Topology Optimization Problem Research Articles

A topology-based in-plane filtering technique for the combined topology and discrete fiber orientation optimization

This work proposes a filtering technique for the concurrent and sequential finite element-based topology and discrete fiber orientation optimization of composite structures. The proposed filter is designed to couple the morphology with the topology of the structural domain throughout the optimization process in a way such that it suppresses the impact of the close-to-void finite elements’ morphology on the overall morphology of the structure while steering at the same time the local fiber orientation towards the neighboring topologically dense areas of the geometry. The functionality of the filter becomes crucial at the boundaries of the optimized structure, where the fiber orientation is forced to conform to the morphology of the local boundaries. The developed filtering technique is incorporated into both the concurrent and sequential finite element-based topology and discrete fiber orientation optimization problem, and the respective optimization problems are formulated for the compliance minimization of the composite structure. To assess the efficacy of the filter, it is demonstrated in the benchmark academic case studies of the 2D Messerschmitt-Bölkow-Blohm and cantilever beams when different state-of-art interpolation techniques are employed for modeling the discrete fiber orientation optimization problem.

Quasi-Newton methods for topology optimization using a level-set method

The ability to efficiently solve topology optimization problems is of great importance for many practical applications. Hence, there is a demand for efficient solution algorithms. In this paper, we propose novel quasi-Newton methods for solving PDE-constrained topology optimization problems. Our approach is based on and extends the popular solution algorithm of Amstutz and Andrä (J Comput Phys 216: 573–588, 2006). To do so, we introduce a new perspective on the commonly used evolution equation for the level-set method, which allows us to derive our quasi-Newton methods for topology optimization. We investigate the performance of the proposed methods numerically for the following examples: Inverse topology optimization problems constrained by linear and semilinear elliptic Poisson problems, compliance minimization in linear elasticity, and the optimization of fluids in Navier–Stokes flow, where we compare them to current state-of-the-art methods. Our results show that the proposed solution algorithms significantly outperform the other considered methods: They require substantially less iterations to find a optimizer while demanding only slightly more resources per iteration. This shows that our proposed methods are highly attractive solution methods in the field of topology optimization.

A Hilbertian projection method for constrained level set-based topology optimisation

We present an extension of the projection method proposed by Challis et al. (Int J Solids Struct 45(14–15):4130–4146, 2008) for constrained level set-based topology optimisation that harnesses the Hilbertian velocity extension-regularisation framework. Our Hilbertian projection method chooses a normal velocity for the level set function as a linear combination of (1) an orthogonal projection operator applied to the extended optimisation objective shape sensitivity and (2) a weighted sum of orthogonal basis functions for the extended constraint shape sensitivities. This combination aims for the best possible first-order improvement of the optimisation objective in addition to first-order improvement of the constraints. Our formulation utilising basis orthogonalisation naturally handles linearly dependent constraint shape sensitivities. Furthermore, use of the Hilbertian extension-regularisation framework ensures that the resulting normal velocity is extended away from the boundary and enriched with additional regularity. Our approach is generally applicable to any topology optimisation problem to be solved in the level set framework. We consider several benchmark constrained microstructure optimisation problems and demonstrate that our method is effective with little-to-no parameter tuning. We also find that our method performs well when compared to a Hilbertian sequential linear programming method.

Topology optimization of geometrically nonlinear structures using reduced-order modeling

High computational costs are encountered in topology optimization problems of geometrically nonlinear structures since intensive use has to be made of incremental-iterative finite element simulations. To alleviate this computational intensity, reduced-order models (ROMs) are explored in this paper. The proposed method targets ROM bases consisting of a relatively small set of base vectors while accuracy is still guaranteed. For this, several fully automated update and maintenance techniques for the ROM basis are investigated and combined. In order to remain effective for flexible structures, path derivatives are added to the ROM basis. The corresponding sensitivity analysis (SA) strategies are presented and the accuracy and efficiency are examined. Various geometrically nonlinear examples involving both solid as well as shell elements are studied to test the proposed ROM techniques. Test cases demonstrates that the set of degrees of freedom appearing in the nonlinear equilibrium equation typically reduces to several tenth. Test cases show a reduction of up to 6 times fewer full system updates.

Concurrent Topology Optimization of Multi-Scale Composite Structures Subjected to Dynamic Loads in the Time Domain

This paper presents a concurrent topology optimization of multi-scale composite structures subjected to general time-dependent loads for minimizing dynamic compliance. A three-field density-based method is adopted to implement the concurrent topological design, with macroscopic effective properties of the microstructure evaluated through energy-based homogenization method (EBHM). Transient response is obtained from the two-scale finite element analysis with the HHT-α approach as an implicit time integration procedure. Design sensitivities are formulated employing the adjoint variable method (AVM) based on two main philosophies: “discretize-then-differentiate” and “differentiate-then-discretize” approaches, respectively. The method of moving asymptotes is adopted to update the design variables at two scales. Several benchmark examples are presented to demonstrate that the “discretize-then-differentiate” AVM attains consistent sensitivities in an inherent manner such that the resulting optimal topology is more efficient when compared with the “differentiate-then-discretize” AVM. Moreover, the potential of the proposed method for concurrent dynamic topology optimization problems under general time-dependent loads is also highlighted.

A Transmission-Reliable Topology Control Framework Based on Deep Reinforcement Learning for UWSNs

This paper focuses on the topology optimization problem to decrease transmission delay and prolong network lifetime on the premise of reliable transmission in Underwater Wireless Sensor Networks (UWSNs). This is extremely challenging owing to dynamic ocean current movement, harsh underwater communication channel and increasingly demanding application requirements. With the development of software defined network architectures, the centralized topology control (TC) strategy with a global perspective in UWSNs is expected to become a more effective way to tackle the above challenges compared with the existing distributed and heuristic TC strategies involving local network state information. Therefore, we first transform the topology optimization problem of UWSNs into an integer nonlinear programming (INLP) model and design a centralized TC framework to solve the INLP model. In this framework a TC center is built to periodically generate the network topology for UWSNs according to the current network state information. Further, an efficient topology generation algorithm based on deep reinforcement learning (TGA-DRL) is proposed in the TC center. In TGA-DRL, to reduce computing overhead and improve operational efficiency, we formulate an action-space narrowed Markov decision process suitable for network topology generation and solve it with the aid of the rainbow algorithm which is a deep reinforcement learning model. Finally, the performance of our centralized TC framework is verified in terms of the node out-degree, algorithm convergence and optimization effect.

DL-MSTO+: A deep learning-based multi-scale topology optimization framework via positive definiteness ensured material representation network

A multi-scale approach to topology optimization has recently emerged due to its lightweight, robust, and multi-functional characteristics. Considering material diversity, an increasing number of materials leads to a computational burden due to numerous design variables and homogenization equations. This study proposes DL-MSTO+, a deep learning-based multi-scale topology optimization framework. The framework consists of two distinct deep neural networks for multi-scale topology optimization problems to reduce the dimensionality of design variables and predict homogenized material properties. First, a generator network learns the low-dimensional representation of a material microstructure in an unsupervised manner. Thus, the trained generator produces a micro-structural image from a low-dimensional vector. Second, a predictor network is trained in a supervised manner to predict the homogenized elasticity matrix for a given micro-structural image. Additionally, the predictor is specifically designed to ensure the positive definiteness of the predicted elasticity matrix. Lastly, the networks are integrated into the algorithm to improve the efficiency of multi-scale topology optimization. The numerical experiments demonstrate higher efficiency for the proposed algorithm than the conventional multi-scale approach. Moreover, the proposed method provides connectable multi-scale designs, and the low-dimensional latent representation enables semantic interpolation between solutions.

A Temporal Coherent Topology Optimization Approach for Assembly Planning of Bespoke Frame Structures

We present a computational framework for planning the assembly sequence of bespoke frame structures. Frame structures are one of the most commonly used structural systems in modern architecture, providing resistance to gravitational and external loads. Building frame structures requires traversing through several partially built states. If the assembly sequence is planned poorly, these partial assemblies can exhibit substantial deformation due to self-weight, slowing down or jeopardizing the assembly process. Finding a good assembly sequence that minimizes intermediate deformations is an interesting yet challenging combinatorial problem that is usually solved by heuristic search algorithms. In this paper, we propose a new optimization-based approach that models sequence planning using a series of topology optimization problems. Our key insight is that enforcing temporal coherent constraints in the topology optimization can lead to sub-structures with small deformations while staying consistent with each other to form an assembly sequence. We benchmark our algorithm on a large data set and show improvements in both performance and computational time over greedy search algorithms. In addition, we demonstrate that our algorithm can be extended to handle assembly with static or dynamic supports. We further validate our approach by generating a series of results in multiple scales, including a real-world prototype with a mixed reality assistant using our computed sequence and a simulated example demonstrating a multi-robot assembly application.

Topology Optimization Design of Resonant Structures Based on Antiresonance Eigenfrequency Matching Informed by Harmonic Analysis

Abstract In this article, we present a design methodology for resonant structures exhibiting particular dynamic responses by combining an eigenfrequency matching approach and a harmonic analysis-informed eigenmode identification strategy. This systematic design methodology, based on topology optimization, introduces a novel computationally efficient approach for 3D dynamic problems requiring antiresonances at specific target frequencies subject to specific harmonic loads. The optimization’s objective function minimizes the error between target antiresonance frequencies and the actual structure’s antiresonance eigenfrequencies, while the harmonic analysis-informed identification strategy compares harmonic displacement responses against eigenvectors using a modal assurance criterion, therefore ensuring an accurate recognition and selection of appropriate antiresonance eigenmodes used during the optimization process. At the same time, this method effectively prevents well-known problems in topology optimization of eigenfrequencies such as localized eigenmodes in low-density regions, eigenmodes switching order, and repeated eigenfrequencies. Additionally, our proposed localized eigenmode identification approach completely removes the spurious eigenmodes from the optimization problem by analyzing the eigenvectors’ response in low-density regions compared to high-density regions. The topology optimization problem is formulated with a density-based parametrization and solved with a gradient-based sequential linear programming method, including material interpolation models and topological filters. Two case studies demonstrate that the proposed design methodology successfully generates antiresonances at the desired target frequency subject to different harmonic loads, design domain dimensions, mesh discretization, or material properties.

Topological optimization BI-adhesive double lap adhesive joint. One-dimension model

The aim of the article is to construct a solution to the problem of topological optimization of a symmetric two-shear and bi-adhesive joint. The one-dimensional mathematical model of the stress state of a symmetrical double-shear adhesive joint with a variable thickness of the outer substrate along the lap length is represented in the article. The proposed model is a generalization of the classical Goland-Reissner model. The proposed mathematical model is used to solve the problem of topological optimization of the outer substrate shape as well as to obtain the optimal length of the sections for the rigid and compliant adhesives. The expansion of the outer substrate thickness dependence along the adhesive joint length into a Fourier series in cosines was used to describe the shape of the outer substrate. The desired values in the optimization problem are the lengths of the adhesive joint sections with the compliant and rigid adhesives as well as the Fourier coefficients. The objective function is the cross-sectional area of the outer substrate. Restrictions are imposed on the maximum stresses in the adhesive layer and in the outer substrate. A genetic algorithm was used to solve the optimization problem. The direct problem of determining the stress state of the adhesive joint at specified parameters was solved using the finite difference method. The model problem is solved and the results of the stress state calculation are compared with the results of finite element modeling.

Quasi-Newton corrections for compliance and natural frequency topology optimization problems

Quasi-Newton corrections for compliance and natural frequency topology optimization problems

Application of Bezier curves for description of structure shape in optimization of adhesive joints

The problem of topological optimization of a symmetrical double-shear adhesive joint has been solved. The suggested mathematical model of a joint with variable thickness generalizes the classic Holland–Reissner model. The shape of the doubler is described by means of the Bezier curve. Seeking parameters in the optimization problem are coordinates of reference points of the Bezier curve. Both joint length and doubler cross-section area can be considered an objective function. The restriction is applied on stress in adhesive film and doubler. The direct problem of finding the joint stress state at given geometric parameters was solved using the finite difference method. The genetic algorithm was used to solve the optimization problem. In order to improve the convergence of the genetic algorithm, the island model of evolution is suggested, which ensures quick evolution selection and stability of obtained results. The model problem is solved.

Proportional Topology Optimization algorithm for two-scale concurrent design of lattice structures

In this paper, the Proportional Topology Optimization (PTO) algorithm is extended for the two-scale concurrent topology optimization, in which both the structure and material cellular micro-structure are subject to design. PTO was originally developed on the concept that the amount of material being distributed to an element would be proportional to the contribution of that element in the objective function. Sensitivity analysis is not required. In a two-scale concurrent topology optimization problem, two sets of design variables are defined, one for macro-structure and one for micro-structure. Here, the objective function is reformulated such that the contribution of each micro-scale design variable can be determined, facilitating the employment of PTO. The macroscopic effective elastic tensor is evaluated by the energy-based homogenization method (EBHM), providing a link between micro-structure and macro-structure. Feasibility and efficiency of the proposed PTO approach are demonstrated via several benchmark examples of both two and three dimensional structures.

Enhanced marine predator algorithm for global optimization and engineering design problems

Marine predator algorithm adopts the policy of optimal encounter rate in biological interaction between predator and prey, inspired by the Lévy and Brownian motions commonly used in oceanic predators. However, the Marine predator algorithm faces problems of premature convergence, poor search capability, and stagnation in a local optimum. In order to tolerate these shortcomings, two different strategies are applied to Marine predator algorithm in this study. (1) Taylor-based optimal neighborhood strategy (TNS) provides a more effective update in obtaining optimal individuals, thus reducing the probability of local optimum falling and attempting to overcome premature convergence, (2) Asymmetric search space with dynamic option-based learning significantly increases the probability of the population achieving the global optimum, thus improving the exploitation ability of Marine predator algorithm. The new method introduced with these two strategies added to the Marine predator algorithm is called Marine Predator Algorithm Dynamic Opposition and Taylor neighborhood search (DOTMPA). DOTMPA’s convergence performance is compared with current metaheuristic optimization and different versions of MPA in the literature on CEC2017, CEC2019, and CEC2022 benchmark suites. The results are discussed with convergence curves and box-plot analyses. DOTMPA is tested in Cantilever beam design, Tension/compression spring design, Speed reducer, Car crashworthiness, and Industrial refrigeration system engineering problems. The results are compared with the studies in the literature. Experimental results show that the proposed algorithm has strong competitiveness in terms of convergence accuracy. In addition, the stability of DOTMPA in convergence to the global optimum with trajectory analysis is also mentioned. Finally, the proposed method is applied to three truss topology optimization problems (20-truss, 24-truss, and 72-truss). The experimental results, the applicability of the proposed algorithm in practical applications, and the benchmark problem make it a promising and competitive optimization algorithm for optimization problems.

HBWO-JS: jellyfish search boosted hybrid beluga whale optimization algorithm for engineering applications

Abstract Beluga whale optimization (BWO) algorithm is a recently proposed population intelligence algorithm. Inspired by the swimming, foraging, and whale falling behaviors of beluga whale populations, it shows good competitive performance compared to other state-of-the-art algorithms. However, the original BWO faces the challenges of unbalanced exploration and exploitation, premature stagnation of iterations, and low convergence accuracy in high-dimensional complex applications. Aiming at these challenges, a hybrid BWO based on the jellyfish search optimizer (HBWO-JS), which combines the vertical crossover operator and Gaussian variation strategy with a fusion of jellyfish search (JS) optimizer, is developed for solving global optimization in this paper. First, the BWO algorithm is fused with the JS optimizer to improve the problem that BWO tends to fall into the best local solution and low convergence accuracy in the exploitation stage through multi-stage exploration and collaborative exploitation. Then, the introduced vertical cross operator solves the problem of unbalanced exploration and exploitation processes by normalizing the upper and lower bounds of two stochastic dimensions of the search agent, thus further improving the overall optimization capability. In addition, the introduced Gaussian variation strategy forces the agent to explore the minimum neighborhood, extending the entire iterative search process and thus alleviating the problem of premature stagnation of the algorithm. Finally, the superiority of the proposed HBWO-JS is verified in detail by comparing it with basic BWO and eight state-of-the-art algorithms on the CEC2019 and CEC2020 test suites, respectively. Also, the scalability of HBWO-JS is evaluated in three dimensions (10D, 30D, 50D), and the results show the stable performance of the proposed algorithm in terms of dimensional scalability. In addition, three practical engineering designs and two Truss topology optimization problems demonstrate the practicality of HBWO-JS. The optimization results show that HBWO-JS has a strong competitive ability and broad application prospects.

Constraints on integral measures of stress state in topology optimization problems

Topology optimization (TO) is a computational method of determining material distribution in a given design area to create the optimal shape of a part under given boundary conditions. The increased interest in the development of effective methods of designing parts of the optimal topology testifies to the relevance of these theoretical studies and the important applied value of the obtained results. In the classic formulation of maintenance, the minimization of the flexibility of the part under restrictions on the volume (mass) of the optimization result is chosen as a criterion for finding the specified distribution. Closer to practical application is the formulation of the maintenance problem, which involves minimizing the volume of the part, taking into account the condition of its strength. The inclusion of aggregate functions for the calculation of integral measures of the stress state has a number of advantages over the traditional check of the maximum value of mechanical stress: significant saving of time for solving the maintenance problem, reduction of computational costs and ensuring the stability of the computational process. This work presents and analyzes the specialization of the applied application of aggregate functions, which have been most widely used in modern research on maintenance issues, taking into account the strength of the optimized part. In particular, the P-norm and P-mean functions, the Kreiselmeier-Steinhauser functions, the smoothed Heaviside function, the measure of exceeded stresses, and the measure of uneven distribution of the stress state are described. The large number of options available in the literature for the mathematical formulation of limitations for integral measures of the stress state of designed parts indicates that the issue of developing a universal and effective method of designing parts, taking into account the criterion of its strength, remains open.

A Problem-Independent Machine Learning (PIML) enhanced substructure-based approach for large-scale structural analysis and topology optimization of linear elastic structures

Structural analysis for structures composed of highly heterogeneous materials which often involves the solution of large-scale linear algebraic equations is very time-consuming even within the linear elastic regime. Furthermore, the tremendous computational cost of iterative large-scale finite element analysis also prevents the widespread use of topology optimization as a powerful design tool especially when the desired design resolution is very high. In order to break the bottleneck hindering the efficient solutions of large-scale structural analysis and design optimization problems, in the present article, a general machine learning (ML) enhanced substructure-based framework is proposed. The essential idea is resorting to the classical substructure-based finite element analysis approach and establishing an implicit mapping between the parameters characterizing the material distribution within a substructure and the corresponding condensed stiffness matrix/numerical shape functions through offline trained deep neural networks. In contrast to most of the existing ML enhanced approaches, the proposed framework is truly independent of the forms of structural geometry, boundary condition and external load, and can be applied to solve various boundary value problems governing by the same type of partial differential equation once the offline training is completed. Armed with modern artificial intelligence techniques, the proposed approach in some sense revives the classical substructure approach in finite element analysis. Compared with the traditional paradigm, it can achieve 104–105 times solution efficiency for tested large-scale examples with satisfactory accuracy. The effectiveness of the approach was also validated for the non-adjoint topology optimization problems, i.e., 3D compliant mechanism design. Finally, to demonstrate the proposed approach’s capability in dealing with extremely large-scale three-dimensional problems, a three-dimensional topology optimization problem with about 109 design variables and 3×109 degrees of freedom (Dofs) is solved on a laptop without resorting to any parallel computing techniques.

部材の密集度を考慮した骨組構造物の大域的トポロジー最適化

It is known that the topology optimization (TopOpt) problem of frame structures can be formulated as a mixed-integer second-order cone programming problem (MISOCP), when the compliance is minimized and the cross-section of each member is selected from a set of predetermined candidates. However, when considering the constructability of real-world building structures, it is insufficient to only minimize compliance. In this paper, in order to control the density of members, the constraints of the number of members connected to each node are additionally considered into TopOpt by MISOCP treated in the previous study.

GPU-based mesh reduction strategy utilizing active nodes for structural topology optimization

This paper presents a novel mesh reduction strategy on graphics processing unit (GPU) that aims to reduce the computational complexity of conventional structural topology optimization. In the proposed strategy, the effective number of design variables is reduced by using the concept of active nodes and active elements in the finite element mesh. A novel mesh numbering scheme is also introduced to facilitate parallel identification of active nodes using a proposed GPU-based algorithm. The preconditioned conjugate gradient (PCG) solver is further developed using the proposed strategy and the numbering scheme. The proposed strategy is tested on three structural topology optimization problems using solid isotropic material with penalization method. Results of the proposed strategy demonstrate up to 8× speedup over the standard GPU implementation considering the entire mesh.

Adaptive topology optimization of fail-safe truss structures

Avoidance of disproportionate and progressive collapse, often termed ‘fail-safe design’, is a key consideration in the design of buildings and infrastructure. This paper addresses the problem of fail-safe truss topology optimization in the setting of plastic design, where damage is defined as a moveable circular region in which members are considered to have zero strength for that particular load case. A rigorous and computationally efficient iterative solution strategy is employed in both the dual (member adding) and primal (damage-case adding) problems simultaneously, which allows cases of high complexity and many damage cases (maximum of 16290 potential members and 16291 damage cases) to be solved to the global optimum. Common member-based damage definitions (e.g. damage to any one member) are shown to be highly dependent on the nodal grid; in the limiting case completely negating the effect of the fail-safe constraints. The method proposed in this article does not have such limitations, enabling a more sophisticated and robust treatment of fail-safe design. Moreover, the global minimization and high resolutions create new benchmarks for the least-material designs of ‘fail-safe’ structures using rigid-plastic materials. A number of example structures are considered (short cantilever, square cantilever, multi-span truss), and the effects of damage radius, location, and structure rationalisation are discussed.