Research Reports of National Institute of Technology, Nagaoka College, Vol. 59, 2024

Paper

Optimization of texture design parameters
for friction reduction based on design of experiments
and finite element fluid analysis
Doan Quang THUAN1, Takahiko KURAHASHI2, Hiroaki ARATA3
and Tetsuro IYAMA4

Mechanical Engineering, Master’s Program, Graduate School of Engineering,1

Nagaoka University of Technology
2

Department of Mechanical Engineering, Institute of GIGAKU, Nagaoka University of Technology
Mechanical Engineering, Master’s Program, Graduate School of Engineering,

3

Nagaoka University of Technology (Current affiliation: Takasago Thermal Engineering Co., Ltd.)
4

Department of Mechanical Engineering，National Institute of Technology，Nagaoka College

In this study, we present a new practical optimum design method that consists of two steps: finite element
analysis (FEA) and design of experiments. The design of experiments is used to generate approximate
evaluation functions for controlling the behavior depending on the changes in the design variables of an
object structure by finite element analyses. Here, we used a design of experiments to determine the optimal
combination of design parameters in the texture analysis for friction coefficient reduction. The effects ofthe design variables can be calculated based on an orthogonal array of design variable combinations. The
approximate evaluation functions were then generated by these effects based on the analysis of variance.
The proposed method was found to be an effective and powerful tool for the optimum design of various
practical design problems.
Key Words: Textures, Design Parameters, Friction Reduction, Design of Experiments Method

1. Introduction

performance, and use materials, lubricants, andmanufacturing processes that have lower
environmental impact. Surface modification is areasonable material-creation technology that adds theproperties required for the surface independent of theinterior and improves the performance of the material.
Surface texturing is one of the most prominent
surface modification technologies that creates“texture” by machining grooves or holes in sliding
surfaces. By applying these textures to sliding
surfaces that are adequately lubricated with oil,friction can be reduced. Surface texturing is a simple
and essential surface treatment process. Therefore,various numerical models have been employed in

Against the backdrop of the growing awareness ofenvironmental issues and the strengthening of theinternational competitiveness of the manufacturingindustry, there is a need to improve the thermal
efficiency and reliability of transportation andindustrial machinery and increase their functionality
and added value. The improvement of tribologicalproperties is a fundamental and core technology forsolving these problems and core technology and isrequired to achieve more advanced technologicalrequirements. In the development of tribologicalproperties, it is essential to reduce costs, improve

12

Optimization of Texture Design Parameters for Friction Reduction Based on Design of Experiments andFinite Element Fluid Analysis[7] The in-plane component of the rate of velocity
many studies to determine the optimal texturingchange of the fluid is negligible compared with thatparameters (shape, size, and distribution) for optimalof the out-of-plane component.
performance enhancement in terms of friction andwear. However, the large number of parameters and
𝜕
𝜕𝑝
𝜕
𝜕𝑝
∂ℎ
complexity of their combinations render determining(ℎ3 ) +(ℎ3 ) = 6𝜂𝑈 in Ω
(2)
𝜕𝑥
𝜕𝑥
𝜕𝑦
𝜕𝑦
∂𝑥
the optimum texture a challenging task, leading tocontrary conclusions.
The Reynolds equation was used as a constraint
Hence, in this study, we used a design offunction. The boundary conditions in Eq. (2) areexperiments1) to determine the optimal combination
expressed by Eq. (3). Practically, cavitation isof design parameters in the texture analysis forconsidered to occur in the negative-pressure area offriction coefficient reduction. FreeFEM++ was used
the texture; therefore, the negative pressure was set
in the finite-element fluid analysis to determine theto zero, as shown in Eq. (4).
coefficient of friction2).

2. Formulation
This study assumes fluid-lubricated sliding surfaces
in which fluid is present between two surfaces thatare in relative motion without contact. The
computational model is shown in Fig. 1. The wholedomain is defined as Ω and its boundary as Γ. The
governing equation is calculated for domain Ω, andthe boundary conditions are given in Γ.
The friction coefficient μ is obtained by dividing thefrictional force F by the load, W as shown in Eq. (1)3), 4)(a) Simplified calculation model𝜂𝑈 ℎ 𝜕𝑝∫Ω ( ℎ + 2 𝜕𝑥 ) 𝑑Ω𝐹𝜇==𝑊
∫ 𝑝 𝑑Ω

(1)Ω

where η and U are the viscosity and velocity in thex-direction of the friction surface, respectively, andare constants. h and p are the oil film thickness andpressure, respectively, and are variables.
The Reynolds equation (Eq. (2)) was used as thegoverning equation based on the Reynolds
assumption, as follows:[1] The fluid flow is laminar.
[2] The fluid is incompressible.
[3] Viscosity is constant.
[4] Gravity and inertial forces are negligible
compared to viscous forces.
[5] There is no pressure variation in the directionof oil film thickness.
[6] There is no slip between the fluid and surface.

(b) Finite-element meshFig. 1 Computational model

13

Doan Quang THUAN, Takahiko KURAHASHI, Hiroaki ARATA and Tetsuro IYAMA
Table 1 Initial condition𝑝 = 0 on Γ1(3)

𝑝 = 0 in p < 0(4)

Load W
Sliding speed U
Viscosity 𝜂 [N]Texture spacing in x direction [mm]Texture spacing in y direction [mm]Initial texture depth ℎ𝑑𝑒𝑝 [mm]Area ratio of texture [%]

10
20
0.08
6.33
6.33
0.008
50

3. Computational conditions
The Stribeck curve is shown in Fig. 2. Numericalexperiments were conducted under the conditions of
the reference factor to create a line showing thechange in the coefficient of friction, as indicated by
the blue line in Fig. 3. The vertical axis represents thefriction coefficient and the horizontal axis represents
the bearing constant. The lubrication areas weredivided into three categories: boundary, mixed, andfluid lubrications. Because the friction coefficient
was lower in the mixed lubrication area, a reference(point O) was determined in the mixed lubrication
area. Next, using the factor at point O as the reference,as presented in Table 2, the larger and smaller valuesof the texture parameters were defined as levels 1 and
2, respectively.
The phenomenon or result that is the subject of anexperiment or investigation is known to be acharacteristic. In this study, the coefficient offriction was determined using the texture parameters.
Hence, factors are those that affect these properties.
The conditions set for these factors are called thelevels. As summarized in Table 2, the texture spacingin the x-direction is defined as factor A, texture
spacing in the y-direction is defined as factor B,
texture depth is defined as factor C, and area ratio of
texture is defined as factor D. Next, we estimated the
combination of factors that reduced the coefficient offriction at point O using the design of experiments forlevels 1 and 2.

Factors
A
B

Table 2 Factors and Levels
Level Initial
Definition of Factors
1
level
Texture spacing in
76.33
x direction [mm]Texture spacing in
5.67
6.33
y direction [mm]

Level
2
5.67
7

C

Texture depth [mm]

0.009

0.008

0.007

D

Area ratio of texture
[%]

49

50

51

4. The Design of Experiments Method
To investigate the relationship between factors andcharacteristics, such as which factors affect thecharacteristics, as well as how to improve thecharacteristics by changing the factors, and what isthe value of the characteristic at that time, the factors
were changed and the data obtained were statisticallyanalyzed. However, a proper analysis cannot be
conducted if the data are obtained using a sloppy
setup. The design of experiment is a statistical toolthat provides a method for planning data collection
and analyses to obtain accurate and efficient results.
From Table 2, 128 different combinations can beobtained from multiple placements. By implementing
each of them, the combination that reduces thecoefficient of friction the most can be determined.
However, this approach is time-consuming.
Contrarily, by the design of experiment method, we
can choose eight out of 128 combinations andorganize them efficiently in an orthogonal table,
resulting in reduction in the number of times and timeof analysis.

14

Optimization of Texture Design Parameters for Friction Reduction Based on Design of Experiments andFinite Element Fluid Analysis

Fig. 2 The Stribeck curve

Fig. 3 Stribeck diagram from numerical experiment

Case 1Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8

B
1
1
1
1
2
2
2
2

A
1
1
2
2
1
1
2
2

E
1
1
2
2
2
2
1
1

D
1
2
1
2
1
2
1
2

Table 3 L8 orthogonal array
C Combination
A×D
B×D
1
1
1
A₁B₁C₁D₁
2
2
2
A₁B₁C₂D₂
1
2
2
A₂B₁C₁D₂
2
1
1
A₂B₁C₂D₁
2
1
2
A₁B₂C₁D₂
1
2
1
A₁B₂C₂D₁
2
2
1
A₂B₂C₁D₁
1
1
2
A₂B₂C₂D₂

Friction coefficient
0.023883
0.017587
0.012071
0.011518
0.022609
0.017737
0.013542
0.011952
Summation T=0.130898

Table 4 Sum and average of each level(𝑇[𝑘]1 : Sum of level 1, 𝑇[𝑘]2 : Sum of level 2, [k]: factors)
𝑇[𝑘]1
Average𝑇[𝑘]2
AverageB
0.065058
0.016265
0.065840
0.016460

A
0.081815
0.020454
0.049083
0.012271

E
0.066963
0.016741
0.063935
0.015984

D
0.072105
0.018026
0.058793
0.014698

15

A×D
0.065643
0.016411
0.065255
0.016314

B×D
0.069962
0.017490
0.060936
0.015234

C
0.066678
0.016670
0.0642200.016055

Doan Quang THUAN, Takahiko KURAHASHI, Hiroaki ARATA and Tetsuro IYAMA
averaged the results for the friction coefficient whenfactor A was one (level 1), and similarly summed andaveraged the results for the friction coefficient whenfactor A was two (level 2). In this case, we focused
only on factor A and ignored all other factors. The
following tables and figures present graphicalrepresentations of these sums.
As shown in Fig. 5, the friction coefficient value offactor D at level 2 was lower than that at Level 1,
whereas the friction coefficient value of factor B at
Level 2 was slightly higher than that at Level 1.
Basically, factors B and D are not dependent of eachother but have synergistic or offsetting effects.
However, factors A and D had the same high values
at Level 1 and the same low values at Level 2.
Fig. 6(a) shows the duality chart between B and D,
and Fig. 6(b) shows the duality chart between A andD. As shown, the two lines intersect in the top figure
while they are parallel in the bottom figure. If there
was no interaction between the two factors, the twolines would be parallel; however, with interaction,
they would not be parallel. This is based on the factthat in the absence of interaction, each factor isindependent (they do not influence each other).
Basically, the interaction can be understood as theeffect of one factor changing with the other.
After determining the interactions between factors,
we proceeded with an analysis of variance(ANOVA) to determine whether these factors andinteractions affect friction reduction.

4.1 Assign factors
In this study, there were two levels and four factors:
texture spacing in x-direction (A1, A2), texture
spacing in y-direction (B1,B2), texture depth (C1,C2),
and texture area ratio (D1,D2). In this case, the L8
orthogonal array was appropriate. As shown in Fig.
4, we used L8 linear graphs to assign the factors. In
the L8 experiment, we chose only three factors to
observe all the interactions. Moreover, one column
should be open to residual factor (e) which is thecolumn sum of squares and represents the portion
that cannot be explained by individual factors. The
number of interactions from which to choose waslimited to two. Factors and levels were arrayed andare presented in Table 3. The friction coefficientsobtained from the eight combinations were used as
evaluation values.

Fig. 4 L8 linear graphs

4.2 Data processing and interaction of factors
From Table 4, differences by level for each factor
are calculated. Specifically, we summed andFactors
A
B
C
D
A×D
B×D
E
Total

Table 5 ANOVA table𝑆
𝜙𝑉
0.00013392
10.00013392
7.6382E-08
17.6382E-08
7.5577E-07
1
7.5577E-07
2.215E-05
12.215E-05
1.8789E-08
1
1.8789E-08
1.0182E-05
1
1.0182E-05
1.1466E-06
1
1.1466E-06
0.00016825
7

16

F0
116.79367
0.0666143
0.6591272
19.31766
0.0163862
8.8799553

P
6%
84%
57%
14%
92%
21%

Optimization of Texture Design Parameters for Friction Reduction Based on Design of Experiments andFinite Element Fluid Analysis

Fig. 5 Average chart of each level

B1
B2

D1
0.03540.031278

D2
0.029658
0.034562

4.3 Analysis of variance (ANOVA)
Analysis of variance (ANOVA) is a statistical
hypothesis-testing technique that determines theeffects of factors and their interactions bydecomposing the observed data variation into anerror variation and variation owing to each factor andtheir interactions.
The sum of squares S[k] of the factors [k] wascalculated using the following equation:
2
𝑇[𝑘]1

𝑇 (𝑇[𝑘]1 − 𝑇[𝑘]2 )𝑆[𝑘] =+− =𝑁/2 𝑁/2 𝑁𝑁(a) For B×D

A1
A2

D1
0.041619
0.025059

2

2
𝑇[𝑘]2

(5)

where the total number of data N=8.
There were two levels for all factors; therefore, thedegree of freedom for each factor was one. The
overall number of degrees of freedom was sevenbecause the number of experiments was eight.
The average square V was obtained using Eq. (6),and the equation for F0 value is shown in Eq. (7)D2
0.040196
0.024024(b) For A×D
Fig. 6 Duality table and graph

17

𝑉=

𝑆[𝑘]
𝛷

(6)

𝐹0 =

𝑉[𝑘]
𝑉𝑒

(7)

Doan Quang THUAN, Takahiko KURAHASHI, Hiroaki ARATA and Tetsuro IYAMA

Factors
A
B
C
D
B×D
E
Total

Table 6 ANOVA table after pooling
F0𝑆
𝜙𝑉
0.00013392
10.00013392 229.82143
7.6382E-08
17.6382E-08 0.1310808
7.5577E-07
1
7.5577E-07 2.2970014
2.215E-05
12.215E-05
38.01244
1.0182E-05
1
1.0182E-05 2.0936415
1.1654E-06
2
5.8271E-07
0.00016825

Table 5 summarizes the calculation results. The Pvalue was calculated between 0.000 and 1 and wasused as a criterion to determine whether a hypothesis
should be accepted or rejected. As shown in Fig. 7, a
one-sided test with a significance level of 0.05 wasused to determine if the results of the analysis weresignificant using the P-value. In this study, wehypothesized that these factors would significantly
reduce the coefficient of friction. The hypothesis isrejected only if the P-value is greater than 5%, andvice versa.
From the P-values in Table 5, we cannot identify
any factor that can be said to have a significant effect;
however, this does not mean that there is no effect.
This is because the orthogonal array simultaneously
tests many factors. Therefore, the analysis is repeatedusing a pooling technique which is a method ofsummarizing ineffective factors into residual factor(e).
The guidelines for pooling are as follows:[a] The criterion is loosened when the residual
degrees of freedom are small.
[b] Pools with P-values greater than 20% or F0
values less than 2

P
0%
75%
37%
3%
28%

[c] When interactions are not pooled, thecorresponding main effects are not pooled
We added factors A×D, which have the least effect,
to the residual factors and create a new analysis ofvariance (Table 6). Analysis of variance showed thatthe main effects of A and D were significant while
those of C and B×D, which were not significant,were not pooled because the F-value was greater than
2 [b]. Owing to [c], because the interaction factor
B×D was not pooled, the corresponding main effect
B was not pooled either. Consequently, after pooling,
we found that factors A, B, C, D, and B×D had asignificant effect on reducing the friction coefficient.
4.4 Estimation results
After determining the factors that affect the
reduction in friction coefficient, it is necessary to
determine the level corresponding to each factor. In
this study, the smallest friction coefficient wasconsidered; thus, the smallest value was selected.
Based on Table 4, Level 2 (A2) was chosen for factor
A and Level 2 (C2) for factor C. According to the
B×D duality table, level B1 D2 was chosen as factor
B×D. Consequently, an optimal combination of A2
B1 C2 D2, was obtained.
Eq. (8) was used to calculate the estimated friction
coefficient 𝜇̂ (A2 B1 C2 D2 ) based on the selected
optimal level.
𝜇̂ (A2 B1 C2 D2 ) = 𝜇̂
+ 𝐴2 + 𝜇̂+ 𝐶2̂
+𝜇 + 𝐵1 + 𝐷2 + (𝐵𝐷)12
𝑇[𝐵𝐷]12 𝑇[𝐴]2 𝑇[𝐶]2 𝑇
=
+
+
−
2
4
4
2
=0.01043025

Fig. 7 One-sided test

18

(8)

Optimization of Texture Design Parameters for Friction Reduction Based on Design of Experiments andFinite Element Fluid Analysis

Fig. 8 Stribeck diagram from numerical experiment

5. Resultanalyses by approximately 94%. The number ofexperiments was reduced, which, in turn, reduced thetime required for the study. Consequently, this study
demonstrated the application of design of experiment
method to evaluate the friction coefficient of textures.

The calculated friction coefficient in FreeFEM++
with the optimal combination (A2 B1 C2 D2 ) was
0.0112057. Compared with the results estimated by
the design of experiments, the relative error was 6%,
which indicates that it is a reliable value.
Furthermore, by comparing the estimated andcalculated values with the initial friction coefficients
listed in Table 3, we found that combination A2 B1
C2 D2 had the lowest friction coefficient. In this study,
texture spacing in the x-direction at Level 2, texture
spacing in the y-direction at Level 1, texture depth atLevel 2, and the area ratio of the texture at Level 2
are the optimal combinations.
Under the optimum combination conditions, a line
showing the change in the friction coefficient was
created, as indicated by the yellow line in Fig. 8. It
was confirmed that the friction coefficient at point O
was reduced by the optimal combination from thedesign of experiment method.

Acknowledgement: We would like to thank Editage(www.editage.com) for English language editing.
References
1)

H. Morita: Illustrated introduction Easy-to-understandbasics and mechanism of the latest experimental design
method, Shuwa system, 2019. (In Japanese)

2)

F. Hecht: New development in FreeFEM++, Journal of
Numerical Mathematics, Vol.20, pp.251-265, 2012.

3)

S. Sasaoka and T. Kurahashi: Proposal of texture shape
optimization algorithm under constant load condition andconsiderations on new shape update equation (Texture
shape optimization for minimization of friction coefficient),
Journal of Fluid Science and Technology, Vol.15, No.3,
pp.1-20, 2020.

4)

H. Arata, M. Kihida and T. Kurahashi: Texture shape
optimization analysis using a new acceleration gradient

6. Conclusions

method based on the Taylor expansion and conjugate
direction, Journal of Fluid Science and Technology,

In this study, the design of experiments was used todetermine the optimal combination of design
parameters for texture analysis to reduce thecoefficient of friction. The design of experiments
method reduced the number of experimental

Vol.17, No.4, pp.1-12, 2022.
(Received July 19, 2023)

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