Vendredi 13 septembre 2024 – 11h00, amphithéâtre de l'IRPHE (exceptionnellement)
A number of Continuum Mechanics Boundary Value Problems involve large displacements or even large transformations of history-dependent (e.g., elasto-plastic) materials whose Finite Element Method (FEM) simulations would require remeshing procedures that are both computationally expensive and intricate for what concerns the transport of hardening variables. Among the so-called particle methods that aim at proposing suitable alternatives to FEM in this aspect, the Material Point Method (MPM) stands as one prominent possibility. The MPM actually overcomes mesh distortion issues through its consideration of a double layer of spatial discretization that includes, first, a fixed mesh serving as a history-less computational grid for solving continuum mechanics equations and, second, a set of material points which carry material constitutive behaviour and move freely within the mesh. The MPM discretization procedure then enables one to compute on the mesh a discrete field of nodal acceleration which should eventually serve to displace material points. However such a double discretization let rise two intricate issues that are discussed in the seminar, after MPM principles have been presented. First, the back-and-forth mapping between mesh nodes and material points lacks the possibility for a unique definition and a great number of various motion integration schemes have been proposed in the literature, since the PIC and FLIP precursors. The consequences of using one or another motion integration scheme for what concerns the (non-)conservative nature of the method are then illustrated on various examples, from simple configurations with known reference results to the more realistic setting of the collapse of an initial column of a frictional material. Second, the arbitrary position of material points within the mesh also raise stress quadrature issues since the latter actually act as integration points during the MPM procedure. This point is illustrated based on simple configurations, together with an overview of a possible remedy.