Comparing different tessellations for a dome shape

What happens if you construct the three tessellations based on the same set of points?

Below you find 2 combined drawings. The hexagonal part is devided in two ways, the lower one comes closer in comparisation with the voronoi and delaunay part.
In these two dwawings the amount of points are chosen by using factors of 2. Thats why the structure is more regular. The second one is the best. Because the density of the hexagonal structure is more equal to the others.


For the research it would be interesting to test the three types of structures and compare them. In the next scheme, I defined a try-out set up of the comparisation.

These domes are all constructed based on the same set of points. The amount of points per ring are random. Thats why the Delaunay and Voronoi version are a bit irregular. The errors in the hexagon sample are because it took too long to draw all lines manually and I arrayed one quarter. but it gives an idea. The size of the dome can be a parameter as well!! (unlike i noted on this picture) But to compare them, the size should be the same for all 3 domes.

How to define the circels and points (in GC), which define the tesselations?

In the picture below is illustrated how a plane is projected onto the dome. the set-up size of the plane has a diameter of 32 m, while the dome itselves has a diameter of 20.37 m. I defined the plane first, in order to get a regular division of rings possible.
The overal size can be changed afterwards by adjusting the diameter of the projection plane.

RULES RINGS
The rings can vary in number. They are always linear distributed on the plane.
Number of rings: n_ring
Diameter of projection plane: d_plane
The max distance between the rings is 4 meter.
The min distance between the rings is 0.5 meter.
The formula which defines this relation is: 1/2 = smaller than: (0.25*pi*d_plane)/n_ring = smaller than: 4

RULES POINTS
The points on each ring can vary in number. They are always linear distributed along the ring.
A similar system rules the division of points onto the rings
Number of points: n_point
Diameter of ring: d_ring
The max distance between the points is 4 meter.
The min distance between the points is 0.5 meter.
The formula which defines this relation is: 1/2 = smaller than: (d_ring*pi)/n_point = smaller than: 4.




definition of the projection plane & rings


Diagram of GC hierarchy.

Proposal; More Details

An other tessellation which is worth taking a closer look at, is the previous used hexagon diagram. In the version below, its a bit more compicated, in the sense that the density of the hexagonal structure varies between the circles. In this case, the radial division is set to 132 points in all sections. The circular division varies in density between the different rings.

other options of this one are:
- varying also the radial density between the rings.
- inserting a random factor when defining the points, to get a more unsteady look of the structure.


An other configuration, Hexagon Structure
The parameters are in this case:
n_c (number of main circels)
p_cALL (density of points along all circles)
n_cSEC_1 (circular density of secondary circles in between the 1st and 2nd circel)
n_cSEC_2 (circular density of secondary circles in between the 2nd and 3rd circel)
n_cSEC_n (etc)



The folowing image illustrates what happens if we construct the Voronoi Diagram, based on the same configuration of the model as in previous image of the Delaunay triangulation.


Same configuration, but Voronoi Diagram

Proposal

Okay, here is my proposal: Since the wind trajectories are not steady enough to use as a starting point, I decided to keep the basic set-up a bit more general in order to have some freedom in the model it selves to develop during the optimization. In the way i propose now, I kept a few things in mind:
1. the aesthetic quality of the tessellation
2. the variabiliy of the model set-up
3. the basic structural needs of a dome structure (rings & ribs)
4. the basal connections to the ground

Since rings are one of the basic needs for a dome structure, I take circular distribution of points as a starting point in this proposal.
The bars defined by the Delaunay triangulation will automatically generate sets of "ribs" which will lead the vertical forces to the ground, the idea is that all irregular ribs will work together as a network of ribs.
The density of distributed points along the outer (lower) ring (not drawn in this picture yet), will define the amount of basal supports the dome will have.


Fig. Drawing of possible configuration Delaunay Triangulation:
amount of points circle1:20 (d_c1:20)
d_c2:40
d_c3:20
d_c4:15
d_c5:50
d_c6:80
d_c7:75
d_c8: invisible in this drawing


The setup is arranged in this way:
1. definition of the rings. they are based on regular vertical divisions of the dome, as pointed out in the drawing.
2. along each circle we set a variable amount of points. (Parameters in GC will be: point density circle1, d_c2, d_c3, ..., d_c9).
3. The points will be connected by a Delaunay triangulation. This will create a closed lattice structure. Each possible structure will look different and perform different.
4. We can optimize this model according to different load cases or even different dome shapes.

note: we can try the Voronoi diagram applied on this model as well, I want to keep this open as an option.
we can also try to play a bit with the amount of circels and distances between the circles, but this seems like a proper starting point.

I post some clear images of stresses in a dome structure caused by wind forces, below. This information made me, together with Peter's comments, decide to jump of the idea of using the stress trajectories, as i proposed before. Because it doesn't make too much sense, in a simplified way. It could make sense to try to work out how to adapt a building perfectly to wind forces, but that could be a complete study on its own.


Principle stresses by simple horizontal load - Staad plot


Dynamic stressed due to different speeds of wind

more tessellations

I tried to combine some different types of tessellations for the dome, in order to respond to the wind forces and gravity acting on the structure.

The first three images are the combination of the wind trajectories, as discussed before, and "vertical" ribs to lead the self weight to the ground.




Vertical ribs


Compression ribs following trajectories and connected to get a Delaunay triangulation


Tension side added.

If we want to split the domes into a double layered dome, only vertical ribs won't be sufficient for the part of the dome that supports the weight of the structure, therefore I added rings. The rings are following the cross sections of the stress trajectories of the other dome. I also added a quick sketch of how a cross section could look like.
I want to note here, that I'm not sure if this option is do-able within the time span we have for this research. I expect that the connection of two domes, is quite complex. The connection part it selves probably needs a lot of study in order to get it reasonable realistic and working.
Solution could be to make a simple connection between both structures, by reducing the distance between both. (I'm not so sure about the aesthetic consequenses in this case).
Integrating both options in one shell also might work.






The next three images show the wind trajectories combined with a Delaunay triangulation. I think that a fine triangulation could work to transfer the self weight to the ground, as it acts like a homogenic shell.. only lighter, and optimized by the genetic algorithm.
In the first 3 images are created in this way: marked points on crossings of wind trajectories, added random points, drew the Delaunay triangulation, added the trajectories again.






If we want to integrate these two tesselations into a single dome structure, it would be better to try to create the trajectory lines, within the Delaunay, instead of "on top". I tried this in the next image. BUT it appeared not really possible in this way. You can see that the green marked triangulation is a correct Delaunay pattern. The red part, which I based on the trajectories, are messy and not correct. This system might work if the tesselation is more dense. In that case, the trajectories will be the main beams. In between this beams, will be a secondary and much denser structure created based on Delaunay triangulation. (I'll sketch it in next post).



This one is just a sketch of Voronoi diagram based on the same trajectories. In the right side I added some vertical ribs. The Voronoi adapts to it in a nice way.
But this doesn't seem to be an optimal structure to bear both loads. Since the lines are not following the stress trajectories any more. I expect that this will create unneccessary moments in the structure.

Stress Trajectories Wind on Dome

As discussed in my report on rads, this image shows the schematic stress trajectories of wind in a dome shaped structure:


Source: Structures, D.L. Schodek

To explore the possibilities of adjusting the structure to the specific wind trajectories, i've made some sketches as seen below.

In the first sketch, I took one side of the trajectories (could be tension or compression). I distributed 9 points evenly along each line. I connected the points and created a Delaunay triangulation (red lines upper part). The lowerpart is the Voronoi diagram, based on the Delaunay triangulation. By connecting the points, we see that two additional line-sets are created. They look like the two sides of stress trajectories, but are not the same. This is because I distributed the points evenly and not accourding to the other side of the stress trajectories.



To check how this tesselation would work in 3d, I sketched the same lines onto a plastic dome shape. But distributed the points at the crossings of the two sides of stress trajectories. This causes a symmetrical image again. except that the thicker lines should resist to compression and the other set, to tension. I elongated the lines in order to get an enclosing structure. The way to close the structure can be more dense or adjusted in an other way.



Then I copied this line-set into illustrator, to get it more clear. The green lines, which appear when connecting the points, become clear ellipses.



I'm curious how this set-up would respond to forces at this point. Since I'm doubting if they can transfer the self weight of the building. We probably need to add an other line-set..

Focus on the Continued Research

I mentioned in the report a number of ideas for possible proposals. For this particular research subject, as described in the previously posted abstract, my interest goes out to one main directions:

"IDEA: If a column free dome is the starting point, we could divide the dome structure in two shells, one resisting the wind loads, and one taking the deadweight of the structure. We could design the “deadweight shell” by distributing points onto the dome shape, and connect the points applying a Voronoi algorithm, in order to get a framework as light as possible, while at the same time being one integrated entity. The wind-shell could be constructed out of a frame following the stress trajectories due to wind loads (one main direction). The one shell should be connected to the other in a way, which is parametrically predefined."

To discuss this a bit further, I want to comment that, wind load is a tricky load case to use as a shape giver. Since it moves around and the force flow caused by the wind on the structure, very much depends on the shape of the structure itselves. To use this principle, we should at least stay to one type of shape of the structure.

Let's assume that we use a dome structure as a model. Based on the adaptive qualities of rads to their loadcases, we could consider to optimize the dome to one main wind direction. And make the dome able to rotate towards the wind direction. Other wise, the structure would collapse, as soon as the wind changes its direction. An other solution could be to over-dimension the structure of the dome in order to make it strong enough to stand wind form all directions, but in this case,this is not our aim, since we want to create a light structure which is as much optimized as possible to its surroundings.

The issue of two cooperating shells: In radiolarians we see that they have to deal with 2 kinds of load cases in general: 1) the distibuted pressure of the water surrounding them, and 2)Impact loads caused my predators or other objects in their environment. This way of dealing with both different types of forces, made them able to develop efficiently. The fine tesselation of their shells withstand the evenly distibuted pressure of the water. While the Impact forces are counter acted by the arm of spines connected to the shells.

If we want to translate this principle to a dome shaped structure, we first have to define our load case. The distributed loading acting on a building is it's selfweight, magnified by gravity. The impact force encountered by a building structure is wind force, although its partly distibuted.

The big question is: How can we integrate this into one structure?


Impact forces and distributed loads acting on radiolarian shell

Deeper Research Radiolarians

I've been working to get a bit deeper in the radiolarian principles.. Via this link you can find a report of my work done. This report is my way to archive my findings for my own process. And its also a way to share it with my colleagues Michela and Peter and all others who are interested in the subject.

you can download the radiolaria report via THIS link