Mesh Generation and Deformation Algorithm for Aeroelasticity Simulations

H IGH-FIDELITY flow solvers for aeroelastic applications require the use of computational meshes that deform as the structure is being displaced. High-aspect-ratio wings increase the demands on the robustness of the mesh-deforming algorithm, because these wings are extremely flexible and attain deformations that are a significant fraction of the span of the wing. The mesh deformation algorithm must be not only robust but also computationally inexpensive to avoid penalizing the turnaround time of the aeroelastic computations. Different approaches have been developed to solve the movingmesh problem. For meshes generated by using overlapping grids, a natural way to allow for gridmotion is to slide the overlapping region of the grids [1,2]. The advantage of thismethod is that the body-fitted meshes do not deform during the body motion. A disadvantage of this approach is that the interpolation algorithm that communicates the solution between grids has to be updated for each overlapping position of the grids. The tension spring analogy [3] is one of themostwidely usedmesh deformation strategies. In this approach, each edge of the mesh is represented by a spring for which the stiffness is proportional to the reciprocal of the length of the edge. By replacing the edges with springs, a deformation of the boundary translates into a deformation of the spring network, which adjusts its shape to the equilibrium position of the network. The displacements in each direction are decoupled and the equation that updates the position of the nodes is relatively easy to solve. A disadvantage of this method is that for highly distorted meshes, collapsed or negative volume cells may appear, especially on high-aspect-ratio cells such as those used for viscous flows. An improvement over the tension spring analogy is the torsion spring analogy [4,5]. The torsion spring analogy consists of adding a torsional spring to the tension-spring-analogy technique. The stiffness of the torsional spring is related to the angle between the edges. As the angle tends to zero, the stiffness tends to infinity, thus preventing vertices from crossing over edges and avoiding negativevolume cells. The disadvantage of this method is the higher complexity and computational cost than with the tension spring analogy. The transfinite-interpolation mesh deformation technique is based on the linear interpolation of the boundary motion [6]. The motion of a node located between amoving and a fixed boundary is equal to the motion of themoving boundary times a scale factor. This scale factor, assigned to each node of themesh, depends on the distances from the node to the moving and the fixed surfaces. The scale factor is 1.0 for nodes on the moving boundary and 0.0 for nodes on the fixed boundary. The method guarantees a smooth transition between the moving boundaries and the fixed boundaries. One disadvantage of this method is that it cannot guarantee the mesh orthogonality at deforming surfaces, a condition that is important for viscous flows. Another approach to simulatemesh deformation is to use the linear elasticity equations [7]. The deformed grid is obtained by solving the equilibrium equations for the stress field. Themodulus of elasticity is chosen to be inversely proportional to the cell volume or to the distance from the deforming boundaries. Therefore, the cells close to the moving boundaries have small deformations, and the majority of the mesh deformation is relegated to the regions farther away from the moving boundary. This Note presents a grid generation and deformation algorithm for wings with large deformations. The computational domain was discretized using a hybrid grid that consisted of structured hexahedra around the wing and unstructured triangular prisms elsewhere. The mesh was divided in layers that were topologically identical in the spanwise direction. The mesh deformation algorithm was applied in two steps. First, the spring analogy technique was applied to deform the nodes within a mesh layer. Second, the layers were deformed to be perpendicular to the boundaries of the domain and to the surface of the wing. The Note describes the mesh generation algorithm and the mesh deformation algorithm and shows results for a wing with large tip deformation.