International Conference «Mathematical and Information Technologies, MIT-2016»

28 August – 5 September 2016

Vrnjacka Banja, Serbia – Budva, Montenegro

Gerasimov A.V.   Cherepanov R.O.  

Tensor smooth length for SPH modelling of high speed impact

Reporter: Gerasimov A.V.

We propose a variant of the method of smooth particles, which uses a tensor representation of the smoothing parameter. This view allows you to switch from spherical particles to particles of ellipsoidal shape, oriented according to the main directions of the strain tensor. Calculations with the representation of particle shape allows better control the amount of adjacent particles involved in the calculation of derivatives, resulting in easier to provide the conditions necessary to calculate approximations of spatial derivatives of maintaining order accuracy. Also, this approach avoids the numerical distortion, which is often observed in the simulation of high-speed collision with smooth particle. To integrate the equations of motion using a weak variational formulation, the Lagrangian for this system is recorded using the SPH-approximation of at least first order accuracy, and time derivatives are approximated by finite difference with the first order of accuracy. The resulting explicit numerical scheme conserved energy and momentum, and tested on the solution of a number of model problems of collision and propagation of elastic waves. Main difference between SPH with constant smoothing length and SPH with tensor smooth parameter is that constant smoothing length lead to numerical fracture, and SPH with tensor smooth shows no numerical fracture and gives more accurate results.

Evolution of smoothing parameter is defined via evolution of strain tensor, and main axes of smoothing are rotated with rotation of material. It allows to avoid neighbour loosing during calculation, and therefore a matrix of approximations of test functions are always well defined. This matrix is used to restore particle inconsistency at each point. If neighbour count of particle becomes small, correction matrix can't be found, and we loose order of accuracy. Variable tensor smoothing parameter allow us to avoid such situation and to preserve order of accuracy.

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