The influence of compatibility equations of non‐linear elasticity on convergence for spectral method solutions

In structural mechanics the stresses are of a primary interest. Due to the required regularity of the approximation it is often inconvenient to obtain the stresses from the displacement methods. Mixed and hybrid methods are thus proposed to avoid this obstacle. The compatibility conditions should be fulfilled in order to obtain a unique solution. In this paper the compatibility equations of non‐linear elasticity are considered. Despite its very long history, this issue has not been satisfactorily resolved for non‐simply‐connected bodies until recently [1]. Its well known however, that the compatibility conditions are linearly dependent. The aim of the present contribution is to include the spectral methods [2] in order to improve the finite element solution procedure for the reversible elastodynamics problem [3], formulated for a large strain deformation with the deformation gradient tensor as one of the basic unknowns. In the case of the small strains the components of the deformation gradient tensor are of different magnitude. The main contribution is the derivation of the solution strategy in order to take this information into account. Applying this strategy to the implicit time integration method the exponent decrease of the condition numbers of the iteration matrices was achieved. Consequently the accuracy of the calculations should increase and the convergence for spectral method solutions should considerably improve. The theoretical findings are confirmed by a selected numerical example from structural mechanics.