Numerical solutions to a microcontinuum model using WENO schemes

  • Armando MajoranaEmail author
  • Rita Tracinà
Original Article


In this paper, we consider a one-dimensional Mindlin model describing linear elastic behaviour of isotropic materials with microstructural effects. After introducing the kinetic and the potential energy, we derive a system of equations of motion by means of the Euler–Lagrange equations. A class of exact solutions is obtained. They have a wave behaviour due to a good property of the potential energy. We transform the set of hyperbolic partial differential equations in a particular form, which makes clear how to impose boundary conditions correctly. Next numerical solutions are obtained by using a weighted essentially non-oscillatory finite difference scheme coupled by a total variation diminishing Runge–Kutta method. A comparison between exact and numerical solutions shows the robustness and the accuracy of the numerical scheme. A numerical example of solutions for an inhomogeneous material is also shown.


Mindlin continuum Wave propagation Numerical solutions Finite differences 



The first author was partially supported by the Italian FIR project “Innovative techniques in computational mechanics based on high continuity interpolation for the integrated design of advanced structures” (Principal Investigator: Massimo Cuomo).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of CataniaCataniaItaly

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