Acoustic Simulations with FEM. 1. FE Model 2. Modal Analysis 3. Response Analysis. Physics: P

Transcription

1 Acoustic Simulations with FEM 1. FE Model 2. Modal Analysis 3. Response Analysis Physics: P P( x, t) The ear perceives local oscillations of pressure in the acoustic medium. Mathematics: Pressure is a scalar function scalar wave equation time-harmonic t Helmholtz equation 2 p + k p = 0 HAW/M+P, Ihlenburg, CompA Acoustics with FEM 1

2 1. FE Model The cavity volume is partitioned into 3-D elements (Hexa, Tetra, etc.). Each node is associated with one DOF (pressure). p ( x, y, z) p ( x ) i The acoustic element is actually a simplified version of the solid element. HAW/M+P, Ihlenburg, CompA Acoustics with FEM 2

3 Output of acoustic FRF run to pch file $TITLE = FLUID WITH ACOUSTIC SOURCE 1 $SUBTITLE= 2 $LABEL = Re, Im 3 $DISPLACEMENTS 4 $REAL-IMAGINARY OUTPUT 5 $SUBCASE ID = 1 6 $FREQUENCY = E S E E E CONT E E E CONT E E E CONT E E E p x, ω ( ) i Practical conclusion: do not animate results as displacements! HAW/M+P, Ihlenburg, CompA Acoustics with FEM 3

4 Solid Fluid t 3 pn 2 t 1 σ x τ xy τ xz τ xy σ y τ yz τ xz τ yz σ z t 2 pn 1 P P P P pn 2 The fluid element is a solid element representing the hydrostatic stress state. The nodal unknowns are the pressure values p(x). Since the pressure satisfies a second-order differential equation (wave or Helmholtz equation), linear interpolation can be used inside the elements. HAW/M+P, Ihlenburg, CompA Acoustics with FEM 4

5 Volume Elements in MSC/ Nastran CHEXA EID PID G1 G2 G6 Linear Interpolation 8 corner nodes.. Quadratic interpolation: + 12 edge nodes. HAW/M+P, Ihlenburg, CompA Acoustics with FEM 5

6 Fluid Elements in Nastran: Solid elements are used for fluid modeling. The elements are marked as PFLUID in the PSOLID card. Additional entry -1 to the GRID cards. GRID CHEXA PSOLID 3 3 PFLUID $LUFT MAT The nodal unknowns are the pressure values P(x,t). The.pch output for the fluid nodes is the complex-valued amplitude p(x). Real G E E E CONT E E E CONT E E E CONT E E E+00 2 Imag HAW/M+P, Ihlenburg, CompA Acoustics with FEM 6

7 PSOLID Card in MSC/Nastran PSOLID PID MID CORDM IN. FCTN HAW/M+P, Ihlenburg, CompA Acoustics with FEM 7

8 Input Deck for Modal Analysis (Sol 103) user-generated solution deck $ Modal Analysis SOL 103 CEND $ Write all results into.pch file DISP(PUNCH) = ALL METHOD = 100 BEGIN BULK $ compute all eigenfreqs up to 1000 Hz EIGRL,100,, $ finite element model INCLUDE 'box_fluid.nas' $ PARAM,COUPMASS,1 PARAM,AUTOSPC,YES $ ENDDATA FE-Model inside Nastran (Modal Equation) Mode Shape ( K ω 2 M ) x = 0 (1) Stiffness Matrix Eigenfrequency Mass Matrix HAW/M+P, Ihlenburg, CompA Acoustics with FEM 8

9 Eigenvalue Problems (EVP) Generalized EVP: Specialized EVP: Ax λbx = 0 Ax λx = 0 B = I The modal equation (1) is a generalized eigenproblem with 2 ω = λ Solution: { λ, x, = 1, } x N n i i i N n N The eigenfrequencies are printed out in a table (see next slide). The eigenvectors are collected column-wise in a matrix: [ x ] Φ = x1, x2,, n of dimension n N. The modes are mass-normalized, i.e. x Mx = δ T i j ij It follows that x Kx = ω T 2 i i i HAW/M+P, Ihlenburg, CompA Acoustics with FEM 9

10 Eigenfrequencies in.f06 file (Nastran) E I G E N V A L U E A N A L Y S I S S U M M A R Y (READ MODULE FOR FLUID) BLOCK SIZE USED... 7 NUMBER OF DECOMPOSITIONS... 1 NUMBER OF ROOTS FOUND NUMBER OF SOLVES REQUIRED R E A L E I G E N V A L U E S F O R F L U I D MODE EIGENVALUE RADIANS CYCLES NO. GENERALIZED GENERALIZED MASS STIFFNESS ` E E E E E E E E E E E E E E E E E E E E E E E E E+07 HAW/M+P, Ihlenburg, CompA Acoustics with FEM 10

11 Mode shapes (eigenvectors) Box: Passg. Car cabin: HAW/M+P, Ihlenburg, CompA Acoustics with FEM 11

12 3. Response Analysis Damping Matrix Dynamic load with forcing frequency ω 2 K + iω C ω M p = f Direct solution: Modal Superposition: p = Φ y p 2 1 = K + iω C ω M f K C M y f T 2 T Φ + iω ω Φ = Φ T T Φ KΦ, Φ MΦ are diagonal matrices. If modal damping T Φ CΦ is assumed, then is also diagonal.. HAW/M+P, Ihlenburg, CompA Acoustics with FEM 12

13 Damping effects in cavities Sound energy of is lost (within or from a bounded cavity) by: a) volume damping (internal friction within the cavity volume) b) interface damping (friction with boundary interfaces, particularly porous materials) c) radiation damping and transmission (sound is leaving the cavity) b) a) c) HAW/M+P, Ihlenburg, CompA Acoustics with FEM 13

14 Volume (Material) Damping FEM: η ω 2 K(1 + i ) M p = f Loss factor Nastran: PARAM, GFL HAW/M+P, Ihlenburg, CompA Acoustics with FEM 14

15 Modal Damping (Fluid) Consider the reduced undamped equation T 2 T 2 2 Φ KΦ ω Φ MΦ y = diag( ωi ) ω I y = f diag(2 ξ ω ) and add a modal damping matrix system for damped vibrations in the form 2 2 diag ( ωi ) iω diag (2 ξiωi ) ω I y = f i i to obtain the It is usually not easy to determine the modal damping factors. In practice one often uses frequency-constant modal damping parameters. Relation to material damping: Equations (1) and (2) can be compared formally by setting iη K η ω = iω C ω 2 diag ( i ). Nastran: TAMDMP1 Modal transform then leads to the diagonal damping matrix ξ i Constant (over the volume) material damping is numerically equivalent to constant (over the range of eigenfrequencies) modal damping with the relation η ξi 2 HAW/M+P, Ihlenburg, CompA Acoustics with FEM 15

16 Interface Damping The normal impedance Z n =p n /v n of a plane wave can be written as Z 0 p ' = ik p, Z0 = ρ0c Z n Nastran: generate fluid skin (preprocessor) QUAD, CTRIA CAABSF (editor) introduce material data in PAABSF card HAW/M+P, Ihlenburg, CompA Acoustics with FEM 16

17 p = Z v n n, Z = R + ix n n n Resistance (Damping) Reactance (Mass, Stiffness) R X n n ω ( ) ω ( ) Rn const. X const. n HAW/M+P, Ihlenburg, CompA Acoustics with FEM 17

18 Example $ rho*c for rho=1e-12, c=6e4, Z=20rho*c INCLUDE 'cavity_skin.bdf' $ $ $ $ PID TZREID TZIMID S A B K RHOC PAABSF e-8 6.e-8 HAW/M+P, Ihlenburg, CompA Acoustics with FEM 18

19 Measured Impedances (1/2) Z n = r n + ix n F. Fahy, Foundations of Eng. Acoustics, S. 171ff HAW/M+P, Ihlenburg, CompA Acoustics with FEM 19

20 Summary: boundary conditions for the Helmholtz equation Physical Mathematical FEM Rigid wall p n = reflecting Absorption p = n ik Z 0 Z n p CAABSF reflecting to (almost) nonreflecting Free surface p = 0 SPC reflecting HAW/M+P, Ihlenburg, CompA Acoustics with FEM 20

21 Frequency Response to Acoustic Source, Direct Solution SOL 108 CEND TITLE= FLUID WITH ACOUSTIC SOURCE, DIRECT FREQUENCY RESPONSE FREQ=200 DLOAD = 1000 $ Pressure in all nodes are output DISP(PLOT) = ALL $ BEGIN BULK $ DAMPING PARAM, GFL, 0.05 $ Source in Node 1000 $ SID EXCITEID DELAY DPHASE TP RHO B ACSRCE $ _ _ _ _ _ _ _ TABLED E ENDT $ FRQUENCY RANGE FREQ e3 2.e3 INCLUDE 'box_fluid.bdf' $ PARAM,COUPMASS,1 $ directs output to.op2 file PARAM, POST, -1 $ ENDDATA for plots of operational shapes HAW/M+P, Ihlenburg, CompA Acoustics with FEM 21

22 Frequency Response to Acoustic Source, Modal Solution $ COPIED AND ADAPTED FROM REFERENCE MANUAL SOL 111 CEND TITLE= FLUID WITH ACOUSTIC SOURCE FREQ=200 METHOD(FLUID)=50 DLOAD = 1000 $ Pressure output of selected MP to punch file SET 123=10224,10235,10243,10251 DISP(SORT1,PUNCH) = 123 $ BEGIN BULK PARAM, GFL, 0.05 $ SID EXCITEID DELAY DPHASE TP RHO B ACSRCE TABLED E ENDT $ FRQUENCY RANGE $ _ _ _ _ _ _ _ _ FREQ EIGRL e3 INCLUDE 'box_fluid.bdf' $ PARAM,COUPMASS,1 $ ENDDATA for graphs of frequency response in selected nodes HAW/M+P, Ihlenburg, CompA Acoustics with FEM 22

23 09:18:41 Analysis started. Log, Direct solution 09:18:41 Finite element model generation started. 09:18:41 Finite element model generated 425 degrees of freedom. 09:18:41 Finite element model generation successfully completed. 09:18:41 Application of Loads and Boundary Conditions to the finite element model started. 09:18:41 Application of Loads and Boundary Conditions to the finite element model successfully completed. 09:18:41 Solution of the system equations for frequency response started. 09:18:41 Solution of the system equations for frequency response successfully completed. 09:18:41 Frequency response analysis completed. 09:18:41 29 records read from JID file "c:/home/lehre/compa/nastran/box/fluid_s111/box_fluid_s108.dat" 09:18:41 A total of 969 records were read from 2 files. 09:18:41 NSEXIT: EXIT(0) 09:18:41 Analysis complete 0 11:17:26 Analysis started. Log, Modal solution 11:17:28 Finite element model generation started. 11:17:29 Finite element model generated 425 degrees of freedom. 11:17:29 Finite element model generation successfully completed. 11:17:29 Application of Loads and Boundary Conditions to the finite element model started. 11:17:29 Application of Loads and Boundary Conditions to the finite element model successfully completed. 11:17:29 Solution of the system equations for normal modes started. 11:17:31 Solution of the system equations for normal modes successfully completed. 11:17:32 Solution of the system equations for frequency response started. 11:17:32 Solution of the system equations for frequency response successfully completed. 11:17:33 Frequency response analysis completed. 11:17:33 27 records read from JID file "c:/home/lehre/compa/nastran/box/fluid_acsrce/box_fluid_s111_pch.dat" 11:17:33 A total of 967 records were read from 2 files. 11:17:33 NSEXIT: EXIT(0) 11:17:33 Analysis complete 0 HAW/M+P, Ihlenburg, CompA Acoustics with FEM 23

24 Head of punch file $TITLE = FLUID WITH ACOUSTIC SOURCE 1 $SUBTITLE= 2 $LABEL = 3 $DISPLACEMENTS 4 $REAL-IMAGINARY OUTPUT 5 $SUBCASE ID = 1 6 $FREQUENCY = E S E E E CONT E E E CONT E E E CONT E E E S E E E CONT E E E CONT E E E CONT E E E S E E E CONT E E E CONT E E E CONT E E E S E E E CONT E E E CONT E E E CONT E E E $TITLE = FLUID WITH ACOUSTIC SOURCE 24 $SUBTITLE= 25 $LABEL = 26 $DISPLACEMENTS 27 $REAL-IMAGINARY OUTPUT 28 $SUBCASE ID = 1 29 $FREQUENCY = E HAW/M+P, Ihlenburg, CompA Acoustics with FEM 24

25 Industrial Case Studies, Verification and Validation of FEM = check computational results by alternativemodels or methods of compuation. comparison to physical experiment Do we solve the problem right? Do we solve the right problem? J. Tinsley Oden Verification Fluid Modes (with A. Müller, Audi AG) Validation Fluid Modes (with S. Wegner, A. Kropp, R. Stryczek, BMW) Radiation of Sound (cooperation with BMW, Dept. EA) HAW/M+P, Ihlenburg, CompA Acoustics with FEM 25

26 Influence of mesh quality on eigenfrequencies of fluid cavity Manually generated mesh HAW/M+P, Ihlenburg, CompA Acoustics with FEM 26

27 Manual Mesh, high quality GRID CHEXA CTETRA 147 HAW/M+P, Ihlenburg, CompA Acoustics with FEM 27

28 Automatic mesh generator, fine mesh GRID CHEXA CTETRA Querschnitt HAW/M+P, Ihlenburg, CompA Acoustics with FEM 28

29 Automatic mesh generator, coarse mesh GRID CHEXA 9871 CTETRA 8782 Querschnitt HAW/M+P, Ihlenburg, CompA Acoustics with FEM 29

30 Results of modal analysis Fluid Mesh Mode Manual Automatic Division 13 Automatic, Division HAW/M+P, Ihlenburg, CompA Acoustics with FEM 30

31 Validation Fluid Modes Coupled Response of an automotive body to harmonic excitation of hatch cover (constant pressure on whole area) HAW/M+P, Ihlenburg, CompA Acoustics with FEM 31

32 Setup Excitation Experiment: loudspeaker, pink noise FEM: harmonic excitation of hatch door Microphone Positions HAW/M+P, Ihlenburg, CompA Acoustics with FEM 32

33 Methodical question Which interior parts should be in the model? HAW/M+P, Ihlenburg, CompA Acoustics with FEM 33

34 Pipe Eigenmodes Compare: First three Eigenmodes of Car Cabin HAW/M+P, Ihlenburg, CompA Acoustics with FEM 34

35 Variants Var 1: Vollausstattung ( mit / ohne Luftspalt bei Hutablage) Var 2: ohne Hutablage f1=38.4 / 76 Hz f1=52.0 Hz Var 3: ohne Hutablage, ohne Rücksitz Var 4: ohne Hutablage, ohne Sitze f1=64.4 Hz f1=70.1 Hz HAW/M+P, Ihlenburg, CompA Acoustics with FEM 35

36 Effects pipe Helmholtz resonator HAW/M+P, Ihlenburg, CompA Acoustics with FEM 36

37 Modes for model with four seats, no trunk cover f = 52Hz f = 96Hz HAW/M+P, Ihlenburg, CompA Acoustics with FEM 37

38 Details HAW/M+P, Ihlenburg, CompA Acoustics with FEM 38

39 Variant: four seats, no hatrack Pressure level at microphone positions Computatio nrechnung gute Überseinstimmung Messung Measurement HAW/M+P, Ihlenburg, CompA Acoustics with FEM 39

40 Variant: hatrack as sound barrier in cabin model Rechnung schlechtere Überseinstimmung Messung HAW/M+P, Ihlenburg, CompA Acoustics with FEM 40

41 Validation Radiation of engine vibrations Setup Microphone positions open top microphones Reflecting walls 35cm Concrete socket Engine block HAW/M+P, Ihlenburg, CompA Acoustics with FEM 41

42 FE Model Structural Model Fluid Cavity with absorbing BC above reflecting walls Simulation: excitation of structure, full structure-fluid compling, evaluation at microphone positions in fluid Fluid model: detail HAW/M+P, Ihlenburg, CompA Acoustics with FEM 42

43 Adaptive Approach Parameter 1: Size of fluid domain a = 39.1 cm a = 57.5 cm Parameter 2: Order of radiation condition (1,2) HAW/M+P, Ihlenburg, CompA Acoustics with FEM 43

44 Adaptive Approach: fluid domains h = 77.9cm h = 57.5 cm h = 39.1 cm HAW/M+P, Ihlenburg, CompA Acoustics with FEM 44

45 Computational Result: Frequency Response at Fluid MP Structural modes Verification and Validation HAW/M+P, Ihlenburg, CompA Acoustics with FEM F. Ihlenburg/ La Corunia 2011

46 Computational Result: Fluid Operational 724Hz Verification and Validation HAW/M+P, Ihlenburg, CompA Acoustics with FEM F. Ihlenburg/ La Corunia 2011

47 Comparison of adaptive Approach with measurement (space-averaged pressure) HAW/M+P, Ihlenburg, CompA Acoustics with FEM 47