On various aspects of evolutionary structural optimization for problems with stiffness constraints

The Evolutionary Structural Optimization (ESO) has been applied to find the optimal structural topology for design and construction purposes. However, its efficiency is low due to the imperfect deletion criteria. This paper presents an improved elimination criterion and the Windowed Evolutionary Structural Optimization (WESO) method. The former considers the structural average strain energy as the elimination criteria in the optimization, with the elimination rate set as a self-adaption state by an adjustable window. Thus, the problems of low optimization efficiency and distorted optimization results in the traditional ESO can be solved to a certain extent. The WESO method can then be extended and applied with a complex finite element model. It is found that the method has better optimal ability in handling discrete elements in vast structures. Compared with the other method, the WESO method has high computational efficiency and offers more stable and reliable topology than the Michell theoretical solution. Lastly, upon developing the optimal topology for a double-deck structure, it is shown that the WESO method has a great potential in civil engineering applications.

Topology optimization of the inner reinforcement for a vehicle’s hood has been performed by evolutionary structural optimization (ESO) method. The purpose of this study is to obtain optimal topology of the inner reinforcement for a vehicle’s hood considering static stiffness and natural frequency simultaneously. To do this, the multiobjective design optimization technique was implemented. From several combinations of weighting factors, a Pareto-optimal solution was obtained. Optimal topologies were obtained by the ESO method, i.e., by eliminating the elements having the lowest efficiency from the structural domain. As the weighting factor of the elastic strain efficiency goes from 1 to zero, it is found that the optimal topologies transmits from the optimal topology of static stiffness problem to that of natural frequency problem. Therefore, it was concluded that ESO method is effectively applied to topology optimization of the inner reinforcement of a vehicle’s hood.

This paper examines the evolutionary structural optimisation (ESO) method and its shortcomings. By proposing a problem statement for ESO followed by an accurate sensitivity analysis a framework is presented in which ESO is mathematically justifiable. It is shown that when using a sufficiently accurate sensitivity analysis, ESO method is not prone to the problem proposed by Zhou and Rozvany (Struct Multidiscip Optim 21(1):80---83, 2001). A complementary discussion on previous communications about ESO and strategies to overcome the Zhou-Rozvany problem is also presented. Finally it is shown that even the proposed rigorous ESO approach can result in highly inefficient local optima. It is discussed that the reason behind this shortcoming is ESO's inherent unidirectional approach. It is thus concluded that the ESO method should only be used on a very limited class of optimisation problems where the problem's constraints demand a unidirectional approach to final solutions. It is also discussed that the Bidirectional ESO (BESO) method is not prone to this shortcoming and it is suggested that the two methods be considered as completely separate optimisation techniques.

The evolutionary structural optimization (ESO) method developed by Xie and Steven (Comput Struct 49(5):885–896, 162), an important branch of topology optimization, has undergone tremendous development over the past decades. Among all its variants, the convergent and mesh-independent bi-directional evolutionary structural optimization (BESO) method developed by Huang and Xie (Finite Elem Anal Des 43(14):1039–1049, 48) allowing both material removal and addition, has become a widely adopted design methodology for both academic research and engineering applications because of its efficiency and robustness. This paper intends to present a comprehensive review on the development of ESO-type methods, in particular the latest convergent and mesh-independent BESO method is highlighted. Recent applications of the BESO method to the design of advanced structures and materials are summarized. Compact Malab codes using the BESO method for benchmark structural and material microstructural designs are also provided.

The purpose of this study was to develop a new element removal method for ESO (Evolutionary Structural Optimization), which is one of the topology optimization methods, ESO starts with the maximum allowable design space and the optimal topology emerges by a process of removal of lowly stressed elements. The element removal ratio of ESO is fixed throughout topology optimization at 1 or 2%. BESO (bidirectional ESO) starts with either the least number of elements connecting the loads to the supports, or an initial design domain that fits within the maximum allowable domain, and the optimal topology evolves by adding or subtracting elements. But the convergence rate of BESO is also very slow. In this paper, a new element removal method for ESO was developed for improvement of the convergence rate. Then it was applied to the same problems as those in papers published previously. From the results, it was verified that the convergence rate was significantly improved compared with ESO as well as BESO.

This paper presents a new strategy to distribute two different materials for multi-material topology optimization. Extended from the bi-directional evolutionary structural optimization (BESO) method for a single material, the multi-material bidirectional evolutionary structural optimization (MBESO) method has been developed, which can effectively handle the topology optimization problems involving two materials like steel and concrete. However, in some special cases, overloading of part of the compressed material occurs in the multi-material structures designed by using the MBESO method. Aimed to solve this critical problem, a simple but effective strategy is proposed in this paper. In steel-concrete composite structures, for instance, the overloaded compressed concrete elements with exceedingly high stress are replaced with steel material. The small amount of steel material added to the highly compressed region can effectively reduce the maximum compressive stress of the concrete material to a safe level. The comparison between the original MBESO method and the improved strategy based on a series of two-dimensional and three-dimensional examples clearly demonstrates the effectiveness of the proposed strategy in enhancing the structural safety and strength of the topologically optimized composite structures. This distinctly different material distribution strategy shows its potential and value in multi-material topology optimization research and applications.

This paper deals with simultaneous topology and shape optimization of elastic contact problems. The structural optimization problem for an elastic contact problem is formulated. Shape as well as topological derivatives formulae of the cost functional are provided using material derivative and asymptotic expansion methods, respectively. These derivatives are employed to formulate necessary optimality condition for simultaneous shape and topology optimization and to calculate a descent direction in numerical algorithm. Level set based numerical algorithm for the solution of this optimization problem is proposed. Numerical examples are provided and discussed.

Since efficient material usage in a structure is an essential intention in engineering design, objective of a structural optimization algorithm should be obtaining the lightest and stiffest structure while considering specific design constraints. Evolutionary structural optimization (ESO) method has been introduced by Xie and Steven in 1993 to overcome structural optimization problems by improving and optimizing the design of structures. Although the algorithm depends on a plain inspiration, performance level of the ESO algorithm has increased owing to the contributions of wide-ranging studies in the subsequent years. This heuristic method has appeared on a simple—sometimes defined as intuitive—idea that is eliminating (or removing) the inefficient elements from the initial design region and attaining a fully stressed structure with an updated topology. Such a process is performed iteratively until an optimum layout is achieved, or until a specified volume is acquired; and the decision of process termination is executed over the feasibility of the designs achieved. This optimization algorithm is based on finite element method and has showed its superiorities by subsequent studies in various disciplines. Within this chapter, a brief on the basis and the main improvement steps of evolutionary structural optimization is presented. Different approaches on objectives, constraints, processing and terminating will be summarized. Thereafter, efficiency and capability of the method will be demonstrated over some examples.

Structural topology optimization is a crucial approach for achieving lightweight design. An effective topology optimization algorithm must strike a balance between the objective functions, constraints, and design variables, which essentially reflects the symmetry and tradeoff between the objective and constraints. In this study, a topology optimization method grounded in load path theory is proposed. Element bearing capacity is quantified using the element birth and death method, with an explicit formulation derived via finite element theory. The effectiveness in evaluating structural performance is assessed through comparisons with stress distributions and topology optimization density maps. In addition, a novel evaluation index for element bearing capacity is proposed as the objective function in the topology optimization model, which is validated through thin plate optimization. Subsequently, sensitivity redistribution mitigates checkerboard patterns, while mesh filtering suppresses multi-branch structures and prevents local optima. The method is applied for the lightweight design of a triangular arm, with results benchmarked against the variable density method, demonstrating the feasibility and effectiveness of the proposed method. The element bearing capacity seeks to homogenize the load distribution of each element; the technique in this study can be extended to the optimization of symmetric structures.

A strength-based multiple cutout optimization in composite plates using fixed grid finite element method

In this paper, a topological preform design optimisation method was proposed based on the evolutionary structural optimisation (ESO) method. In this algorithm, new criteria for element elimination and addition were developed and a boundary smooth technique was employed to improve the boundary quality. Two 2D case problems including forging of aerofoil blade and forging of rail wheel are evaluated using the developed method. The results suggest that this topology optimisation method is an efficient approach for preform design optimisation.

Stress-based topology optimization and nonlinear structural topology optimization is gaining increasing attention in order to make topology optimization more realistic. Thus, this paper extends current concepts of topology optimization to the design of structures made of nonlinear materials. An extended bi-directional evolutionary structural optimization (BESO) method for stress minimization topology optimization of material nonlinear structures is proposed in this work. BESO method based on discrete variables can effectively avoid the well-known singularity problem in density-based methods with low-density elements. The maximum von Mises stress is approximated by the p-norm global stress. The sensitivity information for designing variable updates is derived in detail by adjoint method. As for the highly nonlinear stress behavior, the updated scheme takes advantages from two filters respectively of the sensitivity and topological variables to improve convergence. Moreover, the filtered sensitivity numbers are combined with their historical sensitivity information to further stabilize the optimization process. The effectiveness of the proposed method is demonstrated by several 2D benchmark design problems.

A Multi-volume constraint approach to diverse form designs from topology optimization

Detail control strategies for topology optimization in architectural design and development

This paper proposes a new bi-directional evolutionary structural optimization (BESO) method with draw direction constraints. Draw direction constraints, defined by required manufacturing process, are achieved by modifying element removal/addition criteria such that elements are removed from the top surface of the draw direction to the inner design domain. The optimized design with draw direction constraints is free from hollow or closed cavity geometries which are infeasible for manufacturing. A stiffness design of a motor front cover is carried out to show the ability of the proposed method in practical mechanical design problems.