1    Objective

In this third tutorial, we will store the waterline data of the previous tutorial in a new object within the object Ships. We will perform linear and quadratic interpolation to obtain the relative width of unknown frames at the waterline. This will be checked with the exact solution, and the error is to be calculated. Some hydrostatic calculations will be performed: by numerical integration the waterplane area and the moment of inertia of this area are calculated. The stability GM will be calculated and compared to other ships using the least squares method.

Start

For this tutorial, the knowledgebase from tutorial 2 is used. You can either use your own (verified) knowledgebase, or download it here: [Tutorial 3 Start]

2    Creating a new object within a data object

The solution Waterline containing the shape of the dimensionless waterline, has been created in tutorial 2. In order to store this data and use it for further operations, the contents will be placed in a new object within the object Ships.

  In the class Geometry, create a new object called Hull. As we want to store static data in this object, make sure it is determined by Value from Object/Database (as described in tutorial 2). In the workbase, select the object Ships (under Dataset). In the Knowledge Browser, right click the parameter Hull and select Parameter to Dataset (or drag it tot Ship). When you are asked how to add Hull to the Ships object, select the list and continue. The new object is now placed within Ships.

Let's add the content of the Waterline object in the Waterline solution to the object Hull.

 In the solution Waterline in the tree of the Workbase, select the Waterline object (so the tree node in the solution),  right click  in the table of the Workbase and select All to clipboard (or F4). Paste the data (both Frame and Rel_B) in the object Hull (answer Yes to all for including the parameters in the Hull object).

When you are done, you will have a Ships object containing data in a Hull object. The Hull object should be part of the Ships object because the Ships object is the database entry point. And because you want to use all Waterline data as one set, it should be in one Hull object (or as a TeLiTab value as an alternative).

The workbase is full of solutions we'll not use anymore. So now first let's clean it up. 

3    Interpolation 

The relative width Rel_B at every frame was calculated directly using the polynomial formula. In reality, the waterline of a ship will be defined more likely by a number of points than by a formula. To obtain the width at a certain location, you should interpolate between these points. We'll use the LININT() function to perform a linear integration.

 Add the following relation in Top Goals/Undefined:

Make sure Intpol_Rel_B is dimensionless and determined by SYS: System/Equation.  

The syntax of every intrinsic function in Quaestor is described in the wiki, see the functions overview. The syntax above means the following.

• The parameter Inpol_Rel_B will be the linear interpolated dimensionless width at a desired frame number (Frame).
• @Hull is the object containing all the data you want to use for interpolation.
• 2 is the number of dimensions in which the interpolation will take place
• @Frame makes sure the column Frame of object Hull will be used as the parameter X in the interpolation.
• @Rel_B makes sure the column Rel_B of object Hull will be used as the parameter Y in the interpolation
• Frame is the input parameter for which an interpolated value should be obtained

When Quaestor calculates this function, it determines the value Frame (which is user input). This value is then compared to the values of the given column for X (which is the column Frame of the object Hull), and an interpolated value for Rel_B with respect to the input Frame is assigned to the parameter Intpol_Rel_B. Notice that an @ is used to identify the data within the object.

  Run a solution for Intpol_Rel_B using the process manager and selecting the ship object. Make sure that Intpol_Rel_B is in the class Top Goals/Undefined to see it in the Process Manager. The process manager is crucial here, as static data from the Ships object should be used (which contains the object Hull). For frame, enter 3.20.

Quaestor should return 0.26 for Intpol_Rel_B.

To find out how accurate the interpolation is, we introduce an error. The interpolated parameter uses static data from the object Hull, but we also defined the relative width analytically (the parameter Rel_B).

 Add the following relations to your Top Goals:

in which Error is a dimensionless SYS parameter. Run a solution for this parameter, Intpol_Rel_B will be calculated by Quaestor automatically because it is required by Error. Make sure you use the process manager to select the Ships object as dataset. For frame, enter 3.20 again.

The solution in two-digit format (-0.00) will be presented in the list of the Workbase. When you select this solution, more digits are shown in the Frame Viewer (figure 2a) or Explanation window (figure 2b). Note that you can change the number of decimal places presented in the Workbase by changing the value Decimal places in the Slots & Properties window. 

Please, realise that for input values, this number of digits also defines the maximum accuracy accepted for this input. So, when you van defined B as two digits, providing input with three will give a warning to the user followed by rounding the provided number on two digits.

Figure 2a   In the Frame Viewer, more digits of the selected parameter are shown

Figure 2b   In the Explanation window, more digits of the selected parameter are shown 

 Change the relation (find the relation and press F2) for Intpol_Rel_B  in

which uses the same data, but now performs a quadratic interpolation. It's not necessary to initialize existing solutions. 

Editing relations was covered in tutorial 2. Remember that the use, syntax and examples of all quaestor functions are available in the wiki: functionsoverview.

  Run a solution for Error again at the same frame.

The error now turns out to be zero. This makes sense, as DQUAD is a quadratic interpolation method and this waterline is defined by a quadratic function as well.

4    Integration : waterplane area 

By numerical integration, the waterplane area can be calculated. Therefore, the dimensionless length and width should be transformed to the real length and width. For the width we can use the existing relation for B_Frame, for the length in meters we add a relation.

  In the class Geometry, add the following relation

Which is the position in meters along the longitudinal direction of the ship. By now you should know a proper dimension, reference and determined by value should be provided. 

To calculate the waterplane are for a certain ship, we would like Quaestor to integrate the width with respect to X. The dimensionless width table is already present in our object Hull, let's use it in the integration.

 In the Top Goals class, add the following relation

which is the waterplane area in square meters (m^2).

The syntax used may seem a bit confusing at first.

First, we take a look at the arguments after Hull. Data within the object Hull will be used in the integration, but yet it only contains dimensionless frame numbers and dimensionless widths. By putting (@X, @B_Frame, Lpp, B) behind it, you ask Quaestor to calculate X and B_Frame using data from within Hull, using Lpp and B from outside Hull and add all these parameters to the object Hull. You do actually use the object Hull as a function to calculate other parameters. This is a very powerful ability of Quaestor (see also QuaestorSyntax).

The arguments for the INTEGR function are as follows

• HULL(..) is the object from which data will be used, now containing the columns X and B_frame
• 2 is the number of dimensions = always 2 using INTEGR.
• @X refers to the column that will be used as the parameter X in the integration: X
• @B_Frame refers to the column that will be used as the parameter Y in the integration: B_Frame
• 2 is the mode of integration, either Riemann (0), Trapezium (1) or Simpson (2). The latter is chosen here.
• X_from=0 and X_to=Lpp are the parameters between wich will be integrated.
  
   Run a solution for the waterplane area, using the process manager. 

The waterplane area is calculated for every variation in length and breadth of the ship.

 In the solution tree, select the object Hull or open it as TeLiTab text (remember that F2 is the shortkey to open it as TeLiTab text).

The telitab of the object hull is visible now (figure 3). Because we solved a multiple case problem, the object has been used as function and reused for every case. Only the contents of the last case calculated remains in the object.

Figure 3   Contents of the object Hull 

5    Calculating the stability

The initial stability (GM) of the ships can also be calculated using numerical integration.

  Add a new class, called Stability, as sub-class of Geometry. Note that a subclass is created by naming it Geometry.Stability. Add the following relation to this class:

The dimension of all parameters is meters, and they are defined by a relation (which follows). GM is a Top Goal and should be moved to Top Goals/Undefined.

In the class Stability, enter the following relations.

Distance between keel K and center of buoyancy B, select KB and right click  "New Relation..." (or Ctrl+N). By selecting KB it is automatically presented as left side part of the relation:

Distance between keel K and center of gravity G in meters:

Distance between center of buoyancy B and metacenter M in meters:

Moment of inertia of the waterplane area in m^4:

Please note (and you might have experienced this in the previous tutorial too) that the parameter Moment_of_Inertia might still have a red cross and an Error warning in the knowledge base although you have provided the relation and a dimension. When this is the case, for sure you still have the Hull object in focus in a solution of the Workbase. When this is the case, Quaestor communicates that the relation you have provided cannot be used inside this object (and thus you have a red cross for Moment_of_Inertia). When you select a top node of a solution, you will see that the Moment_of_Inertia parameter is properly defined.

Third power of the width at a specified frame number in m^3: 

Do not forget to provide all dimensions and references, and change the Determined by fields to SYS: System/equation

Note that the syntax of the moment of inertia relation is similar to that of the waterplane area relation.

 Run a solution for the initial stability GM of every possible ship, by selecting the ship object and GM as task in the process manager.

If the object Hull in the solution is now opened again, you'll see that the columns X and B3_frame are added to the object, whose values are calculated for every case of Frame and Rel_B within Hull. Note that the parameter B_Frame is also added, as it is used to calculate B3_Frame. See figure 4. Again, only the values of the last ship (last input case) are stored in the object.

Figure 4    Object Hull with added parameters 

6    Comparing the stability to other ships

Finaly, we'd like to compare the obtained values for GM to other ships. Data of the GM value for a certain ship length is available, so we only have to compare them. We'll use the least squares method to obtain an average GM value for a certain length, using the data of some other ships.

   Add the following new relation to your Top Goals:

Enter the following text in the Data slot  in the Expression Editor of the relation GM_Check:

See figure 5. GM_Check is in meters. 

The LEASQ interpolation function now returns an average GM value in meters for a certain Lpp. The syntax TEXTITEM$(nr) is often used in Quaestor, and refers to a template, text item or TeLiTab in the Data slot of the Expression Editor. In this way, data that is not included in the knowledgebase but is only used in a particular function (as in our relation) can be used.

Note that the syntax of the relation uses TEXTITEM$(nr), but in the data slot the syntax TEXTITEM1= is used. The TeLiTab is written between two | characters. Multiple textitems can be available in a relation (, TEXTITEM$(1)). Furthermore, please note that "the devil is in the detail". Small errors in your syntax will always be a problems for software such as Quaestor. Therefore, keep checking the correctness of your syntax.

Figure 5   Including data in the relation by using textitems

        Run a solution using the Process Manager. Select Waterplane_Area, GM and GM_Check as goals. Note that you are able to select serveral goals at once...

To browse through results, it might be convenient to maximize the workbase window. Remember that you can always return to the default lay-out using the menu sequence Window->Standard. The result of the last solution is shown in figure 6.

Figure 6   Output of the solution, if the workbase window is maximized an inspection of the results is more easy 

7    Check

You can verify your results by comparing it to [Tutorial 3 Finished]

<< Back to tutorial 2 -- Continue to tutorial 4 >>