# 数学代写|优化理论作业代写optimization theory代考|МАTH4230

## 数学代写|优化理论作业代写optimization theory代考|Validation by Synthetic Cases and Real PMTs

Then, the same identifications in the PMT and in the excavation tests were conducted again using the proposed algorithm. Table $4.7$ shows the optimization results, and Fig. $4.10$ shows the minimization process with increasing generation numbers, compared with other optimization methods. It can be seen that the preset solution was finally found by the proposed algorithm with a faster convergence speed, which demonstrates the high performance of the proposed algorithm.

Moreover, to further validate the performance of the enhanced algorithm, an inverse analysis of real PMTs investigated by Papon et al. [32] was conducted.

According to Papon et al. [32], two different PMTs (taken at a depth of 2 and $5.9 \mathrm{~m}$ ) were performed in different soil layers under a spread footing. For reproducing the in situ conditions, the initial stress state of the soil was defined under $K_0$ condition with the initial vertical stress equal to $31 \mathrm{kPa}$ and the horizontal stress equal to $22 \mathrm{kPa}$. The imposed PMT loading was displacement-controlled, and, at each step, the same displacement increment was applied all along the probe.

According to Papon et al. [32], the elastic modulus $E$, the friction angle $\phi$, and the cohesion $c$ are obtained by the identification procedure, whereas Poisson’s ratio was taken equal to $0.33$ and the correlation between the friction and the dilatancy angles $\psi=\phi-30^{\circ}$ was consideréd.

For the optimization, the initial population was set at 30 and was randomly generated using the algorithm SOBOL. The search domain for the linear elastic-perfectly plastic Mohr Coulomb model is the same as the one proposed by Papon et al. [32]. All the parameters of NMDE have the recommended values discussed above. The optimal results are shown in Table 4.8, and the simulations of the PMTs using the optimal set of parameters are shown in Fig. 4.11. The poor simulation at the relatively small strain (between 0 and $5 \%$ ) is found, which attributes to that the first part of the experimental curves (up to around $u(a) / a=4 \%$ ) was not taken into account in the calculation of the error function because of the unusual curvature at the beginning of the pressuremeter tests, due to the remolding of the soil along the cavity wall indicated by Papon et al. [32].

## 数学代写|优化理论作业代写optimization theory代考|General EPR Procedure

The evolutionary polynomial regression (EPR) is a data-driven method based on evolutionary computing, aiming to search for polynomial structures representing a system, which was first introduced by Giustolisi and Savic [1] with applications in the hydroinformatics and environment-related problems. A general EPR expression can be mathematically formulated as:

$$y=\sum_{j=1}^m F\left(\mathbf{X}, f(\mathbf{X}), a_j\right)+a_0$$
where $y$ is the estimated vector of output of the process; $a_0$ is an optional bias; $a_j$ is an adjustable parameter for the $j$ th term; $F$ is a function constructed by the process; $\mathbf{X}$ is the matrix of input variables; $f$ is a function defined by the user; and $m$ is the number of terms of the target expression.

According to Giustolisi and Savic [1], the first step in identifying the model structure is to transfer Eq. (5.1) to the following vector form:
$$\mathbf{Y}{N \times 1}(\boldsymbol{\theta}, \mathbf{Z})=\left[\begin{array}{ll} \mathbf{I}{N \times 1} & \mathbf{Z}{N \times m}^j \end{array}\right] \times\left[\begin{array}{llll} a_0 & a_1 & \ldots & a_m \end{array}\right]^{\mathrm{T}}=\mathbf{Z}{N \times d} \times \boldsymbol{\theta}{d \times 1}^{\mathrm{T}}$$ where $\mathbf{Y}{N \times 1}(\boldsymbol{\theta}, \mathbf{Z})$ is the least squares (LS) estimator vector of $N$ target values; $\boldsymbol{\theta}_{d \times 1}$ is the vector of $d(=m+1)$ parameters $a_j$ and $a_O\left(\theta^{\mathrm{T}}\right.$ is the transposed vector); and $\mathbf{Z}_{N \times d}$ is a matrix formed by $\mathbf{I}$ (unitary vector) for bias $a_0$, with $m$ vectors of variables
$\mathbf{Z}^j$. More details about the EPR can be found in Giustolisi and Savic [1].
Figure $5.1$ shows a typical flowchart for the EPR procedure [1]. The general functional structure represented by $f\left(\mathbf{X}, a_j\right)$ in Eq. (5.1) is constructed from elementary functions by EPR using an optimization algorithm strategy (such as genetic algorithm). Note that any optimization algorithm guaranteeing the global optimal solution can be employed in the EPR procedure. The building blocks (elements) of the structure are defined by the user based on understanding of the physical process. The selection of feasible structures to be combined is conducted through an evolutionary process, while the parameters $a_j$ in Eq. (5.2) are estimated by the least squares method.

# 优化理论代考

## 数学代写|优化理论作业代写optimization theory代考|General EPR Procedure

$$y=\sum_{j=1}^m F\left(\mathbf{X}, f(\mathbf{X}), a_j\right)+a_0$$

$\mathbf{Z}^j$. 有关 EPR 的更多详细信息，请参阅 Giustolisi 和 Savic [1]。

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