Christian B Allen

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.

Efficient Mesh Deformation Using Radial Basis Functions on Unstructured Meshes

An efficient mesh-deformation algorithm has been developed within an unstructured-grid computational-fluid-dynamics solver framework based on a radial-basis-function volume-interpolation method. The data-transfer problem between fluid and structural solvers is simplified here using a beam structural representation, with surface mesh deformation given directly via translational and rotational deformations. The volume mesh deformation is then performed using a radial-basis-function method, which requires no mesh-connectivity information and allows straightforward implementation in an unstructured computational-fluid-dynamics solver in a parallel fashion. However, the pure method is impractical for large meshes, and a novel “greedy” data-reduction algorithm is presented here to select an optimum reduced set of surface mesh points, which makes the mesh-deformation method extremely efficient. Several two- and three-dimensional test cases are presented to validate the algorithm performance, including a realisti...

Comparison of Adaptive Sampling Methods for Generation of Surrogate Aerodynamic Models

A surrogate modeling strategy, using effective interpolation and sampling methods, facilitates a reduction in the number of computational fluid dynamics simulations required to construct an aerodynamic model to a specified accuracy. In this paper, two adaptive sampling strategies are compared for generating surrogate models, based on Kriging and radial basis function interpolation, respectively. The relationships between the two model formulations are discussed, and three test cases are considered, including analytic functions and recovery of aerodynamic coefficients for two example applications: longitudinal flight mechanics analysis for the DLR-F12 aircraft and structural loads analysis of an RAE2822 airfoil. For the airfoil example, models of CL, CD, and CM were constructed with the two sampling strategies using Euler/boundary-layer-coupled computational fluid dynamics and a three-dimensional flight envelope of incidence, Mach, and Reynolds number. The two sampling approaches direct some samples toward...

EROS — a common European Euler code for the analysis of the helicopter rotor flowfield

Abstract The helicopter rotor flowfield is one of the most complex and challenging problems in theoretical aerodynamics. Its accurate analysis is essential for the design of rotors with increased performance, reduced vibratory loads and more environmentally friendly acoustic signatures. European rotorcraft manufacturers have an urgent requirement for a rotor aerodynamic prediction tool to be used within the design office on a routine basis and which is capable of capturing rotational phenomena, such as blade tip and wake vortices, and correctly predict the unsteady blade pressures over a range of different flight conditions. The EROS project addresses this requirement by developing a common European rotor aerodynamic system capable of analysing the inviscid rotor flow environment by solving the three-dimensional Euler equations. The method is based on a proven-technology time-accurate Euler formulation on overlapping structured grids (Chimera method). The grid generator provides an all-in-one capability for grid generation guiding the user from the generation of individual component grids to the Chimera domain decomposition through an interactive process which has embedded visualisation and animation capabilities. The cell-centered finite-volume solver adopts a dual-time implicit scheme on deforming grids. Non-conservative interpolation is used to transfer information across grid overlap regions. This article presents the main components of the system and reviews its capabilities through a number of applications.

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...

Aeroelastic computations using algebraic grid motion

Coupling an unsteady flow-solver with a structural model offers the opportunity to simulate aeroelastic behaviour of wings and rotor blades. The moving and deforming surfaces resulting from unsteady simulations require deforming meshes during the simulation, and it is common to use simple interpolation of surface displacements and velocities onto the initial undisturbed mesh. However, aeroelastic simulations can result in large displacements and deformations of solid surfaces, and simple interpolation of perturbations results in poor grid quality and possible grid crossover. A new interpolation technique is presented which is still simple in that it is driven solely by surface motion, but represents rotational effects near the solid surface, to maintain grid quality there. Furthermore, the scheme is fully analytic, so is very cheap computationally and results in grid speeds also being available analytically. Results, in terms of unsteady grid motion and flow solution, show the scheme to be effective and efficient

Multigrid acceleration of an upwind Euler method for hovering rotor flows

The effect of multigrid acceleration implemented within an upwind-biased Euler method for hovering rotor flows is presented. The requirement to capture the vortical wake development over several turns means a long numerical integration time is required for hovering rotors, and the solution (wake) away from the blade is significant. Furthermore, the flow in the region near the blade root is effectively incompressible. Hence, the solution evolution and convergence is different to a fixed wing case where convergence depends primarily on propagating errors away from the surface as quickly as possible, and multigrid acceleration is shown to be less effective for hovering rotor flows