analysis of the resistance due to waves in ships · analysis of the resistance due to waves in ......

Analysis of the resistance due to waves in

ships

Treball Final de Grau

Facultat de Nàutica de Barcelona

Universitat Politècnica de Catalunya

Treball realitzat per:

Rafael Pacheco Blàzquez

Dirigit per:

Julio García Espinosa

Borja Serván Camas

Grau en (GESTN)

Barcelona, 09/07/2014

Departament de CEN

Analysis of the resistance due to waves in ships

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3

Acknowledgments

The author is grateful to Prof. Julio García and Dr. Borja Serván. Without their incessant

support and effort, this project could not have been carried out.


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5

Abstract

Nowadays the state-of-the-art in hydrodynamics has led to software based on numerical methods which

are able to predict the hydrodynamics performance of complex geometry models. However, most of

these software products require long computational times.

This project aims at validating SeaFEM, a software based on the finite element method(FEM), against an

empirical formulation for planing surfaces. This formulation was obtained by Daniel Savitsky, a former

scientist of Davidson Laboratory.

SeaFEM is a time-domain seakeeping software based on potential flow with a tuned free surface

boundary condition that might be used for simulating planing hulls. The main advantage of SeaFEM

compared to other hydrodynamics software is that the SeaFEM approach makes it much faster

computationally speaking.

In this project, a comparison will between Savitsky´s formulation and SeaFEM will be carried out. Then,

the error propagation will be studied to obtain a correction formula. Finally, a discussion on the results

will be provided.


6

Index

ACKNOWLEDGMENT 3

ABSTRACT 5

INDEX 6

NOMENCLATURE 10

CHAPTER 1. STUDY APPROACH. 11

1.1 SCOPE 11

1.2 SAVITSKY’S FORMULATION 11

1.3 INITIAL ASSIGNMENT: STUDY OF THE SAVITSKY’S FORMULATION 14

1.4 APPLICABILITY OF THE FORMULATION 19

1.5 FINAL ASSIGNMENT: RESULTS 20

CHAPTER 2. MODEL SETUP. 21

2.1. STUDY MODEL 21

2.2. MODEL CREATION 22

2.3. BOUNDARIES 24

2.4. GENERAL DIMENSIONS FOR DIFFERENT VERSIONS 25

2.5. PROBLEM DEFINITION 28

CHAPTER 3. MESH STUDY. 30

3.1. MESH PARAMETERS 30

3.2. MESH TYPE 30

3.3. QUALITY 32

CHAPTER 4. MODEL VERSIONS. 38

4.1. VERSION 1 38

4.2. VERSION 2 38

4.3. VERSION 3 41

4.4. VERSION 4 42

CHAPTER 5. CASE MATRIX. 45

5.1. CASE DEFINITION 45

5.2. GEOMETRICAL DISCRETIZATION 45

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5.3. APPLICABILITY DISCRETIZATION: 46

5.4. DISCRETIZED MATRIX. 47

5.5. DATA EXCLUDED 48

5.6. SUBMERGED VOLUME 50

CHAPTER 6. RESULTS. 55

6.1. RESULT STORING 55

6.2. PROCESSOR 55

6.3. SCHEME 56

6.4. RESULT TYPE 56

6.5. EXCLUDED RESULTS 60

6.6. NON-EXCLUDED RESULTS 62

CHAPTER 7. ERROR STUDY. 67

7.1. LEAST SQUARES 67

7.2. REGRESSION MODEL BY MEANS OF INTEGRATION. 70

7.3. CORRELATION COEFFICIENT OF PEARSON 73

7.4. REGRESSION MODEL BY MEANS OF LEAST SQUARES – GAUSS NORMAL EQUATIONS 77

CHAPTER 8. CONCLUSIONS 82

8.1. EXCLUDED CASES OF CV = 1 82

8.2. NON-EXCLUDED CASES (CV = 2,3,4,5) 83

8.3. TIME 84

8.4. HUMAN FACTOR 85

8.5. TECHNOLOGICAL FACTOR 85

8.6. SAVITSKY EMPIRICAL DATA. 85

8.7. TOWING TANK DATA 85

BIBLIOGRAPHY 87

ANNEXES 89

ANNEX A: USER DEFINED FUNCTIONS. 89

1. TDYN – SCRIPT TO RUN CASES AUTOMATICALLY. 89

2. EXCEL – SAVITSKY CRITERIA 89

3. EXCEL – RESULTS STORAGE. 89

4. EXCEL – ERROR EVALUATION, METHOD 1. 89

5. TDYN – RESULT IMAGES 89

ANNEX B: SECTIONS. 89

1. ISOMETRIC 89

2. PLAN 89

3. ELEVATION 89


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ANNEX C: RESULTS. 89

1. STREAMLINE RESULT TABLE. 89

2. FEM RESULT TABLE. 89

3. STREAMLINE ERROR TABLE. 89

4. FEM ERROR TABLE. 89


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Nomenclature

Symbol Units [IS] Significance

Lk m Amidships wetted length

LC m Wetted chine length

λ - Mean wetted length-beam ratio

b m Beam

d m Draft

β º Deadrise

τ º Trim

g m/s2 Gravity, 9.8 m/s

2

ρ kg/m3 Density, 1.025 m/s

2

V m/s Velocity

CV - Speed Coefficient

ε % Error

Chapter 1. Study approach.

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1.1 Scope

The aim of this study is to re-create and validate the formulation of Savitsky by means of finite element

method (FEM) and a posterior analytic study of the results given by the FEM software. Savitsky’s

formulation is focused basically on predicting the power for a planing hull. The study will be carried out

by a finite element method and posteriorly the results obtained by the software will be compared to the

mentioned formulation and re-adjusted by analytic study.

The Fem software allows to simulate the seakeeping of a planing hull by means of potential flow theory.

This theory tries to describe the knimatics behaviour of the fluid based on the mathematic concept of

potential function.

Savitsky’s formulation focuses on the study of the hydrodynamic forces obtained from an empirical data

and a posterior theorizing of the empirical obtained equations. This study was executed in a towing

tank, the study was described as an experiment for various prismatic hulls which had some fixed

parameters such as the deadrise, trim, draft and velocity due to the carriage speed of the towing tank.

This study pretends to select several cases within the formulas’ application range and compare the FEM

results with the formulation ones. This is not the best data to be compared, real data from Savitsky

cases would have been the best data to contrast but due to the impossibility of finding this empirical

data, the Savitsky’s formulation has been used as comparison.

1.2 Savitsky’s formulation

Daniel Savitsky carried out a number of experiments with different fixed parameters. Those experiments

and the posterior study were published on a paper called “Hydrodynamic Design of Planing Hulls” on

1964. The study had the aim to found out some equations which will be able to describe the best they

could the empirical data of those cases. This study is used to calculate the predicted power and the

seakeeping of a planing hull ship.


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In order to use the Savitsky method, there is the need to set some parameters, these are:

Figure 1. Sketch of Savitsky hull design.

T Thrust β Deadrise

ΔΔ Ship’s displacement b Beam

Df Drag’s viscous component Lk Wetted keel length

τ Trim Lc Wetted chine length

LCG Longitudinal gravity centre V Horizontal velocity of planing surface

CG Centre of gravity d Draft from Lk until lower point on the stern

Є Shaft’s tilting compared to the keel Cv Froud number

N Normal force or Lift f Distance between T and CG

a Distance between Df and CG c Distance between N and CG

Table 1.Description of different Savitsky coefficients.

It is important to clarify that the Froude number Cv is obtained as:

��

Equation 1. Speed coefficient.

The Froude number is in function of the beam instead of the length which is what commonly has been

used.


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This formulation is compound by a total of 37 equation, although some of these are just previous steps

to declare the final equation and in other cases is the same equation but simplifying some values of the

equation such as a, f, c o Є which are set null. At the end, from these several equations it can be

obtained some curves which interrelate the different basic parameters allowing to extrapolate this data

to similar ship which is within the range of applicability. Definitely, the formulation can be used as a

chart to found the optimal values or to programme a script to calculate these equations and returns the

desired solution.

Figure 2. Chart interrelationating different parameters


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1.3 Initial assignment: study of the Savitsky’s formulation

In this initial phase of the present project has been started by studying the Savitsky’s paper,

“Hydrodynamic Design of Planing Hulls” from 1964 to have a better understanding about the subject

and creating a spreadsheet to calculate the results for a number of cases by Savitsky’s formulation. To

obtain the results, it is needed to calculate previously some coefficients. These are:

Coefficients:

1.3.1. Mean wetted length – beam ratio (λ): is the quotient between the mean wetted length and the

beam.

� � � �2� � � �sin � � � � tan�2 � � � tan ��

Equation 2. Mean wetted length – beam ratio in function LK , LC , b, d, τ, β.

1.3.2. Subtraction between wetted length and wetted ( Lk – Lc ) : in order to appreciate these two

parameters clearer, it is added the following images. This CAD model represents four different zones.

The grey zone labelled as “Outside” is the one which is dry. The “Water” is the load waterline length.

The Spray is the main feature of a planing hull which is a phenomenon produced near the zone where

the keel is in contact with the water surface and produces a raising of this surface along the chine.

Finally the pink zone labelled “Inside” is the one which is submerged.

Figure 3. Vessel water zones.


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Figure 4. Better look of the previous zones.


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Once seen the 3D model, it is added the definition of LK y LC parameters.

Figure 5. LK and LC definitions.

By means of the following equation it can be calculated the relation between LK – LC :

� � � � � � tan �� tan �

Equation 3. Relation between LK and LC in function of the beam, deadrise and trim.


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1.3.3. Lift coefficient for a null and beta deadrises (CL0 y CLβ ): these coefficients are dimensionless and

are used to extrapolate the data obtained by the Savitsky’s formulation to a design model within the

applicability range.

Savitsky’s study provides these different equations and charts which define CL0 y CLβ :

�� .� �0.00120��/ 0.00055��/��

Equation 4. Lift coefficient for a null deadrise.

Figure 6. Lift coefficient for β = 0.

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4

CL0

/

τ1

,1

λ

Lift coefficient of a planing surface; β=0

Cv 5

Cv 4

Cv 3

Cv 1

Cv 15

Cv 14

Cv 13

Cv 12

Cv 11

Cv 10

Cv 9

Cv 8

Cv 7

Cv1

Cv2

Cv15


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�� . !"#��.�

Equation 5. Lift coefficient for any deadrise value.

Figure 7. Lift coefficient for β ≥ 0.

At the end only CLβ is used because it has CL0 included. This is useful to find out the total Lift which

has the ship which is moving through a fluid. If the density and speed of the fluid are known, it is

possible to find out the required Lift.

Δ � 12 %��

Equation 6. Bernoulli’s equation.

1.3.4. Pressure’s centre and longitudinal position: This is the centre of pressures of the submerged

surface. It can be calculated by the following equation:

�� &�� 0.75 � 15.21�� 2.39

Equation 7. Pressure’s centre equation.

These four coefficients are the basics to determine the hydrodynamic lift for a surface with no weight.

But apart from these four, there are other relevant coefficients which are needed to be calculated for a

ship. The difference in this project remains on the displacement, which is not taken in count because

only the hydrodynamic lift force is evaluated.

0

0.02

0.04

0.06

0.08

0.1

0.00 0.02 0.04 0.06 0.08 0.10 0.12

CLβ

CL0

10 deg

15 deg

20 deg

25 deg

30 deg


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1.4 Applicability of the formulation

Savitsky’s formulation is applicable within a parameter range. This range changes depending on the

deadrise, trim and the Froude number.

There are some ranges which there is no viability to use some equations, because it is out of range, but

there are some options allowing to use another equation. E.g. , “equation 1” is used to evaluate λ in

function of λ1 , which is another parameter described in the formulation, is not possible to use because

is out of range. But it can be use “equation 4” which calculates the same but using other parameters.

Basically the main boundaries for the present study are:

• Equation 3:

This equation allows to calculate the relation between Lk – Lc . the applicability is:

• For whole angles of deadrise and trim and always a Cv equal or larger than 2.

Cv ≥ 2.0

Β All deg

Τ All deg

Table 2 . Equation 3 applicability.

• For Cv equal or larger than 1, whole trim angles but deadrise up to 10 º included.

β ≤ 10.0 deg

Cv ≥ 1.0

Table 3. Equation 3 applicability.

• For Cv equal or smaller than 1, angles up to 4 º and deadrise until 2 º , both included.

Note that the relation will be larger than predicted.

β 2.0 deg

Cv ≤ 1.0

τ ≤ 4.0 deg

Lk - Lc is larger than prediction


• Equation 15 y 16:

These equation allow to calculate CL0 and CLβ , and its range of applicability is:

τ 2.0 deg - 15.0 deg


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λ ≤ 4

Cv 0.6 - 13.0


1.5 Final assignment: Results

The final aim of the present Project is to extract series of results in function of a few input parameters.

These input parameters are: “τ” which is the trim of the ship, “β” which is the deadrise, “d” which is the

maximum draft in the stern and “V” which is the velocity of the ship.

The studied model is just a flat plane which some parameters such as draft, deadrise and trim will be

modified to adapt the surface to different forms. The simulation of different angles and drafts mean

different cases.

Figure 8. Geometrical parameters.

In function of these parameters, the results are calculated. These results are the vertical hydrodynamic

lift and the torque which is just the multiplication between the length of the pressure’s centre and the

vertical lift force

Once it has been find out the results for various combinations of the previous four parameters, they will

be compared to Savitsky’s formulation and analyzed.

The calculating software is a finite element method which is from the suite of Tdyn, particularly the

SeaFem module which allows to perform seakeeping simulations.

τ

Chapter 2. Model setup.

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2.1. Study model

The model is just a flat plate geometrically defined by three parameters: dead rise, draft and beam. The

computational domain is defined by 3 zones. The first one which is pink colour, is the zone where the

planing hull is located, it is represented the half of a total model due to the symmetry of a ship, so it is

only evaluated the half force of lift of the hull. The second zone is close water zone, in cyan colour and it

commonly represents the close interaction water area with the hull. The third zone in red colour is what

commonly is known as beach, it is a zone where the interaction and distortion in the water free surface

is dissipated, becoming null again. Once again, the model is the half of a real ship because based on the

symmetry of a ship there is no need to recreate it entirely, which only would result in more calculus and

more time to spend into it.

Figure 9. Free surface model zones.

L: Length of the ship

L

Close Water

Beach


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Figure 10. Isometric view of the model.

2.2. Model creation

The creation of the model is simple, first of all these layers are created:

Figure 11. Layers which compound the model.

Note: The following distance parameters are explained later.

The free surface contains the following three elements:

1- Hfs: is the flat lamina with parametric geometry.

The flat lamina is a rectangle with a length of “L” and a beam of 1m.

Figure 12. flat lamina.

2- Inner: close water.

Close water is compound by five quadrilaterals. In the bow those quadrilateral are of (B1 or B2 )

m x L2 m, in the stern (B1 or B2 ) m x L3 m which are bigger than in the bow due to a better study

of the zone later, and a rectangle of B2 m x Lm above the lamina.

Depth


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Figure 13. Close water.

3- Outter: Beach.

This is the rest of the surface to complete the horizontal upper surface of the model. It has an

amplitude of B m and the length is the sum of the length of the lamina, close water and the rest

added to the beach. It has a length value of LT m.

Figure 14. Beach.

Outters and others compound the laterals and the bottom of the model:

The depth of P metres depends on the version. Anyway

the calculus is done taking in count the model has infinite

depth.

Last layer is the volume:

The volume is not needed to be created

because only some parameters with no

displacement dependency will be studied.

Apart from that the old versions of the

software needed to define a volume in order

to do the calculus correctly. But in the study

this volume has no properties assigned. Hence,

it is void and is like there was no volume at all.

Figure 15. Surroundings.

Figure 16. Defined volume.


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2.3. Boundaries

The study model is compound by the following boundaries:

• Flat lamina:

The flat lamina which is the pink colour layer and the half of the ship’s hull has assigned the

property “H Free Surface” which allows to parameterize the height of this surface in function of

some input parameters. The equation is: * � + � tan,�- . � tan,�- � �

Equation 8. Height of the parametric surface.

Where :

z: Height of the surface.

y: Coordinate of Y axis.

x: Coordinate of X axis.

β: Deadrise.

τ: Trim.

h: Maximum depth of the ship in the stern.

• Free Surface:

Is the surface compound by the flat lamina, close water and beach. This surface has no height

limit and simulates the surface of the water.

Figure 18. Free surface.

Figure 17. H Free Surface.


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• Outlet:

These are the surfaces which indicate the inlet and outlet of the water .

Figure 19. Outlet and Inlet of the current.

2.4. General Dimensions for different versions

The previous seen parameters of length, depth, height or amplitude are described in the following draw:

Figure 20. Regular model and its parametric dimensions .

P

L

L2

L3

LT

B

B2

B1


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The model version 1 has the following dimensions:

LT 19 m

P 4 m

B 5 m

L 5 m

L2 1 m

L3 2.5 m

B1 1 m

B2 1 m

Table 6. Version 1.

The model version 2 has these:

LT 22 m

P 4 m

B 5 m

L 8 m

L2 1 m

L3 2.5 m

B1 1 m

B2 1 m

Table 7. Version 2.


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LT 27 m

P 4 m

B 5 m

L 8 m

L2 1 m

L3 2.5 m

B1 1 m

B2 1 m

* It has been added 2 m in the stern and 3 m in the bow, both in the beach zone.

Table 8. Version 3.


LT 37 m

P 10 m

B 15 m

L 8 m

L2 4 m

L3 10 m

B1 2 m

B2 3 m

* This model is based on model version 1 and it has been increased its horizontal surface

and depth. It is able to see that the rectangular prism inside the big one is the model

version 1.

Table 9. Version 4.

B2

B1


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2.5. Problem definition

Once the model, version and boundaries are done. It is necessary to set up the study which is going to

be performed by the FEM software. In this case the software is Tdyn and its calculating model is the

SeaFem which allows to calculate and analyze the Seakeeping of a vessel.

First of all, it is necessary to define which sort of simulation is going to be performed and which

parameters are going to be used. In this case the simulation type is Seakeeping analysis and the

parameters are:

- Dimension: 3D, because is a 3D model.

- Environment: Current which means that it will be simulated a water current across the

vessel. It is important to assign the boundaries of inlet and outlet of this current.

- Type of analysis: Seakeeping.

Figure 21. Interface menu.


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Once the initial data is defined, it is needed to fulfil a few fields inside the options chosen before. These

fields would be:

- General Data:

� Water density: 1025 kg/m3.

� Results: Indicates in which save format and which results would be calculated.

� External loads.

� kinematics: To set movements, velocities and acceleration.

� User Defined: It has to be selected two parameter results by introducing a code

which is in the manual of the software. These two results are the vertical lift force

and the torque of this force.

- Problem description

� Depth: Infinite (when the depth is bigger than the length of the waves).

� Wave absorption: Yes.

� Beach: 7m.

- Environment Data:

� Current:

� Velocity.

� Direction.

- Time data:

� Simulation time.

� Time step.

� Time output.

� Recording time.

� Starting time.

- Numerical data:

� Processor.

� Number of CPUs.

� Type of Solver.

� Stability factor.


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Chapter 3. Mesh study.

3.1. Mesh parameters

Mesh depends basically on two main parameters:

- Mesh type/shape. - Quality / Accuracy.

The type is much more associated to the model version and also helps on getting the results.

The quality is more associated to the analysis time and the accuracy of the results. The quality has no

dependency of the model versions.

3.2. Mesh type

Generally, the mesh type used has been:

- Hfs layer or flat lamina: The mesh is structured and non-symmetric. The surface and lines in this layer are structured as

well.

Figure 22. Structured flat lamina.

The fact that it has been used a structured and non-symmetrical mesh remains on advantage of

having less elements. A regular structured and symmetrical mesh has 4 triangles inside a square,

the non-symmetrical option allows to avoid these 4 elements to just 2 elements. The more

elements it has, the more time it lasts to finish the calculus. The advantage of having a

symmetrical mesh would be that it has much more accuracy inside these squares.


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Figure 23. Symmetrical vs non-symmetrical.

The fact being a structured mesh allows to fix a uniform element size and an equal distribution

along this mesh. To create a structured mesh is necessary to define as structured the elements

that compound these structures as well. E.g. , in case of having a surface structured, it would be

necessary to define the lines which shape this surface.

- Inner layer or close water: This mesh is structured in all versions despite the fourth version with the resolution scheme FEM

which is unstructured due to an instability error in the calculus that doing it unstructured the

error disappeared.

Figure 24. Structured close water.

- Outter layer or beach: The mesh is non-structured due to not requiring a lot of accuracy because in this zone the free

surface of the water should have not much distortion and remain calm.

Figure 25. General meshing on the beach.


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- Outlet and Others layers: The mesh is non-structured because it is a regular mesh, it means that it has no special mesh

properties applied on it. Close to structured elements it seems that the mesh becomes

structured but it is not, that is because the transition is quite low and fits perfectly.

Figure 26. Laterals and bottoms, general meshing.

- Volume layer: The volume, although in the newest software version is not necessary to be defined, has been

applied a regular mesh which means no mesh criteria has been applied on it. It is only structured

on the lamina’s zone and close water’s zone. In addition the volume due to his 3D features has

tetrahedrons instead of triangles.

Figure 27. Meshed volume.

3.3. Quality


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The accuracy has been studied once the version 2 was done. In this version, the maximum wetted length

or LK was fixed to 7 m. To perform the quality study, it has been done by modifying the following

parameters:

- Maximum element size in the general meshing (Beach, Outlet, Laterals, Bottom and Volume) - Structured mesh element size of Inner layer or close water. - Structured mesh element size of Hfs layer or flat lamina.

The maximum element size in the general meshing has not huge influence on the accuracy but cannot

be too much bigger than the rest because it will have an enormous transition, leading to errors in the

calculus. The transition has been set up to 0.1 .

Figure 28. General meshing interface.

The element size of the mesh Inner or close water does not affect too much to the results but it has

little importance. It affects directly to the calculus time and the results on this Inner zone but has no

great impact on them. The main problem would be having a huge transition between Inner and Hfs

layers which will lead to problems as well.

Results extracted from modifying Inner mesh are quite similar for different sizes. E.g. in one case which

its Inner mesh has been modified shows:

ELEMENT SIZE

(INNER)

TIME LIFT

(N) s min

0.75 189 3 18740

0.4 252.591 4 18693

0.3 480.413 8.006883333 18672

Table 10. Comparison for the same case and different Inner meshes.

The variation is quite low. The criterion to be applied on this mesh zone should be an intermediate

structured mesh between the general meshing and the Hfs mesh in order to avoid an abrupt transition

and do it the softest it can be.


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To observe the quality of the mesh in function of the Inner and Hfs mesh, it is represented on the

vertical axis the time in minutes of how much a specific case lasts. And in the horizontal axis, the various

values for the Inner mesh for different Hfs meshes:

Figure 29. Time vs mesh size chart.

In the case of an Inner mesh of 0.3 m and an Hfs mesh of 0.025 m, it happens that the time to run and

finish a case lasts 14 hours. Hence, it is clarified that what increase the time a case lasts is nothing else

than the Hfs mesh size.

Although varying the Inner mesh implies more time if Hfs is quite accurate. It does not imply that the

results will be more accurate:

Here it is observed that between an Inner

mesh of 0.75m and 0.4m for an Hfs mesh

of 0.15m . It has only a difference of 47 N

versus a value of 18 kN which means it

has no relevant influence on the

accuracy.

Y axis is the lift force and X axis is the

simulation time.

1

10

100

1000

0 0.2 0.4 0.6 0.8

Tim

e (

min

ute

s)

Max. Element Size (m)

Time vs Mesh Size

Time vs Mesh size

0.15

Time vs Mesh size

0.1

Time vs Mesh size

0.05

Time vs Mesh size

0.025

18500

18550

18600

18650

18700

18750

18800

0 1 2 3

Fz

0.75 0.15

0.4 0.15

Figure 30. Comparison of the result force for different Inner meshes.


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Previous char without omitted cases would be:

Figure 31. Force summary through the time.

It can be observed that the qualitative gap appears when Hfs mesh is reduced rather than Inner’s and

for an Hfs mesh of 0.05 m it has no difference with one of 0.025 m.

Figure 32. No quality upgrade once Hfs reaches 0.05 m .

Here it can be seen better how the variation is almost null for an Hfs mesh of 0.05 m and 0.025 m and

same Inners mesh size (0.3 m) .

Finally comparing the theoretical Savistky’s value which is 17500 N, it can be noticed that Hfs mesh has

to be 0.05 m. Despite the Inner mesh was studied for 0.3 m, in order to avoid a transition of 6 times

bigger, it is set up to 0.1 m to become a transition of 2 times bigger. That means that it will be more time

16000

16500

17000

17500

18000

18500

19000

0 0.5 1 1.5 2 2.5

Fz

0.75 0.15

0.4 0.15

0.4 0.1

0.5 0.1

0.3 0.05

0.3 0.025

17000

17050

17100

17150

17200

17250

17300

17350

17400

17450

17500

0 0.5 1 1.5 2 2.5

Fz

0.3 0.05

0.3 0.025

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in the calculus process but it will reduce the risk of having stability problems due to a softer transition.

The general meshing is 0.75 m because of its poor influence on the model and it has a 0.1 transition.

For a better understanding of the structured and non

below:

Here it can be seen the different zo

And also the size and type mesh:

Inner

Adjacent




of the structured and non-structured zones of the model see this image

This is the version 4 where the general meshing is

0.75 m, Inner of 0.1 m and

zones to this one of 0.05 m.

Here it can be seen the different zones:

And also the size and type mesh:

Inner

Hfs

Adjacent

Figure

Figure 34. Adjacent




structured zones of the model see this image

This is the version 4 where the general meshing is

of 0.1 m and Hfs and adjacent

of 0.05 m.

Figure 33. Version 4 mesh.

. Adjacent water definition.


Finally, here, the search and optimization of the best suitable mesh for

Observe how it has been modified until reaching the perfect one:

Inner -

Hfs

0.75 –

0.15

0.4 –

0.15

0.3 –

0.15

0.5 – 0.1

0.4 – 0.1

0.5 –

0.05

0.3 –

0.025

Table

The most suitable one would be the penultimate. O

the search and optimization of the best suitable mesh for Hfs and the adjacent zones.

until reaching the perfect one:

Image

Table 11. How mesh affects the result quality of the pressure diagram.

The most suitable one would be the penultimate. Only changing the parameter 0.5 Inner

Figure 35. Model mesh size and type.

37

and the adjacent zones.

Comment

There is no

Hfs meshing

difference.

So there is

no

difference

appreciable.

More or less

the calculus

time was the

same.

There is little

change in

high

pressure

zone where

is more

uniform.

Pressure

zone is much

more

uniform.

There is no

big variation

compared to

previous.

. How mesh affects the result quality of the pressure diagram.

Inner mesh to 0.1.

. Model mesh size and type.

38

Chapter 4. Model versions

There are four version of the present model, every version in order to correct previous design errors

which were affecting some cases.

4.1. Version 1

First version was done to see if the model was valid and functional. In addi

Tdyn environment and mechanics along with determine what results were going to be carried out.

Despite the abrupt results due to the

lack of accuracy for this initial version,

it is also possible to see the sort of

wake that a vessel of this type would

do.

4.2. Version 2

Second version was done once it was checked out that the previous mod

decided to fix the length of the study increasing it from 5 meters to 8 meters, this is due to

First reason is because for doing the study

is 1m for the half hull and 2 meters for the entire hull, and then

velocity which is easier to compare cases

greater than 7 meters. In the following chapters it w

fixed to 7 meters.

Secondly, the other reason is because apart from fixing some geometrical parameters such

maximum wetted length. Finite


Model versions.

four version of the present model, every version in order to correct previous design errors

which were affecting some cases.

if the model was valid and functional. In addition with a first approach to


It is an abrupt mesh because

initial version and

be accurate.

Despite the abrupt results due to the

lack of accuracy for this initial version,

it is also possible to see the sort of

wake that a vessel of this type would

Second version was done once it was checked out that the previous model worked well and then it was

decided to fix the length of the study increasing it from 5 meters to 8 meters, this is due to

is because for doing the study, it is needed to fix some parameters such as the beam which

half hull and 2 meters for the entire hull, and then Froud number, C

which is easier to compare cases. Also the LK is fixed as well, and it is imposed that has to be no

greater than 7 meters. In the following chapters it will be explained with more details why it has been

is because apart from fixing some geometrical parameters such

Finite element method software just calculates and does not dis

Figure 37. Regular wake for a


four version of the present model, every version in order to correct previous design errors

tion with a first approach to


brupt mesh because is the

initial version and should not have to

el worked well and then it was

decided to fix the length of the study increasing it from 5 meters to 8 meters, this is due to two reasons.

it is needed to fix some parameters such as the beam which

Cv , depends only on the

and it is imposed that has to be no

ill be explained with more details why it has been

is because apart from fixing some geometrical parameters such as beam or

element method software just calculates and does not discern about

Figure 36. Meshed version 1.

. Regular wake for a planing hull.

Chapter 4. Model versions.

what is lamina’s zone (the vessel) and what is not. So in order to avoid some troubles while calculating

it has an extra margin to cover the lack of length. E.g. if the L

does not take in count the theore

should be 7.1 m, if it did not have this extra margin of length it w

leading to errors.

So if that wants to be avoid an extra margin is the best s

Apart from changing the flat lamina, it was added more length to close

upstream due to some issues with low velocities

current, making its inlet into it quite abruptly. The fastest it penetrates, the easiest and cleanest it does,

as well as it has less turbulences in the steady estate and lasts less time to stabilise.

That was causing in version 1, in which

cases to crash. That is because the outlet in the stern obligates the height on the end of the current

be null. Despite that, if the model had not enough space to dissipate th

a short distance, calculus will crash.

goes by, a resonance phenomena appeared increasing the height on the middle of the distance between

the stern and the outlet of the fluid.

To made it plain and clear, the fluid during its simulation accumulated tension downstream and the

software could not make it disappear. Once the model had more length

disappeared because it had the proper s

and what is not. So in order to avoid some troubles while calculating

it has an extra margin to cover the lack of length. E.g. if the LK imposed is 7 m but th

count the theoretical formulation of the geometry, calculates for itself that the L

this extra margin of length it would have crashed

So if that wants to be avoid an extra margin is the best solution and easy way to fix it.

Apart from changing the flat lamina, it was added more length to close water zone downstream and

upstream due to some issues with low velocities in which the lamina is not penetrating properly into the

making its inlet into it quite abruptly. The fastest it penetrates, the easiest and cleanest it does,


, in which it had more turbulence due to its low speed current, for some

cases to crash. That is because the outlet in the stern obligates the height on the end of the current

be null. Despite that, if the model had not enough space to dissipate the current turbulence

, calculus will crash. At the beginning the calculus module can handle it


outlet of the fluid.


could not make it disappear. Once the model had more length in downstream

disappeared because it had the proper space to dissipate these turbulences.

39

and what is not. So in order to avoid some troubles while calculating,

imposed is 7 m but the software, which

formulation of the geometry, calculates for itself that the LK

or flooded the vessel

olution and easy way to fix it.

Figure 38. Version 2.

zone downstream and

enetrating properly into the

making its inlet into it quite abruptly. The fastest it penetrates, the easiest and cleanest it does,


it had more turbulence due to its low speed current, for some

cases to crash. That is because the outlet in the stern obligates the height on the end of the current to

e current turbulence because of

t the beginning the calculus module can handle it, but when times



downstream, this effect

40


Figure 39. Evolution downstream for version 2.

Here it can be seen that what

happened previous the enlargement

of the length downstream was clearly

a resonance phenomena.

Once it was enlarged, the err

disappeared.


. Evolution downstream for version 2.

Here it can be seen that what

happened previous the enlargement

of the length downstream was clearly

a resonance phenomena.

Once it was enlarged, the error


4.3. Version 3

The third version is more or less like the second one but it had even more length upstream. That is

because in counterpart to the low range velocities, the high speed currents were causing some problem

as well. Regularly when a vessel is introduced in water, it generates a little concave wave upstream

is because the fluid flux is anticipating to be hit by the surface which i

reduce its impact and generate the less turbulence it can.

cases it was noticed that this wave was quite big an

between the inlet of current and the bow

downstream, which was caused because it had not enough space to adapt the fluid to the vessel. Once it

was modified, the problem disappeared as well.

This image is an elevation of the model

(bow), is the water surface in the amidships

the vessel. And then, upstream is seen a little concave wave. That would be the correct situation, this is

once the model was enlarged. Previous this, the wave was a little concave ne

its height being over the waterline and the


he low range velocities, the high speed currents were causing some problem

as well. Regularly when a vessel is introduced in water, it generates a little concave wave upstream

is because the fluid flux is anticipating to be hit by the surface which is penetrating the

the less turbulence it can. Generally this wave is not big, but in some

cases it was noticed that this wave was quite big and not only that. It even created

of current and the bow, some sort of convex wave. It is like the previous problem

was caused because it had not enough space to adapt the fluid to the vessel. Once it

was modified, the problem disappeared as well.

The upstream zon

increased due to a lack of

space problem.

Figure 41

This image is an elevation of the model. The black line, which extends from the left (stern)

amidships gangway. It is a tilted line on the stern due to the de

upstream is seen a little concave wave. That would be the correct situation, this is

once the model was enlarged. Previous this, the wave was a little concave near the bow then increased

height being over the waterline and then when it reached the upstream, because of the boundaries it

41


he low range velocities, the high speed currents were causing some problem

as well. Regularly when a vessel is introduced in water, it generates a little concave wave upstream. That

s penetrating the water in order to

Generally this wave is not big, but in some

d not only that. It even created in some point,

ike the previous problem with

was caused because it had not enough space to adapt the fluid to the vessel. Once it

The upstream zone has been

increased due to a lack of

space problem.

41. Upstream problem.

from the left (stern) to the right

gangway. It is a tilted line on the stern due to the depth of

upstream is seen a little concave wave. That would be the correct situation, this is

ar the bow then increased

when it reached the upstream, because of the boundaries it

Figure 40. Version 3.

42

was reduced until it reached the 0 value

comparing to the simulated cases, but it is useful to make

4.4. Version 4

The fourth version has more beam, depth and length. Indeed it is based on

added more distance on those lengths direction mentioned before. It can be

prism is version 1 which is wrapped by an outside prism that completes

The reasons that made to change the previous model were:

- Beam: In the previous image

or model there was an accumulation of height which created a wave. That was generating

some sort of turbulence on the free surface downstream.

That is due to a wall effect

boundary layer, goes directly to the lateral

Depression quite unusual at

That should have a height of 0


was reduced until it reached the 0 value, which would be the red line. This red line

ulated cases, but it is useful to make an idea of it.

The fourth version has more beam, depth and length. Indeed it is based on version

added more distance on those lengths direction mentioned before. It can be observed

1 which is wrapped by an outside prism that completes version 4.

e to change the previous model were:

In the previous image, figure 41, it could seen that on the laterals of the towing tank

re was an accumulation of height which created a wave. That was generating

some sort of turbulence on the free surface downstream.

Figure 42. Pressure diagram, depression downstream.

That is due to a wall effect, which means that the stream, which

goes directly to the lateral, hits it and comes back generating a turbulence

Depression quite unusual at the ends of the free surface’s laterals.

That should have a height of 0 or near it respect the waterline


This red line is quite exaggerated

version 1 and it has been

observed that the inside

4.

that on the laterals of the towing tank

re was an accumulation of height which created a wave. That was generating

. Pressure diagram, depression downstream.

which is released from the

hits it and comes back generating a turbulence

the ends of the free surface’s laterals.

or near it respect the waterline.


43

downstream. In a real case in the open sea that effect would not happen. This effect can be

an additive or destructive interference.

44

- Depth: because of the

prevent the same thing

from 4 m it became

mesh for this dimension

delay, at least not noticeable

- Length: the length was increased for two

1 it had to be increased like in the

because to give an extra margin as well. Also it would not have increased the calculus as

well, so in order to avoid risks it was oversized. And other remarkable fact is that the

from version one became close

model were the beach on version 4. Also a new area appeared which was close to the vessel

layer which was the adjacent

the adjacent a structured mesh of 0.1 m and the close

Also for the FEM scheme the adjacent

problems that appeared in the calculus.


because of the wall effect problems, that appeared in some cases, just in case to

prevent the same thing happening with the bottom. It was expanded 3 times its

became 12 m. Otherwise these has no real impact in the calculus because the

dimension its only general meshing and its huge and will not add to much

noticeable.

the length was increased for two reasons. First the base for this

be increased like in the version 2 and 3. And it was increased a little more


well, so in order to avoid risks it was oversized. And other remarkable fact is that the

from version one became close water from version 2 and the extra lengths added to the


layer which was the adjacent water to the vessel. Vessel had a structured

the adjacent a structured mesh of 0.1 m and the close water were unstructured.

Also for the FEM scheme the adjacent water are unstructured as well due to some noise

problems that appeared in the calculus.

Figure 43. Relation between version 1 and version 4.

Version 1 beach became close

in 4.

ysis of the resistance due to waves in ships

that appeared in some cases, just in case to

t was expanded 3 times its length. So

n the calculus because the

its only general meshing and its huge and will not add to much

. First the base for this version was version

2 and 3. And it was increased a little more


well, so in order to avoid risks it was oversized. And other remarkable fact is that the beach

from version 2 and the extra lengths added to the


vessel. Vessel had a structured mesh of 0.05 m

were unstructured.

unstructured as well due to some noise

. Relation between version 1 and version 4.

Version 1 beach became close water

Chapter 5. Case matrix.

45


5.1. Case definition

The case matrix is a matrix where every column is a case and every row a mentioned parameter. In this

Project, it has been defined a parametric study model and by means of scripting the cases had been

modified and set up. These parameters are:

- Draft or d : its range is (0.2 , 0.3 , 0.4) metres.

- Trim or τ : its range is (2 , 3 , 4 , 5 , 6) degrees.

- Deadrise or β : its range is (5 , 10 , 15 , 20) degrees.

- Velocity or V : its range is ( 4.42 , 8.86 , 13.3 , 17.7 , 22.1) metres per second or as Cv (1 , 2 , 3

, 4 , 5).

- Stability factor: (0.1 , 0.2 , 0.3)

- Simulation time: ( 2 , 4 , 8 , 10 , 150) seconds.

The total number of cases depends on the first 4 parameters and for its vector dimension:

- d � 3.

- τ � 5.

- β � 4.

- V � 5.

That sums up to a total of 3x5x4x5 = 300 cases. But not all these cases are geometrically possible and

within the range of applicability of the formulation. Hence, it has to be applied a criteria to discretize

these cases.

5.2. Geometrical discretization

Savitsky’s formulation itself set a series of geometrical formulas:

Remembering that Lk y LC , by trigonometry were:

1. � � �

��

2. � � � � ��

Equation 9.Theoretical geometry criteria.

Then for this study Lk ≤ 7 m y LC ≥ 0 m were fixed.

The criteria become:

1. d 2 7 � sin,τ- 2. tan,β- 2 !"#

$"%&��'�

Equation 10.Applied geometry criteria.

Where b is the completely beam, which means 2 meters.

Figure 44. LK and LC graphical description.


46

To do the discretization, it has been used a Visual Basic script which does these two operation and

checks if it fulfilled the criteria.

INPUT VALUES

Deadrise 20

Trim 6

Lk ≤ 7

Lc ≥ 0

d 0.4 ≤ 7 · sin (t) ≤ 0.7316992

tan (B) ≤ d·π / b·cos(t)

0.363970234 CORRECT 0.63177948

Table 12. Geometry criteria in the spreadsheet.

5.3. Applicability discretization:

Moreover, apart from the geometrical discretization, the applicability limits of the formulation have to

be taken in count. Again, another script has been created to verify if these cases fulfilled the applicability

criteria.

The applicability of these equations were:

Equation 1:

Applicability :

τ 2. deg - 24. deg

λ ≤ 4.0

Cv 0.6 - 25

Table 13 . Equation 1 applicability.

Equation 3:

Applicability :

Case 1 Cv ≥ 2.0

β All deg

τ All deg

Case 2 β ≤ 10.0 deg

Cv ≥ 1.0

Case 3 β ≤ 20.0 deg

Cv ≥ 1.0

τ ≤ 4.0 deg

Lk - Lc is larger than prediction



47

Equation 15:

Applicability :

τ 2. deg - 15. deg

λ ≤ 4

Cv 0.6 - 13


Equation 23:

Applicability :

Cv 1 - 13


5.4. Discretized matrix.

From a total of 300 cases, they only remained up to 196 cases. Up to here, the case matrix has only been

defined by 4 parameters: d , τ , β , V . Otherwise it exists two parameters which are in function of the

current speed. These are the stability factor and the simulation time.

The stability factor is an dimensionless factor which allows to omit the time step value in the simulation.

What it does, is to determine, by its own, the most suitable time step in function of the dimensionless

factor which has been introduced. That is the same as a factor which is multiplying the Courant number

to determine the time step. The Courant number is a parameter which measures the solution’s mobility.

The value depends basically on the spacial resolution of the mesh and the Reynolds number which is

related to the velocity.

The simulation time is the one which will be simulated in the calculus in order to converge the results to

a specific value. Hence, the simulation time would be the stabilization time prorated. Approximately, the

margin given to the simulation time was a 30% for low speed cases and a 100% for high speed cases.

The simulation time reduces exponentially as the velocity increases.

VEL TIME BETA

4.43 150 0.3

8.86 10 0.3

13.3 8 0.2

17.7 4 0.1

22.1 2 0.1

t = 4680· V -2.517

0

50

100

150

200

0 5 10 15 20 25

Simulation

time

(s)

Velocity(m/s)

Time vs Velocity

Figure 45. Chart and table with simulation time data.

48

5.5. Data excluded

Although these 196 cases are theoretically

that it had to be reduced from 196 cases to 172. Those 24 excluded correspond to those cases

its velocity was 4.42 m/s or what is the same a

Despite, the application of the third equation is within the applicability range.

much near the limit and subsequently the error increases much more. I.e. , if the case is in the exactly

limit of application has much more error than if it is almost in the limit, that means, C

increased error value rather if it wo

number. Hence, due to the result data obtained for these low speed cases, the error is quite big.

It was observed in the majority of these low speed cases

a more appropriate wake for a semi

Case Veloci

182

According to the parametric equation of the flat lamina’s surface:

* �The beam in the transom is 1 m, so the C

practice due to is not a real planing

which is a unequivocal sign that the vessel is not lifted by hydrodynamic forces. Hence, on the calculus

of the pressure zone, which are

and then Lc is minor than 0.


are theoretically within the applicability range of the formulation, the truth is

hat it had to be reduced from 196 cases to 172. Those 24 excluded correspond to those cases

its velocity was 4.42 m/s or what is the same as Cv = 1.

of the third equation is within the applicability range.

ubsequently the error increases much more. I.e. , if the case is in the exactly

limit of application has much more error than if it is almost in the limit, that means, C

increased error value rather if it would be a Cv = of 1.1 or 1.2 and both are practically

. Hence, due to the result data obtained for these low speed cases, the error is quite big.

It was observed in the majority of these low speed cases, that the vessel was not

a more appropriate wake for a semi-displacement ship or pre-planing hull. E.g. in the case 182:

Velocity (m/s) Dead rise (º) Draft (m) Trim (º)

4.42944692 5 0.2

Table 17

to the parametric equation of the flat lamina’s surface:

� + � tan 35 � �1805 . � tan 36 � �1805 � 0.2

Equation 11.Geometry of flat lamina’s surface.

The beam in the transom is 1 m, so the CV should be comparable to other cases with the same C

planing hull or at least do not behave like that, the “Spray” does not appear,


shown in the next image, it can be seen that the beam is not really 1 m

Figure 46. Error on the beam for case 182.


within the applicability range of the formulation, the truth is

hat it had to be reduced from 196 cases to 172. Those 24 excluded correspond to those cases in which

It is possible that is too

ubsequently the error increases much more. I.e. , if the case is in the exactly

limit of application has much more error than if it is almost in the limit, that means, Cv = 1 has an

= of 1.1 or 1.2 and both are practically the same Froud

. Hence, due to the result data obtained for these low speed cases, the error is quite big.

hat the vessel was not planing because it had

hull. E.g. in the case 182:

Trim (º)

6

17. Case 182 parameters.

Geometry of flat lamina’s surface.

should be comparable to other cases with the same CV. But in

hull or at least do not behave like that, the “Spray” does not appear,


en that the beam is not really 1 m

. Error on the beam for case 182.


A negative LC means that the LC ends in the downstream as it

It can be observed that LC ends beyond the transom

be minor than 0. These phenomena have

those which have an important draft and

Otherwise, Savitsky’s formulation does not take the length of the vessel as a defining parameter of the

CV , and logically in reality a vessel with a L

with the same CV but different LK / b ratio. E.g. the case 182, according to the definition of the mean

wetted length – beam ratio or λ:

According to the theoretical Savitsky’s value and the one calculated by FEM

Method

Savitsky

FEM

This difference between the coefficient for the same case and different calculus methods

case not be really the same. The explanation is that for a C

planing.

ends in the downstream as it is shown in the following image:

Figure 47. L

ends beyond the transom and based on the discretization

These phenomena have been reproduced in several low speed cases. In general in

rtant draft and a big trim and deadrise angles.

does not take the length of the vessel as a defining parameter of the

, and logically in reality a vessel with a LK / b rate quite low cannot be comparable to another one

/ b ratio. E.g. the case 182, according to the definition of the mean

λ � L( L)2b

Equation 12. Mean wetted length

cal Savitsky’s value and the one calculated by FEM, this ratio would be:

b (m) Lc (m) Lk (m) :

2 1.38 1.91 0.82

2 0 1.91 0.48

Table 18. Comparison between Savitsky and FEM software.

een the coefficient for the same case and different calculus methods

case not be really the same. The explanation is that for a Cv = 1 and a semi-beam of 1 m, the hull is not

49

shown in the following image:

. LC ends downstream.

and based on the discretization criteria, LC cannot

been reproduced in several low speed cases. In general in

does not take the length of the vessel as a defining parameter of the

/ b rate quite low cannot be comparable to another one

/ b ratio. E.g. the case 182, according to the definition of the mean

Mean wetted length – beam ratio.

this ratio would be:

0.82

0.48

. Comparison between Savitsky and FEM software.

een the coefficient for the same case and different calculus methods makes the

beam of 1 m, the hull is not

50

5.6. Submerged Volume

In order to demonstrate this hypothesis was

hydrostatic lift was versus the hydrodynamic lift. Despite Savitsky’s method is mainly used to predict the

power for planing hulls, the formulation itself is not designed for a specific type of hulls.

hulls that fulfil the criteria so it will not do difference between a real

To check out if the previous error was not from the own software, it was decided to compare the

hydrostatic lift for a CV = 1 with the theoretical value of

The hydrostatic lift is just the displaced

can be defined as:

In this case what is being calculated is the semi

the depth, basically because is a theoretical calculu

significant value to assume that the density is going to vary with the depth. The previous equation

becomes:

The integral of the differential of z is the equation of the parametric surface:

;�* �*.+. �

To set boundaries, it is needed to divide the integration in parts because the boundaries do not remain

constant along the length. First of all, clarify that the X axis is not th

direction of the trim tilting. This


hypothesis was right, it was ideated a way to check how important the


hulls, the formulation itself is not designed for a specific type of hulls.

the criteria so it will not do difference between a real planing hull and a fake


1 with the theoretical value of Savitsky’s formulation.

The hydrostatic lift is just the displaced volume of water multiplied by the gravity and the density. That

*.+. � % � � � � � <<<%� � �*

Equation

what is being calculated is the semi-volume. Assuming that the density

basically because is a theoretical calculus and the maximum draft is 40 cm

to assume that the density is going to vary with the depth. The previous equation

*.+. � % � � � � � %�<<<�*

Equation 14. Development of the hydrostatic Lift equation.

The integral of the differential of z is the equation of the parametric surface:

� * � + � tan 3� � �

�,�5 . � tan 3� � �

�,�5 � �

� << + � tan 3� � �1805 . � tan 3� � �1805 � �

Equation 15. Development of the hydrostatic Lift equation.

it is needed to divide the integration in parts because the boundaries do not remain

constant along the length. First of all, clarify that the X axis is not the length, the length would be in the

direction of the trim tilting. This displaced volume by the flat lamina would be:

Figure


d a way to check how important the


hulls, the formulation itself is not designed for a specific type of hulls. It is just for the

hull and a fake planing hull.


multiplied by the gravity and the density. That

Equation 13.Hydrostatic Lift.

density does not vary with

draft is 40 cm, which is not a

to assume that the density is going to vary with the depth. The previous equation

f the hydrostatic Lift equation.

Development of the hydrostatic Lift equation.

it is needed to divide the integration in parts because the boundaries do not remain

e length, the length would be in the

Figure 48. Submerged volume.


What has to be found out is L1 for every case. To do so, it has been recurred to some trigonometric

properties and the equation of a line defined by two points.

Figure 49. Dimensions of the submerged volume.

for every case. To do so, it has been recurred to some trigonometric

properties and the equation of a line defined by two points.

The angles which affect L1 are trim and deadrise.

Figure 50. Angles that determine the submerged volu

51

. Dimensions of the submerged volume.

for every case. To do so, it has been recurred to some trigonometric

are trim and deadrise.

. Angles that determine the submerged volume.

52

L1 is the horizontal distance in the integration zone in which its angl

is the draft without the depth due to deadrise tilting

The equation of a line defined by two points can be described as:

... � �

tantanThe boundaries are from 0 to L1

geometrical criteria erased those cases in which its

reached L1 from 0 to �-."� ��

� �� m.


Both integration

in this picture

is simple and the second

be done by means of setting the

boundary in Y axis with the

equation of a line defined by two

points.

is the horizontal distance in the integration zone in which its angle is the trim and opposite cathetus

is the draft without the depth due to deadrise tilting

� � � � tan,�-tan,�-

Equation

The equation of a line defined by two points can be described as:

� .� � .� � + � +�+ � +� ; . � ��tan,�- � � � + � 10 � 1

� tan,�-tan,�-tan,�-tan,�- � �+ 1; + � � � . � tan,�-tan,�-

Equation 17. Equation of the contour life in integrat

1 m and from L1 m to �

� �� . And for Y axis, from

geometrical criteria erased those cases in which its maximum beam was less than 1 m. And once

m.

Figure


integration zones are labelled

n this picture. The first integration

is simple and the second is going to

be done by means of setting the

boundary in Y axis with the

equation of a line defined by two

e is the trim and opposite cathetus

Equation 16.L1 definition.

Equation of the contour life in integration 2 zone.

from 0 to 1 m because the

beam was less than 1 m. And once

Figure 51. Integration zones.


53

The final equation is:

*.+. � < < 3+ � tan 3� � �1805 . � tan 3� � �1805 � �5 �+�. �

�

�-� ��

�

< < 3+ � tan 3� � �1805 . � tan 3� � �1805 � �5 �+�.�-."� ��

�

��

�-� ��

Equation 18. Development of hydrostatic lift.

It has been introduced into wxMaxima, a mathematics software, and its result is:

*.+. � �tan,� � �180- � 3 � d � tan 3� � �1805 3 � �6 � tan,� � �180-

Equation 19. Final equation to find out the hydrostatic lift.

To express the units in Newton the previous equation has to be multiplied %� :

*.+. � �>tan 3� � �1805 � 3 � d � tan 3� � �1805 3 � �6 � tan 3� � �1805 ?%�

@ABCB: %:1.025E�/G/

�: � 9.81G/H

Equation 20. Hydrostatic lift in Newton.

Then, once this is done and it has been calculated for whole low speed cases, it is time to compare the

hydrostatic Lift versus Savitsky’s results:

Vel Dead Trim Sink Lh.e. LSav. LFEM h.e. vs Sav Valid?

4.429 5 2 0.2 3607 2867 - 26% NO

4.429 5 3 0.2 2403 2143 - 12% NO

4.429 5 3 0.3 6361 5048 - 26% NO

4.429 5 4 0.2 1801 1925 - -6% NO

4.429 5 4 0.3 4767 3996 - 19% NO

4.429 5 4 0.4 9171 7355 - 25% NO

4.429 5 5 0.2 1440 1874 - -23% YES

4.429 5 5 0.3 3810 3514 - 8% NO

4.429 5 5 0.4 7330 6074 7585 21% NO

4.429 5 6 0.2 1198 1896 2034 -37% YES


54

4.429 5 6 0.3 3172 3285 3958 -3% NO

4.429 5 6 0.4 6102 5367 6780 14% NO

4.429 10 2 0.2 2174 1845 2314 18% NO

4.429 10 3 0.2 1448 1465 1791 -1% NO

4.429 10 3 0.3 4554 3788 4801 20% NO

4.429 10 4 0.2 1086 1383 1573 -21% NO

4.429 10 4 0.3 3413 3064 3904 11% NO

4.429 10 4 0.4 7178 5926 7366 21% NO

4.429 10 5 0.2 868 1396 1475 -38% YES

4.429 10 5 0.3 2728 2753 3369 -1% NO

4.429 10 5 0.4 5737 4946 6216 16% NO

4.429 10 6 0.2 722 1450 1439 -50% YES

4.429 10 6 0.3 2271 2622 3020 -13% NO

4.429 10 6 0.4 4775 4419 5371 8% NO

Table 19. Low speed cases.

This is the CV = 1 table results. First 4 columns correspond to the parameters of the case, the 5th

column

to the value of the hydrostatic Lift, the 6th

column to the Savitsky’s Lift, the 7th

column to the lift

calculated by the FEM software, the 8th

column to the comparison between the hydrostatic lift and

Savitsky’s one and the 9th

column to the cases which are really planing.

First of all, observe that the three cases which are clearly planing, are the ones with small volume

underwater. This suggests that for vessels which have huge depth and a speed of 4.42 m/s are not

within the planing range, like it was deducted before. In the other hand, vessels with not much volume

submerged with a speed of 4.42 m/s are not able to create the “Spray” layer and consequently the

mean wetted length – beam ratio is less than predicted,

Furthermore, there is relevant data between the three cases which are clearly planing. This is the

percentage of the hydrostatical lift versus Savitsky’s Lift, and it has to be at least superior to 30% to be

clearly in a planing situation. Also there is relevant data for other relations and the lift calculated by the

FEM software:

1. If Lifth.e. > Savitsky’s Lift . LiftFEM error compared to Savitsky is within 20% - 30 %.

2. Si el Lifth.e. < Savitsky’s Lift . LiftFEM error compared to Savitsky is within 10% - 15 %.

3. Si el Lifth.e. < 70% Savitsky’s Lift . LiftFEM error compared to Savitsky is within 0% - 10 %.

Because of having only three proper cases with planing situation, they cannot be added to the total case

matrix because is not a representative sample for the whole conjunct. That is why CV = 1 cases have

been omitted from the results

Chapter 6. Results.

55

Chapter 6. Results.

6.1. Result storing

Results after running the previous described cases are stored by means of a script in a folder labelled

“Sav_” and the case number. This folder contains 3 documents.

The first is the case identifier, which is the one containing the data relative to the case

The second is a result file which is the one storing the graphical data results such as pressure diagram or

total elevation of the free surface. This can be posteriorly visualized in the post-process module of FEM

software.

The third is a file in which the user defined results are stored. These results are the Lift and the Torque,

which are the half of its real value due to calculating only half of a model.

6.2. Processor

In order to do previous studies of the cases before setting up the version 4, which is the definitive, a

regular laptop was used to perform the calculus. It was an old and low powerful laptop, so in order to

avoid having to wait for a long calculus time and not burn off the computer, a computer was facilitated

by the Naval and Maritime CIMNE department. Apart from being much more powerful it had a GPU so it

allowed to run cases with CPU+GPU instead of using just the CPU which is more time-consuming than

the combination of both.

GPU allows to perform calculus much quicker than CPU, which works sequentially, and GPU in parallel. It

saved lot of time, e.g. a case that could be carried out in 14 hours, was carried out in just 5 hours and

with a much more accurate mesh.

Otherwise, GPU has great disadvantage which is the noise it adds to the calculus that could be

sometimes quite harmful if the results are not revised.

The study was carried out having the two calculi running at the same time. The ones for FEM scheme

have less noise because they were run by CPU and the STREAMLINE scheme has quite distortion due to

the noise added by the CPU+GPU.

56

6.3. Scheme

This study did not limit itself to study the relation between

calculation.

It was used two calculus methods or schemes. A scheme is a calculus pattern which is used by the

software in order to solve the problem, i.e. , it determines on how the calculus will be carried out by the

algorithm. The schemes used for the calculations were

new version recently developed for SeaFem called FEM.

STREAMLINE is a good scheme and quite accurate but has much more calculus time than FEM along with

the stability problems it has. FEM

stability factors rather STREAMLINE, although it is quick, is less precise but it has not bad results. FEM

scheme was in test during the development of this project.

An advantage of FEM is that works without problems with

has had some issues with few cases

to the noise, indeed there was no error file

6.4. Result type

Previously it has been stated that insid

The .flavia is the file containing the case parameters. The

file:

And the Ouput.res is the result file of the two parameters defined

force and the torque of this lift. This file is in Binary1 format. To import all the cases to Excel a script was

done, this script generates a spreadsheet form introducing the parameters of the case on

the forces and torque and representing these on a chart.


This study did not limit itself to study the relation between Savitsky’s formulation

ulus methods or schemes. A scheme is a calculus pattern which is used by the


chemes used for the calculations were STREAMLINE which is the implicit from GID and a

new version recently developed for SeaFem called FEM.


e stability problems it has. FEM in the other hand, is quicker and robust, works quite well for low


scheme was in test during the development of this project.

An advantage of FEM is that works without problems with the noise of GPU, meanwhile STREAMLINE

has had some issues with few cases which have had to be done again because the calculus crashed due

to the noise, indeed there was no error file for these cases.

Previously it has been stated that inside the result folder there were 3 files.

is the file containing the case parameters. The .flavia.res correspond to the graphical result

Figure

is the result file of the two parameters defined previously


spreadsheet form introducing the parameters of the case on

the forces and torque and representing these on a chart.


Savitsky’s formulation versus a unique FEM

ulus methods or schemes. A scheme is a calculus pattern which is used by the


h is the implicit from GID and a


robust, works quite well for low


the noise of GPU, meanwhile STREAMLINE

had to be done again because the calculus crashed due

Figure 52. Result data.

correspond to the graphical result

Figure 53. Graphical result file.

by the user. The lifting


spreadsheet form introducing the parameters of the case on it, evaluating

Chapter 6. Results.

57

The stability time depends basically on the case velocity, there are five values for the velocity and so five

for time stability, these values are: 150, 10, 8, 4, 2 seconds. The script searched the mean value near the

stability time with ± 10% of margin. E.g. in case number 1 of STREAMLINE, when it was imported to

Excel this was the data which the result has written in it.

# SeaFEM body loads file v.1.0

DataSets User defined results

SubSets Set 1 Set 2

numResults 10

time[s] Fz_Hfs[N] My_Hfs[Nm]

0.500323 17081.8 -67114.2

0.50125 17076.5 -67084.8

0.502177 17080.8 -67111.4

0.503103 17075.1 -67081.4

0.50403 17081.6 -67113.5

0.504956 17078.3 -67088.5

0.505883 17079.8 -67108.2

0.506809 17073.5 -67081.5

0.507736 17075.1 -67092.7

… … …

Table 20. User defined result without manipulation.

Then the script added the parameters of the case:

Case 1

Sink 0.2

Trim 2

Vel 5

Dead 22.1472346

Time_Fz 2.00037

Fz_Hfs 17133.6

Time_My 2.00037

My_Hfs -67568.9

Table 21.Added data in user defined result in order to identify the case.


58

And it generated two charts:

Figure 54. Regular force and torque charts.

Once the script had finished this process for all the cases, there was the need of revising to check there

was no errors. On a first look, both previous charts seem not to have converged in a stable value but

once the axis is centered in 0 it reveals is quite stable:

Figure 55. A global point of view of the previous chart.

In the other hand the revision allows to check out if the results are logic or not. E.g. for case 91 in

STREAMLINE, it had an error due to GPU noise and this was:

Figure 56. Chart with errors.

17050

17100

17150

17200

0 0.5 1 1.5 2 2.5

Fz_Hfs[N]

-67800

-67600

-67400

-67200

-67000

0 0.5 1 1.5 2 2.5

My_Hfs[Nm]

0

5000

10000

15000

20000

0 0.5 1 1.5 2 2.5

Fz_Hfs[N]

-1E+66

0

1E+66

2E+66

3E+66

4E+66

5E+66

0 0.2 0.4 0.6 0.8 1

Fz_Hfs[N]

Chapter 6. Results.

59

Revising it with more detail and erasing those spikes it could be more or less appreciated the real value.

Results start to be saved at

second 0.5 and for this case

apart from 0.55 s the force

starts to raise from 15 kN up

to 18 kN and in 0.6 s it

stabilizes. Then, after 0.75 s it

starts to increase

vertiginously again.

According to Savitsky’s

formulation the value should

be around 15kN.

Once it was re-calculated and without error the value was 13.5 kN .

Figure 58. Re-calculated chart.

11800

12000

12200

12400

12600

12800

13000

13200

13400

13600

13800

0 2 4 6 8 10

Fz_Hfs[N]

0

5000

10000

15000

20000

25000

30000

0 0.2 0.4 0.6 0.8 1

Fz_Hfs[N]

Figure 57. Previous chart without spikes.


60

6.5. Excluded results

Although 24 cases were excluded, the ones with CV = 1. Previously to version 4, it was calculated some

of these cases in the STREAMLINE scheme. The pattern for the Lift in a planing hull is more or less like

this:

This is the distribution observed for CV > 1.

This chart is exponential negative, a function like:

I,.- � EB-.

In contrast with STREAMLINE scheme cases of CV = 1 , which its pattern before version 4, were:

There are some oscillations

that as time goes by it converge

to a stable value while its

amplitude is reducing through

the time.

Indeed that was basically due

to a wall effect that on version

4 was solved by expanding the

beam of the model.

For the version 4 in STREAMLINE, a few cases were carried out and the pattern that they show is:

It can be observed GPU’s noise.

Here the pattern changes. Here

there is no negative

exponential behaving and it can

be appreciated that on the

beginning near second 0.5 the

graph falls and stabilize in the

final value.

Fz_Hfs[N]

Fz_Hfs[N]

Fz_Hfs[N]

Figure 59. High speed pattern force chart.

Figure 60. Low speed, STREAMLINE scheme for an old version.

Figure 61. Low speed, STREAMLINE scheme for version 4.

Chapter 6. Results.

61

Otherwise in FEM scheme:

Here appears the opposite of

what was happening when in

wall effect.

In this case there is a mean

value but the amplitude

increase through the time.

In both schemes only 3 of the 24 cases have a correct result. So having this diversity of patterns and the

problems about the hydrostatic lifting, make it easier to decide to exclude these cases.

Fz_Hfs[N]

Figure 62. Low speed, FEM scheme.


62

6.6. Non-excluded results

Non-excluded results correspond to Cv = 2, 3, 4 y 5 . The range case is the combination of these

parameters:

Sink 0.2 - 0.3 - 0.4 m

Trim 2 3 4 5 6 º

Vel 22 18 13 9 - m/s

Dead - 5 10 15 20 º

Table 22.case matrix.

This table would correspond to a total of 240 cases which were discretized to 172 cases. These 172 cases

were run two times entirely for the STREAMLINE scheme, the first time with version 3 and the last time

with version 4. And a few cases had to be run two times more due to noise, human errors when setting

data or mesh errors. For FEM scheme it was run together with STREAMLINE second time cases and they

had to be repeated once more for few cases that had some errors.

E.g. for FEM scheme there was the need to re-generate the mesh due to solve some random errors.

There is no real evidence of what was the problem. But it is believed that probably was due to the mesh

structure, which some node was causing stabilities problems.

These random cases had the same problems, first of all the stability factor was reduce from 0.5 to 0.1 in

steps of 0.1. After trying it so many times it was noticed that for different stability factors the calculus

always crashed on a determined time. Then after trying different things, it was suggested to transform

the adjacent water near the lamina into a non-structured mesh. The results, before changing the mesh,

can be was appreciable that just after the transom stern the flux had like a vortex that was increasing

more and more. Once changing the mesh structure, the problem was fixed. It was probably a point

which was affecting the calculus. The chart obtained even reducing the stability factor up to 0.1 was:

Always near a determined time, in this case

0.9 s, the software used to crash.

Then the adjacent water surface was re-

meshed as unstructured keeping the

maximum element size up to 0.1 to avoid

abrupt transitions.

0.4 0.5 0.6 0.7 0.8 0.9

Fz_Hfs[N]

Figure 63. Chart of the cases that crashed near a constant time value.

Chapter 6. Results.

It can be seen that the mesh, in the adjacent

blurring of the error and avoids this

once and again.

STREAMLINE scheme had much more errors than FEM scheme

STREAMLINE scheme is slower than FEM

has more difference with Savitsky’s formulation

be over and under.

To compare FEM software results with

J � �012

This formula is applicable for lift and torque,

value calculated by Savitsky’s formulation

Figure 64. Modified adjacent water

in the adjacent water of the lamina, now is unstructured

this critical error, which even changing the stability factor, was repeating

EAMLINE scheme had much more errors than FEM scheme which can be seen here:

Scheme Fails

STREAMLINE 32

FEM 7

Table 23.Number of failed cases.

than FEM, but more accurate. A special feature of FEM sch

Savitsky’s formulation, but always is under its value. STREAMLINE scheme can

To compare FEM software results with Savitsky’s formulation, it was applied the following formulas

012 � �3��3�� 100; |J| � L�012 ��3��3�� L � 100

Equation 21.Definition of error

This formula is applicable for lift and torque, VFEM is the value calculated by FEM software and

Savitsky’s formulation.

63

water mesh structure.

is unstructured. This makes more

, which even changing the stability factor, was repeating

can be seen here:

Number of failed cases.

e. A special feature of FEM scheme is that it

STREAMLINE scheme can

s applied the following formulas:

Definition of error and absolute error.

FEM software and VSav is the


64

To summarize the errors in function of its velocities and schemes, check this:

STREAMLINE:

Table 24.Error depending on the speed range and its representation.

FEM:

Table 25. Error depending on the speed range and its representation.

0.00%

10.00%

20.00%

30.00%

40.00%

1 3 5 7

Error

CV

Fz_Error

Fz_Max

My_Error

My_Max

20.00%

25.00%

30.00%

35.00%

40.00%

45.00%

50.00%

1 3 5 7

Error

CV

Fz_Error

Fz_Max

My_Error

My_Max

Cv Fz My

Error Max Error Max

2 6.40% 25.75% 8.86% 25.45%

3 7.29% 19.51% 8.61% 28.14%

4 10.96% 23.68% 10.03% 29.19%

5 12.79% 26.92% 10.93% 29.80%

Total 9.36% 26.92% 9.61% 29.80%

Cv Fz My

Error Max Error Max

2 20.70% 38.24% 25.90% 36.22%

3 31.53% 42.43% 31.98% 38.41%

4 35.19% 43.70% 33.27% 37.72%

5 36.64% 45.79% 33.82% 38.94%

Total 31.01% 45.79% 31.24% 38.94%

Chapter 6. Results.

65

In this chart can be observed the results for STREAMLINE scheme charts:

Figure 65. STREAMLINE result charts.

It can be seen that STREAMLINE scheme has much more accurate results than FEM scheme. Notice that

on the Fz and My charts, the tendency of the results is more or less the same as Savitsky’s. That proves,

that despite the error, cases calculated well.

Figure 66. Tendency of Lift and Torque.

The errors are under 30% for Fz and My. Show that for the Lifting the lowest the speed is the lowest the

error. The Torque has not a specific tendency.

0

50000

100000

150000

1 51 101 151

Fz

Savitsky

Fem

-400000

-300000

-200000

-100000

0

1 51 101 151

My

Savitsky

Fem

0.00%

10.00%

20.00%

30.00%

1

23

45

67

89

111

133

155

177

Fz_hfs %

Fz_hfs %

0.00%

10.00%

20.00%

30.00%

40.00%

1

23

45

67

89

111

133

155

177

My_hfs %

My_hfs %

0

20000

40000

60000

1 51 101 151

FzSavitsky

Fem

-200000

-150000

-100000

-50000

0

1 51 101 151

MySavitsky

Fem


66

For FEM scheme results:

Figure 67. FEM result charts.

Notice that on the Fz and My charts, the tendency of the results is more or less the same as Savitsky’s.

That proves, that despite the error, cases calculated well. Although the tendency is not incorrect, it is

more different comparing to STREAMLINE scheme.

Figure 68. Tendency of Lift and Torque.

Errors here are under 40% for Fz and 45% for My, and both do not seem to reduce in function of the

speed.

The result table is added in the Annex.

0

50000

100000

1 51 101 151

Fz

Savitsky

Fem

-300000

-200000

-100000

0

1 51 101 151

My

Savitsky

Fem

0.00%

20.00%

40.00%

60.00%

1

23

45

67

89

111

133

155

177

My_hfs %

My_hfs %

0.00%

20.00%

40.00%

60.00%

1

23

45

67

89

111

133

155

177

Fz_hfs %

Fz_hfs %

0

20000

40000

60000

1 51 101 151

FzSavitsky

Fem

-200000

-150000

-100000

-50000

0

1 51 101 151

MySavitsky

Fem

Chapter 7. Error study.

67


Once obtained the results and comparing these with Savitsky’s formulation, it is going to be carried out

a study of the error propagation in order to know the quality and precision of FEM software and

minimize its errors. This will lead to being able to get a equation that relates the error propagation

within error range.

It has been used two models to study the propagation of these errors.:

1. Regression model by means of integration. 2. Regression model by means of least squares – Gauss normal equations.

First model was much more complex to programme than the second one. Also the second model was

quicker and simple to analyse.

The objective of both models is the reproduction of the error in function of the characteristic

parameters (τ, d, β y V) and minimize the error obtaining a characteristic polynomial: M�,*,�,5 � N� N�� NA N/� N6� N��A N7�� N8�� … N9��A:��;

Equation 22.. Polynomial equation.

O what would be the same:

P,.- � N�P�,.- NP,.- … N�P�,.- �QN9P9,.-�

9<�

Equation 23.Polynomial equation simplified.

7.1. Least squares

Both models use the least squares method. The method has the purpose to find out the polynomial

whose sum square errors are the minimum.


68

E.g. :

If there is a group of points in which can be represented some different lines inside of these points

passing through the centre of this group.

Figure 69. Possible least squares lines.

Those lines represented could be the optimal sum of the minimum square errors. To solve the system is

only needed to have a number of equations. The more range of the polynomial is wanted the most

equations will be needed.

A line can be represented by:

��

Equation 24. Line definition.

The error can be expressed as::

� � � � �� Equation 25.Error definition.

Figure 70. Error representation.

Least squares is based on the optimization of the square error:

�� ∑ ��

Equation 26. Square error.

0

0.05

0.1

0.15

0.2

0.25

0.3

0 10 20 30 40 50 60 70 80 90

0.2

0.22

0.24

0.26

0.28

0.3

60 65 70 75 80 85 90

P = (xi , yi)


69

Where n is the total number of cases. Simplifying the previous expression the mean square

error is defined:

�� 1��

�

� �1��

�

��

�

Equation 27.Mean square error.

To solve the system, it is represented in the following matrix:

1. Lineal regression

N is the total number of cases.

� � � � ∑ �� ∑ � � ∑ ��

��

�� ∑ ��

� �� ∑ ��

� ��

� � ∑ � � ∑ ��

�� ∑ �� ∑ ��

�� ∑ ��

� �� ∑ ��

� ��

Equation 28. Definition of line parameters.

2. Polynomial regression ( xn ; n > 1)

The system has to be solved. With this matrix it

can be obtained a polynomial of n range (xn),

there is only needed n columns and rows to solve

it.

It is a simple method to solve because is based on

multiplying xn by y to get a higher range

polynomial.

On the other hand it has a problem, in case i>N, i.e. , when the group of points is less than

the polynomial range there is no solution.

Figure 71. Least squares for a lineal interpolation.

Figure 72. Least squares for non lineal interpolation.


70

7.2. Regression model by means of integration.

The aim of this model is to represent the evolution of the error in function of (τ, d, β y V) to do so it is

needed to recur to least squares. The objective is to recreate the error in function of 4 regressions.

Every regression adds a parameter in the polynomial. The regression order is: τ � h � β � V.

V β h τ

22.1472346 5 0.2 2

. . . 3

. . . 4

. . . 5

. . . 6

. . 0.3 2

. . … …

. . 0.4 2

. . … …

. 10 0.2 2

. … … …

. 15 0.2 2

. … … ….

. 20 0.2 2

. … … ….

17.7177877 5 0.2 2

… … … ….

13.2883408 5 0.2 2

… … … ….

8.85889384 5 0.2 2

… … … ….

Table 26.Representation of the order of regression.

It has to be recreated a 4D space integrating from a group of points. First this group of points is

transformed to lines by doing the first regression in function of τ:

J�,�,5,�- � R� R � �

Equation 29.Error first regression.

t1 y t2 are coefficient. These coefficients have different values in function of the other three parameters.

Re

gre

ssio

n 4

(V

)

Re

gre

ssion

1 (τ)

Re

gre

ssion

2 (d

)

Re

gre

ssion

3 (β

)


71

It can be observed that there is a line for

every draft (d).

There are some cases where for deadrise

angles of 20 there is no 0.2 draft.

The following regression adds to the expression: R� � N� N� � � N � � R � N/ N6 � � N� � �

Equation 30. Second regression.

From two coefficients, it becomes to six because the range of the regression line here is n = 2; what

means, it is parabolic. Six coefficients come from the 2 coefficients in first regression multiplied by the 3

of the second. � 2x3 = 6.

J�,5,�. �- � R�,�- R,�- � � � ,N� N� � � N � �- ,N/ N6 � � N� � �-�

Equation 31. Second error regression.

Every regression adds one more dimension, from lines to planes:

Figure 74. Representation of the second regression.

y = -0.0305x + 0.203 y = -0.0206x + 0.2816 y = -0.0125x + 0.3191

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

0 2 4 6 8

τ

0.2

0.3

0.4

Figure 73. Linear regresion for first regresion.


72

Third regression is in function of � and in this case the planes would be grouping in to form a solid. The

expression obtained would be: N� � �� / � �

N� � ��=� ��= � � ��=/ � �

J5,�. �, �- � ,N�,�- N�,�- � � N,�- � �- ,N/,�- N6,�- � � N�,�- � �-�

Equation 32. Third error regression.

This expression is without developing “a” coefficients. The regression adjusts better to a parabolic

regression, despite those cases which for lack of coefficients of the value of � it cannot be used a

parabolic and it has to be lineal. From 6 coefficients now appear 18 (6x3), from b0 to b17 .

The solid obtained is the sum of the different surfaces.

To imagine the fourth dimension, think of a solid

whose form and boundaries change during the time.

The last expression contains a total of 18 coefficients multiplied by 3 which is 54 coefficients in total.

This regression is in function of V:

�� T�=� T�= � � T�=/ � � J,�. �, �, �- � T� T�� T� …T�/ � � � � � � � �

Equation 33. Fourth error regression.

Every coefficient has associated a monomial which is the combination of the four initial parameters.

Every parameter except � is up to second order.

Figure 75. Representation of the third regression.


73

All of this process was optimized by means of a script, once finished the error was compared with the

initial:

Case: 143

Initial Reg (1) Reg (2) Reg (3) Reg (4)

9.40% 9.63% 9.63% 9240% 2735308%

Table 27. Example regression case.

It is more or less the same for every case. First two regression show an almost identical error with the

initial. In the third regression the error becomes 1000 times more and the fourth 30 000 times more

than the third regression.

It was decided to change the order of regression to check if that was the problem because, for example

in some cases draft only had two values, what obligated to do a lineal regression instead of a parabolic

one.

It was changed to : d � τ � β � V.

Once done it, the error casuistry remained. Reached this point, the decision was to do what is known as

coefficient of correlation.

7.3. Correlation Coefficient of Pearson

This coefficient determines the dependency between two random quantitative variables. This

coefficient can be positive or negative and is expressed from -1 to 1.

In this table is notated the relation:

Value Meaning

-1 Negative correlation big and perfect

-0,9 a -0,99 Negative correlation very high

-0,7 a -0,89 Negative correlation high

-0,4 a -0,69 Negative correlation moderated

-0,2 a -0,39 Negative correlation low

-0,01 a -0,19 Negative correlation very low

0 Null correlation

0,01 a 0,19 Positive correlation very low

0,2 a 0,39 Positive correlation low

0,4 a 0,69 Positive correlation moderated

0,7 a 0,89 Positive correlation high

0,9 a 0,99 Positive correlation very high

1 Positive correlation big and perfect

Table 28. Correlation coefficient value explanation.

74

This diagram represents the group of

The correlation coefficient of Pearson is defined by the letter “

Where:

N is the total number of cases.

The term X is the monomial, in this case there are 69 monomials. Every monomial is the combination of

hn·τ

m·β

k·V

j , where n,m,k,j are independent variables that goes from 0 to 2

to 1. To perform this study, every monomial has to be replaced by the parameters which define the

case. E.g.:

εF V β h τ

-14.50% 22.15 5 0.2 2

-11.27% 22.15 5 0.2 3

-22.05% 22.15 5 0.2 4

-7.51% 22.15 5 0.2 5

-19.87% 22.15 5 0.2 6

-26.92% 22.15 5 0.3 3

The correlation coefficient has to be calculated for every monomial.


This diagram represents the group of point’s type and their tilting:

Figure 76. Diagram explanation of Correlation Coefficient.

The correlation coefficient of Pearson is defined by the letter “r” and it is:

C � U.>U. � U>

U.> � .+VVV � ∑.+X

U. � Y.VVV � ,.̅- � [∑.X � �∑.X �

U> � Y+VVV � ,+V- � [∑+X � �∑+X �

Equation 34. Correlation coefficient definition.


are independent variables that goes from 0 to 2 except

To perform this study, every monomial has to be replaced by the parameters which define the

τ τ2 h

2 β

2 V

2 τh τβ τV

2 4 0.04 25 491 0.4 10 44.3

3 9 0.04 25 491 0.6 15 66.4

4 16 0.04 25 491 0.8 20 88.6

5 25 0.04 25 491 1 25 111

6 36 0.04 25 491 1.2 30 133

3 9 0.09 25 491 0.9 15 66.4

Table 29.How Correlation coefficient is calculated.

The correlation coefficient has to be calculated for every monomial.


. Diagram explanation of Correlation Coefficient.

Correlation coefficient definition.


except for m that goes from 0

To perform this study, every monomial has to be replaced by the parameters which define the

τ2h τ

2β τ

2V hβ

0.8 20 88.6 1

1.8 45 199 1

3.2 80 354 1

5 125 554 1

7.2 180 797 1

2.7 45 199 1.5

How Correlation coefficient is calculated.


75

Once the correlation factor is calculated for each monomial, it should be chosen the most influencing

monomials of these 69. In these cases the range to be chosen is from 0.4 or above. These are the

monomials for STREAMLINE scheme chose:

The maximum value, minimum and the absolute

maximum are:

Table 30.Maximum and minimum Coefficients.

It can be observed that the correlation is negative and

the maximum is -0.67. More in detail, the correlation

is stronger for those monomials compound by V y τ

also some compounded by d and β has no relevant

importance.

V � τ � d � β

Only the strongest correlation monomials are chosen.

In total 22 out of 69. And then it will be studied for a

correlation equal or greater to 0.4 and other for equal

or greater 0.5.

Table 31. Important monomials.

Term r | r | % MAX %

V -0.6 0.6 90.56% 60%

V2 -0.57 0.57 85.59% 57%

τh -0.4 0.4 59.92% 40%

τV -0.67 0.67 100.0% 67%

τ2h -0.43 0.43 63.90% 43%

τ2V -0.61 0.61 91.42% 61%

hV -0.58 0.58 87.66% 58%

h2V -0.47 0.47 70.31% 47%

V2τ -0.63 0.63 94.87% 63%

V2h -0.57 0.57 85.72% 57%

τhV -0.62 0.62 93.23% 62%

τ2hV -0.6 0.6 90.54% 60%

τh2V -0.52 0.52 78.05% 52%

τhV2 -0.61 0.61 91.34% 61%

τβV2 -0.4 0.4 60.25% 40%

τ2h2V -0.54 0.54 80.52% 54%

τ2hV2 -0.61 0.61 90.78% 61%

τ2βV2 -0.42 0.42 62.75% 42%

τ2h2V2 -0.55 0.55 81.91% 55%

τh2V2 -0.53 0.53 79.82% 53%

τhβV2 -0.4 0.4 60.10% 40%

h2V2 -0.49 0.49 74.14% 49%

MAX: 0.095249624

MIN: -0.667302546

|MAX| : 0.667302546


76

In the case of FEM scheme which had more errors this study show the following tables:

The maximum value, minimum and the absolute

maximum are:

Table 32. Maximum and minimum Coefficients.

In this scheme there are only 15 monomials equal

or greater than 0.5 . Indeed these monomials are

identically the same as STREAMLINE.

Once it is known which monomials have a strong

correlation with the error and checked out that

the first method is no longer functional, it will be

used the second method.

Table 33.Important monomials.

Term r | r | % MAX %

V -0.60 0.60 90.56% 60%

V2 0.60 0.60 90.64% 60%

τh 0.37 0.37 55.80% 37%

τV 0.66 0.66 100.00% 66%

τ2h 0.36 0.36 53.47% 36%

τ2V 0.57 0.57 85.55% 57%

hV 0.63 0.63 94.59% 63%

h2V 0.52 0.52 78.17% 52%

V2τ 0.64 0.64 96.25% 64%

V2h 0.61 0.61 92.08% 61%

τhV 0.62 0.62 93.74% 62%

τ2hV 0.56 0.56 84.62% 56%

τh2V 0.53 0.53 79.78% 53%

τhV2 0.62 0.62 93.16% 62%

τβV2 0.39 0.39 59.37% 39%

τ2h2V 0.50 0.50 75.85% 50%

τ2hV2 0.58 0.58 87.36% 58%

τ2βV2 0.38 0.38 57.89% 38%

τ2h2V2 0.53 0.53 79.28% 53%

τh2V2 0.55 0.55 82.54% 55%

τhβV2 0.39 0.39 58.15% 39%

h2V2 0.54 0.54 81.70% 54%

MAX: 0.663958179

MIN: -0.102979575

|MAX| : 0.663958179


77

7.4. Regression model by means of least squares – Gauss normal equations

This method is based on matrix calculus by mean of least squares. Coming back to the previous study,

the definition of mean square error was:

�� 1��

�

��

�

Equation 35. Mean square error.

This method is based on the optimization of the square error. If the previous equation was the

mean square error, the expression inside the radicand is:

��

�

��

�

Equation 36. Square error.

Omitting n value because it will not affect the optimization, the expression is derivate and equalized to

0, because the aim is to find out that point that makes the expression minimum.

\��\�� Q2]>�Q ��,�-

�

?^��,�-_`�

� 0

��

�

��

�

� ��

�

Equation 37.Optimization of cj coefficients.

Expressed as a matrix would be:

A x c = y

a,P�, P�- ⋯ ,P�, P:-⋮ ⋱ ⋮,P:, P�- ⋯ ,P:, P:-e . fg�⋮g:h � a,P�, +-⋮,P:, +-e

Equation 38.Matrix redistribution.

In this particular case where n>m, the system is over-determined, that means that it will not have a

unique solution. Hence, the error can be approximated by A x c = y , that is because being over-

determined implies that the solution will not be never exact so it is somehow a way to approximate the

error.

a,P�, P�- ⋯ ,P�, P:-⋮ ⋱ ⋮,P� , P�- ⋯ ,P� , P:-e . fg�⋮g:h � a,+�-⋮,+�-e

Equation 39.Simplification.


78

To isolate the matrix term c, it has to be recurred to the transpose due to matrix A is not quadratic and

then once it is quadratic invert it to isolate c. The final expression would be: i � g � + → ,i? � i-g � i? � + → g � ,i? � i--�i? � +

Equation 40.Isolation of c matrix coefficient.

For this method, the results extracted are good enough. Most of the cases only differ in 1-3 % from the

real error. The propagation of the error was analysed as absolute error and regular error (taking in count

the negative or positive sign).

STREAMLINE scheme:

Figure 77. Error distribution for every case.

0.000%

50.000%

100.000%

1 9

17

25

33

41

49

57

65

73

81

89

97

105

113

121

129

137

145

153

161

169

FEM vs Savitsky

|Error|

|Calcuated|

-50.00%

0.00%

50.00%

100.00%

1 8

15

22

29

36

43

50

57

64

71

78

85

92

99

106

113

120

127

134

141

148

155

162

169

FEM vs Savitsky

Error

Calculated


79

FEM scheme:

Figure 78. Error distribution for every case.

0.000%

50.000%

100.000%

1 9

17

25

33

41

49

57

65

73

81

89

97

105

113

121

129

137

145

153

161

169

FEM vs Savitsky

|Error|

|Calcuated|

-50.00%

0.00%

50.00%

100.00%

1 8

15

22

29

36

43

50

57

64

71

78

85

92

99

106

113

120

127

134

141

148

155

162

169

FEM vs Savitsky

Error

Calculated


80

The coefficients obtained are:

STREAMLINE scheme:

Monomial Coef Value

V a1 0.044823

V2 a2 0.000159

τV a3 -0.011121

τ2V a4 0.001440

hV a5 -0.008954

V2τ a6 -0.000255

V2h a7 -0.013002

τhV a8 -0.005155

τ2hV a9 -0.003054

τh2V a10 -0.013682

τhV2 a11 0.003950

τ2h2V a12 0.005700

τ2hV2 a13 -0.000166

τ2h2V2 a14 -0.000883

τh2V2 a15 0.004727

Table 34. Coefficients of the relevant monomials. For monomials greater than 0.4 on the left and for 0.5

on the right.

Monomial Coef Value

V a1 0.446716

V2 a2 -0.002954

τh a3 0.630927

τV a4 -0.170736

τ2h a5 -0.085525

τ2V a6 0.016818

hV a7 -3.334894

h2V a8 6.516900

V2τ a9 0.000321

V2h a10 0.018492

τhV a11 1.238385

τ2hV a12 -0.120398

τh2V a13 -2.629673

τhV2 a14 -0.001721

τβV2 a15 0.000006

τ2h2V a16 0.260706

τ2hV2 a17 -0.000146

τ2βV2 a18 0.000000

τ2h2V2 a19 -0.002438

τh2V2 a20 0.027913

τhβV2 a21 -0.000019

h2V2 a22 -0.073800


81

FEM scheme:

Fem scheme only has 15 monomials and these are equal or

greater to 0.5 of correlation coefficient.

Table 35.

The total sum of the square errors for STREAMLINE and FEM is:

> 0.4 > 0.5

STREAMLINE 0.4853209 0.67571844

FEM - 0.44084841

Table 36

It can be seen that FEM has a sum of the square error smaller compared to STREAMLINE‘s monomials of

0.4 or greater. That means that the difference between the error and the predicted one for FEM is small

than for STREAMLINE. Hence, FEM error propagation study has a better dependency between the error

and the parameters and it is much more constant.

The final equation would be something like:

J,�, �, �, �- � QN9P9,.-�

9<�

Equation 41. Error expression for correlation coefficients.

Replacing n, ai and fi (x) for its value. fi (x) are the monomials.

Monomial Coef Value

V a1 0.125961

V2 a2 -0.005754

τV a3 -0.021244

τ2V a4 0.000306

hV a5 -0.738419

V2τ a6 0.001023

V2h a7 0.038252

τhV a8 0.145279

τ2hV a9 -0.000051

τh2V a10 0.333389

τhV2 a11 -0.006826

τ2h2V a12 -0.060304

τ2hV2 a13 -0.000097

τ2h2V2 a14 0.003289

τh2V2 a15 -0.018288


82

Chapter 8. Conclusions

This chapter summarizes all the problems that have been found out during the project.

8.1. Excluded cases of Cv = 1

Although Savitsky’s formulation allows to work with these cases, in the reality it can be something too

much connected to λ (mean wetted length – beam rate). Probably for those cases which LK is too small

and beam relatively big, λ becomes too small and logically a vessel with such beam and small length is

not going to plane.

Otherwise, after the integral calculus it verified the importance of the hydrostatic lift versus the

hydrodynamic one.

Finally, according to the obtained results it can be observed a clear difference between the cases run by

STREAMLINE scheme, which have more error for these excluded cases than FEM. Indeed, FEM has an

accurate approximation for the cases in which STREAMLINE failed to approach and contrary

STREAMLINE has a good approach for those cases in which FEM fails. That is mainly because FEM

approximates under the value of Savitsky and in the boundaries this approximation becomes so tiny,

which it is almost the same. But even though, it is not something that important, to include this cases

again. Check out few cases for CV = 1 by these two schemes.

FEM STREAMLINE

14.39% 24.89%

22.78% 7.29%

5.80% 20.47%

5.98% 25.41%

0.99% 21.56%

Table 37.Comparison of FEM and STREAMLINE schemes which have opposite accuracy for the same

cases.


83

8.2. Non-excluded cases (Cv = 2,3,4,5)

In a global analysis, it can be stated that the results are good enough and acceptable. STREAMLINE

scheme has no high error and FEM has higher errors.

Taking in count that STREAMLINE scheme has an average error of 9.36% which is not a big error. It is

important to say that the formulation of Savitsky is not 100% reliable and it is based in empirical data

which does not cover exactly the entire possible cases. It is only a formulation that can be used as

guidance and to have an idea of the power that a planing hull will need

It means in fact, that obtaining an error of 9.36% comparing to Savitsky’s formulation not necessarily

means that the FEM software results are wrong, probably it fits better the real problem than Savitsky.

Indeed it has to be clarified that nowadays there are different formulations based on Savitsky, which are

better than it and much more simplified. It is just up to the ship designer, to decide which formulation is

going to use or even a FEM software which is quicker and simpler.

From these non-excluded cases, the error varies more in function of the velocity and the trim rather

than the deadrise and draft. This is a statement that can lead to an idea of which parameters have to be

modified in case the designer is seeking for the most optimal form for his ship.

The propagation of the error study done, allows to calculate the propagation of these error for cases

within the range of sample space. I.e. , the case matrix was:

Sink 0.2 - 0.3 - 0.4 m

Trim 2 3 4 5 6 º

Vel 22 18 13 9 4 m/s

Dead - 5 10 15 20 º

Table 38.Case matrix.

So for the draft which its range is 0.2 , 0.3 and 0.4, it can be studied the error for another draft of 0.25 .

I.e. , selecting the first case (V=22m/s,β=5º,h=0.2,τ=2) and changing its draft for 0.25 m, the error

obtained would be:

-36.17%

It is huge compared to -14.5 %, but it is important not to omit if these case is within the range of

applicability of Savitsky’s formulation which in this case is not:

d 0.25 ≤ 7 · sin (t) ≤ 0.2442965 *INCORRECT

Table 39.Non-fulfiled criteria case.

Changing trim to 3 º the case would be: (V=22m/s,β=5º,h=0.25,τ=3):

d 0.25 ≤ 7 · sin (t) ≤ 0.3663517 *CORRECT

Table 40. Fulfilled criteria case.

And the error:

-20.31%

Which compared to -14.5% is similar. So, it can be calculated for all those cases within the case matrix

range if they always fulfil the criteria.


84

8.3. Time

This project started on September 2013. It is a project with a long time dedication, which is basically

because when somebody is working in a project which involves a FEM software, generally these sort of

software are quite time-consuming and more if this person has not great experience in this subject.

This project features for needing so much time dedication, always needed more. This mainly is due to

two reasons, first one because is a FEM environment and requires time to calculate and simulate.

Secondly the scripting, this project could be divided in five subjects:

- Mechanics of fluids.

- Element finite method calculus.

- Scripting.

- Mathematics.

- Management.

In this project scripting involves more or less 75% of the real work. That is because there was lots of

cases to be run and a script was needed to carried out the repeated work once and again. Scripting

always is quite difficult, not for the syntax which is easy once the person becomes actually an expert, it

is more for fulfil the features that the script has to do. It can be so much time-consuming sometimes

when writing a code because it is possible to lose too much time with simple errors that are difficult to

detect.

To evaluate the time inverted in this project it can be measured taking in count that the project started

on 15/9/2013 until 1/7/2014 which is the date that calculus where finished, that makes a total of 289

days without holidays, it would be about 247 days and prorating that 60% of these days have been

dedicated to the project , which probably is more, it would be a total of 148 days. And dedicating more

or less 6 hours per day, which also probably it is higher round 7 or 8, it makes a total of 889h. According

to the 24 ECTS credits for this project, it would be a 720h project.

It clearly only can conclude that before starting a project, even being an expert on it the estimated time

should be doubled in case of delays.


85

8.4. Human factor

Another factor to take in count is the human factor, the ones who are beside you and help you.

Despite could being so good at those five subjects defined before and being always under the estimated

time, the most determined factor could be the human one. During this project, there have been so

many times that it was not too much clear what to do next. If it is not for the advice of the tutors it

would have been impossible to end this project. Sometimes are easy troubles, but when somebody is

focused in something, possibly is not seeing the alternatives beyond the trouble, a second opinion is

great to end with this situations.

During the project there were some mesh problems that for inexperience of the advantages and

disadvantages of having a structured or non-structured type of mesh, have been a difficult topic to cope

with. With the right advices, it has been easier to understand how a mesh could affect to the results and

how to solve some problems that apparently had no evidence of what was the problem about.

In the field of error propagation study, it has been like for a month that the study had no progress until

it was suggested to use the second method, which finally in less than a week was ended.

8.5. Technological factor

Technological factor is also important as human one. Commonly in engineering having a decent

computer for calculating FEM problems saves lots of time. In this case if the project would have been

carried out by the initial computer, it would not have ended that quickly.

8.6. Savitsky empirical data.

This problem is more a pity than a problem, the fact is that the project would have been better if the

scope was focused to Savitsky empirical data rather than his formulation. The formulation is not 100%

reliable as the empirical data is. Only if being able to compare the results based on empirical data, it

would have been shown the real possibility of simulating a real case.

8.7. Towing tank data

Finally something that has not been discussed and has importance is the fact that it has been impossible

to find out in which conditions the empirical data cases had been carried out. The lack of information

can lead to speculate from the point of view if the towing tank had enough space to avoid the wall effect

that appeared in this project.

Something that reinforce this speculation is for example the cases of CV = 1. Where Savitsky’s

formulation calculated them without taking in count that the hydrostatic lift was greater than the one

calculated by Savitsky’s formulation. It should be added in the formulation a formula to calculate the

displaced volume and then as a criteria, to help to descretize the cases which do not fulfil this criteria.


86

Bibliography

87

Bibliography

� D. Savitsky. Hydrodynamic Design of Planing Hulls . Marine Technology, Vol. 1, No. 1. October, 1964.

� O. M. Faltinsen. Sea loads on ships and offshore structures. Cambridge University Press, Cambridge, UK.

1990.

� R. A. Royce. A rational prismatic hull approach for planing hull analysis. Society of naval architects and

marine engineers, Great lakes and great river section meeting, Cleveland, OH, U.S.A. January 27th

, 1994.

� D. Savitsky, P. W. Brown. Procedures for hydrodynamic evaluation of planing hulls in smooth and rough

water. Marine Techonology, Vol. 13, No.4. October, 1976.

� Compass Ingeniería y Sistemas, SA. SeaFEM reference. Retrieved December 21th

, 2013 from:

http://www.compassis.com/downloads/Manuals/SeaFEMManual.pdf

� Wikipedia, The Free Encyclopedia. Mínimos Cuadrados - Solución del problema de los mínimos

cuadrados. Retrieved June 3rd

, 2014 from:

http://es.wikipedia.org/wiki/M%C3%ADnimos_cuadrados#Soluci.C3.B3n_del_problema_de_los_m.C3.

ADnimos_cuadrados

� Wikipedia, The Free Encyclopedia. Coeficiente de correlación de Pearson. Retrieved June 15th

, 2014

from: http://es.wikipedia.org/wiki/Coeficiente_de_correlaci%C3%B3n_de_Pearson

� Vitutor.com . Coeficiente de correlación, Retrieved June 15th

, 2014 from:

http://www.vitutor.com/estadistica/bi/coeficiente_correlacion.html

� Monografias.com . Coeficiente de correlación de Karl Pearson. Retrieved June 15th

, 2014 from:

http://www.monografias.com/trabajos85/coeficiente-correlacion-karl-pearson/coeficiente-

correlacion-karl-pearson.shtml

� GiD, The Personal Pre And Post Processor. Manual Selection. Retrieved January 29th

, 2014 from:

http://www.gidhome.com/component/manual/


88

Annexes

89

Annexes

Annex A: User Defined Functions.

1. TDYN – Script to run cases automatically.

2. EXCEL – Savitsky criteria

3. EXCEL – Results storage.

4. EXCEL – Error evaluation, method 1.

5. TDYN – Result images

Annex B: Sections.

1. Isometric

2. Plan

3. Elevation

Annex C: Results.

1. STREAMLINE result table.

2. FEM result table.

3. STREAMLINE error table.

4. FEM error table.


90

Annex A: User Defined Functions.

TDYN – Script to run cases automatically.

#file copy -force $ExecCopy $Directory

set wn .window

toplevel $wn

wm title $wn "Show Output"

wm iconname $wn "Show Output"

set Files [list Outputs.res Name.flavia.res ]

set Sink [ list ]

set Trim [ list ]

set Vel [ list ]

set Dead [ list ]

set Beta [ list ]

set Time [ list ]

set in [llength $Sink]

for { set i 0 } { $i < $in} { incr i } {

set gSink [lindex $Sink $i ]

set gTrim [lindex $Trim $i ]

set gVel [lindex $Vel $i ]

set gDead [lindex $Dead $i ]

set gBeta [lindex $Beta $i ]

set gTime [lindex $Time $i ]

label $wn.msg$i -text "Case [expr $i+1] - Slver : \n $gSink \n $gTrim \n $gVel \n $gSink \n"

grid $wn.msg$i

update

set FileId [open $DataFile.flavia r]

set ThisFile [read $FileId]

close $FileId

regsub {%Sink%} $ThisFile $gSink ThisFile

regsub {%Trim%} $ThisFile $gTrim ThisFile

regsub {%Vel%} $ThisFile $gVel ThisFile

regsub {%Dead%} $ThisFile $gDead ThisFile

regsub {%Beta%} $ThisFile $gBeta ThisFile

regsub {%Time%} $ThisFile $gTime ThisFile

set FileId [open $DataFile$i.flavia w+]

puts $FileId $ThisFile

close $FileId

catch {

exec $ExecFile -name $DataFile$i.flavia -seawaves

}

# exec rename.win.bat

file mkdir [file join $Directory $GiDFile$i.gid]

file copy -force $DataFile$i.flavia [file join $Directory $GiDFile$i.gid $GiDFile.flavia]

file copy -force $DataFile$i.flavia.res [file join $Directory $GiDFile$i.gid $GiDFile.flavia.res]

file copy -force Outputs.res [file join $Directory $GiDFile$i.gid $GiDFile.Outputs.res]

#

}

exit 0;

EXCEL – Savitsky criteria

Function Matrix()

Annexes

91

Dim Row As Integer

Dim Column As Integer

Dim j As Integer

Dim i As Integer

Dim n As Integer

Dim Matriz As Integer

Dim Counter As Integer

Dim z As Integer

Dim Counter_M As Integer

Dim Lk As Integer

Dim Lc As Integer

Dim Trim As Integer

Dim Dead As Integer

Dim Vel As Double

Dim Sink As Double

Workbooks("Global.xlsx").Worksheets("Vectores").Activate

Lk = ActiveSheet.Range("C34")

Lc = ActiveSheet.Range("C35")

Row = ActiveSheet.Range("C8")

Column = ActiveSheet.Range("C7")

Range("H11:BBB39").Delete

Range("H9") = "Initial Matrix"

Matriz = Column ^ Row

i = 1

While (i - 1 <= Matriz)

ActiveSheet.Cells(11, i + 7) = i

i = i + 1

Wend

Stop


92

i = 1

While (i <= Row)

Counter_M = 0

Counter = Column ^ (i - 1)

n = 1

While (n <= Column ^ (Row - i)) '4

j = 1

While (j <= Column) '3

z = 1

While (z <= Counter) '1

Cells(i + 12, 8 + Counter_M) = Cells(i + 1, j + 2)

Counter_M = Counter_M + 1

z = z + 1

Wend

j = j + 1

Wend

n = n + 1

Wend

Cells(i + 12, 8 + Counter_M) = "]"

i = i + 1

Wend

Stop

Range("H22") = "Discretized Matrix"

i = 1

j = 1

While (i <= Matriz)

If (Workbooks("Global.xlsx").Worksheets("Vectores").Cells(13, i + 7) = "-" Or

Workbooks("Global.xlsx").Worksheets("Vectores").Cells(14, i + 7) = "-" Or

Workbooks("Global.xlsx").Worksheets("Vectores").Cells(16, i + 7) = "-" Or

Workbooks("Global.xlsx").Worksheets("Vectores").Cells(15, i + 7) = "-") Then

Else

Sink = Workbooks("Global.xlsx").Worksheets("Vectores").Cells(13, i + 7)

Trim = Workbooks("Global.xlsx").Worksheets("Vectores").Cells(14, i + 7)

Dead = Workbooks("Global.xlsx").Worksheets("Vectores").Cells(15, i + 7)

Vel = Workbooks("Global.xlsx").Worksheets("Vectores").Cells(16, i + 7)

Annexes

93

ActiveSheet.Range("C36") = Sink

ActiveSheet.Range("C33") = Trim

ActiveSheet.Range("C32") = Dead

Workbooks("Savitsky_Calculos.xlsx").Worksheets("Problem").Activate

ActiveSheet.Range("D35") = Sink

ActiveSheet.Range("D22") = Trim

ActiveSheet.Range("E34") = Vel

ActiveSheet.Range("D30") = Dead

If (ActiveSheet.Range("D32") <= Lk) And (ActiveSheet.Range("D33") >= 0) Then

Workbooks("Global.xlsx").Worksheets("Vectores").Activate

If (ActiveSheet.Range("C36") <= ActiveSheet.Range("E36")) And (ActiveSheet.Range("B39") <=

ActiveSheet.Range("E39")) Then

ActiveSheet.Cells(24, j + 7) = j

ActiveSheet.Cells(26, j + 7) = ActiveSheet.Cells(13, i + 7)




j = j + 1

End If

Else

End If

End If

i = i + 1

DoEvents

Wend

ActiveSheet.Cells(26, j + 7) = "]"




End Function


94

EXCEL – Results storage.

Sub Save_form()

'

' Macro_graph Macro

'

'

Dim Root_Dir As String

Dim Save_Dir As String

Dim init As Integer

Dim ending As Integer

Dim i As Integer

Dim Time_ As Double

Dim Fz As Double

Dim My As Double

Dim j As Integer

Dim j2 As Integer

Dim Column_ As Integer

On Error Resume Next

Root_Dir = "D:\PFC_RAFA\StreamLine\errors"

Save_Dir = "C:\Users\Rafa\FNB\PFC\Savitsky\Casos\Discretized_Cases2_Streamline"

init = 0

ending = 182

Column_ = 0

While (Column_ < 32)

i = Workbooks("Global_Streamline.xlsx").Worksheets("Abrir").Cells(5, Column_ + 12) - 1

ChDir Root_Dir & "\Sav_0" & Column_ & ".gid"

Workbooks.OpenText Filename:= _

Root_Dir & "\Sav_0" & Column_ & ".gid\Sav_0.Outputs.res", _

Origin:=xlWindows, StartRow:=1, DataType:=xlDelimited, TextQualifier:= _

xlDoubleQuote, ConsecutiveDelimiter:=False, Tab:=True, Semicolon:=False, _

Comma:=False, Space:=False, Other:=False, FieldInfo:=Array(Array(1, 1), _

Array(2, 1), Array(3, 1), Array(4, 1)), DecimalSeparator:=".", ThousandsSeparator _

:="'", TrailingMinusNumbers:=True

Worksheets("Sav_0.Outputs").Activate

Worksheets("Sav_0.Outputs").Range("G8") = Date$

Worksheets("Sav_0.Outputs").Range("F10") = "Sink"

Worksheets("Sav_0.Outputs").Range("F11") = "Trim"

Worksheets("Sav_0.Outputs").Range("F13") = "Vel"

Worksheets("Sav_0.Outputs").Range("F12") = "Dead"

Worksheets("Sav_0.Outputs").Range("F14") = "Time_Fz"

Worksheets("Sav_0.Outputs").Range("F15") = "Fz_Hfs"

Worksheets("Sav_0.Outputs").Range("F16") = "Time_My"

Worksheets("Sav_0.Outputs").Range("F17") = "My_Hfs"

Worksheets("Sav_0.Outputs").Range("F9") = "Case"

Worksheets("Sav_0.Outputs").Range("G10") = WorksheetFunction.HLookup(i + 1,

Workbooks("Global_Streamline.xlsx").Worksheets("Vectores").Range("H24", "BBB30"), 3)


Workbooks("Global_Streamline.xlsx").Worksheets("Vectores").Range("H24:BBB30"), 4)







ActiveSheet.Shapes.AddChart.Select

ActiveChart.ChartType = xlXYScatterSmoothNoMarkers

Annexes

95

j2 = 1

While (j2 <= 2)

j = 1

While (j <= 4)

ActiveChart.SeriesCollection(j).Delete

j = j + 1

Wend

j2 = j2 + 1

Wend

ActiveChart.SeriesCollection.NewSeries

ActiveChart.SeriesCollection(1).XValues = "='Sav_0.Outputs'!$A$6:$A$30000"

ActiveChart.SeriesCollection(1).Values = "='Sav_0.Outputs'!$B$6:$B$30000"

ActiveChart.SetElement (msoElementChartTitleAboveChart)

ActiveChart.ChartTitle.Text = "Fz_Hfs[N] "

ActiveChart.SetElement (msoElementLegendNone)

ActiveChart.Location Where:=xlLocationAsNewSheet, Name:="Fz_Hfs"

If Worksheets("Sav_0.Outputs").Range("G13") < 5 Then

Time_ = 150

Else


Time_ = 10

Else


Time_ = 8

Else


Time_ = 4

Else


Time_ = 2

End If

End If

End If

End If

End If


ActiveSheet.Range("G14") = WorksheetFunction.VLookup(Time_, ActiveSheet.Range("A6:C30000"), 1, True)


ActiveSheet.Shapes.AddChart.Select

ActiveChart.ChartType = xlXYScatterSmoothNoMarkers

j2 = 1

While (j2 <= 2)

j = 1

While (j <= 4)

ActiveChart.SeriesCollection(j).Delete

j = j + 1

Wend

j2 = j2 + 1

Wend

ActiveChart.SeriesCollection.NewSeries

ActiveChart.SeriesCollection(1).XValues = "='Sav_0.Outputs'!$A$6:$A$30000"

ActiveChart.SeriesCollection(1).Values = "='Sav_0.Outputs'!$C$6:$C$30000"

ActiveChart.SetElement (msoElementChartTitleAboveChart)

ActiveChart.ChartTitle.Text = "My_Hfs[Nm] "

ActiveChart.SetElement (msoElementLegendNone)

ActiveChart.Location Where:=xlLocationAsNewSheet, Name:="My_Hfs"





96

ActiveSheet.Range("H15") = (ActiveSheet.Range("G15") - WorksheetFunction.Max(Range("B6:B30000"))) * 100 /

WorksheetFunction.Max(Range("B6:B30000"))

ActiveSheet.Range("H17") = (ActiveSheet.Range("G17") - WorksheetFunction.Min(Range("C6:C30000"))) * 100 /

WorksheetFunction.Min(Range("C6:C30000"))

ActiveSheet.Range("I15") = "%"

ActiveSheet.Range("I17") = "%"

Fz = ActiveSheet.Range("G15")

My = ActiveSheet.Range("G17")

On Error GoTo 0

Workbooks("Global_Streamline.xlsx").Worksheets("Resultados").Cells(i + 2, 6) = Fz

Workbooks("Global_Streamline.xlsx").Worksheets("Resultados").Cells(i + 2, 7) = My

On Error Resume Next

MkDir (Save_Dir)

ChDir (Save_Dir)

ActiveWorkbook.SaveAs Filename:= _

"Sav_0" & i + 1 & ".xlsx", FileFormat:= _

xlOpenXMLWorkbook, CreateBackup:=False

Workbooks("Sav_0" & i + 1 & ".xlsx").Close SaveChanges:=True

Column_ = Column_ + 1

Wend

End Sub

Annexes

97

EXCEL – Error evaluation, method 1.

Sub ERROR_()

Dim i As Integer

Dim j As Integer

Dim z As Integer

i = 1

While (i <= 173) '' copy Error_F data

j = 1

While (j <= 8)

Worksheets("Error_Results").Cells(i, j) = Worksheets("Error_F").Cells(i, j)

j = j + 1

Wend

i = i + 1

Wend

Stop

Call ERROR_R1 ''Primera iteracion

Worksheets("Error_Results").Range("N2") = "Control" '' Poner titulos

j = 1

While (j <= 3)

Worksheets("Error_Results").Cells(2, j + 14) = Worksheets("Error_Results").Cells(1, j + 1)

j = j + 1

Wend

Worksheets("Error_Results").Range("S1") = Worksheets("Error_Results").Range("E1") ''Variable

Worksheets("Error_Results").Range("S2") = "t0"

Worksheets("Error_Results").Range("T2") = "t1"

Stop

Call ERROR_R2 ''Segunda iteracion

Stop

Worksheets("Error_Results").Range("AB2") = "Control" '' Poner titulos

j = 1

While (j <= 2)

Worksheets("Error_Results").Cells(2, j + 28) = Worksheets("Error_Results").Cells(1, j + 1)

j = j + 1

Wend

i = 1

While (i <= 2)

j = 1

Worksheets("Error_Results").Cells(1, 3 ^ (i - 1) + i - 1 + 31) = "t" & i - 1

While (j <= 3) ''Variable

Worksheets("Error_Results").Cells(2, j + ((i - 1) * 3) + 31) = "h" & j + ((i - 1) * 3) - 1

j = j + 1

Wend

i = i + 1

Wend

Call ERROR_R3

Worksheets("Error_Results").Range("AR3") = "Control"

Worksheets("Error_Results").Range("AS3") = "V"

i = 0

While (i <= 18)

Worksheets("Error_Results").Cells(3, 45 + i) = "B" & i

i = i + 1


98

Wend

Call ERROR_R4

Stop

'' Comprobar Errores

Dim i_max As Integer

'' Reg1

i_max = WorksheetFunction.Count(Worksheets("Error_Results").Range("B2:B200"))

i = 1

j = 1

z = 3

While (i <= i_max)

If (Worksheets("Error_Force").Cells(i + 1, 1) = Worksheets("Error_Results").Cells(z, 15) And

Worksheets("Error_Force").Cells(i + 1, 2) = Worksheets("Error_Results").Cells(z, 16) And

Worksheets("Error_Force").Cells(i + 1, 3) = Worksheets("Error_Results").Cells(z, 17)) Then

Worksheets("Error_Force").Cells(i + 1, 7) = Worksheets("Error_Results").Cells(z, 19) * Worksheets("Error_Force").Cells(i

+ 1, 4) + Worksheets("Error_Results").Cells(z, 20)

i = i + 1

Else

z = z + 1

End If

Wend

'' Reg2

i = 1

z = 3

While (i <= i_max)

If (Worksheets("Error_Force").Cells(i + 1, 1) = Worksheets("Error_Results").Cells(z, 29) And

Worksheets("Error_Force").Cells(i + 1, 2) = Worksheets("Error_Results").Cells(z, 30)) Then

j = 1

While (j <= 6)

Worksheets("Error_Force").Cells(12, j + 13) = Worksheets("Error_Results").Cells(z, 31 + j)

j = j + 1

Wend

j = 1

While (j <= 4)

Worksheets("Error_Force").Cells(2, j + 13) = Worksheets("Error_Force").Cells(i + 1, j)

j = j + 1

Wend

Worksheets("Error_Force").Cells(i + 1, 8) = Worksheets("Error_Force").Range("M4")

i = i + 1

Else

z = z + 1

End If

Wend

'' Reg3

End Sub

Sub ERROR_R1()

Dim i As Integer


Dim minC_R As Integer

Dim i_erase As Integer

Dim j As Integer

Dim i_first As Integer

Annexes

99

'' Regression TRIM

Workbooks("Global_Streamline.xlsx").Worksheets("Error_F").Activate

j = 3 '' Row donde guardar parametros a,b,c de MC

i = 2 '' Row donde empieza el contador

i_max = WorksheetFunction.Count(ActiveSheet.Range("B2:B200")) ''Length maxima del contador

i_first = 0 ''Parametro de control

minC_R = 2 ''Row donde empieza el contador de MC

While (i <= i_max)

If (i = 2) Then ''Escribir en MC los valores de Error_Results para la primera vez

Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(minC_R, 1) =

Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i, 5)



i = i + 1

minC_R = minC_R + 1

i_first = 1

End If

Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Activate

If (i_first = 0 And i > 2) Then '' Escribir en MC los valores de Error_Results para cada serie nueva


Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i - 1, 5)



minC_R = minC_R + 1

i_first = 1

End If

While (Cells(i - 1, 5) < Cells(i, 5)) ''Escribir en MC los valores de Error_Results





i = i + 1

minC_R = minC_R + 1

Wend

i_erase = minC_R '' Borrar valores anteriores en MC

While (minC_R <= 15)

Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(i_erase, 1).Clear


i_erase = i_erase + 1

minC_R = minC_R + 1

Wend

''Regresion lineal

Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 14) =

Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("C2")






100




Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("M3")


Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("N3")

j = j + 1

i = i + 1

minC_R = 2

i_first = 0

Wend

End Sub

Sub ERROR_R2()

Dim i As Integer




Dim j As Integer


Dim param As Integer

'' Regression SINK


param = 1 '' Numero de Coeficientes (a,b,c,d...)

While (param <= 2) '' Numero de coeficientes a hacer la regresion



i_max = WorksheetFunction.Count(Worksheets("Error_Results").Range("Q2:Q50")) ''Length maxima del contador



While (i <= i_max + 2)





Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i, 18 + param)

i = i + 1

minC_R = minC_R + 1

i_first = 1

End If






Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i - 1, 18 + param)

Annexes

101

minC_R = minC_R + 1

i_first = 1

End If






i = i + 1

minC_R = minC_R + 1

Wend






minC_R = minC_R + 1

Wend

Workbooks("Global_Streamline.xlsx").Worksheets("MC").Activate

''Regresion Parabolica

If (Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("C2") = 2) Then








Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 32 + (3 ^ (param - 1) + param - 1)) =




Else








Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("L5")





End If

j = j + 1

i = i + 1

minC_R = 2

i_first = 0

Wend

param = param + 1


102

Wend

End Sub

Sub ERROR_R3()

Dim i As Integer




Dim j As Integer



'' Regression DEAD


param = 1 '' Numero de Coeficientes (a,b,c,d...)

While (param <= 6) '' Numero de coeficientes a hacer la regresion



i_max = WorksheetFunction.Count(Worksheets("Error_Results").Range("AC2:AC19")) ''Length maxima del contador



While (i <= i_max)






i = i + 1

minC_R = minC_R + 1

i_first = 1

End If






Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i - 1, 31 + param)

minC_R = minC_R + 1

i_first = 1

End If






i = i + 1

minC_R = minC_R + 1

Wend


Annexes

103





minC_R = minC_R + 1

Wend

Workbooks("Global_Streamline.xlsx").Worksheets("MC").Activate

''Regresion

If (Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("C2") = 2) Then










Else





Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 46 + (3 * (param - 1))) =






Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(2, 46 + (3 * (param - 1))) = "h" & param - 1

End If

j = j + 1

i = i + 1

minC_R = 2

i_first = 0

Wend

param = param + 1

Wend

End Sub

Sub ERROR_R4()

Dim i As Integer




Dim j As Integer




104

j = 4

param = 1

While (param <= 18)

i = 1

While (i <= 4)

Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(i + 1, 1) =

Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i + 3, 45)

Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(i + 1, 2) =

Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i + 3, 45 + param)

i = i + 1

Wend







Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(2, 72 + (3 * (param - 1))) = "B" & param - 1

Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(3, 72 + 3 * (param - 1)) = "V" & 3 * (param - 1)


+ 1


+ 2

param = param + 1

Wend

End Sub

Annexes

105

TDYN – Result images

# Macros file for GiD v1.0

# Created by GiD version 11.0

#[_ "results analysissel H_Free_Surface 60 ContourFill Pressure_(Pa) Pressure_(Pa)\n'HardCopy PNG

C:/Users/Rafa/Desktop/Snapshot.png"]

set macrosdata(Photo,Number) 61

set macrosdata(Photo,Icon) {camera.png imported_images camera.png themed_image}

set macrosdata(Photo,InToolbar) 1

set macrosdata(Photo,Description) {results analysissel H_Free_Surface 60 ContourFill Pressure_(Pa) Pressure_(Pa)

'HardCopy PNG C:/Users/Rafa/Desktop/Snapshot.png}

set macrosdata(Photo,Group) {}

set macrosdata(Photo,ModificationDate) {2014-01-28 21:24:56}

set macrosdata(Photo,CreationDate) {2014-01-22 22:56:27}

set macrosdata(Photo,PrePost) prepost

set macrosdata(Photo,Accelerators) {}

set macrosdata(Photo,Active) 1

set macrosdata(Photo,IsStandard) 0

proc Photo {} {

catch {

GiD_Process 'Rotate Angle 270 90

GiD_Process escape escape escape escape escape escape results analysissel H_Free_Surface 2

ContourFill Pressure_(Pa) Pressure_(Pa) 'HardCopy PNG C:/Users/Rafa/FNB/PFC/Savitsky/Images/Pressure.png

GiD_Process escape escape escape escape escape escape results analysissel Free_Surface 2 ContourFill

Total_elevation_(m)

after [expr {1000 * 1}]

GiD_Process 'HardCopy PNG C:/Users/Rafa/FNB/PFC/Savitsky/Images/Total_elevation.png

GiD_Process escape escape escape escape escape escape results analysissel Free_Surface 2 ContourFill

Velocity_(m/s) |Velocity_(m/s)|

after [expr {1000 * 1}]

GiD_Process 'HardCopy PNG C:/Users/Rafa/FNB/PFC/Savitsky/Images/Velocity.png

GiD_Process Mescape Preprocess y

}

}

set ::icon_chooser::ImportedDates(camera.png) {2014-01-22 22:56:27}

image create photo ::icon_chooser::images::camera.png -data {

iVBORw0KGgoAAAANSUhEUgAAABgAAAAYCAYAAAGXcA1uAAAABGdBTUEAAYagMeiWXwAABMJJ

REFUSInNk39MlHUcx1/3CwiQO0DAH3knGHgcBx7tKLTNWrm55Q8cpZLOlMb8UbrWbJXiZptF

OTH7p2XiP1qzgfaHW8mhQknU1DWTVALix5p6eoAIx3MXeHfPpz9OD5jQZltb7+3zx/M83/fn

8/6+P+9HIyI8gB7gzNk66ejuAhHBnmuTY43dohl7TLNp8ybJzrESGxOHtv9uH6nzVlBffzbM

eVAVH30om7dsFBFBIyKsKnlZrl1tISfXgaIoZMw2j2ekpqZKUlLyKGMiaMc+HD9REzmlzcjI

kLP1Z6S8vFx6+nrYsfMdAdAuK1pC5f5Kbnh62Vj2Ou2d18MMVQ2zLzc381ltOx1dXeFeDxQ1

NDSIKTFRXC7XP6vKzMyU6OhoAoEAbW1tmgnVAhw9ekSWLl0qthwby5YtoeApJ06nU06ePCmR

HezY+a6YEo3ExMRSdegQzy58DuMMCwYEX8BN2vRUmn5qoqioKEzw+/10dnXi9Q4hAucvNfPl

exUowyFWv6rjleLFpKb0jUqqqT5BtjWbjPQ59N+5S8fv11j30mK8/gD7dm+no7WFrvtuTHrp

yfDIhIdcAmhsbBS73S52u/3hbmNj86CsVqvUHK+WmppqcTgcMvbbQxOcTqdkZj3BxQsXudzc

jMfjIT8/PzJJ39jYKMXFxVTu38ecORksW74Uxa8wy2zGYDCwcUsZJqOJ6OhoGRkZ0WjMZrPs

+eB9rrW00Nz8G9PTZuAb9pNXsIDr3d3c6XXzZ1cnN2/cwu12a/Qiwt27A3R3d9PW2oZ3cJCU

abNYvXItIVWldN0qhrzKeJcUZfRFVEwsO/Z8iveeFl9Qzyeff0Vy2nRCoVD4DgD9/f2jrqkh

bg8M4x70ATDTqCUYDHIvEAhPqKysJGGKEVt2DmazhdaWVna/uYaZJh0zjFr27iijq72Nmurq

yffgcrnElJgoRpNJDh78Ytwe/vssPSomzN5kcDgccv7CeQEi5XLVitPpnFzlRB5NVunp6fLC

oufHeTZ/QaFYrVaZjPOQRVVVVXLgwAFUVcXhmEdh4XwsFgtTEuJp+rmJH8+dIykpmZiYaOLi

41AUBZstl5TkFIwmI7WnTlFa+hoLFy7UwP0c2Ww2qfi4goQp8Uybnoo9L4dgMMjj5plc+vUX

CgoKEBH6enuJi4tDZ4hCdFHcC6rEx8czPOKn7kwtQ16FBQueoaSkBLfbTWSA3+/nh+8byJiT

QV9fH4WFT3PlylW++/YUWq2W7W+/hWV2OpYsO+lR8dx230QZ8oEqBNQQf92D6zdu0evxkJWZ

Nc4RPYCqqmHPVKG1tRUAr3eI4eFhYmNj0Wh1qOioP13LoSPfkJZsDPur0XCrb4CytUXoDQaM

RhM+ny/ym0UGAKgi9PT0TBgEkymR9o5ODh87hcc7zB+eXoKqikGnJeExPYe/dlG25kWiY2II

73R0r1oAj8dDytSpeIeGxjXOzc0jx2YnKzOTXLud0pWLGFIUBCGkCqoIXkVhb/kWsq1WTEYj

nZ2dzJ07N9IjkqK6ujrZtm0bIyMjBAIBDAYDW7e+gcUyG5/Ph8/no3zXLpYXrWBgcJBQKIhO

p8cQFUX96TrWbyjlyfw8Nqwv1YwV+f/6k/8N/gb2aLeIn0XiZwAAAABJRU5ErkJggg==


106

}

array set preferences {toolbar_one_col 1}

Annexes

Annex B: Sections.

Isometric

Figure

107

Figure 79. Isometric view.


108

Plan

Figure 80. Plan view.

Annexes

109

Elevation

Figure 81. Elevation view.


110

Annex C: Results.

STREAMLINE result table.

CASE Vel Dead Trim Sink Fz_hfs My_hfs Displ Savitsky Fz_hfs % My_hfs %

1 22.1472346 5 2 0.2 17133.6 -67568.9 3607 20039 14.50% -69986 3.45%

2 22.1472346 5 3 0.2 21612.8 -62234.1 2403 24357 11.27% -58516 6.35%

3 22.1472346 5 3 0.3 25898.9 -110429 6361 33223 22.05% -121607 9.19%

4 22.1472346 5 4 0.2 26405.2 -60142.2 1801 28551 7.51% -52063 15.52%

5 22.1472346 5 4 0.3 30268.4 -103441 4767 37772 19.87% -106523 2.89%

6 22.1472346 5 4 0.4 34380.7 -152690 9171 47042 26.92% -176124 13.31%

7 22.1472346 5 5 0.2 30990.2 -58833 1440 32504 4.66% -47708 23.32%

8 22.1472346 5 5 0.3 34866.7 -99390.6 3810 42360 17.69% -96870 2.60%

9 22.1472346 5 5 0.4 38496 -144680 7330 51729 25.58% -158564 8.76%

10 22.1472346 5 6 0.2 35312 -57744.7 1198 36234 2.55% -44489 29.80%

11 22.1472346 5 6 0.3 39377.8 -96484.9 3172 46821 15.90% -89937 7.28%

12 22.1472346 5 6 0.4 42734.985 -138573.8 6102 56532 24.41% -146380 5.33%

13 22.1472346 10 2 0.2 13962.5 -42411.7 2174 15293 8.70% -45386 6.55%

14 22.1472346 10 3 0.2 18127 -40016.2 1448 19178 5.48% -38821 3.08%

15 22.1472346 10 3 0.3 22481.4 -81274.1 4554 27431 18.04% -91046 10.73%

16 22.1472346 10 4 0.2 22301.4 -38908.9 1086 22849 2.40% -35000 11.17%

17 22.1472346 10 4 0.3 26487.8 -76451.7 3413 31700 16.44% -80699 5.26%

18 22.1472346 10 4 0.4 30685.3 -120362 7178 40362 23.98% -140926 14.59%

19 22.1472346 10 5 0.2 26250.2 -38622 868 26287 0.14% -32368 19.32%

20 22.1472346 10 5 0.3 30659.8 -74045.7 2728 35918 14.64% -73982 0.09%

21 22.1472346 10 5 0.4 34556.4 -114170 5737 44836 22.93% -127813 10.67%

22 22.1472346 10 6 0.2 30087.5 -38315.2 722 29526 1.90% -30397 26.05%

23 22.1472346 10 6 0.3 35069.8 -72545.8 2271 39989 12.30% -69107 4.98%

24 22.1472346 10 6 0.4 38473.2 -109981 4775 49350 22.04% -118648 7.30%

25 22.1472346 15 2 0.2 10467.8 -22640.2 1489 10889 3.87% -26147 13.41%

26 22.1472346 15 3 0.2 13964.4 -21722.4 992 14203 1.68% -23090 5.92%

27 22.1472346 15 3 0.3 18940.2 -55835.2 3218 21933 13.64% -64818 13.86%

28 22.1472346 15 4 0.2 17381.1 -21532 744 17284 0.56% -21209 1.52%

29 22.1472346 15 4 0.3 22559 -52988.4 2412 25843 12.71% -58325 9.15%

30 22.1472346 15 4 0.4 26816 -90647 5518 33937 20.98% -109425 17.16%

31 22.1472346 15 5 0.2 20568.9 -21551.3 594 20161 2.02% -19869 8.47%

32 22.1472346 15 5 0.3 26571.8 -51792.8 1928 29646 10.37% -54026 4.13%

33 22.1472346 15 5 0.4 30386.3 -86475.2 4411 38145 20.34% -100145 13.65%

34 22.1472346 15 6 0.2 23589.7 -21554.5 495 22874 3.13% -18845 14.38%

35 22.1472346 15 6 0.3 29925.3 -50588.1 1605 33297 10.13% -50861 0.54%

36 22.1472346 15 6 0.4 34149.6 -83729.8 3671 42336 19.34% -93600 10.54%

Annexes

111

37 22.1472346 20 3 0.3 15099.7 -34674.2 2395 16693 9.55% -42765 18.92%

38 22.1472346 20 4 0.3 18225.6 -33149.6 1795 20166 9.62% -39294 15.64%

39 22.1472346 20 4 0.4 22827 -64870.1 4211 27735 17.70% -81461 20.37%

40 22.1472346 20 5 0.3 21557.5 -32434.8 1435 23507 8.29% -36918 12.14%

41 22.1472346 20 5 0.4 25989.9 -61767 3366 31623 17.81% -75428 18.11%

42 22.1472346 20 6 0.3 24388.1 -32096.1 1194 26705 8.68% -35126 8.63%

43 22.1472346 20 6 0.4 29126.5 -60001.5 2802 35457 17.85% -71118 15.63%

44 17.7177877 5 2 0.2 12336 -45942.1 3607 13586 9.20% -46102 0.35%

45 17.7177877 5 3 0.2 14541.1 -40832.3 2403 16023 9.25% -37948 7.60%

46 17.7177877 5 3 0.3 18570.2 -74755.3 6361 22637 17.97% -80297 6.90%

47 17.7177877 5 4 0.2 17338.5 -38889.8 1801 18564 6.60% -33571 15.84%

48 17.7177877 5 4 0.3 20686.2 -68213.8 4767 25097 17.57% -69419 1.74%

49 17.7177877 5 4 0.4 24523.2 -102847 9171 32133 23.68% -116432 11.67%

50 17.7177877 5 5 0.2 20063.9 -37723.5 1440 21017 4.53% -30680 22.96%

51 17.7177877 5 5 0.3 23247.6 -64706.3 3810 27788 16.34% -62727 3.15%

52 17.7177877 5 5 0.4 26502.4 -95569.6 7330 34593 23.39% -103663 7.81%

53 17.7177877 5 6 0.2 22790.9 -36908 1198 23357 2.42% -28568 29.19%

54 17.7177877 5 6 0.3 25837.4 -62241.6 3172 30492 15.26% -58033 7.25%

55 17.7177877 5 6 0.4 28729.9 -90466.9 6102 37336 23.05% -95097 4.87%

56 17.7177877 10 2 0.2 9778.98 -28458.2 2174 10236 4.46% -29736 4.30%

57 17.7177877 10 3 0.2 12073 -26093 1448 12531 3.66% -25108 3.92%

58 17.7177877 10 3 0.3 15980 -54947 4554 18536 13.79% -59928 8.31%

59 17.7177877 10 4 0.2 14558.8 -25094 1086 14797 1.61% -22532 11.37%

60 17.7177877 10 4 0.3 17988.4 -50338.3 3413 20949 14.13% -52486 4.09%

61 17.7177877 10 4 0.4 21856.5 -81330.5 7178 27408 20.25% -92974 12.52%

62 17.7177877 10 5 0.2 16994.3 -24676.3 868 16952 0.25% -20794 18.67%

63 17.7177877 10 5 0.3 20343.5 -48056.2 2728 23474 13.34% -47842 0.45%

64 17.7177877 10 5 0.4 23673 -75383.7 5737 29854 20.70% -83439 9.65%

65 17.7177877 10 6 0.2 19313.8 -24463 722 18997 1.67% -19505 25.42%

66 17.7177877 10 6 0.3 22916.9 -46729.8 2271 25973 11.77% -44548 4.90%

67 17.7177877 10 6 0.4 25835.6 -71849.8 4775 32487 20.47% -76999 6.69%

68 17.7177877 15 2 0.2 7049.12 -15054.3 1489 7199 2.08% -17042 11.66%

69 17.7177877 15 3 0.2 9183.41 -14457 992 9224 0.44% -14895 2.94%

70 17.7177877 15 3 0.3 13333.7 -37614.6 3218 14700 9.29% -42533 11.56%

71 17.7177877 15 4 0.2 11196.3 -13832.6 744 11153 0.39% -13633 1.46%

72 17.7177877 15 4 0.3 15224.7 -34821.7 2412 16988 10.38% -37860 8.02%

73 17.7177877 15 4 0.4 18989.6 -61275.6 5518 22909 17.11% -72049 14.95%

74 17.7177877 15 5 0.2 13170 -13753.4 594 12971 1.54% -12751 7.86%

75 17.7177877 15 5 0.3 17391.5 -33453.3 1928 19306 9.92% -34890 4.12%

76 17.7177877 15 5 0.4 20755.9 -57138 4411 25290 17.93% -65284 12.48%


112

77 17.7177877 15 6 0.2 15009.3 -13686.7 495 14693 2.15% -12084 13.27%

78 17.7177877 15 6 0.3 19509.6 -32590.8 1605 21571 9.55% -32755 0.50%

79 17.7177877 15 6 0.4 22801.5 -54611.2 3671 27781 17.92% -60678 10.00%

80 17.7177877 20 3 0.3 10384.2 -23192 2395 11098 6.43% -27976 17.10%

81 17.7177877 20 4 0.3 12129.3 -21743.9 1795 13189 8.03% -25456 14.58%

82 17.7177877 20 4 0.4 16070 -43748.8 4211 18611 13.65% -53532 18.27%

83 17.7177877 20 5 0.3 13925.9 -21023.2 1435 15255 8.71% -23810 11.70%

84 17.7177877 20 5 0.4 17707 -40854.8 3366 20877 15.18% -49100 16.79%

85 17.7177877 20 6 0.3 15737.3 -20522.1 1194 17257 8.80% -22599 9.19%

86 17.7177877 20 6 0.4 19408.6 -39037.3 2802 23193 16.32% -46054 15.23%

87 13.2883408 5 2 0.2 8742.02 -29455.3 3607 8570 2.01% -27520 7.03%

88 13.2883408 5 3 0.2 9183.03 -24400 2403 9541 3.75% -21950 11.16%

89 13.2883408 5 3 0.3 13123.1 -47576 6361 14408 8.92% -48159 1.21%

90 13.2883408 5 4 0.2 10355.8 -22491.7 1801 10797 4.09% -19188 17.22%

91 13.2883408 5 4 0.3 13428.6 -41274.9 4767 15239 11.88% -40556 1.77%

92 13.2883408 5 4 0.4 17336.2 -65288.4 9171 20542 15.60% -69994 6.72%

93 13.2883408 5 5 0.2 11640.8 -21431.7 1440 12083 3.66% -17437 22.91%

94 13.2883408 5 5 0.3 14282.2 -37850.5 3810 16454 13.20% -36171 4.64%

95 13.2883408 5 5 0.4 17492.3 -57961.3 7330 21268 17.75% -60958 4.92%

96 13.2883408 5 6 0.2 12987.7 -20739.6 1198 13341 2.65% -16185 28.14%

97 13.2883408 5 6 0.3 15376.5 -35760 3172 17792 13.58% -33218 7.65%

98 13.2883408 5 6 0.4 18036 -53406.8 6102 22407 19.51% -55208 3.26%

99 13.2883408 10 2 0.2 6526.03 -17635.8 2174 6304 3.52% -17559 0.44%

100 13.2883408 10 3 0.2 7373.11 -15328.6 1448 7362 0.15% -14442 6.14%

101 13.2883408 10 3 0.3 11057.4 -34705.8 4554 11623 4.86% -35716 2.83%

102 13.2883408 10 4 0.2 8552.34 -14386.2 1086 8535 0.20% -12835 12.08%

103 13.2883408 10 4 0.3 11465 -30277.1 3413 12588 8.92% -30539 0.86%

104 13.2883408 10 4 0.4 15173.8 -51327 7178 17339 12.48% -55662 7.79%

105 13.2883408 10 5 0.2 9769.26 -13941.3 868 9692 0.80% -11791 18.24%

106 13.2883408 10 5 0.3 12372.4 -28074.8 2728 13797 10.32% -27509 2.06%

107 13.2883408 10 5 0.4 15387.4 -45618.4 5737 18205 15.48% -48920 6.75%

108 13.2883408 10 6 0.2 10984.7 -13748.4 722 10808 1.64% -11033 24.61%

109 13.2883408 10 6 0.3 13516.6 -26757.7 2271 15071 10.32% -25447 5.15%

110 13.2883408 10 6 0.4 16111.7 -42376.3 4775 19373 16.84% -44602 4.99%

111 13.2883408 15 2 0.2 4421.41 -9219.25 1489 4330 2.11% -9957 7.41%

112 13.2883408 15 3 0.2 5366.12 -8185.02 992 5351 0.28% -8521 3.94%

113 13.2883408 15 3 0.3 8924.48 -23416.9 3218 9078 1.69% -25191 7.04%

114 13.2883408 15 4 0.2 6428.69 -7865.98 744 6384 0.70% -7741 1.62%

115 13.2883408 15 4 0.3 9527.71 -20721.8 2412 10103 5.70% -21940 5.55%

116 13.2883408 15 4 0.4 13036.8 -38610.4 5518 14339 9.08% -42963 10.13%

117 13.2883408 15 5 0.2 7447.51 -7733.06 594 7378 0.94% -7215 7.18%

118 13.2883408 15 5 0.3 10450.7 -19404.8 1928 11264 7.22% -20006 3.00%

119 13.2883408 15 5 0.4 13347.7 -34478.5 4411 15296 12.74% -38163 9.65%

Annexes

113

120 13.2883408 15 6 0.2 8417.94 -7640.52 495 8329 1.07% -6825 11.95%

121 13.2883408 15 6 0.3 11461.3 -18647.6 1605 12450 7.94% -18672 0.13%

122 13.2883408 15 6 0.4 14097 -32149 3671 16462 14.36% -35070 8.33%

123 13.2883408 20 3 0.3 6720.25 -14415.3 2395 6749 0.43% -16468 12.46%

124 13.2883408 20 4 0.3 7407 -12859.1 1795 7763 4.59% -14691 12.47%

125 13.2883408 20 4 0.4 10883.4 -27494.8 4211 11521 5.53% -31795 13.52%

126 13.2883408 20 5 0.3 8261.52 -12135.1 1435 8837 6.51% -13613 10.86%

127 13.2883408 20 5 0.4 11303.1 -24642.8 3366 12521 9.73% -28617 13.89%

128 13.2883408 20 6 0.3 9098.19 -11650.3 1194 9908 8.18% -12856 9.38%

129 13.2883408 20 6 0.4 11955.9 -23034.3 2802 13655 12.45% -26557 13.26%

130 8.85889384 5 2 0.2 6278.51 -17859.1 3607 4993 25.75% -14236 25.45%

131 8.85889384 5 3 0.2 5486.62 -12744.3 2403 4913 11.68% -10520 21.14%

132 8.85889384 5 3 0.3 10007.7 -29719.7 6361 8538 17.21% -25187 17.99%

133 8.85889384 5 4 0.2 5448.29 -10783 1801 5250 3.78% -8914 20.97%

134 8.85889384 5 4 0.3 8732.8 -22619.1 4767 8202 6.47% -19934 13.47%

135 8.85889384 5 4 0.4 13681.5 -41170.7 9171 12274 11.46% -36802 11.87%

136 8.85889384 5 5 0.2 5684.78 -9785.68 1440 5701 0.29% -7976 22.68%

137 8.85889384 5 5 0.3 8143.77 -19080.7 3810 8361 2.60% -17199 10.94%

138 8.85889384 5 5 0.4 11962.8 -32544.7 7330 11756 1.76% -30442 6.91%

139 8.85889384 5 6 0.2 6053.61 -9192.11 1198 6187 2.16% -7341 25.22%

140 8.85889384 5 6 0.3 8075.25 -17069.4 3172 8722 7.42% -15492 10.18%

141 8.85889384 5 6 0.4 10941.8 -27911.3 6102 11747 6.86% -26710 4.50%

142 8.85889384 10 2 0.2 4175.54 -9906.29 2174 3502 19.23% -8850 11.94%

143 8.85889384 10 3 0.2 4016.49 -7576.72 1448 3672 9.40% -6821 11.09%

144 8.85889384 10 3 0.3 7742.03 -20521.7 4554 6697 15.60% -18398 11.54%

145 8.85889384 10 4 0.2 4250.27 -6691.57 1086 4063 4.62% -5908 13.26%

146 8.85889384 10 4 0.3 6967.15 -16000.8 3413 6621 5.22% -14853 7.72%

147 8.85889384 10 4 0.4 11128.1 -30891.5 7178 10165 9.48% -28977 6.61%

148 8.85889384 10 5 0.2 4603.41 -6239.19 868 4506 2.16% -5360 16.39%

149 8.85889384 10 5 0.3 6781.97 -13839.5 2728 6887 1.52% -12983 6.60%

150 8.85889384 10 5 0.4 9892.73 -24913 5737 9894 0.01% -24246 2.75%

151 8.85889384 10 6 0.2 5015.43 -6018.08 722 4959 1.15% -4982 20.81%

152 8.85889384 10 6 0.3 6896.87 -12578.8 2271 7286 5.34% -11802 6.59%

153 8.85889384 10 6 0.4 9389.47 -21679.9 4775 10012 6.21% -21452 1.06%

154 8.85889384 15 2 0.2 2545.45 -5076.24 1489 2285 11.41% -4890 3.80%

155 8.85889384 15 3 0.2 2678.06 -3946.04 992 2586 3.57% -3967 0.53%

156 8.85889384 15 3 0.3 5796.88 -13302.2 3218 5075 14.23% -12783 4.06%

157 8.85889384 15 4 0.2 2994.03 -3564.48 744 2978 0.53% -3532 0.93%

158 8.85889384 15 4 0.3 5473.77 -10656.8 2412 5190 5.47% -10560 0.91%

159 8.85889384 15 4 0.4 9059.85 -22734 5518 8237 9.99% -22150 2.64%


114

160 8.85889384 15 5 0.2 3350.57 -3405.13 594 3384 0.98% -3261 4.43%

161 8.85889384 15 5 0.3 5504.75 -9343.79 1928 5522 0.32% -9371 0.29%

162 8.85889384 15 5 0.4 8263.85 -18534.6 4411 8167 1.18% -18773 1.27%

163 8.85889384 15 6 0.2 3716.13 -3288.38 495 3784 1.79% -3068 7.17%

164 8.85889384 15 6 0.3 5705.1 -8607.94 1605 5936 3.90% -8611 0.04%

165 8.85889384 15 6 0.4 7992.99 -16208.1 3671 8383 4.65% -16769 3.34%

166 8.85889384 20 3 0.3 4089.84 -8145.53 2395 3652 12.00% -8233 1.06%

167 8.85889384 20 4 0.3 4012.94 -6535.32 1795 3892 3.12% -6998 6.61%

168 8.85889384 20 4 0.4 7161.12 -15801.6 4211 6475 10.59% -16235 2.67%

169 8.85889384 20 5 0.3 4157.21 -5790.02 1435 4255 2.29% -6328 8.51%

170 8.85889384 20 5 0.4 6729.07 -13026.8 3366 6563 2.53% -13972 6.76%

171 8.85889384 20 6 0.3 4389.18 -5348.21 1194 4660 5.82% -5895 9.28%

172 8.85889384 20 6 0.4 6613.38 -11478.9 2802 6848 3.43% -12623 9.06%

PROMEDIO: 9.36% 9.61%

MAXIMOS: 26.92% 29.80%

Table 41. STREAMLINE result table.

Annexes

115

FEM result table.

CASE Vel Dead Trim Sink Fz_hfs My_hfs Displ Savitsky Fz_hfs % My_hfs %

1 22.1472346 5 2 0.2 12655.4 -45979 3491 20039 36.85% -69986 34.30%

2 22.1472346 5 3 0.2 17000 -43000 2326 24357 30.21% -58516 26.52%

3 22.1472346 5 3 0.3 20295 -78791.7 6238 33223 38.91% -121607 35.21%

4 22.1472346 5 4 0.2 18344.5 -38501.4 1743 28551 35.75% -52063 26.05%

5 22.1472346 5 4 0.3 21821.5 -69612.3 4675 37772 42.23% -106523 34.65%

6 22.1472346 5 4 0.4 26775.5 -108272 9059 47042 43.08% -176124 38.53%

7 22.1472346 5 5 0.2 21137.1 -37047.5 1393 32504 34.97% -47708 22.35%

8 22.1472346 5 5 0.3 24616.2 -65446.3 3737 42360 41.89% -96870 32.44%

9 22.1472346 5 5 0.4 28552.4 -99082.3 7241 51729 44.80% -158564 37.51%

10 22.1472346 5 6 0.2 23671.4 -35531.4 1160 36234 34.67% -44489 20.13%

11 22.1472346 5 6 0.3 27297.7 -62584.1 3110 46821 41.70% -89937 30.41%

12 22.1472346 5 6 0.4 30645.8 -92411.4 6027 56532 45.79% -146380 36.87%

13 22.1472346 10 2 0.2 10253.8 -29352.1 2153 15293 32.95% -45386 35.33%

14 22.1472346 10 3 0.2 13006.8 -27067.5 1434 19178 32.18% -38821 30.28%

15 22.1472346 10 3 0.3 17100.2 -58152.1 4385 27431 37.66% -91046 36.13%

16 22.1472346 10 4 0.2 15477.4 -25308.6 1075 22849 32.26% -35000 27.69%

17 22.1472346 10 4 0.3 19706.5 -53214.8 3286 31700 37.83% -80699 34.06%

18 22.1472346 10 4 0.4 24009.4 -86651 6945 40362 40.52% -140926 38.51%

19 22.1472346 10 5 0.2 17777.2 -24038.6 859 26287 32.37% -32368 25.73%

20 22.1472346 10 5 0.3 22324.3 -50429 2626 35918 37.85% -73982 31.84%

21 22.1472346 10 5 0.4 25799.5 -79919 5551 44836 42.46% -127813 37.47%

22 22.1472346 10 6 0.2 20053.4 -23376.5 715 29526 32.08% -30397 23.10%

23 22.1472346 10 6 0.3 24869.8 -48361.4 2186 39989 37.81% -69107 30.02%

24 22.1472346 10 6 0.4 28628.5 -77084.8 4620 49350 41.99% -118648 35.03%

25 22.1472346 15 2 0.2 7325.67 -16067.6 1091 10889 32.73% -26147 38.55%

26 22.1472346 15 3 0.2 9316.23 -14512.4 727 14203 34.41% -23090 37.15%

27 22.1472346 15 3 0.3 14729.2 -41210.4 3192 21933 32.84% -64818 36.42%

28 22.1472346 15 4 0.2 11346.2 -14037.8 545 17284 34.35% -21209 33.81%

29 22.1472346 15 4 0.3 17026.4 -37682.8 2392 25843 34.12% -58325 35.39%

30 22.1472346 15 4 0.4 21066 -67346.4 5331 33937 37.93% -109425 38.45%

31 22.1472346 15 5 0.2 12866.26 -13277.1844 436 20161 36.18% -19869 33.18%

32 22.1472346 15 5 0.3 19376.023 -35951.6541 1912 29646 34.64% -54026 33.46%

33 22.1472346 15 5 0.4 23277.1 -62888.8 4261 38145 38.98% -100145 37.20%

34 22.1472346 15 6 0.2 14572.4 -12819.7 363 22874 36.29% -18845 31.97%

35 22.1472346 15 6 0.3 21564.5 -34058 1592 33297 35.24% -50861 33.04%

36 22.1472346 15 6 0.4 25546.5803 -59108.0377 3547 42336 39.66% -93600 36.85%

37 22.1472346 20 3 0.3 11574.1623 -26154.9639 2203 16693 30.67% -42765 38.84%


116

38 22.1472346 20 4 0.3 13537.7016 -24108.682 1651 20166 32.87% -39294 38.65%

39 22.1472346 20 4 0.4 18388.1787 -49738.759 4185 27735 33.70% -81461 38.94%

40 22.1472346 20 5 0.3 15453.6689 -23002.582 1319 23507 34.26% -36918 37.69%

41 22.1472346 20 5 0.4 20335.123 -46252.4082 3345 31623 35.70% -75428 38.68%

42 22.1472346 20 6 0.3 17217.7891 -22031.7594 1098 26705 35.53% -35126 37.28%

43 22.1472346 20 6 0.4 22431.3723 -43828.9802 2785 35457 36.74% -71118 38.37%

44 17.7177877 5 2 0.2 9427.54505 -31681.2089 3491 13586 30.61% -46102 31.28%

45 17.7177877 5 3 0.2 10596.0881 -27253.1257 2326 16023 33.87% -37948 28.18%

46 17.7177877 5 3 0.3 14655.0842 -53143.2713 6238 22637 35.26% -80297 33.82%

47 17.7177877 5 4 0.2 12048.1723 -24853.3099 1743 18564 35.10% -33571 25.97%

48 17.7177877 5 4 0.3 15015.2733 -45674.7089 4675 25097 40.17% -69419 34.20%

49 17.7177877 5 4 0.4 19789.4178 -74178.3406 9059 32133 38.41% -116432 36.29%

50 17.7177877 5 5 0.2 13730.4495 -23782.9317 1393 21017 34.67% -30680 22.48%

51 17.7177877 5 5 0.3 16399.9564 -42517.7564 3737 27788 40.98% -62727 32.22%

52 17.7177877 5 5 0.4 20096.7485 -65525.9525 7241 34593 41.91% -103663 36.79%

53 17.7177877 5 6 0.2 15279.5911 -22740.2277 1160 23357 34.58% -28568 20.40%

54 17.7177877 5 6 0.3 18035.9832 -40357.9545 3110 30492 40.85% -58033 30.46%

55 17.7177877 5 6 0.4 21021.0099 -60956.2129 6027 37336 43.70% -95097 35.90%

56 17.7177877 10 2 0.2 7213.37752 -19915.3485 2153 10236 29.53% -29736 33.03%

57 17.7177877 10 3 0.2 8545.93891 -17371.9851 1434 12531 31.80% -25108 30.81%

58 17.7177877 10 3 0.3 12133.3168 -38348.4396 4385 18536 34.54% -59928 36.01%

59 17.7177877 10 4 0.2 10100.5373 -16326.9257 1075 14797 31.74% -22532 27.54%

60 17.7177877 10 4 0.3 13351.3891 -34746.0584 3286 20949 36.27% -52486 33.80%

61 17.7177877 10 4 0.4 17481.3455 -59106.3099 6945 27408 36.22% -92974 36.43%

62 17.7177877 10 5 0.2 11499.4129 -15337.5545 859 16952 32.17% -20794 26.24%

63 17.7177877 10 5 0.3 14811.5861 -32633.6713 2626 23474 36.90% -47842 31.79%

64 17.7177877 10 5 0.4 17584.7772 -52311.7693 5551 29854 41.10% -83439 37.31%

65 17.7177877 10 6 0.2 12885.4 -14948.8465 715 18997 32.17% -19505 23.36%

66 17.7177877 10 6 0.3 16278.3139 -31161.2574 2186 25973 37.33% -44548 30.05%

67 17.7177877 10 6 0.4 18983.4149 -49227.1911 4620 32487 41.57% -76999 36.07%

68 17.7177877 15 2 0.2 5078.77545 -10864.704 1091 7199 29.45% -17042 36.25%

69 17.7177877 15 3 0.2 6164.17178 -9515.39733 727 9224 33.17% -14895 36.12%

70 17.7177877 15 3 0.3 10153.1 -27493.4188 3192 14700 30.93% -42533 35.36%

71 17.7177877 15 4 0.2 7262.81158 -8844.77079 545 11153 34.88% -13633 35.12%

72 17.7177877 15 4 0.3 11382.5168 -24761.3119 2392 16988 33.00% -37860 34.60%

73 17.7177877 15 4 0.4 14790.1733 -45177.6465 5331 22909 35.44% -72049 37.30%

74 17.7177877 15 5 0.2 8296.15752 -8537.08752 436 12971 36.04% -12751 33.05%

75 17.7177877 15 5 0.3 12620.7743 -23178.5644 1912 19306 34.63% -34890 33.57%

76 17.7177877 15 5 0.4 15996.9545 -41715.8713 4261 25290 36.75% -65284 36.10%

77 17.7177877 15 6 0.2 9218.6995 -8130.25248 363 14693 37.26% -12084 32.72%

78 17.7177877 15 6 0.3 14023.0911 -21860.496 1592 21571 34.99% -32755 33.26%

79 17.7177877 15 6 0.4 17183.6198 -38704.8119 3547 27781 38.15% -60678 36.21%

80 17.7177877 20 3 0.3 8023.77495 -17692.6743 2203 11098 27.70% -27976 36.76%

Annexes

117

81 17.7177877 20 4 0.3 9091.04317 -15972.5653 1651 13189 31.07% -25456 37.25%

82 17.7177877 20 4 0.4 12859.9129 -33589.8644 4185 18611 30.90% -53532 37.25%

83 17.7177877 20 5 0.3 10168.1874 -14996.8356 1319 15255 33.34% -23810 37.01%

84 17.7177877 20 5 0.4 13837.9158 -30669.9891 3345 20877 33.72% -49100 37.54%

85 17.7177877 20 6 0.3 11219.0634 -14263.9277 1098 17257 34.99% -22599 36.88%

86 17.7177877 20 6 0.4 14977.4723 -28683.4901 2785 23193 35.42% -46054 37.72%

87 13.2883408 5 2 0.2 6865.81634 -20719.0366 3491 8570 19.88% -27520 24.71%

88 13.2883408 5 3 0.2 6662.74475 -16097.3772 2326 9541 30.17% -21950 26.66%

89 13.2883408 5 3 0.3 10514.1713 -33143.503 6238 14408 27.02% -48159 31.18%

90 13.2883408 5 4 0.2 6989.47248 -13838.0485 1743 10797 35.26% -19188 27.88%

91 13.2883408 5 4 0.3 10137.9416 -26719.6257 4675 15239 33.47% -40556 34.12%

92 13.2883408 5 4 0.4 17275 -51870 9059 20542 15.90% -69994 25.89%

93 13.2883408 5 5 0.2 7824.50436 -12971.3059 1393 12083 35.24% -17437 25.61%

94 13.2883408 5 5 0.3 9472.84713 -23062.5119 3737 16454 42.43% -36171 36.24%

95 13.2883408 5 5 0.4 13244.903 -39446.8788 7241 21268 37.72% -60958 35.29%

96 13.2883408 5 6 0.2 8723.56139 -12726.8218 1160 13341 34.61% -16185 21.37%

97 13.2883408 5 6 0.3 10534.2149 -22490.8129 3110 17792 40.79% -33218 32.29%

98 13.2883408 5 6 0.4 13170.7525 -35245.8653 6027 22407 41.22% -55208 36.16%

99 13.2883408 10 2 0.2 4924.08257 -12503.5822 2153 6304 21.89% -17559 28.79%

100 13.2883408 10 3 0.2 5200.17881 -10165.2772 1434 7362 29.36% -14442 29.61%

101 13.2883408 10 3 0.3 8815.33069 -25102.5812 4385 11623 24.16% -35716 29.72%

102 13.2883408 10 4 0.2 5912.96257 -9303.35634 1075 8535 30.72% -12835 27.52%

103 13.2883408 10 4 0.3 8364.54624 -20130.9089 3286 12588 33.55% -30539 34.08%

104 13.2883408 10 4 0.4 12327.2802 -36067.9485 6945 17339 28.90% -55662 35.20%

105 13.2883408 10 5 0.2 6619.97941 -8680.65178 859 9692 31.69% -11791 26.38%

106 13.2883408 10 5 0.3 8813.41267 -18439.5069 2626 13797 36.12% -27509 32.97%

107 13.2883408 10 5 0.4 11168.8505 -30130.905 5551 18205 38.65% -48920 38.41%

108 13.2883408 10 6 0.2 7296.34267 -8325.40366 715 10808 32.49% -11033 24.54%

109 13.2883408 10 6 0.3 9629.92772 -17752.5416 2186 15071 36.10% -25447 30.24%

110 13.2883408 10 6 0.4 11857.9812 -28788.802 4620 19373 38.79% -44602 35.45%

111 13.2883408 15 2 0.2 3310.38554 -6863.48871 1091 4330 23.55% -9957 31.07%

112 13.2883408 15 3 0.2 3667.23792 -5547.69 727 5351 31.47% -8521 34.89%

113 13.2883408 15 3 0.3 6927.9901 -17242.3386 3192 9078 23.69% -25191 31.55%

114 13.2883408 15 4 0.2 4202.80069 -5064.59356 545 6384 34.17% -7741 34.57%

115 13.2883408 15 4 0.3 7066.87673 -14722.7515 2392 10103 30.05% -21940 32.89%

116 13.2883408 15 4 0.4 10498.4891 -28929.8416 5331 14339 26.78% -42963 32.66%

117 13.2883408 15 5 0.2 4748.38446 -4842.99881 436 7378 35.64% -7215 32.88%

118 13.2883408 15 5 0.3 7584.90693 -13333.203 1912 11264 32.66% -20006 33.35%

119 13.2883408 15 5 0.4 10349.6941 -25007.4218 4261 15296 32.34% -38163 34.47%

120 13.2883408 15 6 0.2 5223.78614 -4583.10663 363 8329 37.28% -6825 32.85%


118

121 13.2883408 15 6 0.3 8150.1697 -12303.396 1592 12450 34.54% -18672 34.11%

122 13.2883408 15 6 0.4 10696.2822 -22905.2545 3547 16462 35.02% -35070 34.69%

123 13.2883408 20 3 0.3 5305.81228 -11185.0554 2203 6749 21.38% -16468 32.08%

124 13.2883408 20 4 0.3 5618.90792 -9621.47198 1651 7763 27.62% -14691 34.51%

125 13.2883408 20 4 0.4 8658.00653 -21169.396 4185 11521 24.85% -31795 33.42%

126 13.2883408 20 5 0.3 6029.3002 -8681.15703 1319 8837 31.77% -13613 36.23%

127 13.2883408 20 5 0.4 8822.1502 -18546.0861 3345 12521 29.54% -28617 35.19%

128 13.2883408 20 6 0.3 6571.28337 -8161.03545 1098 9908 33.68% -12856 36.52%

129 13.2883408 20 6 0.4 9095.39851 -16719.8545 2785 13655 33.39% -26557 37.04%

130 8.85889384 5 2 0.2 5087.33154 -12820.6211 3491 4993 1.90% -14236 9.94%

131 8.85889384 5 3 0.2 4144.45577 -8468.69332 2326 4913 15.64% -10520 19.50%

132 8.85889384 5 3 0.3 8510.68584 -22306.3414 6238 8538 0.32% -25187 11.44%

133 8.85889384 5 4 0.2 3867.79515 -6837.01626 1743 5250 26.32% -8914 23.30%

134 8.85889384 5 4 0.3 7036.32317 -15801.3869 4675 8202 14.21% -19934 20.73%

135 8.85889384 5 4 0.4 11857.0366 -30942.1182 9059 12274 3.40% -36802 15.92%

136 8.85889384 5 5 0.2 3928.24178 -6013.53939 1393 5701 31.10% -7976 24.61%

137 8.85889384 5 5 0.3 6354.28703 -12777.799 3737 8361 24.00% -17199 25.71%

138 8.85889384 5 5 0.4 9267.48099 -22499.7172 7241 11756 21.17% -30442 26.09%

139 8.85889384 5 6 0.2 4033.78069 -5531.37333 1160 6187 34.80% -7341 24.65%

140 8.85889384 5 6 0.3 6143.37416 -11377.1646 3110 8722 29.56% -15492 26.56%

141 8.85889384 5 6 0.4 9294.28 -20325.7 6027 11747 20.88% -26710 23.90%

142 8.85889384 10 2 0.2 3223.61574 -7007.20838 2153 3502 7.95% -8850 20.82%

143 8.85889384 10 3 0.2 2896.38089 -5017.02939 1434 3672 21.11% -6821 26.44%

144 8.85889384 10 3 0.3 6334.35495 -15070.3414 4385 6697 5.42% -18398 18.09%

145 8.85889384 10 4 0.2 2918.70485 -4237.08343 1075 4063 28.16% -5908 28.28%

146 8.85889384 10 4 0.3 5473.2404 -11178.0758 3286 6621 17.34% -14853 24.74%

147 8.85889384 10 4 0.4 9694.6495 -23540.5333 6945 10165 4.63% -28977 18.76%

148 8.85889384 10 5 0.2 3082.70386 -3841.10909 859 4506 31.59% -5360 28.34%

149 8.85889384 10 5 0.3 5122.41455 -9351.44313 2626 6887 25.62% -12983 27.97%

150 8.85889384 10 5 0.4 8096.18455 -17953.7444 5551 9894 18.17% -24246 25.95%

151 8.85889384 10 6 0.2 3301.11139 -3612.22434 715 4959 33.43% -4982 27.49%

152 8.85889384 10 6 0.3 5026.85446 -8282.77162 2186 7286 31.01% -11802 29.82%

153 8.85889384 10 6 0.4 7593.20604 -15195.5485 4620 10012 24.16% -21452 29.16%

154 8.85889384 15 2 0.2 2031.1999 -3901.13879 1091 2285 11.10% -4890 20.23%

155 8.85889384 15 3 0.2 1906.94356 -2744.28859 727 2586 26.25% -3967 30.82%

156 8.85889384 15 3 0.3 4664.04267 -9965.19515 3192 5075 8.09% -12783 22.05%

157 8.85889384 15 4 0.2 2033.1696 -2361.15535 545 2978 31.73% -3532 33.14%

158 8.85889384 15 4 0.3 4170.02792 -7572.17434 2392 5190 19.65% -10560 28.30%

159 8.85889384 15 4 0.4 7661.87228 -17325.2414 5331 8237 6.98% -22150 21.78%

160 8.85889384 15 5 0.2 2141.7002 -2123.07687 436 3384 36.71% -3261 34.89%

161 8.85889384 15 5 0.3 4042.5302 -6425.29939 1912 5522 26.79% -9371 31.43%

162 8.85889384 15 5 0.4 6719.35762 -13720.9152 4261 8167 17.73% -18773 26.91%

163 8.85889384 15 6 0.2 2336.90723 -1966.20859 363 3784 38.24% -3068 35.92%

Annexes

119

164 8.85889384 15 6 0.3 4094.0803 -5773.33596 1592 5936 31.03% -8611 32.95%

165 8.85889384 15 6 0.4 6356.05792 -11764.7788 3547 8383 24.17% -16769 29.84%

166 8.85889384 20 3 0.3 3309.42535 -6422.69869 2203 3652 9.37% -8233 21.99%

167 8.85889384 20 4 0.3 3091.31921 -4868.31879 1651 3892 20.56% -6998 30.43%

168 8.85889384 20 4 0.4 5967.40099 -12394.6202 4185 6475 7.85% -16235 23.66%

169 8.85889384 20 5 0.3 3097.43832 -4183.46667 1319 4255 27.20% -6328 33.89%

170 8.85889384 20 5 0.4 5358.46574 -9864.90374 3345 6563 18.35% -13972 29.39%

171 8.85889384 20 6 0.3 3189.69426 -3759.79152 1098 4660 31.56% -5895 36.22%

172 8.85889384 20 6 0.4 5155.19188 -8636.91 2785 6848 24.72% -12623 31.58%

Table 42. FEM result table.


120

STREAMLINE error table.

V β h τ yi Error yi - Error

22.1472346 5 0.2 2 -14.50% -13.13% 1.36%

22.1472346 5 0.2 3 -11.27% -13.25% -1.98%

22.1472346 5 0.2 4 -22.05% -14.79% 7.26%

22.1472346 5 0.2 5 -7.51% -17.74% -10.23%

22.1472346 5 0.2 6 -19.87% -22.11% -2.24%

22.1472346 5 0.3 3 -26.92% -19.54% 7.38%

22.1472346 5 0.3 4 -4.66% -12.45% -7.80%

22.1472346 5 0.3 5 -17.69% -15.29% 2.40%

22.1472346 5 0.3 6 -25.58% -28.04% -2.46%

22.1472346 5 0.4 4 -2.55% -5.50% -2.95%

22.1472346 5 0.4 5 -15.90% -5.94% 9.95%

22.1472346 5 0.4 6 -24.41% -25.23% -0.83%

22.1472346 10 0.2 2 -8.70% -11.55% -2.85%

22.1472346 10 0.2 3 -5.48% -10.57% -5.09%

22.1472346 10 0.2 4 -18.04% -10.80% 7.24%

22.1472346 10 0.2 5 -2.40% -12.25% -9.85%

22.1472346 10 0.2 6 -16.44% -14.91% 1.54%

22.1472346 10 0.3 3 -23.98% -18.22% 5.76%

22.1472346 10 0.3 4 -0.14% -10.29% -10.15%

22.1472346 10 0.3 5 -14.64% -12.07% 2.57%

22.1472346 10 0.3 6 -22.93% -23.56% -0.64%

22.1472346 10 0.4 4 1.90% -5.16% -7.06%

22.1472346 10 0.4 5 -12.30% -5.00% 7.30%

22.1472346 10 0.4 6 -22.04% -23.49% -1.45%

22.1472346 15 0.2 2 -3.87% -9.97% -6.10%

22.1472346 15 0.2 3 -1.68% -7.89% -6.21%

22.1472346 15 0.2 4 -13.64% -6.82% 6.82%

22.1472346 15 0.2 5 0.56% -6.76% -7.32%

22.1472346 15 0.2 6 -12.71% -7.70% 5.01%

22.1472346 15 0.3 3 -20.98% -16.91% 4.08%

22.1472346 15 0.3 4 2.02% -8.13% -10.15%

22.1472346 15 0.3 5 -10.37% -8.86% 1.51%

22.1472346 15 0.3 6 -20.34% -19.09% 1.25%

22.1472346 15 0.4 4 3.13% -4.82% -7.94%

22.1472346 15 0.4 5 -10.13% -4.06% 6.06%

22.1472346 15 0.4 6 -19.34% -21.75% -2.41%

22.1472346 20 0.3 3 -9.55% -15.59% -6.04%

22.1472346 20 0.3 4 -9.62% -5.96% 3.66%

22.1472346 20 0.3 5 -17.70% -5.64% 12.06%

Annexes

121

22.1472346 20 0.3 6 -8.29% -14.62% -6.32%

22.1472346 20 0.4 4 -17.81% -4.47% 13.34%

22.1472346 20 0.4 5 -8.68% -3.12% 5.55%

22.1472346 20 0.4 6 -17.85% -20.01% -2.15%

17.7177877 5 0.2 2 -9.20% -3.43% 5.77%

17.7177877 5 0.2 3 -9.25% -5.70% 3.55%

17.7177877 5 0.2 4 -17.97% -8.38% 9.59%

17.7177877 5 0.2 5 -6.60% -11.46% -4.86%

17.7177877 5 0.2 6 -17.57% -14.96% 2.61%

17.7177877 5 0.3 3 -23.68% -13.35% 10.34%

17.7177877 5 0.3 4 -4.53% -11.65% -7.11%

17.7177877 5 0.3 5 -16.34% -14.31% 2.02%

17.7177877 5 0.3 6 -23.39% -21.34% 2.04%

17.7177877 5 0.4 4 -2.42% -8.93% -6.50%

17.7177877 5 0.4 5 -15.26% -12.53% 2.74%

17.7177877 5 0.4 6 -23.05% -23.14% -0.09%

17.7177877 10 0.2 2 -4.46% -2.42% 2.04%

17.7177877 10 0.2 3 -3.66% -3.98% -0.33%

17.7177877 10 0.2 4 -13.79% -5.83% 7.97%

17.7177877 10 0.2 5 -1.61% -7.95% -6.34%

17.7177877 10 0.2 6 -14.13% -10.35% 3.78%

17.7177877 10 0.3 3 -20.25% -12.51% 7.75%

17.7177877 10 0.3 4 0.25% -10.26% -10.51%

17.7177877 10 0.3 5 -13.34% -12.26% 1.08%

17.7177877 10 0.3 6 -20.70% -18.48% 2.22%

17.7177877 10 0.4 4 1.67% -8.71% -10.37%

17.7177877 10 0.4 5 -11.77% -11.92% -0.16%

17.7177877 10 0.4 6 -20.47% -22.03% -1.55%

17.7177877 15 0.2 2 -2.08% -1.41% 0.67%

17.7177877 15 0.2 3 -0.44% -2.27% -1.83%

17.7177877 15 0.2 4 -9.29% -3.28% 6.02%

17.7177877 15 0.2 5 0.39% -4.43% -4.82%

17.7177877 15 0.2 6 -10.38% -5.74% 4.64%

17.7177877 15 0.3 3 -17.11% -11.66% 5.44%

17.7177877 15 0.3 4 1.54% -8.88% -10.42%

17.7177877 15 0.3 5 -9.92% -10.20% -0.28%

17.7177877 15 0.3 6 -17.93% -15.62% 2.31%

17.7177877 15 0.4 4 2.15% -8.49% -10.64%

17.7177877 15 0.4 5 -9.55% -11.32% -1.77%

17.7177877 15 0.4 6 -17.92% -20.91% -2.99%


122

17.7177877 20 0.3 3 -6.43% -10.82% -4.39%

17.7177877 20 0.3 4 -8.03% -7.50% 0.54%

17.7177877 20 0.3 5 -13.65% -8.14% 5.51%

17.7177877 20 0.3 6 -8.71% -12.75% -4.04%

17.7177877 20 0.4 4 -15.18% -8.27% 6.91%

17.7177877 20 0.4 5 -8.80% -10.72% -1.92%

17.7177877 20 0.4 6 -16.32% -19.80% -3.48%

13.2883408 5 0.2 2 2.01% 4.53% 2.52%

13.2883408 5 0.2 3 -3.75% 1.61% 5.36%

13.2883408 5 0.2 4 -8.92% -1.41% 7.50%

13.2883408 5 0.2 5 -4.09% -4.53% -0.44%

13.2883408 5 0.2 6 -11.88% -7.74% 4.14%

13.2883408 5 0.3 3 -15.60% -4.99% 10.61%

13.2883408 5 0.3 4 -3.66% -6.59% -2.93%

13.2883408 5 0.3 5 -13.20% -9.27% 3.93%

13.2883408 5 0.3 6 -17.75% -13.04% 4.71%

13.2883408 5 0.4 4 -2.65% -5.53% -2.89%

13.2883408 5 0.4 5 -13.58% -11.18% 2.40%

13.2883408 5 0.4 6 -19.51% -16.70% 2.81%

13.2883408 10 0.2 2 3.52% 5.10% 1.58%

13.2883408 10 0.2 3 0.15% 2.57% 2.42%

13.2883408 10 0.2 4 -4.86% 0.02% 4.89%

13.2883408 10 0.2 5 0.20% -2.55% -2.76%

13.2883408 10 0.2 6 -8.92% -5.15% 3.77%

13.2883408 10 0.3 3 -12.48% -4.52% 7.96%

13.2883408 10 0.3 4 0.80% -5.81% -6.61%

13.2883408 10 0.3 5 -10.32% -8.11% 2.21%

13.2883408 10 0.3 6 -15.48% -11.43% 4.05%

13.2883408 10 0.4 4 1.64% -5.41% -7.05%

13.2883408 10 0.4 5 -10.32% -10.84% -0.53%

13.2883408 10 0.4 6 -16.84% -16.08% 0.76%

13.2883408 15 0.2 2 2.11% 5.67% 3.56%

13.2883408 15 0.2 3 0.28% 3.53% 3.25%

13.2883408 15 0.2 4 -1.69% 1.45% 3.15%

13.2883408 15 0.2 5 0.70% -0.58% -1.27%

13.2883408 15 0.2 6 -5.70% -2.56% 3.14%

13.2883408 15 0.3 3 -9.08% -4.05% 5.03%

13.2883408 15 0.3 4 0.94% -5.03% -5.97%

13.2883408 15 0.3 5 -7.22% -6.95% 0.27%

13.2883408 15 0.3 6 -12.74% -9.82% 2.91%

13.2883408 15 0.4 4 1.07% -5.29% -6.35%

13.2883408 15 0.4 5 -7.94% -10.50% -2.56%

13.2883408 15 0.4 6 -14.36% -15.45% -1.08%

Annexes

123

13.2883408 20 0.3 3 -0.43% -3.57% -3.15%

13.2883408 20 0.3 4 -4.59% -4.25% 0.34%

13.2883408 20 0.3 5 -5.53% -5.80% -0.26%

13.2883408 20 0.3 6 -6.51% -8.21% -1.70%

13.2883408 20 0.4 4 -9.73% -5.16% 4.57%

13.2883408 20 0.4 5 -8.18% -10.17% -1.99%

13.2883408 20 0.4 6 -12.45% -14.82% -2.38%

8.85889384 5 0.2 2 25.75% 10.75% -15.00%

8.85889384 5 0.2 3 11.68% 8.66% -3.02%

8.85889384 5 0.2 4 17.21% 6.10% -11.11%

8.85889384 5 0.2 5 3.78% 3.06% -0.72%

8.85889384 5 0.2 6 6.47% -0.45% -6.92%

8.85889384 5 0.3 3 11.46% 5.52% -5.94%

8.85889384 5 0.3 4 -0.29% 2.74% 3.03%

8.85889384 5 0.3 5 -2.60% -0.15% 2.45%

8.85889384 5 0.3 6 1.76% -3.13% -4.89%

8.85889384 5 0.4 4 -2.16% 4.67% 6.83%

8.85889384 5 0.4 5 -7.42% -1.91% 5.51%

8.85889384 5 0.4 6 -6.86% -5.92% 0.94%

8.85889384 10 0.2 2 19.23% 11.00% -8.23%

8.85889384 10 0.2 3 9.40% 9.09% -0.30%

8.85889384 10 0.2 4 15.60% 6.74% -8.87%

8.85889384 10 0.2 5 4.62% 3.94% -0.68%

8.85889384 10 0.2 6 5.22% 0.70% -4.52%

8.85889384 10 0.3 3 9.48% 5.73% -3.74%

8.85889384 10 0.3 4 2.16% 3.08% 0.92%

8.85889384 10 0.3 5 -1.52% 0.37% 1.88%

8.85889384 10 0.3 6 -0.01% -2.42% -2.40%

8.85889384 10 0.4 4 1.15% 4.73% 3.58%

8.85889384 10 0.4 5 -5.34% -1.76% 3.58%

8.85889384 10 0.4 6 -6.21% -5.64% 0.57%

8.85889384 15 0.2 2 11.41% 11.26% -0.15%

8.85889384 15 0.2 3 3.57% 9.52% 5.95%

8.85889384 15 0.2 4 14.23% 7.37% -6.85%

8.85889384 15 0.2 5 0.53% 4.82% 4.29%

8.85889384 15 0.2 6 5.47% 1.85% -3.62%

8.85889384 15 0.3 3 9.99% 5.94% -4.05%

8.85889384 15 0.3 4 -0.98% 3.43% 4.41%

8.85889384 15 0.3 5 -0.32% 0.88% 1.20%

8.85889384 15 0.3 6 1.18% -1.70% -2.88%


124

8.85889384 15 0.4 4 -1.79% 4.78% 6.58%

8.85889384 15 0.4 5 -3.90% -1.61% 2.29%

8.85889384 15 0.4 6 -4.65% -5.36% -0.72%

8.85889384 20 0.3 3 12.00% 6.15% -5.85%

8.85889384 20 0.3 4 3.12% 3.78% 0.66%

8.85889384 20 0.3 5 10.59% 1.40% -9.19%

8.85889384 20 0.3 6 -2.29% -0.98% 1.31%

8.85889384 20 0.4 4 2.53% 4.84% 2.30%

8.85889384 20 0.4 5 -5.82% -1.46% 4.36%

8.85889384 20 0.4 6 -3.43% -5.08% -1.65%

Table 43. STREAMLINE error table for correlation coefficients over 0.4.

Annexes

125


22.1472346 5 0.2 2 -14.50% -13.73% 0.76%

22.1472346 5 0.2 3 -11.27% -11.45% -0.18%

22.1472346 5 0.2 4 -22.05% -11.20% 10.85%

22.1472346 5 0.2 5 -7.51% -12.98% -5.47%

22.1472346 5 0.2 6 -19.87% -16.81% 3.06%

22.1472346 5 0.3 3 -26.92% -19.49% 7.42%

22.1472346 5 0.3 4 -4.66% -12.10% -7.45%

22.1472346 5 0.3 5 -17.69% -12.80% 4.89%

22.1472346 5 0.3 6 -25.58% -21.58% 4.00%

22.1472346 5 0.4 4 -2.55% -6.70% -4.15%

22.1472346 5 0.4 5 -15.90% -7.79% 8.10%

22.1472346 5 0.4 6 -24.41% -24.25% 0.16%

22.1472346 10 0.2 2 -8.70% -13.73% -5.03%

22.1472346 10 0.2 3 -5.48% -11.45% -5.97%

22.1472346 10 0.2 4 -18.04% -11.20% 6.85%

22.1472346 10 0.2 5 -2.40% -12.98% -10.59%

22.1472346 10 0.2 6 -16.44% -16.81% -0.37%

22.1472346 10 0.3 3 -23.98% -19.49% 4.48%

22.1472346 10 0.3 4 -0.14% -12.10% -11.96%

22.1472346 10 0.3 5 -14.64% -12.80% 1.84%

22.1472346 10 0.3 6 -22.93% -21.58% 1.35%

22.1472346 10 0.4 4 1.90% -6.70% -8.60%

22.1472346 10 0.4 5 -12.30% -7.79% 4.51%

22.1472346 10 0.4 6 -22.04% -24.25% -2.21%

22.1472346 15 0.2 2 -3.87% -13.73% -9.86%

22.1472346 15 0.2 3 -1.68% -11.45% -9.76%

22.1472346 15 0.2 4 -13.64% -11.20% 2.45%

22.1472346 15 0.2 5 0.56% -12.98% -13.55%

22.1472346 15 0.2 6 -12.71% -16.81% -4.10%

22.1472346 15 0.3 3 -20.98% -19.49% 1.49%

22.1472346 15 0.3 4 2.02% -12.10% -14.13%

22.1472346 15 0.3 5 -10.37% -12.80% -2.43%

22.1472346 15 0.3 6 -20.34% -21.58% -1.24%

22.1472346 15 0.4 4 3.13% -6.70% -9.83%

22.1472346 15 0.4 5 -10.13% -7.79% 2.33%

22.1472346 15 0.4 6 -19.34% -24.25% -4.91%

22.1472346 20 0.3 3 -9.55% -19.49% -9.95%

22.1472346 20 0.3 4 -9.62% -12.10% -2.48%

22.1472346 20 0.3 5 -17.70% -12.80% 4.90%


126

22.1472346 20 0.3 6 -8.29% -21.58% -13.29%

22.1472346 20 0.4 4 -17.81% -6.70% 11.12%

22.1472346 20 0.4 5 -8.68% -7.79% 0.88%

22.1472346 20 0.4 6 -17.85% -24.25% -6.40%

17.7177877 5 0.2 2 -9.20% -1.04% 8.16%

17.7177877 5 0.2 3 -9.25% -2.21% 7.04%

17.7177877 5 0.2 4 -17.97% -3.92% 14.04%

17.7177877 5 0.2 5 -6.60% -6.19% 0.41%

17.7177877 5 0.2 6 -17.57% -9.02% 8.55%

17.7177877 5 0.3 3 -23.68% -9.01% 14.67%

17.7177877 5 0.3 4 -4.53% -6.64% -2.10%

17.7177877 5 0.3 5 -16.34% -8.70% 7.64%

17.7177877 5 0.3 6 -23.39% -15.20% 8.18%

17.7177877 5 0.4 4 -2.42% -5.05% -2.63%

17.7177877 5 0.4 5 -15.26% -7.60% 7.67%

17.7177877 5 0.4 6 -23.05% -19.17% 3.88%

17.7177877 10 0.2 2 -4.46% -1.04% 3.42%

17.7177877 10 0.2 3 -3.66% -2.21% 1.45%

17.7177877 10 0.2 4 -13.79% -3.92% 9.87%

17.7177877 10 0.2 5 -1.61% -6.19% -4.58%

17.7177877 10 0.2 6 -14.13% -9.02% 5.11%

17.7177877 10 0.3 3 -20.25% -9.01% 11.24%

17.7177877 10 0.3 4 0.25% -6.64% -6.89%

17.7177877 10 0.3 5 -13.34% -8.70% 4.64%

17.7177877 10 0.3 6 -20.70% -15.20% 5.50%

17.7177877 10 0.4 4 1.67% -5.05% -6.72%

17.7177877 10 0.4 5 -11.77% -7.60% 4.17%

17.7177877 10 0.4 6 -20.47% -19.17% 1.31%

17.7177877 15 0.2 2 -2.08% -1.04% 1.04%

17.7177877 15 0.2 3 -0.44% -2.21% -1.77%

17.7177877 15 0.2 4 -9.29% -3.92% 5.37%

17.7177877 15 0.2 5 0.39% -6.19% -6.59%

17.7177877 15 0.2 6 -10.38% -9.02% 1.36%

17.7177877 15 0.3 3 -17.11% -9.01% 8.10%

17.7177877 15 0.3 4 1.54% -6.64% -8.17%

17.7177877 15 0.3 5 -9.92% -8.70% 1.22%

17.7177877 15 0.3 6 -17.93% -15.20% 2.72%

17.7177877 15 0.4 4 2.15% -5.05% -7.21%

17.7177877 15 0.4 5 -9.55% -7.60% 1.96%

17.7177877 15 0.4 6 -17.92% -19.17% -1.25%

17.7177877 20 0.3 3 -6.43% -9.01% -2.58%

17.7177877 20 0.3 4 -8.03% -6.64% 1.40%

17.7177877 20 0.3 5 -13.65% -8.70% 4.95%

Annexes

127

17.7177877 20 0.3 6 -8.71% -15.20% -6.49%

17.7177877 20 0.4 4 -15.18% -5.05% 10.13%

17.7177877 20 0.4 5 -8.80% -7.60% 1.21%

17.7177877 20 0.4 6 -16.32% -19.17% -2.85%

13.2883408 5 0.2 2 2.01% 6.68% 4.67%

13.2883408 5 0.2 3 -3.75% 3.56% 7.31%

13.2883408 5 0.2 4 -8.92% 0.83% 9.75%

13.2883408 5 0.2 5 -4.09% -1.50% 2.58%

13.2883408 5 0.2 6 -11.88% -3.45% 8.43%

13.2883408 5 0.3 3 -15.60% -1.82% 13.78%

13.2883408 5 0.3 4 -3.66% -2.69% 0.96%

13.2883408 5 0.3 5 -13.20% -5.37% 7.83%

13.2883408 5 0.3 6 -17.75% -9.86% 7.89%

13.2883408 5 0.4 4 -2.65% -3.56% -0.91%

13.2883408 5 0.4 5 -13.58% -6.72% 6.86%

13.2883408 5 0.4 6 -19.51% -14.20% 5.31%

13.2883408 10 0.2 2 3.52% 6.68% 3.16%

13.2883408 10 0.2 3 0.15% 3.56% 3.41%

13.2883408 10 0.2 4 -4.86% 0.83% 5.70%

13.2883408 10 0.2 5 0.20% -1.50% -1.70%

13.2883408 10 0.2 6 -8.92% -3.45% 5.47%

13.2883408 10 0.3 3 -12.48% -1.82% 10.66%

13.2883408 10 0.3 4 0.80% -2.69% -3.49%

13.2883408 10 0.3 5 -10.32% -5.37% 4.95%

13.2883408 10 0.3 6 -15.48% -9.86% 5.62%

13.2883408 10 0.4 4 1.64% -3.56% -5.20%

13.2883408 10 0.4 5 -10.32% -6.72% 3.60%

13.2883408 10 0.4 6 -16.84% -14.20% 2.63%

13.2883408 15 0.2 2 2.11% 6.68% 4.57%

13.2883408 15 0.2 3 0.28% 3.56% 3.28%

13.2883408 15 0.2 4 -1.69% 0.83% 2.53%

13.2883408 15 0.2 5 0.70% -1.50% -2.20%

13.2883408 15 0.2 6 -5.70% -3.45% 2.25%

13.2883408 15 0.3 3 -9.08% -1.82% 7.26%

13.2883408 15 0.3 4 0.94% -2.69% -3.63%

13.2883408 15 0.3 5 -7.22% -5.37% 1.85%

13.2883408 15 0.3 6 -12.74% -9.86% 2.88%

13.2883408 15 0.4 4 1.07% -3.56% -4.62%

13.2883408 15 0.4 5 -7.94% -6.72% 1.22%

13.2883408 15 0.4 6 -14.36% -14.20% 0.16%


128

13.2883408 20 0.3 3 -0.43% -1.82% -1.40%

13.2883408 20 0.3 4 -4.59% -2.69% 1.90%

13.2883408 20 0.3 5 -5.53% -5.37% 0.16%

13.2883408 20 0.3 6 -6.51% -9.86% -3.34%

13.2883408 20 0.4 4 -9.73% -3.56% 6.17%

13.2883408 20 0.4 5 -8.18% -6.72% 1.46%

13.2883408 20 0.4 6 -12.45% -14.20% -1.76%

8.85889384 5 0.2 2 25.75% 9.42% -16.33%

8.85889384 5 0.2 3 11.68% 5.85% -5.83%

8.85889384 5 0.2 4 17.21% 3.07% -14.14%

8.85889384 5 0.2 5 3.78% 1.09% -2.69%

8.85889384 5 0.2 6 6.47% -0.09% -6.56%

8.85889384 5 0.3 3 11.46% 2.08% -9.39%

8.85889384 5 0.3 4 -0.29% -0.27% 0.02%

8.85889384 5 0.3 5 -2.60% -2.81% -0.21%

8.85889384 5 0.3 6 1.76% -5.54% -7.30%

8.85889384 5 0.4 4 -2.16% -2.22% -0.06%

8.85889384 5 0.4 5 -7.42% -5.16% 2.26%

8.85889384 5 0.4 6 -6.86% -9.35% -2.49%

8.85889384 10 0.2 2 19.23% 9.42% -9.81%

8.85889384 10 0.2 3 9.40% 5.85% -3.55%

8.85889384 10 0.2 4 15.60% 3.07% -12.53%

8.85889384 10 0.2 5 4.62% 1.09% -3.52%

8.85889384 10 0.2 6 5.22% -0.09% -5.31%

8.85889384 10 0.3 3 9.48% 2.08% -7.40%

8.85889384 10 0.3 4 2.16% -0.27% -2.43%

8.85889384 10 0.3 5 -1.52% -2.81% -1.29%

8.85889384 10 0.3 6 -0.01% -5.54% -5.53%

8.85889384 10 0.4 4 1.15% -2.22% -3.37%

8.85889384 10 0.4 5 -5.34% -5.16% 0.18%

8.85889384 10 0.4 6 -6.21% -9.35% -3.14%

8.85889384 15 0.2 2 11.41% 9.42% -1.99%

8.85889384 15 0.2 3 3.57% 5.85% 2.28%

8.85889384 15 0.2 4 14.23% 3.07% -11.15%

8.85889384 15 0.2 5 0.53% 1.09% 0.57%

8.85889384 15 0.2 6 5.47% -0.09% -5.56%

8.85889384 15 0.3 3 9.99% 2.08% -7.91%

8.85889384 15 0.3 4 -0.98% -0.27% 0.71%

8.85889384 15 0.3 5 -0.32% -2.81% -2.50%

8.85889384 15 0.3 6 1.18% -5.54% -6.73%

8.85889384 15 0.4 4 -1.79% -2.22% -0.43%

8.85889384 15 0.4 5 -3.90% -5.16% -1.26%

8.85889384 15 0.4 6 -4.65% -9.35% -4.71%

Annexes

129

8.85889384 20 0.3 3 12.00% 2.08% -9.93%

8.85889384 20 0.3 4 3.12% -0.27% -3.39%

8.85889384 20 0.3 5 10.59% -2.81% -13.40%

8.85889384 20 0.3 6 -2.29% -5.54% -3.25%

8.85889384 20 0.4 4 2.53% -2.22% -4.75%

8.85889384 20 0.4 5 -5.82% -5.16% 0.66%

8.85889384 20 0.4 6 -3.43% -9.35% -5.92%

Table 44. STREAMLINE error table for correlation coefficients over 0.5.


130

FEM error table.


22.1472346 5 0.2 2 36.85% 36.48% -0.36%

22.1472346 5 0.2 3 30.21% 34.72% 4.51%

22.1472346 5 0.2 4 38.91% 34.59% -4.32%

22.1472346 5 0.2 5 35.75% 36.09% 0.35%

22.1472346 5 0.2 6 42.23% 39.23% -3.00%

22.1472346 5 0.3 3 43.08% 39.21% -3.88%

22.1472346 5 0.3 4 34.97% 36.15% 1.18%

22.1472346 5 0.3 5 41.89% 36.53% -5.35%

22.1472346 5 0.3 6 44.80% 40.36% -4.45%

22.1472346 5 0.4 4 34.67% 33.90% -0.77%

22.1472346 5 0.4 5 41.70% 34.98% -6.71%

22.1472346 5 0.4 6 45.79% 42.42% -3.37%

22.1472346 10 0.2 2 32.95% 36.48% 3.53%

22.1472346 10 0.2 3 32.18% 34.72% 2.54%

22.1472346 10 0.2 4 37.66% 34.59% -3.07%

22.1472346 10 0.2 5 32.26% 36.09% 3.83%

22.1472346 10 0.2 6 37.83% 39.23% 1.40%

22.1472346 10 0.3 3 40.52% 39.21% -1.31%

22.1472346 10 0.3 4 32.37% 36.15% 3.78%

22.1472346 10 0.3 5 37.85% 36.53% -1.31%

22.1472346 10 0.3 6 42.46% 40.36% -2.10%

22.1472346 10 0.4 4 32.08% 33.90% 1.82%

22.1472346 10 0.4 5 37.81% 34.98% -2.82%

22.1472346 10 0.4 6 41.99% 42.42% 0.43%

22.1472346 15 0.2 2 32.73% 36.48% 3.76%

22.1472346 15 0.2 3 34.41% 34.72% 0.31%

22.1472346 15 0.2 4 32.84% 34.59% 1.74%

22.1472346 15 0.2 5 34.35% 36.09% 1.74%

22.1472346 15 0.2 6 34.12% 39.23% 5.12%

22.1472346 15 0.3 3 37.93% 39.21% 1.28%

22.1472346 15 0.3 4 36.18% 36.15% -0.03%

22.1472346 15 0.3 5 34.64% 36.53% 1.89%

22.1472346 15 0.3 6 38.98% 40.36% 1.38%

22.1472346 15 0.4 4 36.29% 33.90% -2.39%

22.1472346 15 0.4 5 35.24% 34.98% -0.25%

22.1472346 15 0.4 6 39.66% 42.42% 2.76%

22.1472346 20 0.3 3 30.67% 39.21% 8.54%

22.1472346 20 0.3 4 32.87% 36.15% 3.28%

22.1472346 20 0.3 5 33.70% 36.53% 2.83%

Annexes

131

22.1472346 20 0.3 6 34.26% 40.36% 6.10%

22.1472346 20 0.4 4 35.70% 33.90% -1.79%

22.1472346 20 0.4 5 35.53% 34.98% -0.54%

22.1472346 20 0.4 6 36.74% 42.42% 5.69%

17.7177877 5 0.2 2 30.61% 27.64% -2.97%

17.7177877 5 0.2 3 33.87% 30.26% -3.61%

17.7177877 5 0.2 4 35.26% 32.42% -2.84%

17.7177877 5 0.2 5 35.10% 34.13% -0.97%

17.7177877 5 0.2 6 40.17% 35.38% -4.79%

17.7177877 5 0.3 3 38.41% 30.49% -7.92%

17.7177877 5 0.3 4 34.67% 34.35% -0.32%

17.7177877 5 0.3 5 40.98% 36.76% -4.22%

17.7177877 5 0.3 6 41.91% 37.74% -4.17%

17.7177877 5 0.4 4 34.58% 36.45% 1.87%

17.7177877 5 0.4 5 40.85% 39.25% -1.60%

17.7177877 5 0.4 6 43.70% 39.48% -4.21%

17.7177877 10 0.2 2 29.53% 27.64% -1.89%

17.7177877 10 0.2 3 31.80% 30.26% -1.55%

17.7177877 10 0.2 4 34.54% 32.42% -2.13%

17.7177877 10 0.2 5 31.74% 34.13% 2.39%

17.7177877 10 0.2 6 36.27% 35.38% -0.88%

17.7177877 10 0.3 3 36.22% 30.49% -5.72%

17.7177877 10 0.3 4 32.17% 34.35% 2.18%

17.7177877 10 0.3 5 36.90% 36.76% -0.14%

17.7177877 10 0.3 6 41.10% 37.74% -3.36%

17.7177877 10 0.4 4 32.17% 36.45% 4.28%

17.7177877 10 0.4 5 37.33% 39.25% 1.93%

17.7177877 10 0.4 6 41.57% 39.48% -2.08%

17.7177877 15 0.2 2 29.45% 27.64% -1.81%

17.7177877 15 0.2 3 33.17% 30.26% -2.92%

17.7177877 15 0.2 4 30.93% 32.42% 1.49%

17.7177877 15 0.2 5 34.88% 34.13% -0.75%

17.7177877 15 0.2 6 33.00% 35.38% 2.38%

17.7177877 15 0.3 3 35.44% 30.49% -4.95%

17.7177877 15 0.3 4 36.04% 34.35% -1.69%

17.7177877 15 0.3 5 34.63% 36.76% 2.13%

17.7177877 15 0.3 6 36.75% 37.74% 0.99%

17.7177877 15 0.4 4 37.26% 36.45% -0.81%

17.7177877 15 0.4 5 34.99% 39.25% 4.26%

17.7177877 15 0.4 6 38.15% 39.48% 1.34%


132

17.7177877 20 0.3 3 27.70% 30.49% 2.79%

17.7177877 20 0.3 4 31.07% 34.35% 3.28%

17.7177877 20 0.3 5 30.90% 36.76% 5.86%

17.7177877 20 0.3 6 33.34% 37.74% 4.39%

17.7177877 20 0.4 4 33.72% 36.45% 2.73%

17.7177877 20 0.4 5 34.99% 39.25% 4.26%

17.7177877 20 0.4 6 35.42% 39.48% 4.06%

13.2883408 5 0.2 2 19.88% 19.57% -0.32%

13.2883408 5 0.2 3 30.17% 24.55% -5.62%

13.2883408 5 0.2 4 27.02% 27.87% 0.85%

13.2883408 5 0.2 5 35.26% 29.53% -5.73%

13.2883408 5 0.2 6 33.47% 29.53% -3.94%

13.2883408 5 0.3 3 15.90% 22.22% 6.31%

13.2883408 5 0.3 4 35.24% 29.83% -5.41%

13.2883408 5 0.3 5 42.43% 33.22% -9.21%

13.2883408 5 0.3 6 37.72% 32.39% -5.33%

13.2883408 5 0.4 4 34.61% 34.33% -0.28%

13.2883408 5 0.4 5 40.79% 37.89% -2.91%

13.2883408 5 0.4 6 41.22% 33.77% -7.45%

13.2883408 10 0.2 2 21.89% 19.57% -2.32%

13.2883408 10 0.2 3 29.36% 24.55% -4.81%

13.2883408 10 0.2 4 24.16% 27.87% 3.72%

13.2883408 10 0.2 5 30.72% 29.53% -1.19%

13.2883408 10 0.2 6 33.55% 29.53% -4.02%

13.2883408 10 0.3 3 28.90% 22.22% -6.69%

13.2883408 10 0.3 4 31.69% 29.83% -1.86%

13.2883408 10 0.3 5 36.12% 33.22% -2.90%

13.2883408 10 0.3 6 38.65% 32.39% -6.26%

13.2883408 10 0.4 4 32.49% 34.33% 1.84%

13.2883408 10 0.4 5 36.10% 37.89% 1.78%

13.2883408 10 0.4 6 38.79% 33.77% -5.02%

13.2883408 15 0.2 2 23.55% 19.57% -3.98%

13.2883408 15 0.2 3 31.47% 24.55% -6.91%

13.2883408 15 0.2 4 23.69% 27.87% 4.19%

13.2883408 15 0.2 5 34.17% 29.53% -4.63%

13.2883408 15 0.2 6 30.05% 29.53% -0.52%

13.2883408 15 0.3 3 26.78% 22.22% -4.56%

13.2883408 15 0.3 4 35.64% 29.83% -5.81%

13.2883408 15 0.3 5 32.66% 33.22% 0.56%

13.2883408 15 0.3 6 32.34% 32.39% 0.05%

13.2883408 15 0.4 4 37.28% 34.33% -2.95%

13.2883408 15 0.4 5 34.54% 37.89% 3.35%

13.2883408 15 0.4 6 35.02% 33.77% -1.25%

Annexes

133

13.2883408 20 0.3 3 21.38% 22.22% 0.83%

13.2883408 20 0.3 4 27.62% 29.83% 2.21%

13.2883408 20 0.3 5 24.85% 33.22% 8.37%

13.2883408 20 0.3 6 31.77% 32.39% 0.62%

13.2883408 20 0.4 4 29.54% 34.33% 4.79%

13.2883408 20 0.4 5 33.68% 37.89% 4.21%

13.2883408 20 0.4 6 33.39% 33.77% 0.38%

8.85889384 5 0.2 2 1.90% 12.27% 10.38%

8.85889384 5 0.2 3 15.64% 17.61% 1.97%

8.85889384 5 0.2 4 0.32% 20.96% 20.63%

8.85889384 5 0.2 5 26.32% 22.32% -4.01%

8.85889384 5 0.2 6 14.21% 21.69% 7.48%

8.85889384 5 0.3 3 3.40% 14.38% 10.98%

8.85889384 5 0.3 4 31.10% 22.60% -8.50%

8.85889384 5 0.3 5 24.00% 25.91% 1.91%

8.85889384 5 0.3 6 21.17% 24.32% 3.15%

8.85889384 5 0.4 4 34.80% 27.55% -7.25%

8.85889384 5 0.4 5 29.56% 30.89% 1.32%

8.85889384 5 0.4 6 20.88% 25.29% 4.40%

8.85889384 10 0.2 2 7.95% 12.27% 4.32%

8.85889384 10 0.2 3 21.11% 17.61% -3.50%

8.85889384 10 0.2 4 5.42% 20.96% 15.54%

8.85889384 10 0.2 5 28.16% 22.32% -5.84%

8.85889384 10 0.2 6 17.34% 21.69% 4.35%

8.85889384 10 0.3 3 4.63% 14.38% 9.75%

8.85889384 10 0.3 4 31.59% 22.60% -8.99%

8.85889384 10 0.3 5 25.62% 25.91% 0.30%

8.85889384 10 0.3 6 18.17% 24.32% 6.15%

8.85889384 10 0.4 4 33.43% 27.55% -5.87%

8.85889384 10 0.4 5 31.01% 30.89% -0.12%

8.85889384 10 0.4 6 24.16% 25.29% 1.13%

8.85889384 15 0.2 2 11.10% 12.27% 1.18%

8.85889384 15 0.2 3 26.25% 17.61% -8.64%

8.85889384 15 0.2 4 8.09% 20.96% 12.86%

8.85889384 15 0.2 5 31.73% 22.32% -9.42%

8.85889384 15 0.2 6 19.65% 21.69% 2.04%

8.85889384 15 0.3 3 6.98% 14.38% 7.39%

8.85889384 15 0.3 4 36.71% 22.60% -14.11%

8.85889384 15 0.3 5 26.79% 25.91% -0.88%

8.85889384 15 0.3 6 17.73% 24.32% 6.59%


134

8.85889384 15 0.4 4 38.24% 27.55% -10.69%

8.85889384 15 0.4 5 31.03% 30.89% -0.15%

8.85889384 15 0.4 6 24.17% 25.29% 1.11%

8.85889384 20 0.3 3 9.37% 14.38% 5.01%

8.85889384 20 0.3 4 20.56% 22.60% 2.04%

8.85889384 20 0.3 5 7.85% 25.91% 18.07%

8.85889384 20 0.3 6 27.20% 24.32% -2.88%

8.85889384 20 0.4 4 18.35% 27.55% 9.20%

8.85889384 20 0.4 5 31.56% 30.89% -0.67%

8.85889384 20 0.4 6 24.72% 25.29% 0.57%

Table 45. FEM error table for correlation coefficients over 0.5.

analysis of the resistance due to waves in ships · analysis of the resistance due to waves in ......

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