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Generic wraparound aerodynamic shape optimization technology is presented and applied to a modern commercial aircraft wing in transonic cruise. The wing geometry is parameterized by a novel domain-element method, which uses efficient global interpolation functions to deform both the surface geometry and corresponding computational fluid dynamics vo...
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... simple example is presented to demonstrate the approach. A cylindrical surface is taken as the design surface, and six square slices are used as the domain element to control this surface (see Fig. 2a). Figure 2b shows a local parameter (i.e., movement of a single domain-element point). Figures 2c and 2d show two global para- meters: a sweep (in fact, this is actually a shear) parameter and a linear twist parameter. Figure 2e shows these two global parameters combined, and Fig. 2f shows these two global parameters and the local ...
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... cylindrical surface is taken as the design surface, and six square slices are used as the domain element to control this surface (see Fig. 2a). Figure 2b shows a local parameter (i.e., movement of a single domain-element point). Figures 2c and 2d show two global para- meters: a sweep (in fact, this is actually a shear) parameter and a linear twist parameter. ...
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... cylindrical surface is taken as the design surface, and six square slices are used as the domain element to control this surface (see Fig. 2a). Figure 2b shows a local parameter (i.e., movement of a single domain-element point). Figures 2c and 2d show two global para- meters: a sweep (in fact, this is actually a shear) parameter and a linear twist parameter. Figure 2e shows these two global parameters combined, and Fig. 2f shows these two global parameters and the local parameter combined. ...
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... 2c and 2d show two global para- meters: a sweep (in fact, this is actually a shear) parameter and a linear twist parameter. Figure 2e shows these two global parameters combined, and Fig. 2f shows these two global parameters and the local parameter combined. This shows both the flexibility of the parameterization scheme and the effectiveness of the surface defor- mation scheme; the volume mesh would be deformed by the same motion, and a mesh deformation example is demonstrated later for the real mesh. ...
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... are used as the domain element to control this surface (see Fig. 2a). Figure 2b shows a local parameter (i.e., movement of a single domain-element point). Figures 2c and 2d show two global para- meters: a sweep (in fact, this is actually a shear) parameter and a linear twist parameter. Figure 2e shows these two global parameters combined, and Fig. 2f shows these two global parameters and the local parameter combined. This shows both the flexibility of the parameterization scheme and the effectiveness of the surface defor- mation scheme; the volume mesh would be deformed by the same motion, and a mesh deformation example is demonstrated later for the real ...
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... that, as shown in Fig. 2, the parameterization is fully three- dimensional, such that deformations due to a movement of the domain element in the plane of a two-dimensional slice smoothly extend in the spanwise direction as well, and so no linear inter- polation is required between slices. An example deformation of the CFD volume mesh to a change in a design ...
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Citations
... A variety of shape control methods are encountered in aerodynamic shape optimisation including: spline surfaces [12]; volume splines [13][14][15]; and various other shape functions derived analytically (Hicks-Henne [16] and shape-class functions [17]), from physical considerations (fictitous loads [18] ) or numerically (orthogonal modes [9]). All such methods reduce the dimensionality of the continuous shape problem and introduce some smoothing, an advantage that was emphasised by Braibant and Fleury when advocating the use of B-Splines for shape optimisation [12]. ...
Orthogonal modes are an effective method for aerodynamic shape optimization due to their excellent design space compactness; however, all existing methods are generated from a database of representative geometry. Moreover, application to high-fidelity design spaces is not possible because high-frequency shape components are insufficiently bounded, leading to nonsmooth and oscillatory geometries. In this work, a new generic methodology for generating orthogonal shape modes is presented based on a purely geometric derivation, eliminating the need for geometric training data. The new method is a further development of the gradient-limiting method developed previously for constraining the design space in a geometrically meaningful way to reduce the effective degrees of freedom and improve optimization convergence rate and final result. Here, the gradient-limiting methodology is reformulated by transforming the constraints directly onto design variables to produce orthogonal shape modes with equivalent constraints for ensuring smooth and valid iterates. The new generic methodology requires no training data, can be applied to arbitrary topologies using different boundary conditions, and naturally includes translational modes as part of the orthogonal basis. When applied to two standard aerodynamic test cases, the new method has superior performance compared to library-derived modes. Importantly, the optimization convergence rate is independent of the number of design variables, and the optimized objective at high design fidelities is greatly improved by avoiding local minima corresponding to spurious geometries. A nonstandard test case is demonstrated, for which traditional library modes are not useable due to nontrivial topology, and it is shown to benefit from the high-fidelity design space and translational mode made possible with the novel methodology.
... The algorithm iterates until the Kuhn-Tucker conditions are satisfied, which then represent a converged solution using a constrained gradient-based optimizer. For computational efficiency, the sensitivity evaluation has been parallelized based on the number of design variables such that the evaluation of the sensitivity of the objective function and constraints with respect to the design variables is split between the number of CPUs available (Morris et al. , 2009). This is necessary as within the ASO environment, an objective function evaluation represents a CFD solution, so this formulation allows parallel evaluation of the required sensitivities; second-order finite-differences are used for the sensitivities. ...
Gradient-based optimization is a calculus-based point-by-point technique that relies on the gradient (derivative) information of the objective function with respect to a number of independent variables. The nature in which gradient-based methods (GBM) operate make them well suited to finding locally optimal solutions but may struggle to find the global optimal. With gradient-based algorithms an understanding of the design space is assumed, as an appropriately pre-conceived starting design point must be given. If large changes in topology are expected lower fidelity panel codes can facilitate useful optimization procedures.
... The algorithm iterates until the Kuhn-Tucker conditions are satisfied, which then represent a converged solution using a constrained gradient-based optimizer. For computational efficiency, the sensitivity evaluation has been parallelized based on the number of design variables such that the evaluation of the sensitivity of the objective function and constraints with respect to the design variables is split between the number of CPUs available (Morris et al. , 2009). This is necessary as within the ASO environment, an objective function evaluation represents a CFD solution, so this formulation allows parallel evaluation of the required sensitivities; second-order finite-differences are used for the sensitivities. ...
The use of computational simulation to scan many alternative designs has proved extremely valuable in practice, but it is also evident that the number of possible design variations is too large to permit their complete evaluation. Thus it is very unlikely that a truly optimum solution can be found without the assistance of automatic optimization procedures. As an vital “ingredients” in gradient base optimization, the sensitivities may now be estimated by making a small variation in each design parameter in turn and recalculating the flow. The gradient can be determined directly or indirectly by number of available methods, including Direct Differentiation (DD), Adjoin Variable(AV), Symbolic Differentiation (SD), Automatic Differentiation (AD), and Finite Difference (FD). Once the gradient has been calculated, a descent method can be used to determine a shape change that will make an improvement in the design.
... and Nadarajah [171] demonstrated the proposed RBF-based grid deformation scheme through several 3D cases within an adjoint-based ASO framework, in which the grid sensitivity is based on a combination of both the primary and secondary grid movement algorithms. In addition, Morris et al. [68,69] and Allen and Rendall [172] presented an integrated approach for geometry parameterisation, surface control, and grid deformation; and they also applied this approach to shape optimisation of aerofoils [68], aircraft wings [69], and rotor blades [172]. Poole et al. [125] further demonstrated this unified approach through investigating ADODG benchmark problems. ...
... and Nadarajah [171] demonstrated the proposed RBF-based grid deformation scheme through several 3D cases within an adjoint-based ASO framework, in which the grid sensitivity is based on a combination of both the primary and secondary grid movement algorithms. In addition, Morris et al. [68,69] and Allen and Rendall [172] presented an integrated approach for geometry parameterisation, surface control, and grid deformation; and they also applied this approach to shape optimisation of aerofoils [68], aircraft wings [69], and rotor blades [172]. Poole et al. [125] further demonstrated this unified approach through investigating ADODG benchmark problems. ...
Computational fluid dynamics (CFD) has become the method of choice for aerodynamic shape optimisation of complex engineering problems. To date, however, the sensitivity of the optimal solution to numerical parameters has been largely underestimated. Meanwhile, aerodynamic shape optimisation based on high-fidelity CFD remains a computationally expensive task. The thesis consists of two research streams aimed at addressing each of the challenges identified, namely revisiting the optimal solution and developing an efficient optimisation framework. This work primarily focuses on the assessment of optimal design sensitivity and computational efficiency in gradient-based optimisation of aeronautical applications.
Two benchmark cases for NACA0012 and RAE2822 aerofoil optimisation are investigated using the open-source SU2 code. Hicks-Henne bump functions and free-form deformation are employed as geometry parameterisation methods. Gradients are computed by the continuous adjoint approach. The optimisation results of NACA0012 aerofoil exhibit strong dependence on virtually all numerical parameters investigated, whereas the optimal design of RAE2822 aerofoil is insensitive to those parameter settings. The degree of sensitivity reflects the difference in the design space, particularly of the local curvature on the optimised shape. The closure coefficients of Spalart-Allmaras model affect the final optimisation performance, raising the importance of quantifying uncertainty in turbulence modelling calibration. Non-unique flow solutions are found to exist for both cases, and hysteresis occurs in a narrow region near the design point.
Wing twist optimisations are conducted using two aerodynamic solvers of different levels of fidelity. A multi-fidelity aerodynamic approach is proposed, which contains three components: a linear vortex lattice method solver, an infinite swept wing solver, and a coupling algorithm. For reference, three-dimensional data are obtained using SU2. Two optimisation cases are considered, featuring inviscid flow around an unswept wing and viscous flow around a swept wing. A good agreement in terms of lift distribution and aerodynamic shape between the multi-fidelity solver and high-fidelity CFD is obtained. The numerical optimisation using the multi-fidelity approach is performed at a negligible computational cost compared to the full three-dimensional CFD solver, demonstrating the potential for use in early phases of aircraft design.
... For suitable mesh deformation, an efficient domain element shape parameterization method has been developed by the authors and presented previously for CFD-based shape optimization [6,42]. The parameterization technique, surface control and volume mesh deformation all use radial basis functions (RBFs), wherein global interpolation is used to provide direct control of the design surface and the CFD mesh, which is deformed in a high-quality fashion [43,44]. ...
The most common aerofoil optimization problem considered is lift-constrained drag minimization at a fixed design point; however, shock-free solutions can result, which can lead to poor off-design performance. As such, this paper presents a study of the construction of the aerofoil optimization problem and its effect on the performance over a range of operating conditions. Single-point and multipoint optimizations of aerofoils in transonic flow are considered, and an improved range-based optimization problem subject to a constraint on fixed nondimensional wing loading with a varying design point is formulated. This problem is more representative of the aircraft design problem, though similar in cost to single-point drag minimization. An analytical treatment using an approximation of wave drag, which demonstrates that the optimum Mach number for a fixed shape is supercritical if the required loading is above a critical threshold, is also presented. This paper presents optimizations that show that to define an effective objective function three-dimensional effects modelled via an induced drag term must be introduced. The general trend is to produce solutions with higher Mach numbers and lower lift coefficients, and shocked solutions perform better when considering the performance in a range over the operating space. Furthermore, the resulting aerofoil shapes are supercritical in nature, a particularly promising result.
... Numerous advanced optimizations using compressible computational fluid dynamics (CFD) as the aerodynamic model have previously been performed (Hicks and Henne 1978;Qin et al. 2004;Nielsen et al. 2010;Lyu et al. 2015;Choi et al. 2014). The authors have also presented work in this area, having developed a modularised, generic optimization tool, that is flow-solver and mesh-type independent, and applicable to any aerodynamic problem (Morris et al. 2008(Morris et al. , 2009Allen and Rendall 2013). ...
... For computational efficiency, the sensitivity evaluation has been parallelised based on the number of design variables such that the evaluation of the sensitivity of the objective function and constraints with respect to the design variables is split between the number of CPUs available (Morris et al. 2008(Morris et al. , 2009). This is necessary as within the ASO environment, an objective function evaluation represents a CFD solution, so this formulation allows parallel evaluation of the required sensitivities; second-order finite-differences are used for the sensitivities. ...
... In previous work the authors have applied a 16-point off-surface domain element for an aerofoil, and a set of section-based domain elements for a wing, which has been shown to be very effective (Morris et al. 2008(Morris et al. , 2009. Hence, an improved version of this approach, implementing the localization method, is used here as a comparison with the new method; the 24-point on-surface set of control points used in two dimensions is again used here. ...
Aerodynamic shape optimization of a transonic wing using mathematically-extracted modal design variables is presented. A novel approach is used for deriving design variables using a singular value decomposition of a set of training aerofoils to obtain an efficient, reduced set of orthogonal ‘modes’ that represent typical aerodynamic design parameters. These design parameters have previously been tested on geometric shape recovery problems and aerodynamic shape optimization in two dimensions, and shown to be efficient at covering a large portion of the design space; the work is extended here to consider their use in three dimensions. Wing shape optimization in transonic flow is performed using an upwind flow-solver and parallel gradient-based optimizer, and a small number of global deformation modes are compared to a section-based local application of these modes and to a previously-used section-based domain element approach to deformations. An effective geometric deformation localization method is also presented, to ensure global modes can be reconstructed exactly by superposition of local modes. The modal approach is shown to be particularly efficient, with improved convergence over the domain element method, and only 10 modal design variables result in a 28% drag reduction.
... For suitable surface control and mesh deformation, an efficient domain element shape parameterization method has been developed by the authors and presented previously for CFD-based shape optimization [8,51]. The parameterization technique, surface control and volume mesh deformation all use radial basis functions (RBFs), wherein global interpolation is used to provide direct control of the design surface and the CFD mesh, which is deformed in a high-quality fashion [52,53]. ...
An investigation into an aerodynamic optimization benchmark case that exhibits mul-timodality is presented using a global optimization approach. The recently suggested case 6 of the AIAA Aerodynamic Design Optimization Discussion Group involves the drag minimization of a rectangular NACA0012 wing subject to lift and root bending moment, as well as other geometric constraints. In this paper, optimization of that case using a state-of-the-art constrained population-based global optimization framework is presented. Shape control is achieved via a hierarchical application of the radial basis function domain element approach. Three different fidelity of optimization cases are performed to investigate the multimodality of various aspects of the full problem, with four independent runs of each case being performed. When considering chord optimization, the optimizer converges to two independently identifiable minima with very similar performance. When adding dihedral and sweep variation, further multimodality is introduced, though this mul-timodality is reasonably well defined, with minima including forward and rearward sweep, and an upward and downward winglet. Introducing thickness variables leads to a highly multimodal problem due to the coupling between the chord and thickness variables. The theoretical minimum drag for this case is 24.7 counts, which a number of the runs get close to achieving.
... The , shows a complex dependency which is dictated by the mesh movement to be differentiated. Morris et al. [3] proposed to use finite differences for [ / ], whereas Hicken and Zingg [4] used the more sophisticated complex differences in conjunction with the mesh-adjoint approach by Nielsen and Park [5]. ...
This paper presents an investigation of the influence of a nonconsistent approach in terms of mesh movement and mesh sensitivity calculation in a discrete adjoint-based optimization. Some mesh movement methods are more robust or of higher quality, whereas others can be more efficient for calculating mesh sensitivity. It is found that a nonconsistent approach gives comparable results when compared to a consistent approach. Therefore, an appropriate combination of nonconsistent approaches can be achieved for efficient adjoint optimization. This paper investigates and compares various consistent and nonconsistent combinations by using linear elasticity, Delaunay graph mapping, and radial basis function mesh movement methods. An investigation is presented, using a lift-constrained drag minimization, to assess which step of the chain introduces a deviation, if any, and to which degree this affects the final result.
... For suitable surface control and mesh deformation, an efficient domain element shape parameterization method has been developed by the authors and presented previously for CFD-based shape optimization [8,41]. The parameterization technique, surface control and volume mesh deformation all use radial basis functions (RBFs), wherein global interpolation is used to provide direct control of the design surface and the CFD mesh, which is deformed in a high-quality fashion [42,43]. ...
An investigation into a multimodal aerodynamic optimization benchmark case using an approximate global optimization approach is presented. A new benchmark case of the AIAA Aerodynamic Design Optimization Discussion Group that involves the drag minimization of a rectangular NACA0012 wing subject to lift and root bending moment, as well as other geometric constraints, that is currently believed to be multimodal has been suggested. In this paper, optimization of that case using a constrained population-based global optimization framework is presented, with three independent runs of the optimizer (each with different starting locations) having been performed. The runs are all coarsely-converged to give approximately converged solutions such that rapid run-times can be achieved and used to demonstrate any pitfalls that may be encountered when using a gradient-based optimizer. All three of the runs converged onto drag values close to the theoretical optimum, however with substantially different final shapes. This likely indicates the presence of a flat optimum region within the design space, where a substantial number of wing shapes are able to obtain the theoretical optimum span loading.
... For suitable mesh deformation, an efficient domain element shape parameterization method has been developed by the authors and presented previously for CFD-based shape optimization. 6,32 The parameterization technique, surface control and volume mesh deformation all use radial basis functions (RBFs), wherein global interpolation is used to provide direct control of the design surface and the CFD mesh, which is deformed in a high-quality fashion. 33,34 A small set of control points placed on the aerodynamic surface are deformed using the modal design variables, which subsequently deforms the CFD mesh; this is shown in figure 2 for an exaggerated deformation of the fourth mode. ...
