Accurate simulations of large deformation hyperelastic materials by the finite element method is still a challenging problem. In a total Lagrangian formulation, even when using a very fine initial mesh, the simulation can break down due to severe mesh distortion. Error estimation and adaptive remeshing on the initial geometry are helpful and can provide more accurate solutions (with a smaller number of degrees of freedom) but are not sufficient to attain very large deformations. The updated Lagrangian formulation where the geometry is periodically updated is then preferred. Remeshing may still be necessary to control the quality of the elements and to avoid too severe mesh distortion. It then requires frequent data transfer from the old mesh to the new one and this is a very delicate issue. If these transfers are not done appropriately, accuracy can be severely affected.
In this paper, we present an updated Lagrangian formulation where the error on the finite element solution is estimated and adaptive remeshing is performed in order to concentrate the elements of the mesh where the error is large, to coarsen the mesh where the error is small and at the same time to control mesh distortion. In this way, we can reach high level of deformations while preserving the accuracy of the solution. Anisotropic remeshing pushes this idea one step further by allowing the presence of elements with large aspect ratio in certain directions compatible with the solution. This also reduces the number of DoF needed to obtain a given accuracy. The mesh is thus adapted in order to both improve the accuracy of the numerical solution and avoid degenerate elements, while also decreasing the computational burden. In this work, the fully optimal anisotropic mesh