MS1.6 - Advances in Computational Structural Dynamics

One-dimensional periodic media have been extensively studied in the past, on the one hand to determine the dispersion curves linking the frequencies to the wavenumbers and on the other hand for the calculation of bounded periodic structures. In the latter case, effective methods have been obtained, in particular based on methods such as the Wave Finite Element (WFE) and allow the rapid determination of the response of a structure to a given excitation from computations made mainly on one substructure. For two-dimensional periodic media, the situation is much less favourable. Dispersion curves can be obtained as in the one-dimensional case and can account for certain properties of these media. For the calculation of structures and obtaining the response of two-dimensional periodic structures to an excitation, the results mainly concern infinite media. For finite media, few results and effective methods are available. This communication aims to treat this last case of response of finite two-dimensional periodic structures to an excitation. One will be limited to structures having symmetries compared to two orthogonal planes parallel to the edges of a substructure. One bases oneself on the WFE by modeling a rectangular substructure by a finite element model. By imposing the wavenumber in one direction, we can numerically calculate the wavenumbers and mode shapes associated with propagation in the perpendicular direction. We can thus build a set of waves propagating respectively along the two directions parallel to the sides of the rectangle which will serve as the basis on which we will decompose the solution. By taking appropriately chosen solutions, we can decouple the waves in the two directions parallel to the sides of the rectangle. The solution of each of these two problems is obtained by a fast Fourier transformation which gives the amplitudes associated with the waves. We can then calculate the global solution at selected points by summing the contributions of all these waves. We thus obtain the global solution for a two-dimensional periodic medium with a low cost in computing time. Examples are given for the case of a two-dimensional membrane.

The use of equivalent mechanical models, of the mass-spring or pendulum type, to represent the forces due to the sloshing of liquids in tanks on the dynamics of the main structure is a method still widely used today in many industrial fields (aeronautics, space, road transport, etc.). The determination of the parameters of these mechanical oscillators (mass, inertia, frequency, position) was initially established from analytical solutions of the linearized Euler equations of incompressible potential fluids in domains of simple shapes (parallelepipedic or axisymmetric), assuming that the axis of (apparent) gravity is aligned with one of the axes of symmetry of the volume considered. These restrictions are limiting for the use of these models because, if we are interested for example in a space launcher, its tanks have in practice sometimes more complex shapes (with internal equipments which break the axial symmetry) and are not necessarily aligned with the axis of the acceleration of the structure. This is why an approach based on a numerical resolution of the sloshing equations has been developed. It is based on a discretization by the Finite Element Method (FEM) or Boundary Element Method (BEM) in three dimensions, which allows to treat any fluid domain geometry. Although simpler to implement, FEM requires meshing the volume of the fluid which can be tedious for complex shaped tanks with internal equipment, especially if several filling levels are envisaged. The BEM more generally used for external potential fluids has been adapted for internal fluids with free surface. The resulting matrices are full but their construction requires only surface meshes of the fluid domain boundaries (fluid-structure interface and free surface). This approach is all the more interesting if we want to model the internal walls that exist inside the tanks (anti-slosh baffles): although they introduce numerical difficulties in the calculation of the operators, taking them into account in the meshing of the interfaces is much easier than with a volume mesh. The developments made around these methods will be illustrated by applications in the aerospace field, in particular for the piloting of reusable space launchers during their atmospheric re-entry phase.

MultiFEBE is an open-source solver that implements different Boundary-Element and Finite-Element models and strategies to tackle static and time-harmonic problems within the field of computational mechanics. Even though the code is able to address single-domain problems through both the Boundary Element Method or the Finite Element Method, its main strength lies in the possibility of addressing multi-domain coupled problems in which the response of some domains is better represented through some Boundary Element Model, while the response of other domains is best represented through some Finite Element Model, as can be the case in crack propagation problems or in mixed-dimensional problems involving beam or shell structural elements that are embedded in bounded or unbounded two-dimensional or three-dimensional continuous domains. The study of Soil-Foundation problems (such as the computation of impedance and kinematic interaction functions in problems involving pile or suction caisson foundations in viscoelastic or poroelastic stratified soils) or of Soil-Structure Interaction problems such as the computation of the dynamic response of structures founded on flexible soils, are some examples of applications of the latter type. The three types of continuum 2D and 3D regions implement are: inviscid fluids, viscoelastic solids and poroelastic solids, which can interact with discrete translational and translational-rotational springs and dashpots, bars, Euler-Bernoulli and Timoshenko beams, and Reissner-Mindlin structural shells (see [1] and [2] for more details). The code is written in Fortran 2003 and includes parallelization capabilities. User documentation and different tutorials are available together with the source code at a github repository [3]. All the material is released under a GPL 2.0 license, so the community is invited to explore the code, make use of it and take part in its development. This work has been developed with the support of research projects: PID2020-120102RB-I00, funded by the Agencial Estatal de Investigación of Spain, MCIN/AEI/10.13039/ 501100011033; ProID2020010025, funded by Consejerı́a de Economı́a, Conocimiento y Empleo (Agencia Canaria de la Investigación, Innovación y Sociedad de la Información) of the Gobierno de Canarias and FEDER; and BIA2017-88770-R, funded by Subdirección General de Proyectos de Investigación of the Ministerio de Economı́a y Competitividad (MINECO) of Spain and FEDER. REFERENCES [1] J.D.R. Bordón. Coupled model of finite elements and boundary elements for the dynamic analysis of buried shell structures. Doctor Thesis. Universidad de Las Palmas de Gran Canaria. 2018. [2] J.D.R Bordón, J.J. Aznárez, O. Maeso. Dynamic model of open shell structures buried in poroelastic soils. Computational Mechanics 60, 269-288, 2017 [3] J.D.R. Bordón, Á.G. Vega-Artiles, G.M. Álamo, L.A. Padrón, J.J. Aznárez, O. Maeso. MultiFEBE. https://github.com/mmc-siani-es/MultiFEBE (2022).