Keywords: discrete element method, multi-scale method, dynamic response, concrete structure, impact, spurious waves.

The general framework of this study deals with the prediction of reinforced concrete structure response under severe dynamic loads such as impacts due to natural hazards or human factors. Reliable and efficient design of structures under such loading needs to take into account local discontinuities due to impact as well as the global response of the structure that may have a linear elastic response far from impacted area. Locally, the discrete element method (DEM) is used to analyze discontinuous phenomena such as failures, fragmentation and compaction. The finite element method (FEM) is applied to the remaining structure to reduce both times of computation and modelling. With this coupled method, different structural responses may be predicted such as the missile penetration, damage of the structure and global displacement or first natural frequencies. The 3D DEM approach consists of rigid spheres with complex interaction laws; it has been validated in previous studies by means of simulations of quasi-static and dynamic tests performed on plain and reinforced concrete structures [1]. The applications of the DEM to large scale structures were limited because of computing costs. The number of elements increase reduces drastically computational efficiency. To improve that point, the region without any assumed damage is modeled by means of the FEM. Both methods use the explicit central difference scheme to solve the equations of motion.

The methodology for multi-scale simulation must take into account different potential problems. The variables can be different on the two domains and it is important to understand the relations that relate them. The process must satisfy the conservation of the physical properties and for dynamic simulations, the interface must not produce spurious mode reflections. For the first difficulty, in our case, the problem occurs from the rotation of the DE. Different tests show that this rotation can be related to the anti-symmetrical part of the displacement gradient. So on the overlapped domains both the displacements and the rotations of the DE model can be written with linear relations of the FE displacements. Different methods have been developed in order to couple DE and FE domains. Some have edge to edge interfaces [2], others use overlapping domains. Our DE model for concrete structures requires heterogeneous (size and position) repartitioning of the DE With such a repartition, an edge to edge interface is not easy to realize so we use an overlapping domain. The starting point of our method uses a bridging sub-domain where the Hamiltonian is taken to be a linear combination of discrete and continuum Hamiltonians. This method has been developed [3] in order to couple the atomistic model and the continuum one. Inside the overlapping domain, we enforce the relations between on the one hand the discrete rotations and displacements and on the other hand the continuum displacements by Lagrange multipliers. For the time discretization, we use the central difference method. At each time step, the displacements are computed without taking into account the Lagrange multipliers. Then, these Lagrange multipliers are computed and they are the solution of a linear problem. In [2], the authors propose to simplify the problem by using a diagonalized matrix in order to reduce the computational cost. In the last step the displacements are modified by taking into account the constraints enforced by the Lagrange multipliers. Due to the variation of discretization sizes, from coarse for the finite elements to fine for the discrete ones, high frequency waves resulting from the impacted zone, modeled with the DE, cannot be transmitted to the FE mesh and are reflected at the interface. The method presented does not suppress these reflections. With the diagonalized matrix, they decrease weakly. We propose a process in which the constraints enforced by the Lagrange multipliers are under estimated. The better result is obtained by dividing these constraints by the number of DE layers of the overlapping domain. The Table 1 gives the percentage of attenuation of the high frequency on a one dimensional problem. We can see that the relaxed multipliers dramatically reduce the wave reflection.

1

S. Hentz, L. Daudeville, F. Donze, "Discrete element modelling of concrete submitted to dynamic loading at high strain rates.", Computers and Structures, 82, 2509-2524, 2004. doi:10.1016/j.compstruc.2004.05.016

2

P. Cundall, M. Ruest, A. Guest, R. Chitombo, "Evaluation of schemes to improve the efficiency of a complete model of blasting and rock fracture", In Numerical modelling in micromechanics Via Particl methods, 107-115, 2003.

3

S. P. Xiao, T. Belytschko, "A bridging domain method for coupling continua with molecular dynamics", Computer Methods in Applied Mechanic Engineering, 193, 1645-1666, 2004. doi:10.1016/j.cma.2003.12.053

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