File size: 22,182 Bytes
b91a8a7
 
80ecb73
b91a8a7
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
0317dde
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
80ecb73
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
b91a8a7
 
 
 
 
0317dde
 
 
 
80ecb73
 
 
 
65b2a0a
 
 
 
 
 
 
 
b91a8a7
65b2a0a
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
---
dataset_info:
- config_name: geometry
  features:
  - name: sample_id
    dtype: int32
  - name: points_x
    list: float32
  - name: points_y
    list: float32
  - name: points_z
    list: float32
  - name: cells
    list:
      list: int32
  - name: edge_index
    list:
      list: int32
  - name: constraint_mask_x
    list: int32
  - name: constraint_mask_y
    list: int32
  - name: constraint_mask_z
    list: int32
  - name: constraint_value_x
    list: float32
  - name: constraint_value_y
    list: float32
  - name: constraint_value_z
    list: float32
  - name: boundary_id
    list: int32
  - name: node_type
    list: int32
  splits:
  - name: total
    num_bytes: 8280604
    num_examples: 3
  download_size: 8191127
  dataset_size: 8280604
- config_name: metadata
  features:
  - name: sample_id
    dtype: int32
  - name: valid
    dtype: bool
  - name: sample_name
    dtype: string
  - name: sample_dir
    dtype: string
  - name: geo_path
    dtype: string
  - name: mesh_path
    dtype: string
  - name: result_dir
    dtype: string
  - name: mpi_np
    dtype: int32
  - name: solver_executable
    dtype: string
  - name: gmsh_executable
    dtype: string
  - name: solution_file_type
    dtype: string
  - name: n_solution_files
    dtype: int32
  - name: first_solution_file
    dtype: string
  - name: last_solution_file
    dtype: string
  - name: E
    dtype: float64
  - name: nu
    dtype: float64
  - name: lambda
    dtype: float64
  - name: mu
    dtype: float64
  - name: rho
    dtype: float64
  - name: c_damp
    dtype: float64
  - name: F0
    dtype: float64
  - name: traction_type
    dtype: string
  - name: spatial_profile
    dtype: string
  - name: traction_amplitude_y_nominal_uniform
    dtype: float64
  - name: frequency_hz
    dtype: float64
  - name: phase
    dtype: float64
  - name: load_start_time
    dtype: float64
  - name: load_end_time
    dtype: float64
  - name: load_center_x
    dtype: float64
  - name: load_center_y
    dtype: float64
  - name: load_center_z
    dtype: float64
  - name: load_sigma_x
    dtype: float64
  - name: load_sigma_y
    dtype: float64
  - name: load_sigma_z
    dtype: float64
  - name: load_center_x_rel
    dtype: float64
  - name: load_center_z_rel
    dtype: float64
  - name: load_sigma_x_rel
    dtype: float64
  - name: load_sigma_z_rel
    dtype: float64
  - name: moving_direction
    dtype: int32
  - name: load_velocity
    dtype: float64
  - name: load_velocity_rel
    dtype: float64
  - name: impact_time
    dtype: float64
  - name: impact_duration
    dtype: float64
  - name: L
    dtype: float64
  - name: H
    dtype: float64
  - name: B
    dtype: float64
  - name: A
    dtype: float64
  - name: I
    dtype: float64
  - name: nx
    dtype: int32
  - name: ny
    dtype: int32
  - name: nz
    dtype: int32
  - name: lc
    dtype: float64
  - name: dirichlet_boundary_ids
    list: int32
  - name: neumann_boundary_ids
    list: int32
  - name: dt
    dtype: float64
  - name: t_end
    dtype: float64
  - name: output_dt
    dtype: float64
  - name: n_nodes
    dtype: int32
  - name: n_cells
    dtype: int32
  - name: n_edges
    dtype: int32
  - name: n_saved_times
    dtype: int32
  - name: snapshot_storage_format
    dtype: string
  splits:
  - name: total
    num_bytes: 2437
    num_examples: 3
  download_size: 86195
  dataset_size: 2437
- config_name: snapshot
  features:
  - name: sample_id
    dtype: int32
  - name: displacement_x
    list:
      list: float32
  - name: displacement_y
    list:
      list: float32
  - name: displacement_z
    list:
      list: float32
  - name: velocity_x
    list:
      list: float32
  - name: velocity_y
    list:
      list: float32
  - name: velocity_z
    list:
      list: float32
  - name: body_force_x
    list:
      list: float32
  - name: body_force_y
    list:
      list: float32
  - name: body_force_z
    list:
      list: float32
  - name: traction_x
    list:
      list: float32
  - name: traction_y
    list:
      list: float32
  - name: traction_z
    list:
      list: float32
  splits:
  - name: total
    num_bytes: 3305606460
    num_examples: 3
  download_size: 3305745573
  dataset_size: 3305606460
configs:
- config_name: geometry
  data_files:
  - split: total
    path: geometry/total-*
- config_name: metadata
  data_files:
  - split: total
    path: metadata/total-*
- config_name: snapshot
  data_files:
  - split: total
    path: snapshot/total-*
license: other
language:
- en
tags:
- deal
- neural_operators
- graph-neural-networks
pretty_name: Beam3D Elastic Dynamics Dataset
---



# Beam3D Elastic Dynamics Dataset

## Dataset Details

### Dataset Description

This dataset contains synthetic 3D beam simulations generated with a finite element solver based on `deal.II`.

Each sample represents one dynamic simulation of a 3D elastic beam. The simulations include randomized geometry, material properties, damping parameters, and loading conditions.

The dataset is intended for scientific machine learning tasks involving elastic dynamics, including surrogate modeling, graph neural networks, neural operators, reduced-order modeling, and spatio-temporal prediction.

The dataset is organized into three Hugging Face configurations:

| Configuration | Content |
|---|---|
| `geometry` | Mesh connectivity, graph connectivity, and static node-level information |
| `snapshot` | Time-dependent physical fields stored component-wise |
| `metadata` | Simulation-level scalar parameters |

- **Curated by:** FAST Computing
- **Shared by:** FAST Computing
- **Language(s):** English
- **License:** Other

---

## Uses

### Direct Use

This dataset can be used for:

- training surrogate models for 3D elastic dynamics;
- training graph neural networks on finite element meshes;
- training neural operators or sequence models for displacement and velocity prediction;
- learning the response of elastic beams under different loading conditions;
- testing reduced-order modeling pipelines;
- benchmarking scientific machine learning methods on structured simulation data.

The dataset is especially suited for methods that use mesh information, graph connectivity, node-level physical quantities, and simulation metadata.

### Out-of-Scope Use

This dataset should not be used as a validated engineering benchmark for safety-critical structural design.

The simulations are synthetic and depend on the numerical assumptions, mesh resolution, material model, and loading conditions used during generation. Any engineering use requires independent verification.

---

## Dataset Structure

The dataset has three configurations:

```python
from datasets import load_dataset

repo_id = "fastcomputing/first_beam3d_test_single_split"

geometry = load_dataset(repo_id, name="geometry", split="total")
snapshots = load_dataset(repo_id, name="snapshot", split="total")
metadata = load_dataset(repo_id, name="metadata", split="total")
```

Each configuration contains one row per simulation sample.

---

## Configuration: `geometry`

The `geometry` configuration stores mesh-related quantities and static node-level information.

Each row corresponds to one simulation sample.

| Field | Meaning | Expected shape | Type |
|---|---|---:|---|
| `sample_id` | Simulation identifier | scalar | `int32` |
| `points_x` | Node coordinates in x direction | `(N, )` | `float32` |
| `points_y` | Node coordinates in y direction | `(N, )` | `float32` |
| `points_z` | Node coordinates in z direction | `(N, )` | `float32` |
| `cells` | Hexahedral cell connectivity | `(C, 8)` | `int32` |
| `edge_index` | Directed graph edges extracted from hexahedral cells | `(2, E)` | `int32` |
| `constraint_mask_x` | Mask identifying constrained displacement components in x| `(N, )` | `int32` |
| `constraint_mask_y` | Mask identifying constrained displacement components in y| `(N, )` | `int32` |
| `constraint_mask_z` | Mask identifying constrained displacement components in z| `(N, )` | `int32` |
| `constraint_value_x` | Prescribed displacement values for constrained components x| `(N, )` | `float32` |
| `constraint_value_y` | Prescribed displacement values for constrained components y| `(N, )` | `float32` |
| `constraint_value_z` | Prescribed displacement values for constrained components z| `(N, )` | `float32` |
| `boundary_id` | Geometric boundary label associated with each node | `(N,)` | `int32` |
| `node_type` | Semantic node classification | `(N,)` | `int32` |

Where:

```text
N = number of mesh nodes
C = number of hexahedral cells
E = number of directed graph edges
```

### `points`

`points` stores the node coordinates:

```text
points_x[i] = [x_i]
points_y[i] = [y_i]
points_z[i] = [z_i]
```

Shape:

```text
(N, )
```

### `cells`

`cells` stores the hexahedral finite element connectivity.

Each row contains the 8 node indices of one hexahedral cell:

```text
cells[c] = [n0, n1, n2, n3, n4, n5, n6, n7]
```

Shape:

```text
(C, 8)
```

This is not the raw VTK flat cell array.  

### `edge_index`

`edge_index` stores graph connectivity derived from the hexahedral cells.

For each hexahedral cell, the 12 standard hexahedron edges are extracted. Both directions are stored for each edge, so the graph is directed:

```text
edge_index[:, e] = [source_node, target_node]
```

Shape:

```text
(2, E)
```

The local hexahedral edges used to build the graph are:

```text
(0, 1), (1, 2), (2, 3), (3, 0),
(4, 5), (5, 6), (6, 7), (7, 4),
(0, 4), (1, 5), (2, 6), (3, 7)
```

For each edge `(i, j)`, both `(i, j)` and `(j, i)` are added. Duplicate edges are removed.

### `constraint_mask`

`constraint_mask_x` identifies which displacement components in x are constrained.

Shape:

```text
(N, )
```

Examples:

```text
[1] -> fixed in x node
[0] -> free node
```

### `constraint_value`

`constraint_value_x` stores the prescribed displacement in x value for constrained components.

Shape:

```text
(N, )
```

For a homogeneous fixed boundary condition:

```text
constraint_value_x[i] = [0]
```

The `constraint_mask` tells whether a component is constrained.  
The `constraint_value` tells the imposed value.

### `boundary_id`

`boundary_id` identifies the geometric boundary region associated with each node.

Shape:

```text
(N,)
```

It answers:

```text
Which mesh boundary does this node belong to?
```

Example:

```text
boundary_id = 1 -> left beam end
boundary_id = 2 -> right beam end
boundary_id = 3 -> loaded surface
```

The exact meaning depends on the mesh labeling used during data generation.

### `node_type`

`node_type` gives the semantic role of the node in the simulation.

Shape:

```text
(N,)
```

Current convention:

```text
0 = internal node
1 = Dirichlet boundary node
2 = Neumann boundary node
3 = boundary node without explicitly assigned boundary condition
```

In short:

```text
boundary_id tells where the node is.
node_type tells what role the node has.
```

---

## Configuration: `snapshot`

The `snapshot` configuration stores time-dependent fields.

Each row corresponds to one simulation sample and contains the full temporal evolution of the saved physical quantities.

The dynamic vector fields are stored component-wise. Acceleration is not stored in the current Hugging Face dataset.

| Field | Meaning | Expected shape | Type |
|---|---|---:|---|
| `sample_id` | Simulation identifier | scalar | `int32` |
| `time` | Saved output times | `(T,)` | `float32` |
| `displacement_x` | x-component of nodal displacement | `(T, N)` | `float32` |
| `displacement_y` | y-component of nodal displacement | `(T, N)` | `float32` |
| `displacement_z` | z-component of nodal displacement | `(T, N)` | `float32` |
| `velocity_x` | x-component of nodal velocity | `(T, N)` | `float32` |
| `velocity_y` | y-component of nodal velocity | `(T, N)` | `float32` |
| `velocity_z` | z-component of nodal velocity | `(T, N)` | `float32` |
| `body_force_x` | x-component of nodal body force | `(T, N)` | `float32` |
| `body_force_y` | y-component of nodal body force | `(T, N)` | `float32` |
| `body_force_z` | z-component of nodal body force | `(T, N)` | `float32` |
| `traction_x` | x-component of nodal surface traction | `(T, N)` | `float32` |
| `traction_y` | y-component of nodal surface traction | `(T, N)` | `float32` |
| `traction_z` | z-component of nodal surface traction | `(T, N)` | `float32` |

Where:

```text
T = number of saved output times
N = number of mesh nodes
```

Examples:

```text
displacement_x[k][i] = x-displacement of node i at time step k
displacement_y[k][i] = y-displacement of node i at time step k
velocity_y[k][i]     = y-velocity of node i at time step k
traction_y[k][i]     = y-component of the surface traction at node i and time step k
```

To reconstruct a full vector field:

```python
import numpy as np

u = np.stack(
    [
        dyn["displacement_x"],
        dyn["displacement_y"],
        dyn["displacement_z"],
    ],
    axis=-1,
)

print(u.shape)
# (T, N, 3)
```

The same convention can be used for velocity, body force, and traction.

---

## Configuration: `metadata`

The `metadata` configuration stores scalar simulation parameters and bookkeeping information.

Each row corresponds to one simulation sample.

### Execution and file information

| Field | Meaning |
|---|---|
| `sample_id` | Simulation identifier |
| `valid` | Whether the simulation sample is valid |
| `sample_name` | Sample folder name |
| `sample_dir` | Sample directory |
| `geo_path` | Path to the `.geo` geometry file |
| `mesh_path` | Path to the mesh file |
| `result_dir` | Directory containing solver outputs |
| `mpi_np` | Number of MPI processes used |
| `solver_executable` | Solver executable path or name |
| `gmsh_executable` | Gmsh executable path or name |
| `solution_file_type` | Type of solution file used, for example `.pvtu` |
| `n_solution_files` | Number of solution files found |
| `first_solution_file` | First solution file |
| `last_solution_file` | Last solution file |

### Material parameters

| Field | Meaning |
|---|---|
| `E` | Young's modulus |
| `nu` | Poisson's ratio |
| `lambda` | First Lamé parameter |
| `mu` | Second Lamé parameter |
| `rho` | Density |
| `c_damp` | Damping coefficient |

### Loading parameters

| Field | Meaning |
|---|---|
| `F0` | Nominal force amplitude |
| `traction_type` | Type of applied surface traction |
| `spatial_profile` | Spatial profile of the applied traction |
| `traction_amplitude_y_nominal_uniform` | Nominal uniform traction amplitude in the y direction |
| `frequency_hz` | Loading frequency, if applicable |
| `phase` | Loading phase, if applicable |
| `load_start_time` | Start time of the applied load |
| `load_end_time` | End time of the applied load |
| `load_center_x` | Load center coordinate in x |
| `load_center_y` | Load center coordinate in y |
| `load_center_z` | Load center coordinate in z |
| `load_sigma_x` | Load width in x for Gaussian profiles |
| `load_sigma_y` | Load width in y for Gaussian profiles |
| `load_sigma_z` | Load width in z for Gaussian profiles |
| `load_center_x_rel` | Relative load center coordinate in x |
| `load_center_z_rel` | Relative load center coordinate in z |
| `load_sigma_x_rel` | Relative Gaussian width in x |
| `load_sigma_z_rel` | Relative Gaussian width in z |
| `moving_direction` | Direction of motion for moving loads |
| `load_velocity` | Physical velocity of the moving load |
| `load_velocity_rel` | Relative velocity of the moving load |
| `impact_time` | Central time of the impact or pulse load |
| `impact_duration` | Duration of the impact or pulse load |

Some parameters may be unused depending on the selected `traction_type`. They are still stored to keep a fixed schema across all samples.

### Geometry and mesh parameters

| Field | Meaning |
|---|---|
| `L` | Beam length |
| `H` | Beam height |
| `B` | Beam width |
| `A` | Cross-sectional area |
| `I` | Second moment of area |
| `nx` | Nominal number of mesh divisions in x |
| `ny` | Nominal number of mesh divisions in y |
| `nz` | Nominal number of mesh divisions in z |
| `lc` | Nominal mesh size |

### Boundary, time, and storage information

| Field | Meaning |
|---|---|
| `dirichlet_boundary_ids` | Boundary IDs with Dirichlet conditions |
| `neumann_boundary_ids` | Boundary IDs with Neumann conditions |
| `dt` | Time step size |
| `t_end` | Final simulation time |
| `output_dt` | Output time interval |
| `n_nodes` | Number of nodes read from the output mesh |
| `n_cells` | Number of cells read from the output mesh |
| `n_edges` | Number of directed graph edges |
| `n_saved_times` | Number of saved output times |
| `snapshot_storage_format` | Storage format used for snapshot fields |

---

## Dataset Creation

### Curation Rationale

The dataset was created to provide simulation data for machine learning models that learn the dynamic response of 3D elastic structures.

The goal is to expose models to different combinations of geometry, material properties, damping, and loading conditions, while keeping a consistent data structure across samples.

### Source Data

The data are fully synthetic. They are generated by numerical finite element simulations of 3D elastic beams.

#### Data Collection and Processing

For each simulation:

1. A set of input parameters is sampled.
2. A 3D beam mesh is generated or loaded.
3. The linear elastodynamic problem is solved with `deal.II`.
4. The mesh and physical fields are exported to `.pvtu` or `.vtu` files.
5. The exported simulation data are converted into Hugging Face datasets.

The governing equation is the linear elastodynamic equation:

```text
  c * du/dt - div(sigma(u)) = f
```

where:

| Symbol | Meaning |
|---|---|
| `u(x,t)` | Displacement field |
| `du/dt` | Velocity field |
| `rho` | Material density |
| `c` | Damping coefficient |
| `f(x,t)` | Body force |
| `sigma(u)` | Linear elastic stress tensor |

The material is linear, isotropic, and elastic.

The stress tensor is:

```text
sigma(u) = lambda * tr(epsilon(u)) * I + 2 * mu * epsilon(u)
```

with:

```text
epsilon(u) = 0.5 * (grad(u) + grad(u)^T)
```

The Lamé parameters `lambda` and `mu` are computed from Young's modulus `E` and Poisson's ratio `nu`.

Continuous parameters are sampled using Latin Hypercube Sampling in a normalized space `[0, 1]^d`. Each sampled value is then mapped to its physical range using either a uniform or log-uniform transformation.

#### Who are the source data producers?

The source data are produced automatically by the simulation pipeline. No human-generated text, personal data, or user-generated content is included.

---

## Personal and Sensitive Information

This dataset does not contain personal, sensitive, or private information.

All samples are generated synthetically from numerical simulations.

---

## Bias, Risks, and Limitations

The dataset is limited by the numerical model and simulation setup used to generate it.

Main limitations include:

- the material model is linear elastic and isotropic;
- the results depend on the mesh resolution;
- the loading profiles are limited to the implemented traction models;
- the data are synthetic and may not represent experimental noise or real structural uncertainty;
- the dataset should not be treated as a certified engineering benchmark;
- acceleration may be computed by the solver but is not stored in the current Hugging Face dataset.

### Recommendations

Users should verify the assumptions of the dataset before using it for engineering or scientific conclusions.

For machine learning research, users should consider:

- checking the distribution of geometry, material, and loading parameters;
- normalizing physical quantities before training;
- validating models on held-out simulations;
- avoiding extrapolation far outside the sampled parameter ranges;
- verifying mesh consistency when using graph-based models.

---

## Loading Profiles

Possible values of `traction_type` include:

| `traction_type` | Meaning |
|---|---|
| `uniform` | Uniform surface traction |
| `gaussian_step` | Spatial Gaussian load active over a time window |
| `gaussian_harmonic` | Spatial Gaussian load with harmonic time dependence |
| `gaussian_pulse` | Spatial Gaussian load with pulse-like time dependence |
| `moving_gaussian` | Gaussian load moving along a prescribed direction |

Possible values of `spatial_profile` include:

| `spatial_profile` | Meaning |
|---|---|
| `uniform` | No spatial localization |
| `x` | Gaussian localization along x only |
| `xz` | Gaussian localization along x and z |

A negative sigma value may be used to disable localization in one direction. For example:

```text
load_sigma_z < 0
```

means that the load is uniform along the z direction.

---

## Minimal Usage Example

```python
from datasets import load_dataset
import numpy as np

repo_id = "fastcomputing/first_beam3d_test_single_split"

geometry = load_dataset(repo_id, name="geometry", split="total")
snapshot = load_dataset(repo_id, name="snapshot", split="total")
metadata = load_dataset(repo_id, name="metadata", split="total")

sample_idx = 0

geom = geometry[sample_idx]
dyn = snapshot[sample_idx]
meta = metadata[sample_idx]

points_x = np.asarray(geom["points_x"], dtype=np.float32)
points_y = np.asarray(geom["points_y"], dtype=np.float32)
points_z = np.asarray(geom["points_z"], dtype=np.float32)
cells = np.asarray(geom["cells"], dtype=np.int64)
edge_index = np.asarray(geom["edge_index"], dtype=np.int64)

u = np.stack(
    [
        np.asarray(dyn["displacement_x"], dtype=np.float32),
        np.asarray(dyn["displacement_y"], dtype=np.float32),
        np.asarray(dyn["displacement_z"], dtype=np.float32),
    ],
    axis=-1,
)

x = np.stack(
    [
        np.asarray(dyn["points_x"], dtype=np.float32),
        np.asarray(dyn["points_y"], dtype=np.float32),
        np.asarray(dyn["points_z"], dtype=np.float32),
    ],
    axis=-1,
)

print("points:", points.shape)        # (N, 3)
print("cells:", cells.shape)          # (C, 8)
print("edge_index:", edge_index.shape) # (2, E)
print("u:", u.shape)                  # (T, N, 3)
print("metadata keys:", meta.keys())
```

---

## Dataset Card Authors

FAST Computing
System theme
TOS
Privacy
About
Careers
Models