When the applet loads, a menubar and blank x- and y-axes are displayed. The menubar is shown at left. The applet requires at least Java JRE 1.4 (aka Java 4) to run; if it does not load, this is a likely source of the problem. The JRE 1.4 plugin is available for most platforms, including Windows, Linux, and Mac. The menubar provides means to enter the coordinates of any number of nodes, set a predefined pattern of nodes, remove nodes one-by-one or all at once, choose for which node the shape function should be calculated, choose a shape function formulation, and calculate the value of the shape function at any given point inside the element. There are also toggle buttons to choose whether to display a plot, the axes, and node labels, and whether to allow node placement by clicking in the plot area. |

Form a square element by inputting the coordinates of its nodes. We enter it as shown, node by node. Each node is displayed as it is inputted.

Turn on plotting by clicking the Plot button. The shape function is plotted, by default for the first node we entered.

Now click "Remove all nodes." Then use "Auto nodes" to form a regular pentagon, as shown.

Our pentagonal element is plotted.

Click the button for no node labeling. Note that we may use the Detail slider to adjust the fineness of the plot. However, finer plots require much more computing power to display and rotate, and may make the applet run very slowly.

Dragging the mouse around the plot area allows the perspective to be changed. The mousewheel controls zooming.

Click "Remove all nodes" again. This time, use "Auto nodes" to form a quadtree-type element. Alternatively, the random nodes button would form a square element with the given number of nodes placed at random inside it.

We form an element with 9 regularly-spaced edge nodes.

The "Select a node" area allows us to cycle through all of the nodes and choose the one for which to form the shape function. We choose one of the side nodes. Note that the "Remove a node" area works very similarly.

The resulting plot.

Click "Remove all nodes" again. If "Click Nodes" is enabled, we can input node positions by clicking on the plot itself. We can remove nodes by right-clicking them.

We enter our nodes in this way, and when plotting is enabled the shape function for the first node we formed is plotted.

We change node labeling and choose a different node.

The shape function for this interior node is plotted.

Choosing the MVC formulation, we can form concave polygonal elements.

The Function Value button allows us to find the value of the shape function at any given point inside the element. We choose (0,0). The value here is about halfway between zero and one, as expected from the plot.

Observe that, because the polygon has more than 3 nodes and is not convex, many of the formulations are disabled. If we were to remove node 6 and place it outside the element, making the polygon convex, the Wachspress formulation would be available to us again.

Note also that the MVC formulation is only available for convex polygons or simple nonconvex polygons. Shown above is a node configuration that is neither convex nor simple (since the path 1-2-3-4-5-6-7-8-9-1 crosses itself) and therefore not suited for MVC.

The "Generate EPS" button creates the contents of an EPS graphics file from the current plot. The contents are displayed in a pop-up window, from which you can copy everything (CTRL-A, CTRL-C) and paste it into a file (CTRL-V) to be saved with the .eps extension.

This work was supported in part by the National Science Foundation under Grant #DMS-0135345.