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In Gmsh mesh option list ,there is a smoothing concept, what does this mean and what is the effect of its number to the mesh quality?

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Meshing algorithms can place the vertices of triangles and tetrahedra in a wide variety of ways, but they are us usually essentially constructive (i.e new vertices are introduced, existing vertices stay where they are). Disappointingly, this can cause the meshes which are generated to be unsuitable for numerical calculation. Mesh smoothing, at least in the case of gmsh, is essentially a post-processing step, which takes an existing mesh and iteratively deforms the positions (but not connectivity) of the vertices such that the coordinates better satisfy the numerical discretization of an elliptic PDE, effectively spreading out jumps in the anisotropy across a finite area, reducing the extrema in mesh quality. For more details, see for example here for a talk given by one of the GMSH authors about elliptic meshing. This is part of a wider class of problems called moving mesh partial differential equations (MMPDES).

Since the algorithm is iterative, you would hope that it converges to a fixed state, for a give mesh resolution. How many steps it will take to get there will depend on the original mesh and the other constraints you apply.

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