The stacking sequence optimisation of a filament wound composite bicycle frame using the data-driven evolutionary algorithm EvoDN2

ABSTRACT

This work focusses on identifying the optimal stacking sequence for composite tubes in mountain bike frames using a data-driven model combined with evolutionary algorithms. The objective is to find a frame that is sufficiently stiff while meeting the requirements of weight, strength, and minimum tube wall thickness. The decision variables are the ply winding angles and the ply thicknesses of each tube. The study performs designs for two load cases – Starting and Uphill – and explores two types of winding: the gradual winding of individual layers (1ply) and the winding of layers between predefined inner and outer layers with variable thicknesses (TW). Additionally, the design process is applied to frames made of isotropic materials, such as steel, aluminium, and titanium, using the same methodology to allow for comparison of results. The article demonstrates the successful application of this methodology to common sports equipment, suggesting its potential for beneficial use in other common composite frame structures.

KEYWORDS:

• bicycle
• composite
• data-driven
• optimisation
• evolutionary

1. Introduction

In the dynamic world of cycling technology, the quest for the ultimate bicycle frame that combines lightness, strength, and enough stiffness has led to innovative materials and manufacturing techniques. At the forefront of this evolution is the utilisation of composite materials, notably carbon fibre, whose superior mechanical properties have revolutionised bicycle design and performance. The advent of composite materials in bicycle frame production marks a pivotal shift, enabling the creation of bikes that are not only lighter and stronger but also more aerodynamically efficient [1, 2].

Historically, the cycling industry has seen a gradual transition from traditional materials such as steel and aluminium to advanced composites. This transition is well documented in the literature, where studies underscore the advantages of composites in terms of their high strength-to-weight ratio and the ability to tailor mechanical properties to specific design requirements. These characteristics make composites particularly suited for high-performance bicycles, catering to the needs of professional racers and cycling enthusiasts alike [3].

Furthermore, research has highlighted the innovative manufacturing techniques associated with composite materials, such as lay-up and resin transfer moulding [4]. These methods offer unprecedented flexibility in fibre orientation and layer thickness, allowing for a customised approach to frame design that optimises structural integrity and performance characteristics. The unique ability of composites to dampen vibrations also contributes significantly to improved ride quality, an essential factor in endurance cycling and off-road applications [5].

Filament winding technology represents a significant advancement in composite bicycle frame manufacturing. It is an integral part of the manufacturing of high-performance bicycles, enabling precise control over fibre orientation and material properties. Additionally, filament winding technology stands out for its capability to manufacture composite structures with variable winding angles, allowing for customised mechanical properties tailored to specific bicycle frame requirements. This flexibility in the design process is important for optimising performance characteristics such as stiffness and strength [6].

Previous studies focusing on composite optimisation have been published. A genetic algorithm was introduced in [7] for optimising the stacking sequence to achieve desired strength in filament wound cylindrical composite shell structures under internal pressure. Similarly, half shaft optimisation was performed in [8], where the effectiveness of genetic algorithms in identifying optimal winding angles was highlighted. In [9], the genetic algorithm was used for optimising cylinders manufactured by Variable-Angle Filament Winding (VAFW) technology, which enables fibre placement at variable angles along the cylinder's axial direction. The cylinder was divided into several segments with constant winding angles and optimised for maximum axial buckling load, demonstrating significant improvements in buckling strength, stiffness, and energy absorption capacities. Study [10] aims to make cylinders, produced using VAFW, lighter while still adhering to specific manufacturing constraints and design loads. To achieve this, a particle swarm optimisation combined with Kriging-based metamodels is employed.

Our work contributes to the ongoing exploration of bicycle frame optimisation by integrating the Finite Element (FE) beam model method for mechanical analyses with the utilisation of machine learning techniques and evolutionary algorithms for the surrogate modelling and optimisation process. FE analysis has been instrumental in advancing our understanding of material behaviour under various loading conditions, providing a robust framework for predicting the structural performance of bicycle frames. By utilising machine learning algorithms, as discussed in [11], we aim to refine the design process further, facilitating the development of surrogate models that can efficiently navigate the complex design space to identify optimal configurations.

Specifically, this study focusses on utilising machine learning to raise insights from FE-generated datasets, paving the way for the identification of the Pareto front set [12]. This approach allows an efficient balance between competing design objectives, such as minimising weight while maximising strength and stiffness. By the union of composite materials mechanics, FE analysis, machine learning, and evolutionary algorithm-based optimisation, this article offers a novel perspective on the design and optimisation of bicycle frames.

It should be noted that here we are trying to solve a Pareto optimal problem requiring simultaneous optimisation of three conflicting objectives. There are gradient based methods for solving such problems as elaborated in the text of Miettinen [12]. Those methods, however, tend to converge on one Pareto solution at a time and resolving the entire Pareto set remains a difficult and time-consuming act. Even the newer and advanced interactive Multi-component decision making (MCDM) algorithms [13] require lengthy sessions with a decision maker and computing the entire Pareto front is still not in their purview. The evolutionary approaches [14, 15] on the other hand. are capable of resolving the complete Pareto set as they converge. In our strategy, we needed the information of the entire Pareto set in a computing time efficient manner, so that the decision maker can select the suitable solutions out of them for the next round of cyclic optimisation that we have adopted here. Therefore, an evolutionary approach was preferred here to come up with the surrogate models and the deep learning capability of the EvoDN2 algorithm that we employed in this study, added further to learning capability of the surrogate models that were ultimately selected.

2. Methods

2.1 Ply stresses, strength coefficients, and stiffness determination

Our methodology, which was described in detail in [11] and [16], utilises the theory of minimum complementary energy and Hooke's law for orthotropic materials to calculate the equivalent stiffnesses of a composite laminate. Four key equations (as shown for example in [17]) determine tensile, bending, torsional, and shear stiffness by incorporating ply properties and geometry. The details are provided in our earlier publications [11, 16].

Beam theory (described, for example, in texts such as [18]) is applied to approximate the behaviour of composite structures, using finite elements to model beams. The 2-node Timoshenko beam element in 3D space is specifically considered, with its stiffness matrix dependent on equivalent stiffnesses and element length. This facilitates the evaluation of deformations, stresses, and other parameters for structural optimisation. The detailed formulation is already provided in our earlier work [11].

Ply stresses are calculated using internal forces and a transformation matrix to pinpoint ply strength coefficients through the maximum stress criterion. This criterion evaluates potential failure directions and material usage rates, leading to the calculation of a safety factor from the highest stress coefficient. When this coefficient reaches the value of unity, it signals damage to the material.

The methodology is extended to include the evaluation of stiffness as a function of the product of force and displacement, embodying the total potential energy concept. This approach enriches the analysis by quantifying the stiffness of the composite structure, pivotal for understanding its mechanical behaviour under load.

Further details and equations related to the methodology employed can be found in reference [11].

It should be noted that the comparison between FE 1D and 3D beam models is shown in earlier studies [19], for example. The general finding for our type of problem is that the 1D models yield to much faster computing without any significant loss of accuracy. Therefore, the usage of FE 1D beam model to optimise a frame structure is convenient and is adopted here. Many studies highlight the effectiveness of the two-node beam models for modelling and optimising frame structures as shown in [20] for example, where a comparison of the experimental results with the numerical calculation of the frame in space using two-node beam elements shows excellent suitability of these models towards the optimisation processes.

2.2 Application of the EvoDN2 algorithm for composite frame design

In this study, an optimisation approach, utilising the Evolutionary Deep Neural Network (EvoDN2) algorithm [21], is applied to design composite frame structures. This method combines a data-driven deep neural network with a bi-objective evolutionary algorithm, refined further by a Linear Least Square (LLSQ) process for final convergence. Additionally, it incorporates a many-objective optimisation module using a constraint-based Reference Vector guided Evolutionary Algorithm (cRVEA) [22] to manage complex trade-offs between model accuracy and complexity. This innovative approach effectively handles non-linear and noisy data, ensuring robustness without relying directly on underlying physical principles. The EvoDN2 algorithm, complemented by cRVEA's reference vectors, enhances population diversity and optimises convergence, even for a many objective situations, addressing common multi-objective optimisation challenges, which often requires deep learning [15]. Further details of the EvoDN2 algorithm and its application, including its operational parameters, instance of its effectiveness in optimising material or geometrical parameters are available elsewhere [23–25].

2.3 Refining design solutions through cyclic optimisation

The concept of cyclic optimisation (also mentioned in [16]) for iterative design improvement employs multiple rounds of optimisation, using the outcomes of each cycle as the starting point for the next. This process allows for in-depth exploration of the design space, with each iteration aiming to enhance performance objectives progressively. Through cyclic optimisation, the design process's effectiveness is improved.

Key to the success of this optimisation strategy is the initial data preparation and the careful setting of training parameters to achieve optimal fitting values, indicating a well-tuned surrogate model. The article incorporates decision-making rules for selecting candidates for subsequent cycles and criteria for terminating the process when improvements cease. The application of these methods to selected geometries will be detailed further.

2.4 Optimisation of a bicycle frame: methodology and objectives

The optimisation process is applied to a bicycle frame, focusing on three primary objectives to enhance its design and performance:

• F1: Total Potential Energy – This objective is associated with the structure's stiffness and is quantified by the total potential energy of the system, indicative of the frame's ability to resist deformation under load.

• F2: Structure Strength – This is measured by the maximum value of the ply strength coefficients. It reflects the maximum stress the material can withstand before failure.

• F3: Structure Weight – This objective considers the thickness of the tubes, aiming to minimise both weight and material costs.

These objectives are pursued in a cyclic optimisation process, where each cycle iteratively refines the design by adjusting ply thicknesses (t_i) and ply angles (α_i) – the decision variables. The initial cycle sets the potential ranges for these variables, aiming for physically achievable and necessary objective values.

As the cyclic optimisation progresses, the decision parameter ranges, especially thicknesses, are dynamically adjusted based on previous outcomes, ensuring the surrogate model built is accurate and reflective of the design goals. This method ensures a systematic approach to optimising each ply and adjusting tube thicknesses for improved performance. The optimisation concludes when further cycles fail to significantly improve the target functions.

This study's methodology is applied to a single geometry – the bicycle frame. This geometry is selected for its relevance to practical design challenges, emphasising the need for a structure that combines optimal stiffness, strength, and weight for performance efficiency. The specific parameters, boundary conditions, and loading details related to the bicycle frame optimisation are described in the next section of the article.

It should be noted that in our previous work [16] we have used the configuration of a curved beam and in this case the bicycle is represented as a frame structure which is mechanistically different. Also, the boundary conditions used in these studies are substantially different, warranting a fresh investigation.

Unlike the displacement that was used in our earlier work [16] as an objective for the Gantry-type machine or a crane structure, here we have used the total potential energy, as it better describes the stiffness and behaviour of this sports equipment. Displacement only considers deformation of just one loaded node, while the potential energy takes into account the displacement, as well as the rotation in the entire structure described by a large number of nodes, rendering this objective far more suitable than the displacement for the bicycle structure.

3. Problem description

As outlined in the previous section, this study employs an established methodology to optimise the enduro bicycle frame, a type of bicycle designed for efficient cycling in mountainous terrain. Enduro bicycles emphasise excellent downhill performance while maintaining good climbing capabilities. Therefore, their geometry is tailored to meet these demands. The parameters of the bicycle analysed in this study are based on the CDURO [26] frame produced by Compotech s.r.o.[27], details of which can be found in previously published work [28], for example. Additionally, the load cases are established based on measurements reported in an article [29]. This section will detail two aspects: the geometry, which is grounded in existing literature, and the uniquely defined load cases, providing a comprehensive foundation for our analysis. Finally, the decision variables and manufacturing and other constraints connected to the given task will be described at the end of the section.

3.1 Bicycle frame geometry

The frame geometry, as depicted in Figure 1 and described in [28], is based on a frame manufactured using filament winding technology by Compotech s.r.o. This technology enables the production of frames with variable ply winding angles and thicknesses. It is crucial to note that our methodology allows for the selection of any variable optimal winding angles and thicknesses, which corresponds to the mentioned production technology.

Figure 1. Bicycle frame geometry 3D.

As illustrated in Figure 2, a simplified beam model has been generated. The front triangle, represented by blue beams and optimised in our study, incorporates node positions, inner diameters, and materials from the original bicycle model. The rear assembly, shown as black beams, is analysed together with the front triangle as a single model and represents the original bicycle's rear assembly. Its parameters, which influence the overall stiffness and strength of the structure, are integral to our analysis. The element XI – the damper – is modelled as a rod element with no bending stiffness, replaced in our model by a steel tube for simplicity. In Table 1, details such as the nodes, inner diameter, names, and materials of each element are provided.

Figure 2. Beam model based on 3D geometry.

Table 1. Parameters of the bicycle frame beam model.

Element	Nodes	d [mm]	Name	Material
I	1,9	56	Down Tube	UMS40_55
II	9,7	56
III	7,2	56
IV	2,3	58	Head Tube	UMS40_53
V	3,4	48	Top Tube	t700_55
VI	4,1	40	Seat Tube	t700_55
VII	5,9	22	Chain Stay Right	UMS40_50
VIII	6,9	22	Chain Stay Left	UMS40_50
IX	5,8	18	Seat Stay	UMS40_50
X	6,8	18	Seat Stay	UMS40_50
XI	7,8	10	Damper	Steel
XII	8,9	18	Seat Stay	UMS40_50
XIII	8,9	18	Seat Stay	UMS40_50

For detailed information on the material properties, please see Table 2 (adapted from [27] and [28]). The density of the composites used in the total weight calculations was set at a value of 1 500 kg/m³.

Table 2. Composite material mechanical properties.

Material property	Symbol	Unit	UMS40_55	UMS40_53	UMS40_50	t700_55
Longitudinal Young’s modulus	E1	MPa	218600	210760	199000	130600
Transversal Young’s modulus	E2	MPa	7376	7184	6907	7376
Shear modulus	G12	MPa	5672	5419	5071	5672
Poisson’s ratio	ν12	–	0.345	0.347	0.35	0.345
Longitudinal tensile strength	XT	MPa	1950	1950	1950	1950
Longitudinal compressive strength	XC	MPa	1480	1480	1480	1480
Transversal tensile strength	YT	MPa	48	48	48	48
Transversal compressive strength	YC	MPa	200	200	200	200
Shear strength	S	MPa	80	80	80	80

3.2 Load case and objective functions definition

The load case definition is a pivotal aspect for comprehending the forces and stresses imposed on the bicycle frame. Initially, we present an illustrative Figure 3 that delineates the frame, with key positions marked from A to H. These locations are strategically selected to denote crucial components such as the pedals, handlebars, and the connection of the fork to the front wheel. This depiction is instrumental in illustrating the locations of load points in relation to the frame's geometry. The transforming of the load acting in points A to H into external forces acting at specific nodes of the bicycle frame, specifically nodes 1, 2, 3, and 5 is used to reach relevant loading of the analysed frame. This transformation process is integral to our analysis as it allows for a more accurate representation of load distribution and its consequent impact on the frame. The approach for this recalculation is based on the seminal work [29] where the loading is based on experimental measurements.

Figure 3. Position of loaded points.

The objective functions applied in our study are the first, F1, addresses the structure’s compliance, gauged through the total potential energy, shedding light on the frame’s flexibility and resilience under load. The second, F2, focuses on the strength coefficient (smax), conceptualised as a function of stress. This metric is pivotal in assessing the maximum strength capacity of the frame. The third, F3, targets the minimisation of the frame’s mass, balancing structural cost against performance efficiency.

Finally, we delve into the application of these objective functions for two distinct load cases: the Uphill and Starting load cases (see Tables 3 and 4). The boundary conditions for both load cases are the same – nodes 5 and 6 are fixed and node 2 cannot move in z-direction. Each load case undergoes a separate optimisation process, culminating in individual tasks for each load case. This approach guarantees that the optimisation is tailored to meet the unique requirements and challenges posed by each loading scenario.

Table 3. Starting load case definition (Forces Fx, Fy, Fz [N] and moments Mx, My, Mz [Nmm]).

Table 4. Uphill load case definition (Forces Fx, Fy, Fz [N] and moments Mx, My, Mz [Nmm]).

3.3 Decision variables

As described in Section 2, the decision variables in our study are winding angle (α_i) and ply thickness (t_i). The winding angle, α_i, is considered within the range of 0 to 90 degrees. Constraints of ply thickness, t_i, are determined during the optimisation cycle. A ply wound at angle α_i with thickness t_i comprises four subplies, each with a thickness of t_i/4, and alternates in angles +α_i, -α_i, -α_i, and +α_i. The minimum tube thickness is established based on technological constraints, set at 1 mm for composite material and 0.5 mm for metals.

4. Results

4.1 Composite bicycle frame optimisation

The stacking sequence of the tubes in the bicycle frame’s front triangle was optimised for two load cases: Starting and Uphill. This optimisation was conducted for two types of winding. The first type, referred to as 1ply, involves a single ply, where each tube ply is wound cycle by cycle. The second type, referred to as Technological Winding (TW), incorporates a specific sequence: an inner ply with a minimum thickness of 0.2 mm wound at 45 degrees, and an outer ply with the same minimum thickness wound at 88 degrees. This type is chosen for its technological production advantages in filament winding and the beneficial mechanical properties of the final product. The thicknesses of the inner and outer ply are decision variables. Between these plies is the middle ply of each tube, which is optimised cycle by cycle.

Each load case presents unique challenges and objectives, addressed through a systematic decision-making approach. For the Uphill Load Case, the objectives were set as follows: F1 (the structure's compliance) should be minimised, F2 (the strength coefficient) should not exceed 0.4 to maintain a safety factor of 2.5, and F3 (the mass of the structure) must be no more than 1.25 kg. Additionally, a secondary condition required limiting the displacement of node 3 to 5 mm. The decision-making process involved filtering results to satisfy the conditions for F2, F3, and the secondary displacement condition, and then selecting the solution that offers the minimum F1 value, with the constraint that the minimum tube thickness must be 1 mm.

Similarly, for the Starting Load Case, the objectives were aligned with those of the Uphill case, except for the displacement condition at node 3, which was adjusted to allow up to 10 mm. The decision-making process mirrors that of the Uphill case.

This structured decision-making process ensures that the optimised tube design meets the established criteria of strength, weight, and compliance for both load cases. The results from this process guide the final selection of the optimised layup sequence for the bicycle frame's front triangle tubes.

For each cycle, a Pareto set represented by hundreds of members was sorted out by the decision-making procedure described above. The best one was chosen for the next cycle. The process was stopped when a cycle did not lead to the improvement of at least one objective without impairing others, and the member from the last cycle was chosen as the result of the optimisation process. Figures 4 and 5 show the development of objectives for both load cases and both types of winding (1ply and TW).

Figure 4. Refining objectives through sequential cycles (Uphill): 1ply and TW.

Figure 5. Refining objectives through sequential cycles (Starting): 1ply and TW.

Decision variables representing winding parameters can be further subjected to strategic adjustment, taking into account the manufacturing technology. This results in a more efficient and beneficial composite frame layup, enhancing manufacturing capabilities. This is evident in Table 5, which presents the variables achieved by the optimisation process, and in Table 6, where some layers with the winding of close angles are replaced by one layer with a calculated winding angle based on a weighted average, whose thickness is the sum of the thicknesses of the original layers. Calculated angle and decision variables rounding subtly modify objectives, which are harmonising with the desired objectives.

Table 5. Bicycle frame parameters obtained as a result of cyclic optimisation (Starting LC, 1ply winding).

		Tube 1		Tube 2		Tube 3		Tube 4
		α [DEG]	T [mm]	α [DEG]	T [mm]	α [DEG]	T [mm]	α [DEG]	T [mm]
Selected member parameters	Layer 1	0.1	0.33	0.1	1.10	1.2	0.33	2.1	1.27
	Layer 2	1.4	0.50	89.6	2.20	3.9	0.47	20.4	0.40
	Layer 3	6.5	0.85			0.9	0.41
TOTAL THICKNESS [mm]			1.68		3.30		1.21		1.67
F1	6787	[mJ]	Starting 1ply only
F2	0.21	[-]
F3	1.22	kg

Table 6. Modified parameters and corresponding objectives (Starting LC, 1ply winding).

		Tube 1		Tube 2		Tube 3		Tube 4
		α [DEG]	T [mm]	α [DEG]	T [mm]	α [DEG]	T [mm]	α [DEG]	T [mm]
Selected member parameters	Layer 1	0.8	0.83	0.1	1.10	2.1	1.21	2.1	1.27
	Layer 2	6.5	0.85	89.6	2.20			20.4	0.40
	Layer 3
TOTAL THICKNESS [mm]			1.68		3.30		1.21		1.67
F1	6794	[mJ]	Starting 1ply only – corr.
F2	0.21	[-]
F3	1.22	kg

4.2 Bicycle frame made from isotropic materials

The same methodology was applied to optimise bicycle frame structures manufactured from isotropic materials usually commonly used for this application. Materials were selected based on publications [30] and [31], and their mechanical properties were taken from the material database [32], are shown in Table 7. Shear strength for steel, is not readily available in research papers or supplier's websites, was calculated based on the Von Mises yield criterion [33].

Table 7. Isotropic material mechanical properties.

	Steel	Aluminium	Titanium
Modulus of Elasticity E [GPa]	210	68.9	100
Poissons Ratio [-]	0.3	0.33	0.34
Ultimate Tensile Strength (UTS)[MPa]	900	310	620
Shear Strength [MPa]	540	207	430
Density [g/cm^3]	7.8	2.7	4.48

The resulting values of objectives for both load cases, compared with the 1ply winding, are shown in Tables 8 and 9. The resulting member was selected from the appropriate Pareto set based on comparable stiffnesses. The ratios of the objective functions F2 and F3 between composite and isotropic materials are mostly greater than 1, indicating an improvement over the reference value.

Table 8. Objectives comparison (starting load case).

Objective starting	Composite	Steel		Aluminium		Titanium
	Value	Value	Ratio	Value	Ratio	Value	Ratio
F1	6787	6776		6799		6760
F2	0.21	0.52	2.48	0.36	1.71	0.37	1.76
F3	1.22	1.90	1.56	1.56	1.28	2.05	1.68

Table 9. Objectives comparison (uphill load case).

Objective uphill	Composite	Steel		Aluminium		Titanium
	Value	Value	Ratio	Value	Ratio	Value	Ratio
F1	1172	1158		1178		1178
F2	0.09	0.18	2.02	0.24	2.67	0.17	1.89
F3	1.21	2.04	1.69	2.34	1.93	2.19	1.81

Typically, the ratio of stiffness to density is employed to compare material efficiency. For this bicycle frame structure, due to geometric constraints, we compare materials using ratios between stiffness and mass, as well as strength to mass. In our results, where the objective is to maximise both stiffness and strength, the ratios for comparison are defined as $1/F1\ast F3$ for stiffness and $1/F2\ast F3$ for strength. Figure 6 presents a bar chart for all materials and load cases. Here, the values are normalised with respect to composite material to enhance clarity and comprehension. The chart clearly demonstrates the efficiency of using composite material for this structure.

Figure 6. Comparison of normalised stiffness and strength ratios for various materials.

The presented methodology can be used to find the best stacking sequence for composite frame structures and can advantageously serve as a supplement to other design steps. The next section will compare this method with others.

5. Comparison with the current research

It is important to highlight that the methodology presented in this study is highly beneficial in applications where achieving specific sports equipment parameters such as mass, stiffness, and strength are crucial. The application of this methodology to a bicycle frame can provide athletes with tailor-made solutions specific to an individual sportsman. As demonstrated in [28], the frame featuring the proposed geometry, manufactured by Compotech, s.r.o., confirms that the joints and other structural elements are fully functional for a bicycle produced by the winding technology.

Optimisation processes for bicycle frames typically utilise finite element models in 3D space, as provided by other studies [34–36]. The methodology outlined in [35] involves two distinct tasks: determining the optimal frame geometry and selecting the most suitable material for production. To identify the best solution, five different load cases were examined. A parametric finite element beam model was employed in [36] to optimise the bicycle frame geometry under various load conditions. The use of such models is significantly computationally demanding; therefore, our methodology, which is based on a simpler beam model in 3D space, may offer a more advantageous and practical approach.

Studies such as [37] and [38] often focus on the optimisation process on finding the optimal stacking sequence for a laminate composed exclusively of plies oriented in 0°, +45°, −45°, and 90° directions, which correspond to capabilities of productions technology. Similar to our work, technology constraints in these studies impact the objective function while adhering to the necessary requirements. The advantage of our approach lies in its ability to design and produce layers with any winding angle, subject to minimal constraints on the ply thickness.

The integration of a finite element model with a genetic algorithm-based optimisation has been utilised previously to enhance solutions for the spaceframe chassis of light rail public transport vehicle made from braided composites, employing surrogate models and machine learning techniques [39]. This approach achieved weight reduction and increased stiffness, underscoring the effectiveness of these techniques in optimising composite frame. Additionally, the combination of surrogate model and genetic algorithms was successfully applied for the multi-objective optimisation of thin-walled deployable composite hinges in [40]. The potential of employing advanced models and optimisation techniques in the optimisation of composite structures is described in other studies as well, in references [41] and [42] for example.

Building upon the foundation laid by previous studies, our method advances the use of surrogate models and evolutionary algorithms for optimisation, distinctively focusing on achieving an optimal balance of stiffness, strength, and minimal weight.

6. Conclusion

In the presented work, the optimisation of a bicycle frame was conducted using a finite element beam model, mechanics for orthotropic material, and an optimisation procedure. The reference geometry was derived from a commercially available bicycle, demonstrating the feasibility of producing frames designed with the presented algorithm without issues. This approach enables customisation of the bicycle according to user preferences.

The optimisation was performed for two types of winding: 1ply and TW. The former involves manufacturing the frame from tubes produced solely with plies whose parameters are determined through optimisation. The latter, TW, involves manufacturing the frame from tubes that have inner and outer plies wound at 45 degrees for the inner ply and 88 degrees for the outer ply, with optimally determined plies stacked between them. The thicknesses of the inner and outer plies are considered as decision variables.

Two load scenarios were evaluated to understand how the stacking sequence would vary for different uses of the sports equipment. The results demonstrate that the presented methodology can successfully identify the best solution for the given load case. Furthermore, the multi-objective optimisation method presented here can be advantageously applied to additional load cases to achieve more versatile solutions. It offers manufacturers a tool for designing customised solutions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This study received support from the Grant Agency of the Czech Technical University in Prague, under grant number SGS24/123/OHK2/3T/12.

Notes on contributors

Anna Malá

Anna Malá, PhD candidate at the CTU in Prague, Faculty of Mechanical Engineering, specializes in the multi-criteria optimisation of wound composite frames.

Zdeněk Padovec

Zdeněk Padovec received a PhD in Applied Mechanics in 2016 from the Faculty of Mechanical Engineering, CTU in Prague. His research focuses on composite pressure vessels and composite construction in general.

Tomáš Mareš

Tomáš Mareš is an Associate Professor at Czech Technical University in Prague. He received his PhD from Czech Technical University in Prague in 2004. The major focus of his research has been in the areas of structural optimisation and composite material analysis.

Nirupam Chakraborti

Nirupam Chakraborti is a Visiting Professor at the Czech Technical University in Prague. He received his PhD from the University of Washington in The US. For a number of decades, the major focus of his research has been in the area of Evolutionary Computation and its engineering applications. His comprehensive book in this domain, Data-driven Evolutionary Modeling in Materials Technology, CRC Press USA-UK, came out in 2023.