We present novel methods for finite element ALE simulations involving curved boundaries and internal material interfaces. Free-slip wall BC are enforced by enhancing the variation formulation of [2] by Nitsche-type terms to enforce the BC weakly [3], preserving the structure of the original discretization, i.e., all mass matrices are constant and total energy is conserved. For mesh optimization, we extend the original Target-Matrix Optimization Paradigm formulation [4] with a penalty term that enforces tangential mesh relaxation while taking into account the discrepancies between the analytic curves and the boundary node positions obtained from the Lagrangian motion. The robustness and accuracy of the method are validated on a set of Sedov-type ALE tests involving nontrivial curved boundaries. Furthermore, we discuss a new method for two-material Lagrangian hydrodynamics [1] that relies on an exact (or sharp) material interface representation, that is, it uses the precise location of the material interface. By reformulating the original interface problem over a surrogate interface, located in proximity of the true interface, we utilize the Shifted Interface approach to avoid cut cells and the associated problematic issues regarding implementation, numerical stability, and matrix conditioning. We demonstrate the proposed algorithm on established two-material numerical benchmarks. [1] Nabil M. Atallah, Ketan Mittal, Guglielmo Scovazzi, Vladimir Z. Tomov, “A High-Order Shifted Interface Method for Lagrangian Shock Hydrodynamics” Journal of Computational Physics, 523(113637), 2025. [2] Veselin A. Dobrev, Tzanio V. Kolev and Robert N. Rieben, “High-Order Curvilinear Finite El- ement Methods for Lagrangian Hydrodynamics”, SIAM Journal on Scientific Computing, 34(5), pp. 606–641, 2012. [3] Nabil M. Atallah, Vladimir Z. Tomov and Guglielmo Scovazzi, “Weak Boundary Conditions for Lagrangian Shock Hydrodynamics: a High-Order Finite Element Implementation on Curved Boundaries”, Journal of Computational Physics, 507(112959), 2024. [4] Veselin A. Dobrev, Patrick Knupp, Tzanio V. Kolev, Ketan Mittal, Robert N. Rieben and Vladimir Z. Tomov, “Simulation-Driven Optimization of High-Order Meshes in ALE Hydrodynamics”, Computers and Fluids, 208(104602), 2020. This work performed under the auspices of the U.S. Department of Energy by Lawrence Liver- more National Laboratory under Contract DE-AC52-07NA27344 (LLNL-ABS-862743)