# Topology Optimization and Efficiency Evaluation of Short-Fiber-Reinforced Composite Structures Considering Anisotropy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials and Material Models

_{eq}exceeds the yield stress σ

_{y}, the stress–strain response becomes nonlinear, and plastic deformation occurs. The plastic’s strength is then defined as follows:

_{plastic}= σ

_{y}+ R(ε

_{p}),

_{y}is the yield stress; R(ε

_{p})=kε

_{p}+ R

_{∞}[1 − e

^{−mεp}] represents the isotropic strain exponential and linear hardening law; and ε

_{p}is the accumulated plastic strain. Here, k is the linear hardening modulus in MPa; m is the hardening exponent; and R

_{∞}is the hardening modulus in MPa. The material model parameters were adjusted by minimizing the difference between the tensile strain-stress curves of the composite material and the experimental results.

#### 2.2. Methods

#### 2.2.1. Topology Optimization

_{1}(

**x**) and constraints such as the “minimum member size” g

_{2}(

**x**) and the “pull-out direction” g

_{3}(

**x**). The formulation was defined as follows:

minimize | f(x) = 0.5u^{T}K(ρ(x), A(x))u, |

by varying | ρ(x) ∈ (0, 1], x ∈ Ω, |

subject to | h(x) = K(ρ(x), A(x))u − F = 0, |

g_{1}(x) = ∫ ρ(x)dΩ − V_{ret} ≤ 0, | |

g_{2}(x) = ∫|∇ρ(x)|dΩ − δ ≤ 0, | |

g_{3}(x) = ρ_{i} − ρ_{k} ≤ 0 ∀ x_{i} = x_{k}, y_{i} = y_{k}, |z_{i}| ≥ |z_{k}|, |

**K**represents the global stiffness matrix,

**u**denotes the nodal displacement vector, F is the nodal force vector, and

**x**is the vector containing design domain elements with coordinates x, y, and z. “Minimum member size” constraint g

_{2}(

**x**) limits the change of the gradient of the density field with respect to spatial coordinates over the design domain. Thus, the minimum width of structural members on average becomes limited to a specified value. In addition, it is known [51] that the minimum member size constraint acts as a mesh independence filter for topology optimization. “Pull-out direction” constraint g

_{3}(

**x**) provides a monotone decreasing of density ρ(

**x**) when moving away from a parting plane.

**A**represents the fiber orientation tensor, and δ is related to the minimum structural member size. TO, in this work, was performed using the “Sequential Convex Programming” solver within the Ansys Mechanical Workbench 18.2 software. The relationship between topological density and finite element stiffness matrix is carried out by the SIMP (Solid Isotropic Material with Penalization) interpolation scheme [51] as follows:

**k**(ρ) =

**k**

_{0}ρ

^{p}.

**k**is the stiffness of the solid material, and p = 3 is the penalty factor, which is used for penalizing intermediate densities.

_{0}#### Unidisciplinary Topology Optimization Considering Constant Molding

#### Multidisciplinary Topology Optimization: Considering Variable Molding

_{struct}and injection molding simulation BC

_{injMold}, material properties of the matrix, and fiber MP

_{matrix+fiber}, injection molding material MP

_{injMold}, TO parameters OP

_{topoOpt}, topology density threshold th, design region volume percentage to retain V

_{ret}, interpolation tolerance between the injection molding and structural analysis meshes δ

_{map}, and an optional geo.stp file for geometry. MP

_{matrix+fiber}and OP

_{topoOpt}are saved as materialProperties.txt and topoParameters.txt, respectively, for correct utilization in AnisoTopo [67]. The structural domain region mesh, elements participating in TO, and elements related to boundary conditions are obtained from Ansys Workbench and saved as designRegionMesh.ans, design.txt, and frozen.txt, respectively. TO with variable fiber orientation proceeds after specifying the convergence criterion for the TO algorithm, which is the relative difference ε between the previous and current objective function values, not exceeding the objective relative difference ε

_{obj}. Key moments in TO include:

_{injMold}, and MP

_{injMold}are introduced to Moldflow for calculating the fiber orientation tensor

**A**, which is exported along with the injection molding mesh mesh

_{injMold}and saved in the files meshMoldFlow.pat and fiberOrientMoldFlow.xml, respectively. Fiber orientation mapping from injection molding to structural analysis mesh is performed in DigimatMAP, and the mapped fiber orientation

**A’**is stored in the file fiberOrientAnsys.xml. The mapping step also extrapolates the orientation tensor field to regions not included in the molding simulation design domain regions, which allows the smooth material characteristics field at the part boundaries to be obtained.

**A’**is introduced to calculate the anisotropic stiffness matrix (K) using the Advani-Tucker orientation averaging technique applied to the material stiffness matrix [4]. AnisoTopo’s code employs the Mori–Tanaka homogenization method to compute mechanical properties.

_{ijkl}is the fourth-order fiber orientation tensor, δ

_{ij}is the second-order unit tensor, and the coefficients B are related to the components of the stiffness matrix of the transversely isotropic unidirectional composite [79]. This resulting tensor is linked to each mesh element and exported as apdl_pre.txt to Ansys Mechanical Workbench.

**ρ**(details in [75]). Subsequently, ε is calculated, and the topology is extracted based on V

_{def}and exported as density.topo. If the current ε is less than ε

_{obj}for the number of times specified by the k-iterations criterion K

_{ε}[80] (in this work, three times in a row), TO is stopped, and the last topology is exported as topoOptStruct.stl. Otherwise, the counter g is incremented by 1, and the loop repeats until convergence is achieved.

Algorithm 1. Multidisciplinary topology optimization. |

Input: BC_{struct}, BC_{injMold}, MP_{injMold,} MP_{matrix+fiber}, OP_{topoOpt}, th, δ_{map}, V_{def}, ε_{obj}, geo.stp (Optional)Output: topoOptStruct.stlwrite materialProperties.txt ← MP_{fiber+matrix}write topoParameters.txt ← OP_{topoOpt}mesh _{struct}, design_{elements}, frozen_{elements} = AnsysWorkbench_Mesh(BC_{struct}, geo.stp);write designRegionMesh.ans ← mesh_{struct}, design.txt ← design_{elements}, frozen.txt ← frozen_{elements};g = 1; counter_epsilon = 0; while (counter_epsilon < K_{ε}) doif g == 1 thenreducedMesh.ans = designRegionMesh.ans; elsedomain_mesh_reduced = delete_elements(designRegionMesh.ans, th, (density.txt) _{g-1});write reducedMesh.ans ← domain_mesh_reducedend ifmesh _{injMold}, A = AutodeskMoldFlow(reducedMesh.ans, BC_{injMold}, MP_{injMold});write meshMoldFlow.pat ← mesh_{injMold}, fiberOrientMoldFlow.xml ← A;A’ = DigimatMAP(fiberOrientMoldFlow.xml, meshMoldFlow.pat, designRegionMesh.ans, δ);write fiberOrientAnsys.xml ← A’;K = AnisoTopo(materialProperties.txt, fiberOrientAnsys.xml, topoPararmeters.txt, designRegionMesh.ans, design.txt, frozen.txt);_{EL}write apdl_pre.txt ← K;_{EL}if g == 1 thenW _{g} = AnsysWorkbench_StructuralAnalysis(designRegionMesh.ans, apdl_pre.txt, BC);_{struct}elseW _{g} = AnsysWorkbench_StructuralAnalysis(designRegionMesh.ans, apdl_pre.txt, BC_{struct}, ρ);ε _{g} = |(W_{g} − W_{g-1})/W_{g-1}|if ε_{g} <= ε_{obj} thencounter_epsilon ++ elsecounter_epsilon = 0 end ifend ifρ = AnsysWorkbench_TopologyOptimization_Iteration(designRegionMesh.ans, design.txt, frozen.txt, V_{def}, topoPara, apdl_pre.txt, W_{g})write density.topo ← ρ;Convert density.topo to (density.txt) _{g} with HDFView();g++ end whiletopoOptStruct = delete_elements(designRegionMesh.ans, th, (density.txt) _{g-1});write topoOptStruct.stl ← topoOptStruct |

#### 2.2.2. Metrics for Evaluating the Structure Design Quality of Composite Materials

_{K,}is employed to assess the quality of a structural arrangement [55,56]. Their typical formulations are defined as follows:

_{eq}represents the equivalent stress, V is the volume of the structure, F is the characteristic load in N, and l is the characteristic linear dimension in m (l represents the distance between areas where loads are applied to the locations of the supports). To properly evaluate the quality of the structural arrangement in composite materials, the LCF has been redefined based on stress criteria.

^{UTS}is the ultimate tensile stress, and F

_{eq}= σ

_{V}/ σ

^{UTS}is the maximum stress criterion, defined as the ratio of von Mises stress σ

_{V}to the material’s ultimate tensile stress σ

^{UTS}. The LCF coefficient remains the same as in Equation (3).

_{0}

^{UTS}is the ultimate tensile strength either along the longitudinal direction (along the fiber), and F

_{TH}is the average Tsai–Hill criterion, determined using Advani-Tucker’s averaging procedure [66]. It is defined as follows:

_{1}= ${\mathrm{F}}_{TH}^{ud}$

_{1111}− 2 ${\mathrm{F}}_{TH}^{ud}$

_{1122}+ ${\mathrm{F}}_{TH}^{ud}$

_{2233}− 4 ${\mathrm{F}}_{TH}^{ud}$

_{1212}+ ${\mathrm{F}}_{TH}^{ud}$

_{2323}; D

_{2}= ${\mathrm{F}}_{TH}^{ud}$

_{1122}− ${\mathrm{F}}_{TH}^{ud}$

_{2233}; D

_{3}= ${\mathrm{F}}_{TH}^{ud}$

_{1212}− ${\mathrm{F}}_{TH}^{ud}$

_{2323}; D

_{4}= ${\mathrm{F}}_{TH}^{ud}$

_{2233}; and D

_{5}= ${\mathrm{F}}_{TH}^{ud}$

_{2323}. The values of the Tsai–Hill criteria tensor are determined using the following expression:

_{ij}represents the components of the stress tensor (component 11 corresponds to the fiber’s longitudinal axis, etc.), X is the longitudinal strength limit, Y is the transverse strength limit, and S is the transverse shear strength.

#### 2.2.3. Bracket Manufacturing and Load Testing Technique

## 3. Results

#### 3.1. Topology of Optimal Constant Molding Structures

#### 3.1.1. Topology Optimization and Structural Arrangement Quality Assessment

^{−2}.

_{ret}was set to 12%. The objective relative difference ε

_{obj}value was established as 0.1%. The material properties were defined as follows: for isotropic material, Young’s modulus of 8 GPa and a Poisson’s ratio of 0.25; for orthotropic material, the following elastic constants: E

_{X}= 13 GPa, E

_{Y}= 7 GPa, E

_{Z}= 6.5 GPa, υ

_{XY}= 0.272, υ

_{YZ}= 0.365, υ

_{XZ}= 0.254, G

_{XY}= 1.979 GPa, G

_{YZ}= 1.639 GPa, and G

_{XZ}= 1.763 GPa (the X-axis corresponds to the bracket symmetry axis, the Y-axis corresponds to force direction, and the Z axis is determined by the right-hand rule).

^{F}/

_{m}=70 N/gr, are presented in Figure 7. It can be observed that the von Mises failure criterion underestimates the strength of structural members loaded transversely in the fiber direction. Meanwhile, the Tsai–Hill failure criterion allows for a more accurate estimation of the strength of structures made of short-reinforced composite materials.

#### 3.1.2. Influence of the Relationship between Elastic Moduli E_{1} and E_{2} of Composite Material on the Resulting Topology

_{1}and E

_{2}of composite material on the resulting part topology has been conducted, considering cases with two and four times higher anisotropy than those previously considered (refer to Figure 8). For each material, the normalized specific stiffnesses were calculated using $\overline{\mathrm{k}}$ (9), representing the ratio of brackets from this material with TCA and TCI shapes. This ratio allows the evaluation of the potential for increasing the stiffness of the product by considering the material’s anisotropy in the design.

_{1}to E

_{2}ratio to 7.4 enables a 1.49-fold increase in the stiffness of the structure when considering the material’s anisotropy during the topology optimization process. For further experimental verification, the case with an E

_{1}/E

_{2}= 13/7 is chosen, which corresponds to the actual properties of the available composite material.

#### 3.1.3. Experimental Verification

#### 3.2. Topology-Optimal Variable Molding Structures

#### 3.2.1. Topology Optimization and Topology Assessment

#### 3.2.2. Topology Reconstruction

^{F}/

_{M}= 70 N/gr. Table 5 displays the corresponding C

_{K}values of both topologies and provides a comparison between them.

^{UTS}for PA6 30CF topologies is 169.35 MPa, while for D16T topologies, it is 476 MPa. The percentage change is calculated with respect to the correspondent baseline topology.

#### 3.2.3. Experimental Validation

## 4. Discussion

_{K}is more suitable for structures made of isotropic materials. The classic C

_{K}does not convey any information regarding anisotropy, making it incapable of accurately estimating the quality of structures made from anisotropic material. For example, the classic C

_{K}indicates the advantage of TCI made from anisotropic materials over TCA, despite the latter having lower specific stiffness (see Table 2 and Table 3). The C

_{K}

^{TH}incorporates the effect of anisotropy on the stress state of the structure by optimizing the placement of structural elements to achieve more efficient mold filling. The minimal C

_{K}

^{TH}values of the TCA and TCI correspond to the maximal specific stiffness of these topologies. However, a significant drawback arises from the fact that the integral over the volume of the failure criteria is the product of the average criteria values over the volume, making LCF insensitive to both under- and over-stressed elements. We recommend evaluating the structure’s topology not only using any of the possible formulations of the LCF coefficient but also employing a metric based on the coefficient of variance of the failure criteria (where the average and deviation are calculated from the failure criteria value over the volume). Moreover, reformulating the objective function to minimize both the average and deviation of the total strain energy of the topology should lead to a more equally strong structure.

_{1}to E

_{2}ratio to 7.4 enables a 49% increase in the stiffness of the structure in a constant molding case, which shows the possibility of increasing the effect in the presence of materials with greater anisotropy. In this work, all resulting topologies correspond to truss structures, which consist mostly of rods. In the variable molding case, all rods have almost the same elastic properties due to the alignment of fibers along their axes during injection molding. The impact of considering anisotropy during TO is more pronounced in the constant molding case. It can be assumed that the contrast in the normalized specific stiffness between the constant and variable molding cases is due to the regular orientation of the flow—and the fibers within it—along the structural elements in the variable molding case, causing the mechanical characteristics in these elements to be closer to those of the 0°-oriented material. In the case of thin-walled structures, the effect of increasing stiffness, considering variable molding anisotropy, can be more prominent.

_{K}

^{TH}of the baseline and reconstructed topologies, suggesting that C

_{K}

^{TH}and the normalized specific stiffness are correlated and that C

_{K}

^{TH}serves as an indicator of the degree of pristine (flawless) structure compared to the baseline topology. The experimental results showed that the normalized specific stiffness increased by 4.84–5.63% and 5.66–7.30% in the constant and variable molding cases, respectively (see Section 3.1.3 and Section 3.2.3). This indicates that two important points in this work—making the stiffness matrix dependent on the fiber orientation tensor and obtaining the fiber orientation tensor by solving the molding equations along with the Folgar–Tucker’s continuity equation—allow us to achieve a stiffer structure (since the 2D case does not account for these points). The deviation of the numerical results from the experimental results can be attributed to differences in the loading scheme, the material model, and the omission of the weld lines during the numerical calculation. During the experiment with TVA and TVI brackets, it was noted that the poor adhesion between the aluminum bushing and the SFRP structure generated a weak joint interfacial strength, which could be the reason for the lower normalized specific stiffness than that predicted by the numerical model. Therefore, the investigation of the adhesion of aluminum and titanium alloys to SFRP can be part of future studies.

^{−1}. Second, the effect of boundary conditions should be minimized to achieve a higher contrast between solutions; alternatively, different boundary conditions should be investigated. For instance, increasing the design region, placing the loads further from the support, or analyzing complex loading schemes such as the geometry presented in the work [82]. Third, changing the type of material model used during TO from linear to non-linear should be considered, as previous works [83] have demonstrated that this leads to stronger structures.

## 5. Conclusions

_{K}

^{TH}, in terms of the Tsai–Hill failure criterion, can be effectively employed in the design of fiber-reinforced polymer-based composite structures. It is worth noting that the C

_{K}approach can be formulated using other failure criteria as well. For instance, the Tsai–Wu failure criterion may be more appropriate when distinguishing between tension and compression strengths is crucial.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Stress–strain curve: (

**a**) PA6 50GF; (

**b**) PA6 30CF; and (

**c**) D16T. Solid lines—model, dashed line—experiment. Angle between polymer flow and tension load: 0°—red, 45°—green, and 90°—blue.

**Figure 7.**Failure criteria in the field of constant molding topologies: (

**a**) von Mises-based failure criterion field of TCA; (

**b**) von Mises-based failure criterion field of TCI; (

**c**) Tsai–Hill failure criterion field of TCA; (

**d**) Tsai–Hill failure criterion field of TCI.

**Figure 8.**Influence of the relationship between E

_{1}and E

_{2}of composite material on the resulting topology.

**Figure 10.**TCA and TCI experimental loading curves: (

**a**) TCA made in PA6 50GF, (

**b**) TCA made in PA6 30CF, (

**c**) TCA made in D16T, (

**d**) TCI made in PA6 50GF, (

**e**) TCI made in PA6 30CF, (

**f**) TCI made in D16T.

**Figure 11.**Specific force-normalized deformation curves of numerical and experimental TCA and TCI made of: (

**a**) PA6 50GF; (

**b**) PA6 30CF; and (

**c**) D16T.

**Figure 14.**Obtained topologically optimal variable molding structures by using (

**a**) fixed fiber orientation and (

**b**) variable fiber orientation.

**Figure 15.**Topology reconstruction divided into three stages: the result of topology optimization (baseline), baseline and reconstructed topology overlaying, and reconstructed geometry: (

**a**) TVA and (

**b**) TVI.

**Figure 16.**Flow fields: (

**a**) internal-exported TVA; (

**b**) baseline TVA; (

**c**) reconstructed TVA; and (

**b**) reconstructed TVI. Fiber orientation tensors: (

**d**) internal-exported TVA; (

**e**) baseline TVA; and (

**f**) reconstructed TVA.

**Figure 17.**Tsai–Hill failure criterion fields: (

**a**) baseline TVA; (

**b**) reconstructed TVA; (

**c**) baseline TVI; and (

**d**) reconstructed TVI. Equivalent stress (von Mises-based) failure criterion fields: (

**e**) baseline TVA; (

**f**) reconstructed TVA; (

**g**) baseline TVI; and (

**h**) reconstructed TVI.

**Figure 19.**Flow fields: (

**a**) simulated and experimental flow of TVA; and (

**b**) simulated and experimental flow of TVI.

**Figure 22.**TVA and TVI bracket-loading experiments: (

**a**) TVA made in PA6 50GF, (

**b**) TVA made in PA6 30CF, (

**c**) TVA made in D16T, (

**d**) TVI made in PA6 50GF, (

**e**) TVI made in PA6 30CF, (

**f**) TVI made in D16T.

**Figure 23.**Specific force-normalized deformation curves of numerical and experimental TVA and TVI made of: (

**a**) PA6 50GF; (

**b**) PA6 30CF; and (

**c**) D16T.

Characteristics | Material | ||
---|---|---|---|

PA 50GF | PA 30CF | D16T | |

Matrix phase | |||

Matrix density, kg/m^{3} | 1148 | 2770 | |

Young’s modulus, E_{m} (MPa) | 4911 | 3994 | 66,059 |

Poisson’s coefficient, υ_{m} | 0.372 | 0.372 | 0.330 |

Yield stress, σ_{y} (MPa) | 17.21 | 14.5 | 294.48 |

Hardening modulus, R_{∞} (MPa) | 37.1 | 37.00 | 109.51 |

Hardening exponent, m | 371.21 | 458.30 | 75.72 |

Linear hardening modulus, k (MPa) | 313.02 | 188.40 | 1107.60 |

Reinforcement phase | |||

Fiber density, kg/m^{3} | 2550 | 1780 | - |

Young’s modulus, E_{f} (MPa) | 72,000 | 230,000 | - |

Poisson’s coefficient, υ_{f} | 0.22 | 0.20 | - |

Fibers’, AR | 13.58 | 16.54 | - |

Wt. % | 30 | 50 | - |

Material’s ultimate tensile strength | |||

Longitudinal, X (MPa) | 153.31 | 169.35 | 476 |

Transverse, Y (MPa) | 97.82 | 85.07 | - |

Transverse shear strength, S (MPa) | 83.90 | 66.33 | - |

Topology | m, g | f, N | C_{K} | C_{K}^{TH} |
---|---|---|---|---|

PA6 50GF | ||||

TCA | 4.655 | 326.8 | 5.2928 | 5.6994 |

TCI | 4.658 | 326.1 | 5.2482 | 5.8285 |

PA6 30CF | ||||

TCA | 3.779 | 264.5 | 5.3407 | 6.7438 |

TCI | 3.781 | 264.7 | 5.2874 | 7.1127 |

D16T | ||||

TCA | 8.146 | 570.3 | 5.1964 | - |

TCI | 8.152 | 570.6 | 5.2287 | - |

Topology | Normalized Specific Stiffness, N/gr | Percentage Change from TCA to TCI, % | |
---|---|---|---|

TCA | TCI | ||

PA6 50GF | 3883 | 3661 | 6.06 |

PA6 30CF | 4621 | 4194 | 10.18 |

D16T | 11,474 | 11,654 | −1.54 |

Material | Average, N/gr | Standard Deviation, N/gr | Coefficient of Variation, % | Percentage Change from TCA to TCI, % | |||
---|---|---|---|---|---|---|---|

TVA | TVI | TVA | TVI | TVA | TVI | ||

PA6 50GF | 1551 | 1479 | 140 | 167 | 9.03 | 11.26 | 4.87 |

PA6 30CF | 1833 | 1737 | 52 | 96 | 2.84 | 5.54 | 5.53 |

D16T | 7192 | 7584 | 76 | 406 | 1.06 | 5.36 | −5.17 |

Topology | Baseline | Reconstructed | Percentage Difference between Baseline and Reconstructed | |||||||
---|---|---|---|---|---|---|---|---|---|---|

m, g | f, N | C_{K}^{eq} | C_{K}^{TH} | m, g | f, N | C_{K}^{eq} | C_{K}^{TH} | C_{K}^{eq} | C_{K}^{TH} | |

PA6 50GF | ||||||||||

TVA | 22.94 | 1606 | 3.3183 | 3.4341 | 25.59 | 1791 | 3.5230 | 3.6541 | 5.98% | 6.21% |

TVI | 23.24 | 1627 | 3.3214 | 3.4437 | 26.17 | 1832 | 3.5481 | 3.6926 | 6.60% | 6.98% |

PA6 30CF | ||||||||||

TVA | 18.62 | 1304 | 3.3225 | 3.6881 | 20.76 | 1448 | 3.5273 | 3.9025 | 5.98% | 5.65% |

TVI | 18.86 | 1320 | 3.3238 | 3.7103 | 21.24 | 1481 | 3.5539 | 3.9799 | 6.69% | 7.01% |

D16T | ||||||||||

TVA | 40.15 | 2810 | 3.3122 | - | 47.34 | 3314 | 3.5707 | - | 7.51% | - |

TVI | 40.66 | 2846 | 3.3176 | - | 48.36 | 3386 | 3.5892 | - | 7.86% | - |

**Table 6.**Normalized specific stiffness and mass characteristics of the baseline and reconstructed TVA and TVI.

Material | Topology | Normalized Specific Stiffness, N/gr | Percentage Change from TVA to TVI, % | |
---|---|---|---|---|

TVA | TVI | |||

PA6 50GF | Baseline | 7250 | 7187 | 0.88 |

Reconstructed | 5500 | 5260 | 4.56 | |

PA6 30CF | Baseline | 8993 | 8924 | 0.77 |

Reconstructed | 7209 | 6893 | 4.58 | |

D16T | Baseline | 19,737 | 19,869 | −0.66 |

Reconstructed | 11,838 | 11,613 | 1.92 |

Material | Average, N/gr | Standard Deviation, N/gr | Coefficient of Variation, % | Percentage Change from TVA to TVI, % | |||
---|---|---|---|---|---|---|---|

TVA | TVI | TVA | TVI | TVA | TVI | ||

PA6 50GF | 3529 | 3289 | 179 | 267 | 5.09 | 8.13 | 7.30 |

PA6 30CF | 4533 | 4290 | 216 | 187 | 4.77 | 4.36 | 5.66 |

D16T | 7293 | 7875 | 775 | 1150 | 10.63 | 14.60 | −7.39 |

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**MDPI and ACS Style**

Kurkin, E.; Espinosa Barcenas, O.U.; Kishov, E.; Lukyanov, O.
Topology Optimization and Efficiency Evaluation of Short-Fiber-Reinforced Composite Structures Considering Anisotropy. *Computation* **2024**, *12*, 35.
https://doi.org/10.3390/computation12020035

**AMA Style**

Kurkin E, Espinosa Barcenas OU, Kishov E, Lukyanov O.
Topology Optimization and Efficiency Evaluation of Short-Fiber-Reinforced Composite Structures Considering Anisotropy. *Computation*. 2024; 12(2):35.
https://doi.org/10.3390/computation12020035

**Chicago/Turabian Style**

Kurkin, Evgenii, Oscar Ulises Espinosa Barcenas, Evgenii Kishov, and Oleg Lukyanov.
2024. "Topology Optimization and Efficiency Evaluation of Short-Fiber-Reinforced Composite Structures Considering Anisotropy" *Computation* 12, no. 2: 35.
https://doi.org/10.3390/computation12020035