A Robust Unstructured Grid Movement Strategy using Three-Dimensional Torsional Springs

The ability to move a grid during a simulation is of critical importance to many types of simulations, including aero-elasticity simulations for wings, gradient-based design optimization, dynamic movement of control surfaces for maneuvering, and surface tracking methods for free surface flows. For structured grids, transfinite interpolation is a robust method for propagating deformations on the surfaces into the volume grid, and its two-dimensional version is readily extendible to three-dimensions. For unstructured grids, two methods that have been used within two-dimensional grids are the use of linear springs and torsional springs to propagate the changes on the surface into the volume mesh. The use of linear springs provides an efficient method to move unstructured grids, but it is not very robust, in that moderate to large deformations will result in invalid meshes, where some of the elements have been inverted. Farhat developed the torsional spring method for two-dimensional grids, and has demonstrated that this method is much more robust, even for large deformations. Farhat and Murayama have attempted to extend the two-dimensional torsional springs method to threedimensions, and their results are impressive. However, both of their methods reduces the three-dimensional tetrahedra into simpler components without analyzing the dynamics of the three-dimensional object, and as such, their methods do not fully represent a three-dimensional extension of the two-dimensional torsional springs method. In this paper, a three-dimensional extension of the torsional springs method is derived by analyzing the equations of volume and area for a tetrahedron and its faces and edges.