Use of Mini-Max Approximation [611979]

Use of Mini-Max Approximation
for the Determination of the Dependency
Between the Input and Output Parameters
for the Automotive Parts Obtained by Fused
Deposition Modeling (FDM)
Nicolae-Doru St ănescu(&), Maria-Luiza Be șliu-Gherghescu,
and Ștefan Tabacu
University of Pite ști, 110040 Pite ști, Romania
[anonimizat]
Abstract. Fused Deposition Modeling (FDM) is one of the most used 3D
technology for rapid prototyping of different parts. The approach consists in
determination of the statistical signi ficance of different parameters using the
Analysis of Variance (ANOVA) technique followed by the determination of thecorrelation laws between the input and output data. These laws are usually
described by polynomials obtained with the aid of the Least Squares Method.
The use of this method is preferred due to its simplicity, but it has the disad-vantage that the error is not known (the only thing we know is that the obtained
polynomial leads to the smallest error in the sense of the Least Squares). In our
paper we purpose a new approach by using the Tchebyshev polynomials whichlead to the mini-max approximation of the law. In this way, the error is always
zero in the division points and, moreover, the great advantage is that the error is
the smallest one for the entire interval of the input data. The theoretical dif ficulty
of this approach consists in the determination of the division points, while thepractical disadvantage is the use of irrational division points for the input data.
The formula obtained for the law of variation of the output with respect to the
input is the most accurate one in the class of certain polynomials.
Keywords: FDM
/C1Tchebyshev polynomials /C1Mini-max approximation
1 Introduction
Nowadays, Fused Deposition Modeling (FDM) became one of the most used 3D
techniques for rapid prototyping [ 1]. The general approach for the determination of the
statistical signi ficance of different parameters consists in using the Analysis of Variance
(ANOVA) [ 3,4,8–10,13].
The laws which describe the dependencies between input and output parameters are
obtained by using equidistant interpolation nodes and the Least Squares Method,resulting polynomials of different degrees (usually second degree) [ 2,3,14], Central
©Springer Nature Switzerland AG 2019
N. Burnete and B. O. Varga (Eds.): AMMA 2018, PAE, pp. 1 –7, 2019.
https://doi.org/10.1007/978-3-319-94409-8_45

composite design [ 10], Finite element method [ 7], Finite difference method [ 6],
gradient-based optimization algorithm Method of Moving Asymptotes [ 5] etc.
In this paper we will present another approach based on Tchebyshev polynomials
[12], in which the interpolation nodes are not equidistant, using the mini-max
approximation. These polynomials have the less deviation from the exact function on
the interval. Based on this approach new polynomials laws between the input and
output parameters are determined. For these polynomials the deviation between the real
and approximate values is the smallest one. A numerical example will conclude the
paper.
The paper is structured as follows: in the second paragraph we introduce the
Tchebyshev polynomials, the deviation from zero for a polynomial and we recall the
Tchebyshev theorem; third paragraph is dedicated to mini-max principle. Further on,
we present a numerical example. The paper ends with conclusions.
2 The Tchebyshev Polynomials
We consider the polynomials
TnxðȚ ¼ cosnarccos x ðȚ : ð1Ț
In these conditions one may prove the following statements:
–TnxðȚis a polynomial of nth degree for which the dominant coef ficient is equal to
2n/C01;
–the equation TnxðȚ ¼ 0 has nreal distinct equations and these equations are all
situated in the interval /C01;1 ½/C138 ;
–the minimum value of the polynomial TnxðȚ, for x2/C0 1;1 ½/C138 is equal to /C01 and it is
obtained for
xl¼cos2lț1 ðȚ p
n/C18/C19
;l¼0;1;… ;n/C01
2/C20/C21
ț1; ð2Ț
–the maximum value of the polynomial TnxðȚ, for x2/C0 1;1 ½/C138 is equal to 1 and it is
obtained for
xk¼cos2kp
n/C18/C19
;k¼0;1;… ;n
2hi
ț1: ð3Ț
In the expressions ( 3) and ( 4) the symbol a½/C138means the integer part of a, that is, the
greatest integer value which is less or equal to a.
The Tchebyshev polynomials are de fined by the relation
KnxðȚ ¼1
2n/C01TnxðȚ ;x2/C0 1;1 ½/C138 ;n2N: ð4Ț391 N.-D. St ănescu et al.

We call the deviation from zero of the polynomial QxðȚon the interval a;b½/C138 , the
expression
dev
x2a;b½/C138QxðȚ ¼ max
x2a;b½/C138QxðȚjj : ð5Ț
The Tchebyshev theorem states:
(i) The deviation from zero of the polynomial QxðȚ ¼ xnța1xn/C01ț…anon the
interval 0 ;1½/C138 cannot be less than1
2n/C01and it is equal to1
2n/C01for the Tchebyshev
polynomial KnxðȚ;
(ii) there exists a unique polynomial of nth degree with the dominant coef ficient
equal to 1, the deviation of which from zero on the interval /C01;1 ½/C138 is equal to1
2n/C01.
Obviously, this polynomial is the Tchebyshev polynomial KnxðȚ.
Some Tchebyshev polynomials and their roots are given below:
–K1xðȚ ¼ cos arccos x ðȚ ¼ x, for which the root is x1¼0;
–K2xðȚ ¼1
2/C2cos 2 arccos x ðȚ ¼ x2/C01
2, with the roots x1¼/C0ffiffi
1
2q
andx2¼ffiffi
1
2q
;
–K3xðȚ ¼1
4/C2cos 3 arccos x ðȚ ¼ x3/C03
4x, the roots of which being x1¼/C0ffiffi
3
4q
,
x2¼0,×3¼ffiffi
3
4q
;
–K4xðȚ ¼1
8/C2cos 4 arccos x ðȚ ¼ x4/C0x2ț1
8, having the roots x1¼/C0ffiffiffiffiffiffiffiffiffiffi
2țffiffi
2p
4q
,
x2¼/C0ffiffiffiffiffiffiffiffiffi
2/C0ffiffi
2p
4q
,x3¼ffiffiffiffiffiffiffiffiffi
2/C0ffiffi
2p
4q
,x4¼ffiffiffiffiffiffiffiffiffiffi
2țffiffi
2p
4q
.
3 Mini-Max Principle
Let us consider a function f:a;b½/C138 ! R, where a;b½/C138 is an interval of the real axis. In
addition, let PnxðȚbe a nth degree polynomial which is an approximate for the function
f. We want to determine the polynomial PnxðȚsuch that
max
x2a;b½/C138fxðȚ /C0 PnxðȚ jj ¼min : ð6Ț
In addition, we ask the polynomial PnxðȚto pass through the interpolation nodes xi,
yi¼fxiðȚ,i¼0;n.
If we consider that the interval a;b½/C138 is exactly the interval /C01;1 ½/C138 , then it results
that the interpolation nodes have to be the roots of the Tchebyshev polynomials
Knț1xðȚ, that is,
u0¼cos2nț1ðȚ /C0 1
2nț1ðȚp/C18/C19
;u1¼cos2n/C01ðȚ /C0 1
2nț1ðȚp/C18/C19
;… ;un¼cosp
2nț1ðȚ/C18/C19
:
ð7ȚUse of Mini-Max Approximation for the Determination 3 92

In the general case of the interval a;b½/C138 the interpolation nodes read
xi¼b/C0a
2uițbța
4 Numerical Example
In the case of fused deposition modeling (FDM) the interval a;b½/C138 can be considered as
0;90/C14½/C138 or 0 ;p
2/C2/C3
. Working, as usual, in degrees, we may take a;b½/C138 ¼ 0;90½/C138 . In this
situation, it results
xi¼45uiț45: ð9Ț
Considering the fourth degree Tchebyshev polynomial, one obtains the roots of this
polynomial as
u1¼/C0ffiffiffiffiffiffiffiffiffiffi
2țffiffi
2p
4q
¼/C00:92388 ;u2¼/C0ffiffiffiffiffiffiffiffiffi
2/C0ffiffi
2p
4q
¼/C00:38628 ;
u3¼ffiffiffiffiffiffiffiffiffi
2/C0ffiffi
2p
4q
¼0:38628 ; u4¼/C0ffiffiffiffiffiffiffiffiffiffi
2țffiffi
2p
4q
¼0:923888
<
:ð10Ț
and the interpolation nodes
x1¼3:42542/C14;x2¼27:77925/C14;x3¼62:22075/C14;x4¼86:57458/C14: ð11Ț
In reference [ 11] the authors obtained a multi-polytropic expression for the
dependency of the Young modulus in function of the raster angle,
E¼1:79/C00:0274 h0:7: ð12Ț
The values given in Table 1are obtained with the aid of this multi-polytropic
formula. The nodes are the roots of the fourth order Tchebyshev polynomial and some
equidistant ones, which can be easily obtained on a regular printing device used in
Fused Deposition Modeling.
In Table 2we present the values for Young ’s modulus calculated with the previous
formula ( E), using Lagrange fourth degree polynomial with equidistant nodes, and with
the aid of the fourth degree Tchebyshev polynomial. The DELandDETare the errors.
The formulae used for Table 2are
E¼1:79/C00:0274 h0:7;DEL¼EL/C0E;DET¼ET/C0E: ð13Ț
The maximum deviation for the Lagrange polynomial with equidistant interpolation
nodes (0/C14,3 0/C14,6 0/C14, and 90/C14)i s dev L¼0:02197 GPa ½/C138 , obtained for an angle
hL¼4:80016/C14, while the maximum deviation for the interpolation with the fourth393 N.-D. St ănescu et al.
: ð8Ț2

degree Tchebyshev polynomial is dev T¼0:02167 GPa ½/C138 , corresponding to the angle
hT¼0/C14. As one expected, the minimum deviation is obtained for Tchebyshev
polynomials.
5 Conclusions
This paper presents a new method by which one may determine the dependency
between the input and output parameters in the most general case. The approach
consists in the use of the mini-max principle based on the Tchebyshev polynomials.
The greatest advantage of this method is that it assures the smallest deviation from the
real values in the class of polynomial functions. The major disadvantage is the use of
particular values of the interpolation nodes. These particular interpolation nodes are not
equidistant and, in general, they have irrational values.
If one does not use polynomial expressions, than one may obtain some other
dependency function between the input and output parameters, but one looses the
advantages of the polynomial expressions: easy of calculation, the use of ANOVA, the
possibility of determination of some correlation parameters etc. because the mathematic
apparatus is not developed in the general case of arbitrary function.Table 1. Values for Young ’s modulus depending on the raster angle.
Raster angle h/C14½/C138Young modulus EGPa½/C138
0 1.79
3.42542 1.73
27.77925 1.51
30 1.49
60 1.31
62.22075 1.30
86.57458 1.17
90 1.15
Table 2. Values for calculated Young ’s modulus using Lagrange and Tchebyshev polynomials.
h EE L ET DEL DET
0 1.79 1.79 1.768 0.000 −0.022
3.42542 1.73 1.746 1.73 0.021 0.000
27.77925 1.51 1.507 1.51 −0.002 0.000
30 1.49 1.49 1.494 0.000 0.004
45 1.396 1.391 1.396 −0.005 0.000
60 1.31 1.31 1.312 0.000 0.002
62.22075 1.30 1.299 1.30 −0.001 0.000
86.57458 1.17 1.170 1.17 0.000 0.000
90 1.15 1.15 1.150 0.000 0.000Use of Mini-Max Approximation for the Determination 394

The numerical example presented in the paper highlights these aspects. A greater
degree for the Tchebyshev polynomial leads to a smaller value for the deviation and
also to a more dif ficult process to obtain the experimental results based on which the
correlation expression is deduced. It is known that the usual printing devices used in
FDM offers only limited values for the raster angles. If one wishes a more precise
dependency between the input and output parameters, then the use of Tchebyshev
polynomials is a good candidate.
Acknowledgments. This work was supported by a grant of the Romanian Ministry of Research
and Innovation, CCCDI-UEFISCDI, project number PN-III-P1-1.2-PCCDI-2017-0224 / 77
PCDI/2018 within PNCDI III.
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