^{1}

^{2}

^{3}

^{4}

^{1}

^{2}

^{3}

^{4}

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 (_{flap}), and flap deflection (_{flap}).

The application of multiquadric functions to approximate a given set of sampling points was introduced for the first time in 1968 by Hardy, mainly due to the limitations of the various harmonic and polynomial series to represent topography surfaces. This new analytical method intends to represent irregular surfaces through the sum of quadric surfaces that emerge from certain points of interest within the domain of the function, which are called center points (or abbreviated just by centers), which are radially symmetrical with respect to that same point. The multiquadric function is associated with unknown parameters that need to be calculated to create an approximation with low interpolation error and with a realistic evolution between any two points belonging to the domain, this being the most laborious task of the whole procedure [

There are essentially two parameters that greatly affect the accuracy of the method: the location and number of centers and the choice of a parameter called form factor. As far as the location of centers within the domain is concerned, according to Hardy [

Another recent application of multiquadric functions is associated with surrogate models, an engineering method used to estimate the value of any parameter of interest that cannot be directly measured. Most engineering works, before being implemented on a real scale, go through a phase of experimentation and/or simulation to evaluate if the objectives that they propose are feasible, with the advantage of reducing associated costs if any failure is detected. From the aeronautical point of view, in order to find the optimum airfoil to apply to an aircraft wing, so as to optimize a given flight condition or even a set of flight conditions, an engineer must carry out computational and/or experimental simulations to understand the effect of the shape variation of the airfoil (e.g., chord, camber, thickness, and material) which sometimes requires several days or weeks of waiting. To overcome this barrier, surrogate models [

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. In low fidelity analyses, the high cost of computational fluid dynamics makes this choice poorly competitive with respect to simpler methods: the most common solutions being the Lifting Line Theory (LLT), the Vortex Lattice Method (VLM), and the 3D Panel Method. 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 (_{l}, _{d}, and _{m}). Particularly, in low Reynolds number flows, it turns out to be absolutely crucial to have a high sampling rate of this coefficients as the optimizations are run because every time there is an airspeed change, the respective impact on the aerodynamic coefficients not only is hard to forecast but also exhibits nonlinear behavior in some situations. If the aerodynamic coefficients are not accurate enough, all the aircraft optimization codes will be mistakenly biased which will result in poor optimization results.

In a standard airfoil, it is right to consider that its aerodynamic coefficients are functions of the Reynolds number (Re) and the angle-of-attack (

However, if the goal is the assessment of a continuous morphing flap solution, the same approach could become more costly as the number of variables would obviously increase. The challenge is thus 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 (_{flap}), and flap deflection (_{flap}) and then compare its accuracy and computation time with the multivariable interpolation approach and with direct analysis by XFOIL [

Multiquadric surfaces can be represented generically by the series:

In this equation, _{i}, _{i}), which are associated with an unknown coefficient of the function _{i}, which must be calculated in such a way as to describe the shape and the algebraic sign of the given surface.

There are certain particular examples of multiquadric surfaces, including the summation of a series of circular hyperboloids in two sheets, probably the most common, which is mathematically represented by

Another typical example is a multiquadric surface defined by the summation of a series of circular paraboloids, which is mathematically represented by

After applying equation (

Multiquadric surface representation. (a) Form factor = 0. (b) Form factor = 1.

The multiquadric approximation function

The sum consists of _{i} that must be calculated according to the sample data to be approximated. In this notation, ∅_{i}(

After calculating this parameter, for the selected centers and form factor, the coefficients _{i} are obtained. This is the true challenge behind the multiquadric approximation, whose main objective is to minimize the difference between the approximation value (

The interpolation matrix

If matrix _{i} that minimize the following function:

It can be shown that it is possible to compute the approximation function

However, in order to obtain a good approximation, it is necessary to understand the influence of the placement of the centers and the form factor that composes the function. Depending on the user, the concept of good approximation may have different meanings, whereas some applications emphasize the speed of the function generation (and low processing time to get the approximation values), and others focus more on accuracy in order to obtain an interpolation error below a certain value. In this way, it is necessary to understand the effects behind the parameters that make up the approximation function so that they can be adjusted according to specific needs and objectives.

It is neither practical nor realistic to manually perform the distribution of the centers over the domain under study, especially when the number of sampling points or the number of independent variables increases. Therefore, a method that automatically calculates the centers regardless of the domain and/or number of variables is needed. It is known from the outset that the greater the number of centers used, the more complex the multiquadric function becomes and, consequently, the greater the computational effort required to generate the function. As such, the user must stipulate a compromise between having greater accuracy (and greater computation time) or faster processing time (with less computational effort and with the subsequent decreased accuracy).

If the function is defined in a two-dimensional space, or in a higher dimension, it is not enough to identify the number of centers that establish the function, since these can be distributed in countless ways along the domain. Therefore, whenever it is necessary to identify the number of centers applied, taking for example a two-dimensional space, the number of centers will be selected in a line parallel to the _{flap}, _{flap}) when applied to flapped wing airfoils. Variables _{flap} and _{flap} (in degrees), respectively.

In order to denote the number of centers that make up the function, the following notation will be used in the two-dimensional space, taking as an example C(5,4) when placing five centers in a line parallel to the

Example of the automatic collocation of center points throughout the domain. (a) Sample points. (b) Automatic center collocation method.

In one of his more recent works, Hardy states that the best and simplest solution is to place the centers exactly in the same positions where the sample points are, since each center point generates a focus of inclination in that position, which is not beneficial to occur in arbitrary domain positions [

As an example, a random two-dimensional domain, with the sample points

The idea of implementing this optimization method came as a corollary due to the typical evolution of the curves of the average relative error versus the form factor, in order to find, in an automated way, without user intervention, the best possible form factor for the combination of predefined centers, that is, the smallest percentage average relative error (or abbreviated by REL.P) is obtained. This method aims to be applied to convex functions to find the value for which its derivative is (approximately) zero in the vicinity of the initial point under study. Despite not being possible to guarantee that this point is in fact an absolute maximum or minimum of the function, it will always be at least a local maximum or minimum in the study region. Considering the case study, a function minimization is required.

FFSQP 3.7 (

In the course of the study, the parameters shown in Table

Parameters applied to the FFSQP 3.7 method.

FFSQP 3.7 (FORTRAN feasible sequential quadratic programming-Version 3.7) | |
---|---|

Step size | 0.1E-6 |

Required precision (%) | 0.001 |

Initial form factor | 0 |

Maximum number of iterations | 100 |

Estimating function gradients by | Forward finite differences |

There are mainly two parameters that influence the value of convergence of the method: the step size and the initial value where the process starts. As for the step size, it affects not only the speed of convergence but also has an impact on the quality of the solution. Having a high step size leads to a larger iteration step, which may allow the desired solution to be reached more quickly, but it can also be a factor that leads the solution to zones of unstable solution, exceeding the desired local minimum. The initial form factor can also have an impact on the solution since one may already be in the vicinity of another local minimum (when the evolution of the function is nonmonotonic), which may be beneficial or harmful, depending on the convex shape acquired by the defined objective function. It was decided to define the step size equal to 0.1E-6 and a form factor whose initial value equals zero.

Another important aspect that mainly affects the convergence time of the method is the required accuracy of the solution, since the greater the precision, the greater the number of iterations required for the method to converge. Therefore, an accuracy of 0.001% was adopted for the mean relative error. By reducing accuracy, it is possible to achieve similar (but less accurate) solutions with lower convergence time; thus, it is up to the user to adjust it to each problem.

The implemented algorithm is made to work regardless of the dataset. This includes datasets with different dimensions, number of points, and variables’ magnitude. Therefore, a normalization to all points belonging to the data set in a range of [−1,1] for all input variables is applied, to be able to always have a relative perception of the zone of the domain that the user is interested in, without investigating the limits of the domain of the variable, always having the guarantee that it is within the defined limits. This approach aims to standardize the datasets, regardless of their size. It is known that the value of the form factor is naturally related to the set of sample points and to the absolute distance between them, so this approach also tries to standardize, as far as possible, the range of form factor values that enables a good approximation to the function. Since typically the computational methods have difficulty in operating with values of great magnitude, it is also intended with this procedure to reduce the absolute value of the large scales.

In the same way that the data points are normalized, the particular centers along the domain will also be normalized, in order to keep the solution consistent. In Figure

Example of normalization. (a) Dataset P(19,7), (b) C(11,5) before normalization, and (c) C(11,5) after normalization.

A more conventional approach to estimate the value of a function between two known points is the linear interpolation method. If the function under analysis is linear, then the linearly interpolated value will be its exact value. If the array has more than two dimensions (multivariable), the value sought will be at a point within the interior of the corresponding polytope, with the number of mathematical operations to be performed depending on the number of problem dimensions. The quality of the interpolated solution depends not only on the number of scattered points for which the function is known but also on the function behavior. The more linear behavior it exhibits, the better will be the fitting between the actual function and the interpolated values.

Before generalizing the multilinear interpolation approach, let us look at the simplest case of a linear interpolation. Let the superscript (

Linear interpolation.

Equation (

If the goal were to obtain a bilinear interpolation, the problem to solve would be something like the scheme shown in Figure

Bilinear interpolation.

Contrary to the approach adopted for the linear interpolation, the bilinear interpolation is divided into two steps, each of which representing a linear interpolation. The first step consists of computing the function value at points _{0} and _{1}, each being obtained via a linear interpolation:

The second step is to use these two new points to estimate the function value at point

As the number of variables increases, the graphical representation becomes harder to understand and it is therefore important to generalize what has been stated for linear and bilinear interpolation to multivariable interpolation,

The first step is to determine the coordinates of the known points which are closer to the point

Following a similar approach to the one presented for the linear and bilinear interpolation and beginning by interpolating variable

Interpolating variable

Interpolating variable

The number of vertices of the aforementioned multidimensional polytope is equal to (2^{n}), with (

This methodology can be applied to a set of data, representing a multidimensional non-linear function, by fitting linear functions inside the polytope made up by consecutive data points. The result is a nonlinear global representation of the data by local piecewise linear multivariable interpolation.

This section begins by approximating the aerodynamic coefficients of airfoils as a function of Re and

Two different airfoils: UBI_03_016 and DAE-21, represented in Figures

UBI-03_106 airfoil.

DAE-21 airfoil.

To understand the behavior of the multiquadric approximations with increasing number of centers, the variation of the REL.P as a function of the number of centers is represented for a constant form factor equal to zero, as shown in Figures

REL.P versus centers’ combination C(Re,

The first important aspect to highlight is the greater difficulty in approximating the data corresponding to the drag coefficient, a trend that is not limited to these airfoils. Irregularity and nonstandard behavior mean that more centers are required to allow reasonable approximations with multiquadric functions. Considering the other two aerodynamic coefficients, it is possible to note some similarities in the approach behavior depending on the number of centers applied when comparing the two wing airfoils. It should be noted that the goal is, however, to optimize the value of the form factor as far as possible to minimize the associated errors.

To have a perception of the value of the form factor from which an unstable behavior occurs in the solution, resulting from the ill-conditioning associated with the interpolation matrix, the evolution of the REL.P with a growing form factor is represented in Figure

Form factor increase: REL.P versus centers combination C(Re,

To demonstrate the benefits of applying this optimization method, a random set of centers’ combinations, associated with the multiquadric function that approximates the lift coefficient (since the trend is the same for the other aerodynamic coefficients), is selected, and then, the method is applied to find the form factor that minimizes the percentage relative error. The results obtained are shown in Tables

Results obtained through the FFSQP method (1).

Centers’ combination C(5,5) | |||
---|---|---|---|

Airfoil | UBI_03_016 | DAE-21 | |

FFSQP 3.7 | Form factor | 1.45 | 0.28 |

REL.P | 4.40% | 6.36% | |

GT [s] | 4.125 | 1.09 | |

Form factor = 0 | REL.P | 16.51% | 8.91% |

GT [s] | 0.0156 | 0.0156 |

Results obtained through the FFSQP method (2).

Centers’ combination C(5,15) | |||
---|---|---|---|

Airfoil | UBI_03_016 | DAE-21 | |

FFSQP 3.7 | Form factor | 0.185 | 0 |

REL.P | 1.823% | 2.22% | |

GT [s] | 4.63 | 0.375 | |

Form factor = 0 | REL.P | 2.04% | 2.22% |

GT [s] | 0.047 | 0.047 |

Results obtained through the FFSQP method (3).

Centers’ combination C(5,30) | |||
---|---|---|---|

Airfoil | UBI_03_016 | DAE-21 | |

FFSQP 3.7 | Form factor | 0 | 0.007 |

REL.P | 0.73% | 1.28% | |

GT [s] | 4.86 | 23.19 | |

Form factor = 0 | REL.P | 0.73% | 2.03% |

GT [s] | 0.094 | 0.094 |

To verify the benefits of the application of the FFSQP method, Figure

REL.P versus form factor. (a) UBI_03_016-C(5,5), (b) UBI_03_016-C(5,15), (c) UBI_03_016-C(5,30), (d) DAE-21-C(5,5), (e) DAE-21-C(5,15), and (f) DAE-21-C(5,30).

The FFSQP method makes it possible to achieve the local minimum of the function with the cost of increasing the required processing time, since several iterations are required until the method converges to the desired final solution. It is up to the user to assess the benefits of having greater precision at the cost of increased generation time. Note that once the function is generated it can provide infinite approximate solutions of the aerodynamic coefficients, depending on the input data entered, since the function needs to be generated only once.

Since the problem has a finite set of data points dispersed over a predefined domain, there is no continuity between any two consecutive points, so it is up to the applied approximation function to estimate its shape. In most situations, it is not enough to obtain a low interpolation error, that is, a low error in the known data points, and it is also necessary to have a realistic evolution between any two consecutive points. To assess the quality of the approximation resulting from the dataset, the concentration of the data points increased within the previously defined domain, again using XFOIL to get P(13,251), or 3263 sampling points, with a uniform step of 50000 and 0.1 for Re and

By knowing the exact value (in some locations) of the neighboring data points used to generate the functions, it is possible to compare those with the values obtained using the approximations methods and find out whether the methods provide or not a good approximation. In Figure

Representation of the known data points.

While the set of known points has increased, the points to generate the functions have not. Therefore, it is required to change the adopted nomenclature. When analyzing the mean relative error for all points represented in yellow in Figure

Looking at the results obtained and only considering the multiquadric functions with form factors equal to 0, 0.1 and 0.15, it was observed that the methods of piecewise linear multivariable interpolation and multiquadric functions provided very similar approximations. While the piecewise linear multivariable interpolation approach yielded a 4.9%, 16.1%, and 1.9% REL.P.A, the multiquadric functions generated minimum values around 5.7%, 15.5%, and 3.5% for the lift, drag, and pitching moment coefficients, respectively.

For certain multiquadric functions with certain centers combinations (keeping a constant form factor equal to zero), the absolute error (ABS) across the domain between the multiquadric and the multivariable interpolation functions was compared. In order to identify the regions of the domain where one of the errors prevailed, the difference between the modulus of the absolute error of the multiquadric function and the modulus of the absolute error of the multivariable interpolation function (ABSD) was calculated, and the value obtained through the predefined domain is represented in Figure

Difference between the ABS of the multiquadric function and the ABS of multivariable interpolation. (a) _{l}_{l}_{l}_{d}_{d}_{d}_{m}_{m}_{m}

There is a clear dominance of the piecewise linear multivariable interpolation function compared to the generated multiquadric functions, since the latter exhibits larger areas of reddish colors that reflect the larger approximation errors obtained with the multiquadric functions. The only zone of the domain where the multiquadric functions have some dominance corresponds to the range of low Re between [75000, 225000], where the need to increase the concentration of points to define the function was identified, since in this zone the largest absolute errors occur in both functions. The advantage of the multiquadric functions (regarding the accuracy obtained) compared to the piecewise linear multivariable interpolation is clear when there are few sample points in areas that typically require a larger point concentration. It is not clear that the increase in the number of centers used to define the multiquadric function improves the approximation along the domain, though they typically improve the approximation at the interpolation points used to define the function.

In order to verify if the results obtained in the previous analysis are in fact a characteristic of the representation of the aerodynamic coefficients of airfoils, the set of points used to define the multiquadric and the piecewise linear multivariable interpolation function (abbreviated by M.I.) was modified in Table

Changing the input data P(Re,

DAE-21 multiquadric functions versus piecewise linear multivariable interpolation | |||||||||
---|---|---|---|---|---|---|---|---|---|

Data | Lift coefficient | Drag coefficient | Pitching moment coefficient | ||||||

Multiquadrics | M.I. | Multiquadrics | M.I. | Multiquadrics | M.I. | ||||

C(Re, | REL.P.A(%) | REL.P.A(%) | C(Re, | REL.P.A(%) | REL.P.A(%) | C(Re, | REL.P.A(%) | REL.P.A(%) | |

P(9,26) | C(9,15) | 2.60 | 2.62 | C(9,17) | 11.88 | 11.72 | C(9,25) | 1.76 | 1.59 |

P(5,51) | C(5,44) | 5.28 | 4.89 | C(5,30) | 15.71 | 16.07 | C(5,13) | 3.58 | 1.88 |

P(13,26) | C(13,16) | 1.75 | 1.76 | C(13,25) | 10.19 | 10.10 | C(13,25) | 1.19 | 1.19 |

P(9,51) | C(9,33) | 2.59 | 2.61 | C(9,32) | 11.09 | 11.19 | C(9,42) | 3.45 | 3.23 |

With the change of the data points used to define the multiquadric and multivariable interpolation functions, as shown in Table

Once the centers’ combinations and form factor influence on the accuracy of the multiquadric functions has been analyzed, some combinations regarding their required processing time will be tested and compared with the piecewise linear multivariable interpolation methods and through a data collection directly from XFOIL.

First of all, it is necessary to define some parameters that aim to assist this analysis. The time for collecting and reading the data used to define the approximation functions will be abbreviated by CT and RT (in seconds), the generation time of the multiquadric functions is defined by GT (this step is not performed by the piecewise linear multivariable interpolation method), and finally the time required to actually obtain solutions of the aerodynamic coefficients (depending on the input data associated with Re and

To compare the computation time associated with the three approaches under analysis, the DAE-21 airfoil was selected, with the domain and the set of points previously used (P(5,51)) to define the functions, selecting three combinations of random centers and complemented with a form factor equal to zero, to define the multiquadric functions. The time required for the multiquadric functions, piecewise linear multivariable interpolation function and XFOIL, to perform

The use of XFOIL, besides not guaranteeing the convergence for certain situations, shows a very high calculation time when compared to the proposed alternatives. Although this method does not require initial time to collect and read data, it becomes time consuming for optimization problems where many function calculations are required. It was possible to verify that, for approximately 225 different calculations, the use of both the piecewise linear multivariable interpolation method and the multiquadric defined functions became beneficial when compared to the XFOIL approach.

In Table

Results obtained from the approximation functions.

DAE-21 airfoil-CL/CD/CM-form factor = 0 | |||
---|---|---|---|

Function | REL.P | CT (s) | GT (s) |

Function MQ C(5,5) | 8.9%/24.2%/7.7% | 25.28 | 0.047 |

Function MQ C(5,25) | 1.9%/1.1%/0.9% | 25.28 | 0.141 |

Function MQ C(5,35) | 0.4%/0.7%/0.4% | 25.28 | 0.281 |

Multivariable interpolation | ∼0% | 25.28 | Not applicable |

Number of calculations versus computation time with two variables for all studied methods. (a) Overall view. (b) Detailed view.

The aerodynamic coefficients of airfoils depend on set parameters to be estimated. In wing optimization studies, it is common to analyze different flap sizes and deflections at different design points, so it becomes a slow process to adjust these parameters in XFOIL each time a change is requested. The implemented alternative is the creation of continuous functions whose solution (value of the aerodynamic coefficients) is dependent on the following input variables for the whole domain:

The dataset P(5,51,5,5) was selected and applied to the UBI_03_016 airfoil to define the multiquadric and multivariable interpolation functions within the following limits:

Thus, increments of 150,000, 0.5, 5, and 2.5 for Re, _{flap}, and _{flap}, respectively, were used to define the known data points in the given intervals.

In order to represent the influence of the number of centers in the REL.P parameter of the given multiquadric function and taking into account the obtained and analyzed data, only the situation where C(Re,

REL.P versus centers’ combination C(Re,

The time required to perform _{flap}, and _{flap} were randomly selected for each performed calculation using the approximation methods. However, when using XFOIL, due to interface limitations the values of _{flap}, and _{flap} were kept constant while different Re were assigned within the bounds of its domain.

Since the set of sampling points used to define each function increased to 6375 points, which makes up a total of 19125 sample points for the three approximation functions associated with the three different aerodynamic coefficients, the time required to collect all the points using the XFOIL (CT) also increased, as well as the time for reading the data (RT). Since this problem has a higher space dimension and more data points, the size of the support matrices needed to create the multiquadric functions increases, which requires more time to perform all the operations needed to obtain the function coefficients, which in turn implies an increase in GT. Once again, both CT and RT are only performed once to obtain and read the 19,125 sample points, both in multivariable interpolation and in multiquadric functions. The latter has also to complete GT relative to the time required to generate the three approximation functions, which is performed only once before the start of the ST phase. The results are shown in Table

Results for the four-variable case obtained from the approximation functions.

UBI_03_016 airfoil-CL/CD/CM | ||||
---|---|---|---|---|

Function | REL.P | CT [s] | RT [s] | GT [s] |

Function MQ C(3,3,5,5) | 16.1%/55.9%/12.2% | 646.4 | 1.797 | 7.359 |

Function MQ C(5,5,5,5) | 11.3%/21.2%/4.8% | 646.4 | 1.797 | 57.93 |

Function MQ C(5,10,5,5) | 5.4%/6.6%/1.9% | 646.4 | 1.797 | 237.93 |

Multivariable interpolation | ∼0% | 646.4 | 1.797 | Not applicable |

In this situation, since CT and RT are higher and more input data exists, the use of the approximation methods only becomes beneficial after approximately 6000 calculations, as can be seen from Figure

Number of calculations versus computation time with four variables for all studied methods. (a) Overall view. (b) Detailed view.

If the user is only interested in calculating a single aerodynamic coefficient, the sampling points associated with that same function are read and only a single function associated with that specific aerodynamic coefficient is generated, thus requiring less computing time. However, XFOIL always calculates all three aerodynamic coefficients for a given set of input data, regardless of the user’s settings. Another feature that favors the use of the approximation functions over XFOIL direct analysis is that the latter sometimes does not guarantee convergence for certain input data, unlike the approximation functions which are defined for the whole domain.

For the two-dimensional case, the slopes of linear regressions in Figure

This work studied the application of multiquadric functions to represent the aerodynamic lift, drag, and pitching moment coefficients of airfoils. The method can approximate a given dataset of any spatial dimension, automatically calculating the location of the center points (as many as required) and also calculating the form factor according to the chosen method. It was concluded that, with the increase of center points, it was possible to obtain better approximations, with only a few exceptions. It was also realized the enormous influence of the form factor, which also plays a relevant role in the quality of the approximation, since it is directly responsible for the ill-conditioning of the interpolation matrix used in the generation of the multiquadric function. The method using FFSQP was able, in most situations, to converge to the best possible form factor value (with lower associated REL.P), although this requires that the user specifies the number of centers beforehand, without the guarantee of obtaining the best solution for this combination, since function REL.P versus form factor may have a local minimum in the starting point neighborhood.

To assess the precision of the multiquadric approximation, it was compared with the piecewise linear multivariable interpolation method, since this is a method widely applied in the approximate representation of the aerodynamic coefficients. Considering the quality of the approximation at points outside the applied dataset, but within the selected domain, it was concluded that, by varying the set of points used to define the functions, there was no clear advantage of any of the approaches. This analysis was performed in a two-dimensional space, but similar results are expected to be obtained for larger dimensions.

The processing time required to perform a certain number of calculations (to obtain the value of the three aerodynamic coefficients) based on a random set of input data was evaluated. For the two-dimensional case, it was observed that the use of the approximation functions is preferred over XFOIL after about 240 calculations, whereas for the four-dimensional case, the break-even point increases to approximately 6000 calculations. Regarding the computation time, the multivariable interpolation approach has shown to be faster compared to multiquadric functions. The increase in the spatial dimension of the problem caused the difference in ST between the multiquadric functions and the multivariable interpolation approach to increase, in favor of the latter, even with a reduced percentage of centers applied to develop the multiquadric function (which in turn worsens the quality of the generated approximation).

Multiquadric functions are still experiencing increased widespread application across many areas of science and engineering. In order to improve the process of defining the multiquadric functions with increased precision, some open issues still need to be addressed: new techniques for placement of center, application of a variable form factor for each center to which it is applied, methods of direct calculation of the form factor, methods that aim to lower the conditioning number of the interpolation matrices with the gradual increase of the form factor, and understanding the best techniques to define the domain and the set of points to apply in the problem.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this article.

This work was supported by the Universidade da Beira Interior.