On the Development of Lattice-Boltzmann Methods for Solid Mechanics

  • Müller, Henning (TU Darmstadt)
  • Schlüter, Alexander (RPTU Kaiserslautern-Landau)
  • Faust, Erik (RPTU Kaiserslautern-Landau)
  • Steinmetz, Felix (RPTU Kaiserslautern-Landau)
  • Müller, Ralf (TU Darmstadt)

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We present a joint project, that works towards the development of Lattice-Boltzmann methods (LBMs) for solid mechanics. The first approach we implemented is derived from Murthy et al. [1] and Escande et al. [2]. This model uses a moment-chain representation, formulated in accordance with [3], to emulate the balance laws for mass and linear momentum, and a constitutive equation linear elastic solids. This way, mass density, linear momentum density, and stress represent the moments of the LBM. Lastly, the displacement field is obtained via time integration of the momentum. For now, the method has been mostly restricted to 2D applications, with a D2Q9 disctretization, and it is based on the simple BGK-collision scheme, i.e. with a single relaxation time. Further, we present boundary conditions, see also [4], to prescribe the displacement (Dirichlet-type) or the traction (Neumann-type), as it is common in engineering. These are relatively simple bounce- back schemes. Results for benchmark problems show great performance for the LBM in terms of computational cost, as well as a favorable scaling, when compared to Finite Element (FE) simulations. We also present, how the LBM can be applied to different dynamical problems, including wave propagation or fracture mechanics. These first result are promising. However, the convergence rate and the stability of the method, as well as the boundary conditions, still require further improvement. Nonetheless, this indicates successful first steps towards a highly efficient numerical method for elastodynamics. [1] J. Murthy, P. K. Kolluru, V. Kumaran, S. Ansumali, J. Narayana Surya, CiCP. 23 (2018), doi:10.4208/cicp.OA-2016-0259. [2] M. Escande, P. K. Kolluru, L. M. Cléon, P. Sagaut, Lattice Boltzmann Method for wave propagation in elastic solids with a regular lattice: Theoretical analysis and validation (2020), doi:10.48550/arXiv.2009.06404. [3] G. Farag, S. Zhao, G. Chiavassa, P. Boivin, Physics of Fluids. 33, 037101 (2021). [4] E. Faust, A. Schlüter, H. Müller, R. Müller, Dirichlet and Neumann boundary conditions in a Lattice Boltzmann Method for Elastodynamics (2022), doi:10.48550/arXiv.2208.04088.