Efficient outdoor sound propagation modelling in time-domain

Transcription

1 Efficient outdoor sound propagation modelling in time-domain Timothy Van Renterghem Acoustics Group Department of Information Technology Ghent University/iMinds CEAS, X-Noise : Atm. and ground effects on aircraft noise Sevilla, 213

2 Why time-domain models? Sound propagation is essentially a timedomain process Response over broad frequency range possible with a single run Including of non-linear effects Modelling realistic sources (moving, transient) Fluid-flow acoustics coupling can be treated more easily

3 Outline Finite-difference time-domain (FDTD) method Solving LEE Numerical discretisation strongly influences modelling efficiency computational cost numerical accuracy numerical stability In absence/presence of flow Finite absorbers Long distance sound propagation moving frame approach hybrid modelling

4 Sound propagation equations Linear continuous sound propagation equation in still air Navier-Stokes equations reduced to Momentum equation (velocity equation) Continuity equation (pressure equation) + (linear) pressure-density relation Assumptions Linearization in acoustical quantities Non-moving propagation medium No thermal, viscous effects and molecular relaxation No gravity v 1 p t p 2 c v t p c p t p p ldt l idx, jdy, kdz i, j, k

5 FDTD spatial aspects Lowest possible spatial stencil Two options for central differences Collocated-in-place (CIP) Staggered-in-place (SIP) p p dx p dx p dx p i1, j, k pi, j, k 2 3 1! x i, j, k 2! x 3! x i, j, k i, j, k dx p dx p dx p i1, j, k pi, j, k 2 3 1! x i, j, k 2! x 3! x i, j, k i, j, k... p p p x dx 2 3 i1, j, k i1, j, k 3 i, jk, 2dx 3! x i, j, k p... p p dx p dx p dx p i1, j, k pi.5, j, k ! x i.5, j, k 2.2! x 2.3! x i.5, j, k i.5, j, k dx p dx p dx p i, j, k pi.5, j, k ! x i.5, j, k 2.2! x 2.3! x i.5, j, k i.5, j, k p x i.5, j, k p p 2 3 i1, j, k i, j, k dx p 3 dx 4. 3! x i.5, j, k... dx i-1 i i+1 i i+.5 i+1

6 p x Extended spatial stencil x a i Taylor DRP a -3 =-a a -2 =-a a -1 =-a a Take more neighbouring cells to better approach the gradient E.g. Involve 6 neighbouring cells (7-point stencil, CIP, central differences) Dispersion-relation preserving (DRP) schemes Numerically optimize values of a i to further decrease phase error Drawbacks FDTD spatial aspects p( x 3dx) 9 p( x 2dx) 45p( x dx) 45p( x dx) 9 p( x 2dx) p( x 3dx) 6 O( dx ) 6dx Point source representation difficult Complicated boundary treatment Reduced time steps for numerical stability p x x 1 dx n in a p( x i idx)

7 p v l-1 FDTD temporal aspects Lowest-order schemes dt Two options for explicit schemes p v Collocated-in-time (CIT) Staggered-in-time (SIT) p v l l+1 p p 2dtc v l l2 2 l1 SIT is advantageous v l-1.5 v v p p p l-.5 l+.5 l+1.5 l l+1 Higher numerical accuracy with lowest order central difference scheme (see spatial discretisation) Halves memory use compared to CIT (in-place computation possible) Doubles time step compared to CIT (stability) l-1 dt l l p p dtc v 1 2 l-.5 v

8 FDTD temporal aspects High-order schemes By Taylor expansion In general : improves accuracy Strongly increases memory cost Advanced schemes Runge-Kutta Crank-Nickolson Implicit scheme Stability guaranteed

9 FDTD stability Numerical stability Time-delay system Update equations can be written as a discrete timedelay system (z-transform of SIT scheme) l l1 l.5 MP M 1P RV i i i NV N V SP l.5 l1.5 l1 i i i, 1.5 i i X m1 AX m X l P l l.5 V i M M M RN S M RN N A i i i i i i, Ni Si Ni Ni, 1.5 Poles of system should have a modulus smaller than or equal to one (or abs of eigenvalues of A should be smaller than 1)

10 FDTD stability Numerical stability Time-delay system : pole plots (SIT,SIP) CN=.5 CN=1 1 1 CN cdt dx dy CN=1.1

11 Numerical accuracy Two aspects FDTD accuracy Phase error Amplitude error SIP/SIT p-v FDTD is amplitude-error free At all Courant Numbers FDTD results in phase errors Phase error decreases with finer spatial discretisation Phase error vanishes when CN=1 Propagation along the diagonal of square cells sin ( k / 2) sin ( k dy / 2) xdx y sin ( kzdz / 2) 2arcsin c dt dx dy dz

12 FDTD accuracy Numerical accuracy

13 time-domain in OSP Including meteorological effects Concept of background flow Most relevant interactions between wind and acoustics in outdoor applications near the ground included Convection in uniform flows Refraction in non-uniform flows Scattering of sound No generation of sound The acoustics will not influence the macro-fluid flow p 2 c v+ v p t Inhomogeneous atmosphere No additional cost v 1 v v v v p t Highest sound speed determines stability criterion

14 time-domain in OSP Numerical discretisation Spatial discretisation : SIP still interesting Temporal discretisation needs care SIT is moderately unstable CIT is fully stable but computationally costly PSIT weakly unstable

15 time-domain in OSP Numerical discretisation PSIT scheme Second order terms in the flow speed are neglected during discretisation as wind speeds are typically low Explicit, efficient scheme still possible Numerical error» Small amplitude-errors appear» Phase error is not affected see Van Renterghem et al.(appl. Acoust., 27) Can be efficiently implemented l l1 2 l-.5 l.5 p p dtc v dtv p noflow l+.5 l-.5 1 l l l v v dt p dt v vnoflow dt vnoflow v ρ

16 time-domain in OSP Numerical discretisation

17 Finite absorbers Impedance boundary condition Locally reacting surfaces Models reflection at surfaces only Including a second medium in the simulation domain Extended reaction (non-local reaction) Models both reflection at surfaces, absorption inside, and transmission through materials Spatially heterogeneous materials

18 Impedance boundary cond. Direct convolution Frequency domain impedance definition: P ZV Each frequency domain signal or function has a time domain analogy Zt 1 Z In time domain, we need a convolution which is a computationally costly operation * p t Z t v t t t p t Z t v d Z t v d

19 Using exp. decaying time-domain functions Efficient direct convolution Z Recursive approach Series in j Easy time-domain equivalent Mass-spring-damper system Pade approximants Examples of application Attenborough 4-parameter model modified Zwikker and Kosten model Digital filters Impedance boundary cond. a 1 j t 1 Za 1 a a1j j t a t t e Z t p t a v t dta v t a 1 1 Efficient IIR filters Highly flexibility to approach any w-z curve Z( z) n i m i t dv t az i bz i dt i i

20 Including porous medium Poro-rigid frame model : Zwikker and Kosten Only the air in between the material matrix is allowed to vibrate Reasonable when density of the frame and the stiffness is significantly larger than those of air 3-parameter model (flow resistivity) (porosity) k s (structure factor) k v p t 2 p c v t s v Z Z ks j 2

21 Including porous medium Poro-elastic models : M.A. Biot Coupled movement of frame and air inside the porous medium included slightly adapted version Parameters Tortuosity: m t Porosity: a, f = 1- a Flow resistivity: Bulk modulus of frame: K f Frame density: f Frame damping coefficient: R f p t a Ka ava Ka P f v f va m a pa a f a t t a t v v 1 2 va v f p p K f f a f t a t v v f R v p v v m v v t 1 2 f f f f f a f a a f t t a

22 time-domain in OSP Long distance propagation Volume discretisation techniques not well suited for long-distance propagation Solutions Moving-frame FDTD Use of short, broadband pulses Mainly software challenge» Allocate and de-allocate memory in an efficient way Mainly efficient if propagation is essentially in onedirection Region a Region b Region c

23 Long distance propagation Solutions time-domain in OSP Hybrid modelling Many techniques highly efficient in particular cases Coupling in an attempt to combine best of both worlds Coupling in same domain» BEM-PE Cross-domain coupling» Raytracing-analytical formulae (e.g. diffraction)» BEM-raytracing» FDTD-PE

24 time-domain in OSP Long distance propagation Green s Function Parabolic Equation method PE-type model : one-way sound propagation, effective sound speed approach, range-dependent impedance and profiles, inclusion of terrain possible (CMM,GTPE,rGFPE), diffraction over thin screen, etc. Works with vertical array of acoustic pressures Extrapollation towards next array based on Green s function Discretisation in vertical direction : strong discretisation in forward direction : stepping at several wavelengths without loss in accuracy Efficiently uses FFT

25 Hybrid modelling Long distance propagation FDTD-GFPE Typical road traffic noise application Complex source region, less complex receiver zone

26 Long distance propagation FDTD-PE Hybrid modelling One-way coupling Interface vertical array of receivers in FDTD Time signal at each receiver recorded Fourier transform gives starting function for PE for all frequencies of interest Should be well chosen 1FDTD, multiple PE calulations

27 Hybrid modelling Long distance propagation FDTD-PE Example : evaluation of T-noise barriers in wind

28 Hybrid modelling Long distance propagation

29 Hybrid modelling Long distance propagation 2 FDTD-PE Starting fields magnitude (Pa) Hz magnitude (Pa) Hz height z (m) height z (m) magnitude (Pa) Hz height z (m)

30 Hybrid modelling location downwind noise barrier location starting field FDTD 2 m FDTD 4 m FDTD+PE 2 m FDTD+PE 4 m

31 Conclusions Time-domain modelling in outdoor sound propagation has become mature Low-order schemes are well suited in outdoor sound propagation when carefully choosing the numerical discretisation scheme Hybrid modelling for long-distance sound propagation Current trends in time-domain modelling Parallellisation (by using GPU) Pseudo-spectral time-domain technique (PSTD)