Geometric Objects#

There are several types of geometric objects currently implemented in pymead that can be useful in creating airfoil objects: points, lines, Bézier curves, polylines, airfoils, and multi-element airfoils.

Points#

Points are the most fundamental geometric object in pymead, consisting only of \(x\) and \(y\) parameters. Points are used as the basis for all other types of objects. For example, lines are always drawn between two points in pymead’s geometry module.

Creation#

To create a point, first either press the P key or left-click on the “Point” button in the toolbar (see the image below). Then, left-click on the geometry canvas to place the point. You can continue clicking on the canvas to create additional points. Press the Esc key to stop creating points.

To add points from a file, select File → Import → Points from File from the menubar. Then, click the “Choose File” button to select a text file storing a set of points. The points should be stored row-wise, with the two columns representing the \(x\) and \(y\) value for each point. The text file must have a .txt, .dat, or .csv file extension and be space-delimited or comma-delimited.

Note

Adding points from a file is designed for convenient construction point or Bézier control point import, not for creating points on an airfoil surface. To import an airfoil directly from a set of coordinates in a text file, use the “Web Airfoil” tool (the W shortcut).

../_images/point_dark.svg — Adding a point#

../_images/point_light.svg — Adding a point#

Modification#

There are several ways to change the location of a point object:

Click and drag: Hold down left-click on the point in the geometry canvas. Then, move the mouse to the desired location while still holding left-click.
Arrow keys: To make small changes to the point’s position, left-click once on the point. Then, press or hold down any of the arrow keys to move the point in the corresponding direction. To make larger changes, hold the Shift key while pressing/holding the arrow keys.
Number keys: To directly the specify the value of the point’s \(x\) or \(y\) position, first double-click on the point’s name in the parameter tree (left-hand side of the figure below). Then, press the button corresponding either to the \(x\) or \(y\) value in the dialog that appears. In the final dialog, modify the value in any of these ways:
- Click the up/down arrows on the right-hand side of the value spin box.
- Click inside the value spin box and use the up/down arrows on the keyboard for small changes or the Page Up/Page Down keys for larger changes.
- Select the numerical value by either triple-clicking it or by clicking inside the value spin box and pressing Ctrl+A. Then, use the number keys on the keyboard to specify a value.

../_images/point_mod_dark.svg — Specifying a point’s \(x\)-value#

../_images/point_mod_light.svg — Specifying a point’s \(x\)-value#

Deletion#

To delete a single point, select the point by either left-clicking on the point in the geometry canvas or by left-clicking on the point’s name in the parameter tree. Then, delete the object by either pressing the Delete key or by right-clicking on the point’s name in the parameter tree and left-clicking the “Delete” option.

To delete multiple points at once, first select the points by either left-clicking on one point at a time in the geometry canvas or by holding Shift or Ctrl and clicking the names of the points in the parameter tree. Then, delete the points by either pressing the Delete key or by right-clicking on any of the selected points’ names in the parameter tree and left-clicking the “Delete” option.

Other Important Bits#

To prevent the parameter/design variable space from becoming cluttered, the \(x\)- and \(y\)-values of each point do not show up under “Parameters” in the parameter tree by default. To expose the \(x\) and \(y\) parameters of a particular point, right-click on the point’s name in the Parameter Tree and click “Expose x and y Parameters”. For a point named “Point-1,” this will add “Point-1.x” and “Point-1.y” to the “Parameters” sub-container in the parameter tree.

To allow the optimizer to change the value of either or both of these parameters, right-click on the newly created parameters in the parameter tree and click “Promote to Design Variable.” To remove the point’s \(x\) and \(y\) parameters from the parameter/design variable space, right-click on the \(x\) or \(y\) parameter’s name in the parameter tree and click “Cover x and y Parameters.”

Lines#

Lines serve two major purposes in pymead: blunt trailing edge construction and flat airfoil surface section construction. In fact, pymead requires that lines be used to close blunt trailing edges for Airfoil objects to be created. These are created by default when adding an airfoil from Airfoil Tools; take the NASA supercritical airfoil sc20012 (generated with the sc20012-il code in the “Web Airfoil” tool) for example:

../_images/te_line_sc20012_light.svg — Trailing edge line for the sc20012 airfoil#

Notice that two lines are created: one from the trailing edge point to the trailing edge upper surface point, and another from the trailing edge point to the trailing edge lower surface point. More information about airfoil trailing edges can be found in the “Airfoil” section.

Creation#

To create a line, first either press the L key or left-click on the “Line” button in the toolbar (see the image below). Then, left-click two different points in the geometry canvas to add a line between them. Continue to select pairs of points to add more lines, or press the Esc key to stop generating lines.

../_images/line_dark.svg — Adding a line#

../_images/line_light.svg — Adding a line#

../_images/lines_dark.gif — Adding/deleting multiple lines#

../_images/lines_light.gif — Adding/deleting multiple lines#

Deletion#

To delete a single line, select the line by left-clicking on the line’s name in the parameter tree. Then, delete the object by either pressing the Delete key or by right-clicking on the line’s name in the parameter tree and left-clicking the “Delete” option. Lines can also be deleted by right-clicking on the line in the geometry canvas and selecting Modify Geometry → Remove Curve from the context menu that appears.

To delete multiple lines at once, first select the lines by holding Shift or Ctrl and clicking the names of the lines in the parameter tree. Then, delete the lines by either pressing the Delete key or by right-clicking on any of the selected lines’ names in the parameter tree and left-clicking the “Delete” option.

If either of the points associated with the line are no longer needed, the line can also be deleted by deleting either of the points to which the line is attached (see the GIF above).

Bézier Curves#

The core of pymead’s geometric parametrization code is built around Bézier curves. Bézier curves are a subclass of B-splines, which are in turn a subclass of Non-Uniform Rational B-Splines (NURBS). More specifically, Bézier curves are uniform, non-rational, clamped B-splines. Uniform means that Bézier curves have uniform knot vectors. Non-rational means that Bézier curves can be represented by non-rational polynomials (e.g., \(t^2 + 1\) as opposed to \(\frac{t^2+1}{t^3+2}\)). Clamped means that Bézier curves have the useful property that they start at their starting control point and end at their ending control point. In addition, the local curve tangent at the endpoints is equal to slope of the line connecting the first and second or last and second-to-last control points. This property is shown visually in the figure below.

../_images/cubic_bezier_dark1.png — Cubic Bézier curve#

../_images/cubic_bezier_light1.png — Cubic Bézier curve#

Bézier curves also have the property that the control points have global control over the shape of the curve, which is not generally the case with NURBS curves. Global control means that changing the location of a control point changes the shape of the entire curve, except at the curve endpoints. This is illustrated in the animation below.

../_images/cubic_bezier_animated_dark.gif — Cubic Bézier curve animation#

../_images/cubic_bezier_animated_light.gif — Cubic Bézier curve animation#

The degree of the Bézier curve (one less the number of control points) influences the amount of global control each individual control point has. Increasing the number of control points further localizes the control of each individual control point.

Mathematical Description#

Bézier curves are described by the following mathematical formula:

\[\vec{C}(t)=\sum_{i=0}^n \vec{P}_i B_{i,n}(t)\]

where \(n\) is the order of the Bézier curve (equal to one less the number of control points); \(\vec{P}_i\) is the control point vector at the ith index of the form \([x_i,y_i]^T\); \(t\) is a parameter, generally in the range \([0,1]\) that describes the position on the Bézier curve; and \(B_{i,n}(t)\) is the Bernstein polynomial, given by

\[B_{i,n}(t)={n \choose i} t^i (1-t)^{n-i}\]

Note that the shape of this Bernstein polynomial is influenced only by the curve degree. The curve itself is affected by a combination of the curve degree as well as the location of the control points given by \(\vec{P}_i\), which are effectively weighting values for the Bernstein polynomial.

Creation#

To create a Bézier curve, first either press the B key or left-click on the “Bézier” button in the toolbar (see the image below). Then, left-click at least three different points in the geometry canvas to and press the Enter key to add a line between them. Continue to select sets of three or more points to add more Bézier curves, or press the Esc key to stop generating curves.

Note

Creating a Bézier curve with only two control points (a linear Bézier curve) is effectively the same as generating a line! Thus, this is not allowed in pymead for simplicity; please use a line instead. Lines can be used in conjunction with Bézier curves to produce Airfoil objects.

../_images/bezier_dark.svg — Adding a Bézier curve#

../_images/bezier_light.svg — Adding a Bézier curve#

Inserting & Removing Control Points#

To insert a control point to an existing Bézier curve, first create a new point. Then, right-click on the curve in the geometry canvas and select Modify Geometry → Insert Curve Point from the context menu that appears. Now, select first the new control point to be added, then the control point that should precede the new control point in the control point sequence. The curve should now be updated to include the new control point, with the curve’s order increased by one. To remove a control point from the curve, simply delete the point using any of the options in the Deletion section.

../_images/add_control_point_dark.gif — Adding a Bézier curve and inserting a control point#

../_images/add_control_point_light.gif — Adding a Bézier curve and inserting a control point#

Deletion#

To delete a single Bézier curve, select the Bézier curve by left-clicking on the curve’s name in the parameter tree. Then, delete the object by either pressing the Delete key or by right-clicking on the curve’s name in the parameter tree and left-clicking the “Delete” option. Curves can also be deleted by right-clicking on the curve in the geometry canvas and selecting Modify Geometry → Remove Curve from the context menu that appears.

To delete multiple curves at once, first select the curves by holding Shift or Ctrl and clicking the names of the lines in the parameter tree. Then, delete the lines by either pressing the Delete key or by right-clicking on any of the selected curves’ names in the parameter tree and left-clicking the “Delete” option.

If only two or fewer of the points associated with the curve are no longer needed, the curve can also be deleted by deleting at least all but two points to which the curve is attached.

Airfoils#

An airfoil in pymead is simply defined as any closed set of curves containing a leading edge point, a trailing edge point, and, optionally, a trailing edge upper surface point and a trailing edge lower surface point. Airfoils can be created in several different ways in pymead.

Creation#

The primary method which gives full control over the shape of the airfoil is started using the F key. This method creates an airfoil from any closed set of lines, polylines, or Bézier curves. After clicking the “Airfoil” button or pressing the F key, first left-click on the leading edge point for the airfoil (either in the geometry canvas or parameter tree). Ideally, this point is the endpoint of two of the curves in the airfoil. Then, left-click on the trailing edge point of the airfoil.

For an airfoil with a thin trailing edge, this point must be the endpoint of the airfoil curves. For an airfoil with a blunt trailing edge, this is not the case. In either case, the trailing edge and leading edge points define the chord line for the airfoil, which also defines the angle of attack. If you are designing a thin airfoil, simply press the Enter key after selecting both the leading edge and trailing edge. If the points are selected properly and the airfoil is valid, the airfoil will now be shaded, and you can hover over the shaded region to see the name of the airfoil.

For the case of a blunt trailing edge airfoil, select both the trailing edge upper surface point and trailing edge lower surface point, in that order. Note that there must be curves connecting the trailing edge point to both of these two points. These curves are normally lines, but they do not have to be. If the points are selected properly and the airfoil is valid, the airfoil will now be shaded (without having to press the Enter key), and you can hover over the shaded region to see the name of the airfoil.

../_images/airfoil_dark.svg — Adding an airfoil with a blunt trailing edge#

../_images/airfoil_light.svg — Adding an airfoil with a blunt trailing edge#

Airfoils can also be added as polylines from coordinates. These coordinates can originate from Airfoil Tools or from a text/dat file. To access either of these methods for creating an airfoil, press the “Web Airfoil” button in the toolbar or press the W key. This will pop up a dialog window that looks like this:

../_images/web_airfoil_dark.svg — Adding an airfoil from the web#

../_images/web_airfoil_light.svg — Adding an airfoil from the web#

Type in the tag for the airfoil (the identifier of the airfoil as shown on Airfoil Tools, not the full name of the airfoil) in the “Web Airfoil” field. Then, press the Enter key. An Airfoil object should now be present in the geometry canvas representing this airfoil. This airfoil contains only a polyline (a series of lines connecting each subsequent pair of airfoil coordinates) and two Line objects if the airfoil has a blunt trailing edge.

To load an airfoil from a .txt, .dat, or .csv file, press the “Web Airfoil” button in the toolbar or press the W key. Now, select the “Coordinate File” option from the drop-down menu. Then, press the “Select Airfoil” button to select a file. The file has to have one of the aforementioned extensions, and the coordinates should be listed row-wise, starting at the trailing edge upper surface point and moving counter-clockwise to the trailing edge lower surface point. The file must also be space-delimited.

../_images/airfoil_from_file_dark.svg — Adding an airfoil from a text file#

../_images/airfoil_from_file_light.svg — Adding an airfoil from a text file#

Deletion#

An airfoil can be deleted by left-clicking the airfoil’s name in the parameter tree and pressing the Delete key or by right-clicking the airfoil’s name in the parameter tree and clicking the Delete option from the context menu that appears. Alternatively, an airfoil can be deleted by deleting any of its associated points or curves.

Multi-Element Airfoils#

Multi-element airfoils do not have any inherent geometric representation, but are simply unordered collections of Airfoil objects. These multi-element airfoils must be created, even in the case of a single airfoil, to run an MSES analysis or optimization.

Creation#

To create a multi-element airfoil, select the “Multi-Element Airfoil” button from the toolbar or press the M key. Then, hold Shift or Ctrl while left-clicking each of the airfoil names from the parameter tree. Then, click anywhere on the geometry canvas and press the Enter key.

../_images/mea_dark.svg — Adding a multi-element airfoil#

../_images/mea_light.svg — Adding a multi-element airfoil#

Deletion#

A multi-element airfoil can be deleted by left-clicking the multi-element airfoil’s name in the parameter tree and pressing the Delete key or by right-clicking the multi-element airfoil’s name in the parameter tree and clicking the Delete option from the context menu that appears.