Introduction
A vessel design process is a very complex and demanding task. During the Conceptual design phase, which is the first stage of design process, the main information about the vessel is decided. Characteristics like length, beam, draft, cargo volume, deck area and service speed are defined at this first stage and should meet the shipowner expectation. Once the conceptual design is approved, the process goes on through the other design phase (basic design, classification, detail design and so on).
Having the most information the possible at the conceptual design phase is important in order to give to the shipowner a best foresight about what the vessel is going to be in the future. An early 3D visualization can aid the decision making process for both designers and shipowner, while helping with volumes, areas and centers calculation. Despite a 3D model being very useful, a hull design is a time consuming task and not always used as a decision making tool at early design phases. However, whit a tool which automates the hull design process, it can add great value to the conceptual design phase.
One possible way to performer this tool implementation involves the use of parametric design. In this application, this approach will be applied following a methodology which is constructed based on parametric models for the ship curves. A parametric Sectional Area Curve (SAC) will be used together with parametric curves for the keel and the design waterline to define the submerged hull shape. Additional ship lines, such as deck and side curves, bow and stern lines, will be used to define the hulls full geometry. The final geometry will be represented by a set of parameters, which will be chosen by the user to define the vessel. These parameters can be stored in a ship object, which can be carried out during the whole design process as an entity which concentrates all the vessel information, such as dimensions, properties and analysis results. The hull shapes will be inspired by Ulstein XBow vessels, but the application can be modified in order to contemplate different hull shapes.
Methodology
On this application, the PSV hull design tool will be developed following the workflow stated on Figure 1. At first, a user input is needed. It can be obtained by direct input of the parameters or by the use of the tool First Guess, which derives all the required parameters from the vessel's main dimensions.
The vessel's profile is defined with different spline curves for each of the hull's sections. The SAC will be defined, based on the Istanbul Technical University Method. Since the Method of Taylor was developed to design military vessels, it will be adapted to the PSV problem. The side curve is defined similarly to the vessel profile, defining splines for each section of the hull using important point on the geometry to control the curve shape. The design waterline and the main deck curves will be also developed following the Istanbul Technical University approach. The cross sectional curves will be defined using different mathematical curves according to the section needed characteristics.
Figure 1 – Methodology Workflow
The main idea behind this design tool is to make the preliminary hull design simple and fast, but it still necessary to have the user participation during the design process, since an evaluation about the obtained results in each step is important to ensure that the desired results are being reached. If the result of any step is unsatisfactory, the user should refeed the tool, changing some of the input parameters in order to improve the obtained result.
In the following sections, each of the steps on the project workflow will be discussed in deeper details, addressing main concepts, parameters and boundary conditions.
Hull Profile
The hull profile will be an important piece to define the hull shape, since it will be relevant to define the cross sections shapes, the deck length, after and fore body shapes and so on.
The profile curve follows, empirically, the shape of the Ulstein XBow vessels, with the most noticeable characteristic being the famous bow shape. In order to simply define the bow, an elliptical curve was used, with the small axis being the vessel's draft and the big axis being a specific user input (bow_big_axis). The complete hull profile curve was defined using key points of the geometry, such as, the position of the upturn point and the depth of the transom stern. On Figure 2, it is possible to see an example of a profile curve generated by this application.
Figure 2 – Lateral view example: Profile curve (pink), Side curve (green), Water line (blue) and Deck (brown)
Side Curve
The side curve is responsible for defining the flat side of the hull. Everything on the hull above the side curve has the same breath as the maximum breadth at the midship section. This curve plays an important role in the above water sections definition.
The height of the side curve at the parallel middle body is defined by the bilge radius, which can be easily obtained from the midsection area and from the vessel's beam and draft.
The curve's after and fore sections are defined using 3rd order polynomials. For each section an equation as stated on Equation 1 need to be solved.
y = 
a + bx + cx^{2} + dx^{3} 
(1) 
In order to solve these equations, the boundary conditions are based on relevant points on the hull geometry such as the intersections of the side curve with the waterline curve. These boundary conditions area as follow, for the after section:
 The first derivative of the run portion of the side curve at the Run length should be zero;
 The height of the side curve at the Run length is the bilge radius;
 The side and waterline curves intersections at the after section;
 The height of the side curve at the transom stern.

And for the fore section:
 The first derivative of the entrance portion of the side curve at the Entrance length should be zero;
 The height of the side curve at the Entrance length is the bilge radius;
 The side and waterline curves intersections at the fore section;
 The intersection of the side and deck curves at the fore section.
On Figure 2, it is possible to see an example of a side curve generated by this application.
Sectional Area Curve
The sectional area curve (SAC) is a fundamental drawing during a vessel design. It plays an important role specially regarding on the vessel resistance [PNA vol1]. One example of SAC can be seen on Figure 3. It represents the longitudinal cross sectional area distribution below the design waterline along the vessel length. The curve is expressed as a position, in length unit, along the vessel waterline length (xaxis) versus a cross section area, in length square units (yaxis). It can also be made nondimensional using the vessel waterline length and maximum cross sectional area. The curve is usually divided into two main segments. One of them represents the vessel afterward portion (after body) and the other one represents the vessel forward portion (fore body). The maximum value of the SAC represents the vessel section with the maximum cross sectional area. If the vessel has any parallel middle body, it is represented by the continuous portion of the curve, parallel to the xaxis. In this case, the afterward length discounting the parallel middle body of the after body section is called Run and the forward length discounting the parallel middle body of the fore body section is called Entrance.
Figure 3 – Sectional Area Curve Example
The SAC has two very important characteristics. Firstly, the centroid of the SAC is located at the same position of the vessel's longitudinal center of buoyance (LCB). Secondly, the ratio of the area under the curve to the area of a circumscribing rectangle is the vessel's prismatic coefficient.
In order to define the vessel's SAC, we are going to apply an approach attributed to Istanbul Technical University. This method suggest that the vessel's SAC can be described as a 5th order polynomial for a ship without parallel middle body or as a 7th order polynomial for a ship with parallel middle body. For a vessel like a PSV, which has a medium to high prismatic and block coefficients and a considerable parallel middle body, the 7th order polynomial curve will be applied.
The polynomial curve will have the form shown in equation 2.
y = 
a + bx + cx^{2} + dx^{3} + ex^{4} + fx^{5} + gx^{6} + hx^{7} 
(2) 
In order to define the unknown coefficients, we need to apply some boundary conditions related to the wanted SAC form. In order to better define the SAC, we are going to remove the parallel middle body section, considering the curve only composed by the Run and Entrance lengths. This will avoid misbehavior on the middle body section due to lack of boundary conditions at the curve definition. After defining the Run and Entrance curve shapes, it is possible to add the middle body length and have a smoother result. Since we need to define a 7th order polynomial, we need eight boundary conditions. These boundary conditions are:
 The transom area, if the vessel has a submerged transom stern;
 The slope of the run curve at its origin;
 The curve should have the maximum area at the Run length;
 The first derivative of the sac curve at the Run length should be zero;
 The bow area, if the vessel has a submerged bow volume;
 The slope of the entrance curve at its origin;
 The total area under the curve should be equal to the vessel displacement;
 The position of the center of the area under the SAC should be equal to the vessel LCB.
The resulting curve quality is very sensible to the input parameters coherent. So, it is important that the user pay closer attention to the parameters relations. For example, if the vessel has a very long parallel middle body it is expected that the vessel will have a high prismatic coefficient. On Figure 4, it is possible to see an example of a SAC generated by this application.
Figure 4 – Sectional Area Curve example
Design Waterline Curve
The design waterline (DWL) curve is the one which represents the boundary between the underwater and the emerged portions of the hull. The approach to develop the DWL curve will follow the Istanbul Technical University Method, using one 7th order polynomial to define the vessel half waterline. As it occurred during the SAC development, in order to ensure good continuity and avoid misbehaviors of the curve at the parallel middle body, we are going to remove the parallel middle body section, considering the curve only composed by the Run and Entrance lengths.
As an important characteristic of the DWL curve we can highlight that the ratio of the area limited by the curve to the area of a circumscribing rectangle is the vessel's waterline coefficient.
One important thing here is that the parallel body length of the DWL curve is defined by the Side curve, possible being bigger than the vessel parallel middle body. The Parallel Body term is used here just to emphasize that this section of the curve is parallel to the hull's longitudinal symmetry line. We are going to calculate only half of the DWL curve, since it is symmetric in relation to the vessel's longitudinal symmetry line.
The 7th order polynomial has the same form as stated in Equation 2. In order to define the unknown coefficients, we need to apply some boundary conditions related to the wanted DWL curve form. Since we need to define a 7th order polynomial, we need eight boundary conditions. These boundary conditions are:
 The transom beam, if the vessel has a transom stern;
 The slope of the run curve at its origin;
 The curve should have the half Beam value at the Run length;
 The first derivative of the DWL curve at the Run length should be zero;
 The curve should have the value zero at the bow;
 The slope of the entrance curve at its origin;
 The total area under the curve should be equal to half of the water plane area;
 The position of the longitudinal center of the area under the curve should be equal to the vessel LCF.
Once again, all the discussion about the SAC result quality should be done for the DWL curve. On Figure 5, it is possible to see an example of a DWL curve generated by this application.
Figure 5 – Top view example: Base curve (pink), DWL curve (blue) and Deck curve (brown)
Main Deck Curve
The main deck curve will follow the same shape as the DWL curve. For this curve, the parallel body length will be also defined by the side curve and will be bigger than the one in the DWL curve. For a PSV, the breadth at the transom stern at the main deck will be the same as the maximum one, in order to maximize the cargo area on the deck. The entrance angle will be also bigger than the one on the DWL curve. The main deck curve (together with the DWL curve) will be important to define the above water hull shape, since with the SAC we have only information to define the underwater body shape.
For the main deck curve, we can use a slightly different approach to define the shape of the curve. Since the after portion of the hull, as the parallel body, will be parallel to the longitudinal symmetry line, we can consider these two curves as one. So the main deck curve can be divided in two curves, a Parallel Body and an Entrance body.
For the Entrance curve we will need to fulfill the following boundary conditions:
 The curve should have the half Beam value at the Entrance length;
 The first derivative of the curve at the Entrance length should be zero;
 The curve should have the value zero at the bow;
 The slope of the Entrance curve at its origin.
Since we have four boundary conditions, a 3rd degree polynomial will be used in this case.
The same discussions made for the SAC and DWL curve should be applied for the main deck curve. Additionally, it is important to notice that the vessel length at the waterline will, most probably, be different from the length at the main deck, although the dimensionless curves have the same length. These two lengths are defined by the vessel profile. On Figure 5, it is possible to see an example of a DECK curve generated by this application.
Base Curve
The base curve plays an important role into the vessel's cross sections definition. It represents the flat portion at the hull's keel. In order to define this curve, it will be divided into three portions: Run, Middle Body and Entrance. The Run curve will start at the profile's Upturn point and the Entrance curve will start at the profile's Bow Length.
The middle body will be parallel to the side curve. Its width is defined by the bilge radius, which can be easily obtained from the midsection area and from the vessel's beam and draft. The half base width will be equal to the vessel's breadth minus the bilge radius.
The Run and Entrance curve will be defined using 2nd degree polynomial curves. The boundary conditions for both curves are as follow:
 The curve width at the first point on the Run (Entrance) is equal zero;
 The curve width at the last point on the Run (Entrance) is equal to half base curve width;
 The curve first derivative at the Run (Entrance) length should be zero.
Solving these equations and putting the three sections together, the resultant curve will be smoother than if the base curve was defined as a single section. On Figure 5, it is possible to see an example of a BASE curve generated by this application.
Cross Section Curves
The cross section curves represent transversal cut along the length of the hull. The longitudinal position of each cross section is defined based on important points along the vessel length, such as, stern position, upturn point position, intersection between side and waterline positions and so on.
The curves for both after and fore bodies are developed using the vessel's main curves as boundary conditions. In order to define each cross section, it is used one point from the base / profile curves, one point from the waterline curve, one point from the side curve, one point from the deck curve and one point on the sectional area curve. In order to have good results during the cross sections definitions, different equations were used.
For the middle body section, the bilge profile was defined using a simple circle equation.
For sections located astern from the intersection between water line and side curve at the after body and for sections ahead from the intersection between water line and side curve at the fore body it was used a parabolic equation to define the cross section. It can be seen on Equation 3.
For sections located ahead from the intersection between water line and side curve at the after body and for sections astern from the intersection between water line and side curve at the fore body it was used a hyperbolic equation to define the cross section. It can be seen on Equation 4.
y = 
ax + b + d / (x + c) 
(4) 
On Figure 6, it is possible to see an example of the cross section curves generated by this application. At the left side are the after body sections and at the right side the fore body sections.
Figure 6 – Cross Section Curves example: Afterward Sections (left) and Foreward Sections (rigth)