Novosibirsk, Russia, May, 30 – June, 4, 2011

International Conference
"Modern Problems of Applied Mathematics and Mechanics: Theory, Experiment and Applications", devoted to the 90th anniversary of professor Nikolai N. Yanenko

Kuropatenko V.  

Simulation of dynamic processes in continuous media: models and methods

The era of mathematical modeling started in 1950 when D. Neumann and R. Richtmyer published a paper describing a shock simulation method which was developed and used for US nuclear weapons calculations. The first Soviet computer STRELA worked at a speed of 2×103 operations per second and had a very small memory. Only very accurate numerical methods could help attain a high fidelity of mathematical modeling with such poor resources. The paper tracks the evolution of continuous mechanics methods and models applied to a wide variety of dynamic processes including
• Shock and detonation waves, and their interaction with one another and with interfaces and weak shocks, rarefaction and compression waves;
• Heat and radiation effects on the environment;
• Elastic and plastic deformation and fracture of initially solid materials;
• Polymorphic phase transitions, melting, evaporation, ionization;
• Mixing from interface instability.

The paper discusses principles that govern the construction of Equations Of State (EOS) for metals and rocks, or for explosives, and presents EOSs which ensure high accuracy in the reproduction of material behavior under pressures to 103 GPa, temperatures to 103KK and compressions to 15-20. Kinetic equations are added to continuum mechanics equations to describe non-equilibrium phase transitions and relaxation in metastable regions (negative pressures and elastic shears).
The paper compares shock simulation methods which differ in the energy dissipation mechanism. The methods replace strong discontinuities by shock waves smeared over several cells. They were for the first time subjected to a comparative analysis in the book by B.L. Rozhdestvensky and N.N. Yanenko in 1968. In the mid 90-s the dissipative properties of these difference schemes across rarefaction waves were investigated. At the beginning of this century their distraction properties across shock waves were established. It was shown that distraction depended on shock amplitude and weak shocks smeared wider.
The paper also considers a non-uniform method in which shocks do not smear but propagate through a structured mesh as surfaces of strong discontinuities. Weak and contact discontinuities are also included. The method helps completely eliminate entropy trails.
A multi-component flow model is proposed for simulating material mixing and separation. Conservation laws for components transforms to conservation laws for the mixture when summed. A necessary condition for such a transformation is the cluster interaction of components with the mixture. The mixture model that includes cluster interaction is unique.
The work was done under the support of Russian Basic Research Foundation Grant 10-01-00032.

Abstracts file: Abstr-ru.doc
Full text file: Kuropatenko.pdf

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