Abstract
Given an array (or matrix) of values for a function of one or more variables, it is often desired to find a value between two given points. Multivariable interpolation and approximation by radial basis functions are important subjects in approximation theory that have many applications in Science and Engineering fields. During the last decades, radial basis functions (RBFs) have found increasingly widespread use for functional approximation of scattered data. This research work aims at benchmarking two different approaches: an approximation by radial basis functions and a piecewise linear multivariable interpolation in terms of their effectiveness and efficiency in order to conclude about the advantages and disadvantages of each approach in approximating the aerodynamic coefficients of airfoils. The main focus of this article is to study the main factors that affect the accuracy of the multiquadric functions, including the location and quantity of centers and the choice of the form factor. It also benchmarks them against piecewise linear multivariable interpolation regarding their precision throughout the selected domain and the computational cost required to accomplish a given amount of solutions associated with the aerodynamic coefficients of lift, drag and pitching moment. The approximation functions are applied to two different multidimensional cases: two independent variables, where the aerodynamic coefficients depend on the Reynolds number (Re) and the angle-of-attack (α), and four independent variables, where the aerodynamic coefficients depend on Re, α, flap chord ratio (cflap), and flap deflection (δflap).
Highlights
Given an array of values for a function of one or more variables, it is often desired to find a value between two given points
Radial basis functions (RBFs) have found increasingly widespread use for functional approximation of scattered data. is research work aims at benchmarking two different approaches: an approximation by radial basis functions and a piecewise linear multivariable interpolation in terms of their effectiveness and efficiency in order to conclude about the advantages and disadvantages of each approach in approximating the aerodynamic coefficients of airfoils. e main focus of this article is to study the main factors that affect the accuracy of the multiquadric functions, including the location and quantity of centers and the choice of the form factor
It benchmarks them against piecewise linear multivariable interpolation regarding their precision throughout the selected domain and the computational cost required to accomplish a given amount of solutions associated with the aerodynamic coefficients of lift, drag and pitching moment. e approximation functions are applied to two different multidimensional cases: two independent variables, where the aerodynamic coefficients depend on the Reynolds number (Re) and the angle-of-attack (α), and four independent variables, where the aerodynamic coefficients depend on Re, α, flap chord ratio, and flap deflection
Summary
Within the context of aircraft design optimization methodologies and computational codes, one of the most important aspects is obviously to compute accurate aerodynamic coefficients for the lifting surfaces for any given flight condition at the lowest possible cost. Nonlinear formulations of LLT and VLM require knowledge of the airfoil aerodynamic coefficient curves at various sections across the span. For using such methods, it is fundamental to have good estimates for the airfoil lift, drag, and pitching moment coefficients (Cl, Cd, and Cm). E challenge is to develop an approximate method, using multiquadric functions, which enable a fast and accurate computation of the aforementioned aerodynamic coefficients as a direct function of Reynolds number (Re), angle-of-attack (α), flap chord ratio (cflap), and flap deflection (δflap) and compare its accuracy and computation time with the multivariable interpolation approach and with direct analysis by XFOIL [29]. For a given set of airfoil operational low-speed conditions, which may comprise airspeed, altitude, air turbulence level, and surface roughness, those parameters were considered the most relevant of all affecting the aerodynamic coefficients of a given fixed airfoil [30]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
