Thomas Rendall

Efficient mesh motion using radial basis functions with data reduction algorithms

Mesh motion using radial basis functions has been demonstrated previously by the authors to produce high quality meshes suitable for use within unsteady and aeroelastic CFD codes. In the aeroelastic case the structural mesh may be used as the set of control points governing the deformation, which is efficient since the structural mesh is usually small. However, as a stand alone mesh motion tool, where the surface mesh points control the motion, radial basis functions may be restricted by the size of the surface mesh, as an update of a single volume point depends on all surface points. In this paper a method is presented that allows an arbitrary deformation to be represented to within a desired tolerance by using a significantly reduced set of surface points intelligently identified in a fashion that minimises the error in the interpolated surface. This method may be used on much larger cases and is successfully demonstrated here for a 10^6 cell mesh, where the initial solve phase cost reduces by a factor of eight with the new scheme and the mesh update by a factor of 55. It has also been shown that the number of surface points required to represent the surface is only geometry dependent (i.e. grid size independent), and so this reduction factor actually increases for larger meshes.

Free-form aerodynamic wing optimization using mathematically-derived design variables

Aerodynamic shape optimizations of aerofoils and wings using mathematically-derived design variables are presented. A novel approach is used for deriving design variables using a proper orthogonal decomposition of a set of training aerofoils to obtain an efficient, reduced set of deformation ‘modes’ that represent typical design parameters such as thickness and camber. A major advantage of this extraction method is the production of orthogonal design variables, and this is particularly important in aerodynamic shape optimization. These design parameters have previously been tested on geometric shape recovery problems and been shown to be efficient at covering a large portion of the design space, hence the work is extended here to consider their use in aerodynamic shape optimization in two and three dimensions. Using these mathematically-extracted design variables allows the use of global search algorithms for the optimization process in two dimensions, since a small number of parameters are required and these are also orthogonal. In three dimensions a parallel gradient-based optimiser is used. It is shown for two-dimensional inviscid compressible test cases, fewer than 10 aerofoil modes are required to obtain shock free solutions from initial strong shock, highly-loaded aerofoils. In three dimensions, a small number of local and global deformation modes are compared to a section-based application of these modes and to a previously-used section-based domain element approach to deformations, and applied to a transonic wing optimisation. The modal approach is shown to be particularly efficient with, again, fewer than 10 design variables required to achieve an effective optimisation.

Domain-element method for aerodynamic shape optimization applied to a modern transport wing

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 volume mesh. The technique also provides a method that allows geometries to be parameterized at various levels, ranging from global three-dimensional planform alterations to detailed local surface changes. Combining all levels of parameterization allows for free-form design control with very few design variables. The method provides an efficient combined shape parameterization and high-quality mesh deformation technique that is totally independent of mesh type (structured or unstructured). Optimization independence from the flow solver is achieved by obtaining sensitivity information for an advanced gradient-based optimizer by finite differences. The entire optimization suite has also been parallelized to allow optimization with highly flexible parameterization in practical times. Results are presented for highly constrained optimizations of the modern aircraft wing in transonic cruise, using three levels of parameterization (number of design variables) to assess the effect of parameterization level on the optimization. The highest-level optimization results in a totally-shock-free geometry with an associated substantial reduction in drag.

Reduced surface point selection options for efficient mesh deformation using radial basis functions

Previous work by the authors has developed an efficient method for using radial basis functions (RBFs) to achieve high quality mesh deformation for large meshes. For volume mesh deformation driven by surface motion, the RBF system can become impractical for large meshes due to the large number of surface (control) points, and so a particularly effective data reduction scheme has been developed to vastly reduce the number of surface points used. The method uses a chosen error function on the surface mesh to select a reduced subset of the surface points; this subset contains a sufficiently small number of points so as to make the volume deformation fast, and a correction function is used to correct those surface points not included. Hence, the scheme is split such that both parts are working on appropriate problems. RBFs are an excellent way of finding smooth orthogonality preserving global deformations, but are less suitable for enforcing an exact geometry for a large number of points, while a simpler approach is ideal for diffusing small changes evenly but has quality (and possibly expense) drawbacks if used for the entire volume. However, alternatives exist for the error function used to select the reduced data set, so here a comparison is made between three different options: the surface error function, the unit function and the power function. Tests run on structured and unstructured meshes show that the surface error function gives the lowest errors, but this also requires a deformed surface shape to be known in advance of the simulation. The unit and power functions both avoid the need for a deformed surface, and the unit function is shown to be superior.

Metric-Based Mathematical Derivation of Efficient Airfoil Design Variables

Within an aerodynamic shape optimization framework, an efficient shape parameterization and deformation scheme is critical to allow flexible deformation of the surface with the maximum possible design space coverage. Numerous approaches have been developed for the geometric representation of airfoils. A fundamental approach is considered here from the geometric perspective; and a method is presented to allow the derivation of efficient, generic, and orthogonal airfoil geometric design variables. This is achieved by the mathematical decomposition of a training library. The resulting geometric modes are independent of a parameterization scheme, surface and volume mesh, and flow solver; thus, they are generally applicable. However, these modes are dependent on the training library, and so a benchmark performance measure, called the airfoil technology factor, has also been incorporated into the scheme to allow intelligent metric-based filtering, or design space reduction, of the training library to ensure eff...

Application of Control Point-Based Aerodynamic Shape Optimization to Two-Dimensional Drag Minimization

An investigation is presented that considers various aspects of an aerodynamic shape optimization framework. A two-dimensional aerofoil transonic zero-lift drag minimization test case is used to investigate the effect of dimensionality, shape deformation parameters, and optimizer on the results from the shape optimization process. A flexible control point-based parameterization is implemented which decouples the design variables from the surface, such that control point deformations determine the surface and volume mesh deformations in a unified manner. A gradient-based optimizer (feasible sequential quadratic programming) and global search algorithm (gravitational search algorithm) are tested on the constrained optimization case. The results show, as expected, that an increase in the number of dimensions produces a greater design space coverage and better optimization results, and the gradient-based method is prone to terminating in local optima or at constraint boundaries, so the global search algorithm is more reliable at locating optima. Efficient, reduced, and orthogonal shape deformation parameters are defined here by singular value decomposition extraction, and are shown to be particularly effective, demonstrating a 99.7% drag reduction for the case considered.

AIAA paper 2015-0761, Proceedings AIAA Science and Technology Forum, Kissemee, Florida

This paper presents a review of aerofoil shape parameterisation methods that can be used for aerodynamic shape optimisation. Six parameterisation methods are considered for a range in design variables: Class function/Shape function Transformations (CST); B-splines; Hicks-Henne bump functions; a domain element approach using Radial Basis functions (RBF); Bezier surfaces; and a singular value decomposition modal extraction method (SVD); plus the PARSEC method. The performance of each method is analysed by considering geometric shape recovery on over 1000 aerofoils using a range of design variables, testing the efficiency of design space coverage. A more in-depth analysis is then presented for three aerofoils, NACA4412, RAE2822 and ONERA M6 (D section), with geometric error and convergence of the resulting aerodynamic properties presented. In the large scale test it is shown that, for all the methods, a large number of design variables are needed to achieve significant design space coverage. For example at least 25 design variables are needed to cover 80% of the design space regardless of the method used; this is often higher than is desired for two-dimensional studies, suggesting that further work may be required to reduce the number of design variables needed.

Computational-Fluid-Dynamics-Based Twist Optimization of Hovering Rotors

Twist optimization of a helicopter rotor in hover is presented using compressible computational fluid dynamics as the aerodynamic model. A domain-element shape parameterization method has been developed, which solves both the geometry control and the volumemesh deformation problems simultaneously, using radial basis function global interpolation. This provides direct transfer of domain-element movements into deformations of the design surface and the computational fluid dynamics volume mesh, which is deformed in a high-quality fashion. The method is independent ofmesh type (structured or unstructured), and it has been linked to an advancedparallel gradient-based algorithm, for which independence from the flow solver is achieved by obtaining sensitivity information by finite differences. This has resulted in aflexible andversatilemodularmethod ofwraparound optimization. Previousfixedwing results have shown that a large proportion of the design space is accessible with the parameterization method, and heavily constrained drag optimization demonstrated significant performance improvements. In the present work, themethod is extended to a rotor blade, and this is optimized forminimum torque in hovering flight with strict constraints. Twist optimization results are presented for three tip Mach numbers, and the effects of different parameterization levels are analyzedusing various combinations of two levels: global and local. Torque reductions of over 12% are shown for a fully subsonic case, and for over 24% for a transonic case, using only three global and 15 local twist parameters.

A geometric comparison of aerofoil shape parameterisation methods

A comprehensive review of aerofoil shape parameterization methods that can be used for aerodynamic shape optimization is presented. Seven parameterization methods are considered for a range of desi...

Multi-dimensional aircraft surface pressure interpolation using radial basis functions

Abstract Multi-dimensional interpolation via radial basis functions is applied to the problem of using aircraft surface pressure data obtained both computationally and experimentally to obtain pressure distribution predictions through parameter space. In the most complicated cases, the data may be a function of spatial position, Mach number, Reynolds number, and angle of attack as well as other more intricate variables such as control surface deflections. Amalgamation of computational fluid dynamics and wind tunnel data for load prediction is currently a time-consuming task, especially given the large number of load cases that need to be evaluated to achieve aircraft certification, so that an efficient tool for making rapid estimates based on all the information available would be of great use. The approach, using radial basis functions, is tested on a combination of simple computational and experimental results and found to offer great flexibility, while still being capable of reproducing relatively detailed features of the pressure distribution.