# 数学代写|优化理论作业代写optimization theory代考|MATH683

## 数学代写|优化理论作业代写optimization theory代考|EPR Modeling of Creep Index

Natural soft clays exhibit significant creep under both laboratory and in situ conditions after primary consolidation, which significantly influences the long-term safety of infrastructures in various fields, such as tunneling [7-10], excavation [11-13], embankment [14-20], urban land subsidence [21-25], etc. Usually, the creep property of soft clays is represented by the creep index $C_\alpha=\Delta e / \Delta \log (t)$, where $e$ is void ratio and $t$ is time during secondary compression. The creep index is a key parameter for most viscoplastic constitutive models applicable to the engineering practice [26-37], which is usually obtained by a conventional oedometer test. According to studies of [29, 30,34], the $C_\alpha$ corresponding to intact clays is not constant because of the effect of destructuration to the creep. In contrast, the $C_\alpha$ of reconstituted clay is an intrinsic property, which is the base for understanding the creep characteristic and thus more suitable to be adopted in practice [38]. Because of this, the attention is paid to the $C_\alpha$ of reconstituted clay in this study.

The creep property should relate to the microstructure of clay [28, 30, 39, 40]. Unfortunately, the microstructure of clay is expensive to measure which may lead to a practical obstacle. Physical properties can somehow reflect the microstructure of clay. Thus, practically, it is very convenient to get ideas of the intrinsic value of $C_\alpha$ only based on physical properties of clay. Some attempts have been made to correlate the $C_\alpha$ to some physical properties of soils (such as water content, void ratio, Atterberg limits) [41-47]. However, these correlations are only applicable for few clays and thus not enough reliable for soft clay engineering practice. Therefore, a robust and effective correlation between $C_\alpha$ and physical properties of clay is worth investigating.

Numerical regression is the most powerful and commonly applied form of regression used to solve the problem of finding the best model to fit the observed data [48-52]. Evolutionary polynomial regression (EPR) is a recently developed hybrid regression method [1] that has advantages in modeling the nonlinear complex problems. Applications in geotechnics include stability prediction of slopes [49, 53, 54], modeling of clay compressibility $[55,56]$, modeling of permeability and compaction characteristics of soils [57], evaluation of liquefaction potential of sand [58, 59], prediction of soil saturated water content [60], settlement prediction of foundations [61-63], evaluation of pile bearing capacity [64-66], pipeline failure prediction [67], and modeling of soil behaviors [68-71]. These successful applications have demonstrated that the EPR technique is superior to other soft computing techniques, such as artificial neural networks (ANNs) [72], genetic programming (GP) [48, 58]. More recently, the development of optimization algorithms $[2,31,55,73-78]$ can improve the EPR technique in a more adaptive way.

## 数学代写|优化理论作业代写optimization theory代考|EPR Modeling Process for Cα

Since selecting $\left(w_{\mathrm{L}}, I_{\mathrm{p}}\right)$ or $\left(w_{\mathrm{p}}, I_{\mathrm{p}}\right)$ is physically the same for evaluating the $C_\alpha$, based on the statistics results of database, four physical properties (CI, $w_{\mathrm{L}}, I_{\mathrm{P}}$, and $e$ ) were selected as the correlating variables of interest to training the EPR model. To attain the nonlinear creep behavior with a consecutively decreasing creep index $C_\alpha$ that fully relates to the soil density [47, 79], a general structure of EPR expression for $C_\alpha$ was proposed as:
$$\ln \left(C_\alpha\right)=\sum_{j=1}^m\left[f\left(\mathrm{CI}, w_{\mathrm{L}}, I_{\mathrm{P}}\right) e\right]+a_0$$
which was further expressed as:
$$\ln \left(C_\alpha\right)=\left(\sum_{j=1}^m\left[a_j(\mathrm{CI})^{\theta_{j 1}}\left(w_{\mathrm{L}}\right)^{\theta_{j 2}}\left(I_{\mathrm{P}}\right)^{\theta_{j 3}}\right]\right) e+a_0$$
where $a_0$ is a constant in the EPR equation; $a_j$ is the coefficient corresponding to $j$ term and $\theta_j$ is the vector of exponent. Note that the use of logarithm in $C_\alpha$ can guarantee the positiveness of $C_\alpha$.

To obtain an accurate and reasonable correlation, 120 data randomly selected in the prepared database were used for training and the remaining data were used for testing. For simplicity, the value of exponent was constrained to $[-2,2]$ with a step size to 1 . Also, the maximum number of terms was set to 8 for restricting the model complexity. For NMDE, the number of initial population was set to ten times of decision variables and the maximum generation was set to 200 . The probability of crossover CR is $0.3$. For NMS, the tolerance for convergence was set to $10^{-4}$. Independent multiple runs were performed to avoid randomness.

# 优化理论代考

## 数学代写|优化理论作业代写optimization theory代考|EPR Modeling Process for Cα

$$\ln \left(C_\alpha\right)=\sum_{j=1}^m\left[f\left(\mathrm{CI}, w_{\mathrm{L}}, I_{\mathrm{P}}\right) e\right]+a_0$$

$$\ln \left(C_\alpha\right)=\left(\sum_{j=1}^m\left[a_j(\mathrm{CI})^{\theta_{j 1}}\left(w_{\mathrm{L}}\right)^{\theta_{j 2}}\left(I_{\mathrm{P}}\right)^{\theta_{j 3}}\right]\right) e+a_0$$

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