By Gunther Nurnberger

ISBN-10: 3540516182

ISBN-13: 9783540516187

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1. 2. 3. 4. 5. 6. Represent the problem domain with particles so that the computational information is known at the discrete particles at an initial instant / with a proper treatment on the boundary conditions. Discretize the derivatives or integrals in the governing equations with proper particle approximations; From the given velocity and/or position, calculate the strain rate and/or strain, and then calculate the stress at each discrete particles at the instant t; Calculate the acceleration at each discrete particles using the calculated stress; Use the acceleration at the instant t to calculate the new velocities and the new positions at time instant t + At, where At is the incremental time step; From new velocities and/or new positions, calculate the new strain rate and/or new strain at time instant t + At , and then calculate the new stress at time instant t + At.

1 The support domain of the smoothing function Wand problem domain.

These modifications lead to various versions of the SPH methods and corresponding formulations. Monaghan (1988; 1982; 1985) proposed symmetrization formulations that were reported to have better effects. , 1996; Johnson and Beissel, 1996) gave an axis symmetry normalization formulation so that, for velocity fields that yield constant values of normal velocity strains, the normal velocity strains can be exactly reproduced. W. K. Liu et al. (Liu and Chen, 1995) presented the reproducing kernel particle method (RKPM) that can result in better accuracy in the particle approximation.

### Approximation by Spline Functions by Gunther Nurnberger

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