Optimisation of offshore wind floater from dynamic cable ...
Aero dynamic - PeopleAero dynamic optimisation r fo complex geometries Michael Giles rd Oxfo y...
Transcript of Aero dynamic - PeopleAero dynamic optimisation r fo complex geometries Michael Giles rd Oxfo y...
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Aerodynamic optimisationfor complex geometries
Michael GilesOxford University Computing LaboratoryApril 25th, 1997
Aerodynamic optimisation for complex geometries 1
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Overview
� grid generation� sensitivity analysis� adjoint approach� applications
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Grid Generation
For design purposes we need to be able tocreate� a base grid for given values of designparameters� perturbed grids for perturbed valuesThe linear/nonlinear perturbed grids have thesame topology as the base grid, so anyobjective function varies smoothly.
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Grid Generation
Base grid generation:� parametric solids-based EPD systemde�nes solid surface as a collection ofsurface patches separated by linesterminated by points� surface is gridded in order ofincreasing dimensionality (point, line,surface patch)� interior grid nodes are then created byadvancing front or Delauneyalgorithms
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Grid Generation
Perturbed grid generation:� parametric perturbation de�nesperturbation to solid surfaces andseparating lines� surface grid node perturbations arede�ned in order of increasingdimensionality (point, line, surfacepatch)� interior grid nodes are perturbed using`method of springs' or elliptic p.d.e.
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Grid Generation
Method of springs:� edges of grid are modelled as springs� base grid is de�ned to be inequilibrium� perturbation to surface points disturbsequilibrium, leading to perturbation tointerior nodes to re-establish it� strength of springs is de�ned to ensureno cross-over in the boundary layer� (similar idea can be used to de�nesurface perturbations in the �rst place)
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Grid Generation
In the elliptic p.d.e. approach, grid nodeperturbations ex(x) are de�ned byr � (k(x)rex) = 0;subject to speci�ed boundary conditions.k(x) is de�ned to ensure no cross-over inboundary layers.
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Nonlinear SensitivityFor a single design variable �, discrete ow equations F (U ; �) = 0;de�ne ow �eld U as a function of �.Gradient of objective function I(U ; �) canbe approximated bydId� � I(U(�+�); �+�)� I(U(�); �)� :Easily generalised to multiple designvariables, at cost of extra calculations.Aerodynamic optimisation for complex geometries 8
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Linear Sensitivity
Linearising discrete ow equations gives@F@U fU + @F@� = 0;where @F@� � @F@X @X@� :i.e. change in � perturbs grid coordinateswhich perturb ux residuals.fU represents ow perturbation as seen byperturbed grid point
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Linear Sensitivity
Gradient of objective function is given bydId� = @I@U fU + @I@�:Generalisation to multiple designparameters requires separate calculationfor each, so no particular bene�tcompared to nonlinear sensitivities.
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Discrete Adjoint
Substituting for fU givesdId� = � @I@U @F@U!�1 @F@� + @I@�;which can be written asdId� = V T @F@� + @I@�;where V satis�es the adjoint equation @F@U!T V + � @I@U�T = 0:
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Discrete Adjoint
The advantage of the adjoint approach isthat the same adjoint solution V can beused for each design variable, since Vdepends on I but not �.The drawback is that because V dependson I a separate calculation must beperformed for each constraint function.Question: in real engineering applications,how many design variables and constraintsare there?Aerodynamic optimisation for complex geometries 12
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Analytic Adjoint
The analytic adjoint is more complicated.Critical �rst step is formulation of linearperturbation equations.Simple linearisation of 2D Euler equations@@xFx(U) + @@yFy(U) = 0;yields @@x(Ax eU) + @@y(Ay eU) = 0;where eU is perturbation at a �xed point.
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Analytic Adjoint
However, linearising the b.c.u�n= 0;gives eu�n+ (ex�ru) � n+ u� en= 0;which is hard to discretise accurately.This is similar to discrete adjointtreatment with no perturbation to interiorgrid points.
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Analytic Adjoint
Start instead with generalised coordinates,@@� Fx@y@� � Fy@x@�!+ @@� Fy@x@� � Fx@y@�! = 0:
Now de�ne perturbed coordinates asx = �+ �X(�; �); y = �+ �Y (�; �);where X(�; �) and Y (�; �) are smoothfunctions which match the surfaceperturbations due to the design variable �.
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Analytic Adjoint
Linearising with respect to � yields@@�(Ax eU) + @@�(Ay eU) = � @@� Fx@Y@� � Fy@X@� !� @@� Fy@X@� � Fx@Y@� ! ;where eU is now the perturbation in the ow variables for �xed (�; �) rather than�xed (x; y).The linearisation of the b.c.'s is simple,and the overall accuracy is much better.
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OGV DesignG.N. ShrinivasP.I. CrumptonM.B. GilesOxford, 1996Aerodynamic optimisation for complex geometries 17
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OGV Design
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OGV Design
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OGV DesignPylonFan
Outlet guide vanes
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OGV Design
Objective is to minimise circumferentialpressure variation upstream of the OGV'sby changing their camber.Optimisation uses� unstructured grid with 560k tetrahedra� Euler equations� multigrid and parallel computing� elliptic p.d.e. for grid perturbation� nonlinear sensitivities andquasi-Newton optimisation
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OGV Design
Leading edge of each OGV is leftunchanged due to uniform ow incidence;camber change varies linearly withdistance from leading edge to change theout ow angle.First design exercise uses a camberchange which varies sinusoidally withcircumferential angle.Only 2 design variables: maximum changeat hub and tip
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OGV DesignOptimisation using sinusoidal variation-1000 -500 0 500 1000
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OGV DesignOptimisation using sinusoidal variation0 1 2
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OGV Design
Drawback of this design is that all OGV'sare di�erent.Second design exercise uses just 3 bladetypes, the original, one with overturningand one with an equal amount ofunderturning.Still only 2 design variables: maximumchange at hub and tip
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OGV Design
DatumUnderturned
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OGV DesignOptimisation using 3 blade types-1000 -500 0 500 1000
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OGV DesignOptimisation using 3 blade types0 1 2 3
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Business JetJ. Elliott and J. Peraire, MIT, 1996
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Business Jet
Objective function is mean-squaredeviation from a target pressuredistribution for a `clean' wing in theabsence of the rear nacelle.6 design variables are used to de�nesmooth perturbations to the wing.
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Business Jet
Optimisation uses� unstructured grid with 860k tetrahedra� Euler equations� multigrid and parallel computing� method of springs for gridperturbation� BFGS optimisation method withdiscrete adjoint formulation forgradients
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Business Jet
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Business Jet
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Conclusions/Future
� aerodynamic optimisation for complexgeometries is becoming a reality� with multigrid and parallel computing,costs are now acceptable for inviscidmodelling; viscous modelling is underdevelopment but will cost up to 5times as much� grid generation for base grids andperturbed grids is a critical component� pros and cons of di�erent optimisationmethods has yet to be properlyinvestigatedAerodynamic optimisation for complex geometries 34