# Introduction

PolyCAD provides a range of hydrostatic and stability analysis of hull forms represented as sections, facets, single or groups of surfaces. The analysis is performed by discretising the hull form into Calculation sections. Upright hydrostatic tables, Section Area Curves and Floating Hydrostatics and Stability calculations are available in separate reports or calculation tools. In addition, Hydrostatics and Section Area Curve calculations analysed at the design waterline are available as tools in the design environment and update dynamically as the design is changed.

# Calculation Sections

For every entity in PolyCAD that supports the calculation of volume and floating characteristics a set of transverse cross-sectional geometry is generated called Calculation Sections. Using Calculation Sections, all the hydrostatic characteristics of a hull surface can be robustly analysed.

This approach is typified by numerical integration techniques such as the
Trapezoidal Rule and
Simpson's Multipliers which provide linear and
quadratic interpolation respectively. These techniques rely on uniformly
spaced sections, something which is not always practical for hull forms.
Often it is desirable to use more closely space sections where there is a
variation in shape, such as the ends of the hull. To support this, PolyCAD
generate more closely spaced Calculation Sections where they are needed.
During calculation, it evaluates the spacing of the Calculation Sections to
determine the Multiplier values in a similar way to the derivation of
Simpsons 1^{st} and 2^{nd} Rules, except for non-uniform
spacing. Where the spacing of sections do not support the use of this
approach, for example, if there are discontinuities in the volume, a
Trapezoidal approach may be used across the affected area.

Calculations Sections are produced using an iterative approach which determines whether the spacing between sections results an accurate enough estimation of volume. Additional sections are generated between adjacent sections until a tolerance is met. Volumes with uniform cross-section will often have a much lower number of sections compared with a volume that has a varying cross-section along the length, and hull forms will often have more sections at the end of the volume compared with midships. Since the spacing of Calculation Sections is dependant on the complete volume, additional points of calculation are introduced at the ends of the waterline for case where the volume is partially immersed.

Calculation Sections should be checked before any analysis to ensure the software has correctly interpreted the volume. The generation process must interpret the surface definition, the implied symmetry and the incorporation of any separate regions of volume into a single section. A volume is considered to be symmetrical if the limit closest to the centre plane (Y = 0) is within 0.1 design units of this plane.

While the quality of the volumetric estimation can be managed, estimation of surface area using this approach suffers a limitation that it can only accounts for the component of area between sections parallel to the integration axis. The leads to inaccuracies especially in shapes where the surface normal is parallel to the integration axis. A sphere is a good example of this scenario where even increasing the number of sections does not lead to an estimate of surface area that may be considered accurate.

An alternative is to evaluate volumetric and hydrostatic properties using surface integration, as it proves an accurate estimation of surface area and better handles hydrostatics at extreme floating attitudes. However, it also requires that the user prepare a full enclosed volume without any gaps. Since this takes extra time, gets in the way of the design process, and requires that the user understand how to prepare their hull form correctly, this expectation is considered unreasonable for the novice designers PolyCAD is tailored for. While the code within PolyCAD does support hydrostatic calculation based on surface integration but there is no available to use it from the application.

# Upright Hydrostatic Curves

The Hydrostatic Curves Calculation lists the hydrostatic properties for upright waterlines of the hull form. These results table can be copied to the Windows Clipboard in plain text or HTML formats, printed or a formatted HTML report generated for copying or printing. Tabulation of additional values is achieved using the selection drop-down on the menu bar. Quantities are fully described in the selection drop-down and on the tooltips of the table header, which are displayed when the mouse cursor is hovered over the top.

Draughts are defined as series of numbers. These can be both a list of comma separated draught values and a numerical sequence definition. This consists of three comma separated values enclosed in round brackets. The values define the first, last and step value respectively, i.e. “(Start, End, Step)”. Multiple individual comma separated draughts and sequence definitions may be mixed in the text. Further information is provided by the tooltip of the Draught definition text box.

Further presentations of the tabulated results can be achieved by adding charts. The quantities presented in each chart are selected in the same way as the tabulated data.

# Cross Curves Curves

This calculation produces the Cross Curve analysis for the selected Hull Volume. Cross (KN) Curves are equivalent to a set of GZ curves generated for a nominal vertical centre of gravity (VCG), usually zero and are used to manually derive a GZ for a loading condition once displacement, trim and vertical centre of gravity. Typically, KN Curves are graphed across displacement for each angle of heel. PolyCAD also provides a chart of the KN Curve data presented in the traditional GZ curve view with a curve for each displacement as this provides insight into change in shape of the curve as displacement is increased.

The calculation takes draughts and heel angles as number sequences, and the nominal VCG as inputs. Tabulated data may be copied to the Windows Clipboard as plain text or HTML, printed or a formatted HTML report produced and copied to the clipboard or printed. Charts may be copied or printed.

# Upright Section Area Curves

The Section Area Curve Report charts the Section Area Curves for the draughts specified in text box. On a second Tab page, the values of the section area curves are listed for each calculation section respectively. The chart can be customised, copied to the Windows Clipboard and Printed using the Right-Click menu. The Section Area Curve Values can be copied to the Windows Clipboard by selecting the range of values desired and pressing Ctrl+C.

As with the Upright Hydrostatics Curves, draughts are defined as a text series. These can be both a list of comma separated draught values and a numerical sequence definition. This consists of three comma separated values enclosed in round brackets. The values define the first, last and step value respectively, i.e. "(Start, End, Step)". Multiple individual comma separated draughts and sequence definitions may be mixed in the text.

# Hydrostatic Docking Window

In addition to the report windows, the upright hydrostatics at the Design Waterline can be calculated, displayed and updated as you are modifying a hull form. The Hydrostatic Docking windows can be found on the Surfaces, X-Topology Surface Modelling and Hull Generation Menu Tabs. It can be Docked to the side of the main application Window. Once the current geometric entity is selected and the Design Draught updated to the desired values the results will be presented every time the geometry is changed. The Design Draught can be edited in the cell in the table.

While using the hull transformations tools, the hydrostatics display will show a comparison of the characteristics between the original parent hull form and the currently selected derived hull surface.

# Section Area Curves Docking Window

The Section Area Docking window functions similarly to the Hydrostatic Calculation displaying the section area curve, updating as the hull geometry is changed. The Calculation can be found on Surfaces, X-Topology Surface Modelling and Hull Generation Menu Tabs. The display is initialised by selecting the entity you want to show the Section Area curve. The Design Draught can be edited on the toolbar and is synchronised with the design draught used by the Hydrostatics Calculation. The chart display can be modified using the Right-Click menu on the chart.

While using the hull transformations tools, the Section Area Curve display will show a comparison of the section area curve between the original parent hull form and the currently selected derived hull surface.

Either of the Hydrostatic and Section Area Curve calculations are active, the respective design waterplane and centre of buoyancy is shown is the design area.

# Free Body Hydrostatics and Stability

The Free Body Hydrostatics and Stability calculation allows the analysis to go beyond the upright condition and look at free floating trimmed and heeled conditions and GZ curve calculation.

The Initial Conditions Tab defines the Floating Condition of the Hull Form, this may be defined using:

- Draught, specifying in addition either upright or trimmed condition and the VCG (Vertical Centre of Buoyancy.
- Displacement, specifying in addition either upright, trimmed or the precise position of the VCG.
- The Parameteric Weight Estimate where weights can be defined as Points or parametrically generated from the Area or Volume of an entity, or using a Structural Weight estimate based on the area of entity accounting for plate thickness and composition of stiffening and bracing (brackets).

The positions of the Aft and Forward Perpendiculars are required to complete the Trim definition. A damage case may be specified to review damage hydrostatics although this functionality is incomplete and currently under development. Through the Initial Conditions definition, the precise position of the centre of gravity can be determined. The hydrostatic relating to the floating position are displayed in the grid and can be copied to the Windows Clipboard.

The GZ Curve may be calculated on the following tab. Heel angles are defined as a text series. These can be both a list of comma separated draught values and a numerical sequence definition. This consists of three comma separated values enclosed in round brackets. The values define the first, last and step value respectively, i.e. "(Start, End, Step)". Multiple individual comma separated draughts and sequence definitions may be mixed in the text.

The floating attitude of the vessel at each heel angle can be visualised by clicking and dragging the mouse cursor across the GZ Curve Chart or by selecting a row in the GZ Table. The columns displayed in the GZ Table can be extended using the Right-Click menu and detailed hydrostatics for the heel angle are displayed on the Floating Position Tab. Copying to the Windows Clipboard and Printing is possible.

The Points of Immersion Tab extends the GZ Calculation addition further information about the immersion angle of the deck edge or points of interest which may be, for example, a downflooding point. The Deck Edge selection is based on named Polylines available in the modelling environment. The edges of surfaces may be extracted using the Extract Edge tool available on most surfaces.

Once the GZ curve is calculated, the angle of immersion and reserve freeboard at the floating equilibrium can be calculated and this values displayed on the Points of Immersion Page. In addition, the immersion angles will be displayed on the GZ Curve and the immersed points illustrated in the 3D View.

# Draught Measurement Definition

The 3D presentation in Free Body Hydrostatics and Stability provides the opportunity to visualise the definition of draught. Draught dimensions are measured perpendicularly from the water plane to the reference position on the base line, the x-axis. Furthermore, the immersed draught dimension can be visualised which represents the perpendicular distance from the waterplane to the deepest part of the hull form, as measured from the Calculation Sections.