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SCIENCE CHINA Technological Sciences, Volume 61 , Issue 12 : 1889-1900(2018) https://doi.org/10.1007/s11431-018-9305-5

Modelling three dimensional dynamic problems using the four-node tetrahedral element with continuous nodal stress

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  • ReceivedMar 1, 2018
  • AcceptedJun 8, 2018
  • PublishedOct 18, 2018

Abstract

A partition of unity (PU) based four-node tetrahedral element with continuous nodal stress (Tetr4-CNS) was recently proposed for static analysis of three-dimensional solids. By simply using the same mesh as the classical four-node tetrahedral (Tetr4) element, high order global approximation function in the Tetr4-CNS element can be easily constructed without extra nodes or nodal DOFs. In this paper, the Tetr4-CNS element is further applied in the analysis of three dimensional dynamic problems. A series of free vibration and forced vibration problems are solved using the Tetr4-CNS element. The numerical results show that, for regular meshes, accuracy obtained using the Tetr4-CNS element is superior to that obtained using the Tetr4 and eight-node hexahedral (Hexa8) elements. For distorted meshes, the Tetr4-CNS element has better mesh-distortion tolerance than both the Tetr4 and Hexa8 elements.


Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant Nos. 51609240, 11572009, 51538001, 51579235 & 41472288) and the National Basic Research Program of China (Grant No. 2014CB047100).


Supplement

Supporting Information

The supporting information is available online at tech.scichina.com and www.springerlink.com. The supporting materials are published as submitted, without typesetting or editing. The responsibility for scientific accuracy and content remains entirely with the authors.


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  • Figure 1

    (Color online) Straight cantilever beam for distortion test [43]. (a) Regular hexahedral mesh; (b) distorted hexahedral mesh generated by longitudinally shifting the associated nodes by a distance d.

  • Figure 2

    (Color online) Sketch of dividing a Hexa8 element into six Tetr4 elements. (a) A Hexa8 element; (b) six Tetr4 elements.

  • Figure 3

    (Color online) Errors in the computed frequencies of the first mode for distortion sensitivity test. (a) Plane-distorted elements; (b) skewed-distorted elements.

  • Figure 4

    Free vibration analysis of a 3D cantilever beam.

  • Figure 5

    (Color online) Regular tetrahedral mesh for the 3D cantilever beam in Figure 4. (a) Mesh A (88 nodes, 180 elements); (b) Mesh B (378 nodes, 1200 elements); (c) Mesh C (1116 nodes, 4320 elements).

  • Figure 6

    (Color online) Regular hexahedral mesh for the 3D cantilever beam in Figure 4. (a) Mesh A (88 nodes, 30 elements); (b) Mesh B (378 nodes, 200 elements); (c) Mesh C (1116 nodes, 720 elements).

  • Figure 7

    (Color online) Convergence of the error in the computed frequency for the first two modes for the 3D cantilever beam. (a) Mode 1; (b) Mode 2.

  • Figure 8

    (Color online) Eigenmodes of 3D cantilever beam problem obtained by using the Tetr4 element. (a) Mode 5; (b) Mode 10; (c) Mode 15.

  • Figure 9

    (Color online) Eigenmodes of 3D cantilever beam problem obtained by using the Tetr4-CNS (LS20) element. (a) Mode 5; (b) Mode 10; (c) Mode 15.

  • Figure 10

    (Color online) One-eighth of a hollow sphere model discretized using four-node tetrahedral elements [42].

  • Figure 11

    (Color online) Convergence of errors in the computed frequency for the first two modes of the hollow sphere. (a) Mode 1; (b) Mode 2

  • Figure 12

    (Color online) Discritized model for a rock slope. (a) Hexahedral mesh; (b) tetrahedral mesh.

  • Figure 13

    (Color online) A very fine mesh for the rock slope (Hexa8 element with 17666 nodes and 15110 elements).

  • Figure 14

    (Color online) A 3D cantilever beam subjected to a harmonic loading on the right end.

  • Figure 15

    (Color online) Dynamic responses of a 3D cantilever beam subjected to a harmonic loading.

  • Figure 16

    (Color online) Displacement errors predicted by different types of elements.

  • Table 1   Computed natural frequencies (Hz) of the first mode through the plane-distorted elements

    2d/W

    Tetr4

    Hexa8

    Tetr10

    Hexa20

    Tetr4-CNS (LS10)

    Tetr4-CNS (LS20)

    Reference

    0.000

    0.2963

    0.2258

    0.1277

    0.1226

    0.1506

    0.1446

    0.1222

    0.025

    0.2971

    0.2259

    0.1277

    0.1227

    0.1506

    0.1446

    0.1222

    0.050

    0.2982

    0.2263

    0.1278

    0.1227

    0.1506

    0.1446

    0.1222

    0.075

    0.2995

    0.2270

    0.1278

    0.1228

    0.1507

    0.1446

    0.1222

    0.100

    0.3011

    0.2279

    0.1279

    0.1228

    0.1507

    0.1446

    0.1222

    0.150

    0.3045

    0.2304

    0.1281

    0.1231

    0.1507

    0.1446

    0.1222

    0.200

    0.3082

    0.2336

    0.1284

    0.1235

    0.1507

    0.1446

    0.1222

    0.250

    0.3119

    0.2371

    0.1288

    0.1240

    0.1508

    0.1446

    0.1222

    0.300

    0.3156

    0.2408

    0.1293

    0.1245

    0.1508

    0.1446

    0.1222

    0.400

    0.3226

    0.2481

    0.1306

    0.1260

    0.1509

    0.1446

    0.1222

    0.500

    0.3290

    0.2546

    0.1323

    0.1276

    0.1510

    0.1446

    0.1222

    0.600

    0.3348

    0.2604

    0.1342

    0.1294

    0.1511

    0.1446

    0.1222

    0.700

    0.3403

    0.2654

    0.1361

    0.1309

    0.1512

    0.1445

    0.1222

    0.800

    0.3456

    0.2699

    0.1379

    0.1323

    0.1514

    0.1445

    0.1222

    0.900

    0.3507

    0.2740

    0.1394

    0.1338

    0.1515

    0.1444

    0.1222

  • Table 2   Computed natural frequencies (Hz) of the first mode through the skewed-distorted elements

    2d/W

    Tetr4

    Hexa8

    Tetr10

    Hexa20

    Tetr4-CNS (LS10)

    Tetr4-CNS (LS20)

    Reference

    0.000

    0.2963

    0.2258

    0.1277

    0.1226

    0.1506

    0.1446

    0.1222

    0.025

    0.2976

    0.2269

    0.1277

    0.1227

    0.1506

    0.1446

    0.1222

    0.050

    0.2995

    0.2284

    0.1277

    0.1227

    0.1506

    0.1446

    0.1222

    0.075

    0.3018

    0.2302

    0.1277

    0.1227

    0.1506

    0.1446

    0.1222

    0.100

    0.3044

    0.2323

    0.1278

    0.1227

    0.1506

    0.1446

    0.1222

    0.150

    0.3098

    0.2370

    0.1279

    0.1228

    0.1506

    0.1446

    0.1222

    0.200

    0.3150

    0.2422

    0.1281

    0.1229

    0.1506

    0.1446

    0.1222

    0.250

    0.3194

    0.2473

    0.1283

    0.1230

    0.1506

    0.1446

    0.1222

    0.300

    0.3232

    0.2522

    0.1286

    0.1232

    0.1506

    0.1446

    0.1222

    0.400

    0.3290

    0.2605

    0.1296

    0.1238

    0.1506

    0.1446

    0.1222

    0.500

    0.3332

    0.2668

    0.1310

    0.1246

    0.1506

    0.1446

    0.1222

    0.600

    0.3365

    0.2715

    0.1328

    0.1255

    0.1507

    0.1446

    0.1222

    0.700

    0.3394

    0.2749

    0.1347

    0.1267

    0.1507

    0.1445

    0.1222

    0.800

    0.3419

    0.2777

    0.1365

    0.1281

    0.1507

    0.1445

    0.1222

    0.900

    0.3443

    0.2799

    0.1382

    0.1296

    0.1507

    0.1445

    0.1222

  • Table 3   Comparison of computed frequencies (Hz) for the 3D cantilever beam using Mesh A ((a) and 6(a))

    Mode

    Tetr4

    (88 nodes, 180 elements)

    Hexa8

    (88 nodes, 30 elements)

    Tetr4-CNS (LS10)

    (88 nodes, 180 elements)

    Tetr4-CNS (LS20)

    (88 nodes, 180 elements)

    Ref. [3]

    1

    48.44

    25.69

    22.96

    22.24

    20.77

    2

    63.48

    52.25

    50.70

    50.54

    49.72

    3

    261.52

    140.64

    146.48

    140.37

    124.47

    4

    309.13

    155.78

    154.43

    140.76

    132.45

    5

    326.98

    268.94

    264.45

    259.89

    252.26

    6

    374.68

    324.24

    323.90

    323.50

    321.94

  • Table 4   Comparison of computed frequencies (Hz) for the 3D cantilever beam using Mesh B ((b) and 6(b))

    Mode

    Tetr4

    (378 nodes, 1200 elements)

    Hexa8

    (378 nodes, 200 elements)

    Tetr4-CNS (LS10)

    (378 nodes, 1200 elements)

    Tetr4-CNS (LS20)

    (378 nodes, 1200 elements)

    Ref. [3]

    1

    31.31

    22.22

    20.94

    20.88

    20.77

    2

    54.10

    50.52

    50.01

    49.89

    49.72

    3

    182.15

    133.94

    126.58

    125.74

    124.47

    4

    218.31

    137.24

    137.25

    135.31

    132.45

    5

    271.80

    257.44

    253.79

    252.81

    252.26

    6

    323.71

    322.78

    322.55

    322.36

    321.94

  • Table 5   Comparison of computed frequencies (Hz) for the 3D cantilever beam using Mesh C ((c) and 6(c))

    Mode

    Tetr4

    (1116 nodes, 4320 elements)

    Hexa8

    (1116 nodes, 720 elements)

    Tetr4-CNS (LS10)

    (1116 nodes, 4320 elements)

    Tetr4-CNS (LS20)

    (1116 nodes, 4320 elements)

    Ref. [3]

    1

    26.45

    21.44

    20.86

    20.82

    20.77

    2

    51.63

    50.10

    49.86

    49.81

    49.72

    3

    156.23

    128.88

    125.25

    124.90

    124.47

    4

    181.24

    134.77

    133.83

    133.07

    132.45

    5

    260.92

    254.65

    252.89

    252.51

    252.26

    6

    322.95

    322.40

    322.28

    322.18

    321.94

  • Table 6   Comparison of computed frequencies (Hz) for the one-eighth of the hollow sphere using Model A (259 nodes, 972 elements)

    Mode

    Tetr4

    Tetr4-CNS (LS10)

    Tetr4-CNS (LS20)

    Reference

    1

    8.74

    8.24

    7.96

    7.63

    2

    9.03

    8.28

    8.09

    7.92

    3

    9.59

    8.30

    8.09

    7.93

    4

    12.12

    10.88

    10.34

    10.11

    5

    12.88

    11.56

    10.87

    10.25

    6

    13.52

    11.74

    10.90

    10.29

  • Table 7   Comparison of computed frequencies (Hz) for the one-eighth of the hollow sphere using Model B (549 nodes, 2304 elements)

    Mode

    Tetr4

    Tetr4-CNS (LS10)

    Tetr4-CNS (LS20)

    Reference

    1

    8.44

    7.96

    7.78

    7.63

    2

    8.57

    8.08

    8.01

    7.92

    3

    8.83

    8.09

    8.01

    7.93

    4

    11.29

    10.39

    10.21

    10.11

    5

    11.90

    10.86

    10.52

    10.25

    6

    12.24

    10.92

    10.53

    10.29

  • Table 8   Comparison of computed frequencies (Hz) for the one-eighth of the hollow sphere using Model C (1001 nodes, 4500 elements)

    Mode

    Tetr4

    Tetr4-CNS (LS10)

    Tetr4-CNS (LS20)

    Reference

    1

    8.29

    7.81

    7.72

    7.63

    2

    8.34

    8.01

    7.98

    7.92

    3

    8.48

    8.02

    7.98

    7.93

    4

    10.89

    10.23

    10.16

    10.11

    5

    11.40

    10.58

    10.40

    10.25

    6

    11.63

    10.62

    10.40

    10.29

  • Table 9   Comparison of computed frequencies (Hz) for the rock slope

    Mode

    Hexa8

    (246 nodes and

    124 elements)

    Hexa20

    (839 nodes and 124

    elements)

    Tetr10

    (1435 nodes and

    744 elements)

    Tetr4-CNS (LS20)

    (246 nodes and

    744 elements)

    Reference

    (17666 nodes and

    15110 elements)

    1

    37.85

    37.66

    37.68

    37.62

    37.68

    2

    54.08

    53.72

    53.80

    53.65

    53.79

    3

    63.73

    62.66

    62.76

    62.75

    62.70

    4

    70.26

    69.12

    69.32

    69.46

    69.23

    5

    88.74

    85.95

    86.32

    86.57

    86.15

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