Master Thesis Report - Ruhr University Bochum · Master Thesis Report Real time surface se˛lements prediction in mechanized tunneling Submi˛ed in Partial Fulfillment of the Requirements

Master Thesis Report

Real time surface se�lements prediction in

mechanized tunnelingSubmi�ed in Partial Fulfillment of the Requirements of the Academic Degree

Master of Science in Computational Mechanics at University of Duisburg-Essen

Author: Syed Aun Abbas

Matriculation Number: DS0-3002307

Internal Supervisors: Dr.-Ing. Dominik BrandsFachgebiet Mechanik / Computational MechanicsUniversity of Duisburg-Essen

Prof. Dr.-Ing. habil. Joachim BluhmFachgebiet MechanikUniversity of Duisburg-Essen

External supervisors: Prof. Dr. Techn. Günther MeschkeLehrstuhl für Statik und DynamikRuhr-Universität Bochum

M.Sc. Ba Trung CaoLehrstuhl für Statik und DynamikRuhr-Universität Bochum

M.Sc. Hoang Giang BuiLehrstuhl für Statik und DynamikRuhr-Universität Bochum

Master Thesis project at: Institute of Static and Dynamic Ruhr Universtät Bochum.

I

Statutory Declaration

Versicherung an Eides sta�

Ich versichere an Eides sta� durch meine untenstehende Unterschri�.

• Dass ich die vorliegende Arbeit - mit Ausnahme der Anleitung durch den Betreuerselbststaendig ohne fremde Hilfe angefertigt habe und

• Dass ich alle Stellen, die wörtlich oder annähernd wortlich aus fremden �ellenentnommen sind, entsprechend als Zitate gekennzeichnet habe und

• Dass ich ausschließlich die angegebenen �ellen (Literatur, Internetseiten, sons-tige Hilfsmi�el) erwendet habe und

• Dass ich alle entsprechenden Angaben nach besten Wissen und Gewissen vorge-nommen habe, dass sie der Wahrheit entsprechen und dass ich nichts verschwie-gen habe.

Mir ist bekannt, dass eine falsche Versicherung am Eides sta� nach §156 und nach§163 Abs. 1 des Strafgesetzbuches mit Freiheitsstrafe oder Geldstrafe bestra� wird.

(Place, Date) (Syed Aun Abbas)

II

Abstract

The aim of thesis is to investigate Proper Orthogonal Decomposition (POD) with RadialBasis Function (RBF) in C++ programming language. This surrogate model can predictresults that would normally require lot of computational time with relatively goodaccuracy.

POD techniques can be employed for many scenarios but for current case study,its application would involve prediction of soil se�lements during tunneling process.This means large sets of numeric data needs to be predicted based provided inputcombination.

In order to employ such model, a snapshot matrix is required that is based on output andinput data obtained from test simulations. Since algorithms are taken from previousresearch, thesis was meant to expand the research my employing di�erent Radialbasis functions, testing with variable range of shape parameter for smoothing factors,employing usage of both eigenvalues and svd singular values and observe the e�ect ofincreasing number of snapshots.

Results were compared based on di�erent error equations using percentage error,mean absolute percentage error, root mean square and normalized root mean squareerror. Most comparisons can be easily analyzed by graphical plots included in theinvestigation.

III

Acknowledgement

I thank my internal supervisors Dr.-Ing. Dominik Brands and Prof. Dr.- Ing. habil.Joachim Bluhm for providing academic support and suggestions during the course ofmy master thesis.

Special gratitude goes to tremendous e�orts by M.Sc. Ba Trung Cao and M.Sc. HoangGiang Bui of Ruhr Universität Bochum who provided me the topic, supervised andguided me through all the obstacles and gave me access to their servers and lecturesto accomplish my tasks. It is due to their guidance and expertise I was able to achieveprogress in my master thesis and would always be grateful for their support.

I acknowledge my gratitude to Prof. Dr.Techn. Guenther Meschke for giving me anopportunity to work with SFB project team at Lehrstuhl für Statik und Dynamik RuhrUniversität Bochum. This provided me great insight of active research scenarios incurrent projects.

Sincere thanks to department of Lehrstuhl für Statik und Dynamik for providing all theresources to achieve my tasks and granting me their valuable time to aid me throughmy master thesis.

Finally I would add thanks to my parents back in Pakistan for being patient andsupportive of all my decisions. I would not be where I am today without their continuoussupport.

Thank you.

IV

Nomenclature

Abbreviations

FEM - Finite Element Method

TBM - Tunnel Boring Machine

υ - Poisson’s ratio

κ - Slope of unload-reload curve

λ - Slope of normal compression, eigen values

r - Spacing ratio, distance between points

n - Shape parameter of yield surface

Φcs - Friction angle at the critical state

e0 - Initial void ratio

POD - Proper orthogonal decomposition

KLD - Karhunen-Loeve decomposition

PCA - Principal component analysis

SVD - Singular value decomposition

Φ - POD basis vector

Ψ - Eigen vectors

Σ - Singular values

V

c - Smoothing factor

N - Shape parameter of smoothing factor

VI

Contents

Nomenclature IV

List of Figures IX

List of Tables XI

Listings XII

1 Introduction 11.1 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Literature Overview 32.1 Surrogate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 KRATOS (Multi Physics) . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Working Principle . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.2 Python interface . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Mechanized Shield Tunneling . . . . . . . . . . . . . . . . . . . . . . . 62.3.1 Working Principle . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 Excavation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.3 Segmented Lining . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.4 Tail void grouting . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.5 Heading face support . . . . . . . . . . . . . . . . . . . . . . . 102.3.6 Mechanized Shield vs Conventional Tunneling . . . . . . . . . 11

2.4 Soil se�lements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Saturated Soil Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5.1 Soil Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5.1.1 Poisson’s Ratio,(υ) . . . . . . . . . . . . . . . . . . . . 142.5.1.2 Slope of unload-reload curve in v - ln p’ space, (κ) . . 152.5.1.3 Slope of normal compression curve in v - ln p’ space,

(λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Contents VII

2.5.1.4 Spacing ratio, (r) . . . . . . . . . . . . . . . . . . . . . 162.5.1.5 Shape parameter of the yield surface,(n) . . . . . . . 172.5.1.6 Slope of critical state line,(M) . . . . . . . . . . . . . 172.5.1.7 Friction angle at the critical state,(Φcs) . . . . . . . . 182.5.1.8 Initial void ratio,e0 . . . . . . . . . . . . . . . . . . . 18

2.5.2 Sensitivity of Soil Parameters . . . . . . . . . . . . . . . . . . . 19

3 Proper orthogonal Decomposition 223.1 POD method of snapshot . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Snapshot Data Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 Node Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.2 Arrangement of Snapshot Data . . . . . . . . . . . . . . . . . . 25

3.3 POD Basis Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.1 Eigenvalue Decomposition . . . . . . . . . . . . . . . . . . . . . 273.3.2 SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.3 Truncated POD Basis Vector . . . . . . . . . . . . . . . . . . . 29

4 Radial Basis Function 304.1 RBF Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2 Function Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2.1 Smoothing Factor . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Result Analysis 345.1 Error Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2 Graphical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3 Investigation 1 : Eigenvalues and SVD . . . . . . . . . . . . . . . . . . . 38

5.3.1 Computation time . . . . . . . . . . . . . . . . . . . . . . . . . 385.4 Investigation 2 : Impact of smoothing factor . . . . . . . . . . . . . . . 395.5 Investigation 3 : Impact of Number of snapshots . . . . . . . . . . . . . 435.6 Investigation 4 :Comparison of di�erent Radial Basis Functions . . . . 445.7 Investigation 5 : Surrogate model for smaller POD basis vector . . . . . 45

6 Surrogate Modeling using C++ 476.1 Library Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.2 License and distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Contents VIII

6.3 Program algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.3.1 POD Basis vectors . . . . . . . . . . . . . . . . . . . . . . . . . 506.3.2 Radial Basis Function . . . . . . . . . . . . . . . . . . . . . . . 536.3.3 User Script . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.3.4 Modified Script Versions . . . . . . . . . . . . . . . . . . . . . . 56

6.3.4.1 SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.3.4.2 Surrogate model for smaller POD basis vector . . . . 57

6.4 Troubleshooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.4.1 Script execution . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.5 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Bibliography 69

IX

List of Figures

2.1 Modular structure of KRATOS (courtesy of Ruhr University BochumIntitute for Structural Mechanics) . . . . . . . . . . . . . . . . . . . . . 5

2.2 Python interface: where simulation script and basic tasks are imple-mented with python script and computational demanding tasks arewri�en in C++ (courtesy of Ruhr University Bochum Institute for Struc-tural Mechanics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Robbins Single Shield TBM . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Cross Section of Single Shield TBM by Robins . . . . . . . . . . . . . . 82.5 The staged excavation (Phase 1) and construction (Phase 2) process in

shield tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 Disc Cu�ers employed by Herrenknecht AG . . . . . . . . . . . . . . . 92.7 Soil se�lement due to loading conditions . . . . . . . . . . . . . . . . . 132.8 Soil response in compression,[BV] . . . . . . . . . . . . . . . . . . . . . 152.9 Figure shows state parameter, critical state constants and reference

state parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.10 Void ratio for stress state point P . . . . . . . . . . . . . . . . . . . . . 182.11 Results for parametric combination of parameter 2 Unload reload slope

κ and parameter 8 initial void ratio e0 . . . . . . . . . . . . . . . . . . . 202.12 Combined displacement plot for previous four cases . . . . . . . . . . . 21

3.1 Optimum nodes selection based on soil displacement . . . . . . . . . . 243.2 Sample coordinates for Nodes selection . . . . . . . . . . . . . . . . . . 253.3 Arrangement of Snapshot Matrix . . . . . . . . . . . . . . . . . . . . . 263.4 Sample snapshot matrix from original data . . . . . . . . . . . . . . . . 27

4.1 Calculating sample mean distance M for smoothing factor . . . . . . . 33

5.1 Nodes selection for graphical analysis . . . . . . . . . . . . . . . . . . 355.2 Plot comparison of horizontal direction for di�erent input cases and

smoothing factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

List of Figures X

5.3 Plot comparison of vertical direction for di�erent input cases andsmoothing factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.4 Screen shot comparison of time taken to compute POD basis and RBFfor eigenvalues and SVD . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.5 Plot comparison for increasing shape parameter for Inverse Multi �ad-rant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.6 Plot comparison for increasing shape parameter for Multi �adrant . 415.7 Plot comparison for increasing shape parameter for Inverse �adrant . 415.8 Plot comparison for increasing shape parameter for Gaussian (division) 425.9 Comparison of plots employing Inverse Multi �adrant for smoothing

factor with increasing number of snapshots and error percentage . . . 43

6.1 Screen-shot for output screen showing selection of number of columnsk and time taken to compute POD basis . . . . . . . . . . . . . . . . . 56

6.2 Screen-shot for script execution . . . . . . . . . . . . . . . . . . . . . . 57

XI

List of Tables

5.1 Time taken for computation of POD basis with Eigen decompositionand svd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2 Comparative table for increasing shape parameter of smoothing factor 425.3 Comparison of error percentage and POD basis time computation with

Inverse multi quadrant for S.M1(38 POD basis)and S.M2 (1 POD basis) 45

6.1 Inverse Multi �adrant for 36 Snapshots . . . . . . . . . . . . . . . . . 596.2 Multi �adrant for 36 Snapshots . . . . . . . . . . . . . . . . . . . . . 606.3 Inverse �adrant for 36 Snapshots . . . . . . . . . . . . . . . . . . . . 616.4 Gaussian for 36 Snapshots . . . . . . . . . . . . . . . . . . . . . . . . . 626.5 Gaussian (Devision) for 36 Snapshots . . . . . . . . . . . . . . . . . . . 636.6 Inverse Multi �adrant for 69 snapshots . . . . . . . . . . . . . . . . . 646.7 Multi �adrant for 69 snapshots . . . . . . . . . . . . . . . . . . . . . . 656.8 Inverse �adrant for 69 snapshots . . . . . . . . . . . . . . . . . . . . . 666.9 Gaussian for 69 snapshots . . . . . . . . . . . . . . . . . . . . . . . . . 676.10 Gaussian (Devision) for 69 snapshots . . . . . . . . . . . . . . . . . . . 68

XII

Listings

1

1 Chapter

Introduction

1.1 Problem

Modeling and simulation are recognized by both science and engineering to tackle realproblems economically without need for actual experimentation. In fact simulationcan give more realistic results as user can configure its inner and outer parameters.Such development lead to need of faster simulated data so proceeding action can beemployed but problem is simulation can be quite time consuming making it restrictedto be used for non real-time applications.

Based on previous available research [Khaledi et al.][Rogersa et al.] and work bySFB-837 Ruhr University Bochum, surrogate modeling for mechanized tunneling usingPOD technique with Radial basis function was re-implemented to predict the soilse�lements. Utilizing the algorithm from previous research, a cheap surrogate modelusing C++ is used to further investigate the research.

1.2 Objective

The main objective of thesis is to investigate Proper orthogonal decomposition (POD)technique with Radial basis functions (RBF) to explore the consequences of relatedfactors leading to di�erence in accuracy of prediction. Results were analyzed basedon number of snapshots, use of di�erent Radial Basis Functions, e�ect of smoothingfactor and comparison of both SVD and Eigen decomposition.

1 Introduction 2

The procedure is programmed in C++ with implementation of a library, making it quiteuser friendly even without a graphic interface. All theoretical aspects will be coveredin the proceeding sections.

3

2 Chapter

Literature Overview

2.1 Surrogate Model

A surrogate model tends to approximate or mimic a complex and higher order model.Since complex models can be computationally expensive where simulations can takeany where between hours to days to complete. This render simulation impractical tobe used in real time environment.

A cheap surrogate model can be used to predict and approximate results faster byreducing the time for computation. This is achieved by constructing a model withlimited sets of sensitive parameters and selection of responsive points on surface. Theobstacle to the course is of course lose of accuracy. Creating a surrogate model withfewer parameters and simulations can be a challenge and is one of the prime focus forscientific community.

Surrogate modeling using any technique would usually involve initially sample selectionthat is in this thesis arranged in form of a matrix known as snapshot. Next step wouldinclude optimizing parameters which for this thesis involves investigating Radial basisfunction and smoothing factors. Finally accuracy will be compared with originalsimulation data to get possible error percentage. In this thesis out of 76 snapshots, 7validation cases were removed from original snapshot to check accuracy of surrogatemodel.

2 Literature Overview 4

2.2 KRATOS (Multi Physics)

KRATOS is an object oriented, modular finite element framework for multiphysicsanalysis. Its aim is to allow a simple and e�icient implementation of various FE al-gorithms and aids the development of new numerical methods. Due to its modulardesign, implementation of element formulations, constitutive models and solutionstrategies becomes uncomplicated.Refer the Kratos wiki link [Multi-physics] for de-tailed documentation and tutorials to aid user.

2.2.1 Working Principle

KRATOS is divided into two main parts, the kernal and the applications. The kernalprovides all prototypes for the classes used such as Element, Node, ConstitutiveLawetc. It also provide access to central database for storage of the mesh, global variablesand the solution step variables. The application part is the actual implementation ofFE algorithms for di�erent types of analysis. Here elements, conditions or constitutivelaws are defined that are specific to the problem.It also consist of several auxiliaryapplications that features parallelization, domain decomposition, collections of linearsolvers and constitutive models. For tunneling problems, an auxiliary application isavailable as well.


Figure 2.1: Modular structure of KRATOS (courtesy of Ruhr University Bochum Intitute

for Structural Mechanics)

2.2.2 Python interface

Programs are usually compiled into a single executable which could have been designedfor special tasks meaning it also consist of restricted set of input data. To overcomethis executable is designed with help of GUI to control work flow. Hence instead ofcompiling single executable, KRATOS is executed inside python script.

Python being a versatile language has numerous applications and is compiled duringrun time. Native compiled libraries like C++ or FORTRAN can be imported intopython scripts. This feature is employed in KRATOS as all its application and kernelare available in Python script. This provides flexibility in designing simulation workflow and making it possible to test di�erent algorithm implementations and solutionstrategies.

Interface is created using Boost.python library which enables so�ware to interpolatebetween both C++ and Python. C++ programs are implemented with Boost.python


interface and then linked into object library. This enable loading of object library withinPython as a module.

Figure 2.2: Python interface: where simulation script and basic tasks are implemented with

python script and computational demanding tasks are wri�en in C++ (courtesy

of Ruhr University Bochum Institute for Structural Mechanics)

In this thesis , simulation model for soil was already available. Input script was used totest parameters and output script was edited to enable only time step output to bewri�en.

2.3 Mechanized Shield Tunneling

Mechanized tunneling as name suggests, refers to any form of tunneling procedurewhich excavates soil mechanically by means of rotating cu�ing disks or wheel.[GEO]


2.3.1 Working Principle

With added shield method, the surrounding soil is usually secured and covered byshield that prevents entry of material into tunnel. Inside the shield, tunnel lining isbuilt which tends to stabilize the ground around tunnel. The lining which consist ofprecast concrete segments are installed in form of rings. Once the ring is completed,the machine advances by means of hydraulic jacks that are connected between shieldand current ring of segmented lining tube. As machine moves forward, the new liningring is pushed against previously existing tube in order to achieve a sealed connection.As the tail of the shield slides onward, it leaves a narrow gap between the lining ringsegments and surrounding ground. Grouting mortar is utilized to refill this gap asit hardens to form a sealed connection between lining and ground. A more detaileddescription mechanized shield tunneling can be found in [Maidl et al., 1996]

Figure 2.3: Robbins Single Shield TBM


Figure 2.4: Cross Section of Single Shield TBM by Robins

Figure 2.5: The staged excavation (Phase 1) and construction (Phase 2) process in shield

tunneling

The shield tunneling process is illustrated in Figure 2.5. In phase 1, the machine is


pushed forward by the length of one ring while the cu�ing wheel excavates the groundin front of the machine (red arrows). At the same time, the soil is excavated andremoved from the excavation chamber by means of a screw conveyor (blue arrow),and the tail gap is grouted by pressurized mortar (yellow arrows). In phase 2, themachine is stopped. The hydraulic jacks are retreated from the lining and a new ringof lining segments is erected by means of a vacuum erector (yellow arrow in phase 2).A�erwards, the excavation recommences with phase 1 in the subsequent excavationstep.

2.3.2 Excavation

Soil properties and particle size defines which excavation method is applicable. Forexample in presence of hard rock, disc cu�ers are used to crack the soil where as so�soil is usually chipped directly by scraping tools. The fact that soil is not homogeneouswhen it comes to practical excavation, that is it carries both rock and so� soil. Hencemost cu�ing wheels feature both types of excavation tools. Disc cu�ers are exposedbefore scraping tool as to prevent its contact with hard rocks. Sometime hydro jets arealso employed to assist the main tool. It should be noted that choice of cu�ing toolma�ers a lot as wrong usage can lead to damaged tool and unexpected delays.

Figure 2.6: Disc Cu�ers employed by Herrenknecht AG


2.3.3 Segmented Lining

As explained in principle working, the shield tunneling is usually constructed withprecast concrete segments that are installed within shield vicinity. This lining ring alsoact as a counter to bearing thrust jack as machine progresses forward. It should benoted that multiple joints between individual segments can lead to possible leakagesource. Since lining segments are subjected to tilt and twist motion in erection phaseas well as loading pressure, it can lead to cracks formations which upon exposure tocorrosive media or vibrations can promote crack propagation. This leads segment toinappropriate sealing and causes leakage or in worst case a safety hazard if segmentsare not properly reinforced.

2.3.4 Tail void grouting

Pressurized grouting mortar is used to fill the gap between the segmented lining and theouter excavated ground that is le� behind the vicinity of shield. This filling helps avoidse�lements and makes sure the lining is securely connected to surrounding ground.Grouting mortar is prevented from entering shield by means of sealings between thelining and shield tail. For detailed information on investigation related to flow ofgrouting mortar can be referred in [Nagel et al., 2009]

2.3.5 Heading face support

While the shield skin secures the material around the excavation area, the headingface, where the actual excavation takes place,is not sealed. In order to prevent materialfrom flowing into the tunnel and to decrease se�lements due to movement of material,supporting measures are taken. The face support aims at creating conditions in theground as close to the undisturbed situation as possible. Depending on the groundconditions, there are several possibilities to realize the face support.

Some of the relevant methods are listed below. Kindly refer to [Maidl et al., 1996] fordetailed description on available support methods.


• Mechanical support

• Compressed air support

• Liquid support

• Earth pressure balance

• Mixed support

2.3.6 Mechanized Shield vs Conventional Tunneling

The process of shielded construction has enabled the excavation of elongated under-ground tunnels without causing much disturbance or significant se�lement on theground surface [Maidl et al., 2011]. Even soil with low load bearing or under thegroundwater table can be excavated with same advantage as compared to conventionalmeans. Shielded tunneling is also practicable for temporarily stable ground as theshield acts to hold surface as head protection. Hence it can be concluded that shieldedtunneling have wider scope of application.

With all such positive features, the shield construction process can not just replaceother tunneling procedures. Although it is still technically more feasible as well aseconomic alternative to other methods in unfavorable geological conditions but itcarries its own set of advantages and disadvantages as summarized below.

Advantages:

• Higher advance or excavation rate,

• High precision and accuracy,

• Minimizes the negative impact on surface building structure,

• Reduced chances of accidents like tunnel collapse and improves miners safety.,

• Economical and environmental friendly,


• Mechanized procedure creates less noise as compared to conventional drill andblast method,

Disadvantages:

• Design, assembly and production phase of shield machine is time consuming,

• Employees need to be trained and familiarize before being given direct control ,

• Supply line and on site facilities can be very expensive especially if operation isdelayed.

• The cross section of machine is usually circular with li�le to no possibility ofvariation

• the lining normally has to be specially designed to resist the thrust forces.

Application is therefore practicable where the advantages can be sensibly exploitedand the disadvantages are taken into account as far as possible in the design andconstruction planning. Experience shows that a shield in the smaller diameter rangecan generally compete with other tunneling methods for tunnel drives up to 2,000 m.For longer tunnels, economic applications of shield machines are possible and evencheaper than using open machines or conventional methods

It can be seen that application of mechanized tunneling is advantageous if its disad-vantages are taken into account that is during the design and assembling phase of plan.It should be noted that usage of shield machine can become cheaper as opposed toopen machines or conventional method if plan involves construction of longer tunnels.

In practical scenario, according to [Philipp, 1985], too many users have chosen thewrong machine or construction concept for the ground conditions and have laterbeen faced with unacceptable se�lement on the surface, unexpectedly slow advancerates, failure of the lining, water ingress or other defects. For the user, only a tunnelconstructed on schedule, of good quality and at reasonable cost, and with as li�leimpact on the environment as possible is of interest.

The designers of shield equipment need to take these natural concerns into consid-


eration. Mechanical engineering issues have to be e�ectively linked to those of thetunnel itself. Constant exchange of experience between mechanical and civil engineersis essential, with the appropriate evaluation of experience from completed projects.

2.4 Soil se�lements

The skeletal soil material and the pore water are usually in-compressible and anychange in volume can only occur due to change in the volume of the voids. For suchchange in volume to occur , pore water must flow into or out of a soil element. Becausethis cannot happen instantaneously when a load is first applied to soil, there cannotbe any immediate change in its volume.

Figure 2.7: Soil se�lement due to loading conditions

Considering a one dimensional condition with no lateral strain or deformation, impliesthat there is no immediate vertical strain and the excess pore pressure is equal to thechange in vertical stress. However, under more general conditions both lateral andvertical strains can occur.

When load is being applied no immediate change in volume will be observed, butthe soil deformations will result in an initial se�lement. This tends to occur underundrained conditions because no pore water has been able to drain from the soil.A�er a period, the excess pore pressures generated during the undrained loading willdissipate and further lateral and vertical strains will occur. Ultimately the se�lementwill reach its long term or drained value. The process by which soils decrease in volume


is called consolidation. [Boeraeve, 2008]

2.5 Saturated Soil Model

Reduced model order specifically Proper Orthogonal Decomposition (POD) was imple-mented using saturated soil model to test the sensitivity of parameters. This modelworks as a unified critical state model for both clay and sand. The CASM (Clay AndSand Model) model for saturated soils is formulated by means of state parameters andcritical state line. Standard Cam-clay model as well as modified cam clay models wereutilized to redefine the state parameters.

A standard cam clay model cannot predict many feature of behavior of sands andover consolidated clay. But by implementing a stress ratio state parametric relation todescribe state boundary surface of soils, a unified model for both clay and sand (CASM)is achieved. One of the main feature of this model is its capability to utilize single setof yield and plastic potential functions to model the the behavior of clay and sandunder both drained and undrained loading conditions. However, it is possible to stilluse the standard cam clay model and modified version with this model by choosingappropriate parameters. [Yu, 1998]

2.5.1 Soil Parameters

Set of parameters tested for application of proper orthogonal decomposition (POD)are defined briefly as followed

2.5.1.1 Poisson’s Ratio,(υ)

By definition, when a material is compressed or is subjected to a tensile force in onedirection, it usually tends to either expand or contract in the other two directionsperpendicular to the applied force. Mathematically, Poisson’s ratio is the negative ratio


of transverse to axial strain.υ =

εtransεlongitudinal

According to [Yu, 1998], typical Poisson’s ratio for clay and sand lies between υ =

0.15− 0.35

2.5.1.2 Slope of unload-reload curve in v - ln p’ space, (κ)

The soil compression curves shows unloading/reloading loops which are usually relatedto viscosity of soil. For instance clayey soils have larger loops as they have relativelyhigher viscosity value as compared sandy soils which have narrower loops due to lowviscosity values.

According to [Yu, 1998], for a isotropic compression test of unload reload cycles, atypical value for sands lie upto κ = 0.005 (In thesis range of 0.001-0.003 was tested)and for clay between κ = 0.01− 0.06

Figure 2.8: Soil response in compression,[BV]


2.5.1.3 Slope of normal compression curve in v - ln p’ space, (λ)

As shown in figure 1.5 for soil response in compression test. Normal compression curveis referred as λ along virgin compression line. According to [Yu, 1998], a typical rangeof values for sand lies between λ = 0.01− 0.05 under low pressure conditions and asfor clay λ = 0.1− 0.2.

2.5.1.4 Spacing ratio, (r)

The spacing ratio defines the shape of yield surface by all critical state models as wellas bounding surface plasticity model. According to [Yu, 1998], it is used to estimatethe reference state parameter which corresponds to the loosest state a soil is likely toreach in practice. By formula spacing ratio r can also be described as relation betweenpreconsolidated pressure and mean stress at critical state with same preconsoladated

pressure r =p′0p′x

For sake of simplicity, the standard cam clay models assume a single constant spacingratio r for all soil types. In original and modified cam clay its fixed at 2.718 and 2.0respectively.

Experimental data states value of r for clays lies between 1.5-3.0 where as it is largerfor sand.


Figure 2.9: Figure shows state parameter, critical state constants and reference state para-

meter

2.5.1.5 Shape parameter of the yield surface,(n)

Shape parameter or stress state coe�icient n, employed in general stress state relationis a new material constant introduce by [Yu, 1998]. In order to determine n for a givensoil sample, it is advised to plot stress paths of fewer triaxial tests for both drained andundrained on soils of di�erent initial conditions in terms of stress ratio η against stateparameter ξ. A value of n could lie in range of 1.0 - 5.0.

A detailed reference can also be found in a book Plasticity and Geotechnics refered in[Yu, 2006]

2.5.1.6 Slope of critical state line,(M)

M refers to one of critical state constants along with λ and Γ. M can be obtain bytriaxial tests on isotropically consolidated soil samples. It is advised to perform teststo large strains to ensure samples are close to critical state. A typical value of of M isbetween 0.8 -1.0 for clay and 1.1 -1.04 for sands.


2.5.1.7 Friction angle at the critical state,(Φcs)

The friction angle is an important parameter used for analyzing the response of sandsto loading. It is usually determined by pouring soil in graduated cylinder filled withwater.

In the model, Φcs controls the di�erence of the strengths between compression andextension triaxial in the deviatoric plane [Sheng et al., 2000]. By default value of Φcs iskept at zero as it implies an independence of Lode’s angle in deviatoric plane.

2.5.1.8 Initial void ratio,e0

Void ratio is defined in terms of porosity and is relation between volume of void space

(fluids) and volume of solids. e =VvVs

Initial void ratio can be more easily expressed by considering the state parameter figureshown.

Figure 2.10: Void ratio for stress state point P

From the figure, when stress state p′ and preconsolidated stress P ′0 are known, initial


void can be calculated with help of ecs = (Γ− 1), κ and λ [Wood, 1990].

Assume from state parameter figure, for space lnp′, intersecting point of CSL andswelling line is given by ex and p′x, represented by X. Here e and p’ represents initialvoid ratio and e�ective mean normal stress respectively, represented by P.

Hence for initial stress state P, initial void ratio is given by e = ecs−(λ−κ).lnp′x−κlnp′)

2.5.2 Sensitivity of Soil Parameters

Sensitivity of all parameters was tested with combination of 6 of the previously definedsoil parameters. These parameters include unload reload slope, slope of normal com-pression curve, spacing ratio, shape parameter of the yield surface, slope of CSL andvoid ratio.Parameters were tested using the saturated soil model by executing pythonscript where for each a minimum and maximum value range was setup to achieve26 = 64 parametric combinations.

A�er test results were compiled using post processing so�ware GID PRO, it wasobserved that unload reload slope was the most sensitive parameter. The secondaryparameter, although not as sensitive as unload reload slope, was initial void ratio.

This comprehends the fact that stress state in soil and its mass directly influences themovement of soil during excavation.

The table show selection of values for both sensitive parameters.

Unload reload slope Void ratio

0.001 0.80.001 0.450.003 0.80.003 0.45


(a) Par-2 min and Par-8 max (b) Par-2 min and Par-8 min

(c) Par-2 max and Par-8 max (d) Par-2 max and Par-8 min

Figure 2.11: Results for parametric combination of parameter 2 Unload reload slope κ and

parameter 8 initial void ratio e0

The figure shows just 4 results out of a total of 64, as e�ect of other parameters wasnot as significant. Details of input combination are included in parameter combinationfile.


Figure 2.12: Combined displacement plot for previous four cases

22

3 Chapter

Proper orthogonalDecomposition

3.1 POD method of snapshot

Proper orthogonal decomposition is one of the widely research technique being dis-cussed for last decades. The original concept was presented in [Pearson, 1901]. PODis also sometime referred by di�erent names depending on its application or proced-ure employed such as Karhunen–Loeve decomposition (KLD) , Principal componentanalysis (PCA) [Hoetelling, 1935] or SVD.

POD is a powerful data analysis method that tends to approximate high dimensionaldata by projecting it in low dimensional space. In this thesis POD is utilized with radialbasis function to predict soil se�lements, a method that has been previously used byKhaledi 2014 [Khaledi et al.] for mechanized tunneling. Hence in order to expand theresearch, POD is tested with both eigen values and SVD to compute the basis system.RBF was tested with five function equations along with e�ect of smoothing factor withhelp of di�erent error equations.

The procedure of POD involves to initially create a snapshot matrix which consist ofall the output data arranged in order by means of time steps and input cases. Next,set of orthonormal vectors are computed also known as POD basis vector. In this caseapplication of both Eigen values and SVD technique is applied to test the method.Details of this method is discussed in further sections along with steps used to extractthe snapshot data.

3 Proper orthogonal Decomposition 23

3.2 Snapshot Data Extraction

Creation of snapshot matrix is the initial step from which data will be interpolated.A snapshot matrix basically contains all the output data corresponding to providedinput parameters. Simulating to retrieve such data can be quite time consuming, hencesimulations were set to run in parallel (in this case using 8 nodes) to achieve fasterresults. It can usually take upto 8 hours to get result of 6 to 7 simulations. Runningmore simulations at a time will directly e�ect the time taken to completion.

In order to verify the correct execution of simulations, log files created for each simula-tions can be checked for errors. Best strategy is to test one simulation for any possibleerrors before proceeding to execute multiple simulations.

Data extraction can be performed by a post processing so�ware. For current task GIDPRO which is preferable for its collaboration with KRATOS was used to extract therequired data. Selection of data type is upto user, it can be based on soil displacementalong surface of model or point evolution for each node. For creation of snapshotmatrix, point evolution was preferred as data can be easily managed and grouped intime steps accordingly.

3.2.1 Node Selection

For each input combination case, 38 time steps file were created. Since parallelizationtechnique was employed, every time step will be divided in eight parts. All files mustbe combined to form complete model with activation of surface layer only. Selectionof nodes for important portion of model is shown in Figure 2.1 corresponding todisplacement of soil.


Figure 3.1: Optimum nodes selection based on soil displacement

To speed up the process of node selection, user can refer to included CoordinateFile. There are 530 nodal points for each surface and graphical data representation ofcombined nodes will be incomprehensible due to too many points but can be read bycreating grf text file.

Process can be automated by directly copying and pasting all coordinates in "combined"text file for point evolution by using macro command line.


Figure 3.2: Sample coordinates for Nodes selection

3.2.2 Arrangement of Snapshot Data

Arrangement of snapshot matrix is based on user preference. A common practice wouldbe to group the data in time steps for each column where every column represents anindividual input case known as Snapshot.


Figure 3.3: Arrangement of Snapshot Matrix

When data is extracted from simulation file, it will be in grf text format consisting oflarge numeric data set. Manual extraction is definitely not an option as it can take lotof time to individually align all data into snapshot matrix.

Instead it is advised to use any programming script to automate the process. For currentcase Java was employed along with use of excel macros to automate the extractionprocess.

The final snapshot will consist of a total of 530×38 = 20140 rows and number columnsbeing equal to number of input cases.


Figure 3.4: Sample snapshot matrix from original data

3.3 POD Basis Vector

Once the snapshot matrix had been compiled, next step will involve computing PODbasis vectors Φ. POD basis vector can be computed either by svd or eigenvalue de-composition. Both cases should give relatively same POD basis but employ di�erentcalculation procedures.

3.3.1 Eigenvalue Decomposition

The term eigenvalue (derived from German word eigenwert) literally means charac-teristic value. Eigenvalues play an important role in situations where the matrix is atransformation from one vector space onto itself

A matrix can be solved by eigenvalue decomposition by converting it into square matrixby means of computing covariance matrix.

Considering snapshot matrix as S, the co-variance matrix will be given by C = ST .S


or S.ST . Snapshot matrix usually consists limited number of columns based on inputcases but can have large number of rows based on number of timesteps and size ofmodel. Considering the covariance matrix by C = ST .S is advisable as square matrixwould be equal to number of columns (current case 76 x 76) rather than by number ofrow (20140 x 20140) which can take very long computing time and may lead to programerror itself.

A�er computing eigenvalue decomposition of matrix C, POD basis vector can becomputed by following equation:

Φ = S.Ψ√λ

where S denotes the output snapshot matrix, Ψ denotes the eigen vectors and λ refersto eigenvalues in diagnolized form.

3.3.2 SVD

Singular values relates to distance between a matrix and the set of singular matrices.Singularvalues play an important role where the matrix is a transformation from one vectorspace to a di�erent vector space, possibly with a di�erent dimension.

Unlike eigenvalue decomposition, snapshot matrix S can be directly employed tocompute svd as it works for rectangular matrices. This means that step for calculatingcovariance matrix will not be required. SVD of snapshot matrix S is decomposed asS = U.Σ.V T , where U and V refers to vector matrices and Σ denotes singular values.

POD basis vector can then be computed by employing following equation.

Φ = (1

Σ).S.V

SVD and eigenvalue decomposition shares some facts which can be used to verify ifcomputation of both matrix decomposition is correct.The eigenvalues λ obtained aresquares of the singular values Σ from SVD


3.3.3 Truncated POD Basis Vector

Truncated POD basis will be selected from the POD basis of whole snapshot matrix.In this case selection of k number of columns will define the accuracy of truncatedPOD basis in calculating the Snapshot matrix S. In order to define k, it is importantthat eigenvalues are arranged or sorted in descending order where the initial value willconsist of highest energy.

Σki=1.λi

ΣMi=1.λi

≥ DesiredAccuracy

k refers to no. of columns for truncated POD basis vector, M refer to total number ofcolumns of POD basis, Desired accuracy can be set accordingly.

POD basis is usually matrix with dimensions equal to number of snapshot or inputcases. In the program, truncated POD basis will be selected based on provided accuracy.

The snapshot matrix S can also be expressed for new basis as

S = Φ.A

From the equation ,matrix A needs to be calculated which represents the amplitudematrix associated with snapshot matrix S. Hence above equation can be rewri�en as

A = ΦT .S

Depending on the POD basis (complete or truncated), amplitude matrix can be calcu-lated accordingly and can be used to approximate the snapshot matrix.

S ≈ Φ.A

30

4 Chapter

Radial Basis Function

4.1 RBF Interpolation

The Radial Basis Function (RBF) method is one of the primary tools for interpolatingmultidimensional sca�ered data. The method’s ability to handle arbitrarily sca�ereddata, to easily generalize to several space dimensions and to provide spectral accuracyhave made it particularly popular in several types of applications [Wreight, 2003].Its principle usually involves describing any any real valued function whose outputdepends exclusively on the distance of its input from some origin or a fixed center.

Scientific Computing with Radial Basis Functions focuses on the reconstruction ofunknown functions from known data. The functions are multi-variant in general, andthey may be solutions of partial di�erential equations satisfying certain additionalconditions. However, the reconstruction of multi-variant functions from data can onlybe done if the space furnishing the trial functions is not fixed in advance, but is datadependent [Mairhuber, 1956].

RBFs can relatively produce good interpolation from large number of data points.Function equations produces good results for gentle variations in data set. However,technique becomes inappropriate if variation is large between short distances in dataset.

4 Radial Basis Function 31

Some of the common RBF types are listed below which were tested in the program.

• Inverse Multiquadrant

φ(c) =1√

r2 + c2

• Multiquadrantφ(c) =

√r2 + c2

• Inverse �adrantφ(c) =

1

r2 + c2

• Gaussian and Modified form

φ(c) = e(−r2.c2)

φ(c) = e−r2

c2

• Thin Splineφ(c) = rk.ln(r), k = 2, 4, 6..

4.2 Function Equations

Application of RBF will require amplitude matrix A and function matrix F to calculatean unknown coe�icient matrix B. The equation that defines the relation is as follows:

Ai = B.Fi

To calculate matrix B, function matrix needs to be computed which is the most im-portant part for radial basis function. Function matrix is based on the RBF equationemployed to interpolate the data. Generally a function matrix can be defined as follows


Fi = Fj(zi − zj) =

f1(z1 − z1) f1(z2 − z1) f1(z3 − z1)

f2(z1 − z2) f2(z2 − z2) f2(z3 − z2)

f3(z1 − z3) f3(z2 − z3) f3(z3 − z3)

− − −fn(z1 − zn) fn(z2 − zn) fn(z3 − zn)

Where z1 − z2 implies the distance between the two coordinate points given by

| z1 − z2 |=√

(x1 − x2)2 + (y1 − y2)2

Type of function equation employed for RBF interpolation will replace Fi accordingly.

4.2.1 Smoothing Factor

Smoothing factor consist of collection of data taken over time that was in some formof random variation. This means smoothing technique tends to reveal the underlyingtrend and cyclic component of data clearly. In the current case scenario [Hardy, 1971],simple moving average type was employed which is one the common method to smoothout short-term fluctuations and highlight longer-term trends in data set.

Smoothing factor is a part of RBF function represented by le�er c in the program.

c = N.(1

M.Σ(Di))

Where Di denotes distance between the ith data point and its neighbor. M refers tomean number of distance for each corresponding point. N is a shape parameter, inthis thesis it was tested with a range of 0.6 to 1.7. In further investigation e�ect of thevalue of shape parameter on error percentage is closely studied.


Figure 4.1: Calculating sample mean distance M for smoothing factor

The figure helps in defining the loop required to calculate smoothing factor c in C++.

34

5 Chapter

Result Analysis

5.1 Error Equations

A detailed analysis was done based on number of snapshots, di�erent types of RBFinterpolating functions and varying smoothing factor. Verification was performed byemploying di�erent error equations.

Four error equations were used to compare accuracy of interpolated data.

• Percentage Error: Percentage Error is normally represented by

Error% =

√ΣN

i=1(Analytical − Interpolated)2

ΣNi=1(Analytical)2

× 100

where N denotes number of total nodes or points in snapshot (in current caseN = 20140)

• Mean Absolute Percentage Error: Also referred as MAPE is given by

MAPE =100

n.ΣN

i=1 |Analytical − Interpolated

Analytical|

• Root Mean Square Error:

RMSE =

√ΣN

i=1(Analytical − Interpolated)2

n

5 Result Analysis 35

• Normalized Root Mean Square Error:

NRMSE =RMSE

Amax − Amin

5.2 Graphical Analysis

For graphical analysis,in order to understand comparative results, 40 points will bechosen to show horizontal movement and 14 points to show vertical movement asshown in the figure 5.1. Selection will be based on varying smoothing factor by fixingtimestep at 21 with two input cases. Inverse Multi-�adrant RBF was selected as itprovides best accuracy related to other functions.

Figure 5.1: Nodes selection for graphical analysis


(a) (b)

(c) (d)

(e) (f)

Figure 5.2: Plot comparison of horizontal direction for di�erent input cases and smoothing

factor


(a) (b)

(c) (d)

(e) (f)

Figure 5.3: Plot comparison of vertical direction for di�erent input cases and smoothing

factor


5.3 Investigation 1 : Eigenvalues and SVD

POD basis is calculated by employing singular values as shown in the equation X. Sincesingular values are closely related to eigenvalues, they can be converted to singularvalues by taking square root of eigenvalues.

SVD is calculated directly for given rectangular matrix where as eigenvalue requirescomputation of covariance matrix and later converted to singular values to computePOD basis. This means, theoretically, eigenvalues goes through two extra steps thatcan yield lose of accuracy when computation can round o� few numbers. This scenariowas tested by verifying result from both cases that yielded almost same result.

Secondly, computation time for SVD will take lot longer when compared to eigenvalues.This may seem odd since eigenvalues goes through more steps but computes answerin far less time as compared to svd that can takes upto 30-60 seconds. This is becauseSVD calculates two more vectors that are le� and right vectors. For a matrix of m× ndimensions, le� matrix is m×m and right being n× n. Hence SVD has to computelarge matrices, in this case the le� matrix have dimension of 20140× 20140 whereaseigenvalues skips this step when computing covariance matrix.

During the investigation it was found that POD basis computed by eigenvalues andsvd were not same. A�er first few columns, the direction and positioning of POD basisvalues did not match each other. Since the first column consist of most energy soeven though POD basis are not same but it leads to accurate prediction. Since theprogram uses library and its way of computation may contain some errors, this leavesroom to further investigate in future to understand why program is flipping signs andpositioning vectors di�erently.

5.3.1 Computation time

POD basis with eigenvalues is computed faster compared to svd. Hence time taken forPOD basis tΦ−Eigen << tΦ−SV D.


Case No. of Snapshots Time for Φ

Eigen 36 1.0sEigen 69 1.9sSVD 36 30sSVD 69 60s

Table 5.1: Time taken for computation of POD basis with Eigen decomposition and svd

(a) Screen-shot of time for eigenvalues, 69 snapshot

(b) Screen-shot of time for SVD, 69 snapshot

Figure 5.4: Screen shot comparison of time taken to compute POD basis and RBF for

eigenvalues and SVD

5.4 Investigation 2 : Impact of smoothing factor

In general, smoothing factor is employed to improve the accuracy of the function. Thisis usually achieved by using shape parameters with smoothing factor to optimize theresult. The value of shape parameter is usually a small number which must be abovezero.

Smoothing factor can help improve accuracy but under certain limit, a�er which theprediction result can have decreased accuracy. This means e�ect of smoothing factoris not linear and must be tested to fix the optimized value.

For current thesis smoothing factor was calculated by multiplying moving averagefactor by 0.6, 0.8, 1.0 ,1.3 , 1.5 and 1.7 with extended range till 10 to verify and under-


stand the impact on results. From the graphical analysis shown under this section,clearly indicates the trend and sensitivity of smooth factor with respect to number ofsnapshots.

For di�erent radial basis function, a similar trend was observed. With increasingnumber of snapshot, function becomes accurate but more sensitive to smoothing factor.Hence a sharp rise in error percentage is observed for higher number of snapshotsas compared to its lower counter part but with optimized value best results can beachieved with higher number of snapshots.

In graphical analysis, higher smoothing factor was removed so plots are appropriatelyscaled. Detailed analysis can be referred in Table 5.2 which consist of compressed meanerror percentage results.

Figure 5.5: Plot comparison for increasing shape parameter for Inverse Multi �adrant


Figure 5.6: Plot comparison for increasing shape parameter for Multi �adrant

Figure 5.7: Plot comparison for increasing shape parameter for Inverse �adrant


Figure 5.8: Plot comparison for increasing shape parameter for Gaussian (division)

Error Percentage %

Shap.Param.

Inverse Multi �ad Multi �ad Inverse �ad Gaussian36 76 36 76 36 76 36 76

0.6 1.8906 1.4482 1.0860 0.4940 16.7538 13.8272 72.0382 65.0990

0.8 1.0825 0.7990 1.0935 0.4758 6.4477 6.3248 27.2674 33.4949

1 0.8885 0.5803 1.0818 0.4570 2.3363 2.7823 4.6885 11.4873

1.3 0.8462 0.4175 1.0439 0.4339 1.6584 1.1267 3.7342 2.0909

1.5 0.8447 0.3831 1.0123 0.4346 1.5397 0.8873 3.6531 2.2522

1.7 0.8500 0.3816 0.9789 0.4406 1.4635 0.7444 3.3565 1.7202

2 0.8622 0.4030 0.9290 0.4488 1.3654 0.6463 2.7489 0.8984

2.3 0.8754 0.4172 0.8824 0.4516 1.2747 0.6308 2.1417 0.4814

2.5 0.8846 0.4228 0.8581 0.4498 1.2204 0.6035 1.8139 0.4233

3 0.9081 0.4283 0.8970 0.4343 1.1139 0.5191 1.2857 6.1200

10 1.0227 16.64157 4.4553 128.3120 1.14223 743.9181 2.0275 6.4607

Table 5.2: Comparative table for increasing shape parameter of smoothing factor


5.5 Investigation 3 : Impact of Number of snapshots

Number of snapshots also directly impact the result by improving the accuracy. Thiscan not only be seen in previous plot comparison for di�erent smoothing factors butcan also be observed in detailed analysis for increasing number snapshots periodically.

Analysis for inverse multi quadrant shows, smoothing factor defines the error thresholdfor the result and adding more snapshots increases the accuracy. From the plot itis clearly visible that a�er approximately 50 to 60 snapshots, the function becomesstable and adding further data does not impact the result a lot. It can also be observedthat increasing shape parameter for smoothing factor to 2.0 starts to increase errorpercentage.

Figure 5.9: Comparison of plots employing Inverse Multi �adrant for smoothing factor

with increasing number of snapshots and error percentage


5.6 Investigation 4 :Comparison of di�erent RadialBasis Functions

Type of RBF interpolating function used ma�ers a lot on the accuracy of final result.From the analysis of five di�erent RBF types employed for case study, shows di�erentbehavior and impact of smoothing factor. This analysis can be verified by referringtables in appendix or examining plots in investigation 1 and 2.

Inverse Multi �adrant : This radial basis function outmatched all the other functionequations in terms of accuracy. Hence, inverse multi quadrant should be selected forfuture case scenarios. Best results were achieved when shape parameter is set between1.5 to 1.7. This can be verified by referring table 5.1 for mean error percentage of76 snapshots or provided graphical analysis as well. Appendix can be referred forindividual validation cases but may not be suitable to achieve conclusive result.

Multi �adrant : This function performs similar to inverse multi quadrant as theyare inverse of each other. The drawback is its error threshold is relatively higher thanthe inverse counter part.

Inverse �adrant : A brief analysis shows that results with this function have higherpercentage when compared with Inverse multi quadrant and may not be suitable to beused as RBF in case scenarios involving parametric data.

Gaussian : Two di�erent Gaussian radial basis function were employed to test accuracy.Since both performed similar to each other, results analysis was based on its inversecounter part. Gaussian proved to be the worst function equation for this case study.With highest error percentage among all the other RBF types, it would be best not touse it in cases involving sca�ered data or parametric analysis.


5.7 Investigation 5 : Surrogate model for smallerPOD basis vector

This investigation involved breaking the snapshot data into smaller sub snapshot, (forexample, in this case snapshot was broken with respect to 38 time steps where 1 timesteps consist of 530 points or number of rows). POD basis will be individually calculatedfor all the sub snapshots to provide truncated version and finally all the results will berearranged to form a single snapshot.

Algorithm for Individual basis :

• Snapshots will be broken into number of time steps, resulting in smaller matrices.

• Breaking point in the program refers to the selection of number of rows by whichdivision takes place.

• POD basis for each matrix will be computed along with its relative functionmatrix.

• Resulting snapshots from individual basis will then be computed

• Finally, resulting individual snapshots will be rearranged into single snapshotmatrix.

Error Percentage %Time POD Basis Φ

Case 0.6 1.0 1.5

S.M1 0.14439 0.04325 0.04451 3s 38 steps

S.M2 0.14464 0.04417 0.04545 1s 1 step

Table 5.3: Comparison of error percentage and POD basis time computation with Inverse

multi quadrant for S.M1(38 POD basis)and S.M2 (1 POD basis)

From the result it can be observed, POD basis computation for case S.M1 wheresnapshot is broken down to compute POD basis provided be�er accuracy as comparedto S.M2 where it computes one POD basis for whole snapshot.


Considering the time factor, computation of 38 POD basis as compared to single PODbasis vector increases by 3 times from 1 second.

47

6 Chapter

Surrogate Modeling using C++

The original goal of thesis was to create a program based on C++, utilizing POD methodwith Radial basis functions to aid interpolating data.For this purpose initial approachwas either to create a system that can deal with large matrices easily or employ freelibraries to ease the complexity in coding. A�er testing phase, Armadillo was selectedas it is free and comes with build in cmake file that makes installation on Linux systemquite easy. As for windows operating system, library has to be uploaded manually[documentation].

6.1 Library Implementation

C++ is a general-purpose programming language. It has imperative, object-orientedand generic programming features, while also providing facilities for low-level memorymanipulation. While the standard C++ library provides many useful algorithms (eg.sorting), in its current form it does not provide direct handling of linear algebra likemathematical matrix applications.

Armadillo is an open source linear algebra library for the C++ language. Its aim is tooptimize for a balance between speed and ease of use. It programming interface orfunction syntax is similar to the widely used Matlab and Octave languages [Eatonet al., 2015]. Due to this expression of mathematical operations and applications canbe done in a familiar and user friendly manner.

Armadillo consists of more than 200 associated functions for manipulating data storedin the objects.Integers, floating point and complex numbers are supported, as well as

6 Surrogate Modeling using C++ 48

dense and sparse storage formats. Various matrix factorization are provided throughintegration with LAPACK [Demmel, 1997], Intel MKL [201, 2016] or OpenBLAS [Xianyiet al., 2016].

In this thesis, Armadillo utilizes LAPACK and BLAS sub libraries. BLAS is a collectionof low-level matrix and vector arithmetic operations. This involves operation likemultiplication of vector by scalar, simple matrix multiplication or addition etc. LAPACKon the other hand is a collection of higher-level linear algebra operations. this involvesoperations like matrix factorization or decomposition (LU, QR, SVD, etc). Usually inpractical application, most user only utilizes LAPACK without any need to be evenaware of BLAS.

For detailed information on library refer to developer’s paper [Sanderson and Curtin,2016].

6.2 License and distribution

As stated, Armadillo versions 7.80 and onward are licensed under the Apache License2.0. Armadillo can be used in proprietary so�ware, without releasing the source code.However, any so�ware that incorporates or distributes Armadillo in source or binaryform must include a readable copy of the a�ributions in the NOTICE.txt file that comeswith Armadillo. The a�ributions can be placed in the documentation and/or othermaterials provided with the so�ware.

6.3 Program algorithm

Following section will contain details related to C++ script. Program originally consistsof one main scripts, one user script and some modified scripts to further enhance theresearch. First script consists two portions that includes finding POD basis vectorsbased on output snapshot matrix and application of selected radial basis functionbased on input data. The second script predict the data based on given user input.


The modified versions of scripts can be used to find POD basis by svd and computePOD along with RBF by breaking the snapshot matrix into number of timesteps. Thismeans for 38 timesteps, a total of 38 POD basis vectors will be computed.


6.3.1 POD Basis vectors

Initially library function for armadillo must be included along with other functions inorder to execute and compile program correctly.

// EIGEN VALUES Determination

mat A; // OUTPUTA.load("/home/aun/POD_RBF_Program/SNAPSHOTS/SNAP

SHOT_69.csv");

mat A_t = A.t();//A_t.save("transpose of A",csv_ascii);//A_t.print("Transpose of the Selected Matrix");

mat C = A_t * A; // Covariant Matrix

// EIGEN VALUES

cx_vec eigval1;cx_mat eigvec1;

eig_gen(eigval1, eigvec1, C);eigval1.save("Eigen_Values.csv",csv_ascii);eigvec1.save("Eigen_Vectors.csv",csv_ascii);

The syntax for loading, saving and printing matrices is shown in the code. Matricescan be saved in ods (LibreO�ice) and xls or xlsx (MS Excel) format. Both files canbe converted into csv text format. Implementing csv with ods format is easier asLibreO�ice can directly open csv document in tabular format without any errors.

Initially covariance matrix is calculated before implementing to solve for eigenvalues.SNAP_SHOT_69

refers the number of snapshot included in that particular file which can be changedbase on number of snapshots required.


Since the program finds eigen vectors in complex form and eigenvalues arranged in 1column matrix. This means next step should involve converting vector values to realform and diagnalizing the eigenvalue matrix.

// Converting eigen vectors to uncomplex form

mat Oe = conv_to<vec>::from(eigval1);mat eigvec = real(eigvec1);

// Diagonalizing Matrix

mat diag_eigen = diagmat(Oe,0.0);mat Oi = inv(diag_eigen);

cx_mat S_Oi = sqrtmat(Oi);mat S_conv = real(S_Oi); // Converting eigen-values into

uncomplex form

From snippet shown above, diagmat( X, k ) converts any given matrix X into a matrixwith k-th diagonal where k-th diagonal contains copy of values of X and all otherelements are set to zero.Hence diagmat(Oe,0.0) generates diagnolized eigenvalues.More details can be referred in o�icial library documentation for Armadillo [document-ation]. Square root of eigenvalues also converts values into complex number so anextra conversion step is added to avoid the problem.

Finally, POD basis vectors can be determined along with amplitude matrix (E).

Φ = S.Ψ√λ


Note that inverse square root of eigenvalues is already compensated in mat S_conv

// POD BASIS CODE

mat P = A * eigvec*S_conv;//mat P = A * V*S_conv;P.save("POD_BASIS.csv",csv_ascii);

mat P_t = P.t(); // transpose of POD Basis

// Mat E (Amplitude Matrix)-mat E = P_t * A;E.save("A Constant.csv",csv_ascii);

Once the POD basis is computed, program automatically creates the truncated versionof the POD basis based on accuracy which is used when executing user script.

mat e_con;mat eigencon = real(eigval1); // Eigen values are

converted to uncomplex formeigencon.save("EigenValuesConv.csv",csv_ascii);

r1=eigencon.n_rows; // Accuracy loop is implementedbased on provided values

for(i=0; i<r1; i++){sum_eigen = sum_eigen + eigencon(i,0);

}

for(i=0; i<r1; i++){local_sum= local_sum + eigencon(i,0);temp = local_sum/sum_eigen*100;

if(temp>=accuracy){k=i;

cout<<"Value of K is: "<<k;cout<<"value of temp: "<<temp;break;

}


}

6.3.2 Radial Basis Function

Input data must be loaded for the same snapshot matrix that was used for POD basisvectors. RBF involves calculation of smoothing factor and computing function matrix.

// Smoothing Factor c

for(i=0; i<r1; i++){for(j=0; j<r1; j++){

sum = sum +sqrt((Z.at(i,0)-Z.at(j,0))*(Z.at(i,0)-Z.at(j,0))+ (Z.at(i,1)-Z.at(j,1))*(Z.at(i,1)-Z.at(j,1)));

}}

sum2 = (r1-1)*r1;c=sum/sum2*1.5; //Change shape parameter accordingly

cout<<",,," << c;

The code shown includes "cout" print command for c exclusively. This is to verify if thevalue of c calculated in main and user script is same. Sometimes its possible loop doesnot work correctly as it might apply loop in addition to previous values. Preventivemeasures were added to set values of c, sum and sum2 to zero before program isre-executed.

Next step will involve normalizing the input data. Normalizing is process of scaling thenumbers of data-set to improve the accuracy of numeric computation. Normalizing isgiven by


Np1 =zi −min(z)

max(z)−min(z)

where Np1 refers to input data taken for first parameter (eg unload reload slope). zirefers to input value of that parameter. min(z) and max(z) as apparent, refers tomaximum and minimum value of the selected parameter. Application of normalizingwill bring all the values of input cases in the range of 0 to 1.

Once the smoothing factor has been calculated and the input data has been normalized, function matrix can be computed.

// Function Matrix// Distance Calculation

mat R = zeros(r1, r1);mat F = zeros(r1, r1);for(i=0; i<r1; i++){

for(j=0; j<r1; j++){x = N.at(i,0)-N.at(j,0);y = N.at(i,1)-N.at(j,1);R.at(i,j)= sqrt(x*x + y*y);

}}for(i=0; i<r1; i++){

for(j=0; j<r1; j++){

F.at(i,j)= 1/sqrt(R.at(i,j)*R.at(i,j)+ c*c); //1.Inverse Multi Quad

//F.at(i,j)= sqrt(R.at(i,j)*R.at(i,j) + c*c); //2.Multi Quad

//F.at(i,j)= 1/(R.at(i,j)*R.at(i,j)+ c*c); //3.Inverse Quad

//F.at(i,j)= exp(-((R.at(i,j)*R.at(i,j))*(c*c))); //4.Gausian

//F.at(i,j)= exp(-((R.at(i,j)*R.at(i,j))/(c*c))); //5.Gausian-Division


//F.at(i,j)= (pow(R.at(i,j),2))*log(R.at(i,j)) ; //6.Thin Plate -- Resulting in NAN

}}

Several function types were use to investigate their e�ect on the accuracy of result. A�ercorrect computation of function matrix, a constant RBF matrix can than be calculatedwith amplitude and function matrix that will aid in computing final truncated snapshotmatrix.

B = E.Fi

where B refers to constant matrix based on function matrix, E refers to previously usedamplitude matrix and Fi is the inverse of function matrix. A complete snapshot matrixcan be obtained although not necessary by following equation.

S = P.B.F

where S denotes complete snapshot matrix and P denotes POD basis found earlier.

6.3.3 User Script

User script is the most important part as it will not only verify the correct computationof function matrix but will also predict the output data based on given user input dataunder the range of max and min variables.

The overall procedure is almost same as that for finding POD basis and radial basisfunction matrix. Only few steps will be altered by taking previously collected data tocompute the truncated snapshot matrix.

Previously calculated POD basis matrix will be used in truncated form. This meansusing complete POD matrix is not required, as same results with optimum accuracy


can be achieved by taking truncated form. The selection of column is based on providedaccuracy. Since first eigenvalue consist of most energy, even one column can provide99.9 % accuracy. For this thesis accuracy was set so program picks around 5 to 6columns from truncated POD basis.

Next amplitude matrix will be recalculated for the truncated version.

Atr = Pt.S

where Atr refers to truncated amplitude matrix, Pt denotes transpose of truncatedPOD basis matrix and S being the output snapshot matrix.

In RBF phase, previous function matrix will be reused along with truncated amplitudematrix to calculate the constant RBF matrix.

B = Atr.Fi−all

Figure 6.1: Screen-shot for output screen showing selection of number of columns k and

time taken to compute POD basis

6.3.4 Modified Script Versions

6.3.4.1 SVD

As explained earlier, modified version employs SVD method to decompose matrix andcompute singular values.

Φ = (1

Σ).S.V


SVD should provide same basis product for POD basis and its transpose as eigen valuedecomposition.

// SVD CODEmat U;vec s;mat V;

svd(U,s,V,A,"std");

s.save("svd_s.csv",csv_ascii);mat diag_s = diagmat(s,0.0); // DIAGNALIZEDdiag_s.save("diag_svd_s.csv",csv_ascii);

// INVERSE OF SVD Smat si = inv(diag_s);

6.3.4.2 Surrogate model for smaller POD basis vector

This script consist two sub scripts where initial program breaks the snapshot intosmaller matrices based on timesteps to create POD basis vectors for each matrix. Oncethe the script is executed, it will require number of rows for each time steps (530)

Figure 6.2: Screen-shot for script execution

The second script uses user input to compute truncated snapshot for each POD basisand re-arrange all sub-snapshots into single one column snapshot.


6.4 Troubleshooting

This section is added to prevent common errors that might occur during compilation ofcode or results in wrong interpolation. Before compiling program user can go throughthese steps to make sure if right command line is activated.

• Check if right number of Snap shot matrix is selected in both main and userscript.

• Same smoothing factor and RBF equation must be employed for both script

If the result still leads to wrong interpolation or formation of void function matrix.Verify the csv file of Snapshot matrix by opening it in text editor. It is a common problemthat occurs if ods file is directly saved in csv a�er modification as it can change thedata type. This can be overcome by saving edited version in ods and renaming it incsv. Another method is to just open ods in csv text format, replace (ctrl-H) the -AC0datatype error with null value.

6.4.1 Script execution

Armadillo uses g++ compiler and following command can be used to execute program.Refer to section example in documentation [documentation]. Here -O2 or -O3 optionsare used to enable optimization when compiling program. In current case O2 wasselected (by default) for all execution since code compiled correctly.

g++ example.cpp -o example -O2 -larmadillo


6.5 Appendices

Inverse Multi �adrant: 36 Snapshots

N of smoothing factor Input Case Error Percentage MAPE RMSD NRMSD

0.6× c

1 1.720 3.2422 2.984E-05 0.00542 1.948 3.4182 3.203E-05 0.00623 1.0371 7.4749 1.583E-05 0.00334 2.005 3.1812 4.227E-05 0.00625 2.129 2.8280 4.212E-05 0.00666 2.270 5.9843 5.685E-05 0.00697 2.121 4.4471 4.970E-05 0.0065

1.0× c

1 0.2418 3.0451 4.196E-06 0.00082 2.0292 2.7058 3.335E-05 0.00643 1.5065 6.0170 2.299E-05 0.00484 0.4264 1.8508 8.985E-06 0.00135 0.7960 1.3041 1.574E-05 0.00256 0.7930 5.5766 1.986E-05 0.00247 0.4270 2.5708 1.000E-05 0.0013

1.5× c

1 0.2967 2.5936 5.148E-06 0.00092 2.3860 3.0502 3.922E-05 0.00753 1.6636 5.9319 2.538E-05 0.00534 0.1121 1.8621 2.362E-06 0.00035 0.6748 1.3372 1.334E-05 0.00216 0.6265 5.7415 1.569E-05 0.00197 0.1531 2.2537 3.587E-06 0.0005

Table 6.1: Inverse Multi �adrant for 36 Snapshots


Multi �adrant: 36 Snapshots


0.6× c

1 2.8510 4.1388 4.946E-05 0.00902 2.3337 3.2242 3.836E-05 0.00743 1.4959 6.5816 2.282E-05 0.00484 0.2630 1.9919 5.542E-06 0.00085 0.1040 0.6370 2.056E-06 0.00036 0.3848 5.5469 9.635E-06 0.00127 0.1698 2.3357 3.978E-06 0.0005

1.0× c

1 2.2526 3.1260 3.908E-05 0.00712 2.2590 3.2796 3.713E-05 0.00713 1.9147 6.8528 2.922E-05 0.00614 0.4232 2.3674 8.920E-06 0.00135 0.1007 0.7867 1.992E-06 0.00036 0.4186 5.7072 1.048E-05 0.00137 0.2034 2.3113 4.765E-06 0.0006

1.5× c

1 1.3990 1.8075 2.427E-05 0.00442 2.0788 3.3127 3.417E-05 0.00663 2.2638 7.0065 3.454E-05 0.00724 0.4677 2.6408 9.855E-06 0.00155 0.2019 0.9781 3.993E-06 0.00066 0.4477 5.8346 1.121E-05 0.00147 0.2275 2.3291 5.329E-06 0.0007

Table 6.2: Multi �adrant for 36 Snapshots


Inverse �adrant: 36 Snapshots


0.6× c

1 18.7011 19.7222 3.245E-04 0.05882 15.7475 17.2234 2.588E-04 0.04983 14.9141 20.4340 2.276E-04 0.04744 16.2865 17.2827 3.432E-04 0.05065 17.3735 18.0845 3.436E-04 0.05426 16.6745 18.9990 4.175E-04 0.05107 17.5795 19.5660 4.118E-04 0.0541

1.0× c

1 0.1875 2.6199 3.252E-06 0.00062 2.8019 3.6249 4.605E-05 0.00893 0.5138 5.8767 7.839E-06 0.00164 2.9860 4.0887 6.293E-05 0.00935 3.5862 4.0011 7.092E-05 0.01126 3.1202 7.1332 7.813E-05 0.00957 3.1584 5.1134 7.398E-05 0.0097

1.5× c

1 3.0432 5.2795 5.280E-05 0.00962 1.5910 2.2438 2.615E-05 0.00503 2.1090 6.2208 3.218E-05 0.00674 0.8561 2.3009 1.804E-05 0.00275 1.5697 2.0949 3.104E-05 0.00496 1.0176 5.9295 2.548E-05 0.00317 0.5913 2.6179 1.385E-05 0.0018

Table 6.3: Inverse �adrant for 36 Snapshots


Gaussian : 36 Snapshots


0.6× c

1 5.6373 18.6000 9.781E-05 0.01772 0.6108 8.1129 1.004E-05 0.00193 10.9017 31.3713 1.663E-04 0.03464 2.3519 9.2985 4.956E-05 0.00735 3.1047 10.1055 6.140E-05 0.00976 1.3729 14.2384 3.437E-05 0.00427 2.2147 8.3635 5.188E-05 0.0068

1.0× c

1 288.0473 432.6345 4.998E-03 0.90642 4.1979 8.7538 6.900E-05 0.01333 548.5152 833.1051 8.369E-03 1.74324 34.0098 58.8754 7.167E-04 0.10565 97.6396 145.6899 1.931E-03 0.30476 102.9766 223.7776 2.578E-03 0.31497 14.3686 26.1344 3.366E-04 0.0442

1.5× c

1 1.6134 6.6250 2.799E-05 0.00512 0.5638 7.9643 9.267E-06 0.00183 1.8153 17.2178 2.770E-05 0.00584 1.7839 8.2819 3.759E-05 0.00555 0.6216 10.6958 1.229E-05 0.00196 1.3283 9.1736 3.326E-05 0.00417 2.3118 9.9657 5.415E-05 0.0071

Table 6.4: Gaussian for 36 Snapshots


Gaussian Devision : 36 Snapshots


0.6× c

1 75.4201 75.5633 1.309E-03 0.23732 67.5506 67.6507 1.110E-03 0.21353 75.2442 75.9590 1.148E-03 0.23914 67.5477 67.7265 1.424E-03 0.20985 75.7392 75.7527 1.498E-03 0.23646 67.1116 68.0241 1.680E-03 0.20527 75.6541 75.9715 1.772E-03 0.2327

1.0× c

1 1.8878 4.3353 3.275E-05 0.00592 3.6858 4.3011 6.058E-05 0.01163 1.8027 6.3010 2.751E-05 0.00574 5.7523 6.7439 1.212E-04 0.01795 7.3786 7.6244 1.459E-04 0.02306 5.8066 9.6964 1.454E-04 0.01787 6.5061 8.2293 1.524E-04 0.0200

1.5× c

1 11.0828 12.5953 1.923E-04 0.03492 0.2805 1.0843 4.611E-06 0.00093 3.9221 7.3423 5.984E-05 0.01254 3.1989 4.5447 6.741E-05 0.00995 4.8497 5.6297 9.591E-05 0.01516 1.3764 6.2250 3.446E-05 0.00427 0.8615 2.8338 2.018E-05 0.0026

Table 6.5: Gaussian (Devision) for 36 Snapshots


Inverse Multi �adrant : 69 Snapshots


0.6× c

1 1.0125 1.7223 1.757E-05 0.00322 0.8526 1.8235 1.401E-05 0.00273 0.7123 6.8699 1.087E-05 0.00234 0.9725 1.3242 2.050E-05 0.00305 2.4834 3.6463 4.911E-05 0.00786 1.5135 3.0900 3.790E-05 0.00467 2.5906 4.8267 6.068E-05 0.0080

1.0× c

1 0.3092 1.2595 5.364E-06 0.00102 0.2655 0.9633 4.364E-06 0.00083 1.3421 5.7759 2.048E-05 0.00434 0.2001 0.9364 4.216E-06 0.00065 0.8110 2.1929 1.604E-05 0.00256 0.3525 1.7926 8.826E-06 0.00117 0.7819 2.7978 1.832E-05 0.0024

1.5× c

1 0.3182 1.0315 5.520E-06 0.00102 0.1316 0.9072 2.163E-06 0.00043 1.6827 6.1745 2.568E-05 0.00534 0.0792 1.1115 1.670E-06 0.00025 0.1889 2.4964 3.736E-06 0.00066 0.1245 1.3725 3.118E-06 0.00047 0.1562 2.7526 3.658E-06 0.0005

Table 6.6: Inverse Multi �adrant for 69 snapshots


Multi �adrant : 69 Snapshots


0.6× c

1 0.9897 1.5150 1.717E-05 0.00312 0.4832 1.1015 7.942E-06 0.00153 1.4533 6.0895 2.217E-05 0.00464 0.0489 0.8931 1.031E-06 0.00025 0.2209 2.0076 4.369E-06 0.00076 0.1539 1.6886 3.854E-06 0.00057 0.1078 2.2062 2.525E-06 0.0003

1.0× c

1 0.7364 1.0734 1.278E-05 0.00232 0.1990 0.9602 3.271E-06 0.00063 1.7748 6.6839 2.708E-05 0.00564 0.0505 1.0824 1.065E-06 0.00025 0.1721 2.5424 3.404E-06 0.00056 0.1405 1.4413 3.519E-06 0.00047 0.1256 2.6308 2.943E-06 0.0004

1.5× c

1 0.4735 0.7883 8.214E-06 0.00152 0.1119 1.0086 1.839E-06 0.00043 1.9771 7.0717 3.017E-05 0.00634 0.0570 1.2153 1.200E-06 0.00025 0.1406 3.0395 2.781E-06 0.00046 0.1344 1.3162 3.366E-06 0.00047 0.1475 3.2538 3.455E-06 0.0005

Table 6.7: Multi �adrant for 69 snapshots


Inverse �adrant : 69 Snapshots


0.6× c

1 11.5445 12.1990 2.003E-04 0.03632 8.9375 9.8592 1.469E-04 0.02823 17.3935 22.3077 2.654E-04 0.05534 8.9323 9.2969 1.882E-04 0.02775 19.4040 20.3222 3.837E-04 0.06066 11.5524 12.7247 2.893E-04 0.03537 19.0264 20.9940 4.457E-04 0.0585

1.0× c

1 0.6290 1.2932 1.091E-05 0.00202 1.3966 2.0346 2.296E-05 0.00443 2.4234 7.1135 3.698E-05 0.00774 1.5514 1.9636 3.270E-05 0.00485 5.5699 6.5366 1.102E-04 0.01746 2.3470 3.6430 5.877E-05 0.00727 5.5585 7.3359 1.302E-04 0.0171

1.5× c

1 1.3789 2.0557 2.392E-05 0.00432 0.2454 0.9773 4.034E-06 0.00083 1.8597 6.1804 2.838E-05 0.00594 0.2650 1.1663 5.585E-06 0.00085 1.0420 2.8081 2.061E-05 0.00336 0.2537 1.5375 6.352E-06 0.00087 1.1663 3.3568 2.732E-05 0.0036

Table 6.8: Inverse �adrant for 69 snapshots


Gaussian : 69 Snapshots


0.6× c

1 12.3310 87.9700 2.139E-04 0.03882 13.0289 94.2494 2.142E-04 0.04123 1.0569 25.1122 1.613E-05 0.00344 2.3934 19.4497 5.044E-05 0.00745 2.8973 34.1486 5.730E-05 0.00906 2.8450 22.5797 7.123E-05 0.00877 1.9400 34.8354 4.544E-05 0.0060

1.0× c

1 12.7504 88.5496 2.212E-04 0.04012 6.9428 28.8518 1.141E-04 0.02193 1.2633 15.4382 1.928E-05 0.00404 2.3453 38.7364 4.942E-05 0.00735 3.1739 33.7096 6.277E-05 0.00996 2.7238 14.4238 6.820E-05 0.00837 1.8348 24.0412 4.298E-05 0.0056

1.5× c

1 23.3529 192.5797 4.052E-04 0.07352 11.7358 34.2750 1.929E-04 0.03713 7.4852 61.6103 1.142E-04 0.02384 6.4142 74.0536 1.352E-04 0.01995 7.8025 87.1711 1.543E-04 0.02446 8.3725 28.4331 2.096E-04 0.02567 8.8034 56.4330 2.062E-04 0.0271

Table 6.9: Gaussian for 69 snapshots


Gaussian Division : 69 Snapshots


0.6× c

1 45.4361 45.5403 7.883E-04 0.14302 44.9794 45.0170 7.393E-04 0.14213 86.7322 86.9324 1.323E-03 0.27564 44.7124 44.7832 9.423E-04 0.13895 87.2232 87.2395 1.725E-03 0.27226 59.2984 59.5194 1.485E-03 0.18137 87.3110 87.4038 2.045E-03 0.2685

1.0× c

1 2.1639 2.5308 3.754E-05 0.00682 3.7990 4.2417 6.244E-05 0.01203 16.5641 19.2465 2.527E-04 0.05264 3.9499 4.4582 8.324E-05 0.01235 23.0728 23.6583 4.563E-04 0.07206 5.9930 6.9406 1.501E-04 0.01837 24.8687 26.1057 5.825E-04 0.0765

1.5× c

1 3.7214 4.2004 6.457E-05 0.01172 2.0414 2.7262 3.355E-05 0.00653 5.6250 10.2007 8.583E-05 0.01794 0.4902 1.4837 1.033E-05 0.00155 1.5189 4.1131 3.004E-05 0.00476 0.7464 1.7584 1.869E-05 0.00237 1.6218 5.9128 3.799E-05 0.0050

Table 6.10: Gaussian (Devision) for 69 snapshots

69

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