The Determination of the Dependency between the Input [611983]

293
The Determination of the Dependency between the Input
and Output Parameters in Fused Deposition Modeling
(FDM) Using Multi Polytropic Functions
Nicolae – Doru Stănescu1, Maria – Luiza Beșliu – Gherghescu, Ștefan Tabacu1, Dinel
Popa1, Alin Rizea1 and Monica Iordache1
1 University of Pitești, Pitești 110040, Romania
[anonimizat]
Abstract. In the process of determination of the input-output correlations for
FDM technology, researchers usually consider polynomial ( 
m
iiixay
0) or
simple polytropic functions (pCxy). For these functions the method of least
squares leads to linear systems from which the unknowns (ia in the first case,
C and p in the second one) are easily determined. The approximations given
by polynomial and polytropic functions do not offer the possibility for the ex-
trapolation of the results for other intermediate or extrapolated points, or the ob-
tained results are very different comparing to the experimental ones. For these
reasons in this paper we purpose the use of the multi polytropic functions
(
m
ip
iixC Cy
10 ), in which the polytropic exponents have real non-zero
values. The obtained system from which the polytropic coefficients and expo-
nents is no longer a linear one. Because of the convergence conditions of the
Newton-Raphson method, this method is not a valid one for the solving of this
system in the general case. The solving of the system is performed with the aid
of the gradient method. An example highlights the theory.
Keywords: FDM, Polytropic Functions, Nonlinear System of Equations, Gra-
dient Method.
1 Introduction
The understanding of the mechanical properties of a part obtained by FDM is a com-
plex task, because this is influenced by many production parameters. Some of these
parameters are: layer thickness, part build orientation, raster angle, raster width, air
gap etc.
In [1] is described the mechanical behavior of FDM parts using the classical lami-
nate theory (CLT). The results are compared to those experimentally measured. Other
authors [2] investigate the effect of manufacturing parameters on dynamic mechanical

294
performance of the processed parts by FDM using computer-generated optimal design
based on I-optimal design. It is proved [3] that properties of FDM built parts exhibit
high dependence on process parameters. It is important to study the effect of process
parameters to the resistance to compressive loading. The study develops a statistically
validated predictive equation, while the compressive stress is predicted using artificial
neural network. Reference [4] proved that on-edge and flat orientation show the high-
est strength and stiffness, ductility decreases as layer thickness and feed rate increase,
the mechanical properties increase as layer thickness increases and decrease as the
feed rate increases for the upright orientation, and the variations in mechanical prop-
erties with layer thickness and feed rate are of slight significance for on-edge and flat
orientations, except in the particular case of low layer thickness. Costa et al. [5] de-
veloped an analytical solution to the transient heat conduction developing during fil-
ament deposition which is used to estimate whether contiguous filament segments
adhere adequately to each other prior to solidification. The predicted and experimen-
tal data are compared and show a very good agreement. Other researchers [6] measure
the residual stress in FDM parts made of ABS employing the hole-drilling method. In
[7] is presented a procedure to select the correct parameters in order to reduce the
build time and to reduce feedstock material consumption while maintaining high dy-
namic mechanical properties. The authors use Q-optimal response surface methodol-
ogy.
Usually, the dependencies between different parameters are approximated by poly-
nomial functions using the least squares method [5, 8, 9, 10, 11], the unknown being
determined by classical methods or by genetic algorithms.
2 The Gradient Method
Let us consider de system 0xf, where
  T
1 x x fsf f  , s
sxx R x  T
1 , sR 0  T00 (1)
and let 0x be an approximation of the solution of this equation.
For a pair of vectors x and y in sR, we define the scalar product , by
yxyxT,  . (2)
Let qx an approximation of the solution obtained at the step q. The new ap-
proximation of the solution, corresponding to the step 1q is given by [12]
       q q
qq qxfxJ x xT12, (3)
where

295
        
                                
q q q q q qq q q q
q
xfxJxJxfxJxJxfxJxJxf
T TT
,,2 (4)
and J is the Jacobian
 








ss ss
xf
xfxf
xf

11
11
J . (5)
3 Case of a Simple Polytropic Function
This case considers the approximation in the form pCxy, in which x and y are
positive real values, C is real, while is real non-zero exponent. The previous expres-
sion may be written as xpCy ln lnln  .
Being given the sets of values 0ix and 0iy, ni ,1, 3n, the least
squares method implies the minimization of the function
 
 n
ii i xpCy f
12lnlnln (6)
or, denoting CDln,
 
 n
ii i xpDy f
12ln ln . (7)
Equating the partial derivatives with 0,
 
  


    


, 0lnln ln2, 0ln ln2
11
n
iii in
ii i
xxpDypfxpDyDf
, (8)
one gets the linear system

  n
iin
ii y xpnD
1 1ln ln , 
   n
iiin
iin
ii yx xpxD
1 12
1lnln ln ln , (9)
with the solution

296


   

n
iin
iin
iin
iin
iiin
iin
ii
x xx nx yxx y
CD
12
1112
11 1
lnlnlnln lnlnln ln
ln ,

  

n
iin
iin
iin
iiin
iin
ii
x xx nyxxy n
p
12
111 11
lnlnlnlnlnlnln
. (10)
4 Case of Sum of a Constant and a Polytropic Function
In this situation, we may write 110pxCCy , with x positive real number, 1p
non-zero real exponent, while 0C, 1C and y are real numbers.
We have to minimize the expression
 
 n
ip
i i xCCyf
12
101. (11)
Considering the partial derivatives
 
  
  



   


n
ip
ip
i oin
ip
ip
i oin
ip
i i
xxCCyCpfxxCCyCfxCCyCf
11
1 1
111
1110
0
11111
2, 2, 2
(12)
and equating them to zero, we obtain the following system of three non-linear equa-
tions with three unknowns (0C, 1C and 1p)
 
 
 



       



   
. 0 ,,, 0 ,,, 0 ,,
11
1
1122
1
11
10 11031 12
1
10 11021 110 1101
1 1 11 1 11
n
ip
iin
ip
in
ip
in
ip
iin
ip
in
ip
in
iin
ip
i
xyCxCxCCpCCfxy xCxCpCCfy xCnCpCCf
(13)
For the calculation of the Jacobian we need to calculate

297
nCf
01, 
n
ip
ixCf
1 11 1, 
n
ip
ixpCpf
11
11
11 1, (14)




  
 


 
, 2, ,
11
1
112
11
11
1
1212
12
1 02
1 1 11 1
n
ip
iin
ip
in
ip
i on
ip
in
ip
i
xyp xpC xpCpfxCfxCf
(15)
    
 


     
  





. 112 1, 2 ,
12
11112
12
1
12
110
1311
112
1
11
0
13
11
1
03
11 11 1 1 1
n
ip
iin
ip
in
ip
in
ip
iin
ip
in
ip
in
ip
i
xy pCx pCx pCCpfxy xC xCCfxCCf
. (16)
It results the matrices
 












13
13
0312
12
0211
11
01
pf
Cf
Cfpf
Cf
Cfpf
Cf
Cf
J .  





321
fff
f (17)
5 The General Case of Multi Polytropic Functions
In this case 
m
jp
jjxCCy
10 , where x is a real positive number, 0C, 1C, …,
mC are real constants, while 1p, …, mp are non-zero real exponents. In addition
nm.
We have to minimize the function
 
 



 n
im
jp
ij ijxCCy f
12
10 . (18)
Equating to zero the partial derivatives

298












 







 



 
   

  
, ,1 , 0 2, ,1 , 0 2, 0 2
11
101 101 10
0
mk xxCCyppfmk xxCCyCfxCCyCf
n
ip
im
jp
ij i k
kn
ip
im
jp
ij i
kn
im
jp
ij i
kjkjj
(19)
one obtains a system of 12m non-linear equations with 12m unknowns (0C,
1C, …, mC, 1p, …, mp),




       
  

 
     
. ,1 ,0, ,1 , 0, 0
11
1 11
11
0 ,31 1 1 10 ,21 1 10 1
mk xy xxC xCfmk xy xxC xCfy xC nCf
n
ip
iin
im
jp
ip
ijn
ip
i kn
ip
iin
im
jp
ip
ijn
ip
i kn
iin
im
jp
ij
k kj kk kj kj
(20)
The partial derivatives are ( mk ,1, ml ,1)
nCf
01, 
n
ip
i
llxCf
11, 
n
ip
ill
llxCppf
11 1, (21)

 


   
   




, , ,
,
11
11
1,1
0,21,2
1 0,2
kln
ip
iiln
ipp
i klln
iklp
il
lkn
ipp
il
lkn
ip
ik
l kl lkl k
xyp xppC xpCpfxCCfxCf
(22)
   
 


  
  



, 21 1, ,
1,212
,
12
0,311 ,3
11
0,3
n
iklp
ii ln
ipp
i kllkln
ip
i l
lkn
ipp
il
lkn
ip
ik
lkl lkl k
xy px ppC x pCpfxCCfxCf
(123)
where kl, is the Kronecker symbol with the values 1 for kl and 0 for kl.

299
The matrices J and f read
 


























mm m
mm m mm mmm m
mm m mm mm m
pf
pf
Cf
Cf
Cfpf
pf
Cf
Cf
Cfpf
pf
Cf
Cf
Cfpf
pf
Cf
Cf
Cfpf
pf
Cf
Cf
Cf
,3
1,3,3
1,3
0,31,3
11,31,3
11,3
01,3,2
1,2,2
1,2
0,21,2
11,21,2
11,2
01,21
11 1
11
01
     
J ,  





mm
fffff
,31,3,21,21

f . (24)
6 Example
The experimental values obtained for a sample of ABS (acrylonitrile-butadiene-
styrene) are given below [1]
Table 1. Values for Young's modulus depending on the raster angle.
Raster angle 0  Young modulus GPa E
0 1.79
30 1.49
60 1.31
90 1.15
Choosing a dependency in the form of 110pCCE  , one gets the system (13).
We start with the approximation of the solution described pin 20
0C ,
1.00
1C , and 10
1p . The solution given by the gradient method is as fol-
lows 79.10C , 0274.01C , 7.01p.
7 Conclusions
This paper proposes a new approach for the determination of the approximate expres-
sion of variation of the output function of input data in FDM. The approach is based
on the use of multi polytropic functions which offers a better approximation of the
experimental results. The parameters are determined with the aid of the gradient
method. This method is not conditioned by the convergence conditions as the New-
ton-Raphson method. The disadvantages of the method consist in a more complicate

300
formula to pass from one step to the next one, and the number of iteration which is
much larger than the number in Newton-Raphson method.
In fact, the gradient method can be used for any formula which approximates the
experimental data, including here the rational functions of multi polytropic functions.
The obtained formula has a better precision than those offered by the references
which are limited to second order polynomials.
Acknowledgment
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
References
1. Casavola, C., Cazzato, A., Moramarco, V., Pappalettere, C.: Orthotropic mechanical prop-
erties of fused deposition modelling parts described by classical laminate theory. Materials
and Design 90, 453–458(2016).
2. Mohamed, O.A., Masood, S.H., Bhownik, J.L.: Characterization and dynamic mechanical
analysis of PC-ABS material processed by fused deposition modelling: An investigation
through I-optimal response surface methodology. Measurement 107, 128–141 (2017).
3. Sood, A.K., Ohdar, R.K., Mahapatra, S.S.: Experimental investigation and empirical mod-
elling of FDM process for compressive strength improvement. Journal of Advanced Re-
search 3, 81–90 (2012).
4. Chacón, J.M., Caminero, M.A., García-Plaza, E.,. Núñez, P.J.: Additive manufacturing of
PLA structures using fused deposition modelling: Effect of process parameters on me-
chanical properties and their optimal selection. Materials and Design 124, 143–157 (2017).
5. Costa, S.F., Duarte, F.M., Covas, J.A.: Estimation of filament temperature and adhesion
development infused deposition techniques. Journal of Materials Processing Technology
245, 167–179 (2017).
6. Casavola, C., Cazzato, A., Moramarco, V., Pappalettere, C.: Residual stress measurement
in Fused Deposition Modelling parts. Polymer Testing 58, 249–255(2017).
7. Mohamed, O.A., Masood, S.H., Bhownik, J.L.: Mathematical modeling and FDM process
parameters optimization using response surface methodology based on Q-optimal design.
Applied Mathematical Modelling 40, 10052–10073 (2016).
8. Gomez-Gras, G., Jerez-Mesa, R., Travieso-Rodriguez, J.A., Lluma-Fuentes, J.: Fatigue
performance of fused filament fabrication PLA specimens. Materials & Design (in press).
9. Béraud, N., Vignata, F., Villeneuve, F., Dendievel, R.: Improving dimensional accuracy in
EBM using beam characterization and trajectory optimization. Additive Manufacturing 14.
1–6 (2017).
10. Domingo-Espin, M., Puigoriol-Forcada, J.M., Garcia-Granada, A.A., Llumà, J., Borros, S.,
Reyes, G.: Mechanical property characterization and simulation of fused deposition mod-
eling Polycarbonate parts. Materials & Design 83, 670–677 (2015).
11. Venkata Rao, R., Rai, D.P.: Optimization of fused deposition modeling process using
teaching learning-based optimization algorithm. Engineering Science and Technology, an
International Journal 19, 587–603 (2016).
12. Teodorescu, P., Stănescu, N.-D., Pandrea, N.: Numerical Analysis with Applications in
Mechanics and Engineering, John Wiley & Sons, Hoboken (2013).