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Just for my interest I'm thinking how to make some cheap-and-dirty simulation of processes such as:

I can imagine how to do it in Lagraginan frame ( something like particle-in-cell ). But I have feeling that eulerian simulation (just grid, no particles) would be much faster.

But the problem I don't see how to avoid is mixing of components. When some amount of steel influx into cell full of air, I loose information where was the steel and where was the air (it instantly spread homogeneously over whole cell). It will therefore spread to next cell much faster than if it would be simulated as particles with finite velocity. It means there will be fast diffusion of the steel.

are there any tricks how to avoid this mixing ?

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One potential option is the Material Point Method (MPM). It is a successor of the PIC methods that you mentioned, but I do not believe it suffers from the same problems. Implementation is non-trivial, but it might do what you need. It can be described as a "mesh-free" finite element method that stores its information at Lagrangian markers, but updates its motion over the course of a single time step using an ephemeral background finite element grid. I like to plug my friends' work, so I recommend you check out an article from my lab-mate on using MPM for granular flows. http://www.sciencedirect.com/science/article/pii/S0022509616306901

Another potential approach would be to use a level-set function to track your interfaces. My advisor came up with an interesting method a few years back to do nonlinear solid mechanics and fluid-structure interaction on an Eulerian grid. He calls this the "Reference Map" method. It may not be exactly the right thing for you, since the particular way it handles multi-body contact makes it difficult to have topology changes, but it is an interesting take on thermodynamically-consistent nonlinear solid mechanics in an Eulerian frame. http://www.sciencedirect.com/science/article/pii/S0022509612001135

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In terms of purely Eulerian methods for interface capturing multi-material simulations, two current techniques are the level-set approach Tyler Olsen refers to in his answer and Piecewise Linear Interface Capturing (PLIC) for volume of fluid methods.

In the level set approach a scalar field is created which has its zero contour surface lying on the interface. This implicit interface is then displaced by modifying the scalar field under the imposed motion, with material properties at a given point defined by which side of the contour line they lie.

In the PLIC method a linear interface is reconstructed in the cells from the available volume fraction data, so that if, for example, a half filled cell lies next to a full cell, the interface passes through the centre of the cell. This reconstruction relies on assumptions about the smoothness of the interface to use information from neighbouring cells in estimating the interface location in a given cell.

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