# The Influence of Mesh Density on the Results Obtained by Finite Element Analysis of Complex Bodies

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Description

#### 2.1. Preliminary Evaluation

#### 2.2. Reference Complex Body

^{3}. The frame is made of cast steel St 50-2 according to DIN, for which the following values were considered: Young’s modulus E = 210,000 N/mm

^{2}, shear modulus G = 80,000 N/mm

^{2}, and ν = 0.3 for the Poisson’s ratio.

_{N}= 250 kN, evenly distributed on the support surfaces of the main shaft bores, and on the work surface of the table, on the other hand, as shown in Figure 1. The elastic deformation was determined in the direction of the pressing axis between the bore axis of the main shaft and the plane of the workbench of the frame. For this frame, it was not possible to elaborate an analytical expression that would allow the determination of the exact solution of the elastic deformation of the frame.

_{i}of deformation y

_{i}from the deviation indicated for the last iteration (y

_{7}), can be found in Table 1. In Table 2, the average size of the elements for each discretisation level are presented.

_{i}being relevant. The variation of the deformation according to the discretization levels suggests an equilateral hyperbole law.

- -
- a law can be established to link the deformations to the levels of the discretization;
- -
- for various bodies, the most probable value of the elastic deformation can be determined;
- -
- a coefficient of proportionality can be established between the theoretically exact deformation and the deformation that corresponds to a certain level of discretization, regardless of the complexity of the studied body;
- -
- it is possible to obtain particularly good results in a short time, using common software and hardware resources, as a result of a single FEA analysis for a reasonable mesh density, related to the elastic deformation of complex geometry and large bodies.

## 3. Method and Results

#### 3.1. Dependence of Deformation Values According to the Level of Discretization

- x—the number of elements of the discretization network of the body subject to finite element analysis,
- y = f(x)—the deformation of the body (under the action of the force F) corresponding to the discretization level to which corresponds the number x of elements of the network, and
- $a=\underset{x\to \infty}{\mathrm{lim}}f\left(x\right)$ is the most probable value of the studied body deformation resulting from the action of force F.

_{1}; y

_{1}); (x

_{2}; y

_{2}), and (x

_{3}; y

_{3}).

_{i}, y

_{i}), for the constant a, a value a

_{r}

_{-s-t}(r < s < t, r ≥ 1, t ≤ k = i

_{max}) is determined.

_{min}= 0.1824 mm and a

_{max}= 0.23309 mm are identified, and the average value a

_{med}= 0.19375 mm is determined. Analysing the values in Table 3 and their evolution trend, most likely a = f(x→∞) = 0.191 … 0.192 mm. The following values are noticeable a

_{1-2-7}= 0.19185 mm, a

_{1-3-6}= 0.1915 mm and a

_{1-4-6}= 0.19183 mm, but also a

_{2-3-6}= 0.19209 mm, a

_{2-4-6}= 0.19206 mm, a

_{3-4-6}= 0.19205 mm.

_{r}

_{-s-t}values can be less than y

_{7}= 0.189 mm, value of the elastic deformation indicated by the study with the greatest level of discretization. As a result, the following values should not be considered a

_{1-2-3}= 0.18124, a

_{1-2-4}= 0.18745, a

_{1-2-5}= 0.18724, a

_{1-3-5}= 0.18803, a

_{1-4-5}= 0.18710, a

_{2-3-5}= 0.18860, a

_{2-4-5}= 0.18708 and a

_{3-4-5}= 0.18666. High values that are significantly above average, such as those over 0.196 mm (namely a

_{1-5-6}= 0.19690, a

_{2-5-6}= 0.19802, a

_{2-5-7}= 0.19653, a

_{3-5-6}= 0.19987, a

_{3-5-7}= 0.19749, a

_{3-6-7}= 0.19537, a

_{4-5-6}= 0.23309, a

_{4-5-7}= 0.20598 and a

_{4-6-7}= 0.19630), can also be excluded.

_{med-1}= 0.192688 mm against which the new limits a

_{min-1}= 0.19011 mm and a

_{max-1}= 0.19588 mm deviate by −1.3379% respectively +1.6565%.

#### 3.2. The Study of Simple Bodies in Which the Elastic Deformation Is Analytically Determinable

#### 3.2.1. Deformation Study of a Simple Pole-Type Body

^{2}, and a simple compression force F = 250 kN evenly distributed, Figure 3.

_{1-5-6}= 30.211 µm and a

_{4-5-7}= 6.9119 µm they are obviously aberrant, and value a

_{2-5-7}= 10.003 µm is over the limit. These are not considered. The values a

_{2-5-6}= 8.3598 µm or a

_{3-5-6}= 8.3222 µm, both lower than any of the y values based on which the values of a

_{2-5-6}are a

_{3-5-6}were determined, cannot be accepted. The average value of the 30 acceptable values of the deformation is a

_{med}= 8.9006 µm, and extreme values are a

_{min}= 8.4796 µm and a

_{max}= 9.6634 µm (which deviates from the average value by −4.73% and respectively by +8.57%). It should be noted that the average value a

_{med}= 8.9006 µm determined this way is practically identical with the deformation y

_{7}= 8.902 µm resulting for a discretization in 123,673 elements of the studied body (Table 4), its deviation from y

_{7}being only +0.02%. Compared to the theoretical elastic deformation δ = 8.2443 µm, the average a

_{med}value deviates by +7.96%.

_{r}

_{-s-t}ϵ [8.902 μm; 9.5 µm], then the corrected average value a

_{med-1}= 9.084 µm is greater by 10.185% compared to theoretical elastic deformation δ = 8.2443 µm.

#### 3.2.2. Deformation Study of a Pole with Arm in the Console

^{3}, a square section with side s

_{2}= 300 mm was adopted for the studied body. The relevant geometric features are lengths l

_{1}= 950 mm and l

_{2}= 650 mm, areas A

_{1}= A

_{2}= 300 × 300 = 90,000 mm

^{2}, with axial moments of inertia I

_{1}= I

_{2}= (s

_{2})

^{4}/12 = 675 × 10

^{6}mm

^{4}. The external force has the same value F = 250 kN.

_{i}of deformation y

_{i}from the deviation indicated for the last iteration (y

_{7})) are presented in Table 6.

_{2-5-6}= −0.85313 mm is aberrant, and the values a

_{3-5-6}= 0.79783 mm and a

_{3-5-7}= 1.17208 mm are exaggerated. These are eliminated and the average for the remaining values a

_{med}= 0.959 mm determined. This is higher than the deformation y

_{7}= 0.95 mm, resulting for the finest discretization of the studied body, with only +0.92%. The minimum and maximum values are a

_{min}= a

_{4-5-7}= 0.90398 mm and a

_{max}= a

_{1-5-6}= 1.05624 mm, respectively. Compared to the theoretical elastic deformation δ = 0.881808 mm ≈ 0.882 mm, the values a

_{min}, a

_{med}, and a

_{max}deviate by +2.51%, +8.71%, and +19.78% respectively.

_{r}

_{-s-t}ϵ [0.95 mm; 1.0 mm], then the corrected average value is a

_{med-1}= 0.9633 mm, being 9.24% higher than the theoretical elastic deformation δ = 0.882 mm.

#### 3.2.3. Deformation Study of a Pole with Double Arm in the Console

_{3}= 280 mm, lengths l

_{1}= 940 mm, l

_{2}= 500 mm, and l

_{3}= 400 mm, areas A

_{1}= A

_{2}= A

_{3}= 280 × 280 = 78,400 mm

^{2}) ensure that the body volume is 0.144 m

^{3}. Axial moments of inertia are I

_{1}= I

_{2}= I

_{3}= (s

_{3})

^{4}/12 = 512.2 × 10

^{6}mm

^{4}and the moment of polar inertia is (I

_{p})

_{2}= (s

_{3})

^{4}/6 = 1024.43 × 10

^{6}mm

^{4}. The body is loaded (compression, bending and torsion) with a force F = 250 kN evenly distributed at the end of the arm.

_{i}≈ 100,000 elements, the value of elastic deformation indicated in the FEA analysis (y

_{i}= 1.35 mm) is 3.85% higher than the elastic deformation analytically determined using the Relation (8).

_{3-5-6}= 0.94147 mm, a

_{3-5-7}= 0.47699 mm, a

_{4-5-6}= 1.22024 mm, a

_{4-5-7}= 1.18606 mm and a

_{4-6-7}= 0.65967 mm are small, and the value of a

_{3-6-7}= 2.40260 mm is exaggerated. By removing these values, the minimum and maximum values, a

_{min}= a

_{3-4-5}= 1.29288 mm, a

_{max}= a

_{5-6-7}= 1.57798 mm are identified, and the average value a

_{med}= 1.37979 mm is determined. The latter is higher than the deformation y

_{7}= 1.35 mm, resulting for the finest discretization level of the studied body, by +2.21%. The values a

_{2-5-6}= 1.55263 mm, a

_{2-5-7}= 1.55865 mm, a

_{2-6-7}= 1.56498 mm, and a

_{5-6-7}= 1.57798 mm are also unrealistically high. Excluding these values, the value of a

_{max-1}= a

_{1-6-7}= 1.48448 mm becomes the maximum, and the corrected average value is a

_{med-1}= 1.35038 mm, extremely close (deviation of only +0.03%) to the deformation corresponding to the finest mesh considered in the study, y

_{7}= 1.35 mm. Compared to the theoretical elastic deformation δ = 1.300476 mm, the values a

_{min}, a

_{med}, a

_{med-1}, a

_{max}, and a

_{max-1}deviate by −0.58%, +6.10%, +3.84%, +21.34%, and +14.15%, respectively.

#### 3.3. Relevant Proportionality Coefficients

_{i}(from Table 4, Table 6 and Table 8) obtained from the finite element analysis for each level i = 1 … 7 of discretization, the theoretical elastic deformation δ, the mean a

_{med}value of the reasonable values a

_{r}

_{-s-t}and an estimated value as the most probable for the deformation of the body studied under the action of the force F, for example a

_{med-1}(corrected average elastic deformation), for each level i of discretization can be highlighted values of the proportionality coefficients (k

_{δ})

_{i}= δ/y

_{i}, (k

_{m})

_{i}= a

_{med}/y

_{i}and (k

_{e})

_{i}= a

_{med-1}/y

_{i}. For the three simple cases presented, the values of the proportionality coefficients mentioned are given in Table 10, Table 11 and Table 12.

_{δ})

_{i}, (k

_{m})

_{I}, and (k

_{e})

_{i}. presented in Table 10, Table 11 and Table 12.

_{δ})

_{i}become subunit, i.e., the analytically determined deformation is smaller than the one resulting from the FEA, regardless of whether the body load is simple or more complex. For low levels of discretization, the differences between the values of the coefficients (k

_{δ})

_{i}are relatively large regardless of whether the body is subject to simple (e.g., only compression) or more complex (e.g., compression, bending, and torsion) loads. However, for high discretization levels (10,000 elements or more), the differences between the values of the coefficients (k

_{δ})

_{i}fade, becoming less than 4%.

_{m})

_{i}and (k

_{e})

_{i}decrease with the increase of the discretization level of the studied bodies, with an asymptotic variation towards 1 being evident. For low discretization levels (characterized by x

_{i}≈ 1000 elements), the values of the coefficients (k

_{m})

_{i}and (k

_{e})

_{i}are significantly higher than the asymptotic limit, even by more than 40%, the magnitude of the deviation being even as the complexity of the body is rising.

_{δ})

_{i}, (k

_{m})

_{I}, and (k

_{e})

_{i}determined according to the level of discretization of some bodies with relatively simple geometry, bodies for which it is possible to analytically determine the elastic deformation corresponding to a certain external load, it is sufficient to determine by FEA the elastic deformation for a certain discretization level to be able to estimate with sufficient precision values of interest of the respective body deformation, such as theoretical elastic deformation δ, average elastic deformation a

_{med}or corrected average elastic deformation a

_{med-1}. They are obtained simply as a product of the value of the elastic deformation determined through FEA for the level of discretization adopted and the value of the coefficient k

_{δ}, k

_{m}, or k

_{e}corresponding to that level of discretization.

_{i}values of the elastic deformation determined using FEA for different levels of discretization being known (Table 1). The frame mentioned is subject to complex load and, as a result, the values of the coefficients (k

_{δ})

_{i}, (k

_{m})

_{I}, and (k

_{e})

_{i}shown in Table 12 will be taken into account. The values thus estimated for the elastic deformation δ

_{i}, the average elastic deformation (a

_{med})

_{i}, and the corrected average elastic deformation (a

_{med-1})

_{i}, corresponding to each of the discretization levels are given in Table 13.

_{7}determined for the finest discretization, to the average elastic deformation a

_{med}and to the corrected average elastic deformation a

_{med-1}, the deviations of the values δ

_{i}, (a

_{med})

_{i}and respectively (a

_{med-1})

_{i}, determined using the proportionality coefficients k

_{δ}, k

_{m}, and k

_{e}, are shown in Table 14.

_{5}≈ 50,000 elements and x

_{6}≈ 75,000 elements respectively. These are accessible levels of discretion for common software and hardware resources and allow particularly good results in a short time in terms of the value of elastic deformation of large and complex bodies.

## 4. Conclusions

_{med}of the reasonable values of the numerical solutions and the corrected average elastic deformation a

_{med-1}(value estimated to be the most probable for deformed body studied).

_{med}value of reasonable values and corrected average elastic deformation a

_{med-1}), the proportionality coefficients k

_{δ}, k

_{m}, and k

_{e}, were determined for each i level of discretization for which the study was performed. As the level of discretization increases, the values of the coefficient k

_{δ}become subunit, i.e., the analytically determined deformation is smaller than that the one resulting from the FEA, regardless of complexity of force loading. For high mesh density (10,000 elements or more) the differences between the values of the coefficient k

_{δ}are diminishing, becoming less than 4%. The values of coefficients k

_{m}and k

_{e}are decreasing with the increase of the discretization level of the studied bodies, being evident an asymptotic variation towards 1.

^{2}+ c).

_{δ}, k

_{m}, and k

_{e}determined according to the discretization level of some bodies with relatively simple geometry for which the analytical solution is easily determined, it is sufficient to determine by FEA the elastic deformation for a certain discretization level to be able to estimate with sufficient precision, by similarity, values of interest of the deformation of a complex body, such as δ, a

_{med}, or a

_{med-1}. They are obtained simply as a product of the value of the elastic deformation determined through FEA for the discretization level adopted and the value of the coefficient k

_{δ}, k

_{m}, or k

_{e}corresponding to that discretization level.

_{δ}, k

_{m}, and k

_{e}become small, within a maximum range of 4%, acceptable for many practical applications. Discretion levels of 20,000–100,000 elements are accessible for common software and hardware resources and allow in a reasonable time to obtain particularly reliable results for the elastic deformation of large and geometrically complex bodies.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

x_{i} | effective number of discretisation items, for discretization level i |

y_{i} | the indicated elastic deformation (numerical solution), for level i of discretization |

i_{max} | i level for maximum discretisation |

a_{r-s-t} (r < s < t, r ≥ 1, t ≤ k = i_{max}) | any value of the probable theoretical elastic deformation determined based on a presumed law of variation |

a_{min}, a_{max}, a_{med} | the minimum, maximum and average values for ^{a}_{r-s-t} set of values |

a_{min-1}, a_{max-1}, a_{med-1} | minimum, maximum, and corrected average value (estimated value as the most probable) resulting from disregarding values a_{r-s-t} that are aberrant and unrealistic |

δ | analytically determined theoretical deformation |

(k_{δ})_{i}, (k_{m})_{i}, (k_{e})_{i} | proportionality coefficient for the theoretical elastic deformation δ determined analytically, for the mean a_{med} value of the numerical solutions a_{r-s-t}, and respectively for the mean a_{med-1} value of the numerical solutions a_{r-s-t}, for discretization level i |

ζ_{i} | deviation for discretization level i |

k | maximum level of discretization |

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**Figure 2.**Example of 3D model discretized into tetrahedral elements and stress state for the reference body.

**Figure 4.**Compression and bending deformation of a pole with arm in the console. Schematic representation (

**a**) and state of deformations (

**b**).

**Figure 5.**Compression, bending and torsion deformation of a pole with double arm in the console. Schematic representation (

**a**) and state of deformations (

**b**).

No. (i) | No. of Proposed Items (Required) | No. of Nodes | No. of Resulted Elements, (x_{i}) | Indicated Elastic Deformation, (y_{i}) [mm] | Deviation, ζ _{i} = [(y_{7} − y_{i})/y_{7}] × 100, [%] |
---|---|---|---|---|---|

1 | 1000 | 851 | 2309 | 0.0937 | 50.504 |

2 | 5000 | 1731 | 5007 | 0.14 | 25.926 |

3 | 10,000 | 3258 | 9810 | 0.16 | 15.344 |

4 | 20,000 | 6744 | 22,182 | 0.176 | 6.878 |

5 | 50,000 | 13,925 | 48,728 | 0.182 | 3.370 |

6 | 75,000 | 20,957 | 76,129 | 0.187 | 1.058 |

7 | 100,000 | 27,721 | 103,410 | 0.189 | 0 |

No. of Proposed Elements | 1000 | 5000 | 10,000 | 20,000 | 50,000 | 75,000 | 100,000 |

Average size of the elements, [mm^{3}] | 144 | 28.8 | 14.4 | 7.2 | 2.88 | 1.92 | 1.44 |

**Table 3.**The set of a

_{r}

_{-s-t}values of the probable theoretical elastic deformation of the PAI 25 press frame, determined based on k = 7 FEA analyses with different discretization levels (in mm).

a_{1-2-3} = 0.18124 | a_{1-3-4} = 0.19011 | a_{1-4-6} = 0.19183 | a_{2-3-4} = 0.19222 | a_{2-4-6} = 0.19206 | a_{3-4-5} = 0.18666 | a_{3-6-7} = 0.19537 |

a_{1-2-4} = 0.18745 | a_{1-3-5} = 0.18803 | a_{1-4-7} = 0.19283 | a_{2-3-5} = 0.18860 | a_{2-4-7} = 0.19305 | a_{3-4-6} = 0.19205 | a_{4-5-6} = 0.23309 |

a_{1-2-5} = 0.18724 | a_{1-3-6} = 0.19150 | a_{1-5-6} = 0.19690 | a_{2-3-6} = 0.19209 | a_{2-5-6} = 0.19802 | a_{3-4-7} = 0.19310 | a_{4-5-7} = 0.20598 |

a_{1-2-6} = 0.19074 | a_{1-3-7} = 0.19246 | a_{1-5-7} = 0.19588 | a_{2-3-7} = 0.19294 | a_{2-5-7} = 0.19653 | a_{3-5-6} = 0.19987 | a_{4-6-7} = 0.19630 |

a_{1-2-7} = 0.19185 | a_{1-4-5} = 0.18710 | a_{1-6-7} = 0.19487 | a_{2-4-5} = 0.18708 | a_{2-6-7} = 0.19511 | a_{3-5-7} = 0.19749 | a_{5-6-7} = 0.19370 |

No. (i) | No. of Proposed Items (Required) | No. of Nodes | No. of Resulted Elements, (x_{i}) | Indicated Elastic Deformation, (y_{i}) [µm] | Deviation, ζ _{i} = [(y_{7} − y_{i})/y_{7}] × 100, [%] |
---|---|---|---|---|---|

1 | 1000 | 296 | 1014 | 8.402 | 5.617 |

2 | 5000 | 791 | 3153 | 8.588 | 3.527 |

3 | 10,000 | 2243 | 8921 | 8.592 | 3.482 |

4 | 20,000 | 4855 | 20,400 | 8.672 | 2.584 |

5 | 50,000 | 12,746 | 55,517 | 8.744 | 1.775 |

6 | 75,000 | 16,724 | 69,660 | 8.831 | 0.798 |

7 | 100,000 | 26,878 | 123,673 | 8.902 | 0 |

**Table 5.**The set of a

_{r}

_{-s-t}values of the probable theoretical elastic deformation at the compression of a loaded pole, determined based on k = 7 FEA analyses with different discretization levels (in µm).

a_{1-2-3} = 8.5935 | a_{1-3-4} = 8.7823 | a_{1-4-6} = 8.9604 | a_{2-3-4} = 8.5787 | a_{2-4-6} = 9.3085 | a_{3-4-5} = 8.8074 | a_{3-6-7} = 9.0575 |

a_{1-2-4} = 8.6880 | a_{1-3-5} = 8.7977 | a_{1-4-7} = 8.9972 | a_{2-3-5} = 8.5459 | a_{2-4-7} = 9.1666 | a_{3-4-6} = 9.0372 | a_{4-5-6} = 8.5925 |

a_{1-2-5} = 8.7561 | a_{1-3-6} = 8.9150 | a_{1-5-6} = 30.211 | a_{2-3-6} = 8.5360 | a_{2-5-6} = 8.3598 | a_{3-4-7} = 9.0476 | a_{4-5-7} = 6.9119 |

a_{1-2-6} = 8.8498 | a_{1-3-7} = 8.9653 | a_{1-5-7} = 9.1949 | a_{2-3-7} = 8.4796 | a_{2-5-7} = 10.003 | a_{3-5-6} = 8.3222 | a_{4-6-7} = 9.0600 |

a_{1-2-7} = 8.9174 | a_{1-4-5} = 8.8030 | a_{1-6-7} = 9.0352 | a_{2-4-5} = 8.8574 | a_{2-6-7} = 9.0784 | a_{3-5-7} = 9.6634 | a_{5-6-7} = 8.9449 |

**Table 6.**Deformation values obtained at compression and bending study of a pole with arm in the console by using finite element analysis.

No. (i) | No. of Proposed Items (Required) | No. of Nodes | No. of Resulted Elements, (x_{i}) | Indicated Elastic Deformation, (y_{i}) [mm] | Deviation, ζ _{i} = [(y_{7} − y_{i})/y_{7}] × 100, [%] |
---|---|---|---|---|---|

1 | 1000 | 338 | 1099 | 0.74 | 22.105 |

2 | 5000 | 1307 | 4922 | 0.867 | 8.737 |

3 | 10,000 | 2696 | 10,804 | 0.891 | 6.211 |

4 | 20,000 | 5129 | 21,575 | 0.919 | 3.263 |

5 | 50,000 | 13,289 | 58,786 | 0.926 | 2.526 |

6 | 75,000 | 17,143 | 74,515 | 0.944 | 0.632 |

7 | 100,000 | 22,519 | 100,429 | 0.95 | 0 |

**Table 7.**The set of a

_{r}

_{-s-t}values of the probable theoretical elastic deformation at the compression and torsion loaded of a pole with arm in the console, determined based on k = 7 FEA analyses with different discretization levels (in mm).

a_{1-2-3} = 0.9121 | a_{1-3-4} = 0.9549 | a_{1-4-6} = 0.9557 | a_{2-3-4} = 1.0103 | a_{2-4-6} = 0.9577 | a_{3-4-5} = 0.9287 | a_{3-6-7} = 0.9728 |

a_{1-2-4} = 0.9378 | a_{1-3-5} = 0.9352 | a_{1-4-7} = 0.9599 | a_{2-3-5} = 0.9388 | a_{2-4-7} = 0.9619 | a_{3-4-6} = 0.9558 | a_{4-5-6} = 0.9141 |

a_{1-2-5} = 0.9323 | a_{1-3-6} = 0.9555 | a_{1-5-6} = 1.0562 | a_{2-3-6} = 0.9637 | a_{2-5-6} = −0.853 | a_{3-4-7} = 0.9605 | a_{4-5-7} = 0.904 |

a_{1-2-6} = 0.951 | a_{1-3-7} = 0.9593 | a_{1-5-7} = 0.9957 | a_{2-3-7} = 0.966 | a_{2-5-7} = 1.0422 | a_{3-5-6} = 0.7978 | a_{4-6-7} = 0.9798 |

a_{1-2-7} = 0.9556 | a_{1-4-5} = 0.9301 | a_{1-6-7} = 0.9691 | a_{2-4-5} = 0.9298 | a_{2-6-7} = 0.972 | a_{3-5-7} = 1.1721 | a_{5-6-7} = 0.9561 |

**Table 8.**Values obtained at FEA study at compression, bending and torsion of a pole with double arm in the console.

No. (i) | No. of Proposed Items (Required) | No. of Nodes | No. of Resulted Elements, (x_{i}) | Indicated Elastic Deformation, (y_{i}) [mm] | Deviation, ζ _{i} = [(y_{7} − y_{i})/y_{7}] × 100, [%] |
---|---|---|---|---|---|

1 | 1000 | 324 | 1027 | 0.953 | 29.407 |

2 | 5000 | 1285 | 4857 | 1.15 | 14.815 |

3 | 10,000 | 2730 | 10,673 | 1.23 | 8.889 |

4 | 20,000 | 4937 | 20,097 | 1.26 | 6.667 |

5 | 50,000 | 12,271 | 50,772 | 1.28 | 5.185 |

6 | 75,000 | 17,364 | 75,224 | 1.32 | 2.222 |

7 | 100,000 | 22,653 | 99,194 | 1.35 | 0 |

**Table 9.**The set of a

_{r}

_{-s-t}values of the probable theoretical elastic deformation at the compression, bending and torsion of a pole with double arm in the console determined based on k = 7 FEA analyses with different discretization levels (in mm).

a_{1-2-3} = 1.3311 | a_{1-3-4} = 1.2983 | a_{1-4-6} = 1.3466 | a_{2-3-4} = 1.2931 | a_{2-4-6} = 1.3502 | a_{3-4-5} = 1.2929 | a_{3-6-7} = 2.4026 |

a_{1-2-4} = 1.3101 | a_{1-3-5} = 1.2948 | a_{1-4-7} = 1.3802 | a_{2-3-5} = 1.2936 | a_{2-4-7} = 1.3874 | a_{3-4-6} = 1.3668 | a_{4-5-6} = 1.2202 |

a_{1-2-5} = 1.2991 | a_{1-3-6} = 1.3387 | a_{1-5-6} = 1.4416 | a_{2-3-6} = 1.3392 | a_{2-5-6} = 1.5526 | a_{3-4-7} = 1.4168 | a_{4-5-7} = 1.1861 |

a_{1-2-6} = 1.3381 | a_{1-3-7} = 1.3697 | a_{1-5-7} = 1.4619 | a_{2-3-7} = 1.3719 | a_{2-5-7} = 1.5587 | a_{3-5-6} = 0.9415 | a_{4-6-7} = 0.6597 |

a_{1-2-7} = 1.3671 | a_{1-4-5} = 1.2938 | a_{1-6-7} = 1.4845 | a_{2-4-5} = 1.2929 | a_{2-6-7} = 1.565 | a_{3-5-7} = 0.477 | a_{5-6-7} = 1.578 |

**Table 10.**Values of the proportionality coefficients (k

_{δ})

_{i}, (k

_{m})

_{i}and (k

_{e})

_{i}for a column (simple geometric body) subject to compression (δ = 8.2443 µm; a

_{med}= 8.9006 µm; a

_{med-1}= 9.084 µm).

i | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

y_{i} (in µm) | 8.402 | 8.588 | 8.592 | 8.672 | 8.744 | 8.831 | 8.902 |

(k_{δ})_{i} = δ/y_{i} | 0.9812 | 0.9600 | 0.9595 | 0.9507 | 0.9429 | 0.9336 | 0.9261 |

(k_{m})_{i} = a_{med}/y_{i} | 1.0593 | 1.0364 | 1.0359 | 1.0264 | 1.0179 | 1.0079 | 0.9998 |

(k_{e})_{i} = a_{med-1}/y_{i} | 1.0812 | 1.0578 | 1.0573 | 1.0475 | 1.0389 | 1.0286 | 1.0204 |

**Table 11.**Values of the proportionality coefficients (k

_{δ})

_{i}, (k

_{m})

_{i}and (k

_{e})

_{i}for a pole with arm in the console, subject to compression and torsion (δ = 8.2443 µm; a

_{med}= 8.9006 µm; a

_{med-1}= 9.084 µm).

i | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

y_{i} (in mm) | 0.74 | 0.867 | 0.891 | 0.919 | 0.926 | 0.944 | 0.95 |

(k_{δ})_{i} = δ/y_{i} | 1.1916 | 1.0171 | 0.9897 | 0.9595 | 0.9523 | 0.9341 | 0.9282 |

(k_{m})_{i} = a_{med}/y_{i} | 1.2959 | 1.1061 | 1.0763 | 1.0435 | 1.0356 | 1.0159 | 1.0095 |

(k_{e})_{i} = a_{med-1}/y_{i} | 1.3018 | 1.1111 | 1.0811 | 1.0482 | 1.0403 | 1.0204 | 1.0140 |

**Table 12.**Values of the proportionality coefficients (k

_{δ})

_{i}, (k

_{m})

_{i}and (k

_{e})

_{i}for a pole with double arm in the console, subject to compression, bending and torsion (δ = 8.2443 µm; a

_{med}= 8.9006 µm; a

_{med-1}= 9.084 µm).

i | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

y_{i} (in mm) | 0.953 | 1.15 | 1.23 | 1.26 | 1.28 | 1.32 | 1.35 |

(k_{δ})_{i} = δ/y_{i} | 1.36461 | 1.13085 | 1.05730 | 1.03212 | 1.01600 | 0.98521 | 0.96332 |

(k_{m})_{i} = a_{med}/y_{i} | 1.44784 | 1.19982 | 1.12178 | 1.09507 | 1.07796 | 1.04530 | 1.02207 |

(k_{e})_{i} = a_{med-1}/y_{i} | 1.41698 | 1.17424 | 1.09787 | 1.07173 | 1.05498 | 1.02302 | 1.00028 |

**Table 13.**Estimated values δ

_{i}, (a

_{med})

_{i}and (a

_{med-1})

_{i}, for PAI 25 press frame determined through amendment of elastic deformation y

_{i}with the values of proportional coefficients (k

_{δ})

_{i}, (k

_{m})

_{i}and (k

_{e})

_{i}.

i | y_{i} | (k_{δ})_{i} | (k_{m})_{i} | (k_{e})_{i} | δ_{i} = y_{i}·(k_{δ})_{i} | (a_{med})_{i} = y_{i}·(k_{m})_{i} | (a_{med-1})_{i} = y_{i}·(k_{e})_{i} |
---|---|---|---|---|---|---|---|

[mm] | - | - | - | [mm] | [mm] | [mm] | |

1 | 0.0937 | 1.36461 | 1.44784 | 1.41698 | 0.127864 | 0.135663 | 0.132771 |

2 | 0.14 | 1.13085 | 1.19982 | 1.17424 | 0.158319 | 0.167975 | 0.164394 |

3 | 0.16 | 1.05730 | 1.12178 | 1.09787 | 0.169168 | 0.179485 | 0.175659 |

4 | 0.176 | 1.03212 | 1.09507 | 1.07173 | 0.181653 | 0.192732 | 0.188624 |

5 | 0.182 | 1.01600 | 1.07796 | 1.05498 | 0.184912 | 0.196189 | 0.192006 |

6 | 0.187 | 0.98521 | 1.04530 | 1.02303 | 0.184234 | 0.195471 | 0.191305 |

7 | 0.189 | 0.96332 | 1.02207 | 1.00028 | 0.182067 | 0.193171 | 0.189053 |

**Table 14.**Deviations of estimated values δ

_{i}, (a

_{med})

_{i}and (a

_{med-1})

_{i}from values δ

_{7}, a

_{med}and a

_{med-1}for PAI 25 press frame.

i | Δδ_{i} = (y _{7} − δ_{i})/y_{7}·100% | Δ(a_{med})_{i} = ((a _{med} − (a_{med})_{i})/a_{med}·100% | Δ(a_{med-1})_{i} = ((a _{med-1} − (a_{med-1})_{i})/a_{med-1}·100% |
---|---|---|---|

1 | 32.347 | 29.981 | 31.095 |

2 | 16.233 | 13.303 | 14.684 |

3 | 10.493 | 7.363 | 8.837 |

4 | 3.887 | 0.525 | 2.109 |

5 | 2.163 | −1.259 | 0.354 |

6 | 2.521 | −0.888 | 0.718 |

7 | 3.668 | 0.299 | 1.887 |

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## Share and Cite

**MDPI and ACS Style**

Pisarciuc, C.; Dan, I.; Cioară, R.
The Influence of Mesh Density on the Results Obtained by Finite Element Analysis of Complex Bodies. *Materials* **2023**, *16*, 2555.
https://doi.org/10.3390/ma16072555

**AMA Style**

Pisarciuc C, Dan I, Cioară R.
The Influence of Mesh Density on the Results Obtained by Finite Element Analysis of Complex Bodies. *Materials*. 2023; 16(7):2555.
https://doi.org/10.3390/ma16072555

**Chicago/Turabian Style**

Pisarciuc, Cristian, Ioan Dan, and Romeo Cioară.
2023. "The Influence of Mesh Density on the Results Obtained by Finite Element Analysis of Complex Bodies" *Materials* 16, no. 7: 2555.
https://doi.org/10.3390/ma16072555