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This article appeared in a journal published by Elsevier. The attachedcopy is furnished to the author for internal non-commercial researchand education use, including for instruction at the authors institution
and sharing with colleagues.
Other uses, including reproduction and distribution, or selling orlicensing copies, or posting to personal, institutional or third party
websites are prohibited.
In most cases authors are permitted to post their version of thearticle (e.g. in Word or Tex form) to their personal website orinstitutional repository. Authors requiring further information
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Analysis and shape optimization in incompressible flows with unstructured grids
Sriram Shankaran a,*, Antony Jameson b, Luigi Martinelli c
a School of Engineering, Stanford University, Durand Building, 496 Lomita Mall, Stanford, CA 94305-4035, United Statesb Department of Aeronautics and Astronautics, School of Engineering, Stanford University, Durand Building, 496 Lomita Mall, Stanford, CA 94305-4035, United Statesc Department of Mechanical and Aerospace Engineering, D 216 E-Quad Princeton, NJ 08544, United States
a r t i c l e i n f o
Article history:Received 23 December 2009Received in revised form 18 April 2010Accepted 19 May 2010Available online 10 June 2010
The aim of this study is to develop and validate numerical methods that perform shape optimization inincompressible flows using unstructured meshes. The three-dimensional Euler equations for compress-ible flow are modified using the idea of artificial compressibility and discretized on unstructured tetra-hedral grids to provide estimates of pressure distributions for aerodynamic configurations.Convergence acceleration techniques like multigrid and residual averaging are used along with parallelcomputing platforms to enable these simulations to be performed in a few minutes. This computationalframe-work is used to analyze sail geometries. The adjoint equations corresponding to the ‘‘incompress-ible” field equations are derived along with the functional form of gradients. The evaluation of thegradients is reduced to an integral around the boundary to circumvent hurdles posed by adjoint-basedgradient evaluations on unstructured meshes. The reduced gradient evaluations provide major computa-tional savings for unstructured grids and its accuracy and use for canonical and industrial problems is amajor contribution of this study. The design process is driven by a steepest-descent algorithm with afixed step-size. The feasibility of the design process is demonstrated for three inverse design problems,two canonical problems and one industrial problem.
� 2010 Elsevier Ltd. All rights reserved.
1. Introduction
With the availability of high performance computing platformsand robust numerical methods to simulate fluid flows, it is possibleto shift attention to automated design procedures which combineCFD with optimization techniques to determine optimumaerodynamic designs. The feasibility of this is by now well esta-blished [1–6], and it is actually possible to calculate optimumthree-dimensional transonic wing shapes in a few hours, account-ing for viscous effects with the flow modeled by the Reynolds aver-aged Navier Stokes (RANS) equations. By enforcing constraints onthe thickness and span-load distribution one can make sure thatthere is no penalty in structure weight or fuel volume. Larger scaleshape changes such as planform variations can also be accommo-dated [7]. It then becomes necessary to include a structural weightmodel to enable a proper compromise between minimum drag andlow structure weight to be determined.
Aerodynamic shape optimization has been successfully per-formed for a variety of complex configurations using multi-blockstructured meshes [8,9]. Meshes of this type can be relatively eas-
ily deformed to accommodate shape variations required in the re-design. However, it is both extremely time-consuming andexpensive in human costs to generate such meshes. Consequentlywe believe it is essential to develop shape optimization methodswhich use unstructured meshes for the flow simulation.
Typically, in gradient-based optimization techniques, a controlfunction to be optimized (the wing shape, for example) is parame-terized with a set of design variables and a suitable cost function tobe minimized is defined. For aerodynamic problems, the cost func-tion is typically lift, drag or a specified target pressure distribution.Then, a constraint, the governing equations can be introduced inorder to express the dependence between the cost function andthe control function. The sensitivity derivatives of the cost functionwith respect to the design variables are calculated in order to get adirection of improvement. Finally, a step is taken in this directionand the procedure is repeated until convergence is achieved. Find-ing a fast and accurate way of calculating the necessary gradientinformation is essential to developing an effective design methodsince this can be the most time consuming portion of the designprocess. This is particularly true in problems which involve a verylarge number of design variables as is the case in a typical three-dimensional shape optimization.
The control theory approach [10–12] has dramatic computa-tional cost advantages over the finite-difference method ofcalculating gradients. With this approach the necessary gradients
0045-7930/$ - see front matter � 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.compfluid.2010.05.013
* Corresponding author. Present address: 1301 Keyes Avenue, Schenectady, NY12309, United States. Tel./fax: +1 518 280 6138.
are obtained through the solution of an adjoint system of equationsof the governing equations of interest. The adjoint method is extre-mely efficient since the computational expense incurred in the cal-culation of the complete gradient is effectively independent of thenumber of design variables.
In this study, a continuous adjoint formulation has been used toderive the adjoint system of equations. Accordingly, the adjointequations are derived directly from the governing equations andthen discretized. This approach has the advantage over the discreteadjoint formulation in that the resulting adjoint equations areindependent of the form of discretized flow equations. The adjointsystem of equations have a similar form to the governing equationsof the flow and hence the numerical methods developed for theflow equations [13–15] can be reused for the adjoint equations.
The gradient is derived solely from the adjoint solution and thesurface displacement, independent of the mesh modification. Thisis crucial for unstructured meshes. If the gradient depends on theform of the mesh modification, then the field integral in the gradi-ent calculation has to be recomputed for mesh modifications corre-sponding to each design variable. This would be prohibitivelyexpensive if the geometry is treated as a free surface defined bythe mesh points. Consequently in order to reduce the computa-tional cost with this approach [16–18], the number of design vari-ables would have to be reduced by parameterizing the geometry.However, this reduced set of design variables could not recoverall possible shape variations.
A steepest-descent method is finally used to improve the initialdesign. In order to guarantee that the shape variations remain suf-ficiently smooth the gradients are redefined so that they corre-spond to an inner product in a Sobolev space. This isaccomplished by an implicit smoothing procedure which also actsas an effective pre-conditioner, with the result that the number ofdesign steps needed to reach an optimum is quite small, of the or-der of 20–50.
The main contributions of this study are on two fronts: (1) dem-onstrate the use of artificial compressibility methods on unstruc-tured meshes for industrial problems like sail geometries used inAmericas Cup, (2) while the general approach of adjoint-basedtechniques for compressible flows has found widespread accep-tance, it is possible, with minimum modifications apply the sametechnique to incompressible flows. We demonstrate the latter ontwo canonical inverse design problems and one industrial problem.
2. Analysis with CFD
2.1. Finite volume discretization of the flow equations
The incompressible flow equations are derived using the idea ofartificial compressibility [19]. These equations are solved using thenow widely used set of algorithms, namely, modified Runge–Kuttaintegrators, switched dissipation algorithms that use a central dis-sipation, residual averaging and multigrid convergence algorithms[20]. As most of these techniques are now commonly used, we re-fer the readers to the references for details. We however detail twoof the algorithms, multigrid and parallel implementation as theyhave some differences from commonly used methods.
2.2. Multigrid
Multigrid techniques are widely used to accelerate the conver-gence of a system of equations to steady state. A general frame-work for the development of full-approximation multigrid meth-ods for non-linear equations can be outlined as follows [21].
For unstructured grids, there are many choices for the nature ofthe coarser grids. In the present study a series of non-nested
meshes was generated by a grid generator (MESHPLANE). The ini-tial solution from a particular mesh is advanced in pseudo-time toobtain new estimates of the flow variables. On transfer to the nextlevel of the multigrid, the solution for the coarse grid mesh pointsis interpolated from the four nodes of the fine mesh tetrahedronthat contains this node. Further, the accumulated residual at eachfine mesh point is distributed to each node of the tetrahedron inthe coarse mesh that that encloses the fine mesh node. The inter-polating factors for each node are computed from weights whichare based on the volume included by a given node and oppositeface of the tetrahedron (Fig. 1). This reduces to a second orderinterpolation scheme on equilateral tetrahedra and has been foundto be sufficient for the present calculations. The estimate of theresiduals from the fine mesh is used to advance the solution onthe coarse mesh where the larger scale errors can be more effi-ciently accomplished on the coarse mesh. Further levels in themultigrid cycle involve the same operations are before, therebyusing grids that are coarser and coarser to convect the error termsout of the computational domain faster. The ascending side of themultigrid cycle estimates a correction from each grid which is theninterpolated to the next finer mesh in the sequence (Fig. 2). Thecorrections from the coarser mesh are transferred using similarinterpolating factors as for the aggregation operations. Multigridcycles which progress in the shape of a W have generally beenfound to provide faster convergence to steady state than the simpleV cycle.
2.3. Parallel implementation of the unstructured multigrid flow solver
To exploit the availability of modern parallel computing plat-forms, a combination of the AIRPLANE [14] and FLO77 [22] compu-tational programs, was parallelized. Due to the unstructurednature of the computational grid, a wide variety of possible datastructures to implement the underlying numerical algorithms ex-ist. The following sections outline the choice of data structuresand algorithms that were made to parallelize the flow solver.
2.3.1. Domain decomposition, load balancingA modified coordinate bisection method was used to recursively
divide a given computational mesh into sub-domains (Fig. 3). Thesub-domains were created such that they contained approximatelythe same number of computational nodes. No effort was made tooptimize the domain decomposition process so as to minimizethe number of edges that are shared between the sub-domains.The sub-domains are distributed among the available processorsin a manner that minimizes a combination of the computationalcost associated with the domains in each processor and the cost
n1
n2
n3
a1
a2
a3
p
Fig. 1. Interpolation coefficients for use in the multigrid cycle.
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of communication among the processors for a given distribution.This methodology was found to result in well balanced loaddistributions.
As the flow solver uses an edge-based data structure to accumu-late the fluxes at each vertex, the edges surrounding the nodes thatlie within a partition are accumulated. If an edge connects a pair ofnodes that lie across processor boundaries, this edge is duplicatedin the two processors and ‘halo’ nodes are constructed for bothprocessors. This idea is illustrated further in Fig. 4.
2.4. Parallel implementation of the multigrid algorithm
The use of multigrid techniques for flow analysis necessitatesthe need to exchange information between fine and coarse gridpoints and vice-versa. While the residuals from the fine grid pointsneed to be accumulated at the coarse grid points, the correctionsfrom the coarse grid points need to be transferred back to the finegrid points. Hence, efficient point location methods are used todetermine the interpolation coefficients associated with each com-putational node in the multigrid cycle. The required search meth-odology has been implemented using an octree-based searchroutine. Each grid in the multigrid cycle is recursively divided intooctants that contain a certain number of points. Using this datastructure, a given point is identified within an octant and the
closest node closest in this octant is determined. The tetrahedraconnected to this node are checked to see if they contain the searchpoint. Once such a tetrahedron is identified, interpolation coeffi-cients to aggregate the residuals and to interpolate the correctionsare constructed. Further, to reduce the communication cost amongprocessors during the transfer of information between the fine andcoarse grids, sub-domains on the coarser grids are constructed anddistributed so as to conform to the division and distribution of thefine mesh. Typical computational times to build the octree and tobuild the interpolation tables are in the range of a few minutesfor a sequence of meshes containing a million nodes. The scalabil-ity of the parallel implementation is shown in Fig. 5.
3. The general formulation of the adjoint approach to optimaldesign
For flow about an airfoil, or wing, the aerodynamic propertieswhich define the cost function are functions of the flow field
fine grid nodes
coarse grid nodes
Fig. 2. Transfer of solution, residuals and corrections between the fine and coarsemesh.
9
8
7
6
54
3
10y
x
16
15
11
12
1314
21
Fig. 3. Domain decomposition of a rectangular region using a modified bisectionmethod.
inter-processor boundary
edges that are shared between processors
Fig. 4. Halo nodes and the distribution of edges along processor boundaries.
2 4 6 8 10 12 14 162
4
6
8
10
12
14
16
Number of Processors
Spee
dUp
Actual SpeedUpIdeal SpeedUp
Fig. 5. Speedup from the parallel implementation.
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variables, w, and the physical location of the boundary, which maybe represented by the function, F, say. Then
I ¼ Iðw;FÞ
and a change in F results in a change
dI ¼ @IT
@wdwþ @IT
@FdF ð1Þ
in the cost function. Using control theory, the governing equationsof the flow field are introduced as a constraint in such a way thatthe final expression for the gradient does not require re-evaluationof the flow field. In order to achieve this, dw must be eliminatedfrom Eq. (1). Suppose that the governing equation R which ex-presses the dependence of w and F within the flow field domainD can be written as
Rðw;FÞ ¼ 0 ð2Þ
Then dw is determined from the equation
dR ¼ @R@w
� �dwþ @R
@F
� �dF ¼ 0 ð3Þ
Next, introducing a Lagrange Multiplier w, we have
dI ¼ @IT
@wdwþ @IT
@FdF� wT @R
@w
� �dwþ @R
@F
� �dF
� �
dI ¼ @IT
@w� wT @R
@w
� �@w
!dwþ @IT
@F� wT @R
@F
� �dF
!
Choosing w to satisfy the adjoint equation
@R@w
� �T
w ¼ @I@w
ð4Þ
the first term is eliminated and we find that
dI ¼ GdF ð5Þ
where
G ¼ @IT
@F� wT @R
@F
� �ð6Þ
This process allows for elimination of the terms that depend on theflow solution with the result that the gradient with respect with anarbitrary number of design variables can be determined without theneed for additional flow field evaluations.
After taking a step in the negative gradient direction, the gradi-ent is recalculated and the process repeated to follow the path ofsteepest-descent until a minimum is reached. In order to avoid vio-lating constraints, such as the minimum acceptable wing thick-ness, the gradient can be projected into an allowable subspacewithin which the constraints are satisfied. In this way one can de-vise procedures which must necessarily converge at least to a localminimum and which can be accelerated by the use of more sophis-ticated descent methods such as conjugate gradient or quasi-New-ton algorithms. There is a possibility of more than one localminimum, but in any case this method will lead to an improve-ment over the original design.
Here the expression for the cost variation depends on the meshvariations throughout the domain which appear in the field inte-gral. However, the true gradient for a shape variation should notdepend on the way in which the mesh is deformed, but only onthe true flow solution.
3.1. Adjoint equations for the Euler equations modified by the artificialcompressibility method
Although the adjoint equation represents a linear set of partialdifferential equations for the adjoint variables, they are of the sameform of the flow equations. The numerical solution proceduresdeveloped for the flow equations are applied to the adjoint systemwith the appropriate boundary conditions. The adjoint co-stateflux terms are modified to account for the introduction of the arti-ficial compressibility terms in the governing flow equations. Themethodology followed here is derived from the work of Cowlesand Martinelli [23]. The adjoint field equations can be expressedas a time dependent system of the form,
@w@t� ½Ai�T
@w@xi¼ 0 ð7Þ
where
w ¼
p
/1
/2
/3
8>>><>>>:
9>>>=>>>;
ð8Þ
Hence, this system can be integrated to steady state using a pre-conditioner similar to that used in the method of artificial com-pressibility. The adjoint ‘continuity’ equation is augmented by atime derivative of the adjoint pressure p.
@p@t� C2 @/i
@xi¼ 0 ð9Þ
The form of C is identical to that used for the flow equations sincethe magnitude of the eigenvalues of the flux Jacobians for the twosystems are identical. Together with Eq. (9), the adjoint system isdiscretized and solved in a manner that is consistent with that usedfor the flow equation.
3.2. Reduced gradient formulations
Continuous adjoint formulations have generally used a form ofthe gradient that depends on the manner in which the mesh ismodified for perturbations in each design variable. To representall possible shapes the control surface should be regarded as a freesurface. If the surface mesh points are used to define the surface,this leaves the designer with a thousands of design variables. Onan unstructured mesh evaluating the gradient by perturbing eachdesign variable in turn, would be prohibitively expensive becauseof the need to determine corresponding perturbations of the entiremesh. This would inhibit the use of this design tool in any mean-ingful design process.
In order to avoid this difficulty an alternate formulation to thegradient calculation is followed in this study. This idea was devel-oped by Jameson and Kim [24] and was validated for two andthree-dimensional problems with structured grids. However, as itis possible to devise mesh modification routines that are computa-tionally cheap on structured grids, the major benefit of this alter-nate gradient formulation is for general three-dimensionalunstructured grids. To complete the formulation of the control the-ory approach to shape optimization, the gradient formulations areoutlined next. The formulation for the reduced gradients in thecontinuous limit is presented in the context of transformation be-tween the physical domain and the computational domain, and iseasily extended to unstructured grid methods where these trans-formations are not explicitly used.
Consider the case of a mesh variation with a fixed boundary.Then,
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dI ¼ 0
but there is a variation in the transformed flux,
dFi ¼ Cidwþ dSijfj
Here the true solution is unchanged. Thus, the variation dw is due tothe mesh movement dx at fixed boundary configuration. Therefore
dw ¼ rw � dx ¼ @w@xj
dxjð¼ dw�Þ
and since
@
@nidFi ¼ 0
it follows that
@
@niðdSijfjÞ ¼ �
@
@niðCidw�Þ ð10Þ
It is verified by Jameson and Kim [24] that this relation holds in thegeneral case with boundary movement. NowZD
wTdRdD ¼ZD
wT @
@niCi dw� dw�ð ÞdD
¼ZB
wT Ci dw� dw�ð ÞdB�ZD
� @wT
@niCi dw� dw�ð ÞdD ð11Þ
Here on the wall boundary
C2dw ¼ dF2 � dS2jfj ð12Þ
Thus, by choosing w to satisfy the adjoint equation and the adjointboundary condition, we have finally the reduced gradient formula-tion that
dI ¼ZBW
wT dS2jfj þ C2dw�� �
dn1dn3
�Z Z
BW
dS21w2 þ dS22w3 þ dS23w4ð Þpdn1 dn3 ð13Þ
3.3. The need for a Sobolev inner product in the definition of thegradient
Another key issue for successful implementation of the contin-uous adjoint method is the choice of an appropriate inner productfor the definition of the gradient. It turns out that there is an enor-mous benefit from the use of a modified Sobolev gradient, whichenables the generation of a sequence of smooth shapes.
When the metric perturbations, dSij, are related to the surfacemotion, Eq. (13) finally yields the cost variation in the form of aninner product defined over the surface
dI ¼ G; dFð Þ ¼Z
BGdFdnB
where dF denotes the surface displacement. Then the update
Fnþ1 ¼Fn � kGn
would result in an improvement
Mesh deformation
Pre−processor
Generate sequence ofmeshes
Repeat until design
Gradient evaluation,camber line changes
Adjoint Solver (Parallel)
Flow Solver (Parallel)
criterion is satisfied
Fig. 6. Flow chart of the overall design process.
24 m
Twisted inflowwith boundary layerprofile
Main Sail
10 m
10 m
2.3 m
Jib
Fig. 7. Sail geometry.
0 50 100 150 200 250 300 350 400 450 50010−14
10−12
10−10
10−8
10−6
10−4
10−2
100
102Convergence History
Number of iterations
Log(
Erro
r)
1 grid2 grid3 grid
Fig. 8. Convergence history for sail geometries with artificial compressibility.
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dI ¼ �kðG;GÞÞ 6 0
It turns out, however, that the gradient is generally a function in alower smoothness class than the initial shape, with the result thatthere is a progressive loss of smoothness with each iteration.
This can be corrected by introducing a modified gradient whichcorresponds to a weighted Sobolev inner product, of the form
hu;vi ¼Zðuv þ �u0v 0Þdn
This is equivalent to replacing G by G where in one dimension
G� @
@n�@G
@n¼ G
with G ¼ 0 at the end points and making a shape change
dF ¼ �kG
This both preserves the smoothness of the redesigned shape, andacts as a pre-conditioner which reduces the number of design stepsneeded to reach an optimum solution.
4. Mesh deformation
The modifications to the shape of the boundary are transferredto the volume mesh using the spring method. This approach hasbeen found to be adequate for the computations performed in thisstudy.
The spring method can be mathematically conceptualized assolving the following equation
AIRPLANE
CP from -1.0000 to -0.5000
Fig. 9. Pressure contours on the Leeward side.
AIRPLANE
CP from -0.6000 to -0.1000
Fig. 10. Pressure contours on the Windward side.
0 5 10 15 20 25−0.2
0
0.2
0.4
0.6
0.8
1
1.2Lift and Drag distribution along the height of the head sail
Height along the span
ClCd
Fig. 11. Spanwise force distributions on the head sail.
0 5 10 15 20 25 30 350
0.2
0.4
0.6
0.8
1
1.2
1.4Lift and Drag distribution along the height of the head sail
Height along the span
ClCd
Fig. 12. Spanwise force distributions on the main sail.
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@Dxi
@tþXN
j¼1
KijðDxi � DxjÞ ¼ 0
where the Kij is the stiffness of the edge connecting node i to node jand its value is inversely proportional to the length of this edge, Dxi
is the displacement of node i and Dxj is the displacement of node j,the opposite end of the edge. The position of static equilibrium ofthe mesh is computed using a Jacobi iteration with known initialvalues for the surface displacements.
5. Results
The overall design process is illustrated in Fig. 6.
Leech is allowed
to move freely
Translational degrees of freedom
suppressed at the mast
Mast is assumed to be rigid
Translational and
rotational degrees of freedom suppressed at the boom
Fig. 13. Boundary conditions for the main sail.
the stayfreedom supressed alongTranslational degrees of
Fig. 17. Pressure distributions along sections at 1%, 25% and 85% of the height ofhead sail after aeroelastic analysis.
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5.1. Aeroelastic computations to predict the flying shape of upwindsails
This section highlights how the incompressible flow solver wasused to obtain force predictions for an industrial problem, namely,sail configurations of high performance yachts for races like theAmericas Cup. Modifications to the inlet boundary condition wereincorporated so as to simulate the boundary layer profile over thesea. Fig. 7 shows a representative head and main sail combinationused in the Americas Cup. Due to the fast turn-around times of theflow solver it was possible to use this computational package toobtain aerodynamic characteristics of the sail configurations for avariety of wind conditions, and also study the effect of varyingtwist, camber and sail trim. Fig. 8 shows a sample convergence his-tory for the incompressible flow solver. Figs. 9–12 show the pres-sure distributions and the forces for the nominal geometry.
Fig. 18. Pressure distributions along sections at 1%, 25% and 85% of the height ofmain sail after aeroelastic analysis.
2 4 6 8 10 12 14−0.5
0
0.5
1
1.5
2
Z = 25
Deformed and original section geometry along the height of the head sail
Z = 18Z = 14Z = 8Z = 3.6
OriginalDeformed
Fig. 19. Original and deformed sail sections for the head sail. At each span location,the lower curve is the original geometry.
10 11 12 13 14 15 16 17 18 19 20−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Z = 5
Z = 8.5
Z = 14
Z = 19
Z = 24
Z = 31
Deformed and original section geometry along the height of the main sail
OriginalDeformed
Fig. 20. Original and deformed sail sections for the main sail. At each span location,the lower curve is the original geometry.
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Further, in order to predict the flying shape of the sail geometries,this flow solver was coupled to NASTRAN. The aeroelastic packageuses an iterative algorithm that transfers the pressure loading ob-tained from the flow solver to the structural model, and uses thedeflections from the structural analysis to modify the computa-tional mesh for the fluid. This process is iteratively carried out untilthe deflections are below a particular threshold.
5.1.1. Structural modelIn the structural model of the sail, the sail cloth is discretized
into quadrilateral membrane finite elements with four nodes (afterneglecting the presence of batten pockets). These elements with-stand all external forces through tension but can resist a smallamount of bending moments. The translational and rotational de-grees of freedom along the foot of the main sail are suppressed.Along the mast, the translational degrees of freedom are inhibitedwhile allowing for rotational motion. For the head sail, the point of
attachment of the foot to the rig is constrained. The leech of themain and head sail are allowed to move freely to induce a geomet-ric twist due to the aerodynamic loading. The mast is assumed tobe rigid during the structural and aeroelastic calculations. Thepresence of battens and tension cables and other structural ele-ments of the sail rig is neglected from this analysis.
The linear system of equations relating the displacements to theforce field is advanced to a steady state by an iterative process thatincrementally adds the load while obtaining a converged displace-ment field for each step. This procedure was introduced in order toallow for large deflections of the sail geometry. Wrinkling of thestructure, which is an important consideration especially aroundthe leading edge (luff) and at the sail tip, is not anticipated by thismodel, but the use of a numerical algorithm to track large deforma-tions allows for wrinkling models to be included at a later stage.Figs. 13 and 14 pictorially depict the boundary conditions usedfor the structural model.
oooooooooooooooooooo o o o o o o o o o o o o o o o o o o
oo
o
Fig. 21. Initial and final pressure distribution, o is the target pressure distribution, x is the computed pressure distribution for the redesigned airfoil.
20 40 60 80 100 120 140 160
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4Gradients
points on the airfoil surface, lower trailing edge to upper trailing edge
syn75syn82
Fig. 22. Comparison of the gradients from SYN75 and SYN82.
0 20 40 60 80 100 120 140 160 180−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8First co−state variable
points on the airfoil surface, lower trailing edge to upper trailing edge
syn75syn82
Fig. 23. Comparison of the first co-state variable from SYN75 and SYN82.
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5.2. Aeroelastic coupling procedure
The pressure loading from the flow solver is fed to the structuralanalysis to estimate the deflected shape of the sail. To enable thetransfer of loads and displacements to be conservative, the fluidmesh on the surface and the structural mesh are identical, elimi-nating the need for interpolation. The deflected shape of the sailis used to deform the computational mesh. The ‘spring-analogy’method has been used to track the mesh deformations. While thismethod restricts the allowable deflections and may impair thequality of the deformed mesh, it provides a simple tool to trackmesh deformations. The deformed mesh is then used to computea new pressure loading for the sail. This iterative process offersno formal guarantee of convergence, but it usually predicts the de-flected shape to reasonable accuracy in a few steps (typically 5 forsail geometries).
5.2.1. Results of aeroelastic simulationsThe deflected shapes of the head and the main sail are shown in
Figs. 19 and 20. It can be seen from these plots that the lower sectionsof the head and the main sail do not undergo appreciable deforma-tion. The largest deflections occur in the mid-sections of the mainsail. As the point of attachment of the main sail to the mast andthe leading edge of the head sail were not allowed to move, the aero-dynamic loading changes the twist of the sail geometry. This has afavorable influence on the pressure distribution, especially on thehead sail (compare Figs. 15 and 16 with Figs. 17 and 18). The pres-sure distribution over the head and sail after the aeroelastic simula-tion highlights the need to perform aeroelastic analysis to obtain theflying shape of these sail geometries. While the lift and the drag dis-tribution of the deformed shape is not significantly different fromthe undeformed shape, the pressure distribution over the sail
0 20 40 60 80 100 120 140 160 180−2
−1.5
−1
−0.5
0
0.5
1Second co−state variable
points on the airfoil surface, lower trailing edge to upper trailing edge
syn75syn82
Fig. 24. Comparison of the second co-state variable from SYN75 and SYN82.
0 20 40 60 80 100 120 140 160 180−2
−1.5
−1
−0.5
0
0.5
1Third co−state variable
points on the airfoil surface, lower trailing edge to upper trailing edge
syn75syn82
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Fig. 27. Attained(+,x) and target(o) pressure distributions at 0% of the wing span.
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sections show that the twist and the camber distribution of the headand the main sail have been altered to provide a smooth entry for theflow over the leading edge of both components.
5.3. Inverse design of airfoils in incompressible flow
In order to validate the design procedure, two-dimensionalproblems were studied first. An inverse design problem that recov-ers the pressure distribution over the Onera M6 airfoil was used tovalidate the gradient formulations and the adjoint solution. A threelevel multigrid cycle was used for obtain steady state solutions forthe flow and the adjoint equations. The grids were generated usinga conformal mapping technique. The initial airfoil shape had a
NACA 0012 profile and the initial pressure distribution is shownin Fig. 21. Around 40 design cycles were required to recover thetarget pressure and shape by the design process (Fig. 21).
A comparison of the gradients from a well documented struc-tured grid adjoint solver (SYN82) and a version which uses thesame numerical schemes and gradient formulations but usingunstructured grids (SYN75) is shown in Fig. 22. These gradientsare for an inverse problem and as can be seen from the plot, theymatch well except neat the leading edge of the airfoil where theunstructured solver predicts a smaller gradient. However, the over-all design process was not affected. The difference between thegradients is attributed to the difference in the flow and adjointsolution near the leading edge. The differences in the adjoint solu-tion are highlighted in Figs. 23–26.
Fig. 31. Initial (o) and final pressure distribution at 15% height on the main sail.
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5.4. Inverse design of wings in incompressible flow
To validate the design process for three-dimensional flows, atest problem similar to the two-dimensional case was used. Theinitial wing had the planform of the Onera M6 but had NACA0012 airfoil sections. The target pressure distribution corre-sponded to the steady state pressure distribution over the OneraM6 wing. Three levels of multigrid were used to obtain steady stateflow and adjoint solutions. The meshes were generated using anautomated grid generator and interpolation coefficients were accu-mulated in a pre-processing step. The parallel implementation ofthe flow and adjoint solvers were used to reduce the computa-tional time of the design process. Modifications to the shape ofthe wing were transmitted to the interior mesh using the springdeformation method which worked well for this problem.
Figs. 27–30 show that the target pressure distribution has beenrecovered in about 50 design cycles. These computations took un-der 30 minutes requiring 8 processors of an SGI Origin 300.
5.5. Inverse design for sail geometries
The results of the flow and aeroelastic simulations, show thatthe interaction of the head sail with the main reduces the develop-ment of sharp pressure gradients around the leading edge. Thisinteraction is crucial to the performance of the main sail as it al-lows the main sail to be set at a higher angle to the center-lineof the boat. These results also show that the region above the headsail has large suction peaks which is a cause of concern. The aero-dynamic shape optimization procedure validated in the previoussections was used to redesign the main sail, with an aim of reduc-ing the pressure gradient around the luff of the main sail. An in-
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Fig. 32. Initial (o) and final pressure distribution at 32% height on the main sail.
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Fig. 33. Initial (o) and final pressure distribution at 75% height on the main sail.
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Fig. 34. Initial (o) and final pressure distribution at 85% height on the main sail.
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Fig. 35. Initial and redesigned camber line at 15% of height.
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verse design procedure was employed and the target pressure dis-tribution was obtained by smoothing the pressure distribution onthe main sail obtained from the aeroelastic analysis. Figs. 31–34show that a significant portion of the leading edge of the main sailhas been redesigned to allow for smooth entry of the flow. Theassociated reduction in sharp suction peaks should have a favor-able affect on the growth of the boundary layer over the upper sur-face. The change to the sections are shown in Figs. 35–38. Theseresults need to be confirmed with a viscous simulations, as themodifications suggested by the shape optimization algorithm oc-cur in a region where the interaction of the mast and main sail in-duces strong viscous effects.
6. Conclusions
This study demonstrates the use of artificial compressibilitytechniques for analysis and shape optimization of aerodynamicflows. We have demonstrated the approach for two academic prob-lems and one industrial problems. While the nested-grid approachto multigrid places the burden of grid generation it can work well
when carefully selected coarser meshes are used. Parallel imple-mentation as outlined here provides good speed-up up to 16 pro-cessors. The compressible adjoint equations can be easilymodified to fit within the artificial compressibility frame-work.The use of gradient formulations that depend only of the surfacemesh allows adjoint-based methods to be used for unstructuredgrids in a computationally efficient manner. Hence, it is now pos-sible to devise a completely automated shape optimization proce-dure for complete aerodynamic configurations. While this studyonly focuses on inviscid flows, extending the approach to viscousflows will enable us to tackle challenging problems in multi-ele-ment airfoil optimization and hydrodynamics (hull and sail shapeoptimization). The second and the third author have made signifi-cant contributions on the latter topic and the current study enablesextension to more complicated geometries.
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Initial and deformed sections at 32 percent height
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Fig. 36. Initial and redesigned camber line at 32% of height.
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Fig. 37. Initial and redesigned camber line at 75% of height.
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Fig. 38. Initial and redesigned camber line at 85% of height.
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