AADRL | Workshop 2

workshop ıı MUSTAFA EL-SAYED

DAHYUN KIM CHENLONG GU ASEEM AHMED MELIS KUCUKTUNC OGULCAN SULUCAY

“Incremental optimization of volume, through uniform and resolved the distribution of density-based voxel types without sacrificing the stability of the system.�

TABLE OF CONTENTs CELLULAR AUTOMATA

1

CONCEPT

5

GEOMETRY

21

NEW RULE SET

37

cellular automata As an initial study, derived from the John Conway’s Game of Life, we used four different seed images and various rule sets modified from the Game of Life. The rule sets we used are created by slightly modifying the rates of the survival, born, and death rules in the Game of Life. Using the seed images and the rule sets, we observed the patterns of the vertical growths of the Game of Life. Throughout the whole experiments, in order to observe the influence of the rule sets on the configuration of the vertical growth, the seed images which sets up the initial state of the voxels and the height of the vertical growth are set in a control group as fixed factors affecting the results.

Having discovered the patterns of the vertical growth, we began to see the fully grown voxel aggregation as a structural body and introduced several concepts that are crucial to the structural stability such as density, clusterization, connectivity, geometries, and gradient. In this booklet, the concepts are individually explained, followed by catalogues using different seed images and the rule sets we developed using these concepts. While trying to optimize the potential structural stability in the voxel aggregation, we also embarked on physical experimentations using Unity’s physical simulation. Using the concepts, the previous rule sets are modified and developed to identify the structural failures that should be avoided. By modifying the neighbouring relationships we tried to achieve the overarching influence on the whole voxel structure and optimized the structure’s potential capability in terms of flexibility and rigidity.

1

NUMBER OF ALIVE NEIGHBOUR VOXELS

A

BORN: 3 SURVIVE: 2,3 DIE: <2, >3

B

BORN: 3 SURVIVE: 3,4 DIE: <3, >4

C

BORN: 2,3 SURVIVE: 2,3 DIE: <2, >3

D

BORN: 3, 4 SURVIVE: 3,4 DIE: <3, >4

E

BORN: 3, 4 SURVIVE: 2, 3 DIE: <2, >3

DIE 2

SURVIVE

GET BORN

current voxel

neighbour voxels

AGGREGATION seed ımage: 32*32 tıme end : 80

RULE A

RULE B

RULE C

RULE D

RULE E

4

concept

density After the 3D aggregation, â&#x20AC;&#x2DC;densityâ&#x20AC;&#x2122; of the alive voxels are calculated, by counting the number of its alive neighbours in x, y and z coordination. The range between the minimum and maximum alive neighbours number is subdivided into four to assign certain color to the voxel accordingly.

dividing voxel into 4 color groups according to their number of neigbourhood

relatively lower density

relatively higher density

number of neÄąghbours

minimum density = 0 6

maximum density = 26 minimum density

maximum density

density catalog RULE A VERTICAL SECTION

frame: 20

frame: 40

frame: 60

frame: 80

density catalog RULE B VERTICAL SECTION

8

frame: 20

frame: 40

frame: 60

frame: 80

density catalog RULE c VERTICAL SECTION

frame: 20

frame: 40

frame: 60

frame: 80

density catalog RULE d VERTICAL SECTION

10

frame: 20

frame: 40

frame: 60

frame: 80

density catalog RULE e VERTICAL SECTION

frame: 20

frame: 40

frame: 60

frame: 80

maxımum densıty voxel After calculating the density of each voxel in the 3D system, the voxel/voxels which have the maximum number of alive neighbours are identified and highlighted.

voxels aggregatıon

12

ıdentıfyıng the voxel wıth maxımum alıve neıghbours

maxımum density voxel catalog seed ımage: 32*32 tıme end : 80

RULE A

RULE B

RULE C

RULE D

RULE E

clusterIZATION Each voxel in the aggregation checks its distance from the maximum density voxels and puts itself to the closest oneâ&#x20AC;&#x2122;s cluster. By this way, we are meaningfully group the voxels in the 3D grid, seperating them into the number of maximum density voxels. Each maximum density voxels create their own cluster, being located at the center.

d1

d2

d1<d2 14

clusterization catalog seed ımage: 32*32 tıme end : 80

RULE A

RULE B

RULE C

RULE D

RULE E

GRADIENT DENSITY Within each cluster, the voxels are seperated into four groups, according to their distance to the center of the cluster. The ones that are closer to the center are assigned relatively higher density then the ones that are further located.

maxımum densıty maxımum densıty

maxımum densıty

according to the distance to closest maxımum density voxel 16

grad覺ent dens覺ty catalog seed 覺mage: 32*32 t覺me end : 80

RULE A

RULE B

RULE C

RULE D

RULE E

AGE CENTROIDS We also tried to change the centroid rules to create the centroids based on ages.

By locating all of the maximum age voxels, we got the groups of ending voxels of the voxel-columns, which ceased growing when getting the maximum age.

Through researching the changes, the maximum age voxels also reflect the density distribution of the voxels, since an amaximum age voxel means a void space above the voxel column, since gaps of the voxels are not only because of CArule, but because of the age-rule.

AGE Columns

Age Centroids 18

Maximum voxel of the voxel column

Typo alert: thoudsands

seed image

centroids

density

gradient

horizontal 20

horizontal 40

horizontal 60

horizontal 80

vertical

20

geometry

Voxel types The aim of was to optimize the flexibility through volume and geometry. Four types of voxels were introduced with different percentages of volumes with reference to the cube voxel.

voxel Type 1 10% volume

22

voxel Type 2 20% volume

voxel Type 3 40% volume

voxel Type 4 75% volume

These geometries were put through tests for flexibility to study their behavior under force and their connections to each other and the other types.

voxel Type 1 flexıbılıty

Under uneven distribution of geometries, the geometries with least volume would break away from the grid when external force was applied, resulting in failure.

voxel Type 2 flexıbılıty

voxel Type 3 flexıbılıty

Under even distribution with most least volume geometries, the grid would wobble under applied force resulting in over flexure. Although intact, it wasnt desirable.

voxel Type 4 flexıbılıty

Under even distribution with equal distribution, the grid behaves as an intact structure with minimal flexural properties, which was the dersired result.

3d prÄąnted Voxel types

24

assıgnıng voxel types accordıng to gradıent densıty Upon assigning the geometry types to the grid we noticed that after applying the gradient with respect to the geometry with the most number of neighbours and clusterisation, Voxel Type 1geometry were dominant with their numbers creating structurally weak connections. To avoid this, we decided to invert the assigment of geometries which would result in more number of Voxel Type 4.

Weak voxels assigned in sparse area Initial assignment of voxel types accroding to gradient density displayed weak/ fragile behaviour due to maximum assignment of type4(weak voxels) at the core.

26

Strong voxels assigned in sparse area Initial assignment of voxel types accroding to gradient density displayed weak/ fragile behaviour due to maximum assignment of type4(weak voxels) at the core.

successfull examples RULE A

28

RULE B

RULE C

RULE D

RULE E

voxels

density

frame: 20

maximum density

frame: 40

clusters

frame: 60

gradient

vertical section

frame: 80

voxels

density

frame: 20

30

maximum density

frame: 40

clusters

frame: 60

gradient

vertical section

frame: 80

voxels

density

frame: 20

maximum density

frame: 40

clusters

frame: 60

gradient

vertical section

frame: 80

unsuccessfull examples RULE A

32

RULE B

RULE C

RULE D

RULE E

Problem:

asymmetrically formed clusters

Problem:

one cluster Unevenly dıstrıbuted gradıent

34

Problem:

lımıted aggregatıon densıty ıs concentrated at the bottom

voxels

density

frame: 20

maximum density

frame: 40

clusters

frame: 60

gradient

vertical section

frame: 80

36

new rule set After identifying the failures in the first set of studies, the preffered growth and aggregation has become more clear.

Even though there were relatively successful examples, the connectivity between the voxels were still weak. A new set of rule is introduced in order to avoid disconnected geometries and have more face to face connections, In this part of the study, the new rule sets and related connectivity concept will be explained.

37

connectivity Concerning structural joints between the current voxel and the neighbouring voxels, there are geometrically two types of connections: adjacent connection and diagonal connection. Adjacent and diagonal connections are shown in 2D and 3D as shown in the diagrams below. Adjacent connections, when compared with diagonal connections, have much larger area attached to the neighbours, thus increase the structural stability of the whole structure.

diagonal connection in 2d

diagonal connection in 3d 38

adjacent connection in 2d

adjacent connection in 3d

Äąsolated voxels As observed from the diagrams above, since diagonal connections are only connected with the current voxel on points, therefore, they weaken structural stability. The voxels that are not well connected to its neighbouring voxels would fall down when the gravity is applied.

pos覺b覺l覺t覺es

40

NUMBER OF ALIVE NEIGHBOUR VOXELS

rule set 01

adjacent

BORN: total= 3, adjacent > 1 SURVIVE: adjacent = 2,3 diagonal > 0

diagonal

adjacent

rule set 02

BORN: total > 2, adjacent > 0 SURVIVE: adjacent = 1

diagonal

adjacent

BORN: total > 2, adjacent > 0 SURVIVE: adjacent = 1

rule set 03 diagonal

current voxel adjacent

BORN: total = 2, 3, adjacent > 0 SURVIVE: total = 3

rule set 04 diagonal

adjacent

BORN: total = 2, 3, 4, adjacent > 1 diagonal > 0 SURVIVE: total = 3

rule set 05 diagonal

DIE

SURVIVE

GET BORN

AGGREGATION seed ımage: 32*32 tıme end : 80

adjacent diagonal

42

density catalog adjacent diagonal

VERTICAL SECTION

frame: 20

frame: 40

frame: 60

frame: 80

density catalog adjacent diagonal

44

VERTICAL SECTION

frame: 20

frame: 40

frame: 60

frame: 80

density catalog adjacent diagonal

VERTICAL SECTION

frame: 20

frame: 40

frame: 60

frame: 80

density catalog adjacent diagonal

46

VERTICAL SECTION

frame: 20

frame: 40

frame: 60

frame: 80

density catalog adjacent diagonal

VERTICAL SECTION

frame: 20

frame: 40

frame: 60

frame: 80

hÄąghest density voxel catalog adjacent diagonal

48

clusterization catalog adjacent diagonal

clusterization catalog adjacent diagonal

50

examples seed Äąmage: 32*32 tÄąme end : 80

adjacent diagonal

52

adjacent diagonal

advantages:

distribution of clusters on different levels

adjacent diagonal

54

advantages:

maximized face to face connections strong connectivity

advantages: strong connections on the horizontal and vertical section symmetrical growth

adjacent diagonal

voxels

density

frame: 20

maximum density

frame: 40

clusters

frame: 60

gradient

vertical section

frame: 80

AGGREGATION seed ımage: 32*32 tıme end : 80

adjacent diagonal

56

voxels

density

frame: 20

maximum density

frame: 40

clusters

frame: 60

gradient

vertical section

frame: 80

maximum density

clusters

gradient

maximum density 126 voxels each have 24 neighbours

58

vertical section

resolution checking number of alive voxels around according to the given resolution

resolution = 1

The original resolution in our coding was 1 which showed a problem of too many centroids.

resolution = 2

To enlarge the range of defining the centroids so as to decrease the amounts of centroids, we changed the resolution as 2.

maximum density

clusters

gradient

maximum density 4 voxels each have 92 neighbours

60

vertical section

tolerence study Tolerence value is a variable introduced in order to optimize the number and distibution of the centroids. After finding the number of maximum neighbourhood, the tolerence value expands the range of number to have more centroids.

maximum density

clusters

resolution value = 2 tolerence value = 5 maximum density

minimum density

tolerence 4

87

92

tolerence value = 5 24 voxels 87 - 92 neighbours

maximum density

clusters

tolerence study resolution value = 2 tolerence value = 9 maximum density

minimum density

tolerence 4

83

92

tolerence value = 9 360 voxels 83 - 92 neighbours

62

feedback loop

As we do our experiments, we go back and forth and adjust the numbers again and again, in order to achieve the optimum number and distribution of the centroids.

to achieve the optimum number and distubution of centroids

maximum density

clusters

resolution value = 2 tolerence value = 7 maximum density

minimum density

tolerence 4

85

92

tolerence value = 7 64 voxels 85 - 92 neighbours

64

assignED voxel geometries

horizontal sections

frame: 20

frame: 40

frame: 60

frame: 80

hor覺zontal sect覺on frame: 20

66

techn覺cal draw覺ng

assignED voxel geometries

vertıcal section

technıcal drawıng

AGGREGATION seed ımage: 32*32 tıme end : 80

adjacent diagonal

68

voxels

density

frame: 20

maximum density

frame: 40

clusters

frame: 60

gradient

vertical section

frame: 80

maximum density

clusters

gradient

maximum density 4 voxels each have 80 neighbours

70

vertical section

maximum density

tolerence study

clusters

resolution value = 2 tolerence value = 3 maximum density

minimum density

tolerence 13

77

80

tolerence value = 3 208 voxels 77 - 80 neighbours

maximum density

tolerence study

clusters

resolution value = 2 tolerence value = 7 maximum density

minimum density

tolerence 4

73

80

tolerence value = 7 544 voxels 73 - 80 neighbours

72

to achieve the optimum number and distubution of centroids

maximum density

clusters

resolution value = 2 tolerence value = 5 maximum density

minimum density

tolerence 4

75

80

tolerence value = 5 400 voxels 75 - 80 neighbours

assignED voxel geometries

horizontal sections

74

frame: 20

frame: 40

frame: 60

frame: 80

hor覺zontal sect覺on frame: 20

techn覺cal draw覺ng

assignED voxel geometries

76

vertıcal section

technıcal drawıng

AGGREGATION adjacent diagonal

78

voxels

density

frame: 20

maximum density

frame: 40

clusters

frame: 60

gradient

vertical section

frame: 80

maximum density

clusters

gradient

maximum density 8 voxels each have 67 neighbours

80

vertical section

maximum density

clusters

tolerence study tolerence value = 3 maximum density

minimum density

tolerence 15

64

67

tolerence value = 3 88 voxels 64 - 67 neighbours

maximum density

clusters

tolerence study tolerence value = 10 maximum density

minimum density

tolerence 15

57

67

tolerence value = 10 1424 voxels 57 - 67 neighbours

82

to achieve the optimum number and distubution of centroids

maximum density

clusters

resolution value = 2 tolerence value = 7 maximum density

minimum density

tolerence 15

60

67

tolerence value = 7 360 voxels 60 - 67 neighbours

assignED voxel geometries horizontal sections

84

frame: 20

frame: 40

frame: 60

frame: 80

hor覺zontal sect覺on frame: 20

techn覺cal draw覺ng

assignED voxel geometries

86

vertıcal section

technıcal drawıng

88