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DEGREE PROJECT, IN STRUCTURAL ENGINEERING AND BRIDGES , SECOND LEVEL

STOCKHOLM, SWEDEN 2014

Optimization of Pile Groups

A PRACTICAL STUDY USING GENETIC

ALGORITHM AND DIRECT SEARCH WITH

FOUR DIFFERENT OBJECTIVE FUNCTIONS

ANN BENGTLARS & ERIK VÄLJAMETS

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Optimization

of Pile Groups

A practical study using Genetic Algorithm and Direct

Search with four different objective functions

A

NN

B

ENGTLARS

E

RIK

V

ÄLJAMETS

Master of Science Thesis

Stockholm, Sweden 2014

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TRITA-BKN. Master Thesis 409, 2014 ISSN 1103-4297 ISRN KTH/BKN/EX--409--SE KTH School of ABE SE-100 44 Stockholm SWEDEN

© Ann Bengtlars & Erik Väljamets 2014 Royal Institute of Technology (KTH)

Department of Civil and Architectural Engineering Division of Structural Engineering and Bridges

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Abstract

Piling is expensive but often necessary when building large structures, for example bridges. Some pile types, such as steel core piles, are very costly and it is therefore of great interest to keep the number piles in a pile group to a minimum.

This thesis deals with optimization of pile groups with respect to placement, batter and angle of rotation in order to minimize the number of piles. A program has been developed, where two optimization algorithms named Genetic Algorithm and Direct Search, and four objective functions have been used. These have been tested and compared to find the most suitable for pile group optimization. Three real cases, two bridge supports and one culvert, have been studied, using the program.

It has been difficult to draw any clear conclusions since the results have been ambiguous. This is probably because only three cases have been tested and the results are very problem-dependent. The outcome depends, for example, on the starting guess and settings for the optimization. However, the results show that the Genetic Algorithm is somewhat more robust in its ability to remove piles than Direct Search and is therefore to prefer in pile group optimization.

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Sammanfattning

Pålning är en dyr men ofta nödvändig åtgärd vid byggande av större konstruktioner såsom exempelvis broar. Vissa påltyper, såsom stålkärnepålar, är mycket kostsamma och det är därför av stort intresse att hålla ner antalet pålar i en pålgrupp.

Denna rapport behandlar optimering av pålgrupper med avseende på placering, lutning och rotationsvinkel med syftet att minimera antalet pålar. Ett program har utvecklats, där två optimeringsalgoritmer, Genetic Algorithm och Direct Search, samt fyra målfunktioner har använts. Dessa har testats och jämförts för att hitta de bäst anpassade för pålgruppsoptimering. Tester med det färdigställda programmet har även utförts på tre olika verkliga fall, två brostöd och en kulvert.

De erhållna resultaten har varit tvetydiga och några tydliga slutsatser har varit svåra att dra. Detta kan förklaras av att endast tre fall har studerats och att resultaten är mycket problemberoende. Resultaten är bland annat beroende av startgissningen och optimeringsinställningar. Dock visar resultaten att Genetic Algorithm är något mer robust i sin förmåga att ta bort pålar än Direct Search och är därför att föredra vid pålgruppsoptimering.

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Preface

This thesis has been written for the Department of Civil and Architectural Engineering, the division of Structural Engineering and Bridges at the Royal Institute of Technology, KTH. First we would like to thank our supervisors Christoffer Svedholm and Majid Solat Yavari for their support and advice. We would also like to thank our examiner Costin Pacoste who has been encouraging during the project.

Our gratitude to ELU Konsult AB for the opportunity to carry out the project there, and for lending us material for our case studies. Thanks to all coworkers at ELU for inspiration and support.

Finally we would like to give special thanks to Christoffer Svedholm for the great enthusiasm he has shown and for his help in developing the computer program.

Stockholm, June 2014

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Contents

Abstract ... i Sammanfattning ... iii Preface ... v List of Abbreviations ... ix 1 Introduction ... 1 1.1 Background ... 1 1.2 Previous Studies ... 2

1.3 Aim and Scope ... 3

1.3.1 Aim ... 3 1.3.2 Scope ... 3 1.4 Outline of Thesis ... 4 2 Theory ... 5 2.1 Optimization ... 5 2.2 Genetic Algorithm ... 7 2.2.1 Overview ... 7 2.2.2 Search method ... 7

2.2.3 Genetic Algorithm in Matlab ... 8

2.2.4 Genetic Algorithm for pile group optimization ... 11

2.3 Direct Search ... 11

2.3.1 Overview ... 11

2.3.2 Direct search in Matlab ... 12

2.3.3 Direct search for pile group optimization ... 13

2.4 Pile Group Analysis ... 14

2.4.1 Forces and deformations ... 14

2.4.2 Resisting earth pressure ... 16

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3 Method ... 19

3.1 Pile Group Optimization Program ... 19

3.1.1 General structure ... 19

3.1.2 Variables ... 20

3.1.3 Objective function ... 21

3.1.4 Constraints ... 25

4 A Simple Two-variable Optimization ... 27

4.1 Method ... 27

4.2 Results ... 30

5 Case Studies ... 35

5.1 General ... 35

5.2 Case I - Intermediary Bridge Support A... 37

5.2.1 Background ... 37

5.2.2 Results ... 40

5.3 Case II – Culvert ... 48

5.3.1 Background ... 48

5.3.2 Results ... 50

5.4 Case III – Intermediary Bridge Support B ... 53

5.4.1 Background ... 53

5.4.2 Results ... 54

5.5 Summary and Comparison ... 58

6 Discussion and Conclusion ... 63

6.1 Discussion ... 63

6.2 Conclusion ... 66

6.3 Future Research ... 66

References ... 69

A Number of Piles with Discrete Values ... 71

B Load Cases – Support A ... 72

C Pile Groups ... 75

Case I ... 76

Case II ... 82

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List of Abbreviations

α weight factor

λi Lagrange multiplier estimates, nonnegative and components of vector λ

ρ the positive penalty parameter ϕj penalty parameter

Aq transformation matrix

ci the nonlinear inequality constraint

ceqi the nonlinear equality constraint

Cq transformation matrix

DX horizontal deformation in x-direction

EI pile bending stiffness

f fitness value

f1-6 forces and moments in pile

fT maximum shear force

fM maximum bending moment

Fq force vector

GA Genetic Algorithm

i number of load combinations kd modulus of soil reaction Kq pile stiffness matrix

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Le buckling length

Nd,max,i maximum design normal force in piles in load combination i

Nd,min,i minimum design normal force in piles in load combination i

Pi ith pile

PCdist,start sum of distances to pile centre in starting guess

PCX x-coordinate of pile centre

PCZ z-coordinate of pile centre

Pq force vector at pile cap origin

PS Pattern Search

R external forces acting on pile cap origin

Rd,c,i design compression capacity of piles in load combination i

Rd,t,i design tension capacity of piles in load combination i

U displacement vector at pile cap origin S global stiffness matrix of pile group

si nonnegative shifts and components of the vector s

VZ rotation around z-axis

x1-6 displacements and rotations in pile

xint,ij x-coordinate of intersection between line i and j

Xq displacement vector

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INTRODUCTION.BACKGROUND

1

Introduction

This chapter starts with a description of the background of the topic and presents previous studies made in the area of pile group optimization. The aims and scope of this thesis are then stated and followed by a brief outline of the report.

1.1 Background

Pile groups are an essential part of many bridge constructions in Sweden. Piling is needed when the soil beneath the bridge supports is too weak to carry the loads, which is often the case. Designing a pile group is a difficult and sometimes very time-consuming task. One of the main reasons is that a pile group has a great number of input parameters and variables. There are geometric parameters such as the pile cap thickness, pile type and diameter, location, angle of rotation and batter (slope) of the piles and pile length. Other important parameters are the pile bearing capacity and external loads. All these parameters can be combined in an almost infinite number of ways. The larger the pile group is, the more complex the task becomes.

A designer is always searching for an optimal solution, which usually means the most cost effective solution. Some pile types are very expensive, costing up to 300.000 SEK per pile. Therefore, finding a solution with as few piles as possible is often the most cost effective. It is difficult however for a designer to find an optimal solution when the problem is this complex. There are no strict rules or guidelines when designing a pile group, instead designers mostly rely on experience and engineering judgement to establish some of the parameters before starting the analysis. Today, most pile groups are designed by calculating the section forces and deformations in the piles for a specific pile group configuration and then slightly adjusting the configuration and recalculating, until the results are satisfactory. Satisfactory meaning, for example, that the pile group is able to carry the design loads and that there are as few piles as possible in the pile group. Even though several of the parameters have been set, the process of finding a satisfactory solution becomes very iterative.

The iterative process and the large number of parameters makes pile group design an ideal target for computer optimization. One of the first attempts at pile group optimization using computers was made in 1981 by James Hill. Since then several more attempts focusing on

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INTRODUCTION.PREVIOUS STUDIES

different aspects have been made, such as those by Hoback and Truman and Chan et. al. (1992a) (2009).

This thesis will examine optimization of pile groups. Different optimization algorithms will be used and compared to develop a computer program that performs the design and optimization of a pile group with respect to location, batter and angle of rotation. Today there are computer programs that calculate the forces and deformations in the piles but no programs that provide a practical design and optimization of pile groups such as the one developed in this thesis project.

1.2 Previous Studies

As mentioned above, one of the earliest attempts at computerized pile group optimization was made by Hill (1981). In his report Hill presented a computer program that optimized the location and batter in order to reduce the cost of the pile foundation. The optimization was divided into several parts. First the pile cap was divided into a grid where piles were placed at all grid points and then the optimal batter was calculated. When the optimal batter had been obtained a search for the optimal spacing was carried out. This was done by a series of deletion passes where the least or most loaded piles were removed. When a set number of piles had been removed the spacing was increased. The process was repeated until the least number of piles possible was found. In the optimization Hill used the Nelder-Mead simplex method to find the optimal batter. The Nelder-Mead simplex method is a deterministic search method for unconstrained optimization. Hill transforms the constrained variables from the pile group optimization to unconstrained variables in order to use the Nelder-Mead method, which is said to be less time-consuming than methods for constrained optimization.

Hoback and Truman presented two papers where they used the Optimality Criteria to optimize the weight of steel in pile foundations (1992a) (1992b). The Optimality Criterion is a numerical gradient-based optimization method, which requires that the problem is modelled mathematically. In their solution, Hoback and Truman varied the pile diameter and batter. Hurd and Truman published a similar research where they optimized the weight of steel for pile foundations varying batter, pile diameter and number of piles using a 3-D computer program (2006). The optimization was performed using the Optimality Criteria. No details are given explaining how the computer program works.

The Genetic Algorithm has been used in different forms to optimize pile groups. A hybrid Genetic Algorithm was used to minimize the material volume of the foundation by mainly varying the pile diameter (Chan, et al., 2009). Liu et. al. used an Automatic Grouping Genetic Algorithm to minimize the cost (2012). They varied the pile diameter and the layout of the pile group. The piles were divided into different modules where all the piles had the same characteristics.

Another example of pile optimization was performed by Kim et. al., who used recursive quadratic programming to minimize differential settlements in a piled raft foundation (2001). Hwang et. al. used a discrete Lagrange multiplier method to minimize the construction cost of a bridge foundation (2011). There are two examples of pile foundation optimization where only pile length is varied in order to minimize differential settlements of the pile cap. (Chow

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INTRODUCTION.AIM AND SCOPE

& Thevendran, 1987) (Leung, et al., 2010). Similar to Hill, Chow and Thevendran used a direct search method for unconstrained optimization, transforming their variables from constrained to unconstrained, however using a different method of transformation.

Several articles have been published where different global optimization algorithms are used to optimize piled grillage foundations (Šešok, et al., 2010) (Belevičius & Šešok, 2008) (Belevičius, et al., 2011). They focus on the optimization methods Genetic Algorithm and Simulated Annealing. Belevičius et al. compared seven different algorithms and found that for grillages Simulated Annealing, Genetic Algorithm and the NEWUOA-algorithm were the best suited (2011).

Several of the authors mentioned above have based their optimization programs on the current national design codes. This allows for a more complete design tool, but limits the use of the program to those specific codes. The program developed in this thesis project is based on basic structural mechanics and does not depend on design codes, which makes it a more general tool and will not need updating if the codes are changed.

1.3 Aim and Scope

1.3.1 Aim

In this master thesis project a computer program will be developed that can design a pile group that is optimized with respect to certain parameters. These parameters are the coordinates in the horizontal plane, angle of rotation, batter and number of piles. Using this program, two main issues will be investigated.

The first issue concerns the optimization algorithm. The two different optimization algorithms Genetic Algorithm and Direct Search will be compared to find out which one is most suitable for the task of optimizing a pile group.

The second issue is the objective function of the algorithms. An objective function is the function that is optimized by the optimization algorithm and is the foundation for the optimization. Four different objective functions have been formulated and will be compared to see which is the most suitable for this problem.

To conclude, this thesis will answer the following two questions. Which optimization algorithm in this thesis is the most suitable for optimizing pile groups? Which objective function gives the best results in pile group optimization? The answers to these questions will aid in the development of a program for practical optimization of pile groups, which is the final aim of the thesis.

1.3.2 Scope

The thesis project will mainly focus on infrastructure foundations with end-bearing piles only. The optimization will be carried out in the programming language Matlab using two different

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INTRODUCTION.OUTLINE OF THESIS

algorithms, Genetic Algorithm and a type of Direct Search called Pattern Search. These algorithms are available in the Matlab optimization toolbox. An external program will be used to analyse the structural mechanics of the pile groups, i.e. no calculations on pile forces and deformations will be performed in the scope of the thesis project. The variables that will be used in the optimization are coordinates in the horizontal plane, angle, batter and number of piles.

To improve the possibility of receiving good and usable results, some restrictions and limitations have been set up. Pile type, pile length, thickness of slab and economics in the form of prices of steel and concrete are not optimized. Including all of these parameters in the optimization would have made the problem too complex and too large to fit into the time frame of the thesis work. Even though the parameters pile type, pile length and the thickness of the slab are not optimized, they are still part of the optimization as constant parameters. The designer using the program will be able to define what type of pile or thickness of slab that is to be used in the current solution.

Constraints are needed to ensure that the pile group obtained from the optimization works in practice. The following constraints are taken into account in the program; pile capacity, distance between piles, distance between piles and the edge of the slab, deformations and moment capacity.

The optimization program will be tested on three different cases. To make the testing of the algorithms and objective functions as general as possible, these cases are different in type, size, number of piles, pile type and loading.

1.4 Outline of Thesis

This thesis consists of six chapters. Chapter 1 introduces the topic and gives an overview of previous studies. The aims and scope of the thesis are also stated here. In chapter 2 a theoretical description of the optimization algorithms are presented. Here the Genetic Algorithm and Direct Search methods are explained in detail. The theory of pile group analysis is also described. Chapter 3 presents the structure of the optimization program. Chapter 4 presents a simple two-variable optimization in order to study the algorithms. Chapter 5 deals with the three case studies. Chapter 6 discusses the results and draws conclusions. Future research is also proposed in the final chapter.

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THEORY.OPTIMIZATION

2

Theory

Chapter 2 starts with an overview of optimization and optimization methods in general and motivates the choice of optimization algorithms used in the thesis. Further, the two algorithms chosen are presented and explained in detail. Finally, the theory of pile group analysis used for the program is described.

2.1 Optimization

Optimization is the process of searching for the optimal value of a function. The optimal value can be a maximum or a minimum value, a local extreme point or a global extreme point. A function can have only one optimum or many, depending on the type of function. There are a large number of different search methods and different ways to categorize them. One way is to divide methods into local and global optimization methods, aiming at finding either local or global optima. In a book on the Genetic Algorithm, the author Goldberg mentions three main types of traditional search methods, namely: calculus-based, enumerative and random search methods (1989). The calculus-based methods are, as the name suggests, based on classic calculus. The idea of the enumerative search methods is to search the whole function space one point at a time. These methods are very time consuming and are not practical in the case of pile group optimization, since the number of variables and the function space is to large. The random search methods search the function space at total random and are, similar to the enumerative methods, too inefficient to work for larger problems. Apart from these, there are natural algorithms, such as the Genetic Algorithm, which is discussed further on.

Search methods can also be divided into gradient- and non-gradient-based depending on whether they use the derivatives of the objective function. The gradient- or calculus-based methods are either direct or indirect (Goldberg, 1989). The indirect methods solve a system of equations based on the fact that the derivative is equal to zero at an extreme point. The direct method uses the concept of “hill climbing” moving in the direction of the steepest gradient to find an extreme point. The gradient-based methods are all local optimization methods, since they only find local optima.

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THEORY.OPTIMIZATION

The non-gradient-based methods are usually global optimization methods. They can be divided into deterministic and stochastic methods (Zabinsky, 1998). The stochastic methods use some form of randomness to aid the search while the deterministic methods use a predetermined procedure, which results in the same solution every time. Some search methods are capable of handling discrete values and others are restricted to continuous values. Figure 1 below describes one example of how Genetic Algorithm and Pattern Search can be categorized. Pattern Search can be either deterministic or stochastic depending on the poll method used.

Figure 1 – One example of categorization of algorithms.

An optimization problem always needs an objective function, in Genetic Algorithm called the fitness function, stating what is to be optimized (Pedregal, 2004). It also needs a clear definition of constraints and boundaries that limit the variables and the objective function itself.

For the problem of pile group optimization an algorithm is required that can handle non-linear constraints and a large number of variables. A non-gradient-based algorithm is preferable since the function space is non-smooth and the objective function cannot be modelled mathematically. It would be possible to use a gradient-based algorithm that approximates the derivatives, but this approximation can lead to errors such as loss of accuracy (Powell, 1998). The algorithm needs to be robust and efficient, since the problem includes many parameters and variables and uses a “black box” function. It is also favourable if the algorithm can handle discrete values, since all variables in a pile group are discrete in practice. Even though it is possible to calculate the coordinate of piles down to the last millimetre, the limited accuracy when constructing makes this impossible to carry out. To reduce the risk of error when constructing, it is also good to keep the coordinates as even as possible and to keep the number of different angles to a minimum.

Table 1 shows the different algorithms available in the Matlab global optimization toolbox. Simulated Annealing is a powerful optimization algorithm, which would have been a good

Optimization algorithms Gradient-based (local) Non-gradient-based (usually global)

Deterministic Pattern Search

Stochastic

Pattern Search

Genetic Algorithm

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THEORY.GENETIC ALGORITHM

option if it was capable of handling nonlinear constraints. There are two algorithms that fulfil all the requirements above, namely the Genetic Algorithm and Pattern Search. That is why they were chosen for this project.

Table 1 - The different algorithms available in Matlab optimization toolbox (The MathWorks, Inc., 2013).

Smooth objective function Non-smooth objective function Linear constraints or

bounds only

Global Search, MultiStart Simulated Annealing

All types of constraints Pattern Search, Genetic Algorithm

Pattern Search, Genetic Algorithm

2.2 Genetic Algorithm

2.2.1 Overview

One of the algorithms chosen for this thesis is the Genetic Algorithm (GA). The GA was developed by John Holland and his colleagues and students at the University of Michigan (Goldberg, 1989). The original aim was to create a search method that was more robust than traditional methods at the time. Inspiration was taken from nature and Darwin’s theory of evolution and survival of the fittest. It has since then been proved both theoretically and empirically to be a robust algorithm even in complex problems. In engineering problems one is often searching for a solution that is good enough within the available time-frame and not the best possible solution. This fits well with what Goldberg writes: The most important goal of optimization is improvement and It would be nice to be perfect: meanwhile, we can only strive to improve. This is one of the GA’s strong points.

The GA is a non-gradient-based global optimization algorithm (The MathWorks, Inc., 2013) (Goldberg, 1989). This means that GA can treat problems that are not clearly mathematically defined and thus have no derivatives. GA only uses the value of the objective function or fitness function called the fitness value.

2.2.2 Search method

When optimizing, GA starts by creating a set of starting points. In GA-terminology this is called a population of individuals (Goldberg, 1989). This is as opposed to other search methods that operate on a single point. From the initial population, individuals, or parents are chosen and are combined to make children or new individuals for the next generation. The choice of parents is based on the fitness values of the individuals. In a simple Genetic Algorithm, the next generation of individuals is created in three different ways, namely reproduction, crossover and mutation, see Figure 2 in section 2.2.3. These three operators are simple in themselves but combined they make a powerful tool for searching through large complex spaces.

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THEORY.GENETIC ALGORITHM

Reproduction, sometimes also called selection, is the process of selecting individuals and placing them in a mating pool (Goldberg, 1989) (The MathWorks, Inc., 2013). This can be done in many different ways. Goldberg describes the roulette process where individuals are given a probability of being selected based on their fitness values, higher fitness value means higher probability of being placed in the mating pool (1989).

Crossover means that parents in the mating pool are combined to produce children for the next generation (Goldberg, 1989). The combination is done by randomly exchanging parts of the individuals to create new ones.

A mutation is a random change of an individual that is passed on to the next generation (Goldberg, 1989). Mutation is described by Goldberg to be a secondary mechanism of evolution, compared to reproduction and crossover which are primary mechanisms. It is, all the same a necessary mechanism that prevents valuable genetic material from being lost.

2.2.3 Genetic Algorithm in Matlab

In Matlab a special version of the reproduction operation, called elite children, is available (The MathWorks, Inc., 2013). Elite children are individuals with the best fitness values that are sent directly to the next generation without undergoing any changes. This ensures that the best fitness values are preserved and not left behind. Apart from the elite children, the rest of the new population is created by crossover and mutation, as described in Figure 2. The ratio between mutation and crossover and the number of elite children can be specified in the algorithm.

Figure 2 - The three different operators, reproduction in the form of elite children, crossover and mutation (The MathWorks, Inc., 2013).

Figure 3 shows a short example, illustrating how GA works. Suppose that one is searching for the optimal combination of patterns on a square. There are three different patterns: stripes, grid pattern and dots. A square is divided in two parts, which can have any of the three patterns. Suppose that each combination of patterns scores a certain number of points, representing a fitness value that one wants to minimize. There are nine different combinations, which are shown in Figure 3, together with their respective fitness values.

Elite child

Crossover child

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THEORY.GENETIC ALGORITHM

The population size is then set to three and three random squares are chosen to be the starting guess for the optimization. The fitness values of these squares are calculated before starting the creation of the next generation. Each new individual will be created using a one of the three operators described above. First of all the square with the lowest fitness value is sent directly to the next generation as the elite child. Secondly, two of the squares are combined to form a crossover child. Finally, one of the squares randomly changes one of its patterns to form a new mutation child. The fitness values of the new generation are calculated and the process is repeated. Since this is a simple example, the optimal solution is found already in generation two.

Figure 3 - Example showing how Genetic Algorithm works.

In a case with more variables and a larger population, individuals with lower fitness values have a higher probability of being chosen to become a parent of crossover children. This means that better genes, or combinations of patterns, are carried on, while weaker ones are left behind. The mutation operator minimizes the risk of completely losing valuable genetic material and can diversify the search by finding new combinations, such as the dotted pattern in the example above.

3

4

7

2

5

9

8

6

1

Possible combinations and

their fitness values

Starting Guess

3

8

4

3

Elite child Crossover child Mutation child

8

2

Elite child Crossover child Mutation child

1

2

Generation 1

Generation 2

9

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THEORY.GENETIC ALGORITHM

The Genetic Algorithm in Matlab can handle both continuous and discrete values of its variables. Using discrete variables is preferable since it makes the algorithm more time efficient and narrows down the number of possible solutions.

Constrain ts

GA handles all types of constraints, such as linear and nonlinear equalities and in-equalities. It also handles bounds, which can be applied to the variables as lower and upper bounds. In a practical problem such as a pile group all variables need to have defined bounds. In the pile group optimization program only bounds and nonlinear inequality constraints are used, therefore only these are described further. The nonlinear inequality constraints are in the form c ≤ 0, where c is a vector containing all the different constraints. The constraints used are presented further in section 3.1.4. The bounds are in the form ub ≤ x ≤ lb, where x is the variable and lb and ub are the lower and upper bounds.

Nonlinear constraints are handled in two different ways depending on whether using discrete variables or not. When using continuous variables only GA uses the Augmented Lagrangian Genetic Algorithm (ALGA) (The MathWorks, Inc., 2013). ALGA formulates a subproblem where the fitness function and nonlinear constraints are combined using the Lagrangian and the penalty parameter.

Equation (2.1) defines a subproblem as:

Θ , , , = − log − + + 2 (2.1)

GA minimizes a subproblem until a required accuracy is achieved and the feasibility conditions are satisfied (The MathWorks, Inc., 2013). Then the Lagrangian estimates are updated and a new subproblem is formulated. This process is repeated until the stopping criteria are met. The structure of subproblems increases the number of function evaluations per generation significantly compared to when using only linear constraints. If no solution is found, the penalty parameter is increased by a penalty factor and a new subproblem is formulated.

If any of the variables in the optimization have been defined as discrete a different method for handling the nonlinear constraints is used. GA then substitutes the fitness function for a penalty function (Deep, et al., 2009). Equation (2.2) describes the fitness function for an individual. ! = " ! , ! $ %& '()* + +,- ! + -, . ℎ 01 (2.2)

This means that when Xi fulfils the constraints, its fitness value is equal the value of the

objective function f (Deep, et al., 2009). Otherwise, the fitness value is the sum of the worst feasible fitness value in the population and a penalty parameter, which is connected to the inequality constraints.

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THEORY.DIRECT SEARCH

2.2.4 Genetic Algorithm for pile group optimization

There are several reasons for using the Genetic Algorithm in pile group optimization. It is an algorithm that handles non-linear problems well, since it is non-gradient-based and the problem of pile group optimization is definitely nonlinear. It has also been used several times before in earlier attempts at pile group optimization such as those by Liu et. al. and Chan et. al. (2012) (2009). In Chan et. al. it is stated that GAs are suitable for pile group optimization since they have many variables that are discrete in nature and the objective function cannot be modelled mathematically (2009).

GA has several advantages. One is its robustness, which allows it to be applicable in many types of problems, including complex pile groups with a large number of variables (Goldberg, 1989). The fact that GA operates on a population of points instead of a single point widens the search area and reduces the possibility of finding a local optimum instead of the global. There are a few downsides with GA. In some cases when working with complex models and when many generations are needed to reach a good result the calculations can become computationally costly, especially if evaluation of the objective function is time-consuming (Javadi, et al., 2005). The GA can also experience convergence problems in cases where the crossover function is too effective and all individuals become the same. The GA then risks getting stuck without finding the optimal solution. Since the GA is specialized at finding global optima, it can have difficulties finding local optima (Chan, et al., 2009). To compensate for these shortcomings attempts have been made to combine GA with other search methods, forming hybrid Genetic Algorithms (Chan, et al., 2009) (Javadi, et al., 2005). It is possible to combine GA with other algorithms in Matlab to form hybrids, however this has not be done in this thesis.

As mentioned above, GA only uses the values of the objective function without having any knowledge about the function itself. Goldberg describes it as being blind (1989). In engineering problems the objective function often acts as a “black box” (Zabinsky, 1998). In this thesis project the analysis of the pile groups has been performed using an external program, functioning as a black box. Therefore, GA is well suited for the optimization task in this project.

2.3 Direct Search

2.3.1 Overview

Direct Search is a method for optimization available in Matlab. The term is, however not clearly defined and is sometimes used as a general expression for all non-gradient-based search methods (Kolda, et al., 2003). According to Kolda et. al. the term originally refers to the method presented by Hooke and Jeeves. Hooke and Jeeves describe the Direct Search as a: sequential examination of trial solutions involving comparison of each trial solution with the "best" obtained up to that time together with a strategy for determining (as a function of earlier results) what the next trial solution will be (1961).

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THEORY.DIRECT SEARCH

Since the definition from Hooke and Jeeves is quite general, many algorithms today fall under this category. Powell published a paper describing seven different algorithms, all under the title of Direct Search (1998). Storn and Price even categorize the Genetic Algorithm as a Direct Search method (1997). In this thesis project the Direct Search method used is the one available in the Matlab Global optimization toolbox and is described below.

2.3.2 Direct search in Matlab

In Matlab the Direct Search method is a non-gradient-based method, which operates from a single point (The MathWorks, Inc., 2013). Direct Search uses three different Pattern Search algorithms. The algorithms create a mesh of points around the current point. The mesh is created by adding the current point to a set of vectors, called pattern, multiplied by a scalar. Then the algorithm searches through the mesh point-by-point to find a point where the objective function has a better value. When a point with a better function value is found, this point becomes the current point at the next iteration.

The search through the mesh is called polling and the three different algorithms are the Generalized Pattern Search (GPS), the Generating Set Search (GSS) and the Mesh Adaptive Search (MADS) (The MathWorks, Inc., 2013). These three polling methods differ in the way the pattern vectors are created and chosen. GPS and GSS are deterministic methods, while MADS is a stochastic method. If the polling is successful, a point with a better function value is found, the mesh size expands. If it the polling on the other hand is unsuccessful, the mesh size shrinks.

In Pattern Search (PS), a search can also be used (The MathWorks, Inc., 2013). A search is an algorithm that runs before the poll, aiming to find a point better than the current point. If the search finds a better point with lower function value no polling is made that iteration. Different search types can be chosen in Matlab, including the three different polling methods presented earlier. However, since the search runs before the poll a search can be comparable to using a different starting guess. Figure 4 below describes how the algorithm works with the help of a flow chart.

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THEORY.DIRECT SEARCH

Figure 4 - Flow chart describing how Pattern Search works (The MathWorks, Inc., 2013).

Pattern Search can only handle continuous values of its variables. This is a minus because by using continuous values, the algorithm risks being less time efficient, since the search area will become infinitely large.

Constrain ts

Pattern Search can handle the same types of constraints as GA such as bounds and linear and nonlinear constraints (The MathWorks, Inc., 2013). Since Pattern Search only uses continuous values for its variables it uses only the Augmented Lagrangian to solve the nonlinear constraints, as described above in section 2.2.3.

2.3.3 Direct search for pile group optimization

No earlier published work on pile group optimization using Direct Search, as defined here, has been found. However, there are several reasons to try. Pattern Search meets all the criteria

Done? Yes Yes Yes Yes No No No No Search enabled? Success? Success? Poll Search Update current point

Expand mesh Refine mesh

Stop Start

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THEORY.PILE GROUP ANALYSIS

mentioned in section 2.1 and the MADS-method is stochastic, which should make it suitable for black-box objective functions as suggested by Zabinsky (1998). Since Pattern Search only operates on one point at a time, it might also be more time efficient than the GA, despite only being able to handle continuous variables.

2.4 Pile Group Analysis

The pile group optimization program uses an external program to analyse the pile groups. The external program calculates normal forces and deformations in the piles, based on input data given by the designer. It is used as a black box, where data is given as input and results come out with no attention given to the process in between. To calculate the forces and deformations in the piles the external program uses a method specified by the Swedish Commission on Pile Research in a report from 1978 (Bredenberg & Broms). A summary of the method is presented below, where all equations and derivations are taken from that report.

2.4.1 Forces and deformations

The method presupposes a rigid pile cap and linear elastic conditions. All coordinate systems are orthogonal and are defined using the right-hand rule. The forces and deformations in a single pile are calculated by assembling a stiffness matrix for that pile. A coordinate system 1-2-3 is defined at the head of the pile with direction 3 along the length of the pile. The forces, moments, rotations and translations are shown in Figure 5.

Figure 5 - Forces, moments, translations and rotations at pile head (Bredenberg & Broms, 1978).

The relationship between the forces and displacements is described by equation (2.3).

23 = 45!3 (2.3)

Where Fq is the force vector, Xq is the displacement vector and Kq is the stiffness matrix as

shown below. f1 f4 f3 f6 f5 f2 x3 x1 x2 x5 x6 x4

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THEORY.PILE GROUP ANALYSIS

A new coordinate system is defined which is parallel to the one of the pile cap. Fq and Xq can

be transformed into this system using a transformation matrix Aq defined as:

where,

and α is the direction of the horizontal projection of the pile and β is the angle between the pile and the vertical plane.

The pile forces and displacements can now be transformed again to the coordinate system of the pile cap using a transformation matrix Cq.

Where z1-3 are the pile coordinates in the pile cap coordinate system and m states whether the

piles are fixed or hinged at the connection with the pile cap, m=0 for hinged and m=1 for fixed connection.

The force vector in the pile head, Fq, is at equilibrium with a force vector at the pile cap

origin, Pq. This gives equation (2.4):

63 = 758523 (2.4)

The displacements at the pile head can be expressed as in equation (2.5): !3 = 8597 59: (2.5) Fq f1 f2 f3 f4 f5 f6

()

= Kq k11 0 0 0 k51 0 0 k22 0 k42 0 0 0 0 k33 0 0 0 0 k24 0 k44 0 0 k15 0 0 0 k55 0 0 0 0 0 0 k66

( )

= Xq x1 x2 x3 x4 x5 x6

()

= Aq A' 0 0 A'

( )

= A' cos β( ) cos α( ) cos β( ) sin α( ) sin β( ) − sin α( ) − cos α( ) 0 sin β( ) cos α( ) sin β( ) sin α( ) cos β( )

(

)

= Cq 1 0 0 0 z3 z2 − 0 1 0 z3 − 0 z1 0 0 1 z2 z1 − 0 0 0 0 m 0 0 0 0 0 0 m 0 0 0 0 0 0 m

( )

=

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THEORY.PILE GROUP ANALYSIS

where U is the displacement vector at the pile cap origin.

The external forces, R, acting on the pile cap origin are equal to the sum of all the pile forces Pq. See equation (2.6).

; = 63

< 3

(2.6)

where n is the number of piles.

The relationship between the forces and movements of the pile cap can be expressed as in equation (2.7):

; = =: (2.7)

where S is the global stiffness matrix assembled from all the individual piles.

By combining equations (2.3) and (2.5) the pile forces can be calculated. This is shown in equation (2.8).

2.4.2 Resisting earth pressure

In some cases it is necessary to take into account a resisting earth pressure on the piles. This can greatly reduce the normal forces but requires that shear forces and moments in the piles are calculated.

Calculating the maximum shear force, fT, in a pile is done by combining the shear forces f1

and f2 obtained from equation (2.8) using equation (2.9).

The maximum bending moment, fM, in a pile is dependent on the connection type between the

pile and pile cap. If the connection is pinned the moment depends on the soil type and the modulus of soil reaction. For cohesion soil the maximum moment occurs 0.8Le below ground

level and is determined by equation (2.10)

where Le is the buckling length of a beam supported by a Winkler bed, calculated in equation

(2.11).

23 = 45859759: (2.8)

> = ? + (2.9)

@ = 0.32 >DE (2.10)

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THEORY.PILE GROUP ANALYSIS

In which kd is the modulus of soil reaction and EI is the pile bending stiffness.

If the connection between pile and pile cap is fixed the maximum moment, fM, occurs at the

connection and is calculated by combining the bending moments f4 and f5 using equation

(2.12).

2.4.3 Pile centre

The pile centre is a geometrical entity, which can help guide the designer when trying to find a solution to a pile group. It is common practice, when designing pile groups, to use the 2D-pile centre. The 2D-pile centre is then considered in two orthogonal planes, usually the transversal and longitudinal directions of the pile cap and is defined as follows. For the 2D-case, if the stiffness matrix of the pile group is assembled in a coordinate system, which has its origin in the pile centre and the correct orientation, then all three rows in the stiffness matrix are independent as can be seen in equation (2.13) (Samuelsson & Wiberg, 1995). If only the position of the pile centre is considered then the rotation is independent of the translations, as in equation (2.14).

When designing a pile group it is favourable to try to place the piles such that most of the applied loads act as close to the pile centre as possible, see Figure 6 below. This is because a load that acts through the pile centre only gives rise to a displacement in the pile group. Having loads that act close to the pile centre reduces the moments in the pile group and thereby the normal forces in the piles.

@ = ? F + L (2.12) (2.13) (2.14) f1 f3 f4

( )

k 0 0 0 k 0 0 0 k

( )

x1 x3 x4

( )

= f1 f3 f4

( )

k k 0 k k 0 0 0 k

( )

x1 x3 x4

( )

=

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THEORY.PILE GROUP ANALYSIS

Figure 6 - Example showing the pile centre and force lines from the applied loads in one direction.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −5 0 5 10 15 X [m] Z [ m ]

Force lines and pile centre ZX

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METHOD.PILE GROUP OPTIMIZATION PROGRAM

3

Method

The Method chapter presents the pile group optimization program. All the objective functions tested are also described in detail.

3.1 Pile Group Optimization Program

3.1.1 General structure

The pile group optimization program is built around the optimization algorithm. First, the program needs a starting guess, supplied by the designer, to begin the optimization. The starting guess can be anything from a wild guess to a pile group that has already been analysed. One of the main concepts of the program is to remove piles to find a pile group with as few piles as possible. The program can remove piles but never add more piles than there are in the starting guess. The designer is able to choose the piles that can be removed from the starting guess.

Another main concept of the program is instructions. The designer defines instructions that describe in which way the pile variables are to vary. Instructions can be given to single piles or sets of piles for the coordinates in the horizontal plane, the batter and angle of rotation. Figure 7 shows the definition of the angle of rotation and batter for a pile. Since instructions can be given to sets of piles the program can keep piles in straight rows and columns with a certain relative distance between themselves. The program can also be instructed to keep the pile group double or single symmetric. If the solution is to be symmetric, only part of the pile group needs to be supplied as input, the rest is created by a mirroring function in the program. This means, if the pile group is to be double symmetric, only one quarter of the pile group is given as input. Symmetric pile groups are desirable since it reduces the risk for errors during construction.

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METHOD.PILE GROUP OPTIMIZATION PROGRAM

Figure 7 - Angle of rotation and batter of piles.

The program consists mainly of a function that receives the starting guess with instructions and converts them into input data for the optimization algorithm, an objective function and a nonlinear constraints function. The objective function calculates the function that is to be optimized and the nonlinear constraints function handles all the constraints and limits of the optimization. After the optimization the best pile group is saved and can be used as a new starting guess or as a final solution.

3.1.2 Variables

The variables in the optimization depend on the instructions given by the designer. If the designer only gives instructions concerning the angle of rotation of the piles, the variables in the optimization will be only the angle of rotation. The more instructions that are given, the more variables are in the optimization. Since every single pile can be given these instructions, the number of variables will most likely increase significantly with increasing number of piles.

It is possible to give any number of instructions to a pile group as long as the instructions do not contradict each other. Defining sets of piles and giving instructions to them is a way to reduce variables since all the piles in a set, for a given instruction, are represented by one variable. When using the number of piles as the objective function another set of variables is added as described below.

When using PS and Number of piles with GA, continuous variables are used. Since the variables need to be discrete in order to follow the instructions, they are converted. This conversion is done before evaluating the fitness values and nonlinear constraints.

Angle of rotation [°] Batter [N:1] y x z Pile

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METHOD.PILE GROUP OPTIMIZATION PROGRAM

3.1.3 Objective function

Four different types of objective functions have been tested for the optimization program. Three of them concern pile forces and are therefore similar. The other is the number of piles, which differs in several ways. The largest difference is the way in which piles are removed and the type of variables used. When the objective function is the number of piles, piles are removed directly by the algorithm and in this implementation, all variables are continuous. In the other three cases, the algorithm searches for a solution that satisfies the constraints. When a solution is found, the least compressed pile is removed. The program continues to remove the least compressed pile until the pile capacities are exceeded, then the algorithm starts over to find a solution with the new number of piles. For GA, the variables in these three cases are discrete. The four objective functions are described in detail below.

Force cap aci ty ratio

The first objective function uses the ratio between the compression and tension normal forces and the pile capacities and is a measure of utilization. The maximum and minimum forces are normalized with respect to their capacities in order to obtain a more even weighting between them. The tension ratio is only used if the minimum force in the pile group is a tension force, i.e. is negative. The sum of the force ratios is used to prevent the algorithm from being stuck with to low or to high forces in the piles. If the tension capacity is close or equal to zero, ten percent of the compression capacity is used instead to stop the second term going towards infinity. Equation (3.1) shows the definition of the force capacity ratio objective function, or Force ratio. = M NN O NN P Q$ RS;J, TU, J,V, W + Q$ RX SJ, <, ;J, , XW , ;J, , < −1 $ [ SJ, <, < 0 Q$ RS;J, TU, J,V, W + Q$ R +SJ, <, + 0.1 ∗ ;J,V,W , ;J, , > −1 $ [ SJ, <, < 0 Q$ RSJ, TU,; J,V, W , SJ, <, > 0 (3.1)

Figure 8 shows a plot of the fitness function. The capacities have been set to ± 1000 kN and the maximum normal force varies from 0 to 200 kN while the minimum normal force varies from -100 to 100 kN. It is clear that the function will strive to minimize both forces, without getting any tension and the minimum value is where both forces are zero.

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METHOD.PILE GROUP OPTIMIZATION PROGRAM

Figure 8 - Fitness function of the objective Force ratio.

Force difference

The second objective function, called the Force difference, gives the absolute value of the difference between the maximum pile force and the minimum force. This gives a measure of the distribution of forces in the pile group. A large difference between the maximum and minimum forces gives a high function value and vice versa. Equation (3.2) shows the definition of the function.

= Q$ S TU, − Q S <, (3.2)

Figure 9 shows a plot of the Force difference function. The forces vary in the same span as in Figure 8. This function surface differs from the Force ratio function since it has its minimum value along a line where the maximum and minimum forces are positive and equal. The function will strive to lower the maximum forces, get rid of tension forces and keep the difference between the maximum and minimum forces as small as possible.

0 50 100 150 200 −100 −50 0 50 100 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.1 0.15 0.2 0.25 0.3 Nmin [kN] Nmax [kN] f [-]

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METHOD.PILE GROUP OPTIMIZATION PROGRAM

Figure 9 - Fitness function of the objective Force difference.

Force ratio and di stance to pi le cen tre

The position of the pile centre is often used by designers to find working pile groups. As mentioned earlier, it is usually preferable that the pile centre is located at the intersection of force lines. In this objective function, the distance from the pile centre to the intersections of the force lines is used as guidance.

The following section describes the distance between the intersection of the force lines and the pile centre. Since the two different directions, ZX and ZY, are calculated analogously, only the ZX-direction is defined below. The equation of the force line i and j respectively can be described by equation (3.6). These equations are derived from the different load cases. Each load case consists of three forces, Fx, Fy and Fz and three moments, Mx, My and Mz. The

slope of the force lines is calculated in equation (3.3). An eccentricity e is calculated in equation (3.4). The intersection with the z-axis is calculated by multiplying ex by kx as is

shown in equation (3.5). The values for the ZY-plane are calculated by exchanging x for y and vice versa. The intersection between the force lines can be calculated as in equation (3.7). ^U =22_ U (3.3) U= @ab` (3.4) 0 50 100 150 200 −100 −50 0 50 100 0 50 100 150 200 250 300 0 50 100 150 200 250 300 Nmin [kN] f [-] Nmax [kN]

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METHOD.PILE GROUP OPTIMIZATION PROGRAM cU = U∗ ^U (3.5) d = ^ ∗ + c , d- = ^-∗ + c- (3.6) < , - =c^ − ^-− c - , d< , - = ^ ∗ e c-− c ^ − ^- f + c (3.7)

The sum of the distance from the intersection between the force lines to the pile centre can then be calculated as shown in equation (3.8) below.

6gJ * ,hi = ? 6gi− < , - + 6gh− d< , -jklm

(3.8)

The objective function is stated as follows in equation (3.9). The function is similar to the force ratio objective function, but here the distance to the pile centre has been added to guide the optimization. A weight factor, α, has been added and set to 0.5. Further on this objective function is written as the Force PC function.

= M NN O NN PQ$ RS;J, TU, J,V, W + Q$ RX SJ, <, ;J, , XW + n 6gJ * 6gJ * ,* T) , ;J, , < −1 $ [ SJ, <, < 0 Q$ RS;J, TU, J,V, W + Q$ R +SJ, <,+ 0.1 ∗ ;J,V,W + n 6gJ * 6gJ * ,* T) , ;J, , > −1 $ [ SJ, <, < 0 Q$ RS;J, TU, J,V, W + n 6gJ * 6gJ * ,* T) , SJ, <, > 0 (3.9) where, 6gJ * = 6gJ * ,hi+ 6gJ * ,ho

The function surface of the Force PC function will probably be similar to the Force ratio function seen in Figure 8. However, the PC-term in the function is very difficult to predict since it is a function of all the piles in the pile group and is therefore hard to visualize.

Nu mb er of pil es

Since the final goal of pile group optimization usually is to reduce the number of piles, it is natural to let one of the objective functions be the number of piles. When the objective function is the number of piles, each removable pile gets a variable, true or false, that is entered into the optimization. An obvious downside to this objective function is that is does not work when the goal is to only find an optimal placement and not to remove any piles. Equation (3.10) shows the objective function.

= 6 (3.10)

The number of piles objective function uses continuous variables instead of discrete. This is because it was shown during testing that discrete variables gave very poor results, see appendix A.

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METHOD.PILE GROUP OPTIMIZATION PROGRAM

3.1.4 Constraints

In the optimization program, there are two types of constraints. All variables have upper and lower bounds that are defined through the instructions given by the designer. Apart from these bounds there are nonlinear inequalities defined in the nonlinear constraints function. The nonlinear inequalities form a vector, c. The optimization algorithm aims to keep the elements of c equal to or below zero. The nonlinear inequalities used are:

• Smallest distance between piles minus the smallest allowed distance • Smallest distance to edge of pile cap minus the smallest allowed distance • Number of piles too close to each other

• Number of piles closer to edge than allowed

• Maximum pile force minus compression pile capacity for each load combination • Minimum pile force minus tension capacity for each load combination

• Maximum moment in piles minus moment capacity (in case of resisting earth pressure)

• Maximum deformations minus allowed deformations (only in Case II)

These constraints are necessary to obtain a practical pile group, since the algorithm only considers a solution successful if the constraints are fulfilled.

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ASIMPLE TWO-VARIABLE OPTIMIZATION.METHOD

4

A Simple Two-variable Optimization

This chapter presents a simple example where the two algorithms are tested in order to visualize how the search procedures work.

4.1 Method

To gain a better understanding of how the two algorithms work, a simple two-variable optimization has been carried out. A double symmetric pile group containing eight piles subjected to one vertical load of 500 kN and two horizontal loads of 100 kN each was set up. Since the pile group was double symmetric, only two piles were needed to start the optimization. Each pile was given only the angle of rotation, ϕ, as its variable, leading to an optimization with two variables. The two piles used to set up the optimization were piles number one and two in Figure 10, which shows the whole pile group. The resultant of the horizontal forces applied at a 45°-angle as shown in Figure 10. Piles 1 and 2 were given the angles 270° and 0° (360°). They were allowed to vary between 270° and 360° in one-degree increments, 91 different angles for each pile. This meant a total of 8281 combinations. The pile capacities were set to 2000 kN in compression and -500 kN in tension.

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ASIMPLE TWO-VARIABLE OPTIMIZATION.METHOD

Figure 10 – First starting guess of the pile group with the applied force, F.

Figure 11 – The second starting guess of the pile group.

The function to be minimized was the Force ratio objective function. Since there were only two variables the function surface could easily be plotted. Combinations where the piles collided or the pile forces became too high or too low were not plotted. Figure 12-Figure 14 show the function surface. The starting point is in the upper left corner (270,360) of Figure 12 and has a function value of 0.94.

Figure 12 – Overview of the function surface.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −1.5 −1 −0.5 0 0.5 1 1.5 X [m] Y [ m ] 1 2 3 4 5 6 7 8 φ1 φ2 45˚ F −2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 X [m] Y [ m ] 1 2 4 3 5 6 8 7 270 280 290 300 310 320 330 340 350 360 270 280 290 300 310 320 330 340 350 360 −1 0 1 2 3 4 5 φ1 [°] φ2 [°]

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ASIMPLE TWO-VARIABLE OPTIMIZATION.METHOD

Figure 13 – The function surface, showing the extreme values of the function.

Figure 14 – The function surface, where the global minimum is visible between high peaks.

260 280 300 320 340 360 260 280 300 320 340 360 0 10 20 30 −1 0 1 2 3 4 5 φ2 [°] φ1 [°] f [-] 260 280 300 320 340 360 260 280 300 320 340 360 0 5 10 15 20 25 −1 0 1 2 3 4 5 φ2 [°] φ1 [°] f [-]

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ASIMPLE TWO-VARIABLE OPTIMIZATION.RESULTS

It is clear that the function is non-smooth with a number of local minima. The global minimum is located at (289,289), as can be seen in Figure 12 and Figure 14. The function value there is 0.26. A pile group with those angles is shown in Figure 15. This solution is obviously very difficult for any algorithm to find since it is located at a single point between two very high ridges as can be seen in Figure 14. This means that at exactly the specified angles the pile forces will be low, but any deviation from those angles due to errors or inaccuracies in construction or variations in the force angle leads to very high forces in the piles and a high risk of failure. Hence, the best available solution would be the point (360,270) which is exactly at the other end of the surface from the starting point, see Figure 12. The function value at that minimum point is approximately 0.49.

Figure 15 – The optimal but unstable solution of the pile group configuration.

Since the initial starting guess happened to be located at a local minimum point so far from the desired minimum point another starting guess was created with the angles 340° and 310°, closer to the desired point. This is to get a comparison between a difficult and an easy starting guess. Figure 11 shows the second starting guess.

The optimization was run with both algorithms on both starting guesses to see how they performed. Settings were chosen to fit this case as well as possible and to show the search paths of the algorithms in a clear way. GA was given a population size of 6 and a maximum number of generations of 5. Pattern Search used the MADSPositiveBasisNp1 poll method and was given an initial mesh size of 150. The function values were plotted on the function surface to give an idea of the search path.

4.2 Results

Figure 16 shows the performance of the Genetic Algorithm on the first starting guess, where number one shows the position of the starting guess. Identical function values have been sorted out to give a clearer picture. Since GA works with a population of individual solutions, it takes the starting guess as the first individual and then randomly creates other individuals to fill the initial population. One can see that after trying several random locations at point 3, 4, 5 and 8, GA is gradually directed towards the best function values. Had it been allowed to

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −1.5 −1 −0.5 0 0.5 1 1.5 X [m] Y [ m ] 1 2 4 3 5 6 8 7

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ASIMPLE TWO-VARIABLE OPTIMIZATION.RESULTS

continue over more generations it would most likely have continued towards the minimum point in the bottom right corner.

Figure 16 – Result for GA, when the first starting guess (270,360) was used.

The performance of the Pattern Search algorithm is shown in Figure 17 and Figure 18. It is clear that even though it finds its way over the first ridge, it gets stuck in the local minimum point (270,305). The performance of PS is very dependent on the initial mesh size. Since the mesh size is decreased if no better solution is found, it is difficult for PS to make large jumps in order to find new areas to search. GA has a clear advantage here, since it can search many points simultaneously over the whole search area. Figure 18 shows that the area around the minimum at (270,305) is thoroughly searched. This, however, does not help the algorithm move towards the desired point at (360,270).

270 280 290 300 310 320 330 340 350 360 270 280 290 300 310 320 330 340 350 360 14 13 12 10 2 9 6 7 11 3 4 5 8 1 −1 0 1 2 3 4 5 φ1 [°] φ2 [°]

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ASIMPLE TWO-VARIABLE OPTIMIZATION.RESULTS

Figure 17 - Result for PS, when the first starting guess (270,360) was used.

Figure 18 – Closer view of the result of PS, first starting guess.

When starting from the easier starting guess, both GA and PS find their way to the minimum rather quickly, GA needing slightly fewer function evaluations than PS. One can also see that GA searches a wider area than PS, with search points over the whole function surface. It is important to point out that, in this case, a gradient-based algorithm would be the best choice

270 280 290 300 310 320 330 340 350 360 270 280 290 300 310 320 330 340 350 360 3 4 5 9 12 7 13 14 15 17 18 1 2 6 8 10 11 16 19 −1 0 1 2 3 4 5 φ2 [°] φ1 [°] 268 270 272 274 276 278 280 298 300 302 304 306 308 310 312 314 316 318 3 4 5 9 12 7 13 14 15 17 18 2 6 8 10 11 16 19 φ1 [°] φ2 [°]

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ASIMPLE TWO-VARIABLE OPTIMIZATION.RESULTS

since the function surface is a smooth slope between the starting point and the minimum. Figure 19 to Figure 22 show the search paths of GA and PS respectively.

Figure 19 - Result for GA, when the second starting guess (340,310) was used.

Figure 20 - Closer view of the result of GA, second starting guess.

270 280 290 300 310 320 330 340 350 360 270 280 290 300 310 320 330 340 350 360 28 29 27 26 20 24 25 23 19 22 13 17 21 14 15 18 6 11 16 10 12 7 1 3 9 2 5 8 4 −1 0 1 2 3 4 5 φ1 [°] φ2 [°] 344 346 348 350 352 354 356 358 360 362 268 269 270 271 272 273 274 275 276 277 28 29 27 26 20 24 25 23 19 22 13 17 21 14 15 18 6 11 16 10 φ1 [°] φ2 [°]

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ASIMPLE TWO-VARIABLE OPTIMIZATION.RESULTS

Figure 21 - Result for PS, when the second starting guess (340,310) was used.

Figure 22 - Closer view of the result of PS, second starting guess.

270 280 290 300 310 320 330 340 350 360 270 280 290 300 310 320 330 340 350 360 24 30 36 27 28 22 23 31 26 29 25 35 18 19 9 16 34 10 12 13 17 21 4 6 7 15 14 33 8 11 2 1 5 20 32 3 −1 0 1 2 3 4 5 φ1 [°] φ2 [°] 348 350 352 354 356 358 360 362 268 269 270 271 272 273 274 275 276 277 278 24 30 36 27 28 22 23 31 26 29 25 35 18 19 9 16 34 10 13 17 21 6 15 φ1 [°] φ2 [°]

References

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