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Uncertain eigenvalue analysis by the sparse grid stochastic collocation method

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Abstract

In this paper, the eigenvalue problem with multiple uncertain parameters is analyzed by the sparse grid stochastic collocation method. This method provides an interpolation approach to approximate eigenvalues and eigenvectors’ functional dependencies on uncertain parameters. This method repetitively evaluates the deterministic solutions at the pre-selected nodal set to construct a high-dimensional interpolation formula of the result. Taking advantage of the smoothness of the solution in the uncertain space, the sparse grid collocation method can achieve a high order accuracy with a small nodal set. Compared with other sampling based methods, this method converges fast with the increase of the number of points. Some numerical examples with different dimensions are presented to demonstrate the accuracy and efficiency of the sparse grid stochastic collocation method.

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References

  1. Sepahvand, K., Marburg, S., Hardtke, H.J.: Stochastic free vibration of orthotropic plates using generalized polynomial chaos expansion. J. Sound Vib. 331, 167–179 (2012)

    Article  Google Scholar 

  2. Qiu, Z., Wang, X.: Interval analysis method and convex models for impulsive response of structures with uncertain-but-bounded external loads. Acta Mech. Sin. 22, 265–276 (2006)

    Article  MathSciNet  Google Scholar 

  3. Zhang, J., Ellingwood, B.: Effects of uncertain material properties on structural stability. J. Struct. Eng. 121, 705–716 (1995)

    Article  MATH  Google Scholar 

  4. Singh, B.N., Iyengar, N., Yadav, D.: Effects of random material properties on buckling of composite plates. J. Eng. Mech. 127, 873–879 (2001)

    Article  Google Scholar 

  5. Stainforth, D.A., Aina, T., Christensen, C.: Uncertainty in predictions of the climate response to rising levels of greenhouse gases. Nature 433, 403–406 (2005)

    Article  Google Scholar 

  6. Palmer, T., Shutts, G., Hagedorn, R.: Representing model uncertainty in weather and climate prediction. Annu. Rev. Earth Planet. Sci. 33, 163–193 (2005)

    Article  MathSciNet  Google Scholar 

  7. Ding, C.T., et al.: Input torque sensitivity to uncertain parameters in biped robot. Acta Mech. Sin. 29, 452–461 (2013)

    Article  MathSciNet  Google Scholar 

  8. Guo, S.-X., Li, Y.: Non-probabilistic reliability method and reliability-based optimal LQR design for vibration control of structures with uncertain-but-bounded parameters. Acta Mech. Sin. 29, 864–874 (2013)

    Article  MathSciNet  Google Scholar 

  9. Wang, J., Qiu, Z.-P.: Fatigue reliability based on residual strength model with hybrid uncertain parameters. Acta Mech. Sin. 28, 112–117 (2012)

    Article  MATH  Google Scholar 

  10. Jia, Y.-H., Hu, Q., Xu, S.-J.: Dynamics and adaptive control of a dual-arm space robot with closed-loop constraints and uncertain inertial parameters. Acta Mech. Sin. 30, 112–124 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fox, R., Kapoor, M.: Rates of change of eigenvalues and eigenvectors. AIAA J. 6, 2426–2429 (1968)

    Article  Google Scholar 

  12. Collins, J.D., Thomson, W.T.: The eigenvalue problem for structural systems with statistical properties. AIAA J. 7, 642–648 (1969)

    Article  MATH  Google Scholar 

  13. Eldred, M., Lerner, P., Anderson, W.: Higher order eigenpair perturbations. AIAA J. 30, 1870–1876 (1992)

    Article  MATH  Google Scholar 

  14. Shinozuka, M., Astill, C.J.: Random eigenvalue problems in structural analysis. AIAA J. 10, 456–462 (1972)

    Article  MATH  Google Scholar 

  15. Székely, G., Schuëller, G.: Computational procedure for a fast calculation of eigenvectors and eigenvalues of structures with random properties. Comput. Methods Appl. Mech. Eng. 191, 799–816 (2001)

    Article  Google Scholar 

  16. Pradlwarter, H., Schuëller, G., Szekely, G.: Random eigenvalue problems for large systems. Comput. Struct. 80, 2415–2424 (2002)

    Article  Google Scholar 

  17. Deif, A.: The interval eigenvalue problem. ZAMM-J. Appl. Math. Mech./Z. Angew. Math. Mech. 71, 61–64 (1991)

    Article  MathSciNet  Google Scholar 

  18. Chen, S.H., Qiu, Z., Liu, Z.: Perturbation method for computing eigenvalue bounds in structural vibration systems with interval parameters. Commun. Numer. Methods Eng. 10, 121–134 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  19. Chen, S.H., Lian, H.D., Yang, X.W.: Interval eigenvalue analysis for structures with interval parameters. Finite Elem. Anal. Des. 39, 419–431 (2003)

  20. Qiu, Z.P., Gu, Y.X., Wang, S.M.: A theorem of upper and lower bounds on eigenvalues for structures with bounded parameters. Acta Mech. Sin. 31, 466–474 (1999)

  21. Qiu, Z., Chen, S., Jia, H.: The Rayleigh quotient iteration method for computing eigenvalue bounds of structures with bounded uncertain parameters. Comput. Struct. 55, 221–227 (1995)

    Article  MATH  Google Scholar 

  22. Wang, C., Gao, W., Song, C.: Stochastic interval analysis of natural frequency and mode shape of structures with uncertainties. J. Sound Vib. 333, 2483–2503 (2014)

    Article  Google Scholar 

  23. Xuedong, G., Jun, X., SuHuan, C.: The uncertain eigenvalues analysis of structures with uncertain parameters. Acta Mech. Solida Sin. 13, 230–236 (2000)

    Google Scholar 

  24. Modares, M., Mullen, R.L., Muhanna, R.L.: Natural frequencies of a structure with bounded uncertainty. J. Eng. Mech. 132, 1363–1371 (2006)

    Article  Google Scholar 

  25. Ghanem, R., Ghosh, D.: Efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition. Int. J. Numer. Methods Eng. 72, 486–504 (2007)

    Article  MathSciNet  Google Scholar 

  26. Bressolette, P., Fogli, M., Chauvière, C.: A stochastic collocation method for large classes of mechanical problems with uncertain parameters. Probab. Eng. Mech. 25, 255–270 (2010)

  27. Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27, 1118–1139 (2005)

    Article  MathSciNet  Google Scholar 

  28. Nobile, F., Tempone, R., Webster, C.G.: A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46, 2309–2345 (2008)

    Article  MathSciNet  Google Scholar 

  29. Smolyak, S.A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Sov. Math. Dokl. 4, 240–243 (1963)

  30. Riesz, F., S.-Nagy, B.: Functional Analysis. Frederick Ungar, New York (1955)

  31. Davis, C.: The rotation of eigenvectors by a perturbation. J. Math. Anal. Appl. 6, 159–173 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  32. Qiu, Z., Wang, X.: Solution theorems for the standard eigenvalue problem of structures with uncertain-but-bounded parameters. J. Sound Vib. 282, 381–399 (2005)

    Article  MathSciNet  Google Scholar 

  33. Massa, F., Ruffin, K., Tison, T., et al.: A complete method for efficient fuzzy modal analysis. J. Sound Vib. 309, 63–85 (2008)

    Article  Google Scholar 

  34. Ganapathysubramanian, B., Zabaras, N.: Sparse grid collocation schemes for stochastic natural convection problems. J. Comput. Phys. 225, 652–685 (2007)

    Article  MathSciNet  Google Scholar 

  35. Berrut, J.-P., Trefethen, L.N.: Barycentric lagrange interpolation. Siam Rev. 46, 501–517 (2004)

    Article  MathSciNet  Google Scholar 

  36. Novak, E., Ritter, K.: High dimensional integration of smooth functions over cubes. Numer. Math. 75, 79–97 (1996)

    Article  MathSciNet  Google Scholar 

  37. Bungartz, H.-J., Griebel, M.: Sparse grids. Acta Numer. 13, 147–269 (2004)

  38. Wasilkowski, G.W., Wozniakowski, H.: Explicit cost bounds of algorithms for multivariate tensor product problems. J. Complex. 11, 1–56 (1995)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The project was supported by the National Nature Science Foundation of China for Distinguished Young Scholars (Grant 11125209), the National Nature Science Foundation of China (Grants 11322215, 51121063), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars.

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Authors and Affiliations

  1. State Key Laboratory of Mechanical Systems and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    J. C. Lan, X. J. Dong, Z. K. Peng, W. M. Zhang & G. Meng

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  1. J. C. Lan
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  2. X. J. Dong
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  3. Z. K. Peng
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  4. W. M. Zhang
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  5. G. Meng
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Correspondence to Z. K. Peng.

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Lan, J.C., Dong, X.J., Peng, Z.K. et al. Uncertain eigenvalue analysis by the sparse grid stochastic collocation method. Acta Mech. Sin. 31, 545–557 (2015). https://doi.org/10.1007/s10409-015-0422-9

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  • DOI: https://doi.org/10.1007/s10409-015-0422-9

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