Copy-paste optimization in creating array of shapes

Assume that you are using a painting/diagraming software and you’ve created certain shape (in our example it is just a square). Also, assume that you want to array this shape for 100 times in one row and the software does not support arraying. So, what are you going to do in this case? Are you going to draw the shape for another 99 times? Or are you going to copy the drawn shape and paste it 99 times? Or what?

I thought about this problem and found that I can copy the original shape and paste it x times to create a group of shapes, then I’m going to copy the group and paste it y times to get the final array. Consider the following example and figure in which we want to create an array of 10 squares. There are many solutions like the following:

Solution	Number of copy-paste/redrawing operations after drawing the original shape
Draw the square 10 times	9
Draw the square once, copy once, and paste it 9 times	9
Draw the square once, copy it, and paste it once. Now, you have a group of two squares; copy the group and paste it 4 times (1)	5
Draw the square once, copy it, and paste it twice. Now, you have a group of three squares; copy the group and paste it 2 times. Finally, copy the square and paste it once. (2)	5
Draw the square once, copy it, and paste it 3 times. Now, you have a group of four squares; copy the group and paste it once. Finally, copy the square and paste it twice. (3)	6
Draw the square once, copy it, and paste it 4 times. Now, you have a group of five squares; copy the group and paste it once. (4)	5

 

So, given those solutions, what is the optimum or minimum time consuming solution. This is the aim of this post. This method can be used for both linear and circular arrays. As we noticed in the previous table that there may be many optimum solutions. In the previous example the minimum number possible of operations is five.

The following relation governs the copy and pasting by groups

To find the best size of the group (number of elements in one group) we should calculate the number of operations for each value ranging from 1 to the half of array size. For extremely large arrays, you may need to construct the group itself using this method.

At the end, I want to say some thing … I’m using Microsoft Office Visio as the drawing/diagraming software and it –in fact- supports two dimensional arraying, but I used this method before I have knew that it does.

The other thing I want to say is that I think that this method is going to be very useful for those who want to create a helical Savonius blade die using discrete (very thin) elements. In this case, you should know blade’s height and specify the thickness of the element and so you get number of elements by division of the previous values.

Helical Savonius, axis of twist and axis of rotation

In a simple Savonius wind turbine there are two types of gaps between the buckets that can be specified based on the design requirements: the primary gap and the secondary gap, but it was common to set the secondary gap to zero. In our graduation project, we were asked to create a two-bladed helical Savonius VAWT (90 degrees twist angle) with different values of the primary gap to know the best dimensionless value (gap distance divided by bucket diameter) of the primary gap that gives highest power coefficient. We thought about creating two helical (twisted) blades and then use them with different primary gap values by sliding them in or out on a straight line (as the same method you can think about when you want to change the primary gap for a conventional Savonius). Since we were in the brain storming phases and not taking what we were saying seriously, we didn’t release that this was a mistake. When we tried to draw the cad model of the blades and bottom and top caps to just test how to draw the blades in Autodesk Inventor, we got assembly errors with a large exclamation mark on our faces !!!!. After a wise thinking we found that we should create a couple of blades for each value of the primary gap. To understand why we reached this conclusion, let’s consider the following figure that shows two cases:

§  In the first case: the axis of twist coincides with the axis of rotation. Therefore, the primary and secondary gaps are constant for all sections of the blade. Simply, axis of rotation, axis of twist of the first blade, and axis of twist of the second blade are coincident.

In the second case: the axis of rotation does not coincide with the axis of twist and the axes of twist of both blades also don’t coincide. In other words, we have 3 axes at different locations. Taking the bottom edge profile (buckets) as a reference, we will find that we increased the primary gap and the secondary gap is zero. Moving up along the axis of rotation, we will find that the values of the primary and secondary gaps are changing from section to section. So, for this reason, the turbine will not perform the same at different phase angles.