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

Outline

• Computational Methodology• Components of the Aeroelastic

Analysis• Results• Conclusions and future work

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

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

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

Mesh Movement

• The edges of the tetrahedral mesh are replaced by springs whose stiffness (Kij) 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

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

Pressure Distribution

Cp distribution, windward and leeward sides

Angle of Incidence = 8 deg, Cl = 0.70, Cd = 0.079, L/D = 8.87

Pressure Distribution

Deflected shape of the sail

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?

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

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)

Parallel Implementation

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

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