DOI: 10.3303/CET23107103 Paper Received: 02 May 2023; Revised: 02 August 2023; Accepted: 03 November 2023 Please cite this article as: Habashneh M., Movahedi Rad M., 2023, Optimizing Topology of Structures Considering Fatigue-Resistance, Chemical Engineering Transactions, 107, 613-618 DOI:10.3303/CET23107103 CHEMICAL ENGINEERING TRANSACTIONS VOL. 107, 2023 A publication of The Italian Association of Chemical Engineering Online at www.cetjournal.it Guest Editors: Petar S. Varbanov, Bohong Wang, Petro Kapustenko Copyright © 2023, AIDIC Servizi S.r.l. ISBN 979-12-81206-07-6; ISSN 2283-9216 Optimizing Topology of Structures Considering Fatigue- Resistance Muayad Habashneh, Majid Movahedi Rad* Faculty of Architecture, Civil Engineering and Transport Sciences, Department of Structural and Geotechnical Engineering, Széchenyi István University, H-9026 Győr, Hungary majidmr@sze.hu The pursuit of reliable design has become increasingly crucial in various engineering disciplines, aiming to minimize environmental impact and enhance resource efficiency. In this context, this study explores the integration of topology optimization techniques with fatigue analysis to develop reliable designs for structural components. Fatigue failure is a critical concern in engineering applications, as it significantly affects the lifespan and reliability of structures. The proposed methodology combines mathematical optimization algorithms, computational modeling, and fatigue analysis techniques. The primary objective of this study is to minimize structural weight by determining the optimal material arrangement within the design domain while also considering fatigue as a constraint within the optimization problem. The bi-directional evolutionary structural optimization (BESO) method is developed to meet the goal of this research. Furthermore, topology optimization of L-shape and U-plate problems are considered as numerical examples to demonstrate the effectiveness of the suggested method. By considering fatigue behavior in topology optimization, engineers can develop lightweight and durable structures that effectively utilize materials while minimizing resource utilization. The integration of these two fields opens up new avenues for reliable design, promoting resource efficiency and contributing to the overall reliability of engineering practices. 1. Introduction Numerous researchers have conducted extensive research on topology optimization (TO) of structures over the past few decades. This discipline aims to find the most efficient arrangement of material within a given design domain while ensuring that all design constraints are satisfied. Several approaches to topology optimization have been proposed, including the Solid Isotropic Material Penalization (SIMP) method (Bendsøe, 1989), evolutionary structural optimization (ESO) method (Xie and Steven, 1993), the developed approach, known as the Bi-Directional Evolutionary Structural Optimization (BESO) method, which focuses on restoring elements that were removed in the previous iteration (Querin et al., 1998), level-set methods (Sethian, 1999), and moving morphable component (MMC) method (Guo et al., 2014). As a pioneering numerical and design technique, topology optimization has undergone significant development and attracted considerable scientific interest. This potent instrument has been exhaustively investigated and implemented in numerous problem domains, such as topology optimization of elasto-plastic materials (Movahedi Rad et al., 2021), geometrically nonlinear problems (Habashneh and Movahedi Rad, 2022) and other developments in the field of TO such as considering thermoelastic analysis (Habashneh and Rad, 2023). Fatigue is a significant concern within the field of structural engineering, as shown by prior studies. An investigation conducted by (Wang et al., 2016) examined the impact of corrosive deterioration on the structural integrity of reinforced concrete highway bridges. By treating fatigue failure as a constraint in the optimization process, optimal designs can be achieved, effectively addressing the challenges posed by real-life applications and ensuring the long-term structural integrity and reliability of engineered systems (Jeong et al., 2015). In light of the growing emphasis on sustainability, it is imperative to establish a strong connection between topology optimization, fatigue considerations, and reliable design principles. (Gao et al., 2021) proposed topological fatigue optimization method, taking into account the effect of defects on fatigue strength and integrating stress constraints. The proposed topology optimization approach by (Desmorat and Desmorat, 613 2008), which focuses on maximizing fatigue lifetime through optimization algorithm considering cyclic plasticity and the Lemaitre damage law, aligns with the objective of reliable design. Additionally, the algorithm developed by (Nabaki et al., 2019), specifically tailored to mitigate the risk of high cycle fatigue failure, offers a promising avenue for integrating sustainability considerations into topology optimization practices. By linking these advancements in topology optimization with sustainability principles, engineers and designers can foster the development of structurally robust, fatigue-resistant, and environmentally responsible solutions for a wide range of applications. This study serves as an extension of the authors' previous work, where they successfully addressed various aspects of topology optimization, including stiffness optimization and other relevant considerations (Movahedi Rad et al., 2021). Therefore, the current study aims to develop a structural optimization algorithm that specifically incorporates fatigue failure as a critical constraint besides the volume constraint. To accomplish this objective, the developed BESO method is utilized within the topology optimization process. Two numerical examples of L- shape and U-plate problems are solved to show the efficiency of the proposed methodology. 2. Theoretical background 2.1 Fatigue analysis In this study, our focus is on the fatigue damage caused by cyclic loadings without considering crack propagation. While our primary objective is not to model crack propagation explicitly, we capture the cumulative effect of these cyclic loadings through a scalar stress state coefficient. As a result, the stress state at every location in the structure and at any point in the cyclic loading may be determined as a scalar factor of the stress state corresponding to the reference loading. Using Basquin's equation, the number of cycles until failure at various effective stress amplitudes is calculated (Dowling, 2004). The cumulative damage from the loading history's stress reversals is then calculated linearly using Palmgren-Miner's rule. Basquin's equation gives the number of cycles 𝑁𝑓𝑖 before failure due to repeated applications of reversal 𝑖 as: 𝑁𝑓𝑖 (𝑥) = 1 2 ( 𝜎𝑎𝑟𝑖 𝜎𝑓 ′ ) 1 𝑏 (1) where 𝜎𝑎𝑟𝑖 is the completely reversed equivalent stress amplitude, 𝜎𝑓 ′ is the coefficient of material’s fatigue strength, 𝑏 is the exponent of material’s fatigue strength. Using Palmgren-Miner's rule, the accumulated fatigue at a position is calculated as the sum of the harm inflicted at that position by every reversal. 𝐷(𝑥) = ∑ 𝑛𝑖 𝑁𝑓𝑖 (𝑥) 𝑁𝑟 𝑖=1 (2) Here, we define 𝑛𝑖 as the count of occurrences of reversals during the cyclic loading, while 𝑁𝑟 represents the overall count of reversals throughout the loading process. 2.2 Definition of the topology optimization problem This section focuses on the analysis of a topology optimization problem aimed at minimizing the mean compliance while integrating constraints, including fatigue and volume. 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒: 𝐶 = 𝑢𝑇𝐾𝑢 (3.a) 𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 𝑉∗ − ∑ 𝑉𝑖𝑥𝑖 𝑁 𝑖=1 = 0 (3.b) 𝑉∗ 𝑉0 − 𝑉𝑓 ≤ 0 (3.c) 𝐷(𝑥) − ∑ 𝑛𝑖 𝑁𝑓𝑖 (𝑥) 𝑁𝑟 𝑖=1 = 0 (3.d) 𝑥𝑖 ∈ {0,1} (3.e) The topology optimization problem aims to minimize the mean compliance (𝐶) while incorporating constraints related to volume and fatigue. The mean compliance depends on several factors, such as displacement vectors (𝑢), and the overall stiffness matrix (𝐾), which are crucial in determining the compliance. To provide further clarification, 𝑁 represents the total number of elements, 𝑉𝑖 signifies the volume of an individual element, and 𝑉∗ denotes the approved volume of the entire structure. Furthermore, the binary design variable 𝑥𝑖 indicates whether an element is present (1) or absent (0) in the design. The total volume of the design domain is denoted 614 as 𝑉0, and 𝑉𝑓 stands for the volume fraction ratio, indicating the proportion of volume occupied in the optimized design. It is important to mention that Eq(3.d) represents the constraint related to fatigue safety. It ensures that the accumulated fatigue damage does not exceed allowable limits. Otherwise, the alternating stress would be negative, leading to undesirable outcomes. Moreover, it is crucial to avoid stresses that are high enough to induce plastic deformation, as this renders the high-cycle fatigue model ineffective. It should be mentioned that the BESO method achieves the optimal design for the structure by iteratively adjusting the number of elements while considering their sensitivity. 3. Numerical examples In this section, the topology optimization problem with constraints associated with fatigue and volume is addressed by utilizing the developed BESO algorithm. As previously mentioned, the proposed work is an extension of our prior work (Movahedi Rad et al., 2021) to propose a structural optimization algorithm that specifically incorporates fatigue failure as a critical constraint. It should be noted that in the figures of the resulting optimized shapes, the black color represents the solid elements of the model. Therefore, the optimized shapes exhibit a reduction in the proportion of solid (black) elements according to the desired volume fraction values. 3.1 L-shape problem The paper's first numerical example takes into account the L-shaped beam optimization problem. The example's geometry, boundary, and loading conditions are shown in Figure 1. The considered material properties are density of 2,800 kg/m3, Young's modulus of 70 GPa, and Poisson's ratio of 0.3. 𝜎𝑓 ′ and 𝑏 are assumed 580 MPa and −0.262. Furthermore, the values of 𝐸𝑅, 𝐴𝑅𝑚𝑎𝑥 , 𝑟𝑚𝑖𝑛 , and 𝜏 for BESO parameters are 1 %, 1 %, 18 mm and 1 %. 𝑉𝑓 is set to 40 %. In the case of linear and geometrically nonlinear optimization, the applied load is represented by 𝐹 = 12 kN . However, for fatigue constraint optimization, the value of 𝐹 is treated as a displacement with a magnitude of 2 mm. Figure 1: Considered L-shaped example A comprehensive comparison of resulting topological shapes is provided, considering different models: linear, geometrically nonlinear, and the proposed fatigue constraint optimization model proposed in the manuscript. The findings presented in Figures from 2 to 4, indicate that when fatigue constraints are taken into consideration, the resulting optimized shape diverges from the shapes obtained through linear and geometrically nonlinear optimization approaches. (a) (b) (c) Figure 2: Topological results in the case of linear designs (a) 𝑉𝑓 = 100 % (b) 𝑉𝑓 = 65 % (c) 𝑉𝑓 = 40 % 615 (a) (b) (c) Figure 3: Topological results of geometrically nonlinear designs (a) 𝑉𝑓 = 100 % (b) 𝑉𝑓 = 65 % (c) 𝑉𝑓 = 40 % (a) (b) (c) Figure 4: Topological results of fatigue constrained designs (a) 𝑉𝑓 = 100 % (b) 𝑉𝑓 = 65 % (c) 𝑉𝑓 = 40 % Table 1 represents another comparison according to the maximal Huber-Mises-Hencky stress (𝜎𝐻𝑀𝐻 𝑚𝑎𝑥 ) values for each design case. It can be noticed that in the fatigue constraint topology optimization the value of 𝜎𝐻𝑀𝐻 𝑚𝑎𝑥 decreased dramatically from the obtained values in the cases of linear and geometrically nonlinear designs. Table 1: Resulted 𝜎𝐻𝑀𝐻 𝑚𝑎𝑥 for each design Model 𝜎𝐻𝑀𝐻 𝑚𝑎𝑥 (𝑀𝑃𝑎) Linear design 314 Geometrically nonlinear design 310 Fatigue constraint design 12.24 3.2 U-plate problem The paper's second numerical example takes into account the U-shaped plate optimization problem. The example's loading conditions, geometry, and boundaries are shown in Figure 5. The material properties are density of 7,800 kg/m3, Young's modulus of 160 GPa, and Poisson's ratio of 0.3. 𝜎𝑓 ′ and 𝑏 are assumed 749 MPa and −0.090 respectively. Furthermore, the values of 𝐸𝑅, 𝐴𝑅𝑚𝑎𝑥 , 𝑟𝑚𝑖𝑛 , and 𝜏 for BESO parameters are 1 %, 1 %, 6 mm and 1 %, respectively. 𝑉𝑓 is set to 50 %. The applied load in the cases of linear and geometrically nonlinear optimization is considered to be 𝐹 = 1 kN while in the case of fatigue constraint optimization, 𝐹 is considered as displacement with magnitude of 1 mm. Figure 5: considered U plate example A comprehensive comparison of the resulting topological morphologies is presented, taking into account various models, including linear, geometrically nonlinear, and the fatigue constraint optimization model. Figures 6 616 through 8 demonstrate that the incorporation of fatigue constraints has a significant impact on the optimal design, resulting in a geometry that is specifically tailored to enhance fatigue resistance and durability. (a) (b) (c) Figure 6: Topological results in the case of linear designs (a) 𝑉𝑓 = 100 % (b) 𝑉𝑓 = 65 % (c) 𝑉𝑓 = 40 % (a) (b) (c) Figure 7: Topological results of geometrically nonlinear designs (a) 𝑉𝑓 = 100 % (b) 𝑉𝑓 = 65 % (c) 𝑉𝑓 = 40 % (a) (b) (c) Figure 8: Topological results of fatigue constrained designs(a) 𝑉𝑓 = 100 % (b) 𝑉𝑓 = 65 % (c) 𝑉𝑓 = 40 % 𝜎𝐻𝑀𝐻 𝑚𝑎𝑥 values for each design case are compared in Table 2. Similar to the findings of the previous example, during fatigue constraint topology optimization, the value of 𝜎𝐻𝑀𝐻 𝑚𝑎𝑥 decreased significantly in comparison to the values obtained for linear and geometrically nonlinear designs. It suggests the fatigue performance of the optimized structure is enhanced. Table 2: Resulted 𝜎𝐻𝑀𝐻 𝑚𝑎𝑥 for each design Model 𝜎𝐻𝑀𝐻 𝑚𝑎𝑥 (𝑀𝑃𝑎) Linear design 289 Geometrically nonlinear design 287 Fatigue constraint design 96 4. Conclusions This study explored the integration of topology optimization techniques with fatigue analysis to develop reliable designs for structural components which minimize fatigue damage by integrating the developed BESO method. Numerical examples show that our technique successfully obtains lightweight structural designs which fulfil fatigue and volume constraints. The results of our investigation have provided compelling evidence for the significance of incorporating fatigue constraints into the topology optimization process. When fatigue considerations were integrated, the resulting optimized shapes exhibited a remarkable departure from those achieved through linear and geometrically nonlinear optimization approaches which was done by (Movahedi 617 Rad et al., 2021). Furthermore, one of the most significant observations was the dramatic decrease in the maximal Huber-Mises-Hencky stress (𝜎𝐻𝑀𝐻 𝑚𝑎𝑥 ) values for designs subjected to fatigue constraint optimization compared to their linear and geometrically nonlinear counterparts. This quantitative finding highlights the improved fatigue performance and durability of the optimized structures. The findings of this study contribute to the advancement of reliable design principles by incorporating fatigue failure as a critical constraint. Engineers can develop lightweight and durable structures that effectively utilize materials while minimizing resource utilization. Future research could delve into the incorporation of advanced materials, such as composites materials within the topology optimization process to Investigate how these materials can further enhance fatigue resistance, or multi-objective optimization that balance not only fatigue performance but also other critical factors, such as cost, environmental impact, and manufacturability. Nomenclature 𝑁𝑓𝑖 – number of cycles, - 𝜎𝑓 ′ – coefficient of material’s fatigue strength, MPa 𝑏 – exponent of material’s fatigue strength, - 𝑁𝑟 – overall count of reversals throughout the loading process, - 𝑛𝑖 – the count of occurrences of reversals during the cyclic loading, - 𝐶 – mean compliance, Nmm 𝑢 – displacement, mm 𝐾 – stiffness matrix, N/mm 𝑁 – total number of elements, - 𝑉𝑖 – element volume, mm3 𝑉∗– approved volume of the entire structure, mm3 𝑥𝑖 – binary design variable, - 𝑉𝑓 – volume fraction, - 𝑉0 – total volume of design domain, mm3 References Bendsøe M.P., 1989, Optimal shape design as a material distribution problem, Structural Optimization, 1, 193– 202, DOI: 10.1007/BF01650949. Desmorat B., Desmorat R., 2008, Topology optimization in damage governed low cycle fatigue, Comptes Rendus Mécanique, 336, 448–453. Dowling N.E., 2004, Mean Stress Effects in Stress-Life and Strain-Life Fatigue, SAE Technical Papers, DOI: 10.4271/2004-01-2227. 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