3d Modeling and simulation of a harpsichord

Abstract

The mathematical characterization of the sound of a musical instrument still follows Schumann’s laws (Schumann in Physik der klangfarben, Leipzig, 1929). According to this theory, the resonances of the instrument body, “the formants”, filter the oscillations of the sound generator (e.g., strings) and produce the characteristic “timbre” of an instrument. This is a strong simplification of the actual situation. It applies to a point source and can be easily performed by a loudspeaker, disregarding the three dimensional structure of music instruments. To describe the effect of geometry and material of the instruments, we set up a 3d model and simulate it using the simulation system UG4 (Vogel et al. in Comput Vis Sci 16(4):165–179, 2013; Reiter et al. in Comput Vis Sci 16(4):151–164, 2014). We aim to capture the oscillation behavior of eigenfrequencies of a harpsichord soundboard and investigate how well a model for the oscillation behavior of the soundboard approximates the oscillation behavior of the whole instrument. We resolve the complicated geometry by several unstructured 3d grids and take into account the anisotropy of wood. The oscillation behavior of the soundboard is modeled following the laws of linear orthotropic elasticity with homogenous boundary conditions. The associated eigenproblem is discretized using FEM and solved with the iterative PINVIT method using an efficient GMG preconditioner (Neymeyr in A hierarchy of preconditioned eigensolvers for elliptic differential operators. Habilitation dissertation, University of Tübingen, 2001). The latter allows us to resolve the harpsichord with a high resolution hybrid grid, which is required to capture fine modes of the simulated eigenfrequencies. We computed the first 16 eigenmodes and eigenfrequencies with a resolution of 1.8 billion unknowns each on Shaheen II supercomputer (https://www.hpc.kaust.edu.sa/content/shaheen-ii). To verify our results, we compare them with measurement data obtained from an experimental modal analysis of a real reference harpsichord.

Appendix

See Figs. 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18.

Fig. 9

Setup of the experimental modal analysis

Fig. 10

Eigenmodes for 5th and 16th Eigenfrequency on models $M_{1}$ and $M_{2}$. Left: 77.9 ${\mathrm{Hz}}$, 113.3 ${\mathrm{Hz}}$ , Right: 175.8, 224.5 ${\mathrm{Hz}}$

Fig. 11

Simulation on $M_{4}$ versus Experiment. Left: EF1 @ 81.9 ${\mathrm{Hz}}$ versus 68.5 ${\mathrm{Hz}}$ , Right: EF2 @ 86.6 ${\mathrm{Hz}}$ versus 75.5 ${\mathrm{Hz}}$

Fig. 12

Simulation on $M_{4}$ versus Experiment. Left: EF3 @ 95.5 ${\mathrm{Hz}}$ versus 97.5 ${\mathrm{Hz}}$ , Right: EF4 @ 101.6 ${\mathrm{Hz}}$ versus 104 ${\mathrm{Hz}}$

Fig. 13

Simulation on $M_{4}$ versus Experiment. Left: EF5 @ 115.3 ${\mathrm{Hz}}$ versus 110 ${\mathrm{Hz}}$ , Right: EF6 @134.7 ${\mathrm{Hz}}$ versus 146.5 ${\mathrm{Hz}}$

Fig. 14

Simulation on $M_{4}$ versus Experiment. Left: EF9 @ 178 ${\mathrm{Hz}}$ versus 198 , Right: EF10 @ 184 ${\mathrm{Hz}}$ versus 202.5 ${\mathrm{Hz}}$

Fig. 15

Simulation on $M_{4}$ versus Experiment. Left: EF11 @ 192.2 ${\mathrm{Hz}}$ versus 203.5 , Right: EF16 @226.11 ${\mathrm{Hz}}$ versus 234.5 ${\mathrm{Hz}}$

Fig. 16

Eigenmodes for 15th and 16th Eigenfrequency on models $M_{3}$ and $M_{4}$. Left: 223.9 ${\mathrm{Hz}}$, 226.05 ${\mathrm{Hz}}$, Right: 228.1 ${\mathrm{Hz}}$, 226.11 ${\mathrm{Hz}}$

Fig. 17

Eigenmodes for 9th and 10th Eigenfrequency on models $M_{2}$ and $M_{3}$. Left: 115.1 ${\mathrm{Hz}}$, 175.5 ${\mathrm{Hz}}$, Right: 125.1 ${\mathrm{Hz}}$, 177.8 ${\mathrm{Hz}}$

Fig. 18

Eigenmodes for 12th and 16th Eigenfrequency on models $M_{2}$ and $M_{3}$. Left: 193.1 ${\mathrm{Hz}}$, 198.8 ${\mathrm{Hz}}$, Right: 224.5 ${\mathrm{Hz}}$, 228.1 ${\mathrm{Hz}}$

1.1 Eigenmodes from experimental modal analysis

See Figs. 19, 20, 21 and 22.

Fig. 19

68.5 ${\mathrm{Hz}}$, 75.5 ${\mathrm{Hz}}$, 97.5 ${\mathrm{Hz}}$, 104 ${\mathrm{Hz}}$

Fig. 20

110 ${\mathrm{Hz}}$, 146.5 ${\mathrm{Hz}}$, 157.5 ${\mathrm{Hz}}$, 167.5 ${\mathrm{Hz}}$

Fig. 21

198 ${\mathrm{Hz}}$, 201.5 ${\mathrm{Hz}}$, 203.5 ${\mathrm{Hz}}$, 112.5 ${\mathrm{Hz}}$

Fig. 22

229.5 ${\mathrm{Hz}}$, 230.5 ${\mathrm{Hz}}$, 234.5 ${\mathrm{Hz}}$

1.2 Eigenmodes from simulation

See Figs. 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37 and 38.

Fig. 23

EF1: 28.5 ${\mathrm{Hz}}$, 81.2 ${\mathrm{Hz}}$, 81.1 ${\mathrm{Hz}}$, 81.9 ${\mathrm{Hz}}$

Fig. 24

EF2: 43.7 ${\mathrm{Hz}}$, 86.5 ${\mathrm{Hz}}$, 86.6 ${\mathrm{Hz}}$, 86.6 ${\mathrm{Hz}}$

Fig. 25

EF3: 57.9 ${\mathrm{Hz}}$, 94.7 ${\mathrm{Hz}}$, 95.4 ${\mathrm{Hz}}$, 95.5 ${\mathrm{Hz}}$

Fig. 26

EF4: 63.8 ${\mathrm{Hz}}$, 98.6 ${\mathrm{Hz}}$, 101.4 ${\mathrm{Hz}}$, 101.6 ${\mathrm{Hz}}$

Fig. 27

EF5: 77.9 ${\mathrm{Hz}}$, 113.3 ${\mathrm{Hz}}$, 114.7 ${\mathrm{Hz}}$, 115.3 ${\mathrm{Hz}}$

Fig. 28

EF6: 85.7 ${\mathrm{Hz}}$, 134.3 ${\mathrm{Hz}}$, 135.3 ${\mathrm{Hz}}$, 134.7 ${\mathrm{Hz}}$

Fig. 29

EF7: 99.9 ${\mathrm{Hz}}$, 145.8 ${\mathrm{Hz}}$, 147.9 ${\mathrm{Hz}}$, 146.9 ${\mathrm{Hz}}$

Fig. 30

EF8: 101.4 ${\mathrm{Hz}}$, 159.7 ${\mathrm{Hz}}$, 160.5 ${\mathrm{Hz}}$,159.3 ${\mathrm{Hz}}$

Fig. 31

EF9: 115.1 ${\mathrm{Hz}}$, 175.5 ${\mathrm{Hz}}$, 178.5 ${\mathrm{Hz}}$, 178 ${\mathrm{Hz}}$

Fig. 32

EF10: 125.1 ${\mathrm{Hz}}$, 177.8 ${\mathrm{Hz}}$, 184.3 ${\mathrm{Hz}}$, 184 ${\mathrm{Hz}}$

Fig. 33

EF11: 127.3 ${\mathrm{Hz}}$, 190.4 ${\mathrm{Hz}}$, 194.1 ${\mathrm{Hz}}$, 192.2 ${\mathrm{Hz}}$

Fig. 34

EF12: 143.1 ${\mathrm{Hz}}$, 193.1 ${\mathrm{Hz}}$, 198.8 ${\mathrm{Hz}}$, 197.2 ${\mathrm{Hz}}$

Fig. 35

EF13: 152.5 ${\mathrm{Hz}}$, 212.1 ${\mathrm{Hz}}$, 217.9 ${\mathrm{Hz}}$, 217.1 ${\mathrm{Hz}}$

Fig. 36

EF14: 156.1 ${\mathrm{Hz}}$, 218.6 ${\mathrm{Hz}}$, 222.9 ${\mathrm{Hz}}$, 222.8 ${\mathrm{Hz}}$

Fig. 37

EF15: 165.5 ${\mathrm{Hz}}$, 222.1 ${\mathrm{Hz}}$, 223.9 ${\mathrm{Hz}}$, 226.05 ${\mathrm{Hz}}$

Fig. 38

EF16: 175.8 ${\mathrm{Hz}}$, 224.5 ${\mathrm{Hz}}$, 228.1 ${\mathrm{Hz}}$, 226.11 ${\mathrm{Hz}}$