Structural vibration analysis with random fields using the hierarchical finite element method

  • A. T. FabroEmail author
  • N. S. Ferguson
  • B. R. Mace
Technical Paper


Element-based techniques, like the finite element method, are the standard approach in industry for low-frequency applications in structural dynamics. However, mesh requirements can significantly increase the computational cost for increasing frequencies. In addition, randomness in system properties starts to play a significant role and its inclusion in the model further increases the computational cost. In this paper, a hierarchical finite element formulation is presented which incorporates spatially random properties. Polynomial and trigonometric hierarchical functions are used in the element formulation. Material and geometrical spatially correlated randomness are represented by the Karhunen–Loève expansion, a series representation for random fields. It allows the element integration to be performed only once for each term of the series which has benefits for a sampling scheme and can be used for non-Gaussian distributions. Free vibration and forced response statistics are calculated using the proposed approach. Compared to the standard h-version, the hierarchical finite element approach produces smaller mass and stiffness matrices, without changing the number of nodes of the element, and tends to be computationally more efficient. These are key factors not only when considering solutions for higher frequencies but also in the calculation of response statistics using a sampling method such as Monte Carlo simulation.


Hierarchical finite element Random field Karhunen–Loève Structural vibration 



The authors gratefully acknowledge the financial support of the Brazilian National Council of Research (CNPq) Process number 445773/2014-6, the Federal District Research Foundation (FAPDF) Process number 0193001040/2015 and the Royal Society for the Newton International Exchanges Fund reference number IE140616.


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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Departmento de Engenharia Mecânica, Faculdade de TecnologiaUniversidade de Brasília – Campus Darcy RibeiroBrasíliaBrazil
  2. 2.ISVRUniversity of SouthamptonSouthamptonUK
  3. 3.Acoustics Research Centre, Department of Mechanical EngineeringUniversity of AucklandAucklandNew Zealand

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