choose the best element size to yield ac

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British J ournal of Engineering and T echnology

ISSN: 2326 – 425X URL: http://www.bjet.baar.org.uk

Vol. 1, No. 7, pp 13-28, MAY 2013

CHOOSE THE BEST ELEMENT SIZE TO YIELD ACCURATE FEA RESULTS

WHILE REDUCE FE MODEL’S COMPLEXITY

Yucheng Liu, PhD, PE

Assistant Professor and Graduate Coordinator

Department of Mechanical EngineeringUniversity of Louisiana at Lafayette

Lafayette, LA 70504, USA

ABSTRACT

n finite element models, element size is a critical

issue which closely relates to the accuracy of the finite element models while directly determines

their complexity level. Thus, a primary problem in creating finite element model is to choose appropriate element size

which can yield correct simulation results while reduces

the model’s complexity as much as possible. This paper presents a systematic study on finding the effects of finiteelement size on the accuracy of numerical analysis results,

based on which brief guidelines of choosing the bestelement size in finite element modeling are provided. Static,

modal, and impact analysis are involved in this study todiscuss the effects of element size in numerical analysis.

Keywords: Element size, mesh density, static analysis,

impact analysis, modal analysis

I


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1. Introduction

In finite element analysis (FEA), the accuracy of the FEA results and requested computing time are

determined by the finite element size (mesh density). According to FEA theory, the FE models with fine

mesh (small element size) yield to highly accurate results but may take longer computing time. On the

contrary, those FE models with coarse mesh (large element size) may lead to less accurate results but do

save more computing time. Also, small element size will increase the FE model’s complexity which is only

used when high accuracy is required. Large element size, however, will reduce the FE model’s size and is

extensively used in simplified models in order for providing a quick and rough estimation of designs. Due to

its importance, in generating FEA models, the foremost problem is to choose appropriate elements size so

that the created models will yield accurate FEA results while save as much computing time as possible. The

objective of this paper is to present guidelines for choosing optimal element size for different types of finite

element analyses. In order to achieve that goal, in this study, a series of static, modal, and impact analyses

were performed on thin-walled beam and plate models to reveal the effects of the element size on the

accuracy of the FEA results. An explicit solver, LS-DYNA, was used for modeling and analyses involved in

this work [1]. The paper is organized as follows: section 2 provides the background for this study, which briefly reviewed previous related literature; section 3 explains the theoretical fundamentals of the applied

finite element; sections 4 to 6 discuss the effects of element size on the accuracy of static analysis, impact

analysis, and modal analysis, respectively; section 7 concludes the whole paper with guidelines of choosing

the best mesh density for different type of FEA presented.

2. Background

A number of investigators have studied the effects of elements size on the accuracy of numerical results of

different types of analysis and important conclusions have been drawn from previous research. Brocca and

Bazant [2] presented a finite element study of the size effect of compressive failure of geometrically similar

concrete columns of different sizes. It was observed from their analyses that the increasing elements size

caused reduction in nominal strength. However, a quantitative analysis showing the relationship between the

elements size and the nominal strength was still needed. Ashford and Sitar [3] evaluated the accuracy of the

computed stress distribution near the free surface of vertical slopes as a function of the element size. A

parametric study was carried out comparing stresses computed using FEA to those obtained from a physical

model composed of photoelastic material. It was found that for a slope height H, an element height of H/10

is adequate for the study of stresses deep within the slope. However, for cases where tensile stresses in the in

the vicinity of the slope face which are critical, element heights as small as H/32 are necessary. Saouma et

al. [4] discussed size effect in nonlinear finite element analysis with a metal-reinforced ceramics composite

material. In their study, a size effect investigation was numerically performed and the range of crack sizes

was presented, for which linear elastic fracture mechanics, nonlinear fracture mechanics or plasticity-basedmodels were applicable. Masakazu [5] conducted a numerical analysis of the size effect on the shear

strength of reinforced concrete (RC) beams through tow-dimensional nonlinear FEA, which was applied to

the simulation of RC members in flexure and shear failure. Three RC beam models with different sizes were

analyzed and the size effects on strain of reinforcement, strain of concrete, descriptive mode, and crack

situation were observed from the simulation and discussed. Perillo-Marcone et al. [6] assessed the effect of

mesh density on material property discretization and the resulting influence on the predicted stress

distribution through analyzing a three-dimensional, quantitative computed tomography based FE model of a

proximal implanted tibia. Significant variations were observed in the modulus distributions between the


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Vol. 1, No. 7, pp 13-28, MAY 2013

coarsest and finest mesh densities. Poor convergence of the material property distribution occurred when the

element size was significantly larger than the pixel size of the source computerized tomography (CT) data.

From those results, they found an optimal element size of 1.4 mm on the contact surfaces which was enough

to properly describe the stiffness, stress and risk ratio distributions within the bone for that particular case.

Zmudzki et al. [7] discussed the influence of mesh density on the results of FE model analysis of mechanical biocompatibility of dental implants. It was found that the increasing of mesh density leads to an

overestimation of loading stresses values and furthermore to an unjustified increase of pillar’s diameter. At

the other hand, too large elements might lead through an underestimation of loading stress level, to

overloading atrophy of bone tissue or to implant loss. From that work it can be found that a guideline of

choosing appropriate element size for certain finite element analysis is highly demanded. Roth and Oudry

[8] touched the influence of element size on the accuracy of dynamic analysis results and they mentioned

that for dynamic analysis, the minimum number of element required for correct simulation is according to

the loading case and material properties. Li et al. [9] investigated the sensitivity of the structural responses

and bone fractures of the ribs to mesh density in order to provide guidelines for the development of FE

thorax models used in impact biomechanics. It was demonstrated in their research that rib FE models

consisting of 2000-3000 trabecular hexahedral elements (weighted element length 2-3 mm) and associated

quadrilateral cortical shell elements with variable thickness more closely predicted the rib structural

responses and bone fracture force-failure displacement relationships observed in the experiments. Based on

the previous work and achievements, a systematic investigation is conducted here to fully discuss the size

effect on simulation accuracy of static, modal, and impact analysis for fundamental structural components

such as plates and beams.

3.

Finite Element Formulation

All finite element models involved in this study were meshed with the full integration shell element: 4-node

Belytschko-Tsay shell element with five integration points through the thickness. Such element is based on acombined co-rotational and velocity-strain formulation, among which the co-rotational portion of the

formulation avoids the complexities of nonlinear mechanics by embedding a coordinate system in the

element, and the choice of velocity-strain formulation facilitates the constitutive evaluation. In that element,

the velocity of any point in the shell is determined as:

3ˆe zvvm (1)

where vm

is the velocity of the mid-surface, θ is the angular velocity vector, and ẑ is the distance along the

fiber direction (thickness) of the shell element.

The velocity strain displacements need to be evaluated at the quadrature points within the element and

standard bilinear nodal interpolation is used to define vm, θ, and the elements coordinates ( x̂ , ŷ , ẑ ). The

velocity strains are found to be

y x x x

N zv

x

N d ̂

ˆˆˆ

ˆˆ

, x y y

y

N zv

y

N d ̂

ˆˆˆ

ˆˆ

,

x y y x xy x

N

y

N zv

x

N v

y

N d ˆ

ˆˆ

ˆˆˆ

ˆˆ

ˆˆ2

y z xy N v x

N d ̂ˆ

ˆˆ2

, x z yz N v

y

N d ̂ˆ

ˆˆ2

(2)

In Eqn. (2), N is the shape function of the element. Constitutive evaluations are then obtained by using the

presented velocity-strain displacement relations.


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4. Static Analysis

Static analyses were performed on a rectangular steel plate with the dimension 300 mm × 200 mm and a

thickness of 3 mm. Material properties of the steel are listed in Table 1. During the analyses, one end of the

plate was fully constrained and a 1 N·m moment was applied at the other end for a duration of 1 second. 10

time steps were used to record the data so that 10 data points were collected during the analysis. A series of

FE models were generated for that plate whose long side was meshed from 2 (coarsest mesh) to 160 (finest

mesh) divisions (Fig. 1). Von mises stress and bending deformation yielded from each model were

calculated and compared to study the influence of element size on the static analysis results. Static analysis

results and comparisons are listed in Table 2. In that table, it is assumed that the FE model with the finest

mesh generated the most accurate results and percentage approximate errors were calculated by comparing

other results to the most accurate ones. It also needs to be mentioned that the using of full integration

Belytschko-Tsay shell element can effectively prevent the shear locking, which usually occurs in lower

order elements when those elements are subjected to bending.

Several observations were made by comparing those results. (1) The errors of bending deformation are farlower than the errors of von Mises stress. According to FEA theory, stresses are not predicted as accurately

as the displacements because they are calculated from the displacements and it is assumed that the stresses

are constant over the element. (2) The difference of von Mises stress generated from the model with 10

elements along the long side of the plate and from the finest mesh model is less than 1%, which is

acceptable in engineering simulation. However the computing time for the coarse mesh model is only 3 sec,

which is less than 1/40 of the time cost by the finest mesh model. It can also be observed from Fig. 2 that

when the number of elements on the long side is higher than 60, the increase of mesh density does not

significantly improve the accuracy of von Mises stress any more. Such phenomenon was also observed in

comparing other static analysis results. (3) Fig. 3 shows the stress distribution and Fig. 4 compares the

bending deformation yielded from the coarsest mesh model (with 2 divisions) and the finest mesh model(with 160 divisions). From Fig. 4 it can be found that even the coarsest mesh model generated a bending

deformation close to the finest mesh model (error = 0.22%), it failed to display a smooth and continuous

bending mode because its less number of elements. An FE model with finer mesh is still needed to correctly

simulate the bending behavior of the steel plate. Similar conclusion can be drawn from the Fig. 3. (4) It can

be concluded that for static analysis, the FE model whose longest side is meshed by 10 elements can give us

optimal combination of accuracy and efficiency.


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Table 1. Steel material properties

Material Properties

Young’s modulus 207GPa

Density 7830kg/m3

Yield stress 200MPa

Ultimate stress 448MPa

Hardening modulus 630MPa

Poisson’s ratio 0.3

Table 2. Static analysis results and comparisons

# of division von Mises stress % Bending deformation % Computing time

2 6.290 MPa 5.08 0.6456 mm 0.22 3 sec

5 6.370 MPa 3.88 0.6457 mm 0.23 3 sec

10 6.580 MPa 0.76 0.6435 mm 0.11 3 sec

20 6.570 MPa 0.85 0.6442 mm 0 3 sec

30 6.607 MPa 0.30 0.6442 mm 0 3 sec

40 6.613 MPa 0.21 0.6442 mm 0 4 sec

50 6.607 MPa 0.30 0.6442 mm 0 6 sec

60 6.620 MPa 0.11 0.6442 mm 0 7 sec

70 6.621 MPa 0.09 0.6442 mm 0 9 sec

80 6.616 MPa 0.16 0.6442 mm 0 13 sec

90 6.624 MPa 0.05 0.6442 mm 0 18 sec

100 6.624 MPa 0.04 0.6442 mm 0 26 sec

120 6.626 MPa 0.01 0.6442 mm 0 40 sec

140 6.623 MPa 0.06 0.6442 mm 0 66 sec

160 6.627 MPa 0.6442 mm 124 sec


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(a) (b)Figure 1. FEA steel plate model (a) coarsest mesh (b) finest mesh

Figure 2. Element size vs accuracy for maximum von Mises stress

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0 20 40 60 80 100 120 140 160 180

E r r o r ( % )

# of Mesh Elements along Plate Length


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(a) (b)Figure 3. Stress distribution of (a) coarsest mesh model (b) finest mesh model

(a) (b)Figure 4. Bending of (a) coarsest mesh model (b) finest mesh model

5.

Impact Analysis

After static analysis, impact analyses were carried out on a thin-walled steel beam with a square cross

section, whose dimension is 120 mm × 120 mm and wall thickness is 3 mm. During the analyses, this beam

impacted a rigid wall at 15 m/s and buckled. A series of FE models were generated for that beam whoseaxial direction was meshed from 2 (coarsest mesh) to 120 (finest mesh) divisions. The crash time was set as

0.01 seconds. Impact force, absorbed energy, and global displacement were computed for each FE model

and compared in Table 3, where the approximate error was calculated based on comparing each result to the

results yielded from the finest-meshed beam model. Fig. 5 displays the crushed model with coarsest mesh (2

divisions), medium mesh (60 divisions), and finest mesh (120 divisions). The effects of elements size on the

accuracy of important impact analysis results are plotted through Figs. 6 to 8. It needs to be mentioned that

hourglass modes (nonphysical, zero-energy modes of deformation that produce zero strain and no stress) did

not occur in these impact analyses. This is because that our models are meshed using Belytschko-Tsay shell


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Vol. 1, No. 7, pp 13-28, MAY 2013

element with five integration points through the thickness, while the hourglass modes occur only in under-

integrated (single integration point) shell elements. A default algorithm provided by LS-DYNA was also

selected to inhibit hourglass modes.

After comparing the listed results, following declaration can be made. (1) The errors generated during the

impact analyses were higher than those yielded from the static analyses. This is because that the impact

analysis dose involve fast, transient loading and it much more complicated than the static analysis. (2)

Similar to the static analysis, the finite element models predicted the displacement more accurate, which was

due to the same reason. (3) In general, the errors in predicting the absorbed energy were far lower than those

in estimating the impact force. A possible explanation is that the absorbed energy is related to the mass and

instantaneous velocity of the beam models, while the impact force is related to their mass and instantaneous

acceleration. There was no error in calculating the mass because different FE models faithfully represent the

volume and density of the steel thin-walled beam model. However, the velocity is calculated as the first time

derivative of the displacement and the acceleration is calculated as the first time derivative of the velocity,

therefore, the errors generated in calculating the velocity are transferred to the step of calculating the

acceleration. Due to this reason, it is understandable that the FE models predicted the absorbed energy moreaccurate than the impact force. (4) In predicting the impact force, the beam model has to be meshed into 80

divisions longitudinally so that the approximate error would drop to below 10%, an accepted level in impact

simulation [10]. However, to reach the same accurate level, the beam model only has to be meshed into 20

divisions for predicting the absorbed energy and maximum displacement. (5) Similar to Fig. 4, it can be seen

from Fig. 5 that in impact analysis, the FE model with fewer number of elements could not correctly reflect

the real progressive buckling mode of the thin-walled steel beam. In other words, a certain number of finite

elements are required to correctly simulate the crash behavior and response of engineering structures during

the impact analysis. (6) In conclusion, in order to correctly simulate the crash process and predict important

impact results while saving as much computing time, the thin-walled beam model has to be meshed into 80

divisions along its axis. Table 3 also reveals that the optimal FE model with 80 divisions along its axis onlytook less than 1/10 of the computing time requested by the finest mesh model.

Table 3. Impact analysis results and comparisons

# of division Impact force % Absorbed energy % Displacement % Computing time

2 90.88 kN 1577 41.48 kJ 119 48.65 mm 48.7 1 sec

5 61.39 kN 1033 39.99 kJ 106 50.03 mm 47.3 1 sec

10 19.02 kN 251 22.88 kJ 21.0 65.67 mm 30.8 1 sec

20 10.09 kN 86.2 20.19 kJ 6.83 85.5 mm 9.88 2 sec

30 9.32 kN 72.1 19.85 kJ 5.00 88.9 mm 6.32 5 sec

40 9.46 kN 74.7 19.85 kJ 5.02 91.36 mm 3.70 9 sec

50 9.65 kN 78.1 17.33 kJ 8.30 93.33 mm 1.62 14 sec60 9.82 kN 81.4 19.97 kJ 5.62 97.37 mm 2.63 21 sec

70 8.85 kN 63.3 19.84 kJ 4.95 96.88 mm 2.11 68 sec

80 5.26 kN 2.86 19.86 kJ 5.08 96.33 mm 1.53 111 sec

90 5.31 kN 2.02 19.45 kJ 2.87 93.44 mm 1.51 198 sec

100 5.35 kN 1.21 19.32 kJ 2.19 94.43 mm 0.47 357 sec

120 5.42 kN 18.90 kJ 94.87 mm 1198 sec


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(a) (b) (c)

Figure 5. Deformed beam models with (a) coarsest mesh, (b) medium mesh, and (c) finest mesh

Figure 6. Element size vs accuracy for maximum impact force

0.00

200.00

400.00

600.00

800.00

1000.00

1200.00

1400.00

1600.00

1800.00

0 20 40 60 80 100 120 140

E r r o

r ( % )

# of Mesh Elements along Beam Model


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Vol. 1, No. 7, pp 13-28, MAY 2013

Figure 7. Element size vs accuracy for maximum absorbed energy

Figure 8. Element size vs accuracy for maximum displacement

0.00

20.00

40.00

60.00

80.00

100.00

120.00

140.00

0 20 40 60 80 100 120 140

E r r o r ( %

)


0.00

10.00

20.00

30.00

40.00

50.00

60.00

0 20 40 60 80 100 120 140

E

r r o r ( % )



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6. Modal Analysis

Finally, modal analyses were conducted on above plate and beam models to determine their natural

frequencies and mode shapes during free vibration. It is common to use the FEA to perform this analysis and

the influences of the element size on the modal analysis results are discussed here. In this study, only the

lowest frequencies are listed and compared for each model because the lowest frequencies are related to the

most prominent modes at which the model will vibrate, dominating all the other higher frequency modes. In

performing the modal analysis, the steel plate was constrained on its short edge and the thin-walled steel

beam was constrained at its end. Table 4 only lists the lowest natural frequency calculated for each finite

element model. Fig. 9 plots the corresponding mode shapes of the coarsest meshed plate model and the

finest meshed plate model. Fig. 10 displays the mode shapes of the coarsest and finest meshed thin-walled

beam model.

From the displayed results it can be seen that developed FEA models correctly predicted the lowest natural

frequencies, with all the errors lower than 10%. The approximation error yielded from coarse models such as

the thin-walled plate model with 5 divisions and the thin-walled box beam model with 10 divisions are below 1%. This is because the modal analysis results such as natural frequencies and mode shapes of a

model during free vibration are only depend on the model’s mass and stiffness matrix. An FEA model can

accurately predict its modal analysis results as long as it faithfully represents the model’s mass and stiffness.

Fig. 9 displays mode shapes of thin-walled plate models with different numbers of meshes. It is found that

the FEA models with different element size correctly plot the mode shape of the thin-walled plate and

predict the deflection of that mode shape. Table 5 lists deflections of the mode shapes respect to the lowest

natural frequency of several FEA models, which are very close to each other.

However, from Fig. 10 it can be seen that even though FEA thin-walled box models with coarse mesh can

correctly compute the natural frequency values, nevertheless, in order to closely simulate the mode shapes,

more finite elements are still required. It also deserves to be mentioned that the computing times for the FEA

models with different mesh density are not listed in Table 4. This is because the finite element modal

analysis is very fast. To finish one modal analysis, the coarsest mesh models took 1 second, the thin-walled

plate model with 140 divisions took 14 seconds and the thin-walled box beam model with 120 divisions took

12 seconds. Therefore, in deciding the optimal element size for the modal analysis, the computing time is

not an important issue.


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Table 4. Modal analysis results and comparisons

Thin-walled steel plate Thin-walled steel box beam

# of division1

st Natural

frequencyDifference % # of division

1st Natural

frequencyDifference %

2 27.0506 Hz 2.6351 2 19.4889 Hz 9.5836

5 27.6131 Hz 0.6105 5 17.5380 Hz 1.3860

10 27.7530 Hz 0.1069 10 17.9067 Hz 0.6871

20 27.7715 Hz 0.0403 20 17.7746 Hz 0.0557

30 27.7758 Hz 0.0248 30 17.7679 Hz 0.0933

40 27.7767 Hz 0.0216 40 17.7726 Hz 0.0669

50 27.7767 Hz 0.0216 50 17.7685 Hz 0.0900

60 27.7773 Hz 0.0194 60 17.7780 Hz 0.0365

70 27.7775 Hz 0.0187 70 17.7295 Hz 0.3093

80 27.7776 Hz 0.0184 80 17.7949 Hz 0.0585

90 27.7780 Hz 0.0169 90 17.8008 Hz 0.0917

100 27.7783 Hz 0.0158 100 17.7731 Hz 0.0641

120 27.7791 Hz 0.0130 120 17.7845 Hz

140 27.7804 Hz 0.0083 140

160 27.7827 Hz 160

Table 5. Deflection of lowest frequency mode shapes of thin-walled plate models

# of division Deflection Difference %

2 1654 mm 2.19

5 1683 mm 0.47

10 1689 mm 0.12

100 1691 mm 0

160 1691 mm


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(a) 2 elements (b) 5 elements

(c) 10 elements(d) 160 elements

Figure 9. Lowest frequency mode of plate with different element size


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(a) 2 elements(b) 5 elements

(c) 10 elements(d) 120 elements

Figure 10. Lowest frequency mode of beam models with different element size

7.

Uniqueness of this Study

Several investigators have performed sound research in revealing the effects of mesh density on the FE

results of computer simulation of linear and nonlinear problems. D’Amours et al. [11] presented an analysis

of crush response of hydroformed aluminum tubes. From the FE simulation, they found that in certain

instances, it may be better to use a fine mesh size for the hydroforming model and remap forming results tocoarser mesh sizes for crashworthiness models to save computational time. Aramayo [12] developed a FE

model of a Ford Explorer SUVwith arbitrary element size and size distribution. The general model was used

for frontal impact analysis with different scenarios, and the simulation results were verified by comparing

with experimental results of crash tests. In developing the FE model, the mesh density was parameterized in

different regions so that a fine mesh was employed in one half of the front of the object and a coarse mesh

elsewhere. Donadon and Iannucci [13] presented an objective algorithm for strain softening material models.

In order to evaluate the performance of the algorithm, a mesh sensitivity study was performed where a

simple coupon test simulation was performed on the virtual coupon models with six different mesh densities.

On comparing the structural response obtained using the different mesh types, it was found that the energy

dissipated in the formation of crack is mesh insensitive. Makino [14] examined the performance on the fine

mesh model and developed a 10-million shell elements car model in order to achieve the good accuracy in

crash analysis by LS-DYNA. Comparing with aforementioned research, the uniqueness of the present study

lies on following areas. (1) Despite the well known fact that fine mesh leads to higher accuracy while the

coarse mesh improves the computing efficiency, this study demonstrates how the element size affect the

accuracy and efficiency through a series of simulations and presents a primary guideline of choosing the

optimal element size for different type of analysis. (2) Instead of only focusing on impact analysis, this

research completely studies the effects of mesh density through static, impact, and modal analysis.


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8. Conclusion

In this study, the effects of element size on accuracy of finite element models and simulation results were

thoroughly investigated through static analysis, impact analysis, and modal analysis. It was found that for

static analysis that assumes steady loading and response conditions, each side of a plate model should be

discretized into 10 divisions in order to obtain satisfied results (with approximation error < 1% in our

example) consuming less computer resources and computing time. For impact analysis which involves fast,

transient loading and considers dynamic structural response, a thin-walled beam model has to be meshed

into 80 divisions along its axis in order to simulate its crash response efficiently and correctly (with

approximation error < 10%). For modal analysis, the selection of the best element size is not very urgent

because (1) the FEA models with less number of elements can correctly predict the natural frequencies (with

approximation error < 1%) and (2) even the FEA models with finest mesh cost no more than 20 seconds to

complete a modal analysis. Nevertheless, it is suggested to apply 10 divisions on the box beam models to

correctly simulate their mode shapes in lowest natural frequencies.

The presented results reveal that different types of structural analysis require appropriate mesh generationschemes. The optimal mesh density for static, impact and modal analysis can be used as guidelines in

creating other finite element models for structural analysis, which will lead to accurate and efficient

computer simulations.

One deficiency of this study is that the authors did not develop mathematical models to visually show the

relationships among element size, accuracy of results, and computing time for different analyses. This is

because that in any type of analysis, more than one type of result will be extracted and studied (this paper

only lists a few type of results as an example). As demonstrated in the paper, the element size has different

influence on different types of results. Therefore, it is neither possible nor necessary to derive mathematical

models for each type of result to show the influences of the element size. Another shortcoming of the FEA

models presented in this paper is that those models use automatic mesh only. Advanced mesh techniques

such as adaptive mesh are not considered. Also, this paper only discusses the structures with regular shape.

The mesh strategy recommended here can be applied to model more complicated structures with irregular

shapes and even engineering assembly for further validation. In the future, effects of adaptive mesh need to

be considered in studying the influences of element size on the accuracy of FEA results. The present study

can be applied to nonstructural problems such as heat transfer problem and fluid flow problem.


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model”, Journal of Structural Engineering, 127(12), 2001, 1382-1390.3. Ashford and Sitar, “Effect of element size on the static finite element analysis of steep slopes”,

International Journal for Numerical Analytical Methods in Geomechanics, 25(14), 2001, 1361-1376.4. V.E. Saouma, D. Natekar and O. Sbaizero, “Nonlinear finite element analysis and size effect study in a

metal-reinforced ceramics-composite”, Materials Science and Engineering A, 323(1-2), 2002, 129-137.5. T. Masakazu, “The finite element analysis about size effect of concrete members”, Memoirs of the

Faculty of Engineering, Fukuyama University, 25, 2001, 49-54.6. A. Perillo-Marcone, A. Alonso-Vazquez and M. Taylor, “Assessment of the effect of mesh density on

the material property discretisation within QCT based FE models: a practical example using theimplanted proximal tibia”, Computer Methods in Biomechanics and Biomedical Engineering, 6(1), 2003,17-26.

7. J. Zmudzki, W. Walke and W. Chladek, “Influence of model discretization density in FEM numerical

analysis on the determined stress level in bone surrounding dental implants”, Information Technologiesin Biomedicine, 47, 2008, 559-567.

8. S. Roth and J. Oudry, “Influence of mesh density on a finite element model under dynamic loading”,Proceedings of 3

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nd – 4

th, 2009,

Ludwigsburg, Germany.9. Z.-P. Li, M.W. Kindig, D. Subit and R.W. Kent, “Influence of mesh density, cortical thickness and

material properties on human rib fracture prediction”, Medical Engineering & Physics, 32(9), 2010, 998-1008.

10. Y.-C. Liu and M.L. Day, “Simplified modeling of thin-walled box section beam”, International Journalof Crashworthiness, 11(3), 2006, 263-272.

11. G. D’Amours, A. Rahem, R. Mayer, B. Williams and M. Worswick, “Effects of mesh size and

remapping on the predicted crush response of hydroformed tubes”, Proceedings of 6 th

European LS- DYNA User’s Conference, May 29 – 30, 2007, Gothenburg, Sweden.

12. G.A. Aramayo and M.H. Koebbe, “Parametric finite element model of a sports utility vehicle –development and validation”, Proceedings of 7

th International LS-DYNA User’s Conference, May 19 –

21, 2002, Dearborn, Michigan.

13. M.V. Donado and L. Iannucci, “An objective algorithm for strain softwening material models”,

Proceedings of 9th

International LS-DYNA User’s Conference, June 2005, Dearborn, Michigan.

14. M. Makino, “The performance of 10-million element car model by MPP version of LS-DYNA onFujitsu PRIMEPOWER”, Proceedings of 10th International LS-DYNA User’s Conference, June 2008,Dearborn, Michigan.

choose the best element size to yield ac

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