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I would like to hear about scientific codes and commercial packages utilizing meshless methods like Element-Free Galerkin based on Moving Least Squares functions. By "serious" I mean they could be used to solve problems comparable e.g. in size to those solved by FEM.

It's already more than fifteen years since their inception. People who developed them considered them to be very promising. I'm simply trying to understand the current state of the art.

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One of the commercial packages that offer element-free Galerkin (EFG) is LS-DYNA, a multiphysics simulation software. You can find several papers that feature numerical examples with timing and accuracy using LS-DYNA's EFG implementation in DYNALOOK. Some of the problems being solved might satisfy your requirements on being comparable to FEM.

For example:

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You could consider Monte Carlo neutral particle transport to be mesh-free, and it's the dominant method for solving particle transport problems of that type, e.g., the Monte Carlo programs on this Wikipedia page.

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Autodyn (SPH) is pretty much the standard for simulating ballistics/impacts/explosions. Used a lot in the industry.

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