Anna

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Design Optimization of a Boom for Drill Rigs

Ida Haglund

Anna Håkansson

EXAM WORK 2011

Mechanical Engineering

Postadress: Besöksadress: Telefon:

Box 1026 Gjuterigatan 5 036-10 10 00 (vx) 551 11 Jönköping

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Postadress: Besöksadress: Telefon: Box 1026 Gjuterigatan 5 036-10 10 00 (vx)

551 11 Jönköping

This exam work has been carried out at the School of Engineering in

Jönköping in the subject area mechanical engineering. The work is a part of the Master of Science programme.

The authors take full responsibility for opinions, conclusions and findings presented.

Examiner: Niclas Strömberg, JTH Supervisors: Niclas Strömberg, JTH

Oskar Sjöholm, Atlas Copco Scope: 30 credits

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Abstract

This thesis describes how a design optimization of a boom structure for drill rigs can be set up and performed where the range is increased and the weight reduced. A program that calculates the coverage area based on the boom angles and

dimensions is connected with a finite element analysis of a boom model that is auto-generated by a macro, using commercial optimization software. The macro was recorded while creating the model and dimensions have been replaced by varying parameters. Workflows are created in the optimization software where suitable design of experiments and optimization algorithms are selected.

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Summary

Atlas Copco Rock Drills AB develops and manufactures a machine called Boomer. It is a series of large, high capacity drill rigs for tunneling and mining, used to drill blast holes in the rock. One or several arms with drills in the ends are mounted on the Boomer. These arms are called booms and this thesis has been focusing on BUT 45 L which is the newest and largest boom.

The two-dimensional area in front of the boomer that can be reached by the boom drill is called coverage area. This area depends on the dimensions and angles of the boom structure, but so does the boom weight and stiffness. One boom can be up to 12 meters and weigh more than 5000 kg and it is Atlas Copcos wish to make a lighter boom. At the same time, a maximized coverage area is preferable since it results in larger drifts.

The purpose with the thesis was to develop the prerequisite to perform a design optimization of the boom structure which maximized the coverage area and minimized the weight. The stresses and deflection in the structure had to be kept within allowed limits.

A program was created in MATLAB to calculate the coverage area dependent on the boom dimensions and angles, using forward kinematics according to Denavit-Hartenberg. To obtain the weight and to make sure that the boom structure would not collapse due to high stresses, a finite element model was created in Abaqus CAE. Macros were recorded while creating the model and the script that was an outcome of the macros where worked through in detail. Ten dimensions on the boom were replaced by parameters and the script could be used to auto-generate and analyze the complete model with given parameters.

The optimization software modeFRONTIER was used to connect the area calculation and FEA-model and to perform the optimizations. Two single-objective and one multi-single-objective workflow were created. The first workflow maximized the coverage area and the resulting dimensions were used as constants in the second workflow where the weight was minimized by reducing the cross-section of the boom body. In the multi-objective optimization, the coverage area was maximized and the weight minimized in the same workflow. Suitable design of experiments and optimization algorithms for each optimization were selected in modeFRONTIER.

By running the optimizations, it was identified that the coverage area could be increased while the weight was reduced, with allowed stresses and deflections.

Keywords

Forward kinematics Denavit-Hartenberg Design optimization

Multi-objective optimization

Finite element analysis Metamodeling

Design of experience

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Preface

This thesis work has been carried out by two students at the School of Engineering at Jönköping University as a part of the Master of Science programme in Product Development and Materials Engineering.

We would like to thank our supervisors for their help and support throughout this thesis work

Niclas Strömberg – for his objectivity and discretion. Oskar Sjöholm – for his enthusiasm, commitment and faith in us.

Furthermore, we would also like to thank Peter Öberg and Atlas Copco Rock Drills AB for giving us the opportunity to realize this project. Last but not least, we would like to thank Björn Ryttare for his patience and for devotion of time despite the lack of it.

Ida Haglund Anna Håkansson

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1 Introduction

The introduction section discusses the background of the project, the regarded product as well as the defined problem.

1.1 Background

Due to a constant wish of shortened product development cycles, fast ways to evaluate design concepts at an early phase in the design process are requested. Advanced Finite Element Analysis software and computer support provides the possibility of performing so called virtual prototyping. By adding optimization to the process, the work can be made even more efficient as an optimal design of a product can be retrieved faster. (Jurecka, 2007)

Atlas Copco Rock Drills AB in Örebro has realized the benefits of optimization and expects to implement this in their design process.

1.2 Atlas Copco Rock Drills AB

Atlas Copco Rock Drills AB is a part of the Atlas Copco Group, founded in Sweden in 1873. Atlas Copco Group is a world leading company within development and manufacturing of equipment for heavy industry. The company is divided into several divisions which act within three different business areas - Compressor Technique, Construction and Mining Technique, and Industrial Technique (Atlas Copco, 2010).

1.3 The Boomer and the BUT 45 L

Within the area of “Construction and Mining technique” various equipment and machines for heavy construction work are developed – one is the Atlas Copco Boomer. The Boomer, which is a series of large, high capacity drill rigs for mining and tunneling, is used for drilling blast holes in the rock, see Figure 1.1. The machine is placed in the end of the drift where one or several booms mounted on the rig allow the operator to reach a large area of the gallery. These booms are named BUT and a following model number. This thesis has focused on the BUT 45 L, which is the newest boom in the BUT product family. A drawing can be seen in Appendix A: Drawing of a BUT 45 L.

The BUT 45 L is a five-axis boom and the largest boom in the BUT-series. It is stiffer than its predecessors and consequently offers more precise positioning and a larger service range. (Atlas Copco, 2007)

Since a 165 kg drill with a rotational motor with 650 Nm torque are mounted to the end of the boom stiffness is of great importance as large forces arise in the up to twelve meter long boom structure while drilling. (Atlas Copco, 2009)

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Figure 1.1: A Boomer with three booms.

1.4 Problem description

The two-dimensional area which the boom reaches is called coverage area and is an important factor when constructing and developing new booms. The coverage area is linked to the dimensions and deflection angles, which also affects the forces in the boom structure, the total weight and the required stiffness. In fully extended position the BUT 45 L boom is 11,4 meters long and weighs over 5000 kg.

Figure 1.2: A Boomer in action, drilling holes in the rock using the drills mounted on the booms.

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If the boom structure becomes heavier extra weight must be compensated for in the Boomer in order to keep stable and prevent from tipping over. The maximum boom swing angle is set to a fixed value since to large swing angle results in a risk of tipping over. It is desirable to reduce the weight and the arisen stresses in the boom in order to be able to change the configuration of the dimensions, resulting in a larger cover area.

The current boom is not developed by optimization and the possibility that a local optimum were found is large. If a configuration exists where the coverage area of the boom are increased at the same time as the total weight are sustained or even reduced, the theory of a local optimum is plausible. To find out, an optimization routine can be used and a number of concepts will be compared and evaluated by the computer.

In order to create an optimization the design of the boom has to be rigorously documented and examined. The objective function in optimization depends on design variables and constraints. Without explicit answers to what is to be optimized, it will be impossible to fulfill the main purpose. More on how optimizations are carried out will be described in chapter 2: Theoretical background. Depending on the answer to what is to be optimized a suitable optimization algorithm has to be chosen. Based on the knowledge retrieved during the mapping and analyzing process a suitable method will be selected. The linking between the software has to be designed for implementation in the current research and development at Atlas Copco Rock Drills AB. Consequently, it has to be able to communicate with the available software at the department.

1.5 Problem formulation

The problem formulation has throughout the project been;

“How can an optimization of the boom structure, which maximizes the coverage area and minimizes the total weight while the stresses in the structure are kept within allowed limits, be set up and performed?”

To solve the problem, two sub-questions were identified;

“How can the coverage area be calculated so that the boom dimensions can be changed with an accurate outcome as a result?”

and

“How can a FE-analysis of the boom be carried out, that functions realistic for all configurations, which gives the stress results and volume as output?”

1.6 Purpose and aim

The main purpose with the project is to develop the prerequisite that enables an optimization which maximizes the coverage area and minimizes the weight of the BUT 45 L while the stresses in the structure are kept within allowed limits. An optimization would imply a shortened developmental time of upcoming booms.

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The aim is to run an optimization of the boom and to compare the results with the current configuration.

1.7 Delimitations

The BUT 45 L is available in two different versions, one with the feeder mounted on the top and one version with the feeder mounted on the side of the boom body. In this thesis only the top mounted version is managed.

The optimization will concern the boom structure from boom support to gear flange. This delimitation has been made in accordance with the engineers on Atlas Copco since the tripod solution dimensions has proven to be the crucial factor when increasing the coverage area. The angles and dimensional variables can be seen in Appendix B: Angles and variables.

1.8 Outline

Each section of this report deals with the three main areas described in the problem formulation and how they are solved; the calculation of coverage area, the FE-analysis of the boom and the optimization of the boom where the three subjects are connected.

The theoretical background explains the method used to calculate the coverage area and a brief introduction to FE-analysis and element types. The theory behind optimization problems is also explained as well as metamodeling, design of experiments and fast optimizers in the optimization software modeFRONTIER™.

In the method and implementation section there is a description of how the coverage area calculation are carried out and how the current FEA-model and new FEA-model differ. It also deals with the mathematical formulation of the optimization problem and how the problem can be set up in modeFRONTIER™ Under findings and analysis the result of the final optimization will be revealed and compared to the current boom configuration.

The project is concluded with a discussion and further opportunities are presented.

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2 Theoretical background

In the theoretical background, the calculation routine used for the coverage area, forward kinematics, will be explained. It will also concern the theory of

optimization, explaining metamodels, different design of experiments and the fast optimizers available in modeFRONTIER™.

2.1 Forward Kinematics

In order to calculate the coverage area of a boom, the original developer of the calculation program turned to the area of robot control and modeling. A boom is similar to a robot arm in the way that they both consist of links and joints which forms a kinematic chain. To determine the position of an end effector of a kinematic chain, depending on the known values of the joint variables, forward kinematics is used. Within the area of forward kinematics a method called the Denavit-Hartenberg Convention, or DH convention, is commonly used.

In this methodology it is assumed that all joints only have a single degree-of-freedom. Links with more than one degree-of-freedom can be modeled as a sequence of one-degree-joint with zero spacing. By this assumption it is possible to describe the displacement of the joint in terms of one rotation (or one translation for prismatic joints). Every link on the kinematic chain is assigned a rigid coordinate frame. As a joint is being put into action the connected link will move in space. The position of a link is described in terms of the preceding link in a homogenous transformation matrix . This matrix is a function of the rotation of the joint (Spong et. al, 2006).

Introducing a framework and assigning the coordinates frames according to the DH convention makes it possible to represent the motion of each link with four basic transformations. An arbitrary homogeneous transformation matrix generally consists of six parameters. By making the following assumptions the reduction in parameters can be made (Spong et. al, 2006):

1. The axis xi is perpendicular to the axis zi-1 2. The axis xi intersects with the axis zi-1

The rigidly attached frame to the link does not have to lie within the physical link but can exist in free space. The Figure 2.1 shows how coordinate frames which fulfills the assumptions for DH convention will look like.

Due to the DH convention of assigning coordinate frames the transformation matrix now depends on the following four parameters, two rotations, θ and α, and two translations, a and d. The matrix looks like the following:

sin sin sin cos

sin cos cos cos sin sin

0 sin cos

0 0 0 1

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The parameter a is usually named link length, α is named link twist, d is the named

link offset and θ is the joint angle. How these are measured can be seen in Figure 2.1.

The first three parameters are constant for a known link while the joint angle is variable (Spong et. al, 2006).

a

Figure 2.1: Coordinate frames, oi-1 and oi, which satisfies the assumptions made in the DH convention (Spong

et. al, 2006).

There is no unique way of assigning the coordinate frames, derivation may differ but the result will end up the same. A summary of the procedure of assigning coordinate frames according to Denavit-Hartenberg is found in Appendix C

Forward Kinematics.

2.2 FE-analysis

When designing new products, the physical behavior of the geometry is important to analyze since it can predict how the final product will respond when in use. By dividing the investigated geometry into smaller parts, so called finite elements that are connected by shared nodes, the behavior of each element as well as the total performance of the physical structure can be obtained. Every element represents a discrete part of the total geometry and a collection of elements are called mesh (Dassault Systèmes, 2009).

The stress and strain in each element can be determined by computing the displacement of the nodes. Both the magnitude and distribution can be analyzed. The most critical regions can give the engineer a hint of where and how to redesign the object with a more equally distributed stress as a result. When computers are used to simulate the finite elements, a fast and accurate result will be achieved on the condition that suitable selections are made by the engineer throughout the procedure. (Dassault Systèmes, 2009)

d

o

i

o

i-1

y

i

y

i-1

x

i

x

i-1

z

i-1

z

i

α

θ

2

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2.2.1 FE-elements

There are eight commonly used element types for stress analysis available in Abaqus; continuum, beam, truss, shell, rigid, membrane, infinite and connector elements. The three element types used in the new FEA-model are continuum, beam and truss elements which main features are explained below (Dassault Systèmes, 2009).

Beam and truss elements are both one-dimensional line elements which act in two or three dimensions and have been assigned stiffness against deformations. By approximating that a three-dimensional part is slender, the beam or truss can be treated as a one-dimensional element in three dimensions given a constant cross section. This assumption can only be made if the cross-section is small compared to the axial length. The type of deformations that can be supported is the main difference between beam and truss elements. (Dassault Systèmes, 2009)

Truss elements

The only supported deformations in trusses are compression and tension i.e. originated from axial forces. No forces normal to the truss axis or moments are supported. If the cross section is not manually defined in Abaqus, an area will be assumed by the software. A material must be selected but it is not allowed to define a material orientation in trusses. Two types of truss elements are available in Abaqus; 2-node straight elements that have a constant stress and use linear interpolation for positions and displacements, or 3-node curved elements that uses quadratic interpolation which offers a varying strain along the structure (Dassault Systèmes, 2009).

Beam elements

In beam elements deformations can arise from axial forces, shear, torsion and bending forces in contrast to truss elements where only axial forces are supported. The beam section can generally not be deformed in its own plane and the use of beam elements should therefore be considered carefully in each case. An exception to this assumption is for example when open section beams are used and warping occurs (Dassault Systèmes, 2009).

Beams have few degrees of freedom and are geometrically simple which implies that the computational time is reduced, one of the main advantages with beam elements. The required bending moment must be defined as a function of the curvature, the torque as a function of twist, or the axial force as a function of strain. Two types of section definitions are available in Abaqus; general beam section where the cross section properties are established in the preprocessor once or a beam section integrated during the analysis where the behavior is recomputed during the analysis (Dassault Systèmes, 2009).

The beam is created along a wire region and the section profile must be defined as well as the section orientation. The cross-section selection affects the beam behavior and can either be chosen from the list in Abaqus or created manually. They can either be solid or thin-walled and for the later closed or open sectioned (Dassault Systèmes, 2009).

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Continuum elements

Continuum or solid elements are the standard volume elements in Abaqus and can be used when contact, plasticity and large deformations occur but also in linear analysis. There are several solid element types available and it is important to choose a suitable type for the particular case. Some characteristics that can be considered is if the element should include hexahedral/quadrilaterals or tetrahedral/triangles, first or second order, reduced or full integration, normal, hybrid or incompatible mode formulation. To get an accurate result the elements should have as equal size and be as well-arranged as possible. The solid part must be assigned a section which can consist either of one material consistently or several layers of materials in a laminated composite design (Dassault Systèmes, 2009).

2.3 Optimization Theory

By using optimization to a defined problem, a solution can be found that fulfills the given requirements and maximizes or minimizes the outcome.

2.3.1 The Optimization Process

For carrying out an optimization there exist some methodology/process for setting up and solving the assignment, see Figure 2.2. In this section the process and the activities for each step is explained.

First of all the problem has to be identified in order to be simplified and formulated in a mathematical way. The reality is often very complex and reasonable estimations and delimitations have to be made for the problem to be solvable. At the same time it is important that the problem is not too generalized to represent the real conditions in a correct way. It is a careful balance between solubility and accuracy where decisions has to be taken according to the desired level of ambition. The identification process is a good source to acquire knowledge and understanding of the system and the relations within (Lundgren et. al, 2003).

The outcome of the first step is a simplified problem; the way in which the simplified model is described mathematically is called an optimization model. The problem has to be formulated in terms of an objective functiondepending on different variables which are limited by a number of constraints (Lundgren et. al, 2003).

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ACTUAL PROBLEM Identification, delimitation & simplification SIMPLIFIED PROBLEM Mathematical formulation OPTIMIZATION MODEL Optimization method SOLUTION Evaluation RESULT

Figure 2.2: A flow chart of optimization process; the various activities and outcome of each step.

The objective function, or target function, is a function of the design variables. The design variables are parameters which can be changed in order to alter the design. They are generally continuous but might be discrete, e.g. due to the available assortment of cylinder sizes. Design variables can be; dimensions of beam cross-sections or lengths, or other attributes, like holes and fillets. Depending on the design variable type structural optimization can be divided into three different disciplines; sizing, shape optimization and topology optimization (Jurecka, 2007), see Figure 2.3.

Figure 2.3: Illustration of the various disciplines in structural optimization; size, shape and topology

optimization.

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Size optimization is the simplest optimization task and can be used to solve problems concerning the minimum cross-sectional area of truss elements for a fixed geometry. Shape optimization involves geometrical parameters and changes the shape within several limits. In topology optimization material can be added or removed. In structural optimization a problem can concern combinations of the different disciplines (Jurecka, 2007).

The constraints, or design requirements, can also depend on state variables derived from constraints on the state of the system. These state variables can be included in the objective function. The state constraints are often limited to be equal or above/under a certain value while the design variables can change within an interval (Strömberg, 2010). The constraints are formulated as equality

0 or inequality constraints 0, as for the state variables. The upper and lower limit of the design variables defines the design space while the constraints of the variables divide the design space into a feasible and unfeasible domain (Jurecka, 2007), see Figure 2.4.

Figure 2.4: The feasible region, grey area, restricted by the function and the function , and by the constraints 0 and 0.

The objective function is a formulation of what is to be optimized; the objective is a measurement of the performance of the different variable combinations. Often it is formulated as a minimization problem, but this does not cause restrictions; maximization problems are simply re-formulated to minimization problems (Jurecka, 2007).

As the problem has been formulated mathematically and the optimization model is fully defined it is possible to determine what optimization method to use and what algorithm to be implemented. As all the previous steps have been worked thru a suggestion for a solution is generated. The solution has to be evaluated and validated since the generalizations and simplifications of the problem will generate an approximate solution. The end of the optimization process is reached when the solution suggestion has been processed into a satisfactory result (Lundgren et. al, 2003).

Optimization problems with two or more objective functions are dealt with within the area of multi-objective optimization, the subject is treated in the next section of this report.

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2.3.2 Multi-objective optimization and Pareto optimality

For optimization problems with several objectives the functions might be conflicting; e.g. trying to reduce mass while striving for lower stresses. No explicit optimal solution can be distinguished for these problems; all optimal solutions will be compromises between the different objectives. For problems where the objectives are interchangeable, all functions, except one, will be unnecessary. For single-objective problems and problems with several compatible objectives it is fairly easy to determine the optimal solution, in contrast to problems with multiple conflicting objectives (Messac & Mullur, 2007).

One way of dealing with several objective functions is to minimize one of them and treat the other objectives as constraints. A downside with turning a soft preference to a hard constraint is that the final solution might largely depend on the chosen value for the constraint and may lead to an inadequate solution. Instead of re-formulating objectives into constraints, multi-objective problems and their solutions can be managed in a more efficient way – by the concept of Pareto optimality (Messac & Mullur, 2007).

For multi-objective problems two different questions is important to consider; how is the solution defined for this type of problem and how is the problem solved? One answer to the first question is; for Pareto optimality a solution is to be considered optimal if the improvement of one objective will lead to a deterioration of another objective. This means a compromise between the different objectives and compromises are a central subject within multi-objective optimization. An optimal solution will be the one resulting in an optimal compromise, or tradeoff, between the design objectives. Identification of a representative Pareto optimal set is the aim for many of the current methodologies for managing multi-objective problems (Messac & Mullur, 2007). Pareto optimal sets can at times easily be identified in a design space – as a so-called Pareto front, see Figure 2.4. The Pareto front consists of all Pareto optimal points of the dominated region in the design space. No Pareto optimal solution is better than another, mathematically seen - it is up to the Engineer to determine the weight of the different objectives, and this is the answer to the second question. What is provided by the Pareto front is a good visualization of characteristics of the different objectives and guidance in decision making (Messac & Mullur, 2007).

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Figure 2.4: Pareto front for a bi-objective problem.

2.4 Metamodeling

In order to reduce computational time for complex optimization problems metamodels are used for reducing the number of design evaluations. A metamodel is an approximate mathematical expression for a correlation between the different input variables and the outcome of an optimization problem. Briefly described metamodels describes the response of the outcome as the variables alter and provide predictions of the behavior for untested points.

Reliable metamodels are generally created from an existing set of design of experiments, DoE. What characterizes a proper set of test points, or training data, is that they provide as much information as possible in relation to a minimum effort (Jurecka, 2007). How Design of Experiments (DoE) is planned is treated more thoroughly later on in this report.

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Behavior models of an original design are more complex providing more accurate results than a metamodel, on the downside the computational time for these might require numerous amount of time. In comparison, metamodels are cheaper to evaluate and are an efficient way of reducing time (Jurecka, 2007). Nevertheless it must be noticed that metamodels are approximations, the optimal solution for the metamodel might not always be the optimal solution for the original problem. However, the solution of the metamodel can provide a solution close enough to the optimal design to upgrade the original design (Strömberg, 2010).

Figure 2.5: Response surface in MATLAB™.

Aside from reducing computational time there are a number of various benefits with metamodels. One is better understanding of the problem and design space as the engineer investigates the system and correlations. Metamodels can also increase the computational speed as design experiments can be carried out and evaluated simultaneously. Multiple information sources can be included in a metamodel; this enables evaluation and analysis of multi-physical problems (Jurecka, 2007).

Fast optimizers in modeFRONTIER™_{uses four different methods for}

metamodeling; polynomial singular value decomposition, Kriging, radial basis function models and neural networks.

2.4.1 Response surface models

A certain type of metamodels are so called response surface models (RSM), just as other types of metamodels RSM’s provides the user with an analytical “surface” of the predicted behavior. Another name for RSM is polynomial regression models, since polynomials are fitted to the training data by regression functions (Jurecka, 2007).

The aim with RSM is to create an explicit functional relationship by fitting free parameters β of a regression function η(v,β) to the response values. The relationship between input variable v and the output value y is given by

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,

where ε is the inaccuracy of the model and β are so called regression variables. By least square approach the regression coefficients can be estimated, this by minimization of the residuals. Residuals are the noted difference between the observed values and the predicted values. Depending on the propagation of the residual values is possible to determine the quality of the response surface. If the plotting the residuals show no distinct pattern then the regression model can be considered suitable (Jurecka, 2007).

Taylor expansion makes it possible for the regression function to be fitted to higher order problems. A linear function would look like

,

Taylor expansion gives a function suitable for second orders ,

Usually the RSM are sequentially fitted to different sub-regions of the problem in an iterative process, this in order to create as good predictions as possible (Jurecka, 2007).

Polynomial Singular Value Decomposition (SVD) in modeFRONTIER™

functions in the same way as the polynomial regression models, or RSM. Polynomial SVD is described as an effective, but imprecise response surface method for producing metamodels. The square of errors are minimized by producing the best fitting polynomial based on the available training set, as previously stated for RSM. The speed of the training process make SVD suitable for generating reliable guesses and detecting regions of interest rather than providing the strict behavior of the function (Rigoni, 2010).

2.4.2 Kriging

Kriging was originally a statistical method used within the area of geostatistics but is today a popular tool for metamodeling and RSM. Its popularity depends on the ability of interpolating response values in an exact way from a set of training points. The Kriging approximation is formulated as

,

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The term η(v,β) has previously been identified as a polynomial with the free

parameters β and ε is the approximate errors. Z(v) is a Gaussian random process where the mean is zero, variance and non-zero covariance (Jurecka, 2007) Kriging interpolation uses known design values to predict the values of the untested points, this by weighting the sum of the known data and minimizing the mean squared error. All covariance of the sample points in the search area are collected in a matrix. This matrix is then inverted to get the weight, as a result large divergences turns into small weights (Lovison, 2007).

2 σ

Interpolation at the observed values at the sample points is according to Jurecka assured due to “local deviation”. The sample points becomes the same as the optimal values and no residuals remains (Jurecka, 2007).

In modeFRONTIER™ the predicted values are expressed as a linear combination of the values for training: linear Kriging e timator. Mathematically formulated ass

where the weights λi are point-dependent (Rigoni, 2010).

The correlation between the different variables is described by a covariance function. The model function of the covariance is also known as variograms, which expresses the spatial variation. The miscalculation of the predicted values is minimized by estimation of spatial distribution. By the covariance function the smoothness of the model can be controlled, this also determines whether the algorithm is approximate or interpolative (Lovison, 2007).

A covariance function controls the smoothness of the model and determines the correlation as a function between the different points. The covariance function is formulated as

,

where σ is the asymptotical value of the variogram function γ(h). The variogram type of default is Gaussian (Rigoni, 2010).

There are three different parameters which can be varied of the variogram; range, sill and noise. The range decides whether the outcome of two points should be correlated or not; depending on the distance between the tested points. Sill is the asymptotical value and determines the variability function. The noise depends on the standard error of the expected response. If the noise of the variogram is large it will result in smooth response, while exact interpolation will be achieved by the noise set to zero (Rigoni, 2010).

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2.4.3 Radial basis function models

Radial basis function models are metamodels especially well-suited for approximating functions depending on many variables/parameters or depending on numerous data. Radial basis functions (RBF) also manages scattered data which is data sampled independent of a grid (Buhmann, 2003). What scattered data and data from grids are is closer explained in section 2.5 Design of experiments. This type of metamodel combines a polynomial part and a radial basis function for prediction of the response. Radial basis functions are, according to Buhmann, usually approximations of “finite linear combinations of translates of a radially symmetric basis function, say · , where · is the Euclidean norm” (2003, p. 3). Radial symmetry in this case means that possible rotations have no effect on the value of the function, since the function value is dependent only on the Euclidean distance (Buhmann, 2003).

Radial functions can take various forms, for instance a cubic function is r r and the Gaussian function r e where r 0 and a 0 is a constant. While interpolation is ensured by the radial function, the polynomial part of the model provides a global trend of the behavior. The set of regression parameters should not be larger than the training set and the polynomial part should be chosen thenceforth. Some radial functions, like the Gaussian, do not require a polynomial term (Jurecka, 2007).

In ModeFRONTIER™ the RBF interpolant is formulated as /

where · as previously mentioned is the Euclidean norm, and r is a radial function chosen by the user. n is the number of sampling points from the function f x and δ is a scaling parameter (Rigoni, 2007).

The available radial functions in ModeFRONTIER™ are concluded in Table 2.1. For the radial basis function requiring a polynomial term an extra column has been added, where m is the degree of the polynomial. The form of the polynomial term is

where all the real-valued polynomials in d variables of m degrees are enclosed by the linear space . is the basis of the space. The dimension of is denoted q and is defined as

For a number three variables and a second order polynomial the polynomial design space would be 10-dimensional (Rigoni, 2007).

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Table 2.1: Radial basis functions available in modeFRONTIER™ – for functions requiring an additional

polynomial term the degree of the polynomial m for a certain number of d variables has been specified.

Gaussian Duchon’s Polyharmonic Spline log d odd d even /2 1 /2 Hardy’s MultiQuadrics 1 / ₀ Inverse MultiQuadrics 1 / Wendland’s Compactly Supported C2 1 3 1 1 4 1 1 5 1 d 1 d 2, 3 d 4, 5 2.4.4 Neural Networks

Neural networks is a metamodeling type which imitates the human brain for processing data. A human brain consist of over a 100 billion neurons connected by 10 000 synapses forming a complex network able of reasoning and storing of data (Jurecka, 2007).

The nodes in a neural network are a mathematical model of a neuron. In the network information is passed through an input layer to an output layer via a hidden layer. All components in the network is uniquely numbered in order to be identified, nodes are continuously number and denoted index i. Connectors are denoted by a double index ij where the first letter i stands for the receiving unit and the second letter j the emitting unit. A single unit i only accept one-dimensional data which is transformed by an activation function ηi into nodal output oi. Net input, which is the scalar input of node i, is symbolized by neti (Jurecka, 2007).

The net input is defined as

where Ai is the set of nodes that are anterior to unit i and wij are characteristic properties defined as individual weights. The output value from node i is passed forward to all posterior nodes after being evaluated of the activation function neti (Jurecka, 2007). The data flow is illustrated in Figure 2.6.

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o₁

Figure 2.6: Data processing in a neural network at a node i with four anterior nodes and three posterior nodes.

In modeFRONTIER™_{the data transfer process is given by net input u of the}

scalar input xi, the weights wi, but the free parameters of the model is also taken in account. The free parameters are represented by bias b. The net input u is processed by the activation function f giving output y (Rigoni, 2010).

Mathematically formul ed as at

As previously mentioned, neural networks imitates the brain and its ability to learn from examples. The network’s weights and biases are altered to minimize prediction errors by a given training set and a target for the output (Rigoni, 2010). If the number of neurons in a single non-linear hidden layer is sufficiently high it is possible to represent any arbitrary function with a linear output by a neural network. ModeFRONTIER™ _{uses feedforward networks where the output from}

a node only is passed on through the subsequent layers, see Figure 2.7. The opposing to such networks is so called recurrent networks where the output from one node can be passed as input on another node on the same layer (Jurecka, 2007).

Figure 2.7: Neural network with one hidden layer where variables are transferred through the different layers.

y v1 v2 v3 v4 v5

Input layer Hidden layer Output layer

w_i1 o₃ o₄ net w_i4 o₂ wi3 w_i2 oi o_i o_i 14

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The neural network in modeFRONTIER™ the hidden layer has a sigmoid (s-shaped) function and the output has a linear activation function. The training algorithm is carried out with a fixed number of iterations and initially the network’s weights are set to random small values. The number of neurons in the hidden layer is also automatically set in the software (Rigoni, 2010).

2.4.5 Advantages and disadvantages of different metamodels

Since several different methods for metamodeling are used in modeFRONTIER™

the imperfections of the different models are compensated for – a metamodel having weaknesses concerning one type of problem might prove to be advantageous for other types. By sequentially execution of the different models and applying the one with best result, the computational effort is utilized where required the most. In the following section some of the advantages and disadvantages with the different metamodeling types are reviewed.

According to Jurecka, polynomial regression models have shown to be especially suitable for problems with less than 10 variables and problems where the problem is governed by linear or quadratic effects. Since RSMs are smoothing they also are suitable for problems where the original response “suffer” from large amounts of noise. They are also attractive for their cheap fitting process and the computational effort for predictions is insignificant (Jurecka, 2007).

One of the most attractive characteristics of Kriging models are their flexibility. There is a large variety of correlation formulations that are feasible. The combination of a Gaussian process and a polynomial function makes the model suitable for universal applications, since the Gaussian compensates for possible deficiencies of the polynomial term. One disadvantage with Kriging is that if the sample points lay to close to each other the correlation matrix will be ill-conditioned and the predictions of the response will be unreliable. Another disadvantage is the sensitivity to noise, this due to its interpolative capability (Jurecka, 2007).

In modeFRONTIER™_{this can be avoided with the right covariance function, see}

closer explanation in section 2.4.2 Kriging. Compared to RSM the Kriging approximations are much more demanding concerning computational time since the fitting to sample points is optimized. Kriging is as a result of the computational demand not suitable for problems concerning more than 50 input variables (Jurecka, 2007).

Just as Kriging, the RBF models are very flexible and have a wide range of application. This due to the wide diversity of radial basis functions available. The lowest order of the polynomial term has to be regarded in order to produce an exclusive solution. The limit of 50 variables is not an issue for RBF models since they are cheaper in computational time compared to Kriging which has to incorporate optimization of the fitting to sample points. Given enough amount of proper training points the RBF will produce accurate predictions even for higher non-linear problems (Jurecka, 2007).

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Neural networks ability to combine hidden layers and number of nodes in various ways make them very adaptable to different types of problems. The versatility put demands on the experience of the user and it is necessary to make conscious decisions between the different alternatives when setting up a neural network (Jurecka, 2007).

Common for all metamodels above is that they all require a proper set of sample points in order to create accurate predictions for the response. How the Design of Experiments should be planned and set up is treated in next section in this report.

2.5 Design of Experiments

In this report it has been previously stated that the best training points are the points providing as much information as possible in relation to a minimum effort. How these training points should be selected is a subject within the area of Design of Experiments DoE and the methodologies associated. In this section some different methods is closer described. The selection of techniques has been made according to the methods available in the software used in this thesis work.

An experimental design is a set of experiments to be carried out and the experiment is expressed by a number of input variables, or factors. The factors are generally design variables and noise factors of the problem. Once it has been established what variables and noise parameters to take in consideration for the experiment it is possible to select a suitable DoE-technique. The proper DoE chosen depends on some different factors. First, what is the aim of the experiment? The prerequisites are different whether the aim is to create a training set for a metamodel or to obtain information about factor relations. Second, the understanding and knowledge of the problem to be analyzed. It is relevant to recognize whether the behavior of the response, if is linear or non-linear, and to identify regions of particular interest or non-interest. Third, and last, is the number of experiments desired or required. The conflict between the amount of information obtained and the calculation time of the experiment must assessed (Jurecka, 2007).

Generally for Design of Experiments the design variables are scaled, or normalized, to an interval of 1 1 (Strömberg, 2010). If a design domain of the variables , 1,2 is taken into consideration the mapping into the -plane will be defined as

The mapping process is illustrated in Figure 2.8

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Figure 2.8: Illustration of the normalization and mapping of designs.

2.5.1 Sobol Design

Consider a case where a number of points x should be generated and well-distributed in a design space D 0, 1 with dimensions. How well the points are scattered over the space is measured in terms of discrepancy. A definition of the discrepancy can be formulate in td he following way (Bratley & Fox, 1988): For a set of points x , x , … , x D and a subset H D , a counting function C H is defined as the number of points x H. For every x x , x , … , x D , let H be the rectangu r -la d dim nsional region e

, , … ,

with the volume x x … x .The discrepancy of the points x , x , … , x then becomes

, , … , | … |

Sobol and other so-called quasi- or pseudo-random sequences aim to minimize the discrepancy for better spacing and are used for optimization and simulation (Bratley and Fox, 1988), see Figure 2.9 and Figure 2.10 for comparison of the distribution for a random sequence and a Sobol sequence.

-1

1 1

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Figure 2.9: Scatter plot of the distribution of a

pseudo-random sequence in modeFRONTIER™. Figure 2.10: Scatter plot of the distribution of a pseudo-random Sobol sequence in modeFRONTIER™.

Quasi-random sequences are finite or infinite sequences with a set of points with very small discrepancies. Pseudo-random points are in fact not random at all, but are generated by simple arithmetic algorithms. For defining a random sequence of random point the following principles has been used as criteria’s; first, the distribution of the sequence must satisfy some distribution properties, and second, the properties should be unchanged for given selection rules for subsequences. Nevertheless, pseudo-random sequences pass as random in several statistical tests (Niederreiter, 1978).

In modeFRONTIER™_{a Sobol sequence is generated by simply entering the}

number of desired designs, see Figure 2.11.

Figure 2.11: ModeFRONTIER™ interface for Sobol.

2.5.2 Latin Hypercube Designs

Latin hypercube designs (LHD) a type of space-filling designs where the factor space is normalized and every factor range divided by m strata, resulting in a factor space divided into mn_{cells, n being the dimensions. Every strata is addressed once}

by selecting a subset of cells randomly in such way, see Figure 2.12. A sampling point is commonly placed in the middle of the cell, but can be positioned within the cell by random sampling according a uniform distribution.

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The result is a LHD where the observations are equally distributed over the range of the individual factors. The segmentation of the factor space assures that no replicates are generated. However, the space-filling with respect to the total factor space may be poor (Jurecka, 2007), see Figure 2.13.

2 2

Figure 2.12: Latin Hypercube Design with 8 strata. Figure 2.13: Latin Hypercube Design with 8

strata with poor space-filling properties.

ModeFRONTIER™_{provides two types of LHD – Uniform Latin Hypercube,}

Figure 2.14, and Latin Hypercube - Monte Carlo, Figure 2.15.

Figure 2.14: Available settings for Uniform Latin Hypercube Designs in modeFRONTIER™.

LHD – Monte Carlo combines LHD and Monte Carlo sampling (MC) for design of experiments. MC methods solves an integral based on a number of sampling points at which the function is evaluated. The sampling points are chosen with respect to probability density of the noise variables (Jurecka, 2007).

The solution of the integral is formulated as

where is a function, is the probability density of the noise variables Z and is a weight.

1 1

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While Plain MC sampling determines the sampling points randomly in consistency with probability density of the noise variables, Stratified MC sampling use partitioning of the entire noise space. This partitioning resembles the proceeding of Latin Hypercube Design and its division of the factor space into stratums. For Stratified MC, in contrast to Plain MC, the sampling is carried out in a systematic way; exploring subsets of noise space that may be of importance but otherwise might have been left out. That is, regions with high influence on the investigated statistics but with small probability (Jurecka, 2007).

Figure 2.15: Available settings for Latin Hypercube –Monte Carlo in modeFRONTIER™.

2.5.3 Incremental Space Filler

One of the easiest ways to create an even distribution of sampling point is by placing them in a n-dimensional grid within the factor space, see Figure 2.16. One disadvantage with this type of sampling is that the designer cannot freely decide which arbitrary points to be included and the segmentation of the factor space determine the total amount of sampling points. Another weakness is that “a projection of the design onto a subspace with reduced dimensionality would yield many replicated points” (Jurecka, 2007, p. 146), which is undesirable in cases where irrelevant factors has been identified (Jurecka, 2007).

Figure 2.16: Design sequence sampled

from a grid. Figure 2.17: Latin Hypercube Design with distance criterion

1 2 min d 2 1 20

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Instead of sampling point from a grid a distance-based criterion can be added for better space-filling properties (Jurecka, 2007), see Figure 2.17. For example, a

minimum distance are so called maximin distance designs, modeFRONTIER™ provides the Incremental Space Filler (ISF) which is

a uniform way. The ISF algorithm is among other

Figure 2.18: Illustration of settings for Incremental Space Filler in modeFRONTIER™.

2.5.4

If a design sequence is to be considered orthogonal if the scalar product of any of the column vectors being evaluated is zero, or in other words, XT_{X is diagonal}

matrix. An important characteristic of orthogonal arrays is the resolution R, which explains the independent estimation of main and the interaction effects.

minimum criterion for the Euclidian distance d between the sampling points vk

and vl

Designs which maximizes the such a design, see Figure 2.18.

The points are added sequentially with a specified radius from an existing design point, filling the design space in

things used for extending the design database with points around the Pareto front when running Fast optimizers in modeFRONTIER™ (Rigoni, 2010), see description of fast optimizers in Section 2.6: Fast optimizers in modeFRONTIER™. Two variants of the algorithm are available in modeFRONTIER™, the Genetic Algorithm Optimization (GOA) and Voronoi-Delaunay Tessellation (VDT).The GOA implements a genetic algorithm for internal optimization of the distance criterion. The method is approximative but works in a fast and robust way, in contrary to the VDT, which is a time-consuming method that will generate an exact solution for maximization of the minimum distance (Rigoni, 2010).

Taguchi Orthogonal Array Designs

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The method might seem limited but the result is less computational effort. This due to the fact that interaction factors of interest are introduces as independent

f noise factors are the environment the

treatment combinations, factors, (Jurecka, 2007). As a result the number of sampling points will be reduced, see Table 2.2 and Table 2.3.

In modeFRONTIER™_{it is possible to design experiments by a technique called}

Taguchi Orthogonal Arrays (TOA). TOA are orthogonal arrays which also considers both noise and design factors. Design factors, or Control factors, are parameters of the design that are effortless and cheap to control. Noise factors are factors that are hard or impossible to control but that might affect the performance of the product. Examples o

product will function in, called an external factor, and material defects or variation, also termed internal factor (Dean et. al, 1999).

Two types of designs exist for TOA, mixed and product arrays. Product arrays consist of one fractional factorial experiment for the design factors and one for the noise factors. All combinations of design factors are considered in coordination with every possible combination of noise factors. Robustness is said to be achieved if the product work consistently well regardless of unrestrained variation of noise factor levels. Mixed arrays consider

which are level combinations of design and noise factors. The robust settings for the design factors are obtained by investigation of the interaction between the noise and design factors (Dean et. al, 1999).

NO. OF SAMPLING POINTS

Full Factorial Orthogonal Array

23_{→ 8} ₄

Full Factorial Orthogonal array

-1 -1 -1 -1 -1 1 28_{→ 256} ₇ ₁ _{-1 -1} _{-1 1 -1} -1 1 -1 1 -1 -1 210_{→ 1 024} ₁₁ 1 1 -1 1 1 1 2 → 32 768 15 ₁₆ 1 -1 1 1 -1 1 313 _{→ 1 594 323} ₂₇ 45_{→ 1 024} ₆ -1 1 1 1 1 1

Table 2.2: Comparison of the num f experiments generated by Full Factorial Design

ogonal Array D n.

Table 2.3: Design sequence of the va les v1. v2 and v3 by Full Factorial Design

omp d to Orth gonal Array Design.

H rent settings fo OA designs in od RO TIER can be applied is seen in Figure 2.19.

ber o and Orth esig

riab

c are o

ow the diffe r T m eF N ™

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Figure 2.19: Possible settings for Taguchi Orthogonal Arrays in the modeFRONTIER™ interface.

2.5.5 Full factorial designs

ill be mn_{distributed evenly over a region of interest (Strömberg, 2010). Mathematically}

e; a case of 2 design vari les varied over 3 levels would generate 9 test points and would be called a 9 experiment. The formation of 9-factorial design is illustrated in Figure 2.20 and Table 2.4.

A factorial design is design where a number of n design variables are varied over a predetermined number of m levels. The generated number of experiments w expressed as As an exampl ab -factorial Point n1 n2 1 1 -1 -1 2 -1 0 3 -1 1 4 0 -1 -1 1 -1 1 0 5 0 6 0 1 7 1 -1 8 1 0 9 1 1

Table 2.4: Set up of a 9-factorial design

with the two variables n1 and n2.

Figure 2.20: A 9-factorial design – 2 variables varied over

3levels

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L oria

estimated by 3-level als. As rule of thumb, a le sed

to fit a polynomial of degree d. The drawback with full factorial experiments is the

r of experiments is to apply the method

less than 4 levels, see Figure 2.21.

Figure 2.21: In modeFRONTIER™ the user decides the level of every single variable. It is possible to select

higher levels for some variables and a lower for another variable. 2

While full factorial generates m

onal factorial sequence contains mn-p

efining the reduction of the sequence

exp ill be reduced (Dean et.

to be governed mainly by only a few of them and by low-order interaction. Another argument is the projection of the design onto other levels; if a number of p factors is excluded from the design sequence due to irrelevance the consistent dimensions of the design space is removed (Jurecka. 2007). The original fraction factorial design becomes a full factorial design, see Figure 2.22.

inear effects are estimated by 2-level fact

factori ls while quadratic behavior isvel of d+1 has to be asses fact that the experimental effort increases rapidly when the number of levels or variables is increased (Jurecka, 2007).

A 3 level factorial with 9 variables would result in 19 683 experiments. Running a sequence with 39_{experiments would fast yield a great amount of computational}

time. A way to decrease the numbe Fractional Factorial instead.

In modeFRONTIER™_{it is possible for the user to set the level of each variable.}

A brief summary points out the fact that the full factorial is suitable for problems with less than 8 variables and

.5.6 Fractional Factorial Designs

n_{experiments a reduced factorial produces a subset}

of the full factorial sequence. A fracti experiments, where p>0 is an integer d

compared to the full factorial. The expression 1/ is used to calculate the fractional of the full design (Jurecka, 2007).

Due to the fact that only that only a fraction of the behavior is observed, each main-effect and interaction contrast cannot be estimated independently. On the other hand, the computational effort for the eriment w

al, 1999).

Fractional Factorial Designs are motivated by some different arguments. One is the fact that in systems which depends on a large number of input variables are expected

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Figure 2.22: Illustration of the projection of a 23-1 fractional factorial design into three 22 full factorial designs.

Fractional factorial is in modeFRONTIER™_{referred to as “reduced factorial” and}

is based on a 2-level factorial sequence, see Figure 2.23. By selecting the largest number in the “Number of Designs”-window will result in a full factorial. For a problem with 8 variables the largest number available would be 256.

2.5.7 Central Composite Designs

For fitting second-order polynomials Central Composite Design (CCD) is one of the most popular techniques. It combines points from a two-level full factorial (or fractional factorial) sequence with a center point and so called star points. Star

points a from

the origin. On the axis

at , see Figure 2.24. The number of star points will be 2p for a number of

p factors (Dean et. al, 1999).

Figure 2.23: Available settings for a Reduced Factorial Design in modeFRONTIER™

re points which are located on all coordinate axes with a distance

two points will be positioned; one at and another

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Figure 2.24: Illustration of distribution of points for a factorial desi

The value of α is chosen depending on the properties required of n the expression

gn, the center point and the star points.

the design, ofte

is used for establishing a suitable value for , where

factorial points (Dean et. al, 1999). For this expression the resulting CCD will be

rota portant the predictio cy. The

prediction discrepancy e

de

contribute largely to tion the linear terms and they are the only points which contribute to of the teraction terms. The center point provides information ab of the s m and estimation of

α is given by are the orthogonal table, which is an im condition for n discrepan

will be equal on a sphere around the center point if th sign is rotatable (Jurecka, 2007).

The three types of points play different roles in providing information. The

factorial points the estima

computat

out curvatuionre inyste

quadratic terms. It also provides the “pure error” of the design. If curvature is found in the system the star point will estimate the quadratic terms in an efficient way (Meyers & Montgomery, 2002)

Two types of regions can be taken in consideration for CCD, spherical and cuboidal regions. If the region interest is spherical a suitable value for

√

where n is the number of factors. This way of choosing α will not result in an exact rotatable design instead the value will create better solution from a prediction point of view. A spherical CCD will put all the star points and factorial points on a sphere with radius √ (Montgomery, 2005).

For cuboidal regions of interest it is suitable to use face-centered CCD, where is of no greater importance when t of interest is obviously cuboidal 1. In this case the star points will be located at the centers of the faces of the factorial cube, see Figure 2.25. This type of design is not rotatable, but rotatability

he region (Meyers & Montgomery, 2002).

For sequential experimentation CCD is an efficient design which not involve a too large number of design points while still providing a reasonable amount of data for testing lack of fit. It is also a flexible design in its use of star points; it can

Full Factorial Points Center Point Star Points α

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accommodate a spherical region by √ and with five levels for each factor, and a cuboidal region for 1 and a three-level design. The CCD can also provide for situation where √ or 1 cannot be used, the value of α can be adopted depending on the circumstances (Meyers & Montgomery, 2002). For central composite design for three variables, see Table 2.5.

In modeFRONTIER™_{it is possible to choose between two} _{es C D, see figure}

2.26. The first type, Cubic Face Centered, places points at all vertexes of the cube, as in Figure 2.25, the second type, Inscribed Composite D n, p

the vertexes of a scaled hypercube within the cube

Point No. 1 -1 -1 -1 2 -1 -1 3 -1 1 -1 4 -1 1 1 typ C

esig laces points at . 5 1 -1 -1 6 1 -1 1 7 1 1 -1 8 1 1 1 9 -α 0 0 10 α 0 0 11 0 -α 0 12 0 α 0 13 0 0 -α 14 0 α 15 0 0 0

Table 2.5: Central Comp te

Design sequence e factors.

Figure 2.25: Illustration of a face-centered CCD sequence for three

factors and 1, the 8 corner points are full factorial points, the 6

points on the axes are star points and the middle point is the center point.

osi for thre

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Figure 2.26: Parameter settings for Central Composite Designs in modeFRONTIER™.

2.5.8 Box-Behnken Designs

For Box-Behnken Designs (BBD) is a special methodology which constructs balanced incomplete block designs, see Table 2.6. The variables are treated in pairs, for an example of three variables: in the first block variable 1 and 2 is paired together in a 22_{factorial while variable 3 is fixed at the center. In the second block}

variable 2 is fixed at the center and instead variable 1 and 3 is paired together. This proceeding holds for a number of variables 2 < n < 6 (Meyers & Montgomery, 2002). Treatment Block 1 2 3 1 X X 2 X X 3 X X

Table 2.6: Demonstration of the order in which

the variables are paired together.

For problems with 6 or more variables the procedure is slightly different, for a closer description see Meyers & Montgomery, 2002.

The resulting design sequence of Box-Behnken with three variables is shown in Table 2.7 and Figure 2.27.

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BBD is closely related to Central Composite Designs but has the advantage of avoiding extreme values of the factors by not taking the corners of the factor space in consideration. On the other hand, the same characteristic of avoiding extreme factor settings result in BBD not being suitable for prediction of behavior in the corners of the factor space. As a conclusion this type of design should be used for cases where there is no interest of predictions of the corner points. Since BBD is a spherical design it is rotatable, or near rotatable, and all sampling points will have same distance to the center point (Jurecka, 2007).

Table 2.7: Design sequence for three variables generated

by Box-Behnken. The last row is the center point

Point No. 1 -1 -1 0 2 -1 1 0 3 1 -1 0 4 1 1 0 5 -1 0 -1 6 -1 0 1 7 1 0 -1 8 1 0 1 9 0 -1 -1 10 0 -1 1 11 0 1 -1 12 0 1 1 13 0 0 0

Figure 2.27: Illustration of Box-Behnken Design for

three variables.

Box-Behnken Design is available in modeFRONTIER™_{without any settings for}

the user to configure, see Figure 2.28.

Figure 2.28: Box-Behnken Designs in modeFRONTIER™.

Anna

Design Optimization of a Boom for Drill Rigs

Ida Haglund

Anna Håkansson

EXAM WORK 2011

Mechanical Engineering

Abstract

Summary

Preface

Contents

1 Introduction

1.1 Background

1.2 Atlas Copco Rock Drills AB

1.3 The Boomer and the BUT 45 L

1.4 Problem description

1.5 Problem formulation

1.6 Purpose and aim

1.7 Delimitations

1.8 Outline

2 Theoretical background

2.1 Forward Kinematics

a

2.2 FE-analysis

d

o

o

y

y

x

x

z

z

α

θ

2.3 Optimization Theory

2.4 Metamodeling

2.5 Design of Experiments