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The finite and spectral cell methods for smart structure applications: transient analysis

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Abstract

This article introduces a robust and efficient numerical tool that is well suited for the simulation of ultrasonic guided waves and can be especially helpful when dealing with heterogeneous materials. The proposed method is based on a combination of high-order finite element methods (FEM) and the fictitious domain concept. If hierarchic shape functions, which are familiar from the p-version of the finite element method (p-FEM), are deployed the method is referred to as the finite cell method (FCM). Where Lagrange polynomials through Gauß–Lobatto–Legendre points are used, we refer to it as the spectral cell method (SCM). The name, SCM, derives from the fact that the deployed shape functions are commonly utilized in the spectral element method. To model smart structure applications such as shape control problems, noise cancelation devices and the excitation/sensing of ultrasonic guided waves, a coupling between electrical and mechanical variables is also taken into account. In this context, we focus on including the linear theory of piezoelectricity in the variational formulation of the FCM and the SCM, respectively. Several numerical benchmark problems are then used to validate the proposed approach. The simulations show promising results with respect to the accuracy of the method and the computational effort. We observe similar convergence properties for the proposed high-order fictitious domain methods as with “conventional” high-order finite element approaches. Implementing the proposed method in existing finite element software is, moreover, a straightforward process. These properties make the method an efficient tool for practical applications in structural health monitoring problems and smart structure applications in general.

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References

  1. Abedian A., Parvizian J., Düster A., Khademyzadeh H., Rank E.: Performance of different integration schemes in facing discontinuities in the finite cell method. Int. J. Comput. Methods 10, 24 (2013)

    Article  Google Scholar 

  2. Abedian A., Parvizian J., Düster A., Rank E.: The finite cell method for the J 2 flow theory of plasticity. Finite Elem. Anal. Des. 69, 37–47 (2013)

    Article  Google Scholar 

  3. Ahmad Z.A.B., Gabbert U.: Simulation of Lamb wave reflections at plate edges using the semi-analytical finite element method. Ultrasonics 52, 815–820 (2012)

    Article  Google Scholar 

  4. Allik H., Hughes T.J.R.: Finite element method for piezoelectric vibration. Int. J. Numer. Methods Eng. 2, 151–157 (1970)

    Article  Google Scholar 

  5. Bazilevs, Y., Hsu, M.C., Scott, M.A.: Isogeometric fluid-structure interaction analysis with emphasis on non-matching discretizations, and with application to wind turbines. Comput. Methods Appl. Mech. Eng. 249–252, 28–41 (2012)

  6. Benjeddou A.: Advances in piezoelectric finite element modeling of adaptive structural elements: a survey. Comput. Struct. 76, 347–363 (2000)

    Article  Google Scholar 

  7. Boller, C., Chang, F.-K., Fujino, Y. (eds.): Encyclopedia of Structural Health Monitoring. Wiley (2009)

  8. Bugrov A.N., Smagulov S.: Fictitious domain method for Navier–Stokes equation. Math. Model Fluid Flow 1, 79–90 (1978)

    Google Scholar 

  9. Cho H., Lissenden C.J.: Structural health monitoring of fatigue crack growth in plate structures with ultrasonic guided waves. Struct. Health Monit. 11, 393–404 (2012)

    Article  Google Scholar 

  10. Ciampa F., Meo M., Barbieri E.: Impact localization in composite structures of arbitrary cross section. Struct. Health Monit. 11, 643–655 (2012)

    Article  Google Scholar 

  11. Cohen, E., Martin, T., Kirby, R.M., Lyche, T., Riesenfeld, R.F.: Analysis-aware modeling: Understanding quality considerations in modeling for isogeometric analysis. Comput. Methods Appl. Mech. Eng. 199, 334–356 (2010)

  12. Dodson, J.C., Inman, D.J.: Thermal sensitivity of Lamb waves for structural health monitoring applications. Ultrasonics 53, 677–685 (2013)

  13. Duczek S., Gabbert U.: Anisotropic hierarchic finite elements for the simulation of piezoelectric smart structures. Eng. Comput. 30, 21 (2013)

    Google Scholar 

  14. Duczek, S., Joulaian, M., Düster, A., Gabbert, U.: Simulation of Lamb waves using the spectral cell method. In: Proceedings of SPIE: Smart Structures & NDE (2013)

  15. Duczek S., Joulaian M., Düster A., Gabbert U.: Numerical analysis of Lamb waves using the finite and spectral cell methods. Int. J. Numer. Methods Eng. 99, 26–53 (2014)

    Article  Google Scholar 

  16. Duczek S., Willberg C., Schmicker D., Gabbert U.: Development, validation and comparison of higher order finite element approaches to compute the propagation of Lamb waves efficiently. Key Eng. Mater. 518, 95–105 (2012)

    Article  Google Scholar 

  17. Düster A., Parvizian J., Yang Z., Rank E.: The finite cell method for three-dimensional problems of solid mechanics. Comput. Methods Appl. Mech. Eng. 197, 3768–3782 (2008)

    Article  MATH  Google Scholar 

  18. Düster A., Sehlhorst H.G., Rank E.: Numerical homogenization of heterogeneous and cellular materials utilizing the finite cell method. Comput. Mech. 50, 413–431 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Geller, S., Tölke, J., Krafczyk, M.: Lattice Boltzmann methods on quadtree-type grids for fluid-structure interaction. In: Fluid-Structure Interaction, Modelling, Simulation and Optimization, vol. 53, Lecture Notes in Computational Sciences and Engineering, pp. 270–293 (2006)

  20. Giurgiutiu V.: Structural Health Monitoring with Piezoelectric Active Wafer Sensors: Fundamentals and Applications. Elsevier, Amsterdam (2008)

    Google Scholar 

  21. Gordon W.J.: Blending-function methods of bivariate and multivariate interpolation and approximation. J. Numer. Anal. 8, 158–177 (1971)

    Article  MATH  Google Scholar 

  22. Ha S., Chang F.K.: Optimizing a spectral element for modeling PZT-induced Lamb wave propagation in thin plates. Smart Mater. Struct. 19, 1–11 (2010)

    Google Scholar 

  23. Hosseini, S.M.H., Gabbert, U.: Numerical simulation of the Lamb wave propagation in honeycomb sandwich panels: a parametric study. Compos. Struct. (2012). doi:10.1016/j.compstruct.2012.09.055

  24. Hosseini, S.M.H., Kharaghani, A., Kirsch, C., Gabbert, U.: Numerical simulation of Lamb wave propagation in metallic foam sandwich structures: a parametric study. Compos. Struct. (2012). doi:10.1016/j.compstruct.2012.10.039

  25. Huang C.H., Lin Y.C., Ma C.C.: Theoretical analysis and experimental measurement for resonant vibration of piezoceramic circular plates. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 51, 12–24 (2004)

    Article  Google Scholar 

  26. Ikeda T.: Fundamentals of Piezoelectricity. Oxford Science Publications, Oxford (1990)

    Google Scholar 

  27. Joulaian, M., Duczek, S., Gabbert, U., Düster, A.: Finite and spectral cell method for wave propagation in heterogeneous materials. Comput. Mech. 54, 661–675 (2014). doi:10.1007/s00466-014-1019-z

  28. Joulaian M., Düster A.: Local enrichment of the finite cell method for problems with material interfaces. Comput. Mech. 52, 741–762 (2013)

    Article  MATH  Google Scholar 

  29. Kollmannsberger S., Geller S., Düster A., Tölke J., Sorger C., Krafczyk M., Rank E.: Fixed-grid fluid-structure interaction in two dimensions based on a partitioned lattice Boltzmann and p-FEM approach. Int. J. Numer. Methods Eng. 79, 817–845 (2009)

    Article  MATH  Google Scholar 

  30. Komatitsch D., Tromp J.: Spectral-element simulations of global seismic wave propagation i—validation. Int. J. Geophys. 149, 390–412 (2002)

    Article  Google Scholar 

  31. Komatitsch D., Tromp J.: Spectral-element simulations of global seismic wave propagation ii—three-dimensional models, oceans, rotation and self-gravitation. Int. J. Geophys. 150, 303–318 (2002)

    Article  Google Scholar 

  32. Komatitsch D., Vilotte J.P., Vai R., Castillo-Covarrubias J.M., Sanchez-Sesma F.J.: The spectral element method for elastic wave equations—application to 2-d and 3-d seismic problmes. Int. J. Numer. Methods Eng. 45, 1139–1164 (1999)

    Article  MATH  Google Scholar 

  33. Krommer M.: The significance of non-local constitutive relations for composite thin plates including piezoelastic layers with prescribed electric charge. Smart Mater. Struct. 12, 318–330 (2003)

    Article  Google Scholar 

  34. Kudela P., Krawczuk M., Ostachowicz W.: Wave propagation modelling in 1D structures using spectral elements. J. Sound Vib. 300, 88–100 (2007)

    Article  Google Scholar 

  35. Kudela P., Ostachowicz W.: 3d time-domain spectral elements for stress waves modelling. J. Phys.: Conf. Ser. 181, 1–8 (2009)

    Google Scholar 

  36. Kudela P., Ostachowicz W., Zak A.: Damage detection in composite plates with embedded PZT transducers. Mech. Syst. Signal Process. 22, 1327–1335 (2008)

    Article  Google Scholar 

  37. Lamb H.: On waves in an elastic plate. Proc. R. Soc. Lond. Ser. A 93, 114–128 (1917)

    Article  MATH  Google Scholar 

  38. Leung A.Y.T., Zhu B.: Comments on free vibration of skew Minidlin plates by p-version of FEM. J. Sound Vib. 278, 699–703 (2004)

    Article  Google Scholar 

  39. Li F., Peng H., Sun X., Wang J., Meng G.: Wave propagation analysis in composite laminates containing a delamination using a three-dimensional spectral element method. Math. Probl. Eng. 1, 1–19 (2012)

    MathSciNet  Google Scholar 

  40. Martin, T., Cohen, E., Kirby, R.M: Volumetric parameterization and trivariate B-spline fitting using harmonic functions. Comput. Aided Geom. Des. 26, 648–664 (2009)

  41. Ostachowicz W., Kudela P., Krawczuk M., Zak A.: Guided Waves in Structures for SHM: The Time-Domain Spectral Element Method. Wiley, New York (2011)

    Google Scholar 

  42. Parvizian J., Düster A., Rank E.: Finite cell method: h- and p-extension for embedded domain problems in solid mechanics. Comput. Mech. 41, 121–133 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  43. Patera A.T.: A spectral element method for fuid dynamics: Laminar flow in a channel expansion. J. Comput. Phys. 54, 468–488 (1984)

    Article  MATH  Google Scholar 

  44. Pavlopoulou S., Soutis C., Manson G.: Non-destructive inspection of adhesively bonded patch repairs using Lamb waves. Plast. Rubber Compos. 41, 61–68 (2012)

    Article  Google Scholar 

  45. Peng H., Meng G., Li F.: Modeling of wave propagation in plate structures using three-dimensional spectral element method for damage detection. J. Sound Vib. 320, 942–954 (2009)

    Article  Google Scholar 

  46. Peskin C.S.: The fluid dynamics of heart valves: experimental, theoretical and computational methods. Annu. Rev. Fluid Mech. 14, 235–259 (1981)

    Article  Google Scholar 

  47. Piefort, V.: Finite Element Modelling of Piezoelectric Active Structures. Ph.D. thesis, University of Brussels (2001)

  48. Pohl J., Willberg C., Gabbert U., Mook G.: Experimental and theoretical analysis of Lamb wave generation by piezoceramic actuators for structural health monitoring. Exp. Mech. 52, 429–438 (2012)

    Article  Google Scholar 

  49. Rank, E.: Adaptive remeshing and h-p domain decomposition. Comput Methods Appl Mech Eng 101, 299—313 (1992)

  50. Rank, E., Ruess, M., Kollmannsberger, S., Schillinger, D., Düster, A.: Geometric modeling, isogeometric analysis and the finite cell method. Comput. Methods Appl. Mech. Eng. 249–252, 104–115 (2012)

  51. Ringwelski S., Gabbert U.: Modeling of a fluid-loaded smart shell structure for active noise and vibration control using a coupled finite element-boundary element approach. Smart Mater. Struct. 19, 13 (2010)

    Article  Google Scholar 

  52. Ringwelski S., Luft T., Gabbert U.: Piezoelectric controlled noise attenuation of engineering systems. J. Theor. Appl. Mech. 49, 859–878 (2011)

    Google Scholar 

  53. Ruess M., Tal D., Trabelsi N., Yosibash Z., Rank E.: The finite cell method for bone simulations: verification and validation. Biomech. Model. Mechanobiol. 11, 425–437 (2012)

    Article  Google Scholar 

  54. Saul’ev, V.K.: A method for automatization of the solution of boundary value problems on high performance computers. Doklady Akademii Nauk SSSR 144, 497–500 (1962) (in Russian); English translation in Soviet Mathematics Doklady 3, 763–766 (1963)

  55. Saul’ev V.K.: On solution of some boundary value problems on high performance computers by fictitious domain method. Sibirian Math. J. 4, 912–925 (1963)

    MathSciNet  Google Scholar 

  56. Schillinger, D., Dede, L., Scott, M.A., Evans, J.A., Borden, M.J., Rank, E., Hughes, T.J.R.: An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces. Comput. Methods Appl. Mech. Eng. 249–252, 116–150 (2012)

  57. Schillinger D., Düster A., Rank E.: The hp-d-adaptive finite cell method for geometrically nonlinear problems of solid mechanics. Int. J. Numer. Methods Eng. 89, 1171–1202 (2012)

    Article  MATH  Google Scholar 

  58. Schillinger D., Ruess M., Zander N., Bazilevs Y., Düster A., Rank E.: Small and large deformation analysis with the p- and B-spline versions of the finite cell method. Comput. Mech. 50, 445–478 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  59. Scholz, D., Kollmannsberger, S., Düster, A., Rank, E.: Thin solids for fluid-structure interaction. Fluid-Structure Interaction, Modelling, Simulation and Optimization, vol. 53 Lecture Notes in Computational Sciences and Engineering, pp. 294–335 (2006)

  60. Sehlhorst, H.G.: Numerical Homogenization Strategies for Cellular Materials with Applications in Structural Mechanics. VDI Fortschritt-Berichte Reihe 18 Nr. 333 (2011)

  61. Staszewski, W.J., Boller, C., Tomlinson, G. R. (eds.): Health Monitoring for Aerospace Structures. Wiley, New York (2003)

  62. Szabó B., Babuška I.: Finite Element Analysis. Wiley, New York (1991)

    MATH  Google Scholar 

  63. Szabó B., Babuška I.: Introduction to Finite Element Analysis: Formulation, Verification and Validation. Wiley, New York (2011)

    Book  Google Scholar 

  64. Thompson L.L., Pinsky P.M.: Complex wavenumber Fourier analysis of the p-version finite element method. Comput. Mech. 13, 255–275 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  65. Viktorov I.A.: Rayleigh and Lamb Waves. Plenum Press, New York (1967)

    Book  Google Scholar 

  66. Willberg, C., Duczek, S., Vivar Perez, J.M., Schmicker, D., Gabbert, U.: Comparison of different higher order finite element schemes for the simulation of Lamb waves. Comput. Methods Appl. Mech. Eng. 241–244, 246–261 (2012)

  67. Willberg C., Gabbert U.: Development of a three-dimensional piezoelectric isogeometric finite element for smart structure applications. Acta Mech. 223, 1837–1850 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  68. Yang, Z.: The Finite Cell Method for Geometry-Based Structural Simulation. Ph.D. thesis, Technical University Munich (2011)

  69. Yang Z., Kollmannsberger S., Düster A., Ruess M., Grande Garcia E., Burgkart R., Rank E.: Non-standard bone simulation: interactive numerical analysis by computational steering. Comput. Vis. Sci. 14, 207–216 (2011)

    Article  MathSciNet  Google Scholar 

  70. Yang Z., Ruess M., Kollmannsberger S., Düster A., Rank E.: An efficient integration technique for the voxel-based finite cell method. Int. J. Numer. Methods Eng. 91, 457–471 (2012)

    Article  Google Scholar 

  71. Yongqiang L., Jian L.: Free vibration analysis of circular and annular sectorial thin plates using curve strip Fourier p-element. J. Sound Vib. 305, 457–466 (2007)

    Article  Google Scholar 

  72. Yosibash, Z.: p-FEMs in biomechanics: bones and arteries. Comput. Methods Appl. Mech. Eng. 249–252, 169–184 (2012)

  73. Zak A., Radzieǹski M., Krawczuk M., Ostachowicz W.: Damage detection strategies based on propagation of guided elastic waves. Smart Mater. Struct. 21, 1–18 (2012)

    Article  Google Scholar 

  74. Zander N., Kollmannsberger S., Ruess M., Yosibash Z., Rank E.: The finite cell method for linear thermoelasticity. Comput. Math. Appl. 64, 3527–3541 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Institute of Mechanics, Otto-von-Guericke-University Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    S. Duczek, S. Liefold & U. Gabbert

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Duczek, S., Liefold, S. & Gabbert, U. The finite and spectral cell methods for smart structure applications: transient analysis. Acta Mech 226, 845–869 (2015). https://doi.org/10.1007/s00707-014-1227-9

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  • DOI: https://doi.org/10.1007/s00707-014-1227-9

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