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Synchronous exploration for the control of real-time external noise suppression in a three-dimensional subdomain

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Abstract

A mathematical (difference) model is proposed for a real-time active shielding device that shields an acoustic field in a given subdomain from the influence of sound sources located in an additional subdomain. An algorithm for computing the current control ensuring a prescribed process is based on information produced by the author’s technique of synchronous weak noise exploration. This information can be measured in real time. Active control problems for nonstationary solutions of linear difference equations in a three-dimensional domain consisting of two subdomains are studied using the difference potential method. The shape of the domain and the boundary conditions may depend on time, while the coefficients may depend on time and spatial coordinates. If the difference problem is a mathematical model of sound propagation, the goal of control is to change the acoustic field in the given subdomains, for example, to shield the acoustic field in one subdomain from the undesirable influence (noise) of sources located in the other subdomain.

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References

  1. G. D. Malyuzhinets, “Nonstationary Diffraction Problem for the Wave Equation with a Contact Right-Hand-Side Support,” Tr. Akust. Inst. Akad. Nauk SSSR 15, 124–139 (1971).

    Google Scholar 

  2. M. V. Fedoryuk, “Nonstationary Problem of Noise Suppression,” Akust. Zh. 22, 339–443 (1976).

    Google Scholar 

  3. P. A. Nelson and S. J. Elliott, Active Control of Sound (Academic, San Diego, 1999).

    Google Scholar 

  4. V. S. Ryaben’kii, “Finite-Difference Problem of Noise Shielding,” Funkts. Anal. Ego Prilozh. 29(1), 70–71 (1995).

    Article  MathSciNet  Google Scholar 

  5. V. S. Ryaben’kii, “Nonlinear Problem of Noise Shielding,” Usp. Mat. Nauk 50(4), 146 (1995).

    Google Scholar 

  6. V. S. Ryaben’kii, “A Nonlinear Difference Shielding Problem,” in Analysis, Numerics, and Applications of Differential and Integral Equations (Longman, Harlow, 1998), pp. 179–182.

    Google Scholar 

  7. E. V. Zinov’ev and V. S. Ryaben’kii, RF Patent No. 6G01K11/16 (01.2.281001), Gos. Inst. Patent. Inf. (1996).

  8. R. N. Veitsman and V. S. Ryaben’kii, “Difference Problems of Shielding and Imitation,” Dokl. Math. 55, 340–343 (1997).

    Google Scholar 

  9. R. N. Veitsman and V. S. Ryaben’kii, “Difference Problems of Imitation,” Tr. Mosk. Mat. Ob-va 58 (1997).

  10. J. Loncaric and S. V. Tsynkov, “Optimization of Acoustic Source Strength in Problems of Active Noise Control,” SIAM J. Appl. Math. 63, 1141–1183 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Loncaric and S. V. Tsynkov, “Quadratic Optimization in the Problems of Active Control of Sound,” Appl. Math. 52, 381–400 (2005).

    MathSciNet  MATH  Google Scholar 

  12. V. S. Ryaben’kii, S. V. Utyuzhnikov, and S. V. Tsynkov, “The Problem of Active Noise Shielding in Composite Domains,” Dokl. Math. 74, 812–814 (2006).

    Article  MATH  Google Scholar 

  13. V. S. Ryaben’kii and S. V. Utyuzhnikov, “Active Shielding Model for Hyperbolic Equations,” IMA J. Appl. Math. 71, 924–939 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  14. V. S. Ryaben’kii, S. V. Tsynkov, and S. V. Utyuzhnikov, “Inverse Source Problem and Active Shielding for Composite Domains,” Appl. Math. Lett. 20, 511–515 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  15. V. S. Ryaben’kii and S. V. Utyuzhnikov, “Differential and Finite-Difference Problems of Active Shielding,” J. Appl. Numer. Math. 57, 374–382 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  16. V. S. Ryaben’kii, S. V. Utyuzhnikov, and A. A. Turan, “On the Application of Difference Potential Theory to Active Noise Control,” J. Adv. Appl. Math. 40, 194–211 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Lim, S. V. Utyuzhnikov, Y. W. Lam, et al., “An Experimental Validation of the Noise Control Methodology Based on Difference Potentials,” AIAA J. 47, 874–884 (2009).

    Article  Google Scholar 

  18. V. S. Ryaben’kii, S. V. Utyuzhnikov, and S. V. Tsynkov, “Difference Problem of Noise Suppression and Other Problems of Active Control of Single-Frequency Sound on a Composite Domain,” Dokl. Math. 79, 240–242 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  19. V. S. Ryaben’kii, S. V. Tsynkov, and S. V. Utyuzhnikov, “Active Control of Sound with Variable Degree of Cancellation,” Appl. Math. Lett. 22, 1846–1851 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  20. S. V. Utyuzhnikov, “Active Wave Control and Generalized Surface Potentials,” J. Adv. Appl. Math. 43, 101–112 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  21. S. V. Utyuzhnikov, “Generalized Calderon-Ryaben’kii Potentials,” IMA J. Appl. Math. 74, 128–148 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  22. V. S. Ryaben’kii, “Use of Weak Noise for Real-Time Control of Strong Noise Suppression in a Shielded Subdomain,” Dokl. Math. 81, 137–138 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  23. V. S. Ryaben’kii, “Model of Real-Time Active Noise Shielding of a Given Subdomain Subject to External Noise Sources,” Comput. Math. Math. Phys. 51, 444–454 (2011).

    Article  Google Scholar 

  24. V. S. Ryaben’kii, Preprint No. 32, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2010).

  25. V. S. Ryaben’kii, Method of Difference Potentials and Its Applications, 3rd ed. (Fizmatlit, Moscow, 2010) [in Russian].

    Google Scholar 

  26. V. S. Ryaben’kii, Method of Difference Potentials and Its Applications (Fizmatlit, Moscow, 2002; Springer-Verlag, Berlin, 2002).

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047, Russia

    V. S. Ryaben’kii

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  1. V. S. Ryaben’kii
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Correspondence to V. S. Ryaben’kii.

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Original Russian Text © V.S. Ryaben’kii, 2011, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2011, Vol. 51, No. 10, pp. 1889–1904.

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Ryaben’kii, V.S. Synchronous exploration for the control of real-time external noise suppression in a three-dimensional subdomain. Comput. Math. and Math. Phys. 51, 1777–1791 (2011). https://doi.org/10.1134/S0965542511100113

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