Analysis of the resistance due to waves in ships

Transcription

1 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

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5 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. 3

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7 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. 5

8 Analysis of the resistance due to waves in ships Index ACKNOWLEDGMENT 3 ABSTRACT 5 INDEX 6 NOMENCLATURE 10 CHAPTER 1. STUDY APPROACH SCOPE SAVITSKY S FORMULATION INITIAL ASSIGNMENT: STUDY OF THE SAVITSKY S FORMULATION APPLICABILITY OF THE FORMULATION FINAL ASSIGNMENT: RESULTS 20 CHAPTER 2. MODEL SETUP STUDY MODEL MODEL CREATION BOUNDARIES GENERAL DIMENSIONS FOR DIFFERENT VERSIONS PROBLEM DEFINITION 28 CHAPTER 3. MESH STUDY MESH PARAMETERS MESH TYPE QUALITY 32 CHAPTER 4. MODEL VERSIONS VERSION VERSION VERSION VERSION 4 42 CHAPTER 5. CASE MATRIX CASE DEFINITION GEOMETRICAL DISCRETIZATION 45 6

9 5.3. APPLICABILITY DISCRETIZATION: DISCRETIZED MATRIX DATA EXCLUDED SUBMERGED VOLUME 50 CHAPTER 6. RESULTS RESULT STORING PROCESSOR SCHEME RESULT TYPE EXCLUDED RESULTS NON-EXCLUDED RESULTS 62 CHAPTER 7. ERROR STUDY LEAST SQUARES REGRESSION MODEL BY MEANS OF INTEGRATION CORRELATION COEFFICIENT OF PEARSON REGRESSION MODEL BY MEANS OF LEAST SQUARES GAUSS NORMAL EQUATIONS 77 CHAPTER 8. CONCLUSIONS EXCLUDED CASES OF C V = NON-EXCLUDED CASES (C V = 2,3,4,5) TIME HUMAN FACTOR TECHNOLOGICAL FACTOR SAVITSKY EMPIRICAL DATA TOWING TANK DATA 85 BIBLIOGRAPHY 87 ANNEXES 89 ANNEX A: USER DEFINED FUNCTIONS TDYN SCRIPT TO RUN CASES AUTOMATICALLY EXCEL SAVITSKY CRITERIA EXCEL RESULTS STORAGE EXCEL ERROR EVALUATION, METHOD TDYN RESULT IMAGES 89 ANNEX B: SECTIONS ISOMETRIC PLAN ELEVATION 89 7

10 Analysis of the resistance due to waves in ships ANNEX C: RESULTS STREAMLINE RESULT TABLE FEM RESULT TABLE STREAMLINE ERROR TABLE FEM ERROR TABLE. 89 8

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12 Analysis of the resistance due to waves in ships Nomenclature Symbol Units [IS] Significance L k m Amidships wetted length L C m Wetted chine length λ - Mean wetted length-beam ratio b m Beam d m Draft β º Deadrise τ º Trim g m/s 2 Gravity, 9.8 m/s 2 ρ kg/m 3 Density, m/s 2 V m/s Velocity C V - Speed Coefficient ε % Error 10

13 Chapter 1. Study approach. Chapter 1. Study approach. 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 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. 11

14 Analysis of the resistance due to waves in ships 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 Δ Δ Ship s displacement b Beam D f Drag s viscous component L k Wetted keel length τ Trim L c Wetted chine length LCG Longitudinal gravity centre V Horizontal velocity of planing surface CG Centre of gravity d Draft from L k until lower point on the stern Є Shaft s tilting compared to the keel C v Froud number N Normal force or Lift f Distance between T and CG a Distance between D f and CG c Distance between N and CG Table 1.Description of different Savitsky coefficients. It is important to clarify that the Froude number C v 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. 12

15 Chapter 1. Study approach. 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 13

16 Analysis of the resistance due to waves in ships 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: Mean wetted length beam ratio (λ): is the quotient between the mean wetted length and the beam. sin tan 2 tan 2 Equation 2. Mean wetted length beam ratio in function L K, L C, b, d, τ, β Subtraction between wetted length and wetted ( L k L c ) : 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. 14

17 Chapter 1. Study approach. Figure 4. Better look of the previous zones. 15

18 Analysis of the resistance due to waves in ships Once seen the 3D model, it is added the definition of L K y L C parameters. By means of the following equation it can be calculated the relation between L K L C : tan tan Figure 5. LK and LC definitions. Equation 3. Relation between L K and L C in function of the beam, deadrise and trim. 16

19 Chapter 1. Study approach Lift coefficient for a null and beta deadrises (C L0 y C Lβ ): 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 C L0 y C Lβ : / / Equation 4. Lift coefficient for a null deadrise. Lift coefficient of a planing surface; β= C Cv 5 Cv 4 Cv 3 Cv 1 C L0 / τ 1, Cv 15 Cv 14 Cv 13 Cv C v2 Cv 11 Cv 10 Cv λ C v15 Cv 8 Cv 7 Figure 6. Lift coefficient for β = 0. 17

20 Analysis of the resistance due to waves in ships.. Equation 5. Lift coefficient for any deadrise value. C Lβ C L0 10 deg 15 deg 20 deg 25 deg 30 deg Figure 7. Lift coefficient for β 0. At the end only C Lβ is used because it has C L0 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 1 2 Equation 6. Bernoulli s equation Pressure s centre and longitudinal position: This is the centre of pressures of the submerged surface. It can be calculated by the following equation: 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. 18

21 Chapter 1. Study approach. 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 λ 1, 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 L k L c. the applicability is: For whole angles of deadrise and trim and always a C v equal or larger than 2. Cv 2.0 Β Τ All deg All deg Table 2. Equation 3 applicability. For C v 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 C v 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 Lk - Lc 4.0 deg is larger than prediction Table 4. Equation 3 applicability. Equation 15 y 16: These equation allow to calculate C L0 and C Lβ, and its range of applicability is: τ 2.0 deg deg 19

22 Analysis of the resistance due to waves in ships λ 4 C v Table 5. Equation 3 applicability. 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. τ 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. 20

23 Chapter 2. Model setup. Chapter 2. Model setup 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. L Close Water Beach Figure 9. Free surface model zones. L: Length of the ship 21

24 Analysis of the resistance due to waves in ships Depth Figure 10. Isometric view of the model 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. 2- Inner: close water. Figure 12. flat lamina. Close water is compound by five quadrilaterals. In the bow those quadrilateral are of (B 1 or B 2 ) m x L 2 m, in the stern (B 1 or B 2 ) m x L 3 m which are bigger than in the bow due to a better study of the zone later, and a rectangle of B 2 m x Lm above the lamina. 22

25 Chapter 2. Model setup. 3- Outter: Beach. Figure 13. Close water. 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 L T 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. Figure 15. Surroundings. 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 16. Defined volume. 23

26 Analysis of the resistance due to waves in ships 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. Figure 17. H Free Surface. 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. 24

27 Chapter 2. Model setup. Outlet: These are the surfaces which indicate the inlet and outlet of the water. Figure 19. Outlet and Inlet of the current General Dimensions for different versions The previous seen parameters of length, depth, height or amplitude are described in the following draw: B L T B 1 B 2 L L 2 P L 3 Figure 20. Regular model and its parametric dimensions. 25

28 Analysis of the resistance due to waves in ships The model version 1 has the following dimensions: L T P B L L 2 L 3 B 1 B 2 19 m 4 m 5 m 5 m 1 m 2.5 m 1 m 1 m Table 6. Version 1. The model version 2 has these: L T P B L L 2 L 3 B 1 B 2 22 m 4 m 5 m 8 m 1 m 2.5 m 1 m 1 m Table 7. Version 2. 26

29 Chapter 2. Model setup. The model version 3 has these: L T P B L L 2 L 3 B 1 B 2 27 m 4 m 5 m 8 m 1 m 2.5 m 1 m 1 m * It has been added 2 m in the stern and 3 m in the bow, both in the beach zone. The model version 4 has these: Table 8. Version 3. L T P B L L 2 L 3 B 1 B 2 37 m 10 m 15 m 8 m 4 m 10 m 2 m 3 m B 1 B 2 * 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. 27

30 Analysis of the resistance due to waves in ships 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. 28

31 Chapter 2. Model setup. 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/m 3. 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. 29

32 Analysis of the resistance due to waves in ships Chapter 3. Mesh study 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 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. 30

33 Chapter 3. Mesh study. 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. 31

34 Analysis of the resistance due to waves in ships - 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 Quality 32

35 Chapter 3. Mesh study. The accuracy has been studied once the version 2 was done. In this version, the maximum wetted length or L K 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) s TIME min LIFT (N) 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. 33

36 Analysis of the resistance due to waves in ships 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: 1000 Time vs Mesh Size Time (minutes) Time vs Mesh size 0.15 Time vs Mesh size 0.1 Time vs Mesh size 0.05 Time vs Mesh size Max. Element Size (m) Figure 29. Time vs mesh size chart. In the case of an Inner mesh of 0.3 m and an Hfs mesh of 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: Fz 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. Figure 30. Comparison of the result force for different Inner meshes. 34

37 Chapter 3. Mesh study. Previous char without omitted cases would be: Fz 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 m. Fz 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 m and same Inners mesh size (0.3 m). Finally comparing the theoretical Savistky s value which is 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 35

38 Analysis of the resistance due to waves in ships 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-structured zones of the model see this image below: This is the version 4 wheree the general meshing is 0.75 m, Inner of 0.1 m and Hfs and adjacent zones to this one of 0.05 m. Figure 33. Version 4 mesh. Here it can be seen the different zones: Inner Adjacent Hfs Figure 34. Adjacent water definition. And also the size and type mesh: 36

39 Chapter 3. Mesh study. Finally, here, the search and optimization of the best suitable mesh for Figure Hfs and 35. the Model adjacent mesh zones. size and type. Observe how it has been modified until reaching the perfect one: Inner - Hfs Image 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. 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 mesh to

40 Analysis of the resistance due to waves in ships Chapter 4. Model versions. There are four version of the present model, every version in order to correct which were affecting some cases. previous design errors 4.1. Version 1 First version was done to see if the model was valid and functional. In addition with a first approach to Tdyn environment and mechanics along with determine what results were goingg to be carried out. It is an abrupt mesh because is the initial version and should not have to be accurate. Figure 36. Meshed version 1. 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 Version 2 Figure 37. Regular wake for a planing hull. 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 two reasons. First reason is because for doing the study, it is needed to fix some parameters such as the beam which is 1m for the half hull and 2 meters for the entire hull, and then Froud number, C v, depends only on the velocity which is easier to compare cases. Also the L K 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 fixed to 7 meters. Secondly, the other reason is because apart from fixing some geometrical parameters such as beam or maximum wetted length. Finite element method software just calculates and does not discern about 38

41 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 K imposed is 7 m but the software, which does not take in count the theoretical formulation of the geometry, calculates should be 7.1 m, if it did not have this extra margin of length it would have crashed leading to errors. So if that wants to be avoid an extra margin is the best solution and easy way to fix it. for itself that the L K or flooded the vessel Figure 38. Version 2. 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 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 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 be null. Despite that, if the model had not enough space to dissipate the current turbulence because of a short distance, calculus will crash. At the beginning the calculus module can handle it, but when times 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 in downstream, this effect disappeared because it had the proper space to dissipate these turbulences. 39

42 Analysis of the resistance due to waves in ships Here it can be seen that what happened previous the enlargement of the length downstream was clearly a resonance phenomena. Once it was disappeared. enlarged, the error Figure 39. Evolution downstream for version 2. 40

43 Chapter 4. Model versions 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. That is because the fluid flux is anticipating to be hit by the surface which is penetrating the water in order to reduce its impact and generate 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 in some point, between the inlet of current and the bow, some sort of convex wave. It is like the previous problem with 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. The upstream zone has been increased due to a lack of space problem. Figure 40. Version 3. Figure 41. Upstream problem. This image is an elevation of the model. The black line, which extends from the left (stern) to the right (bow), is the water surface in the amidships gangway. It is a tilted line on the stern due to the depth of 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 near the bow then increased its height being over the waterline and then when it reached the upstream, becausee of the boundaries it 41

44 Analysis of the resistance due to waves in ships was reduced until it reached the 0 value, which would be the red line. This red line is quite exaggerated comparing to the simulated cases, but it is useful to make an idea of it Version 4 The fourth version has more beam, depth and length. Indeed it is based on version 1 and it has been added more distance on those lengths direction mentioned before. It can be observed that the inside prism is version 1 which is wrapped by an outside prism that completes version 4. The reasons that made to change the previous model were: - Beam: In the previous image, figure 41, it could seen that on the laterals of the towing tank or model there was an accumulation of height which created a wave. That was generating some sort of turbulence on the free surface downstream. 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. Figure 42. Pressure diagram, depression downstream. That is due to a wall effect, which means that the stream, which is released from the boundary layer, goes directly to the lateral, hits it and comes back generating a turbulence 42

45 Chapter 4. Model versions. downstream. In a real case in the open sea that effect would not happen. This effect can be an additive or destructive interference. 43

46 Analysis of the resistance due to waves in ships - Depth: 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 length. So from 4 m it becamee 12 m. Otherwise these has no real impact in the calculus because the mesh for this dimension its only general meshing and its huge and will not add to much delay, at least not noticeable. - Length: the length was increased for two reasons. First the base for this version was version 1 it had to be increased like in the version 2 and 3. And it was increased a little more 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 beach from version one became close water from version 2 and the extra lengths added to the model were the beach on version 4. Also a new area appeared whichh was close to the vessel layer which was the adjacent water to the vessel. Vessel had a structured mesh of 0.05 m 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. Version 1 beach became close water in 4. Figure 43. Relation between version 1 and version 4. 44

47 Chapter 5. Case matrix. Chapter 5. Case matrix 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 C v (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 Geometrical discretization Savitsky s formulation itself set a series of geometrical formulas: Remembering that L k y L C, by trigonometry were: Equation 9.Theoretical geometry criteria. Then for this study L k 7 m y L C 0 m were fixed. The criteria become: 1. d 7 sin τ 2. tan β Equation 10.Applied geometry criteria. Where b is the completely beam, which means 2 meters. Figure 44. L K and L C graphical description. 45

48 Analysis of the resistance due to waves in ships 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 sin (t) tan (B) d π / b cos(t) CORRECT Table 12. Geometry criteria in the spreadsheet 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 deg λ 4.0 C v Equation 3: Table 13. Equation 1 applicability. Applicability : Case 1 C v 2.0 β All deg τ All deg Case 2 β 10.0 deg C v 1.0 Case 3 β 20.0 deg C v 1.0 τ 4.0 deg L k - L c is larger than prediction Table 14. Equation 3 applicability. 46

49 Chapter 5. Case matrix. Equation 15: Applicability : τ 2. deg deg λ 4 C v Table 15. Equation 15 applicability. Equation 23: Applicability : C v 1-13 Table 16. Equation 23 applicability 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. Time vs Velocity VEL TIME BETA Simulation time (s) t = 4680 V Velocity(m/s) Figure 45. Chart and table with simulation time data. 47

50 Analysis of the resistance due to waves in ships 5.5. Data excluded Although these 196 cases are theoretically within the applicability range of the formulation, the truth is that it had to be reduced from 196 cases to 172. Those 24 excluded correspond to those cases in which its velocity was 4.42 m/s or what is the same as C v = 1. Despite, the application of the third equation is within the applicability range. It is possible that is too 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 v = 1 has an increased error value rather if it would be a C v = of 1.1 or 1.2 and both are practically the same Froud 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, that the vessel was not planing because it had a more appropriate wake for a semi-displacement ship or pre-planing hull. E.g. in the case 182: Case Velocity (m/s) Dead rise (º) Draft (m) Trim (º º) According to the parametric equation of the flat lamina s surface: tan tan Table 17. Case 182 parameters. Equation 11.Geometry of flat lamina s surface. The beam in the transom is 1 m, so the C V should be comparable to other cases with the same C V. But in practice due to is not a real planing hull or at least do not behave like that, the Spray does not appear, which is a unequivocal sign thatt the vessel is not lifted by hydrodynamic forces. Hence, on the calculus of the pressure zone, which are shown in the next image, it can be seen that the beam is not really 1 m and then L c is minor than 0. 6 Figure 46. Error on the beam for case

51 Chapter 5. Case matrix. A negative L C means that the L C ends in the downstream as it is shown in the following image: Figure 47. LC ends downstream. It can be observed that L C ends beyond the transom and based on the discretization criteria, L C cannot be minor than 0. These phenomena have been reproduced in several low speed cases. In general in those which have an important draft and a big trim and deadrise angles. Otherwise, Savitsky s formulation does not take the length of the vessel as a defining parameter of the C V, and logically in reality a vessel with a L K / b rate quite low cannot be comparable to another one with the same C V but different L K / b ratio. E.g. the case 182, according to the definition of the mean wetted length beam ratio or λ: λ L L 2b Equation 12. Mean wetted length beam ratio. According to the theoretical Savitsky s value and the one calculated by FEM, this ratio would be: Method Savitsky FEM b (m) L c (m) L k (m) Table 18. Comparison between Savitsky and FEM software. This difference between the coefficient for the same case and different calculus methods makes the case not be really the same. The explanation is that for a C v = 1 and a semi-beam of 1 m, the hull is not planing. 49

52 Analysis of the resistance due to waves in ships 5.6. Submerged Volume In order to demonstrate this hypothesis was right, it was ideated a way to check how important the 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. It is just for the hulls that fulfil the criteria so it will not do difference between a real planing hulll and a fake planing hull. To check out if the previous error was not from the own software, it was decided to compare the hydrostatic lift for a C V = 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 can be defined as:.. Equation 13.Hydrostatic Lift. In this case what is being calculated is the semi-volume. Assuming that the density does not vary with the depth, basically because is a theoretical calculus and the maximum draft is 40 cm, which is not a significant value to assume that the density is going to vary with the depth. The previous equation becomes:.. Equation 14. Development of the hydrostatic Lift equation. The integral of the differential of z is the equation of the parametric surface: tan tan.. tan 180 tan 180 Equation 15. Development of the hydrostatic Lift equation. 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 the length, the length would be in the direction of the trim tilting. This displaced volume by the flat lamina would be: Figure 48. Submerged volume. 50

53 Chapter 5. Case matrix. Figure 49. Dimensions of the submerged volume. What has to be found out is L 1 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 L 1 are trim and deadrise. Figure 50. Angles that determine the submerged volume. 51

54 Analysis of the resistance due to waves in ships Both integration zones are labelled in 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 points. Figure 51. Integration zones. L 1 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 16.L 1 definition. The equation of a line defined by two points can be described as: ; 1 tan 0 1 tan tan 1 ; tan tan The boundaries are from 0 to L 1 m and from L 1 m to geometrical criteria erased those cases in which its maximum beam was less than 1 m. And once reached L 1 from 0 to m. tan tan Equation 17. Equation of the contour life in integration 2 zone.. And for Y axis, from 0 to 1 m because the 52

55 Chapter 5. Case matrix. The final equation is:.. tan 180 tan 180 tan 180 tan 180 It has been introduced into wxmaxima, a mathematics software, and its result is:.. tan d tan tan 180 Equation 18. Development of hydrostatic lift. Equation 19. Final equation to find out the hydrostatic lift. To express the units in Newton the previous equation has to be multiplied : tan d tan tan 180 : : / : 9.81 / 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 L h.e. L Sav. L FEM h.e. vs Sav Valid? % NO % NO % NO % NO % NO % NO % YES % NO % NO % YES 53

56 Analysis of the resistance due to waves in ships % NO % NO % NO % NO % NO % NO % NO % NO % YES % NO % NO % YES % NO % NO Table 19. Low speed cases. This is the C V = 1 table results. First 4 columns correspond to the parameters of the case, the 5 th column to the value of the hydrostatic Lift, the 6 th column to the Savitsky s Lift, the 7 th column to the lift calculated by the FEM software, the 8 th column to the comparison between the hydrostatic lift and Savitsky s one and the 9 th 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 Lift h.e. > Savitsky s Lift. Lift FEM error compared to Savitsky is within 20% - 30 %. 2. Si el Lift h.e. < Savitsky s Lift. Lift FEM error compared to Savitsky is within 10% - 15 %. 3. Si el Lift h.e. < 70% Savitsky s Lift. Lift FEM 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 C V = 1 cases have been omitted from the results 54

57 Chapter 6. Results. Chapter 6. Results 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 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. 55

58 Analysis of the resistance due to waves in ships 6.3. Scheme This study did not limit itself to calculation. study the relation between Savitsky s formulation versus a unique FEM 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 STREAMLINE which is the implicit from GID and a 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 in the other hand, is quicker and robust, works quite well for low 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 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 Result type Previously it has been stated that inside the result folder there were 3 files. The.flavia is the file containing file: Figure 52. Result data. the case parameters. The.flavia.res correspond to the graphical result Figure 53. Graphical result file. And the Ouput.res is the result file of the two parameters defined previously by the user. The lifting 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 it, evaluating the forces and torque and representing these on a chart. 56

59 Chapter 6. Results. 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] Then the script added the parameters of the case: Table 20. User defined result without manipulation. Case 1 Sink 0.2 Trim 2 Vel 5 Dead Time_Fz Fz_Hfs Time_My My_Hfs Table 21.Added data in user defined result in order to identify the case. 57

60 Analysis of the resistance due to waves in ships And it generated two charts: Fz_Hfs[N] My_Hfs[Nm] 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: Fz_Hfs[N] 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: 5E+66 Fz_Hfs[N] 4E+66 3E+66 2E+66 1E E Figure 56. Chart with errors. 58

61 Chapter 6. Results. Revising it with more detail and erasing those spikes it could be more or less appreciated the real value Fz_Hfs[N] 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. Figure 57. Previous chart without spikes. Once it was re-calculated and without error the value was 13.5 kn. Fz_Hfs[N] Figure 58. Re-calculated chart. 59

62 Analysis of the resistance due to waves in ships 6.5. Excluded results Although 24 cases were excluded, the ones with C V = 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: Fz_Hfs[N] This is the distribution observed for C V > 1. This chart is exponential negative, a function like: Figure 59. High speed pattern force chart. In contrast with STREAMLINE scheme cases of C V = 1, which its pattern before version 4, were: Fz_Hfs[N] 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. Figure 60. Low speed, STREAMLINE scheme for an old version. For the version 4 in STREAMLINE, a few cases were carried out and the pattern that they show is: Fz_Hfs[N] 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. Figure 61. Low speed, STREAMLINE scheme for version 4. 60

63 Chapter 6. Results. Otherwise in FEM scheme: Fz_Hfs[N] 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. Figure 62. Low speed, FEM scheme. 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. 61

64 Analysis of the resistance due to waves in ships 6.6. Non-excluded results Non-excluded results correspond to C v = 2, 3, 4 y 5. The range case is the combination of these parameters: Sink m Trim º Vel m/s Dead 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: Fz_Hfs[N] Always near a determined time, in this case 0.9 s, the software used to crash. Then the adjacent water surface was remeshed as unstructured keeping the maximum element size up to 0.1 to avoid abrupt transitions Figure 63. Chart of the cases that crashed near a constant time value. 62

65 Chapter 6. Results. Figure 64. Modified adjacent water mesh structure. It can be seen that the mesh, in the adjacent water of the lamina, now is unstructured. This makes more blurring of the error and avoids this critical error, which even changing the stability factor, was repeating once and again. STREAMLINE 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. STREAMLINE scheme is slower than FEM, but more accurate. A special feature of FEM scheme is that it has more difference with Savitsky ss formulation, but always is under its value. STREAMLINE scheme can be over and under. To compare FEM software results with Savitsky s formulation, it was applied the following formulas: 100 ; 100 Equation 21.Definition of error and absolute error. This formula is applicable for lift and torque, V FEM is the value calculated by FEM software and V Sav is the value calculated by Savitsky s formulation. 63

66 Analysis of the resistance due to waves in ships To summarize the errors in function of its velocities and schemes, check this: STREAMLINE: 40.00% Cv Fz My Error 30.00% 20.00% 10.00% 0.00% C V Fz_Error Fz_Max My_Error My_Max Error Max Error Max % 25.75% 8.86% 25.45% % 19.51% 8.61% 28.14% % 23.68% 10.03% 29.19% % 26.92% 10.93% 29.80% Total 9.36% 26.92% 9.61% 29.80% Table 24.Error depending on the speed range and its representation. FEM: Error 50.00% 45.00% 40.00% 35.00% 30.00% 25.00% 20.00% C V Fz_Error Fz_Max My_Error My_Max Cv Fz My Error Max Error Max % 38.24% 25.90% 36.22% % 42.43% 31.98% 38.41% % 43.70% 33.27% 37.72% % 45.79% 33.82% 38.94% Total 31.01% 45.79% 31.24% 38.94% Table 25. Error depending on the speed range and its representation. 64

67 Chapter 6. Results. In this chart can be observed the results for STREAMLINE scheme charts: Fz My Savitsky Fem Savitsky Fem Fz_hfs % My_hfs % 30.00% 40.00% 20.00% 10.00% Fz_hfs % 30.00% 20.00% 10.00% My_hfs % 0.00% 0.00% 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 Fz Savitsky Fem My Savitsky Fem 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. 65

68 Analysis of the resistance due to waves in ships For FEM scheme results: Fz My Savitsky Fem Savitsky Fem My_hfs % Fz_hfs % 60.00% 60.00% 40.00% 40.00% 20.00% My_hfs % 20.00% Fz_hfs % 0.00% % 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 Fz Savitsky Fem My Savitsky Fem 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. 66

69 Chapter 7. Error study. Chapter 7. Error study. 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:,,, Equation 22.. Polynomial equation. O what would be the same: Equation 23.Polynomial equation simplified 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. 67

70 Analysis of the resistance due to waves in ships 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: The error can be expressed as:: Equation 24. Line definition. Equation 25.Error definition P = (x i, y i ) Figure 70. Error representation. Least squares is based on the optimization of the square error: Equation 26. Square error. 68

71 Chapter 7. Error study. Where n is the total number of cases. Simplifying the previous expression the mean square error is defined: 1 1 To solve the system, it is represented in the following matrix: Equation 27.Mean square error. 1. Lineal regression N is the total number of cases. Equation 28. Definition of line parameters. Figure 71. Least squares for a lineal interpolation. 2. Polynomial regression ( x n ; n > 1) The system has to be solved. With this matrix it can be obtained a polynomial of n range (x n ), there is only needed n columns and rows to solve it. It is a simple method to solve because is based on multiplying x n by y to get a higher range polynomial. Figure 72. Least squares for non lineal interpolation. 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. 69

72 Analysis of the resistance due to waves in ships 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 Regression 4 (V) Regression 1 (τ) Regression 2 (d) Regression 3 (β) 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:,, Equation 29.Error first regression. t 1 y t 2 are coefficient. These coefficients have different values in function of the other three parameters. 70

73 Chapter 7. Error study. y = x y = x y = x It can be observed that there is a line for every draft (d) % 25.00% 20.00% 15.00% 10.00% 5.00% There are some cases where for deadrise angles of 20 there is no 0.2 draft. 0.00% τ Figure 73. Linear regresion for first regresion. The following regression adds to the expression: 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.,. Every regression adds one more dimension, from lines to planes: Equation 31. Second error regression. Figure 74. Representation of the second regression. 71

74 Analysis of the resistance due to waves in ships 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:., 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 b 0 to b 17. 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: Figure 75. Representation of the third regression..,, 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. 72

75 Chapter 7. Error study. 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% % 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 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 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. 73

76 Analysis of the resistance due to waves in ships 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: Where: N is the total number of cases. Equation 34. Correlation coefficient definition. The h n τ m β term k V X is the monomial, in this case there are 69 monomials. Every monomial is the combination of j, where n,m,k,j are independent variables that goes from 0 to 2 except for m that goes from 0 to 1. To perform this study, every ery monomial has to be replaced by the parameters which define the case. E.g.: εf V β h τ τ 2 h 2 β 2 V 2 τh τβ τv % % % % % % τ 2 h τ 2 β τ 2 V hβ Table 29.How Correlation coefficient is calculated. The correlation coefficient has to be calculated for every monomial. 74

77 Chapter 7. Error study. 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: Term r r % MAX % V % 60% V % 57% τh % 40% τv % 67% τ2h % 43% τ2v % 61% hv % 58% h2v % 47% V2τ % 63% V2h % 57% τhv % 62% τ2hv % 60% τh2v % 52% τhv % 61% τβv % 40% τ2h2v % 54% τ2hv % 61% τ2βv % 42% τ2h2v % 55% τh2v % 53% τhβv % 40% h2v % 49% The maximum value, minimum and the absolute maximum are: MAX: MIN: MAX : Table 30.Maximum and minimum Coefficients. It can be observed that the correlation is negative and the maximum is 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. 75

78 Analysis of the resistance due to waves in ships In the case of FEM scheme which had more errors this study show the following tables: Term r r % MAX % V % 60% V % 60% τh % 37% τv % 66% τ2h % 36% τ2v % 57% hv % 63% h2v % 52% V2τ % 64% V2h % 61% τhv % 62% τ2hv % 56% τh2v % 53% τhv % 62% τβv % 39% τ2h2v % 50% τ2hv % 58% τ2βv % 38% τ2h2v % 53% τh2v % 55% τhβv % 39% h2v % 54% The maximum value, minimum and the absolute maximum are: MAX: MIN: MAX : 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. 76

79 Chapter 7. Error study 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. 2 0 Equation 37.Optimization of c j coefficients. Expressed as a matrix would be: A x c = y,,,,,, 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 overdetermined implies that the solution will not be never exact so it is somehow a way to approximate the error.,,,, Equation 39.Simplification. 77

80 Analysis of the resistance due to waves in ships 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: 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: % FEM vs Savitsky % 0.000% Error Calcuated FEM vs Savitsky % 50.00% 0.00% Error Calculated % Figure 77. Error distribution for every case. 78

81 Chapter 7. Error study. FEM scheme: % FEM vs Savitsky % 0.000% Error Calcuated % FEM vs Savitsky 50.00% 0.00% % Error Calculated Figure 78. Error distribution for every case. 79

82 Analysis of the resistance due to waves in ships The coefficients obtained are: STREAMLINE scheme: Monomial Coef Value V a V2 a τh a τv a τ2h a τ2v a hv a h2v a V2τ a V2h a τhv a τ2hv a τh2v a τhv2 a τβv2 a τ2h2v a τ2hv2 a τ2βv2 a τ2h2v2 a τh2v2 a τhβv2 a h2v2 a Monomial Coef Value V a V2 a τv a τ2v a hv a V2τ a V2h a τhv a τ2hv a τh2v a τhv2 a τ2h2v a τ2hv2 a τ2h2v2 a τh2v2 a Table 34. Coefficients of the relevant monomials. For monomials greater than 0.4 on the left and for 0.5 on the right. 80

83 Chapter 7. Error study. FEM scheme: Monomial Coef Value V a V2 a τv a τ2v a hv a V2τ a V2h a τhv a τ2hv a τh2v a τhv2 a τ2h2v a τ2hv2 a τ2h2v2 a τh2v2 a 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 FEM 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:,,, Equation 41. Error expression for correlation coefficients. Replacing n, a i and f i (x) for its value. f i (x) are the monomials. 81

84 Analysis of the resistance due to waves in ships Chapter 8. Conclusions This chapter summarizes all the problems that have been found out during the project Excluded cases of C v = 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 L K 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 C V = 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. 82

85 Chapter 8. Conclusions 8.2. Non-excluded cases (C v = 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 m Trim º Vel m/s Dead 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 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: % It is huge compared to %, 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 sin (t) *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 sin (t) *CORRECT Table 40. Fulfilled criteria case. And the error: % 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. 83

86 Analysis of the resistance due to waves in ships 8.3. Time This project started on September 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. 84

87 Chapter 8. Conclusions 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 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 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 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 C V = 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. 85

88 86 Analysis of the resistance due to waves in ships

89 Bibliography Bibliography D. Savitsky. Hydrodynamic Design of Planing Hulls. Marine Technology, Vol. 1, No. 1. October, O. M. Faltinsen. Sea loads on ships and offshore structures. Cambridge University Press, Cambridge, UK 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 27 th, D. Savitsky, P. W. Brown. Procedures for hydrodynamic evaluation of planing hulls in smooth and rough water. Marine Techonology, Vol. 13, No.4. October, Compass Ingeniería y Sistemas, SA. SeaFEM reference. Retrieved December 21 th, 2013 from: Wikipedia, The Free Encyclopedia. Mínimos Cuadrados - Solución del problema de los mínimos cuadrados. Retrieved June 3 rd, 2014 from: ADnimos_cuadrados Wikipedia, The Free Encyclopedia. Coeficiente de correlación de Pearson. Retrieved June 15 th, 2014 from: Vitutor.com. Coeficiente de correlación, Retrieved June 15 th, 2014 from: Monografias.com. Coeficiente de correlación de Karl Pearson. Retrieved June 15 th, 2014 from: GiD, The Personal Pre And Post Processor. Manual Selection. Retrieved January 29 th, 2014 from: 87

90 88 Analysis of the resistance due to waves in ships

91 Annexes Annexes Annex A: Annex B: Annex C: User Defined Functions. 1. TDYN Script to run cases automatically. 2. EXCEL Savitsky criteria 3. EXCEL Results storage. 4. EXCEL Error evaluation, method TDYN Result images Sections. 1. Isometric 2. Plan 3. Elevation Results. 1. STREAMLINE result table. 2. FEM result table. 3. STREAMLINE error table. 4. FEM error table. 89

92 Analysis of the resistance due to waves in ships 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() 90

93 Annexes 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 91

94 Analysis of the resistance due to waves in ships 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) 92

95 Annexes 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) ActiveSheet.Cells(27, j + 7) = ActiveSheet.Cells(14, i + 7) ActiveSheet.Cells(28, j + 7) = ActiveSheet.Cells(15, i + 7) ActiveSheet.Cells(29, j + 7) = ActiveSheet.Cells(16, i + 7) j = j + 1 End If Else End If End If i = i + 1 DoEvents Wend ActiveSheet.Cells(26, j + 7) = "]" ActiveSheet.Cells(27, j + 7) = "]" ActiveSheet.Cells(28, j + 7) = "]" ActiveSheet.Cells(29, j + 7) = "]" End Function 93

96 Analysis of the resistance due to waves in ships 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) Worksheets("Sav_0.Outputs").Range("G11") = WorksheetFunction.HLookup(i + 1, Workbooks("Global_Streamline.xlsx").Worksheets("Vectores").Range("H24:BBB30"), 4) Worksheets("Sav_0.Outputs").Range("G12") = WorksheetFunction.HLookup(i + 1, Workbooks("Global_Streamline.xlsx").Worksheets("Vectores").Range("H24:BBB30"), 5) Worksheets("Sav_0.Outputs").Range("G13") = WorksheetFunction.HLookup(i + 1, Workbooks("Global_Streamline.xlsx").Worksheets("Vectores").Range("H24:BBB30"), 6) Worksheets("Sav_0.Outputs").Range("G09") = WorksheetFunction.HLookup(i + 1, Workbooks("Global_Streamline.xlsx").Worksheets("Vectores").Range("H24:BBB30"), 1) ActiveSheet.Shapes.AddChart.Select ActiveChart.ChartType = xlxyscattersmoothnomarkers 94

97 Annexes 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$30000" ActiveChart.SetElement (msoelementcharttitleabovechart) ActiveChart.ChartTitle.Text = "Fz_Hfs[N] " ActiveChart.SetElement (msoelementlegendnone) ActiveChart.Location Where:=xlLocationAsNewSheet, Name:="Fz_Hfs" If Worksheets("Sav_0.Outputs").Range("G13") < 5 Then Time_ = 150 Else If Worksheets("Sav_0.Outputs").Range("G13") < 9 Then Time_ = 10 Else If Worksheets("Sav_0.Outputs").Range("G13") < 14 Then Time_ = 8 Else If Worksheets("Sav_0.Outputs").Range("G13") < 18 Then Time_ = 4 Else If Worksheets("Sav_0.Outputs").Range("G13") < 23 Then Time_ = 2 End If End If End If End If End If Worksheets("Sav_0.Outputs").Activate ActiveSheet.Range("G14") = WorksheetFunction.VLookup(Time_, ActiveSheet.Range("A6:C30000"), 1, True) ActiveSheet.Range("G15") = WorksheetFunction.VLookup(Time_, ActiveSheet.Range("A6:C30000"), 2, True) ActiveSheet.Shapes.AddChart.Select ActiveChart.ChartType = xlxyscattersmoothnomarkers j2 = 1 While (j2 <= 2) j = 1 While (j <= 4) ActiveChart.SeriesCollection(j).Delete j = j + 1 Wend j2 = j2 + 1 Wend ActiveChart.SeriesCollection.NewSeries ActiveChart.SeriesCollection(1).XValues = "='Sav_0.Outputs'!$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" Worksheets("Sav_0.Outputs").Activate ActiveSheet.Range("G16") = WorksheetFunction.VLookup(Time_, ActiveSheet.Range("A6:C30000"), 1, True) ActiveSheet.Range("G17") = WorksheetFunction.VLookup(Time_, ActiveSheet.Range("A6:C30000"), 3, True) 95

98 Analysis of the resistance due to waves in ships 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 96

99 Annexes 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 ) = "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

100 Analysis of the resistance due to waves in ships 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 i_max As Integer Dim minc_r As Integer Dim i_erase As Integer Dim j As Integer Dim i_first As Integer 98

101 Annexes '' 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) Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(minC_R, 2) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i, 7) 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("MC").Cells(minC_R, 1) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i - 1, 5) Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(minC_R, 2) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i - 1, 7) 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 Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(minC_R, 1) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i, 5) Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(minC_R, 2) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i, 7) 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 Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(i_erase, 2).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") Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 15) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i - 1, 2) Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 16) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i - 1, 3) 99

102 Analysis of the resistance due to waves in ships Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 17) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i - 1, 4) Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 19) = Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("M3") Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 20) = 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 i_max As Integer Dim minc_r As Integer Dim i_erase As Integer Dim j As Integer Dim i_first As Integer Dim param As Integer '' Regression SINK Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Activate param = 1 '' Numero de Coeficientes (a,b,c,d...) While (param <= 2) '' Numero de coeficientes a hacer la regresion j = 3 '' Row donde guardar parametros a,b,c de MC i = 3 '' Row donde empieza el contador i_max = WorksheetFunction.Count(Worksheets("Error_Results").Range("Q2:Q50")) ''Length maxima del contador i_first = 0 ''Parametro de control minc_r = 2 ''Row donde empieza el contador de MC While (i <= i_max + 2) If (i = 3) 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, 17) Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(minC_R, 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").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("MC").Cells(minC_R, 1) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i - 1, 17) Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(minC_R, 2) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i - 1, 18 + param) 100

103 Annexes minc_r = minc_r + 1 i_first = 1 End If While (Cells(i - 1, 17) < Cells(i, 17)) ''Escribir en MC los valores de Error_Results Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(minC_R, 1) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i, 17) Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(minC_R, 2) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i, 18 + param) 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 Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(i_erase, 2).Clear i_erase = i_erase + 1 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, 28) = Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("C2") Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 29) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i - 1, 15) Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 30) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i - 1, 16) Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Activate Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 32 + (3 ^ (param - 1) + param - 1)) = Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("M3") Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 33 + (3 ^ (param - 1) + param - 1)) = Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("N3") Else Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 28) = Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("C2") Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 29) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i - 1, 15) Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 30) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i - 1, 16) Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 31 + (3 ^ (param - 1) + param - 1)) = Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("L5") Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 32 + (3 ^ (param - 1) + param - 1)) = Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("M5") Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 33 + (3 ^ (param - 1) + param - 1)) = Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("N5") End If j = j + 1 i = i + 1 minc_r = 2 i_first = 0 Wend param = param

104 Analysis of the resistance due to waves in ships Wend End Sub Sub ERROR_R3() Dim i As Integer Dim i_max As Integer Dim minc_r As Integer Dim i_erase As Integer Dim j As Integer Dim i_first As Integer Dim param As Integer '' Regression DEAD Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Activate param = 1 '' Numero de Coeficientes (a,b,c,d...) While (param <= 6) '' Numero de coeficientes a hacer la regresion j = 4 '' Row donde guardar parametros a,b,c de MC i = 3 '' Row donde empieza el contador i_max = WorksheetFunction.Count(Worksheets("Error_Results").Range("AC2:AC19")) ''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 = 3) 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, 30) Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(minC_R, 2) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i, 32 + param) 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("MC").Cells(minC_R, 1) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i - 1, 30) Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(minC_R, 2) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i - 1, 31 + param) minc_r = minc_r + 1 i_first = 1 End If While (Cells(i - 1, 30) < Cells(i, 30)) ''Escribir en MC los valores de Error_Results Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(minC_R, 1) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i, 30) Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(minC_R, 2) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i, 31 + param) i = i + 1 minc_r = minc_r + 1 Wend i_erase = minc_r '' Borrar valores anteriores en MC 102

105 Annexes While (minc_r <= 15) Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(i_erase, 1).Clear Workbooks("Global_Streamline.xlsx").Worksheets("MC").Cells(i_erase, 2).Clear i_erase = i_erase + 1 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 Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 43) = Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("C2") Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 44) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i - 1, 29) Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Activate Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 43 + (3 ^ (param - 1) + param - 1)) = Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("M3") Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 44 + (3 ^ (param - 1) + param - 1)) = Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("N3") Else Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 44) = Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("C2") Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 45) = Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(i - 1, 29) Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 46 + (3 * (param - 1))) = Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("L5") Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 47 + (3 * (param - 1))) = Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("M5") Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 48 + (3 * (param - 1))) = Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("N5") 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 i_max As Integer Dim minc_r As Integer Dim i_erase As Integer Dim j As Integer Dim i_first As Integer Dim param As Integer 103

106 Analysis of the resistance due to waves in ships 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(j, 72 + (3 * (param - 1))) = Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("L5") Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 73 + (3 * (param - 1))) = Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("M5") Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(j, 74 + (3 * (param - 1))) = Workbooks("Global_Streamline.xlsx").Worksheets("MC").Range("N5") 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, * (param - 1)) = "V" & 3 * (param - 1) Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(3, * (param - 1)) = "V" & 3 * (param - 1) + 1 Workbooks("Global_Streamline.xlsx").Worksheets("Error_Results").Cells(3, * (param - 1)) = "V" & 3 * (param - 1) + 2 param = param + 1 Wend End Sub 104

107 Annexes 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) { :24:56} set macrosdata(photo,creationdate) { :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 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) { :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== 105

108 Analysis of the resistance due to waves in ships } array set preferences {toolbar_one_col 1} 106

109 Annexes Annex B: Sections. Isometric Figure 79. Isometric view. 107