Elastic–plastic model identification for rock surrounding an underground excavation based on immunized genetic algorithm
- Wei Gao^{1}Email author,
- Dongliang Chen^{1} and
- Xu Wang^{1}
Received: 2 October 2015
Accepted: 30 June 2016
Published: 11 July 2016
Abstract
To compute the stability of underground engineering, a constitutive model of surrounding rock must be identified. Many constitutive models for rock mass have been proposed. In this model identification study, a generalized constitutive law for an elastic–plastic constitutive model is applied. Using the generalized constitutive law, the problem of model identification is transformed to a problem of parameter identification, which is a typical and complicated optimization. To improve the efficiency of the traditional optimization method, an immunized genetic algorithm that is proposed by the author is applied in this study. In this new algorithm, the principle of artificial immune algorithm is combined with the genetic algorithm. Therefore, the entire computation efficiency of model identification will be improved. Using this new model identification method, a numerical example and an engineering example are used to verify the computing ability of the algorithm. The results show that this new model identification algorithm can significantly improve the computation efficiency and the computation effect.
Keywords
Background
The back analysis method is an important method in underground engineering. Since it was proposed in the 1970s, numerous studies have been performed (Gao and Liu 2009; Wang and Li 1993; Sakurai and Takeuchi 1983). Currently, back analysis can be divided into two types of analyses: parameter identification and model identification (Gao and Liu 2009). In parameter identification, the constitutive model of surrounding rock is assumed to be a simple model, such as the elastic model, the elastic–plastic model and the rheological model. Using these simple models, the mechanical parameters of surrounding rock can be inversed based on the measurement information, such as displacement. For simplicity, parameter identification based on measured displacements has been the most common back analysis method for underground engineering (Feng et al. 2000; Maier and Gioda 1982; Rechea et al. 2008; Sharifzadeh et al. 2013; Yazdani et al. 2012). However, due to its complexity, model identification has developed very slowly. Different from parameter identification, in model identification, the constitutive model of surrounding rock is unknown. Based on the measurement information, such as displacement, the structure and parameters of the constitutive model can be identified. Because it is very complex to select the structure and parameters of the constitutive model at the same time, generally, the structure of the constitutive model is determined by the prior knowledge, such as engineering experience, and then only the parameters of the constitutive model are identified based on measurement information. Thus, the model identification can be simplified as similar as the parameter identification. However, the model identification is more complex than the parameter identification. In model identification, the model parameters and the mechanical parameters can be identified at the same time. And the structure of the constitutive model determined by the prior knowledge is generally more complex than the assumed simple model used in parameter identification. Moreover, different from the mechanical parameters, which have the clear physical and mechanical meaning, and can be determined by the tests easily, the model parameters, which only describe the constitutive model, can not be determined by the tests. In 1987, Gioda and Sakurai proposed that model identification based on displacement should be the main development for back analysis (Gioda and Sakurai 1987). In 1997, Sakurai demonstrated that the identification of a constitutive model is critical (Sakurai 1997). Therefore, researchers have investigated the identification of a constitutive model. Based on the displacement measurements of an underground roadway, Liu (2011) identified the visco-elastic constitutive model of rock mass based on the traditional nonlinear optimum technique. Wang et al. (2007) identified the geo-material constitutive model based on their constitutive model database and several identification algorithms. Yang and Wang (2009) presented a numerical model to identify the unknown equivalent constitutive model in the elastic layered rock mass of an underground opening by the Gauss–Newton technique.
Generalized constitutive law of elastic–plastic model for rock materials
For the non-associative flow rule, the plastic potential G is different from the yield function F. And it is a very hard work to construct the function of G (Nakai 2012). Moreover, there is no generalized form for the function of G (Zheng et al. 2002). However, the elastic–plastic constitutive model with the associative flow rule can describe the main mechanical behaviours of geo-meterials well (Zheng and Kong 2010). Thus, in this study, simply, G = F.
Therefore, only two terms are to be determined in this elastic–plastic constitutive law: the stress yield function F and the elastic matrix [D _{ e }]. The problem of identification for the elastic–plastic constitutive model can be transformed to the problem of identification for the stress yield function F and the elastic parameters E and μ.
Using this generalized constitutive law for an elastic–plastic model, the problem of model identification can be transformed to a parameter identification.
To simply analyze without a loss of generalization, we assume that the parameter β = 0; thus, only the two parameters α and K need to be identified. In addition to the elastic parameters E and μ, four model parameters need to be identified: E, μ, α and K.
Model identification by immunized genetic algorithm
- 1.
Parameter initialization
- 2.
Individual expression
- 3.
Creation of the initial population
- 4.
Fitness function
- 5.
Optimization process
- 6.
Termination condition
In this study, the termination criterion specifies that the difference of the maximum fitness value and the average fitness value is less than 10e-5. To avoid infinite iteration, a maximum number of evolutionary generations is also specified.
Case study
Numerical example
A numerical testing example is employed to test the proposed algorithm. The example is an underground tunnel with a radius of 3 m. The details of this example can be found in reference (Gao 2016).
For computation, the mechanical property of the surrounding rock is assumed as follows:
According to the definition of α and K, we can obtain the values of α and K from the values of C and \(\varphi\), which are as follows: α = 0.165872, K = 0.875726 MPa.
For comparison study, the traditional genetic algorithm (GA), fast genetic algorithm (FGA) (Gao 2007) and the immunized genetic algorithm (IGA) are all applied for this example.
Parameters for three algorithms (GA, FGA and IGA)
n | m | p _{c} | p _{m} | |
---|---|---|---|---|
GA | 100 | 500 | 0.65 | 0.15 |
FGA | 100 | 500 | – | – |
IGA | 100 | 500 | – | – |
Comparison of model identification results
E/MPa | μ | α | K/MPa | |
---|---|---|---|---|
Theory values | 2157 | 0.2 | 0.165872 | 0.875726 |
Identified values by GA | 2135 | 0.182 | 0.171365 | 0.853471 |
Relative error of computing results for GA (%) | 1.02 | 9 | 3.31 | 2.54 |
Identified values by FGA | 2140 | 0.182 | 0.168442 | 0.853532 |
Relative error of computing results for FGA (%) | 0.79 | 9 | 1.55 | 2.54 |
Identified values by IGA | 2143 | 0.188 | 0.167812 | 0.854621 |
Relative error of computing results for IGA (%) | 0.65 | 6 | 1.17 | 2.41 |
As shown in Table 2, the results of the IGA are superior to the results of the GA and FGA; therefore, the computation effect in this study is reasonable.
As shown in Fig. 6, the NOF of IGA is much less than those of other algorithms, and then the computational speed of IGA in this study is very faster than those of other algorithms in the literature. In other words, the computational efficiency of the IGA in this study is the best and is superior to those of other algorithms.
The results of these studies conclude that the IGA method can be used to obtain a suitable model with higher accuracy and less effort.
Engineering example
The Huainan coal mine is located north of Huainan city, of Anhui Province in China. As an old mining, the entire mine has been integrated into the stage of deep mining. The different geological environments range from shallow rock roadway to the deep rock roadway; as a result, the mechanical characteristics are very complicated. To analyze the failure mechanism of the surrounding rock for deep rock roadway, a constitutive model of the surrounding rock based on the displacement measurement results must be identified. In this study, two main types of surrounding rock are identified: types III and II (Liu et al. 2010).
To analyze the stability of the roadway, some site monitoring studies have been performed, including surface displacement measurements and deep displacement measurements.
For the surface displacement measurements, the convergence between both sides and the convergence between the roof and the floor were completely measured. The layout of the monitoring points is illustrated in Fig. 7. The layout of the monitoring points for the deep displacement measurements is also illustrated in Fig. 7. The depth of each monitoring hole was 15 m. Six monitoring points were installed along each monitoring hole, with distances of 1, 3, 5, 7, 10 and 15 m, which are designated point 1, point 2, point 3, point 4, point 5 and point 6. Assuming the point at a depth of 15 m is stationary, the relative displacement of the remaining points can be obtained.
Based on the surface convergence displacement, the constitutive model of the surrounding rock can be performed by this new method.
The parameters of model identification method are as follows:
The number of individuals is 150, and the maximum number of evolutionary generations is 500.
E is between 6 and 15 GPa, μ is almost 0.3, C is between 1.0 and 1.5 MPa and \(\varphi\) is between 30° and 45°.
Thus, the identified values for parameters E and μ are agree with the suggestion values well. However, the two model parameters α and K can not be obtained directly from the tests, and they can be determined from the parameters c and φ based on the yield function. To obtain the model parameters, the yield function of the surrounding rock must be constructed, but it is a very hard work. Because this study is only to analyze the stability of surrounding rock, the hard work to construct the yield function of the surrounding rock is not conducted. Therefore, the model parameters α and K can only be verified by the comparison of measured deep multi-point displacements and the computed deep multi-point displacements by FEM with the identified model.
As shown in Fig. 11, the computation displacements correspond with the measured displacements. Therefore, the identified model for the surrounding rock of the −780 \(C_{13}^{S}\) floor haulage roadway in the Xieyi mine can adequately describe the real rock performance.
The layout of the monitoring points is illustrated in Fig. 12. The layout of the monitoring points for the deep displacement measurements is also illustrated in Fig. 12. The depth of each monitoring hole was 15 m. Six monitoring points were installed along each monitoring hole, with distances of 1, 3, 5, 7, 10 and 15 m, which are designated point 1, point 2, point 3, point 4, point 5 and point 6. Assuming that the point at a depth of 15 m is stationary, the relative displacement of the remaining points can be obtained.
The parameters of model identification method are as follows:
The number of individuals is 150, and the maximum number of evolutionary generations is 500.
As the same studies for surrounding rock of type III, the two mechanical parameters E and μ are verified by the suggestion values of mechanical parameters in previous study and the model parameters α and K are verified by measured deep multi-point displacements.
The suggestion values of mechanical parameters for surrounding rock of type II are as follows (Liu et al. 2010).
E is between 10 and 25 GPa, μ is almost 0.25, C is between 1.2 and 2.0 MPa and φ is between 40° and 55°.
Thus, the identified values for parameters E and μ are agree with the suggestion values well. The model parameters α and K are verified by the comparison of measured deep multi-point displacements and the computed deep multi-point displacements by FEM with the identified model.
As shown in Fig. 16, the computation displacements coincide with the measured displacements. Therefore, the identified model for the surrounding rock can adequately describe the real rock performance.
In this study, the surrounding rock is assumed as one equivalent homogeneous material. Therefore, the computed displacements at both sides are equal. In fact, the measurement displacements at both sides are different. In other words, the inhomogeneous displacement as shown in Figs. 11 and 16 can not be described in this study.
The engineering applications prove that the new model identification method proposed in this study can be used to identify a suitable constitutive model for the surrounding rock of the underground engineering with reasonable efficiency using only surface displacement measurements.
Conclusion
The identification of a rock constitutive model is very important. From the theory analysis of the elastic–plastic constitutive law of rock materials, the generalized law of constitutive model is presented. Using this generalized law, the problem of identification of a constitutive model can be transformed to parameter identification. Therefore, the new immunized genetic algorithm proposed by the author is applied to an elastic–plastic model identification problem. Using a numerical example and an engineering example, this new method is verified. The results indicate that the proposed method can be used to identify a suitable constitutive model with high accuracy and efficiency.
Declarations
Authors’ contributions
WG carried out the new algorithm studies and engineering applications, participated in the sequence alignment and drafted the manuscript. DC carried out the constitutive law studies. XW conceived of the study, and participated in its design and helped to draft the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The financial supports from The Fundamental Research Funds for the Central Universities under Grant Nos. 2014B17814, 2014B07014, 2016B10214 and B15020060 are all gratefully acknowledged.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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