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Numerical techniques for fast dynamics problems Introduction Particle Methods are a class of numerical methods which basic idea is to represent a continuum trough a set of particle with a proper mass, velocity and energy, that carry with themselves the information of the continuum. Such particles are not “rigidly” connected between themselves, in the sense that there is not an underlying mesh or grid. This characteristic permits Particle Methods to avoid numerical problem typical of the Finite Element Methods (i.e., element distortion, spurious numerical errors, etc.), and also make Particle Methods ideal for the description of fast dynamics problems, e.g. explosions and impacts, and also for fluiddynamics and gas dynamics. Goal Given a continuum domain in Ω (Fig. 1), the first step is to discretize it in a certain number of particles (Fig. 2). Then it is necessary a procedure to compute the derivatives on each particle. The general idea developed to achieve the discrete differential operators is to consider the Taylor series expansion of a function u(x) up to the desired order, then to project it on a set of known projection functions. Then we are able to write an algebraic system in the form: A^{i}d^{i}=f^{i}
where A^{i} is a coefficient matrix, d^{i} is the vector containing the derivative values in x_{i}, and f^{i} is a term containing the value of the unknown function in x_{i}. Then we have the expressions of the derivative values depending on the values of the unknown function u, and we can use them for a collocation method. The method has been applied to linear elastostatics and elastodynamics, with results that confirm, in many cases, the expectation of secondorder accuracy (Fig. 3). Figure 1. Plate with a central hole: geometry, symmetry boundary conditions and loads. Figure 2. Discretization of the problem. Figure 3. Convergence diagram of the error of the Stress Intensitiy Facor (SIF) for a plate with a central hole under a remote traction. Figure 4. σ_{xx} distribution in a quarter of perforated plate under a remote traction (obtained with MFPM and 251001 particles). Figure 5. Geometry, boundary conditions and loads in a rectangular bar under a quasiimpulsive traction load. Figure 6. σ_{xx} distribution in a 2D bar under a quasiimpulsive traction load during some time instants, obtained with the MFPM and 2121 particles. References

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