# 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

_{1}P

_{2}P

_{3}P

_{4}}. The global approximation u

^{h}(

**x**) of FE-RPIM QUAD4 element can be simply written as [49]

^{[i]}represent the specified polynomial term number and the number of nodes within Ω

_{i}.

**a**and

**b**are two unknown vectors.

**r**represents the radial basis functions, while

**p**represents the polynomial functions formulated as:

**p**(

**x**) = {1 x y} for M = 3;

**p**(

**x**) = {1 x y xy} for M = 4;

**p**(

**x**) = {1 x y xy x

^{2}y

^{2}} for M = 6.

**r**(x, y) can be formulated with [14,16]:

_{i}, the following equations are obtained:

_{i}, and

^{[i]}) parameters in Equation (8). However, only n

^{[i]}equations are available. Nevertheless, according to the work finished in [14,54], vectors of

**a**and

**b**can be eliminated as proposed in their work and local approximation function in Equation (2) is eventually expressed as [14]:

#### 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

**M**is the global mass matrix,

**K**is the global stiffness matrix, and $f$ is the global load vector, which can be computed as:

**D**is the matrix of the elastic constants of the material, $\overline{\mathit{t}}$ is the specified traction vector applied on stress boundary ${\mathsf{\Gamma}}_{\sigma}^{e}$,

**b**is the body force per unit volume, $\mathit{N}$ represents the matrix of shape function for ${\mathsf{\Omega}}^{e}$, expressed as:

**C**can be formulated into,

#### 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

^{9}, mass density ρ = 2700. Since L/D = 100 > 10, the geometric parameters used assure that the one-dimensional slender rod can be well represented by the two-dimensional model without causing unacceptable error. In the computation, both upside and down sides of the model are fixed in the normal direction.

#### 4.3. An Annulus

_{a}= 0.4, R

_{b}= 0.5, v = 0.33, E = 72 × 10

^{9}, t = 1, ρ = 2700.

#### 4.4. Mesh Distortion Test

^{4}kg/mm

^{2}, $\rho $ = 8.0 × 10

^{−10}kg fs

^{2}/mm

^{4}, t = 1 mm.

- (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

^{9}and ρ = 2700. Left side of the plate is fixed.

#### 4.6. A Cantilever Beam under Harmonic Load

^{3}, D = 1 m, L = 4 m, v = 0.3, E = 1 Pa, $f\left(t\right)=\mathrm{sin}{\omega}_{f}t$, ${\omega}_{f}=0.04rad/s$, ${\beta}_{1}=0.005$ and ${\beta}_{2}=0.272$.

## 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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**Figure 1.**The nodal support domain of first order (${\mathsf{\Omega}}_{i}$) for a central node 1: (

**a**) central node within computational domain; (

**b**) central node on computational boundary.

**Figure 2.**Element support domain: (

**a**) central node within computational domain; (

**b**) central node on computational boundary.

**Figure 3.**A two−dimensional domain (2 m× 2 m) discretized with 4 quadrilateral elements and 9 nodes.

**Figure 4.**Shape functions of FE−RPIM QUAD4 element for different nodes at different locations: (

**a**) node 1; (

**b**) node 2; (

**c**) node 3; (

**d**) node 5.

**Figure 5.**Shape functions of QUAD4 element for different nodes at different locations. (

**a**) node 1; (

**b**) node 2; (

**c**) node 3; (

**d**) node 5.

**Figure 8.**Normalized mesh for slender rod in Figure 3.

**Figure 10.**Triangular meshes for the annulus. (

**a**) Mesh A (20 elements, 20 nodes); (

**b**) Mesh B (80 elements, 60 nodes); (

**c**) Mesh C (320 elements, 200 nodes); (

**d**) Mesh D (1280 elements, 720 nodes).

**Figure 11.**Quadrilaterial meshes for the annulus: (

**a**) Mesh A (10 elements, 20 nodes); (

**b**) Mesh B (40 elements, 60 nodes); (

**c**) Mesh C (160 elements, 200 nodes); (

**d**) Mesh D (640 elements, 720 nodes).

**Figure 15.**Comparison of accuracy of FE−RPIM QUAD4 (CMM) and FE−RPIM QUAD4(DLMM) for mesh distortion sensitivity test.

**Figure 17.**Triangular mesh for the plate with four holes. (

**a**) Mesh A (1584 elements and 871 nodes); (

**b**) Mesh B (2350 elements and 1278 nodes); (

**c**) Mesh C (3170 elements and 1710 nodes); (

**d**) Mesh D (5302 elements and 2814 nodes).

**Figure 18.**Quadrilateral mesh for the plate with four holes: (

**a**) Mesh A (792 elements and 871 nodes); (

**b**) Mesh B (1175 elements and 1278 nodes); (

**c**) Mesh C (1585 elements and 1710 nodes); (

**d**) Mesh D (2651 elements and 2814 nodes).

**Figure 19.**First 6 mode shapes of the plate with four holes using FE-RPIM QUAD4 element (CMM): (

**a**) Mode 1; (

**b**) Mode 2; (

**c**) Mode 3; (

**d**) Mode 4; (

**e**) Mode 5; (

**f**) Mode 6.

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 |

**Table 3.**Computed natural frequencies (Hz) of the first mode for the mesh distortion sensitivity test.

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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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Luo, 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