  The aim of this paper is to present and validate two new procedures to
enforce the Geometric Conservation Law (GCL) on a moving grid for an Arbitrary
Lagrangian Eulerian (ALE) formulation of the Euler equations discretized in
time for either the Non Linear Frequency Domain (NLFD) or Time-Spectral (TS)
methods. The equations are spatially discretized by a structured finite-volume
scheme on a hexahedral mesh. The derived methodologies follow a general
approach where the positions and the velocities of the grid points are known at
each time step. The integrated face mesh velocities are derived either from the
Approximation of the Exact Volumetric Increments (AEVI) relative to the
undeformed mesh or exactly computed based on a Trilinear Mapping (TRI-MAP)
between the physical space and the computational domain. The accuracy of the
AEVI method highly depends on the computation of the volumetric increments and
limits the temporal-order of accuracy of the deduced integrated face mesh
velocities to between one and two. Thus defeating the purpose of the NLFD
method which possesses spectral rate of convergence. However, the TRI-MAP
method has proven to be more computationally efficient, ensuring the
satisfaction of the GCL once the convergence of the time derivative of the cell
volume is reached in Fourier space. The methods are validated numerically by
verifying the conservation of uniform flow and by comparing the integrated face
mesh velocities to the exact values derived from the mapping.
