An adjoint-based optimization procedure is proposed for improving the robustness and extending the range of linear-elasticity-based mesh deformation techniques. Using the values of the modulus of elasticity E defined in each mesh cell as the design variables, the procedure seeks to determine an optimum distribution of E throughout the mesh, in order to minimize a global objective function which reflects the skewness or lack of quality of the deformed mesh. The technique is applied to highly-stretched mixed element meshes in two and three dimensions on complex geometries, and is shown to be capable of recovering a valid mesh in a small number of optimization cycles for cases where the non-optimized linear-elasticity approach fails. However, the solution of the optimization problem remains relatively costly in terms of cpu time, compared to a non-optimized mesh deformation calculation, making this technique best suited for precomputing improved E distributions prior to the simulation, or for use as a plug-in module to be invoked in cases where the non-optimized procedure fails.