Soil compaction optimization with soft constrain

Abstract

The primary benefit of fuzzy systems theory is to approximate system behavior where analytic functions or numerical relations do not exist. In soil mechanics relationship between the causes and effects can be observed with laboratory tests but is difficult to develop analytical functions or numerical relations between input and output. In geotechnical optimization the most usual constraints represent state variables of structural response for each loading case. The aim of this paper is to define the soft constrain with adaptive network-based fuzzy inference system (ANFIS) in the soil mechanics. The developed soft constrain is than applied in non-linear programming (NLP) to obtain optimal solution. In the case of soil compaction the performance of the proposed optimization algorithm is evaluated. The main aim of soil compaction is to define optimal water content at which soil can be compacted to a densest state that improve their mechanical and physical properties.

Keywords

Soil compaction fuzzy relation equations adaptive network-based fuzzy inference system non-linear programming

1 Introduction

For many geotechnical projects, soils have to be compacted to a denser state to improve their mechanical and physical properties [1]. They are compacted with ramming, rolling and vibrating. The water content has the most important effect on soil compaction. The densest soil is obtained at the optimum water content. The laboratory compaction tests are used to determine the relation between water content and dry unit weight and optimum water content. The upper limit of compaction is defined with saturation lines [2].

The majority of current optimization schemes require designers to define crisp-valued constrains even when in the presence of subjective “soft” constraints, thereby ignoring the uncertainty that is inherent during the definition of early stage design constraints [3]. The optimization problem with a linear objective function subjected to the fuzzy relational equations based on the max-average composition was proposed by Khorram and Ghodousian [4]. Loetamonphong et al. [5] used the genetic algorithm and solved the multi-objective optimization problem with fuzzy relation equation constraints with max-min composition. Multi-objective optimization problems with Fuzzy relation equation constraints regarding max-average composition was improved by Khorram and Zarei [6]. Zhang et al. [7] proposed a hybrid multiobjective fireworks optimization algorithm (MOFOA) that evolves a set of solutions to the Pareto optimal front by mimicking the explosion of fireworks. The experimental tests and real-world applications in oil crop production in east China demonstrate the effectiveness and practicality of the algorithm. Solving high dimensional bilevel multiobjective programming problem using a hybrid particle swarm optimization (PSO) algorithm with crossover operator was presented in [8, 9].

Pasdarpour and et al. [10] described the incorporation of genetic algorithm methodology using fuzzy system for determining the optimum design of dynamic compaction. The results show that the genetic algorithm has abilities to optimize dynamic compaction design which can increase the depth of improvement about 20–30% more than traditional design. Soil compaction is also an important component of the land degradation syndrome which is an issue for soil management throughout the world [11]. Qin et al. [12] showed with application and evaluation that fuzzy slope positions effectively predict the spatial distribution of soil properties and can provide useful information for digital soil mapping.

The Adaptive network-based fuzzy inference system (ANFIS) is considered to be one of the intelligent tools to understand complex problems [13]. ANFIS is being successfully used in many industrial areas as well as in research [14 –20].

In this paper two optimization models are presented. The objective function is obtained from laboratory compaction test and constrain function is defined with soil phase relationship. First the optimization model with a nonlinear objective function and crisp-valued constrain is developed. Next the optimization model with fuzzy relation equation constrain is presented. The ANFIS was used to develop soft constrain function. The comparison of optimization models is possible because both crisp and soft constrain are based on soil phase relationship. Therefore the performance of the proposed optimization algorithm is evaluated. The present article aims: (1) to analyze the optimization model subjected to crisp and soft constraints, (2) to evaluate the performance of the proposed algorithm for selected laboratory compaction test and (3) to present the benefit of ANFIS where analytic functions do not exist, but relationship between the causes and effects can be observed with laboratory tests

2 Physical soil properties

Soil is three phase material, composed of solid, liquid and gas. The solid phase is creating from particles, which are different size, shape and from different minerals. The spaces between soil particles are called voids, mostly filled by water (fluid) and air (gas). If all voids are filled by water, the soil is saturated. Otherwise, the soil is unsaturated. If all voids are filled by gas, the soil is fully dry. Commonly known idealization of three phase soil is shown in Fig. 1. The physical and mechanical properties of soil are dependent on the relative proportions of each three phases.

2.1 Soil phase relationships

The total volume of the soil is the sum of the volume of solids (V _s) and volume of the voids (V _v), that are volume of water (V _w) and volume of the air (V _a). $V = V_{s} + V_{v} = V_{s} + V_{w} + V_{a} .$ (1)

The weight of the soil is the sum of the weight of solids (W _s) and the weight of water (W _w). The weight of air is negligible. $W = W_{s} + W_{w}$ (2)

In the interpretation of compaction terms, given below, are used for soil phase relationships. Water content (w) is the ratio of the weight of water (W _w) to the weight of solids (W _s). $w = \frac{W_{w}}{W_{s}} \cdot 100 % .$ (3)

Void ratio (e) is the ratio of the volume of void space (V _v) to the volume of solids (V _s). $e = \frac{V_{v}}{V_{s}} .$ (4)

Specific gravity (G _s) is the ratio of the weight of the solids to the weight of water of equal volume $G_{s} = \frac{W_{s}}{V_{s} \cdot γ_{w}} .$ (5)

The dry unit weight (γ _d), specific gravity (G _s), water content (w), degree of saturation (S _r) and water unit weight (γ _w) are related through the analytical equation $S_{r} = \frac{(1 + {wG}_{s}) \cdot γ_{d}}{G_{s} \cdot γ_{w}} .$ (6)

When S _r, γ _w and G _s are given the saturation lines are defined.

2.2 Soil compaction

Soil compaction is the densification of soil by reduction of void ratio. It is the expulsion of air, if water content remains constant. In saturated soil densification is possible only by reduction of water content. Maximum dry unit weight (γ _d,max) is the maximum unit weight that a soil can attain using a specified means of compaction.

Using (6) dry unit weight of soil is given as a function of water content (w) and degree of saturation (S _r). The theoretical maximum dry unit weight is obtained when S _r = 1. The theoretical maximum dry unit weight (γ _d) decreases with reducing of degree of saturation (S _r). Figure 2 shows relationships maximum dry unit weight (γ _d) and water content (w) for different degrees of saturation (S _r). Optimum water content (w _opt) is the water content allowing a soil to attain its maximum dry unit weight following a specified means of compaction.

2.3 Proctor compaction test

A laboratory test called Proctor test was developed to determine the maximum dry unit weight (γ _d,max) and corresponding optimum water content (w _opt). Using standard ASTM D1557 [21] a dry soil specimen is mixed with water and compacted in a cylindrical mold, internal dimension 101.6 mm in diameter and 116.4 mm high. Specimen is compacted by repeated blows from the mass of hammer, 2.5 kg, falling freely from 305 mm. The soil is compacted in three layers, each of which is subjected to 25 blows. For projects involving heavy loads a modified proctor test was developed. In this test, the hammer has a mass 4.54 kg and falls freely from a height of 457 mm. Four or more tests need to be performed on the soil using different water content. The last test is identified when additional water causes the unit weight of the soil is decrease. The results are expressed as dry unit weight (γ _d) versus water content (w). Figure 2 show an example of a compaction test. The results are presented in the form of a compaction curve: dry unit weight (γ _d) versus water content (w). This curve is obtained using data points for each compacted sample and connecting these points by smooth curve. The curve is expressed as third order polynomial function. The saturation lines are also plotted next to the compaction curve. Usually all the experimental data points fall under the 100% saturation line. Lower saturation lines define limits of possible maximum dry unit weight at select degrees of saturation (γ _d,max (S _r)).

3 Optimization model

The comparison of optimization models is presented. The first optimization model is subjected to crisp constrains and the second optimization model is subjected to soft constrains. As an interface for mathematical modeling and data inputs/outputs GAMS (General Algebraic Modeling System), a high level language [22], was used.

3.1 Optimization model with crisp constrains

As the problem of the soil compaction is non-linear, e.g. the objective function and (in)equality constraints are non-linear, the non-linear programming (NLP) optimization approach is applied. The general NLP optimization problem can be formulated as follows: $\begin{matrix} Maxz = f (x) \\ subjected to : (NLP) \\ g (x) \leq 0 \\ h (x) = 0 \\ x \in X = {x | x \in R^{n}, x^{Lo} \leq x \leq x^{Up}} \end{matrix}$ where x is a vector of the continuous variables defined within the compact set X. Functions f ( x ), g ( x ) and h ( x ) are nonlinear functions involved in the objective function z, inequality and equality constraints, respectively. All functions f ( x ), g ( x ) and h ( x ) must be continuous and differentiable. In the context of optimal water content and maximum dry unit weight determination, the objective function is determined. The objective function is expressed as third order polynomial function obtained from compaction test: $γ_{d} = 0.0014 w^{3} - 0.1003 w^{2} - 2.166 w + 2.3961 .$ (7)

The constrain function is defined with degree of saturation: $γ_{d} \leq \frac{S_{r} \cdot G_{s} γ_{w}}{1 + w \cdot G_{s}} .$ (8) With NLP optimization the optimum water content (w _opt) and maximum dry unit weight (γ _d,max) are obtained for various degree of saturation (S _r). The graphical presentation of optimization problem is given in Fig. 3.

3.2 Optimization model with soft constrains

In order to predict the dry unit weight, the ANFIS-COMP model was build, based on the data set presented in Table 1. The inputs of a model are water content (w), degree of saturation (S _r) and specific gravity (G _s). The output is the dry unit weight (γ _d). The data presented in Table 1 are obtained with (6) for water unit weight γ _w = 9.801 kN/m³. In this paper 45 data sets were used to develop ANFIS-COMP model with fuzzy toolbox [18]. The developed model is then applied in NLP optimization approach as a constrain function and present the saturation lines.

The structure of the ANFIS-COMP model is shown in Fig. 4. While the nodes on the left side represent the input data, the right node stands for the output. In a conventional fuzzy inference system, a number of rules is decided by an researcher/engineer who is familiar with the system to be modeled. There are no simple ways to determine in advance the minimal number of membership functions to achieve a desired performance level. In this attempt, the number of membership functions as-signed to each input variable was chosen empirically by examining the desired input-output data and by trial and error. For model ANFIS-COMP the, we choose two membership functions in each input (Fig. 5).

The calculation procedure of ANFIS-COMP model is as follows: first the membership grade of a fuzzy set is calculated; second the product of membership function for each rule is calculated; third the ratio between the i-th rule’s firing strength and the sum of all rule’s firing strengths is calculated; next the output of each rule is calculated; and final the weighted average of each rule’s output is calculated.

The first membership grade of a fuzzy set (A _i, B _i, C _i) was calculated with (9), (10) and (11): $μ_{A_{i}} (w) = exp [- {(\frac{w - c_{A_{i}}}{\sqrt{2} σ_{A_{i}}})}^{2}]$ (9) $μ_{B_{i}} (S_{r}) = exp [- {(\frac{S_{r} - c_{B_{i}}}{\sqrt{2} σ_{B_{i}}})}^{2}]$ (10) $μ_{C_{i}} (G_{s}) = exp [- {(\frac{G_{s} - c_{B_{i}}}{\sqrt{2} σ_{B_{i}}})}^{2}]$ (11) Where w, S _r and G _s are inputs to Gaussian membership functions and parameters c _Ai, c _Bi σ _Ai, σ _Bi, σ _Ci, σ _Ci are premise parameters (Table 2). In addition, the products between membership functions for every rule were calculated, see (12) and (13): $u_{1} = μ_{A_{1}} (w) \cdot μ_{B_{1}} (S_{r}) \cdot μ_{C_{1}} (G_{s})$ (12) $u_{1} = μ_{A_{1}} (w) \cdot μ_{B_{1}} (S_{r}) \cdot μ_{C_{1}} (G_{s})$ (13) where u _i represents the firing strength of the i-th rule. The weighted average of each rule’s output is defined as a ratio between the i-th rule’s firing strength and the sum of all rule’s firing strengths, see (14) and (15): ${\bar{u}}_{1} = \frac{u_{1}}{u_{1} + u_{2}}$ (14) ${\bar{u}}_{2} = \frac{u_{2}}{u_{1} + u_{2}}$ (15)

The output of each rule was finally determined as a sum of products between the weighted average of each rule’s output and a linear combination between input variables and consequence parameters: $γ_{d} = \sum_{i = 1}^{2} \bar{u_{i}} \cdot (a_{0}^{i} + a_{1}^{i} \cdot w + a_{2}^{i} \cdot S_{r} + a_{3}^{i} \cdot G_{s})$ (16)

Once the dry unit weight is defined the constrain function could be applied in NLP optimization approach, see (17). $γ_{d} \leq \sum_{i = 1}^{2} \bar{u_{i}} \cdot (a_{0}^{i} + a_{1}^{i} \cdot w + a_{2}^{i} \cdot S_{r} + a_{3}^{i} \cdot G_{s})$ (17)

4 Comparison of optimization models

The result of NLP optimization with crisp and soft constrain is given in Table 3. In the optimization with crisp constrain the Equation (8) was used. In optimization with soft constrain the (17) was used. In both cases the objective function is same and is defined in (7). With both optimization approaches the optimal water content w (%) and maximum dry unit weight γ _d,max (kN/m³) are obtained. The maximum error of optimization model with soft constrain depends on the accuracy of the ANFIS-COMP model. In the case of soil compaction problem the maximum error is 0.95% . Prediction accuracy of ANFIS model can be expected to improve as more data sets are utilized. Higher capacity of ANFIS model is important to obtain the exact optimal solution.

5 Laboratory soil testing

In the practice of geotechnical engineering, soil testing addresses real problems. The experiment data for three different soil types are presented in the form of a compaction curve: dry unit weigh γ _d,max (kN/m³) versus water content w (%). According to ASTM standard [21], for each soil type five test points should be obtained with standard compaction test. To obtain five well-placed points on the compaction curve, the water content is selected low for the first test point and is gradually increased for the other points. It should be about 4 to 5% below the optimum water content for the first point, and 4 to 5% above the optimum water content for the fifth and last point. Figure 6 show an example of a compaction test for Gravelly silt (MG). The maximum unit weight and optimum water content are determined by using two different methods referred to as method A and method B. Method A uses a conventional cubic polynomial regression fit to an experimental data set [1]. The optimum of method A is plotted as a solid circle. Method B uses ANFIS to obtain the best-fit curve to the experimental data. The optimum of method B is plotted as a solid triangle in Fig. 6. In this example, method B gives better optimum water content than method A. Figures 7 and 8 shows an example of a compaction test for Clayey sand (SC) and Low plasticity clay (CL).

6 Conclusion

The most important thing in the soil mechanics are laboratory test. With laboratory tests the relationship between the causes and effects can be observed. Usually the results of laboratory test are difficult to describe with analytic functions. In laboratory test the inputs are also vague and ambiguous. The primary benefit of fuzzy systems theory is to approximate system behavior where analytic functions or numerical relations do not exist. Since the solution set of the fuzzy relation equations is in general a non-convex set, when it is not empty, conventional nonlinear programming methods are not ideal for solving such a problem. In proposed optimization algorithm the ANFIS model is used to develop soft constrain. The model contains symmetric Gaussian membership functions which are continuous and differentiable. To evaluate the performance of the optimization with soft constrain the ANFIS model is based on the soil phase relationship and results are than compared with current optimization schemes with crisp-valued constrains. The comparison is made on selected laboratory compaction test. Since the deviation between the optimization models is very small, less than 1% , the optimization with soft constrain was successfully applied in the case of soil compaction. The fuzzy system is very useful in soil mechanic, because the numerical relations based on experimental measurements could be effectively developed. Those numerical relations are than used in the optimization approach as constrain function.

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