Praks Brkic 2020a - Revision history

Rimni: /* Abstract */

2021-05-19T13:47:15Z

‎Abstract

Rimni at 15:41, 1 March 2021

2021-03-01T15:41:44Z

Rimni: /* 3.3. Summary of accuracy and speed tests */

2021-03-01T15:40:46Z

‎3.3. Summary of accuracy and speed tests

Rimni at 15:39, 1 March 2021

2021-03-01T15:39:20Z

Rimni: /* 3. Methods and discussion */

2021-03-01T15:35:33Z

‎3. Methods and discussion

Dejan.brkic at 15:46, 22 January 2021

2021-01-22T15:46:52Z

Dejan.brkic at 19:13, 28 September 2020

2020-09-28T19:13:00Z

Dejan.brkic at 19:06, 28 September 2020

2020-09-28T19:06:50Z

Dejan.brkic at 18:22, 28 September 2020

2020-09-28T18:22:58Z

Dejan.brkic at 14:51, 28 September 2020

2020-09-28T14:51:32Z

@@ Line 18: / Line 18: @@
 Using only a limited number of computationally expensive functions, we show a way how to construct accurate and computationally efficient approximations of the Colebrook equation for flow friction. The presented approximations are based on the asymptotic series expansion of the Wright ω-function and symbolic regression. The results are verified with 8 million of Quasi-Monte Carlo points covering the domain of interest for engineers. In comparison with the built-in “wrightOmega” feature of Matlab R2016a, the herein introduced related approximations of the Wright ω-function significantly accelerate the computation. With only two logarithms and several basic arithmetic operations used, the presented approximations are not only computationally efficient but also extremely accurate. The maximal relative error of the most promising approximation which is given in the form suitable for engineers’ use is limited to 0.0012%, while for a little bit more complex variant is limited to 0.000024%.
-'''Keywords''': Colebrook equation, Hydraulic flow friction, Wright <math>\omega</math>-function, explicit approximations, symbolic regression, computational intelligence
+'''Keywords''': Colebrook equation, hydraulic flow friction, wright <math>\omega</math>-function, explicit approximations, symbolic regression, computational intelligence
 ==1. Introduction==

@@ Line 467: / Line 467: @@
 |  style="text-align: center;"|11394 seconds ~<br/>189.9 minutes
 |-
-|  colspan='4'  style="vertical-align: top;padding:10px;"|Asymptotic expansions
+|  colspan='4'  style="text-align: center;vertical-align: top;padding:10px;"|Asymptotic expansions
 |-
 |  style="text-align: center;vertical-align: top;"|Eq. (8)

@@ Line 460: / Line 460: @@
 !  <math display="inline">y\approx \omega (x)-x</math> !!  Approximation !! Maximal relative error !! Time
 |-
-|  colspan='4'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;padding:10px;"|Exact solution by built-in ''wrightOmega'' function of Matlab
+|  colspan='4'  style="text-align: center;vertical-align: top;padding:10px;"|Exact solution by built-in ''wrightOmega'' function of Matlab
 |-
 |  style="text-align: center;vertical-align: top;"|Eq. (2)

@@ Line 462: / Line 462: @@
 |  colspan='4'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;padding:10px;"|Exact solution by built-in ''wrightOmega'' function of Matlab
 |-
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (2)
+|  style="text-align: center;vertical-align: top;"|Eq. (2)
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
+|  style="vertical-align: top;"|
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0% i.e. exact
+|  style="text-align: center;vertical-align: top;"|0% i.e. exact
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;text-align: center;"|11394 seconds ~
+|  style="text-align: center;"|11394 seconds ~<br/>189.9 minutes
-−
 .9 minutes
 |-
-|  colspan='4'  style="border-top: 1pt solid black;text-align: center;vertical-align: top;padding:10px;"|Asymptotic expansions
+|  colspan='4'  style="vertical-align: top;padding:10px;"|Asymptotic expansions
 |-
 |  style="text-align: center;vertical-align: top;"|Eq. (8)
@@ Line 491: / Line 489: @@
 |  style="text-align: center;vertical-align: top;"|2.4 seconds
 |-
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (12)
+|  style="text-align: center;vertical-align: top;"|Eq. (12)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Figure 2
+|  style="text-align: center;vertical-align: top;"|Figure 2
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.00247%
+|  style="text-align: center;vertical-align: top;"|0.00247%
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|4.4 seconds
+|  style="text-align: center;vertical-align: top;"|4.4 seconds
 |-
 |  colspan='4'  style="text-align: center;vertical-align: top;padding:10px;"|Asymptotic expansions with corrective constant
@@ Line 508: / Line 506: @@
 |  style="text-align: center;vertical-align: top;"|0.4 seconds
 |-
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (15)
+|  style="text-align: center;vertical-align: top;"|Eq. (15)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (24)
+|  style="text-align: center;vertical-align: top;"|Eq. (24)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.00527%
+|  style="text-align: center;vertical-align: top;"|0.00527%
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1.1 seconds
+|  style="text-align: center;vertical-align: top;"|1.1 seconds
 |-
 |  colspan='4'  style="text-align: center;vertical-align: top;padding:10px;"|Asymptotic expansions with corrective function obtained using symbolic regression
 |-
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (16)
+|  style="text-align: center;vertical-align: top;"|Eq. (16)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (25)
+|  style="text-align: center;vertical-align: top;"|Eq. (25)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.000391%
+|  style="text-align: center;vertical-align: top;"|0.000391%
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1.1 seconds
+|  style="text-align: center;vertical-align: top;"|1.1 seconds
 |-
 |  colspan='4'  style="text-align: center;vertical-align: top;padding:10px;"|Symbolic regression
@@ Line 532: / Line 530: @@
 |  style="text-align: center;vertical-align: top;"|0.42 seconds
 |-
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (19)
+|  style="text-align: center;vertical-align: top;"|Eq. (19)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (28)
+|  style="text-align: center;vertical-align: top;"|Eq. (28)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.00337%
+|  style="text-align: center;vertical-align: top;"|0.00337%
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.42 seconds
+|  style="text-align: center;vertical-align: top;"|0.42 seconds
 |-
 |  colspan='4'  style="text-align: center;vertical-align: top;padding:10px;"|Symbolic regression with the optimized coefficients
 |-
-|  style="border-bottom: 1pt solid black;vertical-align: top;text-align: center;"|Eq. (20);
+|  style="vertical-align: top;text-align: center;"|Eq. (20);
 optimized Eq. (19)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (29) – Figure 3
+|  style="text-align: center;vertical-align: top;"|Eq. (29) – Figure 3
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.0012%
+|  style="text-align: center;vertical-align: top;"|0.0012%
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.388 seconds
+|  style="text-align: center;vertical-align: top;"|0.388 seconds
 |-
 |  colspan='4'  style="text-align: center;vertical-align: top;padding:10px;"|Symbolic regression with the optimized coefficients and corrective function
 |-
-|  style="border-bottom: 1pt solid black;vertical-align: top;text-align: center;"|Eq. (20)
+|  style="vertical-align: top;text-align: center;"|Eq. (20)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|Eq. (30)
+|  style="text-align: center;vertical-align: top;"|Eq. (30)
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.000024%
+|  style="text-align: center;vertical-align: top;"|0.000024%
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1.1 seconds
+|  style="text-align: center;vertical-align: top;"|1.1 seconds
 |}

@@ Line 74: / Line 74: @@
-'''<big>3.1. Series expansion of <math display="inline">\omega (x)-x</math></big>'''
+===3.1. Series expansion of <math display="inline">\omega (x)-x</math>===
-'''3.1.1. Asymptotic expansion of <math display="inline">\omega (x)-x</math> about infinity'''
+====3.1.1. Asymptotic expansion of <math display="inline">\omega (x)-x</math> about infinity====
 Eqs. (3)-(7) with <math display="inline">{s}_{i}\left( x\right)</math>  terms (<math display="inline">i=1, 2,\cdots, 5</math>) used for approximating <math display="inline">\omega \left( x\right) -</math><math>x</math> by a truncated series about infinity [41], where generally <math display="inline">{y}_{i}(x)=</math><math>\sum _{i=1}^{+\infty }{s}_{i}(x)</math>. The terms <math display="inline">{s}_{1}\left( x\right) ,\, {s}_{2}\left( x\right) ,\ldots ,\, {s}_{5}\left( x\right)</math>  are given in Eqs. (3)-(7), see [45]. The corresponding first five approximations of <math display="inline">\omega \left( x\right) -</math><math>x</math> are summarized in Eqs. (8)-(12)
@@ Line 238: / Line 238: @@
 The error function <math display="inline">\xi</math>  of Eq. (16) is developed using symbolic regression, while <math display="inline">\omega (x)-</math><math>x</math> will be approximated using the same approach in Section 3.1.2.
-'''3.1.2. Symbolic regression-based expansions of <math display="inline">y=\omega \left( x\right) -x</math> for Colebrook equation'''
+====3.1.2. Symbolic regression-based expansions of <math display="inline">y=\omega \left( x\right) -x</math> for Colebrook equation====
 An additional strategy for increasing the accuracy of <math display="inline">y=\omega \left( x\right) -x</math> for the Colebrook equation is given in Eqs. (17)-(20). These approximations were found by symbolic regression as already explained in [21] and software Eureqa is used to perform analyses [40] (very recently, also novel artificial intelligence - AI software AI Feynman: A physics-inspired method for symbolic regression is available [62]). These approximations are optimized for the domain 7.51< <math display="inline">x</math><619, which is of the interest for the Colebrook equation.
@@ Line 287: / Line 287: @@
 Eq. (17) is with the maximal relative error of no more than <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.0497% and it needs 0.42 seconds to execute the benchmark sample, Eq. (18) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.0105% and 0.42 seconds, Eq. (19) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.00229% and 0.42 seconds while Eq. (20) with <math display="inline">{\left\{ {\delta }_{\%}\right\} }_{max}</math>=0.000024% and 1.1 seconds.
-'''<big>3.2. Approximations suitable for engineering use</big>'''
+===3.2. Approximations suitable for engineering use===
-'''3.2.1. Approximations and procedures based on series expansions of <math display="inline">\omega (x)-x</math> about infinity'''
+====3.2.1. Approximations and procedures based on series expansions of <math display="inline">\omega (x)-x</math> about infinity====
 To use the procedure for solving Eq. (2) based on series expansions given with Eqs. (3)-(7) and Eqs. (8)-(12), one should follow the algorithm illustrated in Figure 2. It can be modified so <math display="inline">s</math> can be given as <math display="inline">s\approx {s}_{1}+</math><math>{s}_{2}</math>, <math display="inline">s\approx {s}_{1}+{s}_{2}+{s}_{3}</math>, etc. Also, Eqs. (8)-(12) can be extended to construct <math display="inline">{s}_{6},\, {s}_{7},\, {s}_{8},</math> etc. [45].
@@ Line 366: / Line 366: @@
 |}
-'''3.2.2. Approximations based on symbolic regression'''
+====3.2.2. Approximations based on symbolic regression====
 Four simple but very accurate approximations based on symbolic regression are presented in Eqs. (26)-(29) where Eqs. (26) and (27) are already given in [21], whereas Eqs. (28) and (29) are novel (they are simple, but very accurate). Although Eureqa allows the only minimization of the absolute error [40], we can use it also for minimizing the relative error as Gholamy and Kreinovich [63] suggested (they showed how to use existing absolute-error-minimizing software to minimize relative errors). Consequently, Eureqa was used for the minimization of the maximal error (worst-case) of <math display="inline">\displaystyle \frac{\left| {y}_{sr}-y\right| }{y}</math>, where <math display="inline">{y}_{sr}</math> denotes a symbolic regression-based approximation of <math display="inline">y</math>. Eqs. (26)-(29) show that this formulation significantly reduces the approximation error
@@ Line 447: / Line 447: @@
 The computation speed of Eq. (30) is the same as Eq. (15), i.e. 1.1 seconds. The maximal relative error of Eq. (15) is 0.00527%, while the maximal relative error of Eq. 30 is only 0.000024%. Thus, the novel symbolic regression approximation of Eq. (30) reduces the relative error by factor 0.00527%/0.000024%≈219, which confirms the advantages of using symbolic regression for flow friction estimations.
-====3.3. Summary of accuracy and speed tests====
+===3.3. Summary of accuracy and speed tests===
 The presented approximations of the Colebrook equation which are presented in this article have been compared in [[#tab-1|Table 1]] with respect of accuracy and speed of execution. The column ‘time in sec’ is given for execution of the Colebrook equation for 8 million samples generated using the Sobol’s Quasi Monte Carlo algorithm, as already given through our paper.

@@ Line 596: / Line 596: @@
 | <math>\epsilon</math>: relative roughness of inner pipe surface (dimensionless) – main input parameter
 |-
-| <math>{\delta }_{\%}</math>: relative error (%); <math display="inline">{\delta }_{\%}=>\displaystyle \frac{f-{f}_{i}}{f}\cdot 100\%</math>
+| <math>{\delta }_{\%}</math>: relative error (%); <math display="inline">{\delta }_{\%}=\displaystyle \frac{f-{f}_{i}}{f}\cdot 100\%</math>
 |-
 | <math display="inline">\xi</math>  and <math display="inline">{\xi }_{1}</math>: error-functions developed using symbolic regression

@@ Line 302: / Line 302: @@
 | colspan="1" style="padding:10px;"| '''Figure 2'''. Algorithm for the procedure for solving Eq. (2) based on series expansion by Eq. (12)
 |}
 Besides, if <math display="inline">y\approx {s}_{1}</math>, Eq. (8), the relative error is up to 0.153% with around 0.2 seconds for the execution of 8 million of Quasi Monte-Carlo tested samples. Based on the algorithm from Figure 2 and using only <math display="inline">y\approx {s}_{1}</math>, Eq. (21) can be constructed (this equation is already presented in [21])
@@ Line 319: / Line 318: @@
 To construct the approximations from Eqs. (22)-(24) based on Eqs. (13)-(14), the notation <math display="inline">\mathrm{C=ln}\,\left( x\right)</math>  should be used to avoid multiple repetitions of the logarithmic function, as <math display="inline">C</math> is computed only once.
 Comparing Eq. (22) with the maximal relative error of 0.129% and Eq. (21) with 0.153%, it is confirmed that adding the carefully chosen constant <math display="inline">\alpha</math>, gives more accurate results for the Colebrook equation

@@ Line 294: / Line 294: @@
 Following the algorithm from [[#img-2|Figure 2]], Eq. (12) can be executed (it introduces a relative error of no more than 0.00247% and that it needs around 4.4 seconds to execute 8 million of Quasi Monte-Carlo tested samples carefully selected from the domain of the Colebrook equation which is of interest for engineering practice).
 Besides, if <math display="inline">y\approx {s}_{1}</math>, Eq. (8), the relative error is up to 0.153% with around 0.2 seconds for the execution of 8 million of Quasi Monte-Carlo tested samples. Based on the algorithm from Figure 2 and using only <math display="inline">y\approx {s}_{1}</math>, Eq. (21) can be constructed (this equation is already presented in [21])
@@ Line 306: / Line 315: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (21)
 |}
 Further accurate approximations based on expansions from Eqs. (13)-(14) are given in Eqs. (22)-(25). Based on the expansions from Eqs. (13)-(14), the appropriate explicit approximations to the Colebrook equation can be constructed: Eq. (13) → Eq. (22), Eq. (14) → Eq. (23), and Eq. (15) → Eq. (24).
@@ Line 312: / Line 320: @@
 To construct the approximations from Eqs. (22)-(24) based on Eqs. (13)-(14), the notation <math display="inline">\mathrm{C=ln}\,\left( x\right)</math>  should be used to avoid multiple repetitions of the logarithmic function, as <math display="inline">C</math> is computed only once.
 Comparing Eq. (22) with the maximal relative error of 0.129% and Eq. (21) with 0.153%, it is confirmed that adding the carefully chosen constant <math display="inline">\alpha</math>, gives more accurate results for the Colebrook equation

@@ Line 451: / Line 451: @@
 The presented approximations of the Colebrook equation which are presented in this article have been compared in [[#tab-1|Table 1]] with respect of accuracy and speed of execution. The column ‘time in sec’ is given for execution of the Colebrook equation for 8 million samples generated using the Sobol’s Quasi Monte Carlo algorithm, as already given through our paper.
 <div id='tab-1'></div>
 <div class="center" style="font-size: 75%;">'''Table 1'''. Comparison of the presented approximations with related error and speed of execution</div>
@@ Line 551: / Line 554: @@
 |  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1.1 seconds
 |}
 ==4. Conclusions and future work==