# Sensitivity Analysis of Key Formulations of Topology Optimization on an Example of Cantilever Bending Beam

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Description of the Optimization Problem

#### 2.1.1. The Optimization Problem

_{0}] is the stiffness matrix with unit elastic modulus and n is the number of elements. Penalization is introduced in Section 2.1.3.

#### 2.1.2. Material Model

#### 2.1.3. Penalization

#### 2.1.4. Finite Element Method

#### 2.1.5. The Optimality Criterion Algorithm

#### 2.1.6. Filtering Methods

#### Density Filter

#### Sensitivity Filter

#### Grayscale Filter

#### 2.1.7. Displacement Solver and Termination Criteria

#### 2.2. Key Formulations

**Formulation of the filter radius**mentioned in Section 2.1.6 is important because the radius defines the element’s neighborhood. If the defined range is too small, the energy is distributed only to a few elements. However, if the neighborhood is too large, the energy is scattered to a point where it is difficult to evaluate the optimum.

**Formulation of the filter type**was already mentioned in Section 2.1.6 but the theory does not provide an answer to the question of which filter should perform best.

**Formulation of the penalization**was also mentioned in Section 2.1.3; the theory, however, is able to provide only the lower boundary, not the upper one.

**Formulation of the element approximation**mentioned in Section 2.1.4 is a necessary step in the initiation of optimization. The element approximations greatly affect the accuracy of the solved displacement and the value of the objective function.

**The formulation of the type of the weight factor**mentioned in Section 2.1.6 is defined by two functions. However, theory does not provide enough evidence to decide, which function offers the better performance.

#### 2.3. Description of Numerical Test

## 3. Results

#### 3.1. Formulation of Filter Radius

#### 3.2. Formulation of Filter Type

#### 3.3. Formulation of Penalization

#### 3.4. Formulation of Element Approximation

#### 3.5. Formulation of Type of Weight Factor

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

3D | Three Dimensional |

2D | Two Dimensional |

AWB | ANSYS Workbench |

KKT | Karush-Kuhn-Tucker |

SIMP | Solid Isotropic Material with Penalty |

TET | Tetrahedral |

HEX | Hexahedral |

DOF | Degrees of Freedom |

OC | Optimality Criterion |

SAO | Sequential Approximate Optimization |

1D | One Dimensional |

$ES$ | Element Size |

NF | No Filter |

D | Density filter |

S | Sensitivity filter |

G | Gray scale filter |

SG | Combination of Sensitivity and Gray scale Filters |

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**Figure 1.**Topology optimization of cantilever bending; the initial geometry (

**left**) and the optimal shape in the form of design variables (

**right**).

**Figure 4.**2D Neighborhood of element (

**left**), examples of the radius dependence on the element size $ES$ in 2D; R = $1.2\phantom{\rule{4pt}{0ex}}ES$ for the red circle, R = $1.5\phantom{\rule{4pt}{0ex}}ES$ for the blue circle, R = $2.0\phantom{\rule{4pt}{0ex}}ES$ for the yellow circle and R = $3.0\phantom{\rule{4pt}{0ex}}ES$ for the purple circle.

**Figure 6.**Results of numerical examples of topology optimization; from left to right-four point bending, L problem, two-loadcase cantilever plate, beam with square cross-section subjected to torsion.

**Figure 8.**Final shapes for linear HEX elements (

**left**) and linear TET elements (

**right**) with various radii, Density filter (D).

**Figure 9.**Results of individual filtering algorithms: Density filter (D), Sensitivity filter (S), Grayscale filter (G), Combination of the Sensitivity and Grayscale filters (SG).

**Figure 10.**The final shape for linear HEX elements (

**left**) and linear TET elements (

**right**) with various filters, HEX radius $R=1.5$ $ES$ and TET radius $R=3.0$ $ES$.

**Figure 11.**The final shape for linear HEX elements (

**left**) and linear TET elements (

**right**) with various filters, radius $R=4$ mm.

**Figure 12.**The final shapes for different values of penalization p, Density filter, radius of filter R = 2 mm.

**Figure 13.**The final shapes for evaluation of element approximations of HEX elements (

**left**) and TET elements (

**right**); Density filter, radius $R=\phantom{\rule{4pt}{0ex}}1.5\phantom{\rule{3.33333pt}{0ex}}Es$ for HEX and radius $R=\phantom{\rule{4pt}{0ex}}3.0\phantom{\rule{3.33333pt}{0ex}}Es$ for TET.

**Figure 14.**The final shape with different weight functions, mesh with element size $ES$ = 1.0 mm (

**left**), mesh with element size $ES$ = 0.5 mm (

**right**), Density filter.

TET Elements | HEX Elements | ||
---|---|---|---|

Coaser mesh | Number of nodes | 686 | 2460 |

Number of elements | 2034 | 1680 | |

Element size [mm] | 1.500 | 2.000 | |

Normal mesh | Number of nodes | 1403 | 6405 |

Number of elements | 4458 | 4800 | |

Element size [mm] | 1.000 | 1.000 | |

Fine mesh | Number of nodes | 10,850 | 44,650 |

Number of elements | 38,740 | 38,400 | |

Element size [mm] | 0.355 | 0.500 |

**Table 2.**Results of optimization with different radius, for meshes with linear HEX elements (first multiplier) or linear TET elements (latter multiplier).

HEX Elements | TET Elements | ||||||
---|---|---|---|---|---|---|---|

Radius | Value | 2.0 mm | 1.0 mm | 0.5 mm | 1.5 mm | 1.0 mm | 0.355 mm |

1.2/2.4 $ES$ | Deformation energy [mJ] | 6.1 | 4.1 | 3.4 | 8.0 | 5.6 | 3.6 |

Normed value [-] | 4.2 | 2.8 | 2.3 | 5.8 | 3.9 | 2.5 | |

Iteration | 110 | 96 | 200 | 200 | 200 | 200 | |

Volume fraction [%] | 63.6 | 56.0 | 46.9 | 75.6 | 67.4 | 54.4 | |

1.5/3.0 $ES$ | Deformation energy [mJ] | 7.2 | 4.6 | 3.7 | 9.8 | 6.7 | 3.9 |

Normed value [-] | 5.0 | 3.1 | 2.5 | 7.1 | 4.8 | 2.7 | |

Iteration | 83 | 200 | 200 | 200 | 200 | 200 | |

Volume fraction [%] | 64.6 | 57.6 | 50.2 | 81.1 | 72.0 | 59.9 | |

2.0/4.0 $ES$ | Deformation energy [mJ] | 9.7 | 5.2 | 4.0 | 12.6 | 9.0 | 4.3 |

Normed value [-] | 6.7 | 3.5 | 2.7 | 9.1 | 6.3 | 3.0 | |

Iteration | 167 | 200 | 200 | 200 | 200 | 200 | |

Volume fraction [%] | 78.2 | 55.3 | 50.2 | 87.4 | 75.7 | 58.5 | |

3.0/6.0 $ES$ | Deformation energy [mJ] | 14.5 | 7.2 | 4.5 | 18.4 | 12.4 | 5.1 |

Normed value [-] | 10.1 | 4.9 | 3.1 | 13.3 | 8.7 | 3.5 | |

Iteration | 200 | 200 | 200 | 139 | 200 | 200 | |

Volume fraction [%] | 89.7 | 66.1 | 51.6 | 98.3 | 87.0 | 62.8 | |

AWB | Deformation energy [mJ] | 5.7 | 4.9 | 3.9 | 2.7 | 5.5 | 4.7 |

Normed value [-] | 3.9 | 3.4 | 2.7 | 2.0 | 3.9 | 3.2 | |

Iteration | 35 | 42 | 33 | 31 | 57 | 33 | |

Volume fraction [%] | 51.7 | 51.3 | 40.3 | 78.4 | 58.6 | 58.5 |

**Table 3.**Results of optimization for various filters for each mesh, dependent radius, filter radius of linear HEX mesh is R = 1.5 $ES$ an filter radius of linear TET mesh is R = 3.0 $ES$.

HEX Elements | TET Elements | ||||||
---|---|---|---|---|---|---|---|

Filter | Value | 2.0 mm | 1.0 mm | 0.5 mm | 1.5 mm | 1.0 mm | 0.355 mm |

S | Deformation energy [mJ] | 6.1 | 3.9 | 3.4 | 7.7 | 9.3 | 3.5 |

Norm value [-] | 4.3 | 2.7 | 2.3 | 5.5 | 6.6 | 2.4 | |

Iteration | 117 | 53 | 200 | 80 | 26 | 167 | |

Volume fraction [%] | 71.2 | 54.1 | 45.5 | 82.4 | 93.4 | 58.8 | |

G | Deformation energy [mJ] | 7.6 | 4.9 | 3.9 | 10.6 | 7.5 | 4.1 |

Norm value [-] | 5.3 | 3.3 | 2.6 | 7.6 | 5.3 | 2.8 | |

Iteration | 35 | 34 | 34 | 34 | 36 | 34 | |

Volume fraction [%] | 63.4 | 59.5 | 51.0 | 82.9 | 72.6 | 61.8 | |

D | Deformation energy [mJ] | 7.2 | 4.6 | 3.7 | 9.8 | 6.7 | 3.9 |

Norm value [-] | 5.0 | 3.1 | 2.5 | 7.1 | 4.8 | 2.7 | |

Iteration | 83 | 200 | 200 | 200 | 200 | 200 | |

Volume fraction [%] | 64.6 | 57.6 | 50.2 | 81.1 | 72.0 | 59.9 | |

SG | Deformation energy [mJ] | 3.8 | 3.3 | 3.1 | 3.7 | 3.3 | 3.0 |

Norm value [-] | 2.6 | 2.2 | 2.1 | 2.7 | 2.4 | 2.1 | |

Iteration | 74 | 45 | 47 | 47 | 51 | 49 | |

Volume fraction [%] | 30.0 | 30.0 | 30.1 | 30.0 | 30.0 | 30.0 | |

AWB | Deformation energy [mJ] | 5.7 | 4.9 | 3.9 | 2.7 | 5.5 | 4.7 |

Norm value [-] | 3.9 | 3.4 | 2.7 | 2.0 | 3.9 | 3.2 | |

Iteration | 35 | 42 | 33 | 31 | 57 | 33 | |

Volume fraction [%] | 51.7 | 51.3 | 40.3 | 78.4 | 58.6 | 58.5 |

**Table 4.**Results of optimization for various filters for each mesh, independent radius, filter radius R = 4 mm.

HEX Elements | TET Elements | ||||||
---|---|---|---|---|---|---|---|

Filter | Value | 2.0 mm | 1.0 mm | 0.5 mm | 1.5 mm | 1.0 mm | 0.355 mm |

S | Deformation energy [mJ] | 9.9 | 9.8 | 9.6 | 7.4 | 8.2 | 5.1 |

Norm value [-] | 6.9 | 6.7 | 6.5 | 5.3 | 5.8 | 3.5 | |

Iteration | 101 | 117 | 135 | 54 | 50 | 24 | |

Volume fraction [%] | 84.9 | 84.5 | 84.3 | 82.8 | 92.1 | 92.7 | |

G | Deformation energy [mJ] | 11.1 | 10.9 | 10.6 | 9.6 | 10.4 | 9.6 |

Norm value [-] | 7.7 | 7.4 | 7.2 | 6.9 | 7.4 | 6.6 | |

Iteration | 41 | 35 | 39 | 36 | 34 | 36 | |

Volume fraction [%] | 78.8 | 79.8 | 79.6 | 80.4 | 80.2 | 80.3 | |

D | Deformation energy [mJ] | 9.7 | 9.6 | 9.5 | 8.7 | 9.0 | 9.7 |

Norm value [-] | 6.7 | 6.6 | 6.44 | 6.3 | 6.3 | 6.7 | |

Iteration | 167 | 158 | 200 | 200 | 200 | 200 | |

Volume fraction [%] | 78.2 | 76.9 | 79.6 | 77.2 | 75.7 | 77.5 | |

SG | Deformation energy [mJ] | 3.9 | 3.8 | 3.7 | 3.7 | 4.1 | 3.0 |

Norm value [-] | 2.7 | 2.6 | 2.5 | 2.7 | 2.9 | 2.0 | |

Iteration | 47 | 48 | 38 | 34 | 37 | 50 | |

Volume fraction [%] | 30 | 30 | 29.2 | 30.0 | 30.0 | 30.0 |

Penalization | $\mathit{ES}$ = 2.0 mm | $\mathit{ES}$ = 1.0 mm | $\mathit{ES}$ = 0.5 mm | AWB, $\mathit{ES}$ = 1.0 mm |
---|---|---|---|---|

1 | 3.02 | 2.90 | 2.85 | 2.90 |

2 | 4.68 | 4.78 | 4.76 | 4.23 |

3 | 5.54 | 5.68 | 5.66 | 4.96 |

4 | 6.78 | 6.88 | 6.90 | 5.44 |

5 | 8.69 | 8.97 | 8.66 | 6.02 |

6 | 9.40 | 11.02 | 10.31 | 6.66 |

7 | 10.18 | 14.16 | 14.39 | 7.82 |

8 | 24.27 | 27.67 | 36.70 | 11.91 |

9 | 30.66 | 26.97 | 24.55 | 21.58 |

10 | 34.16 | 29.08 | 29.98 | 62.78 |

Linear Elements | Quadratic Elements | ||
---|---|---|---|

Number of nodes | 6405 | 23,930 | |

HEX | Number of elements | 4800 | 4800 |

Element size [mm] | 1.0 | 1.0 | |

Number of nodes | 1391 | 8269 | |

TET | Number of elements | 4351 | 4351 |

Element size [mm] | 1.0 | 1.0 |

Linear Elements | Quadratic Elements | ||||
---|---|---|---|---|---|

With Filter | Without Filter | With Filter | Without Filter | ||

HEX elements | Deformation energy [-] | 4.6 | 3.63 | 4.79 | 3.73 |

Norm value [-] | 3.16 | 2.49 | 3.22 | 2.51 | |

Iteration | 100 | 33 | 100 | 57 | |

Volume fraction [%] | 58.1 | 30.2 | 56.4 | 30.1 | |

Solving time [s] | 140 | 51.7 | 4820 | 3260 | |

TET elements | Deformation energy [-] | 6.71 | 2.83 | 10.07 | 4.64 |

Norm value [-] | 4.74 | 2.00 | 4.91 | 2.27 | |

Iteration | 100 | 24 | 100 | 58 | |

Volume fraction [%] | 73.6 | 30.1 | 73.9 | 30.1 | |

Solving time [s] | 50 | 17.1 | 118.9 | 77.1 |

$\mathit{ES}$ = 1.0 mm | $\mathit{ES}$ = 0.5 mm | ||||
---|---|---|---|---|---|

$\mathit{R}\mathbf{=}\mathbf{1.5}\phantom{\rule{4pt}{0ex}}\mathit{ES}$ | $\mathit{R}\mathbf{=}\mathbf{2.0}\phantom{\rule{4pt}{0ex}}\mathit{ES}$ | $\mathit{R}\mathbf{=}\mathbf{1.5}\phantom{\rule{4pt}{0ex}}\mathit{ES}$ | $\mathit{R}=\mathbf{2.0}\phantom{\rule{4pt}{0ex}}\mathit{ES}$ | ||

Linear function | Deformation energy [mJ] | 4.61 | 5.26 | 3.74 | 4.02 |

Norm value [-] | 3.16 | 3.61 | 2.53 | 2.72 | |

Iteration | 100 | 100 | 100 | 100 | |

Volume fraction [%] | 58.15 | 56.2 | 50.2 | 50.3 | |

Gauss function | Deformation energy [mJ] | 4.11 | 4.67 | 3.61 | 3.91 |

Norm value [-] | 2.82 | 3.2 | 2.44 | 2.65 | |

Iteration | 100 | 100 | 100 | 100 | |

Volume fraction [%] | 59.25 | 59.15 | 50.8 | 50.0 |

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**MDPI and ACS Style**

Sotola, M.; Marsalek, P.; Rybansky, D.; Fusek, M.; Gabriel, D.
Sensitivity Analysis of Key Formulations of Topology Optimization on an Example of Cantilever Bending Beam. *Symmetry* **2021**, *13*, 712.
https://doi.org/10.3390/sym13040712

**AMA Style**

Sotola M, Marsalek P, Rybansky D, Fusek M, Gabriel D.
Sensitivity Analysis of Key Formulations of Topology Optimization on an Example of Cantilever Bending Beam. *Symmetry*. 2021; 13(4):712.
https://doi.org/10.3390/sym13040712

**Chicago/Turabian Style**

Sotola, Martin, Pavel Marsalek, David Rybansky, Martin Fusek, and Dusan Gabriel.
2021. "Sensitivity Analysis of Key Formulations of Topology Optimization on an Example of Cantilever Bending Beam" *Symmetry* 13, no. 4: 712.
https://doi.org/10.3390/sym13040712