Lattice Boltzmann for solid mechanics: elastostatics and elastodynamics

Laura De Lorenzis

Computational Mechanics Group, ETH Zurich

Thursday 6th March 2025

14:00:00

TBA

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The talk overviews recent research by the authors on the development of second-order consistent and stable lattice Boltzmann formulations to solve elastostatics and elastodynamics problems. The first proposed scheme [1] solves the quasi-static equations of linear elasticity in two dimensions using a collision operator with multiple relaxation times. In contrast to previous works, our formulation solves for a single distribution function with a standard velocity set and avoids any recourse to finite difference approximations. As a result, all computational benefits of the lattice Boltzmann method can be used to full capacity. The second proposed scheme [2] solves the equations of linear elastodynamics in two dimensions (the extension to three dimensions is currently also available but still unpublished). The construction of the numerical scheme uses an equivalent first-order hyperbolic system of equations as an intermediate step, for which a vectorial lattice Boltzmann formulation is introduced. The only difference to conventional lattice Boltzmann formulations is the usage of vector-valued populations, so that once again all computational benefits of the algorithm are preserved. Both schemes are systematically derived using the asymptotic expansion technique. Stability is assessed for elastostatics with von Neumann analysis, whereas in elastodynamics we exploit the notion of pre-stability structures to prove stability for an arbitrary combination of material parameters under a CFL-like condition. Boundary formulation for various cases are proposed [3]. All theoretical derivations are numerically verified by convergence studies using manufactured solutions and long-term stability tests. [1] O. Boolakee, M. Geier, L. De Lorenzis (2023), A new Lattice Boltzmann scheme for linear elastic solids: periodic problems. Computer Methods in Applied Mechanics and Engineering, 404: 115756. [2] O. Boolakee, M. Geier, L. De Lorenzis (2025). Lattice Boltzmann for linear elastodynamics: periodic problems and Dirichlet boundary conditions. Computer Methods in Applied Mechanics and Engineering, 433: 117469. [3] O. Boolakee, M. Geier, L. De Lorenzis (2023), Dirichlet and Neumann boundary conditions for a lattice Boltzmann scheme for linear elastic solids on arbitrary domains. Computer Methods in Applied Mechanics and Engineering, 415: 116225. Contatti: davide.riccobelli@polimi.it

Laura De Lorenzis received her Engineering degree and her PhD from the University of her hometown Lecce, in southern Italy, where she first stayed as Assistant and later as Associate Professor of Solid and structural mechanics. In 2013 she moved to the TU Braunschweig, Germany, as Professor and Director of the Institute of Applied Mechanics. There she was founding member and first Chair (2017-2020) of the Center for Mechanics, Uncertainty and Simulation in Engineering. Since 2020 she is Professor of Computational Mechanics at ETH Zürich, in the Department of Mechanical and Process Engineering. She was visiting scholar in several renowned institutions, including Chalmers University of Technology, the Hong Kong Polytechnic University, the Massachusetts Institute of Technology (as holder of a Fulbright Fellowship in 2006), the Leibniz University of Hannover (with an Alexander von Humboldt Fellowship in 2010-2011), the University of Texas at Austin and the University of Cape Town. She is the recipient of several prizes, including the RILEM L’Hermite Medal 2011, the AIMETA Junior Prize 2011, the IIFC Young Investigator Award 2012, the Euromech Solid Mechanics Fellowship 2022, the IACM Fellowship 2024, two best paper awards and two student teaching prizes. In 2011 she was awarded a European Research Council Starting Researcher Grant. She has authored or co-authored more than 150 papers on international journals on different topics of computational and applied mechanics. Since 2023 she is Editor of Computer Methods in Applied Mechanics and Engineering.