A Modified Radial Point Interpolation Method (M-RPIM) for Free Vibration Analysis of Two-Dimensional Solids
Abstract
:1. Introduction
2. Formulation of the Original RPIM and the Present M-RPIM
3. Formulation of the Elastodynamics of Two-Dimensional Solids
4. Numerical Example
4.1. Free Vibration Analysis of the Cantilever Beam
4.1.1. Computation Accuracy Study
4.1.2. Convergence Study
4.1.3. Computation Efficiency Study
4.2. Free Vibration Analysis of the Cantilever Beam with Variable Cross-Section
4.3. Free Vibration Analysis of the Cantilever Beam with Holes
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Mode | FEM-Q4 | Error (%) | RPIM | Error (%) | M-RPIM | Error (%) | Ref. |
---|---|---|---|---|---|---|---|
1 | 830.567 | 1.003 | 815.423 | 0.839 | 825.213 | 0.351 | 822.322 |
2 | 4989.034 | 1.132 | 4909.931 | 0.471 | 4952.383 | 0.389 | 4933.177 |
3 | 12,826.933 | 0.022 | 12,826.322 | 0.017 | 12,825.428 | 0.010 | 12,824.145 |
4 | 13,167.852 | 1.336 | 13,035.277 | 0.316 | 13,025.003 | 0.237 | 12,994.215 |
5 | 23,992.489 | 1.604 | 23,772.722 | 0.673 | 23,725.015 | 0.471 | 23,613.775 |
6 | 36,701.000 | 1.910 | 36,492.012 | 1.329 | 36,197.042 | 0.510 | 36,013.226 |
7 | 38,467.305 | 0.059 | 38,461.345 | 0.044 | 38,450.819 | 0.016 | 38,444.488 |
8 | 50,697.394 | 2.248 | 50,570.105 | 1.991 | 49,854.144 | 0.547 | 49,582.799 |
9 | 64,062.116 | 0.225 | 64,045.638 | 0.199 | 63,984.254 | 0.103 | 63,918.563 |
10 | 65,590.504 | 2.524 | 65,609.724 | 2.554 | 64,290.994 | 0.493 | 63,975.503 |
11 | 81,118.869 | 3.012 | 81,334.286 | 3.286 | 79,233.413 | 0.618 | 78,746.943 |
12 | 89,562.090 | 0.252 | 89,514.438 | 0.199 | 89,345.277 | 0.010 | 89,336.686 |
Mode | FEM-Q4 | Error (%) | RPIM | Error (%) | M-RPIM | Error (%) | Ref. |
---|---|---|---|---|---|---|---|
1 | 824.304 | 0.241 | 819.913 | 0.293 | 822.800 | 0.058 | 822.322 |
2 | 4946.701 | 0.274 | 4924.719 | 0.171 | 4936.588 | 0.069 | 4933.177 |
3 | 12,824.618 | 0.004 | 12,824.955 | 0.006 | 12,824.496 | 0.003 | 12,824.145 |
4 | 13,036.557 | 0.326 | 13,008.259 | 0.108 | 13,004.647 | 0.080 | 12,994.215 |
5 | 23,706.655 | 0.393 | 23,657.845 | 0.187 | 23,635.380 | 0.091 | 23,613.775 |
6 | 36,182.573 | 0.470 | 36,152.539 | 0.387 | 36,049.829 | 0.102 | 36,013.226 |
7 | 38,449.523 | 0.013 | 38,447.183 | 0.007 | 38,445.401 | 0.002 | 38,444.488 |
8 | 49,857.999 | 0.555 | 49,874.211 | 0.588 | 49,637.697 | 0.111 | 49,582.799 |
9 | 63,996.004 | 0.121 | 63,990.287 | 0.112 | 63,976.475 | 0.091 | 63,918.563 |
10 | 64,332.241 | 0.558 | 64,422.173 | 0.698 | 63,994.651 | 0.030 | 63,975.503 |
11 | 79,334.621 | 0.746 | 79,524.455 | 0.987 | 78,846.611 | 0.127 | 78,746.943 |
12 | 89,391.481 | 0.061 | 89,377.038 | 0.045 | 89,336.977 | 0.000 | 89,336.686 |
Mode | FEM-Q4 | Error (%) | RPIM | Error (%) | M-RPIM | Error (%) | Ref. |
---|---|---|---|---|---|---|---|
1 | 823.104 | 0.095 | 821.051 | 0.155 | 822.413 | 0.011 | 822.322 |
2 | 4938.559 | 0.109 | 4928.453 | 0.096 | 4933.920 | 0.015 | 4933.177 |
3 | 12,824.010 | 0.001 | 12,823.612 | 0.004 | 12,823.953 | 0.001 | 12,824.145 |
4 | 13,011.299 | 0.131 | 12,992.024 | 0.017 | 12,996.698 | 0.019 | 12,994.215 |
5 | 23,651.752 | 0.161 | 23,631.689 | 0.076 | 23,619.234 | 0.023 | 23,613.775 |
6 | 36,083.214 | 0.194 | 36,074.813 | 0.171 | 36,022.821 | 0.027 | 36,013.226 |
7 | 38,445.687 | 0.003 | 38,444.328 | 0.000 | 38,443.843 | 0.002 | 38,444.488 |
8 | 49,697.499 | 0.231 | 49,714.721 | 0.266 | 49,597.527 | 0.030 | 49,582.799 |
9 | 63,982.840 | 0.101 | 63,979.749 | 0.096 | 63,939.314 | 0.032 | 63,918.563 |
10 | 64,092.160 | 0.182 | 64,149.033 | 0.271 | 63,974.133 | 0.002 | 63,975.503 |
11 | 78,994.780 | 0.315 | 79,104.908 | 0.455 | 78,774.275 | 0.035 | 78,746.943 |
12 | 89,358.561 | 0.024 | 89,351.356 | 0.016 | 89,334.272 | 0.003 | 89,336.686 |
Mode | FEM-Q4 | Error (%) | RPIM | Error (%) | M-RPIM | Error (%) | Ref. |
---|---|---|---|---|---|---|---|
1 | 822.674 | 0.043 | 821.486 | 0.102 | 822.279 | 0.005 | 822.322 |
2 | 4935.637 | 0.050 | 4929.842 | 0.068 | 4932.985 | 0.004 | 4933.177 |
3 | 12,823.747 | 0.003 | 12,823.464 | 0.005 | 12,823.711 | 0.003 | 12,824.145 |
4 | 13,002.246 | 0.062 | 12,991.382 | 0.022 | 12,993.901 | 0.002 | 12,994.215 |
5 | 23,632.124 | 0.078 | 23,621.334 | 0.032 | 23,613.557 | 0.001 | 23,613.775 |
6 | 36,047.788 | 0.096 | 36,044.602 | 0.087 | 36,013.347 | 0.000 | 36,013.226 |
7 | 38,444.192 | 0.001 | 38,443.250 | 0.003 | 38,443.144 | 0.003 | 38,444.488 |
8 | 49,640.417 | 0.116 | 49,653.240 | 0.142 | 49,583.475 | 0.001 | 49,582.799 |
9 | 63,977.974 | 0.093 | 63,975.937 | 0.090 | 63,920.007 | 0.002 | 63,918.563 |
10 | 64,006.958 | 0.049 | 64,044.242 | 0.107 | 63,973.057 | 0.004 | 63,975.503 |
11 | 78,874.387 | 0.162 | 78,944.368 | 0.251 | 78,749.088 | 0.003 | 78,746.943 |
12 | 89,346.670 | 0.011 | 89,342.201 | 0.006 | 89,332.973 | 0.004 | 89,336.686 |
Mode | FEM-Q4 | Error (%) | RPIM | Error (%) | M-RPIM | Error (%) | Ref. |
---|---|---|---|---|---|---|---|
1 | 41.771 | 0.333 | 41.536 | 0.233 | 41.678 | 0.109 | 41.633 |
2 | 147.202 | 0.781 | 146.826 | 0.523 | 146.335 | 0.187 | 146.062 |
3 | 151.597 | 0.058 | 151.511 | 0.002 | 151.532 | 0.015 | 151.508 |
4 | 298.805 | 1.349 | 298.048 | 1.092 | 295.483 | 0.222 | 294.829 |
5 | 412.666 | 0.326 | 412.032 | 0.172 | 411.396 | 0.017 | 411.327 |
6 | 442.931 | 1.685 | 441.428 | 1.340 | 436.366 | 0.178 | 435.592 |
7 | 528.614 | 1.053 | 526.132 | 0.578 | 523.667 | 0.107 | 523.108 |
8 | 601.857 | 2.143 | 598.737 | 1.614 | 590.187 | 0.163 | 589.229 |
9 | 619.528 | 1.005 | 613.227 | 0.023 | 613.441 | 0.012 | 613.365 |
10 | 671.507 | 1.529 | 662.514 | 0.170 | 662.167 | 0.117 | 661.392 |
11 | 710.007 | 2.389 | 705.817 | 1.785 | 695.000 | 0.225 | 693.441 |
12 | 713.997 | 0.802 | 710.025 | 0.241 | 708.647 | 0.046 | 708.320 |
Mode | FEM-Q4 | Error (%) | RPIM | Error (%) | M-RPIM | Error (%) | Ref. |
---|---|---|---|---|---|---|---|
1 | 1626.190 | 0.617 | 1612.353 | 0.239 | 1618.711 | 0.154 | 1616.218 |
2 | 8272.300 | 0.174 | 8246.759 | 0.135 | 8268.312 | 0.126 | 8257.923 |
3 | 11,373.419 | 0.791 | 11,239.656 | 0.395 | 11,302.342 | 0.161 | 11,284.188 |
4 | 19,395.928 | 1.595 | 19,004.812 | 0.454 | 19,101.194 | 0.051 | 19,091.435 |
5 | 33,523.877 | 1.273 | 33,233.174 | 0.395 | 33,231.319 | 0.390 | 33,102.326 |
6 | 33,972.380 | 1.786 | 33,568.275 | 0.575 | 33,472.489 | 0.288 | 33,376.214 |
7 | 37,191.685 | 2.943 | 36,443.890 | 0.873 | 36,333.778 | 0.568 | 36,128.559 |
8 | 52,155.832 | 2.894 | 51,179.337 | 0.968 | 51,042.893 | 0.699 | 50,688.744 |
9 | 52,582.353 | 3.420 | 51,249.812 | 0.799 | 51,117.100 | 0.538 | 50,843.699 |
10 | 55,474.223 | 2.473 | 54,470.879 | 0.620 | 54,276.585 | 0.261 | 54,135.295 |
11 | 67,782.825 | 2.359 | 66,555.530 | 0.505 | 66,471.730 | 0.379 | 66,220.863 |
12 | 75,775.407 | 1.343 | 75,309.818 | 0.721 | 74,789.424 | 0.025 | 74,771.060 |
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Sun, T.; Wang, P.; Zhang, G.; Chai, Y. A Modified Radial Point Interpolation Method (M-RPIM) for Free Vibration Analysis of Two-Dimensional Solids. Mathematics 2022, 10, 2889. https://doi.org/10.3390/math10162889
Sun T, Wang P, Zhang G, Chai Y. A Modified Radial Point Interpolation Method (M-RPIM) for Free Vibration Analysis of Two-Dimensional Solids. Mathematics. 2022; 10(16):2889. https://doi.org/10.3390/math10162889
Chicago/Turabian StyleSun, Tingting, Peng Wang, Guanjun Zhang, and Yingbin Chai. 2022. "A Modified Radial Point Interpolation Method (M-RPIM) for Free Vibration Analysis of Two-Dimensional Solids" Mathematics 10, no. 16: 2889. https://doi.org/10.3390/math10162889
APA StyleSun, T., Wang, P., Zhang, G., & Chai, Y. (2022). A Modified Radial Point Interpolation Method (M-RPIM) for Free Vibration Analysis of Two-Dimensional Solids. Mathematics, 10(16), 2889. https://doi.org/10.3390/math10162889