# 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.

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## References

1. Schumann, E.K.: Physik der klangfarben. Leipzig (1929)

2. Vogel, A., Reiter, S., Rupp, M., Nägel, A., Wittum, G.: Ug 4: a novel flexible software system for simulating PDE based models on high performance computers. Comput. Vis. Sci. 16(4), 165–179 (2013)

3. Reiter, S., Vogel, A., Heppner, I., Wittum, G.: A massively parallel geometric multigrid solver on hierarchically distributed grids. Comput. Vis. Sci. 16(4), 151–164 (2014)

4. Neymeyr, K.: A hierarchy of preconditioned eigensolvers for elliptic differential operators. Habilitation dissertation, University of Tübingen (2001)

5. Shaheen II. Shaheen II. https://www.hpc.kaust.edu.sa/content/shaheen-ii

6. Hackbusch, W.: Elliptic Differential Equations: Theory and Numerical Treatment. Volume 18 of Springer Series in Computational Mathematics. Springer, Berlin (2003)

7. Johnson, C.A., Bilhuber, P.H.: The influence of the soundboard on piano tone quality. J. Acoust. Soc. Am. 11(3), 311–320 (1940)

8. Wogram, K.: Acoustical research on pianos: vibrational characteristics of the soundboard. Das Musikinstrument 24, 694–702 (1980)

9. Wang, I.-C., Kindel, J.: Modal analysis and finite element analysis of a piano soundboard. In: IMAC, pp. 1545–1549 (1987)

10. Giordano, N.: Mechanical impedance of a piano soundboard. J. Acoust. Soc. Am. 103(4), 2128–2133 (1998)

11. Chabassier, J., Chaigne, A., Joly, P.: Modeling and simulation of a grand piano. J. Acoust. Soc. Am. 134, 648–665 (2013)

12. Polytec: http://www.polytec.com

13. Martin, D.: The Art of Making a Harpsichord. Robert Hale, London (2012)

14. Joly, P.: The mathematical model for elastic wave propagation. In: Kampanis, N.A., Dougalis, V.A., Ekaterinaris, J.A. (eds.) Effective Computational Methods for Wave Propagation. CRC Press, New York (2008)

15. Kollmann, F., Côté, W.A.: Principles of Wood Science and Technology. Volume 1: Solid Wood. Springer, Berlin (1968)

16. Rupp, M..: Ein filterndes algebraisches Mehrgitterverfahren mit Anwendungen in der Strukturmechanik. Dissertation, Universität Frankfurt (2017) (Unpublished manuscript)

17. Blender. Blender: https://www.blender.org

18. Reiter, S.: Effiziente Algorithmen und Datenstrukturen für die Realisierung von adaptiven, hierarchischen Gittern auf massiv parallelen Systemen. Dissertation, Universität Frankfurt (2014)

19. Reiter, S.: Promesh3d, meshing of unstructured grids in 1, 2, and 3 dimensions. http://promesh3d.com/. (Accessed July 2019)

20. Piperkova, R.: Diplomarbeit. Dissertation, Universität Frankfurt (2014)

21. Polytec: Psv-400 scanning vibrometer. http://www.polytec.com/us/products/vibration-sensors/vibrometer-accessories/scanning-vibrometer-accessories/psv-400/. (Accessed July 2019)

22. Giordano, N.: Sound production by a vibrating piano soundboard: experiment. J. Acoust. Soc. Am. 104(3), 1648–1653 (1998)

23. Braess, D.: Finite Elemente. Springer, Berlin (2003)

24. Corradi, R., Miccoli, S., Squicciarini, G., Fazioli, P.: Modal analysis of a grand piano soundboard at successive manufacturing stages. Appl. Acoust. 125, 113–127 (2017)

## Acknowledgements

The support by Merzdorf GmbH and Polytec GmbH is gratefully acknowledged. For computing time, this research used the resources of the Supercomputing Laboratory at King Abdullah University of Science and Technology (KAUST) in Thuwal, Saudi Arabia. Furthermore, we would like to thank Rossitza Piperkova for setting up an initial geometry of the soundboard.

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Correspondence to Lukas Larisch.

Communicated by Michael Heisig.

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## Appendix

### Appendix

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

### 1.1 Eigenmodes from experimental modal analysis

See Figs. 19, 20, 21 and 22.

### 1.2 Eigenmodes from simulation

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

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Larisch, L., Lemke, B. & Wittum, G. 3d Modeling and simulation of a harpsichord. Comput. Visual Sci. 23, 6 (2020). https://doi.org/10.1007/s00791-020-00326-1

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• DOI: https://doi.org/10.1007/s00791-020-00326-1