Topology Design Of Structures Research Articles

Discrete variable topology optimization for maximizing single/multiple natural frequencies and frequency gaps considering the topological constraint

AbstractFinding optimized structural topology design for maximizing natural frequencies and frequency gaps of continuum structures is crucial for engineering applications. However, two significant numerical issues must be addressed: non‐smoothness caused by multiple frequencies and Artificial Localized Rigid Motion (ALRM) modes due to the violation of the topological constraint related to isolated islands and point‐connections. The above two issues are solved by employing the discrete variable topology optimization method based on Sequential Approximate Integer Programming (SAIP). First, the directional differentiability and multiple frequencies preservation constraints are formulated as the linear integer constraints. And then, the integer programming with these linear integer constraints is established and solved by the discrete variable linear or quadratic programming solver with multi‐constraint Canonical Relaxation Algorithm (CRA). We also prove that the popular Average Modal Frequencies (AMF) strategy, like the Kreisselmeier‐Steinhauser (KS) function, cannot rigorously tackle this non‐smoothness caused by multiple frequencies. Furthermore, to eliminate the ALRM modes and concentrate on the real structural global modes, the burning method is employed to impose the topological constraint of the first Betti number that represents the number of isolated islands and point‐connections. Numerical examples, including 2D and 3D, two‐fold and three‐fold multiple frequencies, natural frequencies and frequency gaps, are presented to show the effectiveness of the proposed method.

Multi-material structural discrete variable topology optimization with minimum length scale control under mass constraint

Topology optimization of multi-material structure has a larger design space compared with single material optimization, and it also requires efficient material selection methods to provide references for designers. The multi-material additive manufacturing ensures the manufacturability of topology design of multi-material structures, but also puts forward new manufacturability requirements, such as the minimum size of multi-material and the connection interface. This paper focuses on the development of a topology optimization algorithm for multi-material structures under the total mass constraint based on discrete variables. Recursive multiphase materials interpolation (RMMI) scheme is utilized to establish the relationship between discrete design variables and element elastic modulus and density, and sequential approximate integer programming (SAIP) and canonical relaxation algorithms are constructed to update the discrete design variables. It is shown that this method can obtain multi-material topology design with clear interface display, and has the ability of free material coexistence and degradation, which indicates that the discrete variables have more advantages than the continuous variables in the RMMI scheme. Based on this, the material selection under mass constraints is discussed in depth, and it is pointed out that the traditional determination of material coexistence or degradation in design based on specific stiffness has limitations. Thus, the present work expands the necessary conditions of material coexistence. Then, a minimum size control method of multi-material design based on post-processing is proposed to solve the minimum size geometry constraint problem by SAIP. Numerical examples show that this method can not only precisely control the minimum size of each material phase (including void phase), but also resolve the problem of interface and boundary diffusion of the multi-material phase, and significantly improve the manufacturability of multi-material design. Finally, it is applied to optimization problems with complex design domain and irregular meshes which are closer to engineering design.

Optimizing structural topology design through consideration of fatigue crack propagation

This paper presents an advanced approach for structural topology optimization by incorporating fatigue crack propagation analysis. The extended finite element method (X-FEM) is employed to model initial crack propagation, while the Paris model serves as the basis for simulating fatigue crack growth. The proposed methodology aims to optimize the structural design by minimizing compliance while considering volume and fatigue constraints. The proposed method employs the developed bi-directional evolutionary structural optimization (BESO) algorithm. The accuracy of the proposed technique is validated through the solution of benchmark problem and is further demonstrated in its effectiveness and robustness by examining several numerical examples. The optimization process considers various crack conditions, including the absence of cracks, horizontal and vertical cracks of different lengths. The optimized topologies obtained through the proposed algorithm clearly demonstrate the impact of crack presence, crack direction, and crack length on the material distribution. Furthermore, the convergence histories of the objective function, represented by mean compliance, highlight the influence of crack length on the stiffness and converged compliance of the structure. The results demonstrate its ability to adapt the material distribution based on fatigue cracks propagation conditions and achieve optimal topologies that balance structural integrity and performance.

End‐to‐end generation of structural topology for complex architectural layouts with graph neural networks

AbstractCurrent automated structural topology design methods can only deal with limited design spaces or simplified architectural layouts for lack of data or a proper representation of structure topology. To address this, the abundant information of manually designed architectural and structural layouts should be exploited to guide the topology design. To achieve automatic generation of structural topologies according to real‐world architectural layouts, this research introduces StrucTopo‐generative adversarial network (GAN), an end‐to‐end generative model with node and edge generation stages based on proper graph representation. Nodes are generated using an image‐to‐image translation model, and edges are generated with a GAN‐based approach. The model is trained and tested on a dataset of 300 complex architectural and structural layouts. Measured against the manually designed topologies, the results indicate that the proposed model can generate reasonable structural topologies, with a recall of 97% and an intersection‐over‐union of 80% in node generation, with a precision of 92% and a recall of 91% in edge generation. Additionally, the joint generation shows a graph similarity of 72%. The proposed model is the first of its kind to consider complex architectural layout constraints in the generation of structural topology, marking a step forward in applying artificial intelligence to practical structural design.

Incorporating buckling effect into the topology design of 2D continuum structures using isolines

Isolines Topology Design (ITD) is an iterative algorithm for use in the topological design of two-dimensional (2D) continuum structures using isolines. This paper presents an extension to this algorithm for topology design of 2D continuum structures under the influence of buckling. Topology design has been used to obtain the lightest structure that can support the loading conditions without failure, with optimal designs typically consisting of slender members. In many cases, instability (or buckling) of the slender compressive members may occur at load levels below those predicted using a stress-based failure criteria. Although topology optimization is often used in the conceptual phase of the design, the influence of buckling has a significant impact on the features and performance of the final structure. This article presents an alternative approach to incorporate the buckling effect into the ITD algorithm for the design of 2D continuum structures. The concept consists of transforming the buckling topology optimization problem into a conventional von Mises stress-based topology design problem at each iteration using the shape of the buckling mode of the structure obtained by the eigenvalue analysis. Three examples are presented to show the viability and effectiveness of the alternative approach implemented into the ITD algorithm. The effect of the displacement factor ratio value on the first critical load of a resulting designs was studied. The resulting designs presented are in good agreement with those from the literature. The main conclusion is that the alternative approach can maximize the first critical load of a design subject to final volume constraints if the associated stiffness loss can be assumed.

The influence of structural topology of additively manufactured PEGDA monolith on adsorption performance for textile wastewater treatment

AbstractWith the rapid advancement of three dimensional (3D) printing technology that offers flexibility to achieve desired surface area and permeability for wastewater applications, structural topology design has the potential to significantly improve the adsorption performance. This present work involves fabricating the green water technology with four different complex monolithic structural designs (Kagome, gyroid, Fischer, and body‐centered cubic [BCC]) with specific surface areas varied from 19.4 to 39.84 mm2 via digital light processing (DLP) technique. All the monolithic structures with different structural topologies were grafted with chitosan to evaluate their adsorption performance in a batch process. The influence of different structural topology on the adsorption performance and efficiency was evaluated at an optimal temperature of 60°C with optimum pH of 6.3 for a reaction time of 2 h using synthetic methyl orange (MO) dye and textile wastewater. Results indicated that the body‐centered cubic (BCC) design exhibited the highest MO adsorption capacity (12.71 mg/g), with the highest permeability (2.57 10−7 m2) and surface area (39.84 mm2). Additionally, a direct relationship was found between the surface area and the permeability of the adsorbent as well as adsorption strength. Fifty percent of color and 2.46% of chemical oxygen demand (COD) removal efficiencies were achieved to verify the adsorbent feasibility in actual application. The present work mainly establishes the feasibility of 3D printing in fabricating different monolithic structures, which enables future topology optimization for improved wastewater management.

An Improved Evolutionary Structure Optimization Method for Smooth Topology Design of Structures

To meet the needs of industrial production, an improved evolution structural optimization (ESO) method with high efficiency is proposed. The optimized design variables with intermediate density were designed using the windowed evolution structural optimization (WESO) method to increase the stability of the algorithm. The efficient calculation method of the element node sensitivity was established, which realizes the establishment of level set functions, smooth topological design of structures and the updating of design variables. The stability of the proposed algorithm was verified by the Zhou–Rozvany problem, two- and three-dimensional (3D) numerical results. The effectiveness and efficiency of the proposed algorithm was further verified by numerical comparison with other topology optimization frameworks. Lastly, the improved windowed ESO method was applied to the initial configuration design of the double-deck bridge structure, which not only provides guidance for its initial design but also demonstrates the applicability of the method in complex structural systems.

Topology Optimization for Harmonic Excitation Structures with Minimum Length Scale Control Using the Discrete Variable Method

Continuum topology optimization considering the vibration response is of great value in the engineering structure design. The aim of this study is to address the topological design optimization of harmonic excitation structures with minimum length scale control to facilitate structural manufacturing. A structural topology design based on discrete variables is proposed to avoid localized vibration modes, gray regions and fuzzy boundaries in harmonic excitation topology optimization. The topological design model and sensitivity formulation are derived. The requirement of minimum size control is transformed into a geometric constraint using the discrete variables. Consequently, thin bars, small holes, and sharp corners, which are not conducive to the manufacturing process, can be eliminated from the design results. The present optimization design can efficiently achieve a 0–1 topology configuration with a significantly improved resonance frequency in a wide range of excitation frequencies. Additionally, the optimal solution for harmonic excitation topology optimization is not necessarily symmetric when the load and support are symmetric, which is a distinct difference from the static optimization design. Hence, one-half of the design domain cannot be selected according to the load and support symmetry. Numerical examples are presented to demonstrate the effectiveness of the discrete variable design for excitation frequency topology optimization, and to improve the design manufacturability

A generalized objective function based on weight coefficient for topology-finding of tensegrity structures

This paper proposes a generalized objective function for the topology-finding of tensegrity structures to be able to assign selection priorities to different members and efficiently find multiple tensegrity structures through a single ground structure. The generalized objective function is constructed by the sum of the product of member internal forces and weight coefficients. The member weight coefficients can be defined and adjusted freely to change the selection priorities of different members in the topology-finding process. By adjusting the weight coefficients, different tensegrity structures can be generated. The weight coefficients can be determined by the designer according to the practical design requirements and preferences, e.g., member length limitations, member position requirements. In addition, a circular computing strategy is proposed for the weight coefficient adjustment to efficiently obtain a large number of tensegrity structures through a single ground structure. The topology design of typical regular tensegrity structures, as well as irregular ellipsoid tensegrity structures, are carried out to demonstrate the effectiveness of the proposed method. Furthermore, by using the proposed method, multiple novel tensegrity structures based on common Archimedes polyhedrons have been found; detailed information (e.g., member connectives, self-stress) are given as an open database for future investigation and applications.

Structural Topology Design Optimization of Fiber-Reinforced Composite Frames with Fundamental Frequency Constraints

Structural Topology Design Optimization of Fiber-Reinforced Composite Frames with Fundamental Frequency Constraints

Topology design and equivalent mechanical properties of a three-dimensional star-shaped auxetic structure

Topology design is the most important method of obtaining optimal material distribution that is directly related to the mechanical properties of structural materials. Aimed at improving the accuracy and efficiency of topology design, this paper explores an innovative topological technique that integrates a sigmoid-rational approximation of material properties approach, with a modified optimality criteria algorithm. Furthermore, a three-dimensional (3D) star-shaped auxetic structure, based on the sigmoid-rational approximation of material properties approach and a modified optimality criteria technique is proposed. The unit cell structural parameters are extracted, and the mechanical properties of the 3D star-shaped structure, including the relative density, equivalent elastic modulus, and equivalent Poisson's ratio, are investigated using mechanical derivation, numerical, and experimental methods. The subsequent results demonstrate that the sigmoid-rational approximation of material properties approach and a modified optimality criteria technique provides a feasible novel idea for multicellular structure development and that the advantageous 3D star-shaped auxetic structure has great potential within the field of structural safety.

Machine-specified ground structures for topology optimization of binary trusses using graph embedding policy network

This paper proposes the concept of machine-specified ground structures for topology optimization of trusses. Unlike general ground structures with dense and regular connectivity, machine-specified ground structures are sparse stable ground structures with a specified number of members designed by machines. Firstly, the generation process of machine-specified ground structures from a given node-set is formulated as a reinforcement learning task. Graph embedding is used to integrate the structural information into a comprehensive feature matrix to describe the state. By establishing the policy network, the probability of each action, i.e., selecting each node in the node-set, is obtained based on the comprehensive feature matrix. The task is solved using a gradient-based algorithm called REINFORCE. A randomized 4 × 4 node-set is used to train the agent. The policy converges with a high average reward, and generates different yet reasonable structures because a stochastic policy is employed. Besides, the agent can handle different-sized node-sets without re-training. Hence, the machine-specified ground structures generated by the trained agent can be utilized to assist the structural topology design. Subsequently, a method for a typical problem with singular optimal solutions, i.e., topology optimization of binary trusses with stress and displacement constraints, is proposed based on machine-specified ground structures. Finally, through different-sized numerical examples, it is demonstrated that the machine-specified ground structures lead to a variety of optimal solutions, and it is more likely to obtain the global optimum than fully-connected ground structures. It is worth noting that machine-specified ground structures can also be applied to other problems without re-training.

A review on the topology optimization of the fiber-reinforced composite structures

According to the requirements of the aerospace industry for high strength, high stiffness, and lightweight structural parts, topology optimization has been proved to be an effective product design method. As one of the most conceptual and prospective structural optimization design methods, topology optimization intends to seek the optimal layout of materials in an allowed design region under a given load and boundary conditions. Thus, the object of study in the article is the method of topological optimization of aircraft structures. The goal of this article is to analyze the existing approaches, algorithms, as well as application of the method of topological optimization in the aerospace field in applied problems. The tasks are to describe the existing various approaches methods, features, and research directions of topological optimization as well as to study the possibility of application in the manufacturing process of composite structures. The following results were obtained. The optimization methods are briefly explained and compared, and the advantages and limitations of each approach are discussed. The various ways of simultaneous optimization of fiber orientation and structural topology were described and analyzed. The features of different methods of continuous fiber orientation optimization method were reviewed. The discrete fiber orientation optimization methods were represented. The possibility of multi-scale concurrent topological optimization was described. The combination of topology optimization and additive manufacturing was considered. Finally, the topology optimization of FRC structures which have been resolved in literature are reviewed and the potential research fields requiring more investigation are pointed out. Conclusions. In the article, a comprehensive review of the topology optimization design of FRC structures was presented. The promising way is to combine topology optimization with additive manufacturing techniques. However, these proposed methods may not suitable for other more complex problems, such as bucking stability and natural frequency. Hence, the topology optimization design of complex FRC components under complicated conditions is the main challenge in the future. This can be a new trend in the topology design of FRC structures.

High rate capability composite particles with root-inspired hierarchical channel structure

Composite particle with high speed channels for delivering lithium is a promising candidate for the high performance electrode in electric vehicles. However, it still suffers from poor rate capability due to the long diffusion pathway caused by large particle size. Inspired by the efficient transporting ability of the hierarchical structure in tree roots, we design the channel structure to improve the rate capability using a diffuse interface model. This model enables us to simulate the fast lithium diffusion along the channel network and the electrochemical reaction on the particle surface. Our results demonstrate that two microstructure features of the channel network can efficiently enhance the rate capacity: One is the gradient distribution of the channel density with a higher channel density near the surface than that in the center; the other is the orientation of the channel structure parallel to lithium flux from the particle surface. The root-inspired hierarchical channel network displaying these two features can improve the capacity retention at large C-rates. Furthermore, our study can offer an efficient tool that can rationalize the topology design of channel structure in the purpose of achieving a high rate capability.

An Effective Feature Modeling Approach for 3D Structural Topology Design Optimization

An Effective Feature Modeling Approach for 3D Structural Topology Design Optimization

Design of mechanical metamaterial with controllable stiffness using curved beam unit cells

In this paper, a novel mechanical metamaterial with controllable stiffness is proposed based on the structural topology and rational design of curved beam unit cells. First, both straight and curved beams are mechanically analyzed, and the corresponding stiffness of these two beams is compared. Second, various curved beam unit cells are presented based on fractal geometries, and lattice structures assembled from curved beam unit cells are optimally designed and precisely fabricated by means of 3D printing techniques. Finally, multi-layered mechanical metamaterials are proposed and mechanically tested, which exhibit advantages of improved specific stiffness and reduced spatial volume. The theoretical and experimental results are verified using the finite element method (FEM). This study proposes a novel design principle for mechanical metamaterials with adjustable stiffness using curved beam unit cells, which can provide a theoretical basis for the development of mechanical metamaterials and present promising potentials in versatile engineering applications.

Topology optimization of elasto-plastic structures under reliability constraints: A first order approach

The objective of this study is to propose a relatively simple and efficient method for reliability based topology optimization for structures made of elasto-plastic material. The process of determining the optimal topology of elasto-perfectly plastic structures is associated with the removal of material from the structure. Such a process leads to weakening of structural strength and stiffness causing at the same time increase the likelihood of structural failure. An important aspect of engineering design is to track this probability during the optimization process and not allow the structure safety to exceed a certain level specified by the designer. The purpose of this work is to combine the previously developed yield-limited topology optimization method with reliability analysis using first order approach. Effectiveness of the proposed methodology is demonstrated on benchmark problems proposed by Rozvany and Maute, and the elasto-plastic topology design of L-shape structure which is frequently used in different approaches for stress constrained topology optimization.

Spatial zoning for better structural topology design and performance

A new three-dimensional spatial zoning procedure that has been tailored to structural design is presented in this work. Zoning is the grouping of spaces or sub-parts of spaces such that the resulting spatial layout is logical from a disciplinary perspective. Two automated structural design methods, so-called grammars, have been defined to generate structural designs for zoned building spatial designs. The first grammar, here termed stable design grammar, places structural slabs at the boundaries of a zone, and columns at places where a zone intersects with a space. The second grammar, here termed unstable grammar, generates a structural frame that is unstable by definition. Via a new stabilization method, which is also presented in this work, the unstable structural frame is stabilized by considering the boundaries of a zone for the placement of stabilizing measures. In a case study, the zoning procedure is applied in conjunction with each of the design grammars to three distinct building spatial designs. The case study shows—for both design grammars—that structural designs use less material and are stiffer when they are generated based on a zoned building spatial design. Moreover, the zoning procedure has been tailored such that it yields a subset of all possible zoned designs for which, in general, the generated structural designs perform well.

Three-dimensional high resolution topology optimization considering additive manufacturing constraints

This paper studies additive manufacturing oriented structural topology optimization with SIMP approach and aims at 3D high-resolution printable structural topology design with overhang and horizontal minimum length control for minimum compliance.To start with, we construct a hyperplane in ℝ4 by fitting a local element density distribution in the 18 Elements Scheme, use its gradient to estimate the overhang angle, the directional-dependent overhang angle and formulate the corresponding constraints. Next, we propose a Horizontal Square Scheme and four support sets around the concerned element. The horizontal minimum length is controlled by forbidding the concerned element’s density to be larger than the average density of the elements in one of the support sets. By combining these two constraints, the hanging feature is well suppressed.A new implementation scheme with an improved weight function is proposed to meet these element wise AM constraints well. To get high resolution structural boundaries with low computational efforts, this paper applies the multiresolution topology optimization (MTOP) method.The structural TO problem is solved by MMA. A number of numerical examples and AM experiments show the effectiveness of this method. The present approach works efficiently when the building direction is in slight misalignment with the vertical direction.

Topology design of binary structures subjected to design-dependent thermal expansion and fluid pressure loads

The future perspective of using topology optimization to solve challenging design-dependent physics problems motivates the creation of methods with clear structural boundaries and well-defined volume. This paper develops the topology optimization of binary structures (TOBS) method to include design-dependent fluid pressure and constant thermal expansion loads. Topology design in thermoelastic and fluid pressure problems have been only handled separately up to date. To the authors’ best knowledge, this is the first work to consider both type of loads simultaneously within a structural topology optimization framework. The TOBS method uses discrete design variables, sensitivity filtering, and formal mathematical programming (integer linear optimization) to achieve convergent and mesh-independent solutions. The discrete nature of the method presents attractive features when dealing with design-dependent body and surface loads. In this paper, we use the structural mean compliance and volume as functions for optimization. The sensitivity analysis is carried out using the adjoint and semi-analytical methods. Numerous examples are shown to design novel structural designs which perform well under the applied fluid pressure and thermal loads. The observed computational times signify the practicability of integer programming for structural optimization problems.