Topology Optimization of Lightweight Cantilever Beam Structures under Nonlinear Dynamic Impact Loading

Topology Optimization of Lightweight Cantilever Beam Structures under Nonlinear Dynamic Impact Loading | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Topology Optimization of Lightweight Cantilever Beam Structures under Nonlinear Dynamic Impact Loading Mahmoud Fadhel Idan This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7810029/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This paper investigates the topology optimization of lightweight cantilever beam structures subjected to nonlinear dynamic impact loading. A simplified one-dimensional finite element model (FEM) was developed in MATLAB and coupled with the Solid Isotropic Material with Penalization (SIMP) method to determine the optimal material distribution under transient impulsive forces. The objective of the optimization was to minimize the tip displacement of the beam while satisfying a prescribed volume constraint. Through an explicit time integration scheme, the dynamic response of the structure was evaluated iteratively. A cantilever beam was analyzed using numerical modeling and optimization to minimize tip displacement under a nonlinear dynamic impact load, while maintaining a 50% volume limit. The study utilized MATLAB with a one-dimensional finite element model and the SIMP method. The beam, fixed at one end and subjected to a short-term impact load at the free end, was divided into 15 elements. The governing dynamic equilibrium equation is solved using explicit time integration, and the squared tip displacement is minimized using a penalty factor (P) = 3. Design variables were iteratively adjusted based on finite-difference sensitivities to enhance performance during impact. The beam, modeled as a 1.0 m long rectangular cross-section bar, assumed linear stiffness with Young's modulus (E) = 210 GPa and density (ρ) = 7800 kg / m³. The results demonstrate that the optimized topologies concentrate material near high-stress regions, significantly reducing peak displacements after impact. Although this framework provides an efficient proof of concept, future enhancements are needed to capture more realistic behavior, including geometric nonlinearities, contact, and inelastic material responses. This work lays the groundwork for designing crashworthy, lightweight structures in automotive, aerospace, and defense applications. Mechanical Engineering Topology optimization Lightweight structures Nonlinear impact SIMP method MATLAB finite element Structural dynamics Crash worthiness Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 1. Introduction In recent years, the demand for lightweight, yet high-performance structures has increased in engineering fields such as aviation, cars, and defense [ 1 – 8 ]. These industries often require components that can efficiently absorb energy while keeping structural integrity under energetic and impact loading conditions [ 9 – 11 ]. Consequently, optimizing material distribution within a given design space to achieve the best structural performance under such extreme conditions has become a central research topic [ 12 ]. Topology optimization (TO) have emerged as a powerful design methodology to determine the ideal material layout within a predefined space, Subject to boundary conditions and specified loads [ 13 – 20 ]. Among the various approaches, the SIMP method remains one of the most widely adopted techniques due to its simplicity, flexibility, and ease of integration with FEM [ 21 – 25 ]. While TO has been widely applied to passive and direct problems, extending it to nonlinear dynamic impact scenarios poses additional challenges, including high strain rates, geometric nonlinearity, and potential material failure [ 26 – 28 ]. This research focuses on the TO of lightweight structures subjected to nonlinear impact loading. A beam is used as a representative model to investigate how material can be efficiently redistributed to reduce the dynamic response resulting from an impulse force. The study utilizes a simplified 1D finite element model implemented in MATLAB and incorporates the SIMP strategy to iteratively improve the design topology while adhering to a volume limitation [ 29 – 30 ]. The main objective is to investigate how TO can be utilized to enhance impact resistance while ensuring material efficiency. The paper further provides a discussion on limitations and outlines future extensions toward more realistic and complex models. This work aims to enhance the development of efficient, crashworthy structural designs that effectively balance performance and weight. 2. Literature Review TO is highly regarded in structural design for developing weight-efficient structures that meet functional constraints [ 31 – 38 ]. Among the most studied benchmark problems in TO is the beam, which provides a simplified yet insightful test case for evaluating optimization algorithms [ 39 – 40 ]. However, the majority of studies traditionally focused on linear, static loading scenarios [ 41 – 48 ]. Addressing nonlinear impact conditions introduces additional complexities that recent literature has begun to explore. The SIMP strategy has been broadly used for optimizing beams under inactive loads [ 50 – 58 ]. While effective in linear settings, it faces challenges under dynamic and nonlinear impact [ 59 ]. Evolutionary Structural Optimization (ESO) and Bi-directional Evolutionary Structural Optimization (BESO) offer alternative frameworks but are less stable for transient dynamics [ 60 – 64 ]. Later studies have introduced hybrid strategies, such as Enhanced Selective Laser Melting and Hybrid Cellular Automata (ESLM + HCA), and advanced contact modeling techniques, including mortar and phase-field damage methods, to amplify the scope of TO to include nonlinear collision systems [ 64 – 67 ]. In lightweight design, TO methods optimizing stiffness and energy absorption are explored, especially with multi-material or robust frameworks. These approaches are progressively applied in safety-critical spaces, such as cars and aviation [ 67 – 70 ]. 2.1 Classical Topology Optimization for Cantilever Beams The SIMP strategy has been widely adopted for optimizing Cantilever Beams under inactive loads [ 71 – 75 ]. In this strategy, the plan space is discredited by using finite elements, and each element is assigned a pseudo-density variable ranging from 0 (void) to 1 (solid). A penalization factor is applied to suppress intermediate densities and promote clear solid-void boundaries. While SIMP performs well in linear settings, it struggles to find solutions for transient or high-rate loading conditions. Developmental Structural Optimization and Bi-directional ESO [ 76 ] provide alternative frameworks where inefficient elements are systematically removed or added based on stress criteria. Although ESO-based methods offer intuitive implementation, they are sensitive to mesh quality and less efficient for energetic problems. 2.2 Nonlinear Impact Problems in Beam Design When a Cantilever Beam is subjected to high-speed impact — such as a falling mass or projectile — the structural response becomes highly nonlinear. These include large deformations, contact effects, and material nonlinearity. Traditional TO frameworks must be extended to account for time-dependent behavior and strain-rate sensitivity. Dynamic finite element formulations, such as explicit time integration or new mark-beta schemes, are used to simulate these problems. A few studies have investigated such scenarios. Henckel et al. [ 77 ] proposed a hybrid methodology that combines the Equivalent Static Load Method with Hybrid Cellular Automata to optimize thin-walled structures, which can also be adapted for beam configurations. Their method introduces a pseudo-static representation of dynamic loads, thereby simplifying optimization while preserving the key characteristics that impact the dynamic loads. 2.3 Contact Modeling and Material Failure To realistically model contact between the impacting body and the Cantilever Beam, numerical techniques such as Mortar contact methods or Third Medium Contact have been incorporated into frameworks [ 78 ]. These methods allow accurate force transmission and stress wave propagation during impact, which are critical for capturing localized damage or buckling. Moreover, to account for material failure under repeated or extreme impact loading, phase-field damage models or elastic-plastic constitutive laws are embedded in the simulation [ 79 ]. These methods enable TO algorithms to eliminate inefficient material and identify regions prone to failure. 2.4 Lightweight Beam Design and Energy Absorption In lightweight design, the trade-off between stiffness and energy absorption is crucial, especially in safety-critical applications such as crash protection. Optimization studies for Cantilever Beams indicate that including dynamic energy absorption as an objective, alongside compliance or weight, results in more robust structures [ 80 – 81 ]. These designs incorporate geometric features such as holes, fillets, and curvature to effectively dissipate energy during impacts. Probabilistic or robust optimization methods have been connected to Cantilever Beams under uncertain impact conditions, considering variations in impact angle, mass, and velocity [ 82 – 84 ]. These methods improve reliability and reduce the risk of structural failure under unforeseen scenarios. 3. Methodology This section describes the numerical modeling and optimization approach for a Cantilever Beam under nonlinear dynamic impact loading. The aim is to find an ideal material distribution that minimizes tip displacement during an impulsive event while adhering to a 50% volume imperative. The method is implemented in MATLAB using a 1D finite element model with the SIMP technique. The Cantilever Beam is divided into 15 elements and settled at one end, experiencing a short-duration impact load at the free tip. Explicit time integration solves the governing dynamic equilibrium equation, while the optimization objective aims to minimize the squared tip displacement using the SIMP law with a penalization factor of p = 3. Design variables are adjusted iteratively based on finite difference sensitivities and an optimality criteria scheme for improved performance under impact. 3.1 Problem Description A cantilever beam of length L = 1.0 m and rectangular cross-section is modeled as a one-dimensional bar, divided into n = 15 equal-length finite elements. The Cantilever Beam is clamped at the left end and subjected to an external impulsive force at the free right end to simulate an impact event. The material is considered to be linearly elastic, with a Young's modulus of E = 210 GPa, and a density of ρ = 7800 kg / m³. 3.2 Governing Equations The transient behavior of the structure is described by a second-order differential equation [ 85 – 88 ]. $$\:F\left(t\right)=\:K\:u\left(t\right)+\:M\:\ddot{u}\left(t\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(1\right)$$ Where: M is the lumped mass matrix, K is the global stiffness, \(\:u\) (t) is the displacement vector, F (t) is the external force representing impact, and \(\:\ddot{u}\) (t) is the acceleration vector. Time integration is performed using an explicit central difference scheme with a small time step to ensure stability. 3.3 SIMP-Based Topology Optimization Formulation The design domain is represented by a vector of relative densities, with each element corresponding to a specific part of the domain [ 88 – 94 ]: ρ = [ρ₁, ρ₂... \(\:{\rho\:}_{n}\) ], where each ρ i ∈ [0.001, 1] The stiffness of each element is interpolated using the SIMP model as: E i (ρ i ) = \(\:{\rho\:}_{i}^{p}\:\) E 0 Where p ≥ 3 is the penalty factor to encourage binary (0–1) designs, and the optimization problem is formulated as follows: Minimize over ρ: C (ρ) = \(\:{u}_{tip}^{2}\) Subject to: (1/n) ∑ i ρ i ≤ \(\:{V}_{f}\) and 0.001 ≤ ρ i ≤ 1 Where \(\:{u}_{tip}\:\) represents the final displacement of the free end, and \(\:{V}_{f}\) denotes the specified volume fraction (e.g., 50%). 3.4 Optimization Strategy The optimization processor is executed iteratively using the optimization criteria method. At each concentration, the dynamic response is simulated using the current material distribution, the objective work C is evaluated based on the maximum tip displacement, the sensitivities \(\:\frac{\partial\:c}{\partial\:{\rho\:}_{i}}\) are estimated using finite difference, and the design plan variables are updated to reduce the target size requirements. Filtering and regularization techniques (e.g., sensitivity filtering) are omitted in this preliminary study but can be integrated to avoid numerical instabilities. 3.5 Implementation Details 1. Software: MATLAB R2013a, 2. Mesh: One-dimensional linear element, 3. Time integration: Explicit central difference, 4. Iterations: 10 optimization loops, 5. Impact force: Short-duration impulse (1 m/s), and 6. Performance metric: Final displacement of the cantilever beam tip. 3.6 Assumptions and Limitations The material is modeled as linear elastic; plasticity and failure strength are not included, and the structure is assumed to deform only along the longitudinal axis (a one-dimensional simplification). Contact and geometric nonlinearities (such as buckling) are not considered in this version. These simplifications enable a clear and quick prototype, which can later be extended to 2D/3D geometries and more complex material behaviors. 3.7 Prepare an initial MATLAB code for the simulation a- Code Function : The code was written and implemented in MATLAB to simulate cantilever collisions, known as "cantilever collision simulation." This code simulates a vertical cantilever beam subjected to a sudden impact from a mass falling from above. The nonlinear effect represents large motion over time, while the model remains linear in the material. It has been extended to include plastic or damaged materials, using the FEM, incorporating nonlinear effects: large deformation, a linear elastic material as a starting point, and a time solution using explicit time integration. b- Code development to include topology optimization (SIMP) To improve the previous code to include a simple TO (SIMP) algorithm on a vertical Cantilever Beam, while maintaining the following: SIMP model: each element has a density variable (ρ i ∈ [0.001, 1]), the distribution is optimized by minimizing the free end displacement (or energy) under impact, and an optimization algorithm based on design iteration (based on gradient or update rule). A revised version of the code, which includes TO (SIMP), has been developed and implemented in MATLAB, titled "cantilever impact SIMP.” c- Key features of the code Key features of the code include: Simulation of a vertical Cantilever Beam divided into one-dimensional elements, using the SIMP algorithm to optimize the density distribution, and attempts to minimize tip displacement after dynamic collision (which improves collision resistance). 4. Numerical Results and Discussion This section presents and analyzes numerical results obtained from a linear regression (TO) analysis of a cantilever beam under a nonlinear impact load. The SIMP-based optimization framework, described in Section 3 , was implemented using MATLAB for ten iterations, with a 50% volume constraint. 4.1 Dynamic acceleration, velocity, and displacement response distribution, and Topology Distribution before development Figure 1 shows the acceleration versus distance from the fixed edge of a Cantilever Beam under the impact force before optimization. The Cantilever Beam exhibits irregular oscillations due to the elastic rebound of the structure. These oscillations reflect the dynamic elasticity of the Cantilever Beam structure, with peak values ranging from approximately (-2000 m/s² to 1200 m/s²). Figure 2 shows the displacement compared to the distance from the fixed edge of the Cantilever Beam when subjected to an impact force before optimization. The Cantilever Beam exhibits irregular displacement due to elastic rebound, with peak values reaching approximately 2.2 x 10 − 6 m at a distance of 0.86 m from the fixed edge, which gradually decreases beyond that point. Figure 3 shows the velocity against the distance from the fixed edge of the Cantilever Beam under the action of an impact force before optimization. The Cantilever Beam exhibits irregular velocity due to elastic rebound, with peak values reaching approximately 0.074 m/s at a distance of 0.66 m from the fixed edge, which gradually decreases beyond that point. Figure 4 illustrates the response of the Cantilever Beam tip to the temporal displacement upon initial impact. The tip displays a sharp peak displacement of 8.7 x 10 − 6 m shortly after the effects begin (approximately 1 m s). Following this, oscillations occur due to the elastic recoil of the structure. The amplitude of these oscillations reflects the dynamic elasticity of the Cantilever Beam's topology, reaching a maximum of about 6.2 x 10 − 6 m. Figure 5 illustrates the predicted density distribution before optimization. The figure shows that the material concentration increases from the fixed edge until it reaches 1, the maximum concentration along the Cantilever Beam. This indicates a fixed edge weakness, where stresses are concentrated on the Cantilever Beam is subjected to loads or shocks. 4.2 Dynamic acceleration, velocity, and displacement response distribution, and Optimized Topology Distribution Figure 6 illustrates the distribution of acceleration about the distance from the fixed edge of the Cantilever Beam under impact force following optimization. The data reveal that the irregular oscillating acceleration completely disappears up to a distance of 0.86 m from the fixed edge. Beyond this point, the acceleration experiences a sharp increase, reaching 2×10⁻⁹ m / s² near the free edge, before gradually decreasing again. As optimization progresses, a noticeable reduction in peak displacement are observed, indicating enhanced stiffness and improved energy dissipation. Figure 7 shows the displacement relative to the distance from the fixed edge of the Cantilever Beam under the impact force after optimization. The data show that the irregular displacement disappears up to a distance of 0.93 m from the fixed edge. After this point, the displacement increases sharply, reaching 3.8 × 10 ⁻6 m near the free edge. As the optimization progresses, a significant decrease in the peak displacement is observed, indicating improved stiffness and energy dissipation Figure 8 illustrates the relationship between velocity and distance from the fixed edge of the Cantilever Beam after optimization due to the impact force. The data indicate that irregularities in velocity vanish up to a distance of 0.93 m from the fixed edge. Beyond this point, the velocity increases sharply, reaching (4 × 10⁻⁴ m/s) near the free edge. The optimization process results in a significant reduction in the maximum speed value, indicating increased stiffness and energy dissipation. Figure 9 shows the temporal response of the Cantilever Beam tip upon initial impact after optimization. The tip exhibits a steady increase in displacement shortly after impact onset (about 1 ms), reaching a maximum value of about 3.8 × 10 − 6 m. As the optimization process progresses, a clear decrease in peak displacement is observed, indicating improved stiffness and energy dissipation. Figure 10 illustrates the optimized material distribution along the Cantilever Beam, showing even concentration while prioritizing high-stress areas near the fixed support. This aligns with fixed Cantilever Beam theory and highlights the effectiveness of the SIMP model for dynamic optimization. The resulting design exhibits a trade-off between structural mass and impact resistance. Elements with intermediate densities are minimized due to the penalization scheme (p = 3), promoting a quasi-binary design suitable for manufacturing. 4.3 Discussion and Insights The simulation results demonstrate a sharp initial displacement peak after impact, followed by damped oscillations. As optimization progresses, the peak displacement decreases, indicating an improved stiffness. The optimized topology shows material concentrated near the clamped end, consistent with the stress distribution under impact. Material is removed from low-stress areas, achieving volume efficiency. While simplified, the framework shows clear trends that validate the use of SIMP for dynamic impact optimization. Energy Absorption: Although not explicitly modeled, the reduced tip displacement suggests improved energy absorption through stiffness reallocation. Computational Efficiency: The use of a simplified 1D model allows fast iteration cycles, though extensions to 2D/3D are necessary for realistic geometries. Design Interpretation: The final topology shows the potential for lightweight structural components optimized for transient, high-speed loading. Limitations: Contact, plasticity, and large-deformation effects were neglected. Incorporating them would enhance realism but require significantly more complex modeling and sensitivity analysis. 5. Conclusion and Future Work 5.1 Conclusion This study presents a SIMP-based TO framework for lightweight structures subjected to nonlinear impact loads. This framework effectively minimizes dynamic displacement while considering material constraints. Using a Cantilever Beam as a case study, a one-dimensional finite element model was implemented using MATLAB, allowing for iterative optimization of structural design under dynamic loading conditions. The simulation results demonstrate that: The SIMP-based approach is capable of producing optimized material distributions that reduce dynamic tip displacement. The material tends to concentrate near the fixed support, confirming classical Cantilever Beam behavior under dynamic conditions. The framework successfully enforces a volume constraint while improving impact resistance through the redistribution of stiffness. While the methodology is computationally efficient and suitable for early-stage conceptual design, it is subject to several limitations: The model assumes linear elasticity, excluding plasticity and material damage mechanisms common in real impact scenarios. The simulation is restricted to a one-dimensional domain, which limits its applicability to real-world components. 5.2 Future Work To improve realism and extend the applicability of the proposed framework, the following directions are recommended for future research: Extension to 2D and 3D models: Incorporating plane stress or shell elements to model realistic structural geometries such as plates, brackets, or thin-walled Cantilever Beams. Inclusion of material nonlinearity: Implementing elasto-plastic or Viscoelasticity material models to more effectively measure energy absorption and permanent deformation during impact. Modeling of contact and fracture: Integrating contact algorithms (e.g., Mortar methods) and damage models (e.g., phase-field) for improved prediction of failure. Adjoint-based sensitivity analysis: Replacing finite-difference sensitivity estimation with efficient adjoint formulations to enable high-resolution designs with reduced computational cost. Multi-objective and robust optimization: Considering multiple performance criteria such as energy absorption, natural frequency, or robustness under uncertain impact conditions. By addressing these aspects, the proposed methodology can evolve into a powerful tool for the design of crashworthy, lightweight components in automotive, aerospace, and defense industries. Abbreviations Declarations Ethics approval and consent to participate: Springer Nature maintained a neutral position with regard to jurisdictional claims in published maps and institutional affiliations and no official or unofficial entity objected to the publication. Consent for publication : Author has read and approved the manuscript for submission for publication. Availability of data and material: All data generated or analyzed during this study are included in this published article. Competing interests: The author declare that they have no competing interests. Funding: This research received no external funding. Author contribution: The author M.F. completed the following tasks: concept development, methodology design, software development, information and data validation, scientific analysis of the results, result generation, resource provision, writing and original draft preparation, review and editing, supervision, and project management. Acknowledgements: The researcher extends its gratitude to the Deanship of the College and the professors of the Department of Engineering for the great support and encouragement they provided to complete the research. References Zhang, L., Zhang, Y., & Van Keulen, F. (2023). Topology optimization of geometrically nonlinear structures using reduced-order modeling. Computer Methods in Applied Mechanics and Engineering , 416 , 116371.‏ Liu, Y., Oda, Y., & Sasahara, K. (2024). Shape and topology optimization method with generalized topological derivatives. 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Topology optimization for thermal structures considering design-dependent convection boundaries based on the bidirectional evolutionary structural optimization method. Mechanical Sciences , 14 (1), 223-235.‏ Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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04:30:49","extension":"png","order_by":22,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":5845,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage9.png","url":"https://assets-eu.researchsquare.com/files/rs-7810029/v1/727ac90b40c63c53673e1911.png"},{"id":93191246,"identity":"e2878c6f-43e4-4dc8-902c-adb9bc9a3c28","added_by":"auto","created_at":"2025-10-10 04:14:50","extension":"xml","order_by":23,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":165230,"visible":true,"origin":"","legend":"","description":"","filename":"rs78100290structuring.xml","url":"https://assets-eu.researchsquare.com/files/rs-7810029/v1/f8476e61868a9daf72e05fac.xml"},{"id":93191248,"identity":"e385c199-94af-46f4-bf84-3429481ab4ef","added_by":"auto","created_at":"2025-10-10 04:14:50","extension":"html","order_by":24,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":177478,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-7810029/v1/884952245c3a94fa41837bd3.html"},{"id":93191215,"identity":"7a72fc6b-b643-4586-a659-80ba9d5818e8","added_by":"auto","created_at":"2025-10-10 04:14:49","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":38383,"visible":true,"origin":"","legend":"\u003cp\u003eshows the acceleration against the distance from the fixed edge of the Cantilever Beam under the action of an impact force before optimization.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-7810029/v1/f84c9422f3cbbd30d4520295.png"},{"id":93191942,"identity":"ab99fc69-1ad1-48fb-adcb-c8eb73160c61","added_by":"auto","created_at":"2025-10-10 04:30:49","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":39455,"visible":true,"origin":"","legend":"\u003cp\u003eshows the displacementagainst the distance from the fixed edge of the Cantilever Beam under the action of an impact force before optimization.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-7810029/v1/6da8533edcda0d516a37e2c0.png"},{"id":93191216,"identity":"6f44d64d-9587-4df1-9452-26b6f0b28faa","added_by":"auto","created_at":"2025-10-10 04:14:49","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":41357,"visible":true,"origin":"","legend":"\u003cp\u003eshows the velocity against the distance from the fixed edge of the Cantilever Beam under the action of an impact force before optimization.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-7810029/v1/190804e57cf78f332c27dace.png"},{"id":93191501,"identity":"db4b0d29-b8f0-47ba-8e7b-fe5bc72da237","added_by":"auto","created_at":"2025-10-10 04:22:49","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":41737,"visible":true,"origin":"","legend":"\u003cp\u003eshows the displacement-time response of the Cantilever Beam tip under the initial impact before optimization.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-7810029/v1/387649d1c1f8d39e7a2309bb.png"},{"id":93191220,"identity":"ee2ad878-528d-4562-8500-7ac382ccd5c7","added_by":"auto","created_at":"2025-10-10 04:14:49","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":29099,"visible":true,"origin":"","legend":"\u003cp\u003eshows the predicted density distribution plot before optimization.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-7810029/v1/621488d8b990a64ecc305ab2.png"},{"id":93191503,"identity":"0f097c1f-3414-48e8-b4b2-261a4e25591f","added_by":"auto","created_at":"2025-10-10 04:22:49","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":34967,"visible":true,"origin":"","legend":"\u003cp\u003eshows the acceleration against the distance from the fixed edge of the Cantilever Beam under the action of an impact force after optimization.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-7810029/v1/96b6d291c13630734724ccc2.png"},{"id":93191239,"identity":"b6165a95-0b10-4c67-bd47-1533363c66d9","added_by":"auto","created_at":"2025-10-10 04:14:49","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":26027,"visible":true,"origin":"","legend":"\u003cp\u003eshows the displacementagainst the distance from the fixed edge of the Cantilever Beam under the action of an impact force after optimization.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-7810029/v1/da3f9e162b95c4566855a10f.png"},{"id":93191243,"identity":"7cdbabba-22b5-46e6-a2f0-b34feb8236fc","added_by":"auto","created_at":"2025-10-10 04:14:49","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":27217,"visible":true,"origin":"","legend":"\u003cp\u003eshows the velocity against the distance from the fixed edge of the Cantilever Beam under the action of an impact force after optimization.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-7810029/v1/451bfdd5ced733c117a68ce0.png"},{"id":93191229,"identity":"340c6652-9e41-40c0-9c3e-3e93dfbec66c","added_by":"auto","created_at":"2025-10-10 04:14:49","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":31959,"visible":true,"origin":"","legend":"\u003cp\u003eshows the displacement-time response of the Cantilever Beam tip under initial impact after optimization.\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-7810029/v1/3d05e1b7ffa1e4db12a5eb4b.png"},{"id":93191504,"identity":"1ab3de17-34a3-4581-9b22-9266859cf54e","added_by":"auto","created_at":"2025-10-10 04:22:49","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":34614,"visible":true,"origin":"","legend":"\u003cp\u003eshows the predicted density distribution plot after optimization.\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-7810029/v1/9cc0ac83facc8ac27337e5ce.png"},{"id":93192570,"identity":"39784161-711a-4d64-b066-19d15cbd1341","added_by":"auto","created_at":"2025-10-10 04:46:51","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1234990,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7810029/v1/82fac7a1-e8e2-43a3-ae90-3109e9494bea.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003e\u003cstrong\u003eTopology Optimization of Lightweight Cantilever Beam Structures under Nonlinear Dynamic Impact Loading\u003c/strong\u003e\u003c/p\u003e","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eIn recent years, the demand for lightweight, yet high-performance structures has increased in engineering fields such as aviation, cars, and defense [\u003cspan additionalcitationids=\"CR2 CR3 CR4 CR5 CR6 CR7\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. These industries often require components that can efficiently absorb energy while keeping structural integrity under energetic and impact loading conditions [\u003cspan additionalcitationids=\"CR10\" citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]. Consequently, optimizing material distribution within a given design space to achieve the best structural performance under such extreme conditions has become a central research topic [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]. Topology optimization (TO) have emerged as a powerful design methodology to determine the ideal material layout within a predefined space, Subject to boundary conditions and specified loads [\u003cspan additionalcitationids=\"CR14 CR15 CR16 CR17 CR18 CR19\" citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e]. Among the various approaches, the SIMP method remains one of the most widely adopted techniques due to its simplicity, flexibility, and ease of integration with FEM [\u003cspan additionalcitationids=\"CR22 CR23 CR24\" citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]. While TO has been widely applied to passive and direct problems, extending it to nonlinear dynamic impact scenarios poses additional challenges, including high strain rates, geometric nonlinearity, and potential material failure [\u003cspan additionalcitationids=\"CR27\" citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e]. This research focuses on the TO of lightweight structures subjected to nonlinear impact loading. A beam is used as a representative model to investigate how material can be efficiently redistributed to reduce the dynamic response resulting from an impulse force. The study utilizes a simplified 1D finite element model implemented in MATLAB and incorporates the SIMP strategy to iteratively improve the design topology while adhering to a volume limitation [\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e]. The main objective is to investigate how TO can be utilized to enhance impact resistance while ensuring material efficiency. The paper further provides a discussion on limitations and outlines future extensions toward more realistic and complex models. This work aims to enhance the development of efficient, crashworthy structural designs that effectively balance performance and weight.\u003c/p\u003e"},{"header":"2. Literature Review","content":"\u003cp\u003eTO is highly regarded in structural design for developing weight-efficient structures that meet functional constraints [\u003cspan additionalcitationids=\"CR32 CR33 CR34 CR35 CR36 CR37\" citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e]. Among the most studied benchmark problems in TO is the beam, which provides a simplified yet insightful test case for evaluating optimization algorithms [\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e]. However, the majority of studies traditionally focused on linear, static loading scenarios [\u003cspan additionalcitationids=\"CR42 CR43 CR44 CR45 CR46 CR47\" citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e48\u003c/span\u003e]. Addressing nonlinear impact conditions introduces additional complexities that recent literature has begun to explore. The SIMP strategy has been broadly used for optimizing beams under inactive loads [\u003cspan additionalcitationids=\"CR51 CR52 CR53 CR54 CR55 CR56 CR57\" citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR58\" class=\"CitationRef\"\u003e58\u003c/span\u003e]. While effective in linear settings, it faces challenges under dynamic and nonlinear impact [\u003cspan citationid=\"CR59\" class=\"CitationRef\"\u003e59\u003c/span\u003e]. Evolutionary Structural Optimization (ESO) and Bi-directional Evolutionary Structural Optimization (BESO) offer alternative frameworks but are less stable for transient dynamics [\u003cspan additionalcitationids=\"CR61 CR62 CR63\" citationid=\"CR60\" class=\"CitationRef\"\u003e60\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e64\u003c/span\u003e]. Later studies have introduced hybrid strategies, such as Enhanced Selective Laser Melting and Hybrid Cellular Automata (ESLM\u0026thinsp;+\u0026thinsp;HCA), and advanced contact modeling techniques, including mortar and phase-field damage methods, to amplify the scope of TO to include nonlinear collision systems [\u003cspan additionalcitationids=\"CR65 CR66\" citationid=\"CR64\" class=\"CitationRef\"\u003e64\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR67\" class=\"CitationRef\"\u003e67\u003c/span\u003e]. In lightweight design, TO methods optimizing stiffness and energy absorption are explored, especially with multi-material or robust frameworks. These approaches are progressively applied in safety-critical spaces, such as cars and aviation [\u003cspan additionalcitationids=\"CR68 CR69\" citationid=\"CR67\" class=\"CitationRef\"\u003e67\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR70\" class=\"CitationRef\"\u003e70\u003c/span\u003e].\u003c/p\u003e\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\u003ch2\u003e2.1 Classical Topology Optimization for Cantilever Beams\u003c/h2\u003e\u003cp\u003eThe SIMP strategy has been widely adopted for optimizing Cantilever Beams under inactive loads [\u003cspan additionalcitationids=\"CR72 CR73 CR74\" citationid=\"CR71\" class=\"CitationRef\"\u003e71\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR75\" class=\"CitationRef\"\u003e75\u003c/span\u003e]. In this strategy, the plan space is discredited by using finite elements, and each element is assigned a pseudo-density variable ranging from 0 (void) to 1 (solid). A penalization factor is applied to suppress intermediate densities and promote clear solid-void boundaries. While SIMP performs well in linear settings, it struggles to find solutions for transient or high-rate loading conditions. Developmental Structural Optimization and Bi-directional ESO [\u003cspan citationid=\"CR76\" class=\"CitationRef\"\u003e76\u003c/span\u003e] provide alternative frameworks where inefficient elements are systematically removed or added based on stress criteria. Although ESO-based methods offer intuitive implementation, they are sensitive to mesh quality and less efficient for energetic problems.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\u003ch2\u003e2.2 Nonlinear Impact Problems in Beam Design\u003c/h2\u003e\u003cp\u003eWhen a Cantilever Beam is subjected to high-speed impact \u0026mdash; such as a falling mass or projectile \u0026mdash; the structural response becomes highly nonlinear. These include large deformations, contact effects, and material nonlinearity. Traditional TO frameworks must be extended to account for time-dependent behavior and strain-rate sensitivity. Dynamic finite element formulations, such as explicit time integration or new mark-beta schemes, are used to simulate these problems. A few studies have investigated such scenarios. Henckel et al. [\u003cspan citationid=\"CR77\" class=\"CitationRef\"\u003e77\u003c/span\u003e] proposed a hybrid methodology that combines the Equivalent Static Load Method with Hybrid Cellular Automata to optimize thin-walled structures, which can also be adapted for beam configurations. Their method introduces a pseudo-static representation of dynamic loads, thereby simplifying optimization while preserving the key characteristics that impact the dynamic loads.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\u003ch2\u003e2.3 Contact Modeling and Material Failure\u003c/h2\u003e\u003cp\u003eTo realistically model contact between the impacting body and the Cantilever Beam, numerical techniques such as Mortar contact methods or Third Medium Contact have been incorporated into frameworks [\u003cspan citationid=\"CR78\" class=\"CitationRef\"\u003e78\u003c/span\u003e]. These methods allow accurate force transmission and stress wave propagation during impact, which are critical for capturing localized damage or buckling. Moreover, to account for material failure under repeated or extreme impact loading, phase-field damage models or elastic-plastic constitutive laws are embedded in the simulation [\u003cspan citationid=\"CR79\" class=\"CitationRef\"\u003e79\u003c/span\u003e]. These methods enable TO algorithms to eliminate inefficient material and identify regions prone to failure.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\u003ch2\u003e2.4 Lightweight Beam Design and Energy Absorption\u003c/h2\u003e\u003cp\u003eIn lightweight design, the trade-off between stiffness and energy absorption is crucial, especially in safety-critical applications such as crash protection. Optimization studies for Cantilever Beams indicate that including dynamic energy absorption as an objective, alongside compliance or weight, results in more robust structures [\u003cspan citationid=\"CR80\" class=\"CitationRef\"\u003e80\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR81\" class=\"CitationRef\"\u003e81\u003c/span\u003e]. These designs incorporate geometric features such as holes, fillets, and curvature to effectively dissipate energy during impacts. Probabilistic or robust optimization methods have been connected to Cantilever Beams under uncertain impact conditions, considering variations in impact angle, mass, and velocity [\u003cspan additionalcitationids=\"CR83\" citationid=\"CR82\" class=\"CitationRef\"\u003e82\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR84\" class=\"CitationRef\"\u003e84\u003c/span\u003e]. These methods improve reliability and reduce the risk of structural failure under unforeseen scenarios.\u003c/p\u003e\u003c/div\u003e"},{"header":"3. Methodology","content":"\u003cp\u003eThis section describes the numerical modeling and optimization approach for a Cantilever Beam under nonlinear dynamic impact loading. The aim is to find an ideal material distribution that minimizes tip displacement during an impulsive event while adhering to a 50% volume imperative. The method is implemented in MATLAB using a 1D finite element model with the SIMP technique. The Cantilever Beam is divided into 15 elements and settled at one end, experiencing a short-duration impact load at the free tip. Explicit time integration solves the governing dynamic equilibrium equation, while the optimization objective aims to minimize the squared tip displacement using the SIMP law with a penalization factor of p\u0026thinsp;=\u0026thinsp;3. Design variables are adjusted iteratively based on finite difference sensitivities and an optimality criteria scheme for improved performance under impact.\u003c/p\u003e\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\u003ch2\u003e3.1 Problem Description\u003c/h2\u003e\u003cp\u003eA cantilever beam of length L\u0026thinsp;=\u0026thinsp;1.0 m and rectangular cross-section is modeled as a one-dimensional bar, divided into n\u0026thinsp;=\u0026thinsp;15 equal-length finite elements. The Cantilever Beam is clamped at the left end and subjected to an external impulsive force at the free right end to simulate an impact event. The material is considered to be linearly elastic, with a Young's modulus of E\u0026thinsp;=\u0026thinsp;210 GPa, and a density of ρ\u0026thinsp;=\u0026thinsp;7800 kg / m\u0026sup3;.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e\u003ch2\u003e3.2 Governing Equations\u003c/h2\u003e\u003cp\u003eThe transient behavior of the structure is described by a second-order differential equation [\u003cspan additionalcitationids=\"CR86 CR87\" citationid=\"CR85\" class=\"CitationRef\"\u003e85\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR88\" class=\"CitationRef\"\u003e88\u003c/span\u003e].\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\:F\\left(t\\right)=\\:K\\:u\\left(t\\right)+\\:M\\:\\ddot{u}\\left(t\\right)\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left(1\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eWhere: M is the lumped mass matrix, K is the global stiffness, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:u\\)\u003c/span\u003e\u003c/span\u003e (t) is the displacement vector, F (t) is the external force representing impact, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\ddot{u}\\)\u003c/span\u003e\u003c/span\u003e (t) is the acceleration vector.\u003c/p\u003e\u003cp\u003eTime integration is performed using an explicit central difference scheme with a small time step to ensure stability.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e\u003ch2\u003e3.3 SIMP-Based Topology Optimization Formulation\u003c/h2\u003e\u003cp\u003eThe design domain is represented by a vector of relative densities, with each element corresponding to a specific part of the domain [\u003cspan additionalcitationids=\"CR89 CR90 CR91 CR92 CR93\" citationid=\"CR88\" class=\"CitationRef\"\u003e88\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR94\" class=\"CitationRef\"\u003e94\u003c/span\u003e]:\u003c/p\u003e\u003cp\u003eρ = [ρ₁, ρ₂...\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\rho\\:}_{n}\\)\u003c/span\u003e\u003c/span\u003e], where each ρ\u003csub\u003ei\u003c/sub\u003e \u0026isin; [0.001, 1]\u003c/p\u003e\u003cp\u003eThe stiffness of each element is interpolated using the SIMP model as:\u003c/p\u003e\u003cp\u003eE\u003csub\u003ei\u003c/sub\u003e (ρ\u003csub\u003ei\u003c/sub\u003e) = \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\rho\\:}_{i}^{p}\\:\\)\u003c/span\u003e\u003c/span\u003eE\u003csub\u003e0\u003c/sub\u003e\u003c/p\u003e\u003cp\u003eWhere p\u0026thinsp;\u0026ge;\u0026thinsp;3 is the penalty factor to encourage binary (0\u0026ndash;1) designs, and the optimization problem is formulated as follows:\u003c/p\u003e\u003cp\u003eMinimize over ρ: \u003cem\u003eC\u003c/em\u003e (ρ) = \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{u}_{tip}^{2}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\u003cp\u003eSubject to:\u003c/p\u003e\u003cp\u003e(1/n) \u0026sum;\u003csub\u003ei\u003c/sub\u003e ρ\u003csub\u003ei\u003c/sub\u003e \u0026le;\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{V}_{f}\\)\u003c/span\u003e\u003c/span\u003e and 0.001\u0026thinsp;\u0026le;\u0026thinsp;ρ\u003csub\u003ei\u003c/sub\u003e \u0026le; 1\u003c/p\u003e\u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{u}_{tip}\\:\\)\u003c/span\u003e\u003c/span\u003erepresents the final displacement of the free end, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{V}_{f}\\)\u003c/span\u003e\u003c/span\u003e denotes the specified volume fraction (e.g., 50%).\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e\u003ch2\u003e3.4 Optimization Strategy\u003c/h2\u003e\u003cp\u003eThe optimization processor is executed iteratively using the optimization criteria method. At each concentration, the dynamic response is simulated using the current material distribution, the objective work C is evaluated based on the maximum tip displacement, the sensitivities \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\frac{\\partial\\:c}{\\partial\\:{\\rho\\:}_{i}}\\)\u003c/span\u003e\u003c/span\u003e are estimated using finite difference, and the design plan variables are updated to reduce the target size requirements.\u003c/p\u003e\u003cp\u003eFiltering and regularization techniques (e.g., sensitivity filtering) are omitted in this preliminary study but can be integrated to avoid numerical instabilities.\u003c/p\u003e\u003cp\u003e\u003cb\u003e3.5 Implementation Details\u003c/b\u003e\u003c/p\u003e\u003cp\u003e1. Software: MATLAB R2013a, 2. Mesh: One-dimensional linear element, 3. Time integration: Explicit central difference, 4. Iterations: 10 optimization loops, 5. Impact force: Short-duration impulse (1 m/s), and 6. Performance metric: Final displacement of the cantilever beam tip.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec12\" class=\"Section2\"\u003e\u003ch2\u003e3.6 Assumptions and Limitations\u003c/h2\u003e\u003cp\u003eThe material is modeled as linear elastic; plasticity and failure strength are not included, and the structure is assumed to deform only along the longitudinal axis (a one-dimensional simplification). Contact and geometric nonlinearities (such as buckling) are not considered in this version. These simplifications enable a clear and quick prototype, which can later be extended to 2D/3D geometries and more complex material behaviors.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec13\" class=\"Section2\"\u003e\u003ch2\u003e3.7 Prepare an initial MATLAB code for the simulation\u003c/h2\u003e\u003cp\u003e\u003cb\u003ea- Code Function\u003c/b\u003e:\u003c/p\u003e\u003cp\u003eThe code was written and implemented in MATLAB to simulate cantilever collisions, known as \"cantilever collision simulation.\" This code simulates a vertical cantilever beam subjected to a sudden impact from a mass falling from above. The nonlinear effect represents large motion over time, while the model remains linear in the material. It has been extended to include plastic or damaged materials, using the FEM, incorporating nonlinear effects: large deformation, a linear elastic material as a starting point, and a time solution using explicit time integration.\u003c/p\u003e\u003cp\u003e\u003cb\u003eb- Code development to include topology optimization (SIMP)\u003c/b\u003e\u003c/p\u003e\u003cp\u003eTo improve the previous code to include a simple TO (SIMP) algorithm on a vertical Cantilever Beam, while maintaining the following: SIMP model: each element has a density variable (ρ\u003csub\u003ei\u003c/sub\u003e \u0026isin; [0.001, 1]), the distribution is optimized by minimizing the free end displacement (or energy) under impact, and an optimization algorithm based on design iteration (based on gradient or update rule). A revised version of the code, which includes TO (SIMP), has been developed and implemented in MATLAB, titled \"cantilever impact SIMP.\u0026rdquo;\u003c/p\u003e\u003cp\u003e\u003cb\u003ec- Key features of the code\u003c/b\u003e\u003c/p\u003e\u003cp\u003eKey features of the code include: Simulation of a vertical Cantilever Beam divided into one-dimensional elements, using the SIMP algorithm to optimize the density distribution, and attempts to minimize tip displacement after dynamic collision (which improves collision resistance).\u003c/p\u003e\u003c/div\u003e"},{"header":"4. Numerical Results and Discussion","content":"\u003cp\u003eThis section presents and analyzes numerical results obtained from a linear regression (TO) analysis of a cantilever beam under a nonlinear impact load. The SIMP-based optimization framework, described in Section \u003cspan refid=\"Sec7\" class=\"InternalRef\"\u003e3\u003c/span\u003e, was implemented using MATLAB for ten iterations, with a 50% volume constraint.\u003c/p\u003e\u003cdiv id=\"Sec15\" class=\"Section2\"\u003e\u003ch2\u003e4.1 Dynamic acceleration, velocity, and displacement response distribution, and Topology Distribution before development\u003c/h2\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e shows the acceleration versus distance from the fixed edge of a Cantilever Beam under the impact force before optimization. The Cantilever Beam exhibits irregular oscillations due to the elastic rebound of the structure. These oscillations reflect the dynamic elasticity of the Cantilever Beam structure, with peak values ranging from approximately (-2000 m/s² to 1200 m/s²).\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e shows the displacement compared to the distance from the fixed edge of the Cantilever Beam when subjected to an impact force before optimization. The Cantilever Beam exhibits irregular displacement due to elastic rebound, with peak values reaching approximately 2.2 x 10\u003csup\u003e− 6\u003c/sup\u003e m at a distance of 0.86 m from the fixed edge, which gradually decreases beyond that point.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e shows the velocity against the distance from the fixed edge of the Cantilever Beam under the action of an impact force before optimization. The Cantilever Beam exhibits irregular velocity due to elastic rebound, with peak values reaching approximately 0.074 m/s at a distance of 0.66 m from the fixed edge, which gradually decreases beyond that point.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e illustrates the response of the Cantilever Beam tip to the temporal displacement upon initial impact. The tip displays a sharp peak displacement of 8.7 x 10\u003csup\u003e− 6\u003c/sup\u003e m shortly after the effects begin (approximately 1 m s). Following this, oscillations occur due to the elastic recoil of the structure. The amplitude of these oscillations reflects the dynamic elasticity of the Cantilever Beam's topology, reaching a maximum of about 6.2 x 10\u003csup\u003e− 6\u003c/sup\u003e m.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e illustrates the predicted density distribution before optimization. The figure shows that the material concentration increases from the fixed edge until it reaches 1, the maximum concentration along the Cantilever Beam. This indicates a fixed edge weakness, where stresses are concentrated on the Cantilever Beam is subjected to loads or shocks.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec16\" class=\"Section2\"\u003e\u003ch2\u003e4.2 Dynamic acceleration, velocity, and displacement response distribution, and Optimized Topology Distribution\u003c/h2\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e illustrates the distribution of acceleration about the distance from the fixed edge of the Cantilever Beam under impact force following optimization. The data reveal that the irregular oscillating acceleration completely disappears up to a distance of 0.86 m from the fixed edge. Beyond this point, the acceleration experiences a sharp increase, reaching 2×10⁻⁹ m / s² near the free edge, before gradually decreasing again. As optimization progresses, a noticeable reduction in peak displacement are observed, indicating enhanced stiffness and improved energy dissipation.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e shows the displacement relative to the distance from the fixed edge of the Cantilever Beam under the impact force after optimization. The data show that the irregular displacement disappears up to a distance of 0.93 m from the fixed edge. After this point, the displacement increases sharply, reaching 3.8 × 10\u003csup\u003e⁻6\u003c/sup\u003e m near the free edge. As the optimization progresses, a significant decrease in the peak displacement is observed, indicating improved stiffness and energy dissipation\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e illustrates the relationship between velocity and distance from the fixed edge of the Cantilever Beam after optimization due to the impact force. The data indicate that irregularities in velocity vanish up to a distance of 0.93 m from the fixed edge. Beyond this point, the velocity increases sharply, reaching (4 × 10⁻⁴ m/s) near the free edge. The optimization process results in a significant reduction in the maximum speed value, indicating increased stiffness and energy dissipation.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e shows the temporal response of the Cantilever Beam tip upon initial impact after optimization. The tip exhibits a steady increase in displacement shortly after impact onset (about 1 ms), reaching a maximum value of about 3.8 × 10\u003csup\u003e− 6\u003c/sup\u003e m. As the optimization process progresses, a clear decrease in peak displacement is observed, indicating improved stiffness and energy dissipation.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e illustrates the optimized material distribution along the Cantilever Beam, showing even concentration while prioritizing high-stress areas near the fixed support. This aligns with fixed Cantilever Beam theory and highlights the effectiveness of the SIMP model for dynamic optimization. The resulting design exhibits a trade-off between structural mass and impact resistance. Elements with intermediate densities are minimized due to the penalization scheme (p = 3), promoting a quasi-binary design suitable for manufacturing.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec17\" class=\"Section2\"\u003e\u003ch2\u003e4.3 Discussion and Insights\u003c/h2\u003e\u003cp\u003eThe simulation results demonstrate a sharp initial displacement peak after impact, followed by damped oscillations. As optimization progresses, the peak displacement decreases, indicating an improved stiffness. The optimized topology shows material concentrated near the clamped end, consistent with the stress distribution under impact. Material is removed from low-stress areas, achieving volume efficiency. While simplified, the framework shows clear trends that validate the use of SIMP for dynamic impact optimization.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003col\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eEnergy Absorption: Although not explicitly modeled, the reduced tip displacement suggests improved energy absorption through stiffness reallocation.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eComputational Efficiency: The use of a simplified 1D model allows fast iteration cycles, though extensions to 2D/3D are necessary for realistic geometries.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eDesign Interpretation: The final topology shows the potential for lightweight structural components optimized for transient, high-speed loading.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eLimitations: Contact, plasticity, and large-deformation effects were neglected. Incorporating them would enhance realism but require significantly more complex modeling and sensitivity analysis.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003c/ol\u003e\u003cp\u003e\u003c/p\u003e"},{"header":"5. Conclusion and Future Work","content":"\u003ch2\u003e5.1 Conclusion\u003c/h2\u003e\u003cp\u003eThis study presents a SIMP-based TO framework for lightweight structures subjected to nonlinear impact loads. This framework effectively minimizes dynamic displacement while considering material constraints. Using a Cantilever Beam as a case study, a one-dimensional finite element model was implemented using MATLAB, allowing for iterative optimization of structural design under dynamic loading conditions.\u003c/p\u003e\u003cp\u003eThe simulation results demonstrate that:\u003c/p\u003e\u003col\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eThe SIMP-based approach is capable of producing optimized material distributions that reduce dynamic tip displacement.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eThe material tends to concentrate near the fixed support, confirming classical Cantilever Beam behavior under dynamic conditions.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eThe framework successfully enforces a volume constraint while improving impact resistance through the redistribution of stiffness.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003c/ol\u003e\u003cp\u003eWhile the methodology is computationally efficient and suitable for early-stage conceptual design, it is subject to several limitations:\u003c/p\u003e\u003col\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eThe model assumes linear elasticity, excluding plasticity and material damage mechanisms common in real impact scenarios.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eThe simulation is restricted to a one-dimensional domain, which limits its applicability to real-world components.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003c/ol\u003e\u003ch2\u003e5.2 Future Work\u003c/h2\u003e\u003cp\u003eTo improve realism and extend the applicability of the proposed framework, the following directions are recommended for future research:\u003c/p\u003e\u003col\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eExtension to 2D and 3D models: Incorporating plane stress or shell elements to model realistic structural geometries such as plates, brackets, or thin-walled Cantilever Beams.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eInclusion of material nonlinearity: Implementing elasto-plastic or Viscoelasticity material models to more effectively measure energy absorption and permanent deformation during impact.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eModeling of contact and fracture: Integrating contact algorithms (e.g., Mortar methods) and damage models (e.g., phase-field) for improved prediction of failure.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eAdjoint-based sensitivity analysis: Replacing finite-difference sensitivity estimation with efficient adjoint formulations to enable high-resolution designs with reduced computational cost.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eMulti-objective and robust optimization: Considering multiple performance criteria such as energy absorption, natural frequency, or robustness under uncertain impact conditions.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003c/ol\u003e\u003cp\u003eBy addressing these aspects, the proposed methodology can evolve into a powerful tool for the design of crashworthy, lightweight components in automotive, aerospace, and defense industries.\u003c/p\u003e"},{"header":"Abbreviations","content":"\u003cp\u003e\u003cimg 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\" width=\"472\" height=\"471\"\u003e\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eEthics approval and consent to participate:\u003c/strong\u003e Springer Nature maintained a neutral position with regard to jurisdictional claims in published maps and institutional affiliations and no official or unofficial entity objected to the publication.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConsent for publication\u003c/strong\u003e\u003cstrong\u003e:\u0026nbsp;\u003c/strong\u003eAuthor has read and approved the manuscript for submission for publication.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAvailability of data and material:\u003c/strong\u003e All data generated or analyzed during this study are included in this published article.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests:\u003c/strong\u003e The author declare that they have no competing interests.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding:\u003c/strong\u003e This research received no external funding.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contribution:\u003c/strong\u003e The author M.F. completed the following tasks: concept development, methodology design, software development, information and data validation, scientific analysis of the results, result generation, resource provision, writing and original draft preparation, review and editing, supervision, and project management.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgements:\u003c/strong\u003e The researcher extends its gratitude to the Deanship of the College and the professors of the Department of Engineering for the great support and encouragement they provided to complete the research.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eZhang, L., Zhang, Y., \u0026amp; Van Keulen, F. 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Topology optimization for thermal structures considering design-dependent convection boundaries based on the bidirectional evolutionary structural optimization method. \u003cem\u003eMechanical Sciences\u003c/em\u003e, \u003cem\u003e14\u003c/em\u003e(1), 223-235.\u0026rlm;\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"UNIVERSITY OF AL MAARIF","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Topology optimization, Lightweight structures, Nonlinear impact, SIMP method, MATLAB finite element, Structural dynamics, Crash worthiness","lastPublishedDoi":"10.21203/rs.3.rs-7810029/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7810029/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper investigates the topology optimization of lightweight cantilever beam structures subjected to nonlinear dynamic impact loading. A simplified one-dimensional finite element model (FEM) was developed in MATLAB and coupled with the Solid Isotropic Material with Penalization (SIMP) method to determine the optimal material distribution under transient impulsive forces. The objective of the optimization was to minimize the tip displacement of the beam while satisfying a prescribed volume constraint. Through an explicit time integration scheme, the dynamic response of the structure was evaluated iteratively. A cantilever beam was analyzed using numerical modeling and optimization to minimize tip displacement under a nonlinear dynamic impact load, while maintaining a 50% volume limit. The study utilized MATLAB with a one-dimensional finite element model and the SIMP method. The beam, fixed at one end and subjected to a short-term impact load at the free end, was divided into 15 elements. The governing dynamic equilibrium equation is solved using explicit time integration, and the squared tip displacement is minimized using a penalty factor (P)\u0026thinsp;=\u0026thinsp;3. Design variables were iteratively adjusted based on finite-difference sensitivities to enhance performance during impact. The beam, modeled as a 1.0 m long rectangular cross-section bar, assumed linear stiffness with Young's modulus (E)\u0026thinsp;=\u0026thinsp;210 GPa and density (ρ)\u0026thinsp;=\u0026thinsp;7800 kg / m\u0026sup3;. The results demonstrate that the optimized topologies concentrate material near high-stress regions, significantly reducing peak displacements after impact. Although this framework provides an efficient proof of concept, future enhancements are needed to capture more realistic behavior, including geometric nonlinearities, contact, and inelastic material responses. This work lays the groundwork for designing crashworthy, lightweight structures in automotive, aerospace, and defense applications.\u003c/p\u003e","manuscriptTitle":"Topology Optimization of Lightweight Cantilever Beam Structures under Nonlinear Dynamic Impact Loading","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-10-10 04:14:44","doi":"10.21203/rs.3.rs-7810029/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"03faa908-56aa-42e5-8b42-a0dead7f5903","owner":[],"postedDate":"October 10th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":55980751,"name":"Mechanical Engineering"}],"tags":[],"updatedAt":"2025-10-10T04:14:44+00:00","versionOfRecord":[],"versionCreatedAt":"2025-10-10 04:14:44","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7810029","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7810029","identity":"rs-7810029","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}