Modeling Structural Dynamics Using FE-Meshfree QUAD4 Element with Radial-Polynomial Basis Functions
Abstract
:1. Introduction
2. Shape Functions for FE-RPIM QUAD4 Element
2.1. Formulation of Shape Functions
2.2. Properties of Shape Functions
- (i)
- Kronecker-delta character
- (ii)
- Compatibility property at the interface of elements.
- (iii)
- High order completeness, in other words, reproducibility of all the assumed Cartesian terms (Equation (3)).
3. FE-RPIM QUAD4 for Elastodynamic Problems
3.1. FE-RPIM QUAD4 for Dynamic Analysis
3.2. Time Integration Scheme
3.3. Generalized Eigenvalue Problem
3.4. Diagonally Lumped Mass Matrix
4. Numerical Examples
4.1. Cook’s Skew Beam
4.2. A Slender Rod
4.3. An Annulus
4.4. Mesh Distortion Test
- (1)
- First, as distortion parameter’s value increases, the errors based on FE-RPIM QUAD4 element do not change appreciably, while those based on QUAD4 element, TRIG3 element and QUAD8 elements change rapidly. The FE-RPIM QUAD4 element is immune to mesh distortion.
- (2)
- Second, accuracy of FE-RPIM QUAD4 element is always much higher than QUAD4 and TRIG3 elements.
- (3)
- Third, when 2d/D < 0.2, QUAD8 element’s accuracy is higher than QUAD4, FE-RPIM QUAD4 and TRIG3 elements. However, as the value of 2d/D increases, accuracy through QUAD8 element deteriorates quickly. If meshes used are distorted severely, QUAD8 element’s accuracy is much lower than FE-RPIM QUAD4 element.
- (4)
- Fourth, compared to CMM, FE-RPIM QUAD4 element can achieve better results if DLMM is employed.
4.5. A Plate with Four Holes
4.6. A Cantilever Beam under Harmonic Load
5. Conclusions
- (1)
- Based on 4-node quadrilateral mesh, FE-RPIM QUAD4 element’s accuracy is much higher than QUAD4 and TRIG3 elements (Table 2).
- (2)
- Although FE-RPIM QUAD4 element’s accuracy is slightly inferior to QUAD8 element, QUAD8 element requires more nodes than FE-RPIM QUAD4 element to discretize the problem domain. In addition, FE-RPIM QUAD4 element can achieve results closing to the reference solution, even for coarse mesh (Figure 22).
- (3)
- For distorted meshes, FE-RPIM QUAD4 element’s accuracy is always much higher than QUAD4 and TRIG3 elements. Moreover, FE-RPIM QUAD4 element is immune to mesh distortion, but TRIG3, QUAD4 and QUAD8 elements give very bad results as the mesh quality deteriorates (Figure 14).
- (4)
- In the tests associated to the analysis of free vibration, the result based on the DLMM are very close to those based on the CMM in the context of FE-RPIM QUAD4 element. In the test on forced vibration analysis, the result from the DLMM also agrees well with that from the CMM, which means DLMM can supersede the CMM in the context of the FE-RPIM QUAD4 element even for the scheme of implicit time integration.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Mesh | Mode | TRIG3 | QUAD4 | QUAD8 | FE-RPIM QUAD4 (CMM) | FE-RPIM QUAD4 (DLMM) | Analytical Solution [62] |
---|---|---|---|---|---|---|---|
Mesh A (100 × 1) | 1 | 25.820968 | 25.820965 | 25.819889 | 25.820844 | 25.820870 | 25.819889 |
2 | 51.648164 | 51.645511 | 51.639778 | 51.636459 | 51.647617 | 51.639778 | |
3 | 77.487948 | 77.485357 | 77.459667 | 77.333589 | 77.486075 | 77.459667 | |
4 | 103.346621 | 103.353319 | 103.279556 | 103.770772 | 103.341991 | 103.279556 | |
5 | 129.231393 | 129.235285 | 129.099445 | 129.144371 | 129.220982 | 129.099445 | |
6 | 155.144150 | 155.090370 | 154.919334 | 154.271801 | 155.128495 | 154.919334 | |
7 | 181.097287 | 181.093361 | 180.739223 | 179.861768 | 181.069767 | 180.739223 | |
8 | 207.100128 | 207.044472 | 206.559112 | 207.167158 | 207.049784 | 206.559112 | |
9 | 233.139899 | 233.163742 | 232.379001 | 231.620531 | 233.073245 | 232.379001 | |
10 | 259.236693 | 259.205890 | 258.198890 | 257.916223 | 259.144523 | 258.198890 | |
Mesh B (200 × 2) | Mode | TRIG3 | QUAD4 | QUAD8 | FE-RPIM QUAD4 (CMM) | FE-RPIM QUAD4 (DLMM) | Analytical Solution [62] |
1 | 25.820157 | 25.820255 | 25.819884 | 25.819736 | 25.819876 | 25.819889 | |
2 | 51.641819 | 51.643160 | 51.639488 | 51.642793 | 51.639674 | 51.639778 | |
3 | 77.466851 | 77.468030 | 77.460498 | 77.428906 | 77.459316 | 77.459667 | |
4 | 103.296728 | 103.295143 | 103.276931 | 103.223462 | 103.278723 | 103.279556 | |
5 | 129.132375 | 129.129519 | 129.098294 | 129.103269 | 129.097812 | 129.099445 | |
6 | 154.976563 | 154.971476 | 154.916664 | 154.910322 | 154.916499 | 154.919334 | |
7 | 180.830533 | 180.849114 | 180.739424 | 180.695709 | 180.734699 | 180.739223 | |
8 | 206.695973 | 206.729136 | 206.571040 | 206.625255 | 206.552319 | 206.559112 | |
9 | 232.574473 | 232.597669 | 232.373245 | 232.226958 | 232.369264 | 232.379001 | |
10 | 258.459661 | 258.424349 | 258.192677 | 258.548565 | 258.185433 | 258.198890 | |
Mesh C (400 × 4) | Mode | TRIG3 | QUAD4 | QUAD8 | FE-RPIM QUAD4 (CMM) | FE-RPIM QUAD4 (DLMM) | Analytical Solution [62] |
1 | 25.819955 | 25.819955 | 25.819872 | 25.819892 | 25.819889 | 25.819889 | |
2 | 51.640264 | 51.640326 | 51.639657 | 51.639523 | 51.639777 | 51.639778 | |
3 | 77.461418 | 77.461344 | 77.455360 | 77.458972 | 77.459663 | 77.459667 | |
4 | 103.283785 | 103.284022 | 103.280289 | 103.281014 | 103.279545 | 103.279556 | |
5 | 129.107774 | 129.107848 | 129.098050 | 129.100114 | 129.099422 | 129.099445 | |
6 | 154.933575 | 154.933779 | 154.917190 | 154.920049 | 154.919292 | 154.919334 | |
7 | 180.762060 | 180.761938 | 180.731242 | 180.738237 | 180.739152 | 180.739223 | |
8 | 206.593046 | 206.592175 | 206.540550 | 206.564266 | 206.558998 | 206.559112 | |
9 | 232.427179 | 232.427476 | 232.386967 | 232.371678 | 232.378826 | 232.379001 | |
10 | 258.265837 | 258.264934 | 258.234066 | 258.188904 | 258.198630 | 258.198890 |
Mesh | Mode | TRIG3 | QUAD4 | QUAD8 | FE-RPIM QUAD4 (CMM) | FE-RPIM QUAD4 (DLMM) | Reference Solution [61] |
---|---|---|---|---|---|---|---|
Mesh A | 1 | 1069.0 | 764.6 | 331.6 | 465.7 | 459.3 | 307.3 |
2 | 1069.0 | 765.7 | 331.6 | 465.8 | 459.3 | 307.3 | |
3 | 1973.0 | 1917.4 | 945.3 | 1683.8 | 1623.6 | 838.5 | |
4 | 2759.1 | 2346.5 | 945.3 | 1686.7 | 1623.6 | 838.5 | |
5 | 2760.6 | 2350.0 | 1823.4 | 1938.7 | 1937.9 | 1535.4 | |
6 | 2779.7 | 2775.5 | 1823.9 | 2665.5 | 2714.8 | 1535.4 | |
Mesh B | Mode | TRIG3 | QUAD4 | QUAD8 | FE-RPIM QUAD4 (CMM) | FE-RPIM QUAD4 (DLMM) | Reference Solution [61] |
1 | 601.2 | 430.5 | 310.7 | 318.9 | 317.8 | 307.3 | |
2 | 601.2 | 430.5 | 310.7 | 318.9 | 317.8 | 307.3 | |
3 | 1622.2 | 1221.1 | 851.1 | 895.7 | 890.0 | 838.5 | |
4 | 1622.2 | 1221.3 | 851.1 | 895.8 | 890.0 | 838.5 | |
5 | 1869.2 | 1855.6 | 1566.6 | 1689.0 | 1665.2 | 1535.4 | |
6 | 2619.8 | 2351.4 | 1567.6 | 1691.0 | 1665.2 | 1535.4 | |
Mesh C | Mode | TRIG3 | QUAD4 | QUAD8 | FE-RPIM QUAD4 (CMM) | FE-RPIM QUAD4 (DLMM) | Reference Solution [61] |
1 | 402.7 | 340.1 | 308.0 | 308.0 | 307.8 | 307.3 | |
2 | 402.7 | 340.1 | 308.0 | 308.0 | 307.8 | 307.3 | |
3 | 1098.2 | 938.0 | 840.6 | 841.7 | 841.2 | 838.5 | |
4 | 1098.4 | 938.0 | 840.6 | 841.7 | 841.2 | 838.5 | |
5 | 1843.9 | 1742.3 | 1539.8 | 1544.0 | 1542.4 | 1535.4 | |
6 | 2013.4 | 1742.4 | 1539.9 | 1544.0 | 1542.4 | 1535.4 | |
Mesh D | Mode | TRIG3 | QUAD4 | QUAD8 | FE-RPIM QUAD4 (CMM) | FE-RPIM QUAD4 (DLMM) | Reference Solution [61] |
1 | 333.8 | 315.7 | 307.4 | 307.4 | 307.4 | 307.3 | |
2 | 333.8 | 315.7 | 307.4 | 307.4 | 307.4 | 307.3 | |
3 | 911.4 | 863.7 | 839.0 | 839.0 | 839.0 | 838.5 | |
4 | 911.4 | 863.8 | 839.0 | 839.0 | 839.0 | 838.5 | |
5 | 1670.8 | 1587.3 | 1536.3 | 1536.5 | 1536.6 | 1535.4 | |
6 | 1670.8 | 1587.6 | 1536.3 | 1536.5 | 1536.6 | 1535.4 |
2d/D | TRIG3 (CMM) | QUAD4 (CMM) | QUAD8 (CMM) | FE-RPIM QUAD4 (CMM) | FE-RPIM QUAD4 (DLMM) | Reference Solution [61] |
---|---|---|---|---|---|---|
0.000 | 4140.56 | 2623.12 | 868.78 | 1024.59 | 984.12 | 822.13 |
0.025 | 4296.89 | 2709.93 | 871.92 | 1028.00 | 986.58 | 822.13 |
0.050 | 4444.46 | 2888.54 | 880.81 | 1033.54 | 989.62 | 822.13 |
0.075 | 4556.81 | 3052.25 | 894.10 | 1037.37 | 989.81 | 822.13 |
0.100 | 4642.17 | 3168.80 | 910.07 | 1039.62 | 987.54 | 822.13 |
0.150 | 4772.43 | 3294.48 | 947.03 | 1041.76 | 979.18 | 822.13 |
0.200 | 4880.56 | 3350.17 | 999.17 | 1042.67 | 969.09 | 822.13 |
0.250 | 4979.99 | 3382.91 | 1085.88 | 1043.14 | 958.78 | 822.13 |
0.300 | 5074.54 | 3412.02 | 1219.18 | 1043.44 | 948.37 | 822.13 |
0.400 | 5255.27 | 3484.76 | 1593.73 | 1043.84 | 925.26 | 822.13 |
0.500 | 5428.50 | 3586.06 | 1988.16 | 1044.17 | 894.74 | 822.13 |
0.600 | 5596.39 | 3714.14 | 2309.06 | 1044.52 | 853.35 | 822.13 |
0.700 | 5759.40 | 3867.25 | 2551.71 | 1044.91 | 802.31 | 822.13 |
0.800 | 5919.14 | 4040.12 | 2741.70 | 1045.38 | 747.99 | 822.13 |
0.900 | 6073.71 | 4229.27 | 2904.91 | 1045.94 | 698.62 | 822.13 |
Mesh | Mode | TRIG3 | QUAD4 | FE-RPIM QUAD4 (CMM) | FE-RPIM QUAD4 (DLMM) | Reference Solution |
---|---|---|---|---|---|---|
Mesh A | 1 | 49.21 | 48.60 | 48.24 | 48.26 | 47.93 |
2 | 118.15 | 117.29 | 116.73 | 116.72 | 116.25 | |
3 | 129.69 | 128.04 | 126.93 | 126.96 | 126.13 | |
4 | 209.36 | 206.57 | 204.58 | 204.76 | 203.25 | |
5 | 214.32 | 210.56 | 207.54 | 207.42 | 205.34 | |
6 | 235.48 | 232.65 | 230.54 | 230.80 | 229.13 | |
Mesh B | Mode | TRIG3 | QUAD4 | FE-RPIM QUAD4 (CMM) | FE-RPIM QUAD4 (DLMM) | Reference Solution |
1 | 48.85 | 48.39 | 48.11 | 48.12 | 47.93 | |
2 | 117.70 | 116.97 | 116.55 | 116.54 | 116.25 | |
3 | 128.71 | 127.38 | 126.61 | 126.63 | 126.13 | |
4 | 207.63 | 205.38 | 204.02 | 204.10 | 203.25 | |
5 | 211.84 | 208.73 | 206.59 | 206.53 | 205.34 | |
6 | 233.86 | 231.49 | 229.97 | 230.00 | 229.13 | |
Mesh C | Mode | TRIG3 | QUAD4 | FE-RPIM QUAD4 (CMM) | FE-RPIM QUAD4 (DLMM) | Reference Solution |
1 | 48.65 | 48.26 | 48.05 | 48.05 | 47.93 | |
2 | 117.37 | 116.79 | 116.46 | 116.46 | 116.25 | |
3 | 128.19 | 127.06 | 126.46 | 126.45 | 126.13 | |
4 | 206.78 | 204.86 | 203.79 | 203.76 | 203.25 | |
5 | 210.77 | 207.83 | 206.20 | 206.12 | 205.34 | |
6 | 233.10 | 231.02 | 229.73 | 229.83 | 229.13 | |
Mesh D | Mode | TRIG3 | QUAD4 | FE-RPIM QUAD4 (CMM) | FE-RPIM QUAD4 (DLMM) | Reference Solution |
1 | 48.40 | 48.13 | 47.99 | 47.99 | 47.93 | |
2 | 116.92 | 116.55 | 116.34 | 116.34 | 116.25 | |
3 | 127.41 | 126.65 | 126.27 | 126.27 | 126.13 | |
4 | 205.48 | 204.17 | 203.49 | 203.46 | 203.25 | |
5 | 208.73 | 206.80 | 205.72 | 205.68 | 205.34 | |
6 | 231.49 | 230.13 | 229.39 | 229.44 | 229.13 |
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Luo, H.; Sun, G. Modeling Structural Dynamics Using FE-Meshfree QUAD4 Element with Radial-Polynomial Basis Functions. Materials 2021, 14, 2288. https://doi.org/10.3390/ma14092288
Luo H, Sun G. Modeling Structural Dynamics Using FE-Meshfree QUAD4 Element with Radial-Polynomial Basis Functions. Materials. 2021; 14(9):2288. https://doi.org/10.3390/ma14092288
Chicago/Turabian StyleLuo, Hongming, and Guanhua Sun. 2021. "Modeling Structural Dynamics Using FE-Meshfree QUAD4 Element with Radial-Polynomial Basis Functions" Materials 14, no. 9: 2288. https://doi.org/10.3390/ma14092288
APA StyleLuo, H., & Sun, G. (2021). Modeling Structural Dynamics Using FE-Meshfree QUAD4 Element with Radial-Polynomial Basis Functions. Materials, 14(9), 2288. https://doi.org/10.3390/ma14092288