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Mesh deformation techniques
Numerical techniques have been developed at ONERA for 2D and 3D mesh deformation (Dugeai et al., 2000). They are based on a linear structural analogy, with discrete spring networks or continuous elastic analogy. A finite element formulation is used, and special features allow the reduction of the size of the problem, especially in the Navier-Stokes case.
In the case of the spring analogy, two different techniques have been developed. The first one is the method proposed by Batina (Batina, 1989), and the second one is an extension in which the 3 components of the displacement vector are coupled.
In the case of the continuous elastic analogy, 8-node hexahedral finite elements are used to discretize the problem of the deformation of a linear elastic medium. The local stiffness matrix is computed using a numerical Gauss integration procedure with a cheap but not exact one Gauss point integration, which leads to Hour-Glass modes terms. A special procedure is used to remove the singularity of the stiffness matrix, giving satisfactory enough results for the grid deformation purpose.
For both approaches, spring network or elastic material analogy, the static equilibrium of the discretized system leads to the following linear system:
Kii qi Kif qf
where qi and qf are respectively the induced and prescribed displacement vectors. As the stiffness matrix is positive definite, the system is solved using a pre-conditioned conjugated gradient method. The technique has been implemented in the case of multi-block structured grids. The full mesh deformation is defined as a sequence of individual block deformations. Additional conditions are set on the boundaries to impose zero or prescribed displacement values, and to get a continuity of the deformations at block interfaces.
A macro-mesh technique is used for large grid sizes, which is often the case in 3D Navier-Stokes computations. The macro-mesh is defined from the original one by packing several cells, typically 2, 3, or 5 cells, in each direction. In the case of Navier-Stokes meshes, the whole boundary layer region is packed, in normal direction, in a single macro-cell. The coarse macro-mesh is then deformed using the structural analogy techniques, and the inner node displacements are finally interpolated in each macro-cell.
