Optimization of Shear Wave Velocity (Vs) From a Post ...

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Settlement Using a Genetic Algorithm Multi-Objective NSGA II. Kamel Goudjil. 1. , Badreddine Sbartai. 2. Abstract - Several studies applied heuristic methods ...
International Review of Mechanical Engineering (I.RE.M.E.), Vol. 11, N. 3 ISSN 1970 - 8734 March 2017

Optimization of Shear Wave Velocity (Vs) From a Post-Liquefaction Settlement Using a Genetic Algorithm Multi-Objective NSGA II Kamel Goudjil1, Badreddine Sbartai2

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Abstract - Several studies applied heuristic methods such as neuronal networks (ANN), genetic algorithms (GA) and particle swarm optimization (PSO) to predict and optimize soil parameters. Therefore, the aim of the present paper is to optimize shear wave velocity of soil. In this direction, the genetic algorithm NSGAII was used to find the optimum values of shear wave velocity of soil that give an actual value of 0.3m settlement post-liquefaction. The results show that our genetic algorithm has been successfully employed to optimize the shear wave velocity (vs) with acceptable test errors of (R2=0.9999, RMSE=4.029). Moreover, this algorithm can be easily used for prediction and optimization of other geotechnical systems. The genetic algorithm can solve other problems in this field than the presented in this study. Copyright © 2017 Praise Worthy Prize S.r.l. - All rights reserved.

Nomenclature

PSO SPT VEGA

Introduction

The shear wave velocity (vs) is a leading geotechnical property that leads to the evaluation of the shear modulus. The latter is necessary in the dynamic analysis that covers a wide range of geotechnical applications including underground constructions [1], deep foundations [2], soil-structure interaction [3], machinery foundations [4], response of the free field [5]and susceptibility of the soil to liquefaction [6]. Gmax can be experimentally obtained using two methods : resonant column device or Bender elements. However, these data require high quilty and undisturbed samples. Unfortunetly, due to a financial issue driven by high investiment costs of such equipements and technical limitation due to cohesionless soils, it is important to develop data based on high performance models. To do so, several researches have investigated the relationships between Vs and other tests(CPT,SPT and BPT) [7]. Other researchers have used stochastic methods, such as neural networks and simple genetic algorithms to predict the shear wave velocity (vs) according to geotechnical soil parameters [8] and [9]. Even the NSGAII genetic algorithm has been largely applied to predict and optimize different engineering systems such as mechanical, electrical, computer science,…etc. However, this study presents the first attempt to explore the potential of applying this method to optimize shear wave velocity of soil (vs). Therefore, first we apply a single objective genetic algorithm.To do this, we tried to answer the following question:  What are the values of (vs) along the soil profile that induce an overall settlement close to the observed settlement?

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Best non-dominated set of solutions Last front of solutions Objective function First and second objective functions Lower boundary Number of members in population Standard penetration test blow counts with corrected energy Initially population Multi-objective optimization problem. Next population Offspring population Coefficient of determination Root-mean square error Settlement. Upper boundary Shear wave velocity of soil Vector of non-dominated solutions. Lower and upper bounds of (vs).

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P0 POMO Pt+1 Q0 R2 RMSE Sett UB vs X Xlower, Xupper ANN BPT CPT GA NPGA NPGAII NSGA NSGA II

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F1 Flast Fobj Fobj1, Fobj2 LB N N60

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Keywords: Genetic Algorithm (Nsga Ii), Liquefaction, Shear Wave Velocity, Settlement

Artificial neural network Becker penetration test Cone penetration test Genetic algorithm Niched Pareto Genetic Algorithm Niched Pareto genetic algorithm II Non-dominated sorting genetic algorithm Non-dominated sorting genetic algorithm II Particle swarm optimization Standard penetration test Genetic Algorithm for Vector

Copyright © 2017 Praise Worthy Prize S.r.l. - All rights reserved

https://doi.org/10.15866/ireme.v11i3.11226

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Kamel Goudjil, Badreddine Sbartai

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Genetic Algorithm NSGA II

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II.2

II.3

Method of Non-Dominated Sorting

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Individuals of the current population are sorted to form several Pareto fronts. All non-dominated population individuals received the No. 1 rank and defined the front 1. These individuals are removed from the population and the rest is again sorted. Similarly, all non-dominated population individuals receive the rank No. 2 and define the front No. 2. The operation is repeated until all individuals have a rank (figure 1). II.4

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Pareto Optimal Solutions

The definition of Pareto optimal solution directly results from the notion of dominance that means that it is impossible to find a solution that improves performance on a criterion without causing degradation of another criterion. That is why, in the multi-objective optimization, the notion of compromise is always mentioned.

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To solve the multi-objective optimization problem, many geneticalgorithms have been developed [12]. Among the most significant there are:: - The Genetic Algorithm for Vector or VEGA Evaluation, Niched Pareto Genetic Algorithm(NPGA) using tournament selection, based mainly on the Pareto dominance, The NPGAII algorithm, based on the degree of domination of an individual, The NSGA or Not dominated Sorting Genetic Algorithm, The algorithm Micro-GA, referring to algorithms with small populations, Finally the algorithm NSGA-II, based on a classification of individuals in several level The latter uses a procedure based on non-dominated sorting that is faster than its predecessor based on nondominance or Pareto optimal, an elitist approach that preserves population diversity and safeguards the best solutions found in previous generations and a comparison operator based on a calculation of the distance crowding. In this paper, NSGA II was used. Before describing the operating principle of the NSGA II algorithm, the following concepts must be explained first: Genetic operators, Pareto optimal solution, Sort Method non-dominated and distance approximation (Crowding Distance).

one cycle of the genetic algorithm. It is done by applying genetic operators: selection, crossover and mutation. The selectionis to chooseindividualsfromwhich to createthenext generation.The selection of individuals is mostly based on their evaluation function for singleobjective problems and other parameters that will be described later as Pareto rank and crowding distance for multi-objective problems. Several selection operators exist among of these methods:roulette-wheel selection, Ranking Selection and tournament selection. The crossover is the first step in a reproduction process; the genes of the parents are used to form a new chromosome. Several crossover types are available: Single point crossover, two point crossover, Uniform crossover and Arithmetic crossover. The mutation is the modification of one or more genes of the selected individual to introduce variability in the population with a certain probabilityPm between zero and one.

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Nevertheless, with the data that we have, the genetic algorithm method failed to identify the values of (vs) that correspond to the actual values. That is why: the question was reformulated as follows:  What are the values of (vs) along the soil profile that give an overall settlement equal to the observed settlement (0.3m) and that correspond to the actual characteristics of the soil in question? The appropriate response to the above question is to find solutions (values (vs)) that simultaneously satisfy two objectives. The first goal is that the induced settlement of these solutions is equal to the observed settlement (0.3m) and the second goal is that these solutions should be very close to actual values. This makes the problem a multi-objective problem because we opted for the multi-objective genetic algorithm NSGA II [10]. The data usedin this paper is from the reportof Boulangeret al[11]. Thevalue (0.3 m) is the observed settlementoccurredin thesite named'Moss landing, located in MontereyBay, California' after LomaPrietaearthquake1989.

Crowding Distance

In case two individuals have the same rank, Deb. et al [10] conceptualized a criterion called Crowding Distance. The latter represents the average distance of each objective between the two closest points, located on either side of the same solution of the Pareto front.This technique maintains a good diversity throughout the population, and allows exploring a wider space of solutions (figure2). II.5

Working Principle of NSGA II

NSGA II varies from simple genetic algorithm only in the way the selection operator works. The crossover and mutation remain equal. We describe hereafter the working principle of NSGA II as shown in figure 3.

Genetic operators

Reproduction plays a fundamental role in the transition from one generation to another. This represents

Copyright © 2017 Praise Worthy Prize S.r.l. - All rights reserved

International Review of Mechanical Engineering, Vol. 11, N. 3

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Kamel Goudjil, Badreddine Sbartai

defined population size. To fix this issue, the solutions of the last set Flast have been classified based on the crowded-comparison operator in descending order. This process keeps updating until the stopping criteria is reached.

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Fig.1 Illustration of non-dominated sorting method of NSGA II

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Fig.1 Schematic of the NSGA-II procedure

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Problem Formulation

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As we mentioned above, our work is divided in two parts. In the first, the problem is treated as a single optimization problem, while in the second part as a multi-objective optimization problem. The database used to calculate the settlement has been taken from the paper by Idriss and Boulanger [11].

Fig.2 The crowding distance calculation

III.1

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Initially, a random parent population P0 is created using the creation function (the function that creates the initial population, see the options of Matlab toolbox). The population is sorted based on non-domination. A fitness (or rank) equal to its non-domination level (1 is the best level, 2 is the next-best level, and so on) is assigned to each solution. Thereafter, the usual tournament selection, recombination, and mutation operators are used to create an offspring population of size N inscribed by Q0. Then, a combined population Rt=Pt+Qt is formed. The population Rt of size 2N is sorted according to non-domination.To ensure elitism, all the previous and current population members are considered in Rt. In the second step, solutions belong to top non-dominated set F1,which are the best solutions in the combined population and must be classified more than the rest. As long as the size of F1 is smaller than the initial population, then all members of the set F1 can be chosen for the new population Pt+1. The remaining members of this population, can be classied according to their rank. Accordingly, the solutions of F2 set are taken next, then the solutions of F3 set, etc. This process is carried on until no more sets can be hosted. However, it can be noted that the number of solutions from set F1 to set Flast is greater than the

Problem of Single Objective Optimization

In this part, the mathematical formulation of the problem is defined as follows: min F x