Carbon fiber composites, characterized by their high specific strength and low weight, are becoming increasingly crucial in automotive lightweighting. However, current research primarily emphasizes layer count and orientation, often neglecting the potential of microstructural design, constraints in the layup process, and performance reliability. This study, therefore, introduces a multiscale reliabilitybased design optimization method for carbon fiberreinforced plastic (CFRP) drive shafts. Initially, parametric modeling of the microscale cell was performed, and its elastic performance parameters were predicted using two homogenization methods, examining the impact of fluctuations in microscale cell parameters on composite material performance. A finite element model of the CFRP drive shaft was then constructed, achieving parameter transfer between microscale and macroscale through Python programming. This enabled an investigation into the influence of both micro and macro design parameters on the CFRP drive shaft’s performance. The MultiObjective Particle Swarm Optimization (MOPSO) algorithm was enhanced for particle generation and updating strategies, facilitating the resolution of multiobjective reliability optimization problems, including composite material layup process constraints. Case studies demonstrated that this approach leads to over 30% weight reduction in CFRP drive shafts compared to metallic counterparts while satisfying reliability requirements and offering insights for the lightweight design of other vehicle components.
Lured by the high specific strength, lightweight nature, and relatively high resistance to corrosion of carbon fiberreinforced composites, these materials have seen extensive research and attention within the automotive industry [
Abu Talib et al. [
Zhou et al. [
To fully leverage the potential for microstructural design of CFRP, while considering layup process constraints and the reliability of performance indicators, this study proposes a multiscale reliabilitybased design optimization method, applied in realworld CFRP drive shaft development. The structure of this paper is organized as follows:
The multiscale design of unidirectional carbon fiberreinforced composites encompasses both microscale and macroscale levels, as illustrated in
In addition to traditional theoretical analysis methods, the method of simulating the mechanical properties of composite materials through finite element analysis, by imposing appropriate boundary conditions on a unit cell, has gained increasing popularity [
Unidirectional composite materials are typical orthotropic materials, and their constitutive equations are as follows:
where
The macroscopic stress and strain tensors can be obtained through numerical calculations on the unit cell, derived from the average stress and average strain. The mathematical expressions for the average stress tensor
where
To calculate the compliance matrix in the finite element model of the unit cell, a constant nonzero displacement component is sequentially applied on a specific surface, while imposing appropriate compatibility constraints on the other surfaces [
The use of linear perturbation techniques for solving twoscale progressive homogenization method has been reported as feasible and is suitable for addressing multiscale analysis problems in carbon fiberreinforced composites [
In this formula,
To calculate the stiffness matrix as outlined in
In this context,
The approach is executed in the following manner: Periodic boundary conditions (PBC) are established using MultiPoint Constraints. Thermal stress application is achieved through a Fortran subroutine, and a Python script is crafted for the extraction of stressstrain information, thereby aiding in the determination of homogenization coefficients. Given the limitation of space, this document refrains from an indepth exploration of the homogenization techniques; nevertheless, readers seeking additional insights are advised to explore the pertinent academic resources [
To realize the multiscale reliability design of CFRP drive shafts, it is crucial to explore the microscale properties of CFRP. Consequently, this section presented a comprehensive study on the microscale cell, including methods for establishing microscale cells, the impact of microscale parameters on the elastic properties, and the influence of design parameter uncertainty on the elastic properties.
To develop a realistic microscale model, the crosssection of unidirectional CFRP was examined using an electron microscope (model JSMIT300LA), as shown in
This study employed T70012k carbon fibers and chose EPOLAM 5015 RESIN as the matrix material. The mechanical characteristics of both the carbon fibers and the resin have already been determined by experiments in our previous research [
Material  

Fiber  227.0  13.4  6.8  4.8  0.2  0.25 
Matrix  3.3  –  –  –  0.35  – 
To predict the elastic properties of CFRP, the microscale cell model C_{5} was developed in ABAQUS, as shown in
Mesh sizes  Thermal stressbased method  Boundary forcebased method  

0.2  115.12  6.03  3.44  118.22  6.35  3.84 
0.4  115.12  6.03  3.44  118.22  6.34  3.84 
0.6  115.98  6.02  3.20  118.12  6.33  3.69 
0.8  115.80  6.01  3.18  117.95  6.31  6.62 
1.0  115.12  6.01  3.17  117.56  6.30  6.59 
To investigate the influence of microscale cell type on the elastic properties of unidirectional CFRP, a range of cells, specifically C_{1} to C_{5}, were modeled in ABAQUS, each with a fiber volume fraction (
Material properties  

C_{1}  115.75  6.03  3.21  4.10  0.32  0.27 
C_{2}  115.39  6.08  3.21  4.09  0.34  0.26 
C_{3}  115.31  6.02  3.22  4.08  0.36  0.28 
C_{4}  115.23  6.04  3.20  4.10  0.31  0.27 
C_{5}  115.12  6.03  3.44  4.8  0.34  0.25 
Experimental result  113.57  6.10  3.46  –  0.33  – 
To corroborate the precision of the homogenization theory, which is predicated on thermal stress, in forecasting the elastic attributes of microscale cells, this research juxtaposes its derived outcomes with empirical evidence found in the literature [
Carbon fiberreinforced composites are composed of a fiber reinforcement phase and a matrix resin, wherein the ratio of each component plays a crucial role in determining the composite’s overall performance. To investigate the influence of fiber volume fraction on the elastic properties of unidirectional CFRP, this study has examined the impact of various fiber volume fractions by conducting a series of analyses on microscale cells. These analyses employed the boundary forcebased and thermal stressbased homogenization methods, as detailed in
From
In practical engineering design, numerous uncertainties, such as variability in material parameters and fluctuations in characteristic dimensions, exist. These uncertainties can pose risks of failure to the outcomes of deterministic designs, as noted in references [
In evaluating the influence of parameter fluctuations in microscale cells on the elastic properties of composite materials, the cells were designed with a fiber volume fraction (
An analysis of
To investigate the impact of fluctuations in the properties of component materials on the elastic properties of composite materials, literature [
From
This paper focuses on the CFRP drive shaft, which has been experimentally validated as per literature [
During the design process of the drive shaft, a crucial transmission component in automobiles, meeting the torsional stiffness requirements is imperative. Additionally, in operation, the highspeed rotation of the drive shaft can induce vibrations, potentially leading to resonance. The occurrence of resonance not only compromises the vehicle’s overall stability but also shortens the lifespan of the drive shaft. Consequently, in the design of CFRP drive shafts, it is critical to ensure that both the torsional stiffness and the initial natural frequency satisfy the design specifications while also pursuing a lightweight design. The performance indicators for the CFRP drive shaft are listed in
Performance indicators  Firstorder natural frequency  Torsional angle per unit length  TsaiWu strength index 

Threshold  ≥150 Hz  ≤4.5°/m  0 ≤ TWSI ≤ 1 
To explore the influence of fiber volume fraction on the performance of CFRP drive shafts, the comprehensive performance of the drive shaft was calculated by varying the halfside length
Analysis of
To investigate the relationship between the layup angles in composite materials and drive shaft performances, a unidirectional fiber composite material with 51% fiber volume fraction was selected for analysis using the aforementioned finite element model. Layup angles for individual layers were set at 5° intervals within the range of [0°, 90°], and simulations were conducted using a 10layer laminate with uniform layup angles.
An analysis of
To enhance the torsional stiffness of the drive shaft while achieving maximum lightweighting, this study set the mass
where variable
Layup process constraints  Constraint equations  Description of constraints 

Layup orientation principle  –  Discrete integer variables 0, 1, 2, and 3 correspond to layup angles of −45°, 0°, 45°, and 90°, respectively, with N_{u} representing the angle of the 
Balanced symmetry layup principle  The balanced symmetry layup mandates that the laminate is symmetrically structured during the layup process, effectively preventing warping under load. In the constraint equation, if the layup angle of the uth layer falls within the range of 0–3, then the layup angle of the (11 

±45° singleply symmetrical layup principle  To prevent warping deformation of the laminate under load, it is necessary to pair 

Avoiding the consecutive layup of four layers with the same angle  –  In composite laminate designs, an excessive number of identical and consecutive layups heightens the risk of delamination. To mitigate this, the design typically restricts to no more than four consecutive layers of the same layup. This manufacturing constraint is addressed by scrutinizing the design variables during the optimization process to preclude such scenarios. 
Based on the deterministic optimization model previously established, this study additionally developed a multiscale reliability optimization design mathematical model for the CFRP drive shaft, explicitly incorporating the uncertainties in microscale cell parameters and layup angles, as described below:
where design feasibility is defined as the probability (
Based on the unidirectional CFRP multiscale analysis method introduced in
Step 1: Improving the particle generation and updating strategy of MOPSO to consider the layup process constraints of carbon fiberreinforced composites: In the MOPSO algorithm, the layup orientation constraint, along with process constraints such as the balanced symmetry layup principle, the ±45° singleply symmetrical layup principle, and avoiding the consecutive layup of four layers with the same angle, have been effectively integrated and applied through adapted methods for generating and updating particles.
Step 2: CFRP multiscale modeling and elastic parameters prediction: Python programming was utilized for the parametric modeling of microscale cells. Furthermore, the homogenization theory based on thermal stress was applied to accurately predict the cells’ elastic properties.
Step 3: Construction and parameter transfer of the CFRP drive shaft model: Using ABAQUS software, a CFRP drive shaft model was developed, which incorporated material parameters predicted in step 2. Subsequent analyses, including torsional stiffness and modal analyses, were conducted on this drive shaft model to assess its performance. Following these analyses, a parametric model of the CFRP drive shaft was established, leading to the development of a multiscale reliability optimization model.
Step 4: Development of a multiscale reliabilitybased design optimization method for CFRP drive shaft: An innovative multiscale reliability optimization algorithm for the design and analysis of CFRP drive shafts was proposed, integrating the enhanced multiobjective particle swarm algorithm with both micro and macroanalysis techniques for CFRP drive shafts.
In this study, the efficacy of the modified MOPSO algorithm in handling layup process constraints for CFRP drive shafts was assessed through simulations spanning 60, 80, 100, and 120 generations. Stability in convergence was observed at the 100generation mark. The parameters of the modified MOPSO algorithm are detailed in
Parameter  Maximum iterations  Number of particles  Inertial weight  Particle learning coefficient  Global learning coefficient 

Value  100  50  0.9  1.5  1.5 
While the Pareto solution set presents numerous feasible options, selecting a suitable solution that guides engineering design is pivotal in practical scenarios. To effectively balance the conflicting performance indicators of torsional stiffness and mass in the CFRP drive shaft, this study adopts the minimum distance selection method [
Performance indicators  Threshold  Metal drive shaft  CFRP drive shaft 

Firstorder natural frequency  379  374  
0.63  2.1  
Mass/kg  –  15.509  11.969 
The mathematical model described in
To further illustrate, the minimum distance selection method has been used to select the optimal solution from the Pareto set, as presented in
Performance indicators  Metal drive shaft  Deterministic design  Reliability design R = 0.9  Reliability design R = 0.95  Reliability design R = 0.99 

0.628  2.416  2.254  2.338  2.082  
Mass/kg  15.509  10.665  10.729  10.694  10.886 
Lightweight effect  –  31.23%  30.82%  31.05%  29.81% 
Feasibility  –  79.12%  93.47%  95.12%  99.59% 
In this study, a multiscale reliabilitybased design optimization methodology for carbon fiberreinforced composites was developed, and demonstrated through a case study on a unidirectional CFRP drive shaft. The main conclusions are as follows:
(1) The boundary forcebased and thermal stressbased homogenization methods were employed to predict the elastic properties of unidirectional CFRP. The findings suggest a general consistency in the predictions from both methods. Notably, compared to experimental data from the literature, the thermal stressbased homogenization method aligned more closely with experimental outcomes.
(2) Investigations into the influence of uncertainties in microscale cell parameters on the elastic properties of unidirectional CFRP revealed that the uncertainty in the halfside length of microscale cells, reflecting fiber volume content, has a more pronounced effect on these properties as compared to the variability in matrix material properties.
(3) The application of the refined MOPSO algorithm for the design optimization of unidirectional CFRP drive shafts has proven its efficacy in managing constraints including the balanced symmetry layup principle, the ±45° singleply symmetrical layup principle, and the prevention of consecutive layups of four identical angles. This capability significantly enhances the engineering applicability of the multiscale optimization method proposed in this research.
(4) Applying the multiscale reliabilitybased design optimization methodology to a unidirectional CFRP drive shaft, the results indicate that the optimal solutions not only enhance the comprehensive performance of the drive shaft but also meet the reliability requirements of the constraints as per the design specifications.
We would like to thank the authors for their contributions to this manuscript. We also thank the journal of CMES for their supports for publications of this special issue.
This research work was supported by the S&T Special Program of Huzhou (Grant No. 2023GZ09) and the Open Fund Project of the Shanghai Key Laboratory of Lightweight Structural Composites (Grant No. 2232021A406).
The authors confirm contribution to the paper as follows: study conception and design: Huile Zhang, Yurui Wu, data collection: Shikang Li, Huile Zhang, Pengpeng Zhi, analysis and interpretation of results: Huile Zhang, Wei Wang, Zhonglai Wang, draft manuscript preparation: Huile Zhang. All authors reviewed the results and approved the final version of the manuscript.
Not applicable.
The authors declare that they have no conflicts of interest to report regarding the present study.
Monte Carlo Simulation (MCS) is a numerical method used to estimate the probability of complex events by simulating random processes. In the context of reliability engineering, MCS is applied to estimate the probability of failure, defined as
The probability of failure is estimated as follows: