# Gemlab examples
* [block_2d_qua8](#block_2d_qua8) -- Shows how to subdivide a rectangle (Block) into Qua8 cells
* [check_grid_search_performance](#check_grid_search_performance) -- Investigates the performance of Grid Search when checking a very fine mesh
* [find_in_grid_2d](#find_in_grid_2d) -- Uses GridSearch to assist in finding points in the x-y space
* [integ_mom_inertia_disk](#integ_mom_inertia_disk) -- Approximates the second moment of inertia of a disk (circle)
* [integ_mom_inertia_ring](#integ_mom_inertia_ring) -- Approximates the second moment of inertia of a ring
* [integration_tet4](#integration_tet4) -- Computes the stiffness matrix of a Tet4 element
* [integration_tri3](#integration_tri3) -- Performs numerical integrations with a Tri3 element
* [isoparametric_qua4](#isoparametric_qua4) -- Illustrates the isoparametric formula with a Qua4
* [mesh_2d_tri3_qua4](#mesh_2d_tri3_qua4) -- Converts a string into a Mesh of Tri3 and Qua4
* [grid_search_triangles](#grid_search_triangles) -- Fast-search triangles in a mesh using the GridSearchCell tool
## block_2d_qua8
Shows how to subdivide a rectangle (Block) into Qua8 cells.
[Source code](https://github.com/cpmech/gemlab/blob/main/examples/block_2d_qua8.rs)
The output is:
```text
# header
# ndim npoint ncell nmarked_edge nmarked_face
2 21 4 0 0
# points
# id marker x y {z}
0 0 0.0 0.0
1 0 1.0 0.0
2 0 1.0 1.0
3 0 0.0 1.0
4 0 0.5 0.0
5 0 1.0 0.5
6 0 0.5 1.0
7 0 0.0 0.5
8 0 2.0 0.0
9 0 2.0 1.0
10 0 1.5 0.0
11 0 2.0 0.5
12 0 1.5 1.0
13 0 1.0 2.0
14 0 0.0 2.0
15 0 1.0 1.5
16 0 0.5 2.0
17 0 0.0 1.5
18 0 2.0 2.0
19 0 2.0 1.5
20 0 1.5 2.0
# cells
# id marker kind points
0 1 qua8 0 1 2 3 4 5 6 7
1 1 qua8 1 8 9 2 10 11 12 5
2 1 qua8 3 2 13 14 6 15 16 17
3 1 qua8 2 9 18 13 12 19 20 15
```
## check_grid_search_performance
Investigates the performance of Grid Search when checking a very fine mesh.
[Source code](https://github.com/cpmech/gemlab/blob/main/examples/check_grid_search_performance.rs)
Must run with (it will take +7 min):
```bash
cargo run --release --example check_grid_search_performance
```
Or:
```bash
bash zscripts/ex_check_grid_search_performance.bash
```
Results:
```text
......................... nr = 1 na = 2 .........................
grid.ndiv = 5
max vol = Some(4.5)
mesh generation : 471.974µs
check_all : 10.265173ms
check_overlap. : 95.04µs
grid.new : 465ns
grid.insert : 53.236µs
grid.search : 12.491µs
grid.ndiv = 20
max vol = Some(4.5)
mesh generation : 645.302µs
check_all : 27.15µs
check_overlap. : 63.005µs
grid.new : 321ns
grid.insert : 46.117µs
grid.search : 5.483µs
grid.ndiv = 40
max vol = Some(4.5)
mesh generation : 386.923µs
check_all : 12.842µs
check_overlap. : 46.267µs
grid.new : 229ns
grid.insert : 39.291µs
grid.search : 4.482µs
grid.ndiv = 100
max vol = Some(4.5)
mesh generation : 347.954µs
check_all : 12.895µs
check_overlap. : 46.404µs
grid.new : 173ns
grid.insert : 41.44µs
grid.search : 4.489µs
......................... nr = 4 na = 8 .........................
grid.ndiv = 5
max vol = Some(0.0703125)
mesh generation : 3.848926ms
check_all : 366.83µs
check_overlap. : 688.255µs
grid.new : 889ns
grid.insert : 408.958µs
grid.search : 239.396µs
grid.ndiv = 20
max vol = Some(0.0703125)
mesh generation : 3.736197ms
check_all : 272.142µs
check_overlap. : 646.482µs
grid.new : 733ns
grid.insert : 411.393µs
grid.search : 103.026µs
grid.ndiv = 40
max vol = Some(0.0703125)
mesh generation : 3.588525ms
check_all : 266.499µs
check_overlap. : 638.451µs
grid.new : 718ns
grid.insert : 502.294µs
grid.search : 74.603µs
grid.ndiv = 100
max vol = Some(0.0703125)
mesh generation : 3.693314ms
check_all : 254.096µs
check_overlap. : 643.18µs
grid.new : 946ns
grid.insert : 488.086µs
grid.search : 64.748µs
......................... nr = 50 na = 100 .........................
grid.ndiv = 5
max vol = Some(3.6e-5)
mesh generation : 2.808792394s
check_all : 469.93421ms
check_overlap. : 1.072179757s
grid.new : 5.718µs
grid.insert : 355.420318ms
grid.search : 6m16.008905797s ← ← ← ← ← ← ← ← ← ← ← ← ← ← ←
grid.ndiv = 20
max vol = Some(3.6e-5)
mesh generation : 2.646534618s
check_all : 468.895862ms
check_overlap. : 1.042336842s
grid.new : 6.213µs
grid.insert : 336.348302ms
grid.search : 10.027990761s
grid.ndiv = 40
max vol = Some(3.6e-5)
mesh generation : 2.621904809s
check_all : 467.453497ms
check_overlap. : 1.035710922s
grid.new : 5.561µs
grid.insert : 350.267471ms
grid.search : 1.632270955s
grid.ndiv = 100
max vol = Some(3.6e-5)
mesh generation : 2.610841122s
check_all : 467.56941ms
check_overlap. : 1.03700237s
grid.new : 5.751µs
grid.insert : 472.889228ms
grid.search : 451.273946ms
```
Thus, with finer meshes, the performance of GridSearch is significantly better with more grid divisions. Note the line indicate with arrows.
For coarser meshes, the larger number of grid divisions has no significant impact on the performance.
Thus, it seems best to use the NDIV = 100 as the default for GridSearch.
## find_in_grid_2d
[Source code](https://github.com/cpmech/gemlab/blob/main/examples/find_in_grid_2d.rs)
Uses GridSearch to assist in finding points in the x-y space.
## integ_mom_inertia_disk
Approximates the second moment of inertia of a disk (circle).
[Source code](https://github.com/cpmech/gemlab/blob/main/examples/integ_mom_inertia_disk.rs)
First mesh:
Second mesh:
The output is:
```text
mesh 1: second_mom_inertia = 63.61725744867274 (63.61725123519331): err = 6.21e-6
mesh 2: second_mom_inertia = 63.61725006590863 (63.61725123519331): err = 1.17e-6
```
## integ_mom_inertia_ring
[Source code](https://github.com/cpmech/gemlab/blob/main/examples/integ_mom_inertia_ring.rs)
Approximates the second moment of inertia of a ring.
The output is:
```text
second_mom_inertia = 62.83185310895357 (62.83185307179586): err = 3.72e-8
```
## integration_tet4
Computes the stiffness matrix of a Tet4 element.
[Source code](https://github.com/cpmech/gemlab/blob/main/examples/integration_tet4.rs)
The output is:
```text
┌ ┐
│ 149 108 24 -1 6 12 -54 -48 0 -94 -66 -36 │
│ 108 344 54 -24 104 42 -24 -216 -12 -60 -232 -84 │
│ 24 54 113 0 30 35 0 -24 -54 -24 -60 -94 │
│ -1 -24 0 29 -18 -12 -18 24 0 -10 18 12 │
│ 6 104 30 -18 44 18 12 -72 -12 0 -76 -36 │
│ 12 42 35 -12 18 29 0 -24 -18 0 -36 -46 │
│ -54 -24 0 -18 12 0 36 0 0 36 12 0 │
│ -48 -216 -24 24 -72 -24 0 144 0 24 144 48 │
│ 0 -12 -54 0 -12 -18 0 0 36 0 24 36 │
│ -94 -60 -24 -10 0 0 36 24 0 68 36 24 │
│ -66 -232 -60 18 -76 -36 12 144 24 36 164 72 │
│ -36 -84 -94 12 -36 -46 0 48 36 24 72 104 │
└ ┘
```
## integration_tri3
Performs numerical integrations with a Tri3 element.
[Source code](https://github.com/cpmech/gemlab/blob/main/examples/integration_tri3.rs)
```text
a =
┌ ┐
│ 36.00000000000001 │
│ 36 │
│ 36 │
└ ┘
b =
┌ ┐
│ 24.000000000000007 │
│ 24.000000000000007 │
│ 24 │
│ 24 │
│ 24 │
│ 24 │
└ ┘
c =
┌ ┐
│ -2 │
│ -4 │
│ 6 │
└ ┘
d =
┌ ┐
│ -15 │
│ -10 │
│ 12 │
│ 4 │
│ 3 │
│ 6 │
└ ┘
```
## isoparametric_qua4
Illustrates the isoparametric formula with a Qua4
[Source code](https://github.com/cpmech/gemlab/blob/main/examples/isoparametric_qua4.rs)
```text
xm = 4, ym = 4.5
x_interpolated =
┌ ┐
│ 4 │
│ 4.5 │
└ ┘
```
## mesh_2d_tri3_qua4
Converts a string into a Mesh of Tri3 and Qua4
[Source code](https://github.com/cpmech/gemlab/blob/main/examples/mesh_2d_tri3_qua4.rs)
Draws the following mesh:
## grid_search_triangles
Fast-search triangles in a mesh using the GridSearchCell tool.
[Source code](https://github.com/cpmech/gemlab/blob/main/examples/grid_search_triangles.rs)
We run the code with 200, 800, 3200, 7200, and 20000 triangles. The next sections show the results.
The majority of the containers are occupied by 8 triangles.
### 200 triangles
```text
Summary
=======
ncell = 200
xmin = [-0.01, -0.01]
xmax = [15.9932, 8.9918]
side_length = 1.0002
num of non-empty containers = 109
max container num items = 12
Histogram of container num items
================================
[ 0, 1) | 0
[ 1, 2) | 11 ***********
[ 2, 3) | 0
[ 3, 4) | 9 *********
[ 4, 5) | 8 ********
[ 5, 6) | 7 *******
[ 6, 7) | 2 **
[ 7, 8) | 7 *******
[ 8, 9) | 56 ************************************************************
[ 9,10) | 0
[10,11) | 0
[11,12) | 2 **
[12,13) | 7 *******
sum = 109
```
### 800 triangles
```text
Summary
=======
ncell = 800
xmin = [-0.01, -0.01]
xmax = [30.996199999999998, 17.993599999999997]
side_length = 1.0002
num of non-empty containers = 397
max container num items = 12
Histogram of container num items
================================
[ 0, 1) | 0
[ 1, 2) | 19 ****
[ 2, 3) | 0
[ 3, 4) | 19 ****
[ 4, 5) | 18 ****
[ 5, 6) | 15 ***
[ 6, 7) | 2
[ 7, 8) | 15 ***
[ 8, 9) | 270 ************************************************************
[ 9,10) | 2
[10,11) | 0
[11,12) | 2
[12,13) | 35 *******
sum = 397
```
### 3200 triangles
```text
Summary
=======
ncell = 3200
xmin = [-0.01, -0.01]
xmax = [61.0022, 34.997]
side_length = 1.0002
num of non-empty containers = 1463
max container num items = 15
Histogram of container num items
================================
[ 0, 1) | 0
[ 1, 2) | 35 *
[ 2, 3) | 1
[ 3, 4) | 34 *
[ 4, 5) | 37 **
[ 5, 6) | 29 *
[ 6, 7) | 6
[ 7, 8) | 29 *
[ 8, 9) | 1083 ************************************************************
[ 9,10) | 5
[10,11) | 12
[11,12) | 6
[12,13) | 184 **********
[13,14) | 0
[14,15) | 0
[15,16) | 2
sum = 1463
```
### 7200 triangles
```text
Summary
=======
ncell = 7200
xmin = [-0.01, -0.01]
xmax = [91.00819999999999, 52.0004]
side_length = 1.0002
num of non-empty containers = 3140
max container num items = 15
Histogram of container num items
================================
[ 0, 1) | 0
[ 1, 2) | 50 *
[ 2, 3) | 0
[ 3, 4) | 52 *
[ 4, 5) | 0
[ 5, 6) | 42 *
[ 6, 7) | 8
[ 7, 8) | 44 *
[ 8, 9) | 2429 ************************************************************
[ 9,10) | 8
[10,11) | 43 *
[11,12) | 8
[12,13) | 448 ***********
[13,14) | 0
[14,15) | 0
[15,16) | 8
sum = 3140
```
### 20000 triangles
```text
Summary
=======
ncell = 20000
xmin = [-0.01, -0.01]
xmax = [150.02, 86.0072]
side_length = 1.0002
num of non-empty containers = 8634
max container num items = 15
Histogram of container num items
================================
[ 0, 1) | 0
[ 1, 2) | 84
[ 2, 3) | 0
[ 3, 4) | 86
[ 4, 5) | 0
[ 5, 6) | 70
[ 6, 7) | 14
[ 7, 8) | 71
[ 8, 9) | 6852 ************************************************************
[ 9,10) | 15
[10,11) | 70
[11,12) | 14
[12,13) | 1344 ***********
[13,14) | 0
[14,15) | 0
[15,16) | 14
sum = 8634
```
