amp Predicted Sound Pressure at All Receiver Locations (at 4Hz Case 2) Monopole ESM (15,45 regu) Monopole ESM (2,6 regu) Index of the Receivers Prediction from monopole ESMs at 4 Hz A monopole ESM requires at least 26 model parameters to achieve accurate predictions. 12

14 Results from multipole ESMs Regularization does not cause instabilities (more robust). The spatially averaged predictions are similar with different multipole orders (accurate up to 5 Hz). Multipole ESM requires much fewer number of model parameters than monopole ESM a) Pre essure Amplitude (P Pressure Am mplitude (Pa) Overall Prediction Comparison for Case 1 Multipole ESM (3rd) Multipole ESM (6th) Frequency (Hz) 5 Overall Prediction Comparison for Case 2 Multipole ESM (3rd) Multipole ESM (6th) Frequency (Hz) Spatially averaged prediction from multipole ESMs 13

15 Results from multipole ESMs Pressure Amplitude (Pa) Pressure Am mplitude (Pa) Prediction Comparison at 2 Hz (Case 1) Multipole ESM (3rd) Multipole ESM (6th) Index of the Receivers Prediction Comparison at 2 Hz (Case 2) 3 Multipole ESM (3rd) 2 Multipole ESM (6th) Index of the Receivers Prediction from multipole ESMs at 2 Hz Pressure Amplitude (Pa) Pressure Am mplitude (Pa) Prediction Comparison at 4 Hz (Case 1) Multipole ESM (3rd) Multipole ESM (6th) Index of the Receivers Prediction Comparison at 4 Hz (Case 2) Multipole ESM (3rd) Multipole ESM (6th) Index of the Receivers Prediction from multipole ESMs at 4 Hz Increase of multipole order improves the prediction oscillation in space. It is flexible to balance the computational effort and the prediction accuracy. (choose an appropriate truncation order 14

16 Equivalent source models are constructed for room acoustics simulations with finite-size source, arbitrary geometry and non- uniform surface normal impedances. In room acoustics ESMs, both outgoing and incoming waves should be included, and the impedance boundary conditions for the room component sound field are used for parameter estimation. Both monopole ESMs and multipole ESMs can achieve accurate performance up to 5 Hz, but the multipole ESM requires fewer model parameters and is more robust. When using multipole ESM, there is a flexible balance between the computational effort and the prediction accuracy, controlled by choosing an appropriate truncation order. 15