Keywords: global interpolation, radial basis functions, mesh deformation, shape parameterisation, volume data, system reduction, adaptive sampling, computational aerodynamics.

A review is presented of sample applications of multivariate function approximation within computational methods, particularly aeroelastic simulation and optimization. Generic interpolation methods that are entirely flow-solver and mesh type independent are extremely attractive, as they can be developed as stand-alone pre-, co-, and postprocessing tools, to ease the use of numerical methods. The authors have developed interpolation methods based on a universal underlying methodology, which is inherently 'meshless', i.e. can be used on arbitrary point clouds. The approach is also n-dimensional, so can be applied to data space interpolation just as easily as coordinate-based problems. Multivariate function approximation was originally developed for scattered data interpolation within the computer graphics field, but is a very powerful general tool. This type of approach was applied to computational aerodynamics problems, specifically the complex problem of coupling fluid dynamic and structural dynamics solvers to allow aeroelastic simulation. The computational fluid dynamics (CFD) surface mesh and the computational structural dynamics (CSD) structural mesh will not normally occupy the same space, and so the interpolation scheme adopted needs to be able to transfer forces and moments from the CFD surface mesh to the structural model, and displacements in the opposite direction, in a consistent way. A universal global dependence method has been developed to solve this problem, which is totally connectivity free, working purely on point clouds in any order, so can be applied to any mesh type. For any simulation involving a deforming surface, the volume mesh also needs to deform. The method has been further extended to provide very high quality and robust mesh deformation, again for arbitrary point clouds so for any mesh. Computational fluid dynamics codes are also being used increasingly within an optimization loop, but to optimize the shape of a surface, an effective geometry parameterisation and surface control method is required. A domain element approach has been developed, linked with the radial basis function (RBF) mesh deformation, to allow domain element point movements to control the design surface and volume mesh deformation simultaneously. System reduction methods have also been developed, to reduce the number of control points in the system, and have been linked to the global interpolation method, to also produce a general volume interpolation method. Applications include mesh-to-mesh solution interpolation, and velocity field interpolation to compute streamlines. In the aerodynamic design field, it is desirable to have load and moment data available throughout the entire parameter space. However, if this data is obtained using CFD, it is prohibitively expensive, and so an effective data interpolation and adaptive sampling method has been developed to both optimise the positions of the requested CFD data points, and develop an interpolation throughout parameter space. This paper presents the background theory to global interpolation methods, and an overview of all the applications mentioned above.

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