Parameter Determination of the 2S2P1D Model and Havriliak–Negami Model Based on the Genetic Algorithm and Levenberg–Marquardt Optimization Algorithm
Abstract
:1. Introduction
2. Methodology and Optimization Study
2.1. Theory of GA
2.2. Parameter Optimization of 2S2P1D Model
2.2.1. Theory of 2S2P1D Model
2.2.2. Parameter Optimization Process
- (1)
- Parameter optimization independent of the WLF functionIn the first approach, the test data obtained at different temperatures are independently optimized, resulting in the determination of the relaxation time () values for each temperature. Based on Equation (4), we know that six temperature-independent parameters (, , , , , and ) and one temperature-dependent parameter () should be determined. The detailed procedure is given below:
- (1)
- Take , , , , , , and as unknowns and the regularized 2-norm sum of the differences between the calculated (based on Equations (4) and (8)) and experimental dynamic moduli and phase angles as the objective function, and then conduct the optimization using the GA considering the given constraints and upper and lower bounds.
- (2)
- Take the parameters obtained from the GA as the initial values for further optimization using the L–M algorithm.
- (3)
- Substitute the obtained parameters into Equations (4) and (8) to calculate the corresponding dynamic modulus () and phase angle (). The correlation coefficient between the calculated values and the original experimental values is calculated to evaluate the fitting results.
- (2)
- Parameter optimization dependent on WLF functionIn the second method, the test data obtained at different temperatures are shifted to the reference temperature (Tr) using the WLF equation, with the subsequent optimization of only one relaxation time value () at the reference temperature. The specific steps are as follows:
- (1)
- Take , , , , , , , , and as unknowns, and employ the GA to obtain the initial parameter values.
- (2)
- Take the parameter values obtained with the GA as the initial values and further optimize the parameters using the L–M algorithm.
- (3)
- Substitute the obtained parameters into Equations (4) and (8) to calculate the corresponding dynamic modulus () and phase angle (). The correlation coefficient between the calculated values and the original experimental values is calculated to evaluate the fitting results.
2.3. Parameter Optimization of the H–N Model
2.3.1. Theory of the H–N Model
2.3.2. Parameter Optimization Process
- (1)
- Plot the storage modulus and loss modulus obtained from experimental tests on the same coordinate system. The inverted U-shaped Cole–Cole curve formed by the scattered points is approximated using a third-order polynomial, as follows:
- (2)
- Based on the determined Cole–Cole curve using the polynomial, calculate , , , , and .
- (3)
- Calculate and using Equation (10) and determine the peak point of the Cole–Cole curve . Substitute this point into Equation (17) to check if is smaller than 1:
- (4)
- Use the GA to calculate and . If , the objective function can be written as Equation (18):If , the objective function can be written as Equation (19).
- (5)
- Based on the calculated , , , and , determine the optimization interval and apply the GA again to optimize the temperature-independent parameters (, , , and ) based on the objective function, as shown below:
- (6)
- Based on the optimized values of the four parameters (, , , and ), apply the GA again to calculate the relaxation time () at each temperature. Based on the WLF equation, calculate the shift factor at each test temperature with respect to the reference temperature (Tr = 30 ℃). defines the relationship between and , as shown below:According to the shift factors at different temperatures, the reduced frequency can be calculated based on Equation (22), as shown below:
3. Results and Discussion
3.1. Case Study on the Optimization of the 2S2P1D Model
3.1.1. Parameter Optimization Independent of the WLF Function
3.1.2. Parameter Optimization Dependent on the WLF Function
3.2. Case Study on the Optimization of the H–N Model
3.3. Sensitivity Analysis of the Input Variables
4. Findings and Conclusions
- 1.
- Two optimization methods were employed for the 2S2P1D model: one independent of the WLF equation utilizing the GA and one dependent on the WLF equation utilizing the L–M algorithm. A comparison of the fitting results revealed that the optimization method independent of the WLF equation exhibited higher accuracy in parameter fitting, while the method dependent on the WLF equation demonstrated slightly lower accuracy.
- 2.
- For the 2S2P1D model, the fitting accuracy of the parameters optimized using the GA could be further improved by secondary optimization using the L–M algorithm. This suggests that a combination of the GA and L–M algorithm improves the overall fitting accuracy.
- 3.
- The H–N model often encounters complex fractional-order derivatives during the GA fitting, resulting in a loss of physical meaning in the solution domain and optimization failure. To address this issue, the Cole–Cole curve was initially fitted with a third-order polynomial function, followed by further fitting using the GA with given upper and lower bounds. This improved method can increase the accuracy of the calculated results. However, the optimized values are significantly influenced by the experimental data, and it is challenging to improve the accuracy using the L–M algorithm. This may be closely related to the method used to collect the fitting data values.
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Procedure SGA |
begin Initialize P(0); t = 0; while (t ≤ T) do for i = 1 to M do Evaluate fitness of P(t); end for for i = 1 to M do Select operation to P(t); end for for i = 1 to M/2 do Crossover operation to P(t); end for for i = 1 to M do Mutation operation to P(t); end for for i = 1 to M do P(t + 1) = P(t) end for t = t + 1; end while end |
Appendix B
- (1)
- Choose initial point p0; control the constant for termination ; calculate , set k = 0, , and v = 10 (or another number greater than 1).
- (2)
- Calculate the Jacobian matrix , , and construct the increment normal equation .
- (3)
- Solve the increment normal equation to obtain :
- ①
- If , set , and if , stop the iteration and output the results. Otherwise, set , and go to Step 2;
- ②
- If , set , re-solve the normal equation to obtain , and return to Step 1.
Algorithm (-) | TolFun (-) | MaxFunctionEvaluations (-) | MaxIterations (-) |
---|---|---|---|
sqp-legacy algorithm | 1 × 10−40 | 1 × 104 | 1 × 103 |
Input: A vector function with , a measurement vector and an initial parameter estimate . Output: A vector minimizing . Algorithm: begin k := 0; v := 2; and p := p0; A = JTJ; ; and g := JT; Stop := (); ; while (not stop) and (k < kmax) k := k + 1; repeat Solve ; if() stop := true; else pnew := p + if p = pnew; A = JTJ; ; and g := JT; stop := () or (); else end if end if until () or (stop) end while p+ := p; end |
Appendix C
- (1)
- In the H–N model’s Cole–Cole domain, .When , . The following series can be found in Hilfer [32]:Substituting Equation (A1) into Equation (1), the H–N equation is displayed as:The storage modulus and loss modulus are displayed as:Let N be the maximum integer that satisfies . When , Equation (A5) can be approximated as:So, we can obtain:For any positive integer N, Equation (A7) is equal to:
- (2)
- In the H–N model’s Cole–Cole domain [32],When [32]:Applying Equation (A9), the H–N model’s equation can be expanded in series as:The storage modulus and loss modulus should be:Substituting t for 1/, Equation (A13) can be rewritten as:Then, can be attained as:
References
- Gao, Y.; Sarode, A.; Kokoroskos, N.; Ukidve, A.; Zhao, Z.; Guo, S.; Flaumenhaft, R.; Gupta, A.S.; Saillant, N.; Mitragotri, S. A polymer-based systemic hemostatic agent. Sci. Adv. 2020, 6, eaba0588. [Google Scholar] [CrossRef] [PubMed]
- Songhyun, L.; Seung-Jae, S.; Hoyong, B.; Yeonwoo, C.; Kyunglim, H.; Myungeun, S.; Kyunam, K.; Dong-Yeun, K.; Hyungjun, K.; Minkee, C. Dynamic metal-polymer interaction for the design of chemoselective and long-lived hydrogenation catalysts. Sci. Adv. 2020, 6, eabb7369. [Google Scholar]
- Yao, X.; Guo, Z.; Basha, S.H.; Huang, Q. Innovative seismic strengthening of historic masonry walls using polymer mortar and steel strips. Eng. Struct. 2021, 228, 111507. [Google Scholar] [CrossRef]
- Barraj, F.; Mahfouz, S.; Kassem, H.; Khatib, J.; Goulias, D.; Elkordi, A. Investigation of Using Crushed Glass Waste as Filler Replacement in Hot Asphalt Mixtures. Sustainability 2023, 15, 2241. [Google Scholar]
- Banks, H.T. A brief review of some approaches to hysteresis in viscoelastic polymers. Nonlinear Anal. 2008, 69, 807–815. [Google Scholar] [CrossRef]
- Barraj, F.; Khatib, J.; Castro, A.; Elkordi, A. Effect of Chemical Warm Mix Additive on the Properties and Mechanical Performance of Recycled Asphalt Mixtures. Buildings 2022, 12, 874. [Google Scholar]
- Ehsan, B.; Savvas, G.H. Viscoelastic properties and constitutive modelling of bitumen. Fuel 2013, 108, 391–399. [Google Scholar]
- Liyan, S.; Yanan, X.; Hongsen, H.; Nanqi, R. Optimization criterion of viscoelastic response model for asphalt binders. Constr. Build. Mater. 2016, 113, 553–560. [Google Scholar]
- Olard, F.; Di Benedetto, H. General “2S2P1D” Model and Relation Between the Linear Viscoelastic Behaviours of Bituminous Binders and Mixes. Road Mater. Pavement Des. 2003, 4, 185–224. [Google Scholar]
- Hao, D.; Li, D. Determination of dynamic mechanical properties of carbon black filled rubbers at wide frequency range using Havriliak–Negami model. Eur. J. Mech.-A/Solids 2015, 53, 303–310. [Google Scholar]
- Schapery, R.A. Approximate Methods of Transform Inversion for Viscoelastic Stress Analysis. In Proceedings of the Fourth U.S. National Congress of Applied Mechanics, Berkeley, CA, USA, 18–21 June 1962. [Google Scholar]
- Schapery, R.A. On the characterization of nonlinear viscoelastic materials. Polym. Eng. Sci. 1969, 9, 295–310. [Google Scholar] [CrossRef]
- Purohit, V.B.; Pieta, M.; Pietrasik, J.; Plummer, C.M. Recent advances in the ring-opening polymerization of sulfur-containing monomers. Polym. Chem. 2022, 13, 4858–4878. [Google Scholar] [CrossRef]
- Ghosh, A.; Spakowitz, A.J. Active and thermal fluctuations in multi-scale polymer structure and dynamics. Soft Matter 2022, 18, 6629–6637. [Google Scholar] [CrossRef]
- Ni, Y.; Medvedev, G.A.; Curliss, D.B.; Caruthers, J.M. Linear viscoelastic relaxation in the α and α+ regions of linear polymers, crosslinked polymers and small molecules. Polymer 2020, 202, 122745. [Google Scholar] [CrossRef]
- Sun, H.; Liang, Y.; Thompson, M.P.; Gianneschi, N.C. Degradable polymers via olefin metathesis polymerization. Prog. Polym. Sci. 2021, 120, 101427. [Google Scholar] [CrossRef]
- Yue, W.; Chenxin, Z.; Raphael, P.; Hongping, Y.; Lihua, J.; Shucheng, C.; Francisco, M.; Franziska, L.; Jia, L.; Noelle, I.R.; et al. A highly stretchable, transparent, and conductive polymer. Sci. Adv. 2017, 3, e1602076. [Google Scholar]
- Ghosh, A.; Samanta, S.; Ge, S.R.; Sokolov, A.P.; Schweizer, K.S. Influence of Attractive Functional Groups on the Segmental Dynamics and Glass Transition in Associating Polymers. Macromolecules 2022, 55, 2345–2357. [Google Scholar] [CrossRef]
- Konishi, S.; Kashiwagi, Y.; Watanabe, G.; Osaki, M.; Katashima, T.; Urakawa, O.; Inoue, T.; Yamaguchi, H.; Harada, A.; Takashima, Y. Design and mechanical properties of supramolecular polymeric materials based on host-guest interactions: The relation between relaxation time and fracture energy. Polym. Chem. 2020, 11, 6811–6820. [Google Scholar] [CrossRef]
- Li, Y.; Liu, Z. A novel constitutive model of shape memory polymers combining phase transition and viscoelasticity. Polymer 2018, 143, 298–308. [Google Scholar] [CrossRef]
- Zhu, J.B.; Watson, E.M.; Tang, J.; Chen, E.Y.X. A synthetic polymer system with repeatable chemical recyclability. Science 2018, 360, 398–403. [Google Scholar] [CrossRef]
- Liang, H.Y.; de Pablo, J.J. A Coarse-Grained Molecular Dynamics Study of Strongly Charged Polyelectrolyte Coacervates: Interfacial, Structural, and Dynamical Properties. Macromolecules 2022, 55, 4146–4158. [Google Scholar] [CrossRef]
- Zhang, L.; Wang, D.; Xu, L.Q.; Zhang, A.M. A supramolecular polymer with ultra-stretchable, notch-insensitive, rapid self-healing and adhesive properties. Polym. Chem. 2021, 12, 660–669. [Google Scholar] [CrossRef]
- Du, Z.; Lin, Y.; Xing, R.; Cao, X.; Yu, X.; Han, Y. Controlling the polymer ink’s rheological properties and viscoelasticity to suppress satellite droplets. Polymer 2018, 138, 75–82. [Google Scholar] [CrossRef]
- Bai, J.; Hu, K.M.; Zhang, L.Z.; Shi, Z.X.; Zhang, W.M.; Yin, J.; Jiang, X.S. The Evolution of Self-Wrinkles in a Single-Layer Gradient Polymer Film Based on Viscoelasticity. Macromolecules 2022, 55, 3563–3572. [Google Scholar] [CrossRef]
- Potaufeux, J.E.; Odent, J.; Notta-Cuvier, D.; Lauro, F.; Raquez, J.M. A comprehensive review of the structures and properties of ionic polymeric materials. Polym. Chem. 2020, 11, 5914–5936. [Google Scholar] [CrossRef]
- An, X.; Wang, K.; Bai, L.B.; Wei, C.X.; Xu, M.; Yu, M.N.; Han, Y.M.; Sun, N.; Sun, L.L.; Lin, J.Y.; et al. Intrinsic mechanical properties of the polymeric semiconductors. J. Mater. Chem. C 2020, 8, 11631–11637. [Google Scholar] [CrossRef]
- You, W.; Yu, W. Control of the dispersed-to-continuous transition in polymer blends by viscoelastic asymmetry. Polymer 2018, 134, 254–262. [Google Scholar] [CrossRef]
- Ali, N.; Riccardo, L. Genetic algorithms in chemometrics. J. Chemometr. 2012, 26, 345–351. [Google Scholar]
- Nguyen, Q.T.; Di Benedetto, H.; Sauzéat, C. Linear and nonlinear viscoelastic behaviour of bituminous mixtures. Mater. Struct. 2015, 48, 2339–2351. [Google Scholar] [CrossRef]
- Hu, X. Micro- and Macro-Viscohyperelastic Behavior of Carbon Black Filled Rubbers. Ph.D. Thesis, Xiangtan University, Xiangtan, China, 24 April 2013. [Google Scholar]
- Hilfer, R. Analytical representations for relaxation functions of glasses. J. Non-Cryst. Solids 2002, 305, 122–126. [Google Scholar] [CrossRef]
paretoFraction (-) | CrossoverFraction (-) | UseParallel (-) | PopulationSize (-) | Generations (-) | stallGenLimit (-) | TolFun (-) |
---|---|---|---|---|---|---|
0.3 | 0.8 | true | 2000 | 200 | 200 | 10−10 |
(MPa) | (MPa) | (-) | (-) | (-) | (-) | (-) | (-) | (-) | (-) | (-) |
---|---|---|---|---|---|---|---|---|---|---|
2.255 | 1.230 × 103 | 0.367 | 0.349 | 9.99 | 0.325 | 1.879 | 1.409 | 1.487 | 1.983 | 1.935 |
(MPa) | (MPa) | (-) | (-) | (-) | (-) | (-) | (-) | (-) | (-) | (-) |
---|---|---|---|---|---|---|---|---|---|---|
0.176 | 1.062 × 103 | 0.395 | 0.985 | 9.999 | 1.574 | 1.696 | 1.361 | 1.433 | 1.960 | 1.940 |
paretoFraction (-) | CrossoverFraction (-) | UseParallel (-) | PopulationSize (-) | Generations (-) | stallGenLimit (-) | TolFun (-) |
---|---|---|---|---|---|---|
0.3 | 0.8 | true | 500 | 200 | 200 | 10−20 |
(MPa) | (MPa) | (-) | (-) | (-) | (-) | (-) | (-) | (-) | (℃) |
---|---|---|---|---|---|---|---|---|---|
0.0943 | 1.247 × 103 | 0.329 | 0.704 | 3.615 | 99.999 | 2.981 | 40.443 | 298.987 | 7.943 × 10−5 |
(MPa) | (MPa) | (-) | (-) | (-) | (-) | (-) | (-) | (-) | (℃) |
---|---|---|---|---|---|---|---|---|---|
0.095 | 1.247 × 103 | 0.321 | 0.691 | 3.309 | 100 | 2.143 | 40.100 | 298.458 | 0.445 |
(-) | (-) | (-) | (-) |
---|---|---|---|
4.204 × 10−8 | −4.635 × 10−4 | 1.243 | −27.113 |
(MPa) | (MPa) | (-) | (-) |
---|---|---|---|
15.678 | 4.515 × 103 | 1.189 | −0.378 |
(-) | (-) | ||
---|---|---|---|
17.245 | 4.325 × 103 | 0.546 | 0.545 |
(-) | (-) | ||
---|---|---|---|
18.970 | 3.906 × 103 | 0.491 | 0.492 |
Temperature (℃) | −70 | −60 | −50 | −30 | −10 | 10 | 30 | 50 |
---|---|---|---|---|---|---|---|---|
(-) | 5.206 × 102 | 1.975 | 0.663 | 0.616 | 10−8 | 10−8 | 10−8 | 2.390 × 10−9 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Qiu, M.; Cao, P.; Cao, L.; Tan, Z.; Hou, C.; Wang, L.; Wang, J. Parameter Determination of the 2S2P1D Model and Havriliak–Negami Model Based on the Genetic Algorithm and Levenberg–Marquardt Optimization Algorithm. Polymers 2023, 15, 2540. https://doi.org/10.3390/polym15112540
Qiu M, Cao P, Cao L, Tan Z, Hou C, Wang L, Wang J. Parameter Determination of the 2S2P1D Model and Havriliak–Negami Model Based on the Genetic Algorithm and Levenberg–Marquardt Optimization Algorithm. Polymers. 2023; 15(11):2540. https://doi.org/10.3390/polym15112540
Chicago/Turabian StyleQiu, Mingzhu, Peng Cao, Liang Cao, Zhifei Tan, Chuantao Hou, Long Wang, and Jianru Wang. 2023. "Parameter Determination of the 2S2P1D Model and Havriliak–Negami Model Based on the Genetic Algorithm and Levenberg–Marquardt Optimization Algorithm" Polymers 15, no. 11: 2540. https://doi.org/10.3390/polym15112540