Advanced Topology Optimization: Methods and Applications

1. Introduction

Structural topology optimization is a powerful computational design paradigm that seeks the most efficient material distribution within a prescribed design domain to satisfy given performance requirements. Over the past decades, a variety of mature and widely adopted algorithms have been developed, including the Solid Isotropic Material with Penalization (SIMP) method [1], Bi-directional Evolutionary Structural Optimization (BESO) method [2], the level-set method [3], Moving Morphable Components (MMCs) method [4], Floating Projection Topology Optimization (FPTO) method [5], and Smooth-Edged Material Distribution for Optimizing Topology (SEMDOT) approach [6,7]. As a core technique in lightweight structural design, topology optimization has been extensively applied across aerospace, automotive, civil engineering, and defense industries [8,9,10,11,12].

Although topology optimization has made major progress in theory and computation, several challenges still prevent its widespread adoption in complex engineering practice. These challenges include the incorporation of realistic manufacturing constraints, the efficient solution of large-scale and high-dimensional optimization problems, robust coupling with multi-physics phenomena, and the generation of interpretable, CAD-ready geometries suitable for direct downstream manufacturing. Moreover, emerging application scenarios—such as additive manufacturing, structural dynamics, and thermal–fluid devices—place increasing demands on topology optimization frameworks, requiring them to move beyond traditional static and single-physics formulations. Addressing these issues requires not only methodological and algorithmic innovations, but also improvements in numerical efficiency, modeling fidelity, and application-oriented optimization strategies.

Driven by recent advances in theory and computational methods, this Special Issue, “Advanced Topology Optimization: Methods and Applications”, aims to showcase state-of-the-art research and practical developments in the field. The collected papers cover a wide range of topics, including advanced optimization methodologies, improvements in computational efficiency, manufacturing-oriented design, and multi-physics coupling, highlighting the expanding influence of topology optimization across diverse engineering disciplines. A total of twelve peer-reviewed papers were selected, covering both methodological advances and application-oriented studies. These contributions address topics ranging from algorithmic development and numerical strategies to engineering applications in additive manufacturing, structural dynamics, CAD-ready design, and thermal–fluid device design. The following section provides concise summaries of each contribution, outlining their main ideas and key findings within the context of recent advances in topology optimization.

2. Contributions

Huang et al. (Contribution 1) addressed the long-standing challenge of unclear structural boundary identification in meshless topology optimization via the SEMDOT method. The study introduced non-overlapping cell-based density variables within a meshless framework to explicitly describe material presence and absence and further developed a non-penalized SEMDOT formulation using an interpolation-based heuristic sensitivity scheme. The effectiveness of the proposed approach was demonstrated through two- and three-dimensional compliance minimization examples, showing its capability to generate optimized structures with continuous and smooth edges or surfaces without manual post-processing.

Cao et al. (Contribution 2) highlighted that while topology optimization has strong potential to improve the lightweight design, strength, and fatigue performance of rock-cutting tools, its practical application remains constrained by complex dynamic loading conditions, material–structure coupling in severe wear environments, data scarcity for adaptive design, and insufficient validation of manufacturing compatibility and long-term reliability.

Pehnec et al. (Contribution 3) presented a topology optimization approach that combines a parameterized level-set formulation with genetic algorithm-based global optimization. By employing B-spline interpolation to reduce the number of design variables, the method enables the effective use of evolutionary optimization and overcomes the strong dependence of traditional level-set methods on initial solutions. A novel genetic algorithm penalty operator was introduced to improve convergence behavior by penalizing similar individuals within the population. The proposed approach demonstrated robustness against local minima and was shown to be effective for typical two-dimensional benchmark problems.

Miler et al. (Contribution 4) proposed a multi-objective optimization framework with embedded topology optimization for multi-body systems, enabling the simultaneous optimization of component topology and spatial arrangement. To alleviate the high computational cost caused by repeated topology optimization calls within the genetic algorithm, two acceleration strategies were introduced, involving mesh coarsening and iteration reduction during early generations. The results showed that both strategies significantly reduced computational time while maintaining comparable optimization performance, demonstrating their effectiveness in improving the efficiency of multi-objective topology optimization.

Zhang et al. (Contribution 5) proposed a multi-constraint topology optimization strategy for flight control rudder structures by simultaneously incorporating additive manufacturing overhang constraints and flutter-related mass center control within a unified density-based framework. By introducing a build-direction-aware projection filter and a smooth Heaviside mass center constraint, the approach enables the concurrent satisfaction of lightweight design, structural stiffness, aeroelastic stability, and AM manufacturability requirements during the optimization process. Numerical results demonstrated that the proposed method significantly reduces support material usage and residual deformation while improving flutter velocity, highlighting its effectiveness in achieving manufacturable, dynamically stable, and high-performance AM-ready rudder designs without post-processing.

Zhang et al. (Contribution 6) proposed an integrated optimization strategy combining discrete element method simulation and topology optimization to reduce the excessive mass of chain drive systems in long-distance scraper conveyors. The study showed that a non-equally spaced scraper arrangement can maintain conveying capacity while reducing the number of scrapers in chains of equal length, and that further topology optimization of the scraper structure leads to additional mass reduction. The proposed approach achieved a significant overall weight reduction of the chain drive system, resulting in lower no-load energy consumption and improved operational efficiency.

Kyriakidis et al. (Contribution 7) investigated the mechanical performance of topology-optimized shin pad structures incorporating advanced lattice geometries and materials using finite element analysis under impact scenarios representative of real contact sports. By replicating median and extreme stud kick impact conditions, the study evaluated the energy absorption and injury prevention capabilities of the proposed designs. The results showed that lattice geometries, particularly gyroid structures, promote more uniform stress distribution and significantly reduce transmitted forces, while the use of a matrix reinforced with reused ground tire rubber further enhances strength, damping, and absorption performance.

Huang et al. (Contribution 8) addressed the computational inefficiency of filtering operations in variable-density topology optimization by introducing a novel filter implementation based on the k-d tree data structure. By transforming conventional neighborhood searches into highly efficient spatial searches, the proposed method significantly accelerates the optimization process while preserving solution accuracy. The approach supports a wide range of manufacturability constraints, including symmetry, local volume control, periodic patterns, and stamping-oriented overhang constraints, without increasing computational cost. Numerical examples demonstrated that the method is particularly effective for large-scale optimization problems, complex geometric constraints, and unstructured meshes, highlighting its strong potential for scalable and efficient engineering design applications.

Sardone et al. (Contribution 9) introduced a CAD-embedded, time-dependent structural topology optimization framework that integrates an adjoint-based optimization core within a visual programming environment. The framework integrates a parametric CAD interface with a dynamic finite element solver to support topology optimization under harmonic and seismic loading. Case studies on tall building facades subjected to earthquake excitation demonstrated the framework’s ability to reduce displacement at targeted degrees of freedom and dynamically adapt material distributions.

Sun et al. (Contribution 10) presented a multifidelity topology optimization framework for thermal–fluid devices that combines a computationally efficient Darcy–convection model with high-fidelity Navier–Stokes–convection analysis to balance temperature reduction and pressure-loss control. The approach employs SEMDOT to decouple geometric smoothness, enabling the generation of CAD-ready boundaries during low-fidelity optimization. Candidate designs are evaluated using high-fidelity simulations under consistent operating conditions, demonstrating that the optimized layouts achieve simultaneous reductions in peak temperature and pressure drop compared with conventional channel designs. The results further showed that the proposed multifidelity strategy provides a more favorable Pareto distribution than conventional RAMP-based pipelines.

Christensen and Alexandersen (Contribution 11) proposed a gradient-based topology optimization framework for heat sinks incorporating phase-change materials (PCMs) to attenuate temperature oscillations induced by cyclic thermal loading. The method integrates transient thermal analysis with adjoint sensitivity evaluation and the Globally Convergent Method of Moving Asymptotes (GCMMA) optimizer, enabling the optimized distribution of PCM and conductive material through a homogenized material representation. Numerical investigations showed that exploiting latent heat effects can significantly reduce temperature variability, while manufacturability-oriented penalization produces near-discrete designs with only limited loss in performance. The study demonstrates that adjoint-driven PCM topology optimization provides an effective and systematic strategy for thermal oscillation control.

Liu et al. (Contribution 12) provided a comprehensive perspective on the development of the SEMDOT framework, positioning it as an effective solution to the long-standing boundary fuzziness and jagged-edge issues inherent in element-based topology optimization methods. The review systematically traces the theoretical evolution of SEMDOT, covering both penalty-based and non-penalty-based formulations. Open-access implementations have further accelerated its adoption and reproducibility across the research community.

As the Guest Editor and SEMDOT developer, whose primary expertise is in mechanical engineering rather than in mathematics or computational mechanics, I (Yun-Fei Fu) acknowledge that certain aspects of the SEMDOT algorithm have not yet been developed with full mathematical rigor. As discussed in this review, SEMDOT is still evolving, and its theoretical foundations would benefit from deeper mathematical analysis and critical examination. We therefore sincerely encourage more researchers—particularly those with strong expertise in applied mathematics and computational mechanics—to engage with SEMDOT, whether by improving its theoretical formulation, extending its numerical framework, or offering constructive criticism. Such open discussion and collaboration are essential for refining the method and advancing it toward a more rigorous and comprehensive optimization platform.

3. List of Contributions

Contribution 1: Huang, J.; Long, K.; Chen, Y.; Geng, R.; Saeed, A.; Zhang, H.; Tao, T. A Framework of the Meshless Method for Topology Optimization Using the Smooth-Edged Material Distribution for Optimizing Topology Method. Computation 2025, 13, 6. https://doi.org/10.3390/computation13010006

Contribution 2: Cao, Y.; Zhang, Q.; Zhang, S.; Tian, Y.; Dong, X.; Song, X.; Wang, D. Optimization of Rock-Cutting Tools: Improvements in Structural Design and Process Efficiency. Computation 2025, 13, 152. https://doi.org/10.3390/computation13070152

Contribution 3: Pehnec, I.; Sedlar, D.; Marinic-Kragic, I.; Vučina, D. A New Approach to Topology Optimization with Genetic Algorithm and Parameterization Level Set Function. Computation 2025, 13, 153. https://doi.org/10.3390/computation13070153

Contribution 4: Miler, D.; Hoić, M.; Tomić, R.; Jokić, A.; Mašović, R. Simultaneous Multi-Objective and Topology Optimization: Effect of Mesh Refinement and Number of Iterations on Computational Cost. Computation 2025, 13, 168. https://doi.org/10.3390/computation13070168

Contribution 5: Zhang, H.; Shi, S.; Ding, X.; Yang, J.; Xiong, M. Topology Optimization for Rudder Structures Considering Additive Manufacturing and Flutter Effects. Computation 2025, 13, 208. https://doi.org/10.3390/computation13090208

Contribution 6: Zhang, Q.; Liu, W.; Jia, A.; Sun, S.; Li, X.; Song, X. Mining Scraper Conveyors Chain Drive System Lightweight Design: Based on DEM and Topology Optimization. Computation 2025, 13, 225. https://doi.org/10.3390/computation13090225

Contribution 7: Kyriakidis, I.F.; Kladovasilakis, N.; Pechlivani, E.M.; Tsongas, K. Mechanical Evaluation of Topologically Optimized Shin Pads with Advanced Composite Materials: Assessment of the Impact Properties Utilizing Finite Element Analysis. Computation 2025, 13, 236. https://doi.org/10.3390/computation13100236

Contribution 8: Huang, J.; Saeed, A.; Long, K.; Chen, Y.; Geng, R.; Jia, J.; Tao, T. An Efficient Filter Implementation Method and Its Applications in Topology Optimization Utilizing k-d Tree Data Structure. Computation 2025, 13, 262. https://doi.org/10.3390/computation13110262

Contribution 9: Sardone, L.; Sotiropoulos, S.; Fiore, A. A CAD-Integrated Framework for Dynamic Structural Topology Optimisation via Visual Programming. Computation 2025, 13, 267. https://doi.org/10.3390/computation13110267

Contribution 10: Sun, Y.; Fu, Y.-F.; Xu, S.; Guo, Y. Multifidelity Topology Design for Thermal–Fluid Devices via SEMDOT Algorithm. Computation 2026, 14, 19. https://doi.org/10.3390/computation14010019

Contribution 11: Christensen, M.B.M.; Alexandersen, J. Topology Optimisation of Heat Sinks Embedded with Phase-Change Material for Minimising Temperature Oscillations. Computation 2026, 14, 23. https://doi.org/10.3390/computation14010023

Contribution 12: Liu, M.; Hu, W.; Gong, X.; Zhou, H.; Zhao, B. State-of-the-Art Overview of Smooth-Edged Material Distribution for Optimizing Topology (SEMDOT) Algorithm. Computation 2026, 14, 27. https://doi.org/10.3390/computation14010027