CFD-based optimization of hovering rotors using radial basis functions ...

Optim Eng (2013) 14:97–118 DOI 10.1007/s11081-011-9179-6

CFD-based optimization of hovering rotors using radial basis functions for shape parameterization and mesh deformation Christian B. Allen · Thomas C.S. Rendall

Received: 7 September 2010 / Accepted: 28 November 2011 / Published online: 21 December 2011 © Springer Science+Business Media, LLC 2011

Abstract Aerodynamic shape optimization of a helicopter rotor in hover is presented, using compressible CFD as the aerodynamic model. An efficient domain element shape parameterization method is used as the surface control and deformation method, and is linked to a radial basis function global interpolation, to provide direct transfer of domain element movements into deformations of the design surface and the CFD volume mesh, and so both the geometry control and volume mesh deformation problems are solved simultaneously. This method is independent of mesh type (structured or unstructured) or size, and optimization independence from the flow solver is achieved by obtaining sensitivity information for an advanced parallel gradient-based algorithm by finite-difference, resulting in a flexible method of ‘wraparound’ optimization. This paper presents results of the method applied to hovering rotors using local and global design parameters, allowing a large geometric design space. Results are presented for two transonic tip Mach numbers, with minimum torque as the objective, and strict constraints applied on thrust, internal volume and root moments. This is believed to be the first free form design optimization of a rotor blade using compressible CFD as the aerodynamic model, and large geometric deformations are demonstrated, resulting in significant torque reductions, with off-design performance also improved. Keywords Multivariate function approximation · Radial basis functions · Numerical simulation · Numerical optimization · Volume mesh deformation · Shape parameterization · Aerodynamics · Rotors · CFD

C.B. Allen () · T.C.S. Rendall Department of Aerospace Engineering, University of Bristol, Bristol, BS8 1TR, UK e-mail: [email protected] T.C.S. Rendall e-mail: [email protected]

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Nomenclature c CP CT CZ Cmx , Cmy , Cmz C D E f F G i, j, k i, j, k J m M MT ip n n N p P P q q r r R s t t/c u, v, w U V x, y, z x, y, z Xqc , Zqc α β γ δ λ φ ρ σ

C.B. Allen, T.C.S. Rendall

Aerofoil chord Pressure coefficient Thrust coefficient Sectional normal force coefficient Hinge moment coefficients Constraint vector Number of domain element nodes Total energy Scalar function Flux vector Source term vector Matrix and vector component indices Unit vectors Objective function Number of constraints Number of polynomial terms Tip Mach number Number of dimensions Unit normal vector Number of interpolation control points Polynomial term Static pressure Polynomial matrix Polynomial component Velocity vector Euclidean norm Cartesian coordinate vector Support radius Interpolated function Time Sectional thickness to chord ratio Velocity components Conserved quantity vector Cell volume Cartesian coordinates Vectors of Cartesian coordinates Sectional quarter chord coordinates Polynomial weighting coefficients Interpolation weighting coefficients Ratio of specific heats Design variables Lagrange multipliers Basis function Density Rotor solidity

CFD-based optimization of hovering rotors using radial basis functions

θ ω ζ

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Section pitch angle Angular velocity vector n-dimensional coordinate vector Angular velocity

1 Introduction Numerical simulation methods of many fidelity levels are now used routinely in industrial design, and increasing computer power has resulted in their increasing use in optimization. Integrating a numerical optimization scheme with an effective geometry control method and an aerodynamic model can result in an aerodynamic shape optimization approach, and these are invaluable in the design stage. The quality and cost of results gained by numerical optimization are inherently dependent on the fidelity of the analysis tool used, and so developing a modular approach means that the complexity and cost of each module required at various stages of the design process can be changed, in terms of aerodynamic or fluid dynamic fidelity, number of surface shape design degrees of freedom, and the complexity of the optimization scheme. Computational fluid dynamics (CFD) is at the forefront of aerodynamic analysis capabilities, and application of numerical optimization algorithms with such analysis has already produced notable results (Jameson 1988, 2003; Dulikravich 1992). Compressible CFD has been used in numerous optimizations of two-dimensional aerofoil sections (Hicks and Henne 1978; Li et al. 2002), three-dimensional aircraft (Chung and Alonso 2004; Reuther et al. 1996; Wong et al. 2007), and threedimensional aeroelastic aircraft (Gumbert et al. 2001a; Jameson et al. 2007), but little has been achieved in the way of rotor blades, primarily due to high cost. High density and high quality grids are necessary compared to fixed-wing simulations to capture the vortical wake, and long solution integration times are required to develop and capture this wake (Allen 2004; Renzoni et al. 2000), and this can result in huge computational power requirements to optimize such a system. Furthermore, there are two distinct flow regimes, hover and forward flight, and good performance in one is unlikely to be carried over to the other. The requirements of rotor optimization are also fundamentally different to those of fixed-wing due to the large range of rotor operating conditions: detailed minor surface design changes are valuable to minimize fixed-wing drag at the cruise condition; a rotor blade may need to operate with a large range of collective pitch in hover, depending on the payload and operation, and in forward flight numerous criteria need to be satisfied. Hence, in the rotor case an optimum global design is more important, and detailed minor surface changes of less significance. Aerodynamic optimization has been applied previously to rotors in hover but mainly with both low fidelity aerodynamics and only gross geometric changes, see for example Celi (1999), with only a few notable results using compressible CFD, namely those of ONERA (Pape and Beaumier 2005; Dumont et al. 2009), Nadarajah et al. (2008), Choi et al. (2008), and Nielsen et al. (2009). In Pape and Beaumier (2005) a RANS-based optimization of a rotor is presented, with extremely low density grids and incorporating a very small number of design variables. This has been

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extended to a discrete adjoint approach (Dumont et al. 2009) again with a coarse mesh and using few parameters (13 and 25). Nielsen et al. (2009) also use a discrete adjoint approach, using 44 section thickness and camber parameters. A composite objective function is also considered. Nadarajah et al. (2008) presented optimization of the Caradonna and Tung two-bladed rotor (as is considered here) using an adjoint approach, and significant performance benefits were achieved, using detailed aerofoil surface changes, but planform was fixed and there were no moment constraints. Choi et al. (2008) also used an adjoint approach with the UH-60A blade, and this was coupled to a trimming algorithm with constraints applied on thrust, and over 100 design variables were used, including section geometries. Both the latter methods incorporate a frequency domain approach, and so have also been extended to initial forward flight optimization, although only a small number of time instances were used. The research presented here is aimed at the development of a modularized, generic optimization tool that is flow-solver and mesh type independent, and applicable to any aerodynamic problem; rotor blades in hover in this case. The key aspect of a flexible optimization and design process is an effective geometry parameterization technique, that is flexible enough to allow sufficient design space investigation and robust enough to be applicable to any geometry or design surface. Furthermore, a small number of design parameters is desirable, particularly if using a finite-difference gradient evaluation. Related to the surface control is the required volume mesh deformation or regeneration once the design surface has been deformed, and mesh deformation is much preferred, to avoid introducing differing discretization error. To satisfy these requirements, an efficient domain element shape parameterization method has been developed by the authors and presented previously for CFD-based shape optimization in two dimensions (Morris et al. 2008), three-dimensional fixed-wing optimizations (Morris et al. 2009), and initial rotor blade cases (Allen et al. 2010). The parameterization technique, surface control and volume mesh motion all use radial basis functions, wherein global interpolation is used to provide direct transfer of domain element movements into deformations of the design surface and the CFD volume mesh, which is deformed in a high-quality fashion. The method requires very few design variables to allow free-form design, and the properties of the parameterization mean that design parameters of any scale may be used, allowing a hierarchy ranging from detailed surface changes to large scale planform changes, and changes of any size can be made to the design surface without the requirement for regeneration of a CFD volume mesh (Rendall and Allen 2008b, 2009). For the control parameters defined, an effective sensitivity evaluation method is required, along with an optimization algorithm. To achieve modularity and flowsolver independence, a finite-difference approach to parameter sensitivities has been adopted, and so a small number of design parameters is essential. The method has been coupled to an advanced gradient-based constrained optimization algorithm, and the optimization suite has been parallelized to allow optimization of threedimensional bodies in practical times. Previous work (Allen et al. 2010) has considered hovering rotors using only twist parameters, to test the general framework for rotors before increasing the level of design parameters, and to validate the optimization scheme. This paper now extends the methods to allow more geometric flexibility, by using local and global parameters, and results are presented for two transonic tip Mach numbers.

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2 Domain element parameterization and mesh deformation The problems of surface parameterization and control during an optimization process, and the required volume mesh deformation if using a CFD code, are often treated separately, but are treated simultaneously here. There have been numerous parameterization methods presented for CFD-based shape optimization, and these can be split into those that parameterize the aerodynamic mesh or those that parameterize the design geometry from which a mesh is generated. Grid parameterization methods are generally independent of the mesh generation package, so require a mesh deformation algorithm, but allow the use of previously generated meshes for optimization. Methods of this nature include discrete (Jameson 1988; Reuther et al. 1996), analytical, basis vector (Pickett et al. 1973), free form deformation (FFD) (Watt and Watt 1992), domain element methods (Morris et al. 2008)1 and the recent control grid approach (Anderson et al. 2009). Geometry parameterization methods are inherently linked with the mesh generation package, and optimization requires automatic mesh generation tools. Methods of this nature include partial differential equation methods (PDE) (Bloor and Wilson 1995; Smith et al. 1995), polynomial or spline (Braibant and Fleury 1984), CAD and recently CST (Kulfan 2007) methods. Reviews of a range of parameterization methods are presented in Samareh (2001) and Nadarajah et al. (2007). Grid parameterization methods offer most flexibility, by allowing independence from both the mesh generation tools and the mesh structure. Furthermore, mesh regeneration introduces the problem of varying the discretization error. Hence, although a mesh deformation scheme is required, and this is a significant issue, the grid parameterization and deformation approach is much preferred. The efficient parameterization method here links a set of aerodynamic mesh points to a domain element that controls the shape of the design. At the center of this parameterization technique is a multivariate interpolation using radial basis functions (RBFs), and this provides a direct mapping between the domain element, the surface geometry and the locations of grid points in the CFD volume mesh. A global dependence can be evaluated between the control points (the domain element nodes) and the grid nodes, and this has many advantages: the mapping is unique and dependent only on the initial mesh, as the values of the parametric coordinates of the grid points with respect to the domain element remain constant throughout the optimization; updates to the geometry and the corresponding mesh are provided simultaneously by application of multivariate interpolation, and this is extremely fast and efficient and results in very high quality mesh deformation, even for large deformations (Rendall and Allen 2008b, 2009); the approach is meshless, so requires no connectivity and is applicable to any mesh type. Domain element points and volume mesh points are simply treated as independent point clouds; the system is only the size of the number of domain element nodes, and so is not related to the mesh size. RBFs have received much attention in computational and numerical methods, and have been used recently in some optimization approaches. For example, Amoignon 1 http://www.optimalsolutions.us/ (2008).

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(2010) used RBFs for mesh deformation within a two-dimensional aerofoil optimization, using surface mesh points as the design parameters, and Jakobsson et al. (2009b) used RBFs to build a surrogate model used within a multidisciplinary optimization approach. In this paper RBFs are used for both the surface parameterization and control and volume mesh deformation problems simultaneously. 2.1 Interpolation formulation The general theory of RBFs is presented in Buhmann (2005) and Wendland (2005), and the basis of the method used here is detailed in Rendall and Allen (2008b). Let f (ζ ) be the original function to be modeled, and fi be the scalar values at N discrete points ζ i , i = 1, . . . , N , where ζ i is the n-dimensional coordinate vector of the ith sample point. The set of data points X = {ζ 1 , . . . , ζ N } is confined to a domain in n-dimensional space. A RBF model is then a linear combination of basis functions, whose argument is the Euclidean distance between the point ζ at which the interpolation is made and the N points in the known data set. In other words, the interpolation at an untried site is a sum of contributions from all the known function values, the influence of which is controlled by a basis function that depends on the distance they are from the new site. If φ is the chosen basis function and  ·  is used to denote the Euclidean norm, then an interpolation model s has the form s(ζ ) =

N 

βi φ(ζ − ζ i ) + p(ζ ),

(1)

i=1

where βi , i = 1, . . . , N are model coefficients, and p(ζ ) is an optional polynomial. The coefficients are found by requiring exact recovery of the original data, sX = f, for all points in the training data set X . Hence the model is an interpolant, and all original solution information is preserved. It is possible to construct the model to give some smoothing of the data in the event that the data contains numerical noise, and the reader is referred to other works (Fasshauer 2007; Rendall and Allen 2008a) for details on formulation in this case. Inclusion of the polynomial in (1) allows recovery of polynomials up to that order. The model is then a sum of contributions from p(ζ ), which gives a generalized least squares fit to the data, and the basis functions, which form a set of departures. If the training data are from an exact polynomial of equal or lower order than p(ζ ), then any new evaluations are exact. In other words, the basis function coefficients are all zero, and the polynomial coefficients provide exact recovery of the function. The polynomial may be written p(ζ ) =

M 

αk qk (ζ ),

(2)

k=1

where qk (ζ ), k = 1, . . . , M are the monomial components and αk are additional coefficients which must be evaluated. If the order of the polynomial is denoted by m, the number of new unknowns M introduced is given by the binomial coefficient ( m n ).

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A side constraint is introduced to provide the additional equations required to retain the solubility of the system. This ensures that the polynomial is orthogonal to the basis functions in polynomial space, which is the requirement for a least squares approximation to the data, and is written N  i=1

βi = 0,

N 

βi qk (ζ i ) = 0,

k = 1, . . . , M.

(3)

i=1

This problem can be written as a linear system in matrix form, which must be solved before any new approximations can be made. Writing the training data as f, gives a system of the form  M×M    M  P 0 α 0 (4) = β f PT φ where φij = φ(ζ i − ζ j ), Pik = pk (ζ i ),

i, j = 1, . . . , N

(5)

i = 1, . . . , N, k = 1, . . . , M.

(6)

In the above qk,i indicates the kth monomial term at point i and the points ζ j and ζ i in (5) are the evaluation point and basis function center respectively. After solving for the coefficients, the response at each of a set of J arbitrary evaluation points Z = {ζ 1 , . . . , ζ J } may be calculated using (1). Many choices are available for the basis function itself, but a radial conditionally positive definite function is chosen to ensure (4) has a unique solution. Basis functions are said to be compactly supported when they decay to zero at a given distance from the center, known as the support radius R. Compactly supported functions have desirable numerical characteristics and modelling behavior, hence the work presented here concentrates on Wendland’s functions (Wendland 2005) φn,k , which are compact functions of minimal degree for a stated number of continuous derivatives C 2k in n dimensions. Wendland’s C 2 (φn,1 ) in up to three dimensions is given by φ3,1 = (1 − r)4+ (4r + 1), where r =  ·  denotes the norm and (·)+ refers to the cut-off function  1 − r, 1 − r ≥ 0, (1 − r)+ = 0, 1 − r < 0.

(7)

(8)

The support radius used in the function also acts to control the region of influence of each of the centers. A larger support radius allows each sample point to influence the interpolation at a given point from a greater distance away, and in general leads to a smoother interpolation. For a compact function, it defines a hypersphere around each sample point outside of which the value of φ is zero, and inside of which the point has some influence. The support radius scales the basis function argument as follows: r (9) rscaled = , R

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Fig. 1 Initial rotor domain element and surface

hence the distance between two points is then the scaled Euclidean distance   n 1 ζ − ζ i scaled =  (ζj − ζj,i )2 . R

(10)

j =1

For surface and volume mesh deformation, it is sensible to use decaying functions, to give the interpolation a local character and ensure deformation is contained in a region near the moving body, and also to not include polynomial terms, since these will transfer deformation throughout the entire mesh. 2.2 Domain element deformation Given an initial rotor design, a domain element is automatically placed around the blade surface conforming to the local chord and incidence, and the domain element is defined here to consist of 17 two-dimensional slices evenly distributed along the span. The Caradonna and Tung two-bladed rotor (Caradonna and Tung 1981) is considered here, and Fig. 1 shows the domain element parameterization extended to this blade for the one million cell mesh used later. Instead of using the individual locations of each domain element node, an intuitive set of applicable shape deformation design variables have been developed by inverse design techniques (Morris et al. 2008), and the domain element method allows for independent multiple levels of design variables; global, local coarse, and local fine. As mentioned previously, detailed surface section changes are of less significance for rotor blades than wings and, hence, of interest in this paper are global and coarse local parameters. Local coarse parameters are individual section deformations, for example a chord length change, or movement of a section fore or aft (a sweep), or normal to the chord (a dihedral). This has helped determine the number of domain element sections used; no study has been performed, but it is sensible to adopt two or three sections per chord length. More than this allows the possibility of detailed, small wavelength spanwise changes, which are not appropriate, and less may allow the possibility of parts of the blade falling between moving sections and experiencing no deformation at all. The interpolation results in a decaying volume control method, and is used to deform the design surface and volume mesh directly from movements of the domain

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element nodes. Hence, it is sensible to interpolate spatial deformations rather than positions. Given a vector of N D domain element displacement vectors, D D D D T rD = rD N ew − rI nitial = (x , y , z )

(11)

the interpolation weightings are computed for each component (j = 1, 2, 3) by: φβ j = rD j

(12)

where φ is the N D × N D basis function matrix computed using the domain element node positions. This is computed using the initial domain element coordinates. The displacement of any point in space is then computed for each component by: D

rj =

N 

βj,i φ(r − rD i ).

(13)

i=1

3 Optimization scheme The parameterization and surface and volume mesh deformation approach has been developed such that it is applicable to any mesh type, flow solver, and optimization scheme. It is linked here to a non-linear Feasible Sequential Quadratic Programming (FSQP) (Zhou et al. 1997; Zhou and Tits 1993) optimization algorithm, and this choice is important, since it allows enforcement of aerodynamic and geometric constraints, for example minimum thickness, minimum volume, minimum lift, maximum pitching moment etc. Unconstrained optimizations can incorporate constraints by using a penalty function for design parameters that take the design near or beyond the constraint boundary, but these methods are now considered inefficient and have been replaced by methods that focus on the solution of the Kuhn-Tucker equations. The aim of the optimization process is to obtain the values of a set of design variables, δ = (δ1 , δ2 , . . . , δn )T , that in some way can be defined as optimal. An objective function(s), determined by the numeric values of the design variables, J(δ), is to be minimized and can be subject to equality constraints, inequality constraints, and/or parameter bounds. A general constrained problem description is stated as: minimize J(δ) subject to: equality constraints

Ci (δ) = 0,

inequality constraints Ci (δ) ≤ 0, parameter bounds

δl ≤ δ ≤ δu

i = 1, . . . , me , i = me + 1, . . . , m,

(14)

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where the vector function C(δ) is a vector of length m containing the values of the equality and inequality constraints evaluated at δ. The Kuhn-Tucker equations are necessary conditions for optimality for a constrained optimization problem and can be stated as (in addition to the original constraints of (14)) ∇J(δ ∗ ) +

m 

λ∗i .∇Ci (δ ∗ ) = 0,

(15)

i=1

λ∗i .Ci (δ ∗ ) = 0, λ∗i ≥ 0,

i = 1, . . . , m,

i = me + 1, . . . , m

(16) (17)

where δ ∗ refers to the optimum values of δ. Equation (15) expresses a canceling of the gradients of the objective and gradient functions (at the optimum design point). For the gradients to be canceled, Lagrange multipliers (λi , i = 1, . . . , m) are necessary to balance the constraint gradients with the objective gradients. Only active constraints are included in this canceling operation, and so constraints that are not active have Lagrange multipliers set to zero. The solution of these equations forms the basis of the nonlinear programming algorithm. The constrained quasi-Newton method guarantees superlinear convergence by accumulating second-order information relating to the Kuhn-Tucker equations using a quasi-Newton updating procedure, i.e. at each major iteration, an approximation is made of the Hessian of the Lagrangian function. This is then used to generate a quadratic programming (QP) subproblem where the solution is used to form a search direction for a line search procedure. This forms the basis of the sequential quadratic programming (SQP) algorithm. Details of the Hessian update and line search algorithms are available in Zhou et al. (1997) and Zhou and Tits (1993). The method requires the gradients with respect to each parameter, and has been made independent from the flow-solver by obtaining sensitivity information by finitedifferences using a symmetric central-difference stencil, i.e. one positive and one negative perturbation are considered for each design variable. The code is not linked to a flow-solver, but has been written to control its own calls to the solver used, and reads the output back to compute the required gradients. Since each gradient is computed by separate calls to an external flow-solver, the algorithm has been parallelized, in a data sense, such that one parameter (or a group of parameters) is assigned to each available CPU, allowing parallel evaluation of the required sensitivities. Function evaluations and optimizer updates occur on the master process, and each CPU controls the geometry (and CFD volume mesh) perturbation corresponding to a different design variable, and calls the flow solver. Flow-solver results are then returned to the master for optimizer updates. This gradient evaluation module remains independent of the flow-solver, so either a serial or parallel version of the flow solver may be called. The algorithm is structured such that each evolution requires: one flow simulation using a negative perturbation for each parameter; one flow simulation using a positive perturbation for each parameter; (at least one) flow simulation using the proposed

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updated design surface, to test for constraint violations and gradients, and step size. Hence, using one CPU per parameter, each evolution requires clock time of around 3.5 flow simulations. If more CPUs are available, positive and negative perturbations can be run independently, resulting in around 2.5 flow simulations of clock time per evolution.

4 Rotor aerodynamic optimization The domain element parameterization and advanced optimization approaches have been presented previously for two-dimensional heavily constrained aerofoil optimization (Morris et al. 2008), three-dimensional fixed-wing (Morris et al. 2009), and initial rotor optimizations (Allen et al. 2010). The effectiveness and robustness of the scheme has proven, so is now extended to a larger design space to allow planform optimization of a rotor in hover. The Caradonna and Tung two-bladed rotor (Caradonna and Tung 1981) is a rectangular blade, aspect ratio 6, constant NACA0012 section, with no twist or taper. The case chosen was the 8◦ collective pitch case, and optimization performed for two tip Mach numbers. Two transonic cases were chosen, tip Mach numbers of 0.794 and 0.877, since these include the effect of wave drag due to the transonic regions, and the effect of tip speed will be clear. A one million cell structured multiblock mesh was used, generated using advanced transfinite interpolation (Allen 2008). Figure 2 shows the domain and block boundaries, and the grid in the rotor disk and the structure near the blade tip. The farfield is set to approximately 20 chords from the blade above, below, and spanwise. 4.1 Parameters investigated The optimization approach was recently validated (Allen et al. 2010) using twist variables only; global blade pitch, linear root to tip twist, and non-linear 1/y root to tip twist were used along with local individual pitch of each of the domain element slices, and it was shown that the global twist parameters are extremely effective. Hence, it is sensible to include them, combined with detailed local parameters to allow planform changes. The three global twist variables were combined with four parameters for each of the outer 15 domain element slices; an/dihedral (movement normal to the chord), sweep (movement along the chord), chord length, and thickness, giving a total of 63 parameters. Figure 3 shows exaggerated examples of the parameters used here. The deformed domain element and resulting blade surface are shown for a global linear root to tip twist deformation, and a dihedral, sweep, chord, and thickness deformation of the 16th domain element slice. The flexibility of the approach and the smoothness of the deformed surface is clear; the volume mesh is deformed in the same smooth fashion, and so no mesh regeneration is required, see Rendall and Allen (2008b).

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Fig. 2 Domain and block boundaries, rotor disk mesh and mesh near tip

4.2 Flow-solver The flow-solver used is a structured multiblock finite-volume, unsteady, inviscid upwind code (Allen 2007) using the flux vector splitting of van Leer (van Leer 1982; Parpia 1988) and the implicit pseudo-time stepping scheme of Jameson (Alonso and Jameson 1994), and convergence acceleration is achieved through a multigrid approach (Allen 2004, 2001). The method solves the Euler equations in integral form and, for hovering rotor cases, the three-dimensional unsteady Euler equations are transformed to a blade-fixed rotating reference frame. In this frame the hover case is then a steady problem. If the frame rotates with angular velocity ω = [ x , y , z ]T , and the absolute velocity vector in the rotating frame is denoted by qr = [ur , vr , wr ]T , the resulting Euler equations in integral form are then d dt



Ur dVr + Vr

Fr .nr dSr +

∂Vr

Gr dVr = 0 Vr

(18)

CFD-based optimization of hovering rotors using radial basis functions

Fig. 3 Deformed domain element and rotor surface

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where Vr is the control volume, ∂Vr is the control volume boundary, and   ⎡ ⎡ ⎤ ⎤ ρ qr − (ω × r) ρ   ⎢ ⎢ ⎥ ⎥ ⎢ ρur ⎥ ⎢ ρur qr − (ω × r) + P ir ⎥ ⎢ ⎢ ⎥ ⎥   ⎢ ⎢ ⎥ ⎥ ρvr ⎥ , q ρv − (ω × r) + P j ⎢ ⎥, r r r F Ur = ⎢ = r ⎢ ⎢ ⎥ ⎥   ⎢ ρw ⎥ ⎢ ⎥ ⎢ r⎥ ⎢ ρwr qr − (ω × r) + P kr ⎥ ⎣ ⎣ ⎦ ⎦   E E qr − (ω × r) + P qr ⎡

0

⎤

(19)

⎥ ⎢ ⎢ ρ(ω × qr ) · ir ⎥ ⎥ ⎢ ⎥ ⎢ ρ(ω × q ) · j ⎢ r r ⎥. Gr = ⎢ ⎥ ⎢ ρ(ω × q ) · k ⎥ ⎢ r r⎥ ⎦ ⎣ 0 Here Gr is the source term resulting from the transformation, and r = [xr , yr , zr ]T is the coordinate vector. The equation set is closed by   ρ 2 (20) P = (γ − 1) E − qr . 2 4.3 Optimization results The two optimizations were run in parallel on 64 CPUs (one per variable plus one for the master process). The cases were run with strict constraints: Objective:

Minimize torque (Cmz ) with respect to domain element node positions

Constraint 1 (thrust): Constraint 2 (bending moment): Constraint 3 (pitching moment): Constraint 4 (internal volume):

CT ≥ CT (original) Cmx ≤ Cmx (original) |Cmy | ≤ |Cmy |(original) Vint ≥ Vint (original)

Table 1 shows results of the optimizations, and both have resulted in a significant reduction in torque. Two views of the original and optimized blades are shown in Fig. 4 for the two tip speeds, and a third view with upper and lower surface pressure coefficient contours in Fig. 5. Hence, the optimization has produced significant geometric changes in both cases, with twist, sweep, dihedral and thickness changes. Clearly the majority of the local changes are in the transonic region where the sensitivities of the local parameters are large due to the wave drag. The convergence histories and loading distributions are presented in Fig. 6. Considering the convergence history, it is clear the majority of the objective improvement is achieved very quickly; this is due to the global linear twist parameter having a much larger sensitivity than the local parameters, and this rapidly approaches its optimum value. The optimum twist is the

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Fig. 4 Original and optimized surfaces, MT ip = 0.794 (upper) and 0.877 (lower)

minimum induced power case, i.e. linear loading, and the loading distributions show the loading tending towards this case, but with allowance for a decrease in the tip area to reduce the wave drag. More detailed geometry changes are presented in Fig. 7; these are changes to the twist, chord, thickness, and x and z positions of the quarter chord position. The changes presented in these figures are shown in absolute not relative terms, i.e. a 1% change in thickness is a change from 12% to 11%. The twist distribution is as expected, moving the load inboard and hence reducing the root moment. The sweep, dihedral, and thickness change trends are similar in both cases, just scaled for the higher tip speed case. The sweep, chord, and thickness variations are a result of the optimizer attempting to reduce the wave drag, while it is assumed the vertical deformation is an attempt to minimize the effect of the tip vortices. The 0.877 case has almost halved the section thickness in the transonic region, along with an 18% increase in chord and 29% forward movement of the quarter chord position. The increase in chord and decrease in thickness are almost neutral in terms of volume, and so there is no issue satisfying the volume constraint. The reduction of the transonic region also results in a large reduction in bending moment (Cmy ). It is very interesting to note that the optimized blade has deformed towards the Agusta-Westland BERP blade shape, see for example Perry (1987). This was actually designed for forward

5.495 3.873 0.066 0.377

5.495

4.300

0.113

0.460

CT /σ

Cmy /σ

Cmz /σ

5.619 4.420 0.151 0.545

0.00 −9.93 −41.59 −18.03

Initial/10−2

%Difference

Final/10−2

Initial/10−2

MT ip = 0.877 51

MT ip = 0.794

39

Cmx /σ

Evolutions

Table 1 Rotor optimization results. MT ip = 0.794 and 0.877

0.384

0.024

3.964

5.619

Final/10−2

−29.54

−84.04

−10.32

0.00

%Difference

112 C.B. Allen, T.C.S. Rendall

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Fig. 5 Original and optimized pressure coefficient contours, MT ip = 0.794 (upper) and 0.877 (lower)

flight performance, using lower fidelity methods than here, but the primary objective was to reduce transonic effects on the advancing side, which is similar to the requirements here. The results presented demonstrate that the approach is extremely flexible, with significant planform changes possible. The optimizations are single point cases, so off-design performance has also been considered. Figure 8 shows the thrust-torque polars for the two cases, compared to the initial blade. These were computed by running half degree increments, from around −5◦ to a positive pitch at which convergence became difficult. Hence, the blade optimized at Mach 0.794 performs slightly worse at negative and around zero thrust, but performs better at all positive thrusts. The Mach 0.877 blade performance is similar at negative thrust, experiences a small region around zero where performance is slightly worse, then performs significantly better at all positive thrust. The slightly lower performance around zero lift is unavoidable, since the blade twist results in the

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Fig. 6 Convergence history and loading distribution, MT ip = 0.794 (upper) and 0.877 (lower)

tip region being at negative incidence at low and negative lift, and the lower surface transonic effects being significantly stronger than in the untwisted case. 5 Conclusions A generic and efficient domain element method of geometric parameterization has been presented for application to CFD-based aerodynamic optimization. Design surface control and volume mesh deformation are the key issues when considering optimization, and both are solved using this approach. Radial basis function global interpolation allows the domain element to control the grid coordinates and provide simultaneous deformation of the design surface and its corresponding CFD mesh. The approach is totally independent of mesh type, so requires no connectivity, and both the design surface and volume meshes are deformed in a smooth, high quality fashion, regardless of the size of the deformation. The technique also allows the use of design variables of different scales and types with only a few parameters. The surface parameterization and mesh deformation scheme has been linked to an advanced feasible sequential quadratic programming (FSQP) gradient-based optimizer, and this has been parallelized, such that optimization of the rotor can be

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Fig. 7 Geometric deformations, MT ip = 0.794 (upper) and 0.877 (lower)

Fig. 8 Polars, MT ip = 0.794 (left) and 0.877 (right)

performed in a practical run time. The optimization suite has been parallelized in a generic fashion such that the code is still independent of the flow solver. Results for a rotor in hover have been presented, minimizing torque subject with strict constraints, allowing global and local planform geometric changes. 63 parame-

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ters were considered for two transonic tip speeds, and torque reductions of 18% and 29% have been achieved for the two cases, which are very impressive results. Large geometric changes have been demonstrated, due to the optimizer attempting to reduce the induced power, the wave drag, and blade-vortex interaction effects, showing the effectiveness and flexibility of the scheme. Off-design performance has also been considered, and demonstrates improved performance at all positive thrust values. An inviscid solver has been used here, and the results using a viscous solver may be slightly different. However, it is the demonstration of the capability and flexibility of the optimization framework which is the key objective of this paper. Acknowledgements The authors would like to thank Dr Asa Morris for his contribution to the development of the parallel optimizer.

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