# Structural Optimization of Locally Continuous Fiber-Reinforcements for Short Fiber-Reinforced Plastics

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## Abstract

**:**

## 1. Introduction

## 2. Manufacturing

## 3. Optimization of Hybrid Structures

## 4. Isotropic Topology Optimization

## 5. Topology Optimization for Anisotropic Materials

_{i}. This is done by the stiffness matrix K

_{i}(see Equation (4.4)).

## 6. Topology Optimization for Multiphase Structures

## 7. Topology Optimization of Multiphase Anisotropic Reinforcement Materials

- Step 1: The available design space is discretized with an FE mesh and boundary conditions are specified.
- Step 2: BESO parameters V*, ert and p are defined.
- Step 3: Initialization of the model. A list of neighboring elements with elements and element distance r is created. The list of elements and distance between the elements is needed later for filtering sensitivities. This step only occurs at the beginning of the optimization and is not repeated in every iteration.
- Step 4: The FE calculation is carried out in Abaqus.
- Step 5: For every element, the vector of the maximum absolute principal stress is calculated from the result of the previous calculation. The local material coordinate system is oriented in the direction of the calculated vector.
- Step 6: From the result of the FE calculation the element sensitivities are calculated according to Equation (6.1). Afterward, the sensitivities are filtered spatially (Equation (4.5)) and over the optimization history (Equation (4.7)).
- Step 7: The target volume for the next iteration is calculated according to Equation (4.8) depending on ert and ${\mathrm{V}}_{\mathrm{k}}$ as long as V* is not reached.
- Step 8: cFRP and sFRP (which is assumed to be isotropic for simplification) are defined as solid and void material (see Equations (4.9) and (4.10))
- Step 9: The convergence criterion is monitored. If the convergence criterion (4.11) is fulfilled, the calculation is terminated. If it is not fulfilled, the process starts again at step 4.

## 8. Design Problem

_{min}= 6 mm is chosen for the filter function in Equation (5). The evolutionary ratio is set to ert = 1.5.

## 9. Results

## 10. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Side impact door beam made of continuous fiber-reinforced organic sheet and injection molded thermoplast [19].

**Figure 2.**Typical procedure for the optimization of hybrid structures reinforced with cFRP [3]. In a first step, isotropic topology optimization is performed. In a second step, the orientation of the anisotropic material within the resulting volume is optimized.

**Figure 3.**FE-Element with neighboring elements. If the adjacent elements lie within the radius ${\mathrm{r}}_{\mathrm{min}}$, the sensitivities are averaged according to formulas (5) and (6).

**Figure 4.**Summary of the BESO method. First, the design space and the boundary conditions are discretized in an FE model. Then the optimization parameters are defined. Within the iteration loop, the FE calculation is performed, the element sensitivities are calculated and material properties are assigned until the convergence criterion is reached.

**Figure 5.**Qualitative comparison of the strain energy distribution for a cantilever beam consisting of isotropic material (

**a**) and anisotropic, unidirectional reinforced material (

**b**).

**Figure 6.**The procedure of concurrent topology optimization considering the material orientation of anisotropic materials. In contrast to Figure 2, the topology and material optimization are not executed sequentially, but in every iteration a fiber-angle optimization is performed.

**Figure 7.**Summary of the BESO method with integrated optimization of fiber-angle. Additionally, to the method described in Figure 4, in dependence of the max. absolute principal vector the local material orientation is assigned.

**Figure 8.**This method of simultaneous multi-phase topology and material optimization considers the most important properties of the materials used in the hybrid structures under consideration. The stiffness of the sFRP is considered by a multiphase BESO (see 6) and the anisotropy of the cFRP by the fiber angle optimization, which is performed in each iteration of the entire optimization.

**Figure 10.**These examples will be used to investigate the influence of the optimization algorithm for different geometries. Example (

**a**) is an L-bracket, (

**b**) shows the square design domain with a central square rigid support [22] and (

**c**) a cantilever beam with an aspect ratio of 5.

**Figure 12.**Example of optimized material orientation within the reinforcement structure. The blue lines indicate the material orientation which follows the direction of the absolute maximum principal stress.

**Figure 13.**Solution of the optimization problem taking into account the anisotropy of the reinforcement material.

**Figure 14.**Solution of the optimization problem considering the anisotropy of the reinforcement material and the stiffness of the base material.

**Figure 15.**Optimization results for different stiffness of the base material (

**a**) 2500 MPa, (

**b**) 7500 MPa, (

**c**) 12,500 MPa.

**Figure 16.**Stiffness for hybrid structures with different Young’s moduli of the base material. The reference is the optimization without consideration of the anisotropy of the reinforcing material and the stiffness of the base material. It can be shown that with increasing Young’s modulus of the base material the new procedure leads to stiffer results.

**Figure 17.**Results of isotropic and hybrid anisotropic topology optimizations for the examples presented in Figure 10. All examples show that taking into account the stiffness of sFRP and the anisotropy of cFRP during optimization leads to an increase in the resulting stiffness of the structure compared to an isotropic optimization. The amount of increase depends strongly on the particular load case. (

**a**–

**c**) show that hybrid optimization does not always result in a continuous structure between load introduction and fixed constraint.

**Table 1.**Comparison of the performed optimizations. As reference a cantilever beam made of sFRP with an ${\mathrm{E}}_{2}=12,500\mathrm{MPa}$ (

**a**) is used here. A significant increase in stiffness can be achieved by inserting a continuous fiber reinforcement (${\mathrm{E}}_{1\Vert}=100,000\mathrm{MPa}$, ${\mathrm{E}}_{1\perp}=10,000\mathrm{MPa}$). For (

**b**) the reinforcement structure was optimized using a BESO algorithm for isotropic materials. Variant (

**c**) shows the result when the anisotropy of the fiber reinforcement is taken into account. In (

**d**), both the anisotropy of the fiber reinforcement and the stiffness of the injection molding material were included in the optimization. This allows to increase the stiffness even further for the same amount of material.

Material | Optimization | Result | Stiffness Increase Factor |
---|---|---|---|

a sFRP without continuous fiber-reinforcement | - | reference | |

b sFRP with continuous fiber-reinforcement | Standard isotropic BESO algorithm neglecting stiffness of sFRP | approx. 5.5 | |

c sFRP with continuous fiber-reinforcement | Anisotropic BESO algorithm with concurrent fiber-angle optimization neglecting stiffness of sFRP | approx. 5.5 | |

d sFRP with continuous fiber-reinforcement | Anisotropic BESO algorithm with concurrent fiber-angle optimization considering the stiffness of the base-material | approx. 6.1 |

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**MDPI and ACS Style**

Mehl, K.; Schmeer, S.; Motsch-Eichmann, N.; Bauer, P.; Müller, I.; Hausmann, J.
Structural Optimization of Locally Continuous Fiber-Reinforcements for Short Fiber-Reinforced Plastics. *J. Compos. Sci.* **2021**, *5*, 118.
https://doi.org/10.3390/jcs5050118

**AMA Style**

Mehl K, Schmeer S, Motsch-Eichmann N, Bauer P, Müller I, Hausmann J.
Structural Optimization of Locally Continuous Fiber-Reinforcements for Short Fiber-Reinforced Plastics. *Journal of Composites Science*. 2021; 5(5):118.
https://doi.org/10.3390/jcs5050118

**Chicago/Turabian Style**

Mehl, Konstantin, Sebastian Schmeer, Nicole Motsch-Eichmann, Philipp Bauer, Ingolf Müller, and Joachim Hausmann.
2021. "Structural Optimization of Locally Continuous Fiber-Reinforcements for Short Fiber-Reinforced Plastics" *Journal of Composites Science* 5, no. 5: 118.
https://doi.org/10.3390/jcs5050118