Bridging Scales Analysis of Wave Propagation in Heterogeneous Structures with Imperfections

  • Srinivasan Gopalakrishnan
  • Massimo Ruzzene
  • Sathyanarayana Hanagud
Part of the Springer Series in Reliability Engineering book series (RELIABILITY)


The analysis of wave propagation has been extensively used as a tool for non destructive evaluation of structural components. The numerical analysis of wavefield in damaged media can be useful to investigate the problem theoretically and to support the interpretation of experimental measurements. A finite element analysis of non homogeneous media can be computationally very expensive, especially when a fine mesh is required to properly model the geometric and/or material discontinuities that are characteristic of the damaged areas. The computational cost associated with wave propagation simulations motivates the development of the simplified damage models presented in  Chaps. 6 and  7. This chapter presents a different approach whereby the computational cost is reduced through a multi-scale analysis. A coarse mesh is employed to capture the macroscopic behavior of the structure, and a refined mesh is limited to the small region around the discontinuity. The co-existence of two scales in the model is handled through the application of proper bridging relations between the two scales, and the generation of interaction forces at the interfaces according to the Bridging Scales Method. This technique allows a coarse description of the global behavior of the structure while simultaneously obtaining local information regarding the interaction of propagating waves with a localized discontinuity in the domain. Time and frequency domain formulations of the Bridging Scales Method are illustrated through examples on simulations of 1D and 2D waveguides.


Coarse Scale Total Mechanical Energy Spurious Reflection Stiff Inclusion Symmetric Complex Linear System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Gonella S, Ruzzene M (2008) Bridging scales analysis of wave propagation in heterogeneous structures with imperfections. Wave Motion 45:481–497MathSciNetCrossRefGoogle Scholar
  2. 2.
    Kadowaki H, Liu WK (2004) Bridging multi-scale method for localization problems. Comput Methods Appl Mech Eng 193(30–32):3267–3302MATHCrossRefGoogle Scholar
  3. 3.
    Liu WK, Chen Y, Chang C, Belytschko T (1996) Advances in multiple scale kernel particle methods. Comput Mech 18(2):73–111MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Liu WK, Karpov E, Park H (2006) Nano mechanics and materials. Theory, multiscale methods and applications. Wiley, New YorkCrossRefGoogle Scholar
  5. 5.
    Liu WK, Park HS, Qian D, Karpov EG, Kadowaki H, Wagner GJ (2006) Bridging scale methods for nanomechanics and materials. Comput Methods Appl Mech Eng 195(13–16):1407–1421MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    McVeigh C, Vernerey F, Liu WK, Brinson LC (2006) Multiresolution analysis for material design. Comput Methods Appl Mech Eng 195(37–40):4291–4310MathSciNetGoogle Scholar
  7. 7.
    Park HS, Liu WK (2004) An introduction and tutorial on multiple-scale analysis in solids. Comput Methods Appl Mech Eng 193(17–20):1733–1772MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Wagner G, Liu W (2003) Coupling of atomistic and continuum simulations using a bridging scale decomposition. J Comput Phys 190(1):249–274MATHCrossRefGoogle Scholar
  9. 9.
    Weeks W (1966) Numerical inversion of laplace transforms using laguerre functions. J Assoc Comput Mach 13(3):419MathSciNetMATHGoogle Scholar
  10. 10.
    Weideman J (1999) Algorithms for parameter selection in the weeks method for inverting the laplace transform. SIAM J Sci Comput 21(1):111–128MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  • Srinivasan Gopalakrishnan
    • 1
  • Massimo Ruzzene
    • 2
  • Sathyanarayana Hanagud
    • 3
  1. 1.Department of Aerospace EngineeringIndian Institute of ScienceBangaloreIndia
  2. 2.School of Aerospace Engineering, School of Mechanical EngineeringGeorgia Institute of TechnologyAtlantaUSA
  3. 3.School of Aerospace EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations