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If I use very coarse mesh for the pre-deformation (such as pre-bending, gravity loading etc), and then refine the mesh in the later forming analysis. Will it result in obvious numerical error for the final results?

With coarse mesh, it is very efficient to obtain results for the pre-deformation. However, if the mesh is too coarse, and there are big differences between the neighboring element normals, then there might cause unexpected artificial bending in the later deformations.

Even though the mesh can be refined after pre-deformation, the newly added nodes will undergo further deformation to reach the curved rigid body surface. Accordingly, there will be obvious difference between the strains of neighboring elements, and there might cause unexpected deformation localization along the edges of the adapted elements.

It is not a good practice to have a too coarse mesh for the pre-deformation if the curvature of the rigid tool is large. In other words, the initial mesh should be based on the geometry of the rigid tools.