The ESO method revisited

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.

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.

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.

The “hard-kill” optimization methods such as evolutionary structural optimization (ESO) and bidirectional evolutionary structural optimization (BESO) may result in a nonoptimal design (Zhou and Rozvany in Struct Multidisc Optim 21:80–83, 2001) when these methods are implemented and used inadequately. This note further examines this important problem and shows that failure of ESO may occur when a prescribed boundary support is broken for a statically indeterminate structure. When a boundary support is broken, the structural system could be completely changed from the one originally defined in the initial design and even BESO would not be able to rectify the nonoptimal design. To avoid this problem, it is imperative that the prescribed boundary conditions for the structure be checked and maintained at each iteration during the optimization process. Several simple procedures for solving this problem are suggested. The benchmark problem proposed by Zhou and Rozvany (Struct Multidisc Optim 21:80–83, 2001) is revisited, and it is shown that the highly nonoptimal design can be easily avoided.

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 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.

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.

Evolutionary Structural Optimization (ESO) and its later version bi-directional ESO (BESO) have gained widespread popularity among researchers in structural optimization and practitioners in engineering and architecture. However, there have also been many critical comments on various aspects of ESO/BESO. To address those criticisms, we have carried out extensive work to improve the original ESO/BESO algorithms in recent years. This paper summarizes latest developments in BESO for stiffness optimization problems and compares BESO with other well-established optimization methods. Through a series of numerical examples, this paper provides answers to those critical comments and shows the validity and effectiveness of the evolutionary structural optimization method.

The effectiveness and efficiency of the bi-directional evolutionary structural optimization (BESO) method has been demonstrated on the problems of minimizing compliance of structures with f ixed external loads. This paper considers the minimization of the mean compliance of continuum structure subjected to design-dependent gravity loads. Due to the non-monotonous behaviour for this type ofthe optimization problems, the extended BESO method using discrete design variables has difficulty in obtaining convergent solutions for such problems. In this paper, a new BESO method is developed based on the sensitivity number computation utilizing an alternative material interpolation scheme. Several examples are presented to demonstrate the capabilities of the proposed method for achieving convergent optimal solutions for structures with design-dependent gravity loads.

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

Determination of strut-and-tie models using evolutionary structural optimization

Owing to advancement in advanced manufacturing technology, the reinforcement design of concrete structures has become an important topic in structural engineering. Based on bi-directional evolutionary structural optimization (BESO), a new approach is developed in this study to optimize the reinforcement layout in steel-reinforced concrete (SRC) structures. This approach combines a minimum compliance objective function with a hybrid truss-continuum model. Furthermore, a modified bi-directional evolutionary structural optimization (M-BESO) method is proposed to control the level of tensile stress in concrete. To fully utilize the tensile strength of steel and the compressive strength of concrete, the optimization sensitivity of steel in a concrete–steel composite is integrated with the average normal stress of a neighboring concrete. To demonstrate the effectiveness of the proposed procedures, reinforcement layout optimizations of a simply supported beam, a corbel, and a wall with a window are conducted. Clear steel trajectories of SRC structures can be obtained using both methods. The area of critical tensile stress in concrete yielded by the M-BESO is more than 40% lower than that yielded by the uniform design and BESO. Hence, the M-BESO facilitates a fully digital workflow that can be extremely effective for improving the design of steel reinforcements in concrete structures.

ABSTRACTAn underwater glider is a type of autonomous profiling instrument platform used for gathering data to explore the ocean. Having a neutrally buoyant glider hull is one way to improve the glider's endurance with a passive compensation for buoyancy change. This article applies the bi-directional evolutionary structural optimization (BESO) method to the optimization of an underwater glider hull, based on two materials. Firstly, the method for determining the glider's neutral buoyancy is carried out. Secondly, the optimization problem is defined and the optimization procedure is presented. In the BESO procedure, the original design area elements with low strain energy are iteratively switched from high-value materials to low-value materials until a prescribed fraction is reached. Finally, an optimal underwater glider design is generated and the result demonstrates a reasonable material distribution of the neutrally buoyant glider hull. A 26.4% buoyancy adjustment is achieved and the mass of the glider is decreased by 31%.

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