1. Steady Aeroelastic Computations to Predict the Flying Shape of Sails Sriram Antony Jameson Dept. of Aeronautics and Astronautics Stanford University First MIT Conference on Computational Fluid and Solid Mechanics June 12-15 2001

2. Outline Computational Methodology Components of the Aeroelastic Analysis Results Conclusions and future work

3. Computational Methodology Solve the equations of motion of the flow Estimate the deflected shape of the sail Deform the computational mesh Static equ.? Prescribe initial geometry No

4. Inviscid flow Computation Unstructured computational mesh generated using MESHPLANE (Prof. Tim Baker, Princeton University) The Euler equations are integrated in time using a modified Runge-Kutta scheme The convective and diffusive fluxes for each node are efficiently evaluated by traversing the edges of the computational mesh Blended first and third order dissipation terms are constructed from differences along edges Residual averaging is used to accelerate convergence to steady state

5. Finite Element Structural Analysis The pressure loading from the flow solver is interpolated to the structure using cubic splines for each section The sail cloth is approximated as an orthotropic membrane which cannot resist bending The translational degrees of freedom at the boom and the mast were suppressed The deflections were evaluated using a non-linear analysis with the commercial package MSC/Nastran

6. Mesh Movement The edges of the tetrahedral mesh are replaced by springs whose stiffness (K ij ) is inversely proportional to the square of the length of the edge The equation of static equilibrium for each node is solved using a Jacobi iteration scheme To prevent grid cross-over, the change in the shape of the sail is decomposed into smaller steps and the equilibrium position of the nodes of the fluid mesh is computed for each step

7. Results Elliptic planform, parabolic section profiles with no initial twist Maximum camber = 10 %, position of maximum camber = 35-45 % of local chord The mast is elliptic in cross-section and assumed to be rigid The thickness of the sail = 1 mm Boom length = 1 m, Mast height = 3.5 m The aspect ratio of the sail was 2.3 No. of cells in the fluid mesh = 1.3 million No. of nodes in the structural mesh = 4000 5 aeroelastic iterations were required to obtain the steady deflected shape of the sail

8. Pressure Distribution C p distribution, windward and leeward sides Angle of Incidence = 8 deg, C l = 0.70, C d = 0.079, L/D = 8.87

9. Pressure Distribution

10. Deflected shape of the sail

11. Status of the Analysis The basic components of the aeroelastic procedure have been tested for a sample sail geometry Improved modeling of the sail cloth will lead to more accurate predictions of the flying shape Is the turn-around time of this analysis reasonable enough to be used in an automated design environment?

12. Status of the Analysis (cont) Total computational time ≈ 20 hours Time for each flow solution ≈ 4 hours Time for each structural analysis ≈ 10 minutes Time for each mesh perturbation ≈ 3 minutes (All times are for one processor of an SGI Origin 2000) The computational time for the flow solution can be significantly reduced by parallelizing the flow solver

13. Parallel Implementation The computational grid is divided into physical regions which approximately contain the same number of nodes These physical regions are distributed among the available processors taking into account the cost of communication across processor boundaries The nodes within each processor and the edges that surround these nodes are stored for each processor Communication tables which allow each processor to gather information across processor boundaries are pre- processed and stored All parallel communication is handled by MPI (Message Passing Interface)

14. Parallel Implementation

15. Status of the Analysis (cont) Total computational time ≈ 2 hours Time for each flow solution ≈ 15 minutes Time for each structural analysis ≈ 5 minutes Time for each mesh perturbation ≈ 3 minutes (The flow solver was run on a parallel machine and used 8 processors) The computational time for the flow solution can be further reduced by using multigrid techniques to accelerate convergence to steady state

16. Conclusions and Future Work Parallel implementation of multigrid techniques is currently in progress Low mach number corrections which will improve the convergence of the iterative scheme and the accuracy of the solution need to be incorporated Modifying the inlet profile to account for the boundary layer over the sea will allow the analysis to predict the flow physics more accurately This analysis tool will be embedded in an automated design environment where changes to the shape of sail will be predicted to maximize its performance