Bi-directional Evolutionary Structural Optimization Research Articles

Bi-directional Evolutionary Structural Optimization on Advanced Structures and Materials: A Comprehensive Review

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.

An Efficient Method for Topology Optimization of Continuum Structures in the Presence of Uncertainty in Loading Direction

This paper presents a simple yet efficient method for the topology optimization of continuum structures considering interval uncertainties in loading directions. Interval mathematics is employed to equivalently transform the uncertain topology optimization problem into a deterministic one with multiple load cases. An efficient soft-kill bi-directional evolutionary structural optimization (BESO) method is proposed to solve the problem, which only requires two finite element analyses per iteration for each external load with directional uncertainty regardless of the number of the multiple load cases. The presented algorithm leads to significant computational savings when compared with Monte Carlo-based optimization (MCBO) algorithms. A series of numerical examples including symmetric and nonsymmetric loading variations demonstrate the considerable improvement of computational efficiency of the proposed approach as well as the significance of including uncertainties in topology optimization when to design a structure. Optimums obtained from the proposed algorithm are verified by MCBO method.

Concurrent topological design of composite structures and the underlying multi-phase materials

This paper presents a concurrent topology optimization approach for simultaneous design of composite structures and their periodic material microstructures with three or more phases. The effective properties of multi-phase materials are obtained via homogenization technique which serves as a bridge of the finite element models of the macrostructure and the material microstructure. The base materials of periodic microstructures used in each phase of the macrostructure are divided into several groups and sensitivity analysis are carried out on them one by one. Meanwhile, the sensitivity number at the macrostructure is derived which is coupled with the designed material properties. Then, the composite configurations of material microstructures and macrostructures are inversely optimized concurrently based on the bi-directional evolutionary structural optimization (BESO) algorithm. Several 2D and 3D numerical examples are presented to demonstrate the effectiveness of proposed design approach.

A novel two-stage optimization method for beam–plate structure design

In this article, a novel optimization method for beam–plate structure is proposed based on two optimization levels, including the topology optimization and the section optimization. First, the optimal topology of structure with fixed material and dimensions of design domain is obtained using bidirectional evolutionary structure optimization method, and then, the section optimization is made to produce final optimal solution based on response surface model. In order to deal with beam–plate structures, the traditional bidirectional evolutionary structure optimization method is improved using cubic box as the unit cell instead of solid unit to construct periodic lattice structure. To handle irregular structural element layout produced in bidirectional evolutionary structure optimization method, section conversion process is employed to smooth the edges of sections; in this way, the manufacturability of optimal solution will be significantly improved. To determine the section parameters, response surface method is used to perform the section optimization, and the structure mass is further reduced. One example of cantilever beam structure design is provided and discussed. The applied two-stage optimization procedure resulted in 86 iteration steps to obtain the optimal topology, and a further decrease in 2.65% mass on the basis of optimal topology, which shows that the proposed method has advantages on convergence rate and computational cost.

A simple alternative formulation for structural optimisation with dynamic and buckling objectives

Structural topology optimisation has mainly been applied to strength and stiffness objectives, due to the ease of calculating the sensitivities for such problems. In contrast, dynamic and buckling objectives require time consuming central difference schemes, or inefficient non-gradient algorithms, for calculation of the sensitivities. Further, soft-kill algorithms suffer from numerous numerical issues, such as localised artificial modes and mode switching. This has resulted in little focus on structural topology optimisation for dynamic and buckling objectives. In this work it is found that nominal stress contours can be derived from applying the vibration and buckling mode shapes as displacement fields, defined as the dynamic and buckling von Mises stress, respectively. This paper shows that there is an equivalence between the dynamic von Mises stress and the frequency sensitivity numbers for element removal and addition in bidirectional evolutionary structural optimisation. Likewise, it was found that the contours of buckling von Mises stress and buckling sensitivity numbers are analogous; therefore, an equivalence is shown for element removal and addition. The examples demonstrate consistent resulting topologies from the two different formulations for both dynamic and buckling criteria. This article aims to develop a simple alternative, based on visual correlation with a mathematical verification, for topology optimisation with dynamic and buckling criteria.

Design and Fabrication of Structural Connections Using Bi-Directional Evolutionary Structural Optimization and Additive Manufacturing

The need to simplify the construction issues of complex structures leads to definition of SmartNodes project as a research which aims to confine the complexity of structure to a limited area (nodes) in order to decrease processing steps and labor intensity by application of additive manufacturing (AM) techniques. Bi-Directional Evolutionary Structural Optimization (BESO) is used to design efficient and elegant nodal connections of large scale spatial structures and minimise the volume of nodes to be printed and to ultimately replace welded, forged and cast connections by 3D printed connections. The prototypes discussed in this paper demonstrate BESO design process through two generic cases.

Topology optimization for submerged buoyant structures

ABSTRACTThis paper presents an evolutionary structural topology optimization method for the design of completely submerged buoyant modules with design-dependent fluid pressure loading. This type of structure is used to support offshore rig installation and pipeline transportation at all water depths. The proposed optimization method seeks to identify the buoy design that has the highest stiffness, allowing it to withstand deepwater pressure, uses the least material and has a minimum prescribed buoyancy. Laplace's equation is used to simulate underwater fluid pressure, and a polymer buoyancy module is considered to be linearly elastic. Both domains are solved with the finite element method. Using an extended bi-directional evolutionary structural optimization (BESO) method, the design-dependent pressure loads are modelled in a straightforward manner without any need for pressure surface parametrization. A new buoyancy inequality constraint sets a minimum required buoyancy effect, measured by the joint volume of the structure and its interior voids. Solid elements with low strain energy are iteratively removed from the initial design domain until a certain prescribed volume fraction. A test case is described to validate the optimization problem, and a buoy design problem is used to explore the features of the proposed method.

Topology optimization of photonic structures for all-angle negative refraction

This paper proposes a new topology optimization algorithm based on the bi-directional evolutionary structural optimization (BESO) method to design photonic crystals with broad all-angle negative refraction (AANR) frequency range. The photonic crystals are assumed to be two-dimensional periodical structures, which consist of dielectric materials and air. The conditions for the occurrence of AANR are identified and the design objective is to enlarge the AANR frequency range. The BESO algorithm is proposed based on finite element analysis for band diagrams of photonic crystals and the derived sensitivity numbers. Starting from a simple initial design without any AANR, BESO gradually re-distributes the dielectric materials within the periodical unit cell so that the AANR property emerges and its frequency range is enlarged accordingly. The numerical results show that the proposed BESO algorithm can effectively obtain AANR photonic crystals with novel patterns. The effects of dielectric permittivity contrast of two constituent materials, mesh-refinement and filter are discussed.

Concurrent topological design of composite thermoelastic macrostructure and microstructure with multi-phase material for maximum stiffness

A method for the multi-scale design of composite thermoelastic macrostructure and periodic microstructure with multi-phase materials is proposed. A concurrent topology optimization model of macrostructure and microstructure is established, where the objective is to maximize the macrostructural stiffness subject to volume constraints on the macro-material distribution and phase materials. Based on the material interpolation scheme of the solid isotropic material with penalization (SIMP), the sensitivity of the mean compliance of the composite macrostructure with respect to design variables on two scales, i.e., macro and micro scales, is derived. The optimization problem is solved using a bi-directional evolutionary structural optimization (BESO) method and the corresponding optimization procedure for the concurrent topology optimization is proposed. Several examples are presented to demonstrate the effectiveness of the proposed method.

Dynamic response reliability based topological optimization of continuum structures involving multi-phase materials

For the purpose of improving structural dynamic reliability, a topological optimization methodology for maximizing the dynamic response reliability index is proposed based on the bi-directional evolutionary structural optimization (BESO) method. The objective of the present study is to maximize the dynamic response reliability index at specified points on the structure under random excitation, subject to volume constraints on multi-phase materials over the admissible design domain. The sensitivity of the dynamic response reliability index with respect to the design variables is derived. The optimization procedure of the extended BESO method is presented. A series of numerical examples in both 2D and 3D are presented to demonstrate the effectiveness and efficiency of the proposed approach.

Topological Design of Cellular Phononic Band Gap Crystals.

This paper systematically investigated the topological design of cellular phononic crystals with a maximized gap size between two adjacent bands. Considering that the obtained structures may sustain a certain amount of static loadings, it is desirable to ensure the optimized designs to have a relatively high stiffness. To tackle this issue, we conducted a multiple objective optimization to maximize band gap size and bulk or shear modulus simultaneously with a prescribed volume fraction of solid material so that the resulting structures can be lightweight, as well. In particular, we first conducted the finite element analysis of the phononic band gap crystals and then adapted a very efficient optimization procedure to resolve this problem based on bi-directional evolutionary structure optimization (BESO) algorithm in conjunction with the homogenization method. A number of optimization results for maximizing band gaps with bulk and shear modulus constraints are presented for out-of-plane and in-plane modes. Numerical results showed that the optimized structures are similar to those obtained for composite case, except that additional slim connections are added in the cellular case to support the propagation of shear wave modes and meanwhile to satisfy the prescribed bulk or shear modulus constraints.

Evolutionary topological design for phononic band gap crystals

Phononic band gap crystals are made of periodic inclusions embedded in a base material, which can forbid the propagation of elastic and acoustic waves within certain range of frequencies. In the past two decades, the systematic design of phononic band gap crystals has attracted increasing attention due to their wide practical applications such as sound insulation, waveguides, or acoustic wave filtering. This paper proposes a new topology optimization algorithm based on bi-directional evolutionary structural optimization (BESO) method and finite element analysis for the design of phononic band gap crystals. The study on the maximizing gap size between two adjacent bands has been systematically conducted for out-of-plane waves, in-plane waves and the coupled in-plane and out-of-plane waves. Numerical results demonstrate that the proposed optimization algorithm is effective and efficient for the design of phononic band gap crystals and various topological patterns of optimized phononic structures are presented. Several new patterns for phononic band gap crystals have been successfully obtained.

Structural design of stamping die components using bi-directional evolutionary structural optimization method

Nowadays, the casting structure of stamping dies is designed according to die design standards. These standards are usually not based on a structural optimization algorithm and often rely on high safety factors which cause the weight of die components to be more than required. This in turn calls for higher prices of dies and production energy required per part. Therefore, alternative methods to reduce the weight of these components are required. In this paper, a software package is presented which can design an improved structure of stamping dies with a substantial reduction in weight. This package implements Abaqus software and uses the bi-directional evolutionary structural optimization (BESO) method to create a new lighter structure which resembles the shape of the sheet metal part and applied forces in the operation. It obtains the desired optimum design by removing from and adding material to the die component structure. This method involves adding material to that part of the component where the structure is overstressed and simultaneously removing material where the structure is understressed. This procedure is carried out again and again until the objective function is minimized. Finally, the proposed structure can also be reconstructed by the designer to accommodate for a simpler casting method. The operation of the software is demonstrated by an example where the dies for a sheet metal part are studied. The die components are initially designed, analyzed, and compared with the standard die (the die which is in general use today). The final results show a reduction of 31 % of volume while the maximum displacement and stress of the die do not change approximately. This software package is developed in a Microsoft Visual C# programming environment with a link to Abaqus software to analyze finite element simulation processes.

Two-scale dynamic optimal design of composite structures in the time domain using equivalent static loads

In this paper, a method for the concurrent topology optimization of macrostructural material distribution and periodic microstructure under dynamic loading based on the equivalent static loads (ESLs) in the time domain is proposed. The sensitivity analysis of the maximal dynamic strain energy with respect to design variables in two scales, i.e., macro and micro scales, is presented. The corresponding concurrent topology optimization model of macrostructure and microstructure is established, where the objective is to minimize the maximal dynamic strain energy of the macrostructure subject to volume constraints on the macromaterial distribution and phase materials. The optimization problem is solved using a bi-directional evolutionary structural optimization (BESO) method. The results of several examples demonstrate that the proposed method is effective for the concurrent topology optimization under dynamic loading.

A New Approach Based on Strain Sensitivity for Reinforcement Optimization in Slope Stability Problems

The shape of the reinforcement for stabilizing slopes in geotechnical application is usually based on empirical methods, and is often in the shape of a grouted curtain to cover the slope surface to increase the shear strength. The reinforcement optimization is an important measure to achieve the most stable shape with the determined amount of materials. This paper aims at exploring the state-of-the-art design of reinforcement to approach stability in the slope using minimum amount of the reinforcement. In this research, enhancement in the bi-directional evolutionary structural optimization method has been made to optimize the reinforced layers in slope stabilization design. Through an iterative approach, the stability of slope is increased by addition of reinforced material to minimize total deformation. A new analytical technique for the slope reinforcement optimization has been introduced based on strain sensitivity analysis. The obtained results show a reasonable resemblance to the practical slope reinforcement solutions. The process of optimization requires the finite element analysis to assess elements respond which is carried out by finite element package ABAQUS in this research.

Topology optimization of continuum structures with uncertain-but-bounded parameters for maximum non-probabilistic reliability of frequency requirement

A method for the non-probabilistic reliability optimization on frequency of continuum structures with uncertain-but-bounded parameters is proposed. The objective function is to maximize the non-probabilistic reliability index of frequency requirement.The corresponding bi-level optimization model is built, where the constraints are applied on the material volume in the outer loop and the limit state equation in the inner loop. The non-probabilistic reliability index of frequency requirement is derived by the analytical method for the continuum structure with the uncertain elastic module and mass density. Further, the sensitivity of the non-probabilistic reliability index with respect to the design variables is analyzed. The topology optimization in the outer loop is performed by a bi-directional evolutionary structural optimization (BESO) method, where the numerical techniques and the optimization procedure of BESO method are presented. Numerical results show that the proposed BESO method is efficient, and convergent optimal solutions can be achieved for a variety of optimization problems on frequency non-probabilistic reliability of continuum structures.

Concurrent topology optimization for minimizing frequency responses of two-level hierarchical structures

This paper presents a concurrent topology optimization methodology for minimizing the frequency responses of multiscale systems composed of macro and micro phases. Although there is existing research on the topology optimization of structures and optimization of the materials for frequency responses, topology optimization approaches considering both scales simultaneously are relatively limited. The methodology proposed here aims to apply the bi-directional evolutionary structural optimization (BESO) method to find the optimum layout on both macro and micro scales of the structure, with the objective of minimizing the frequency response in the macro structure. For this coupled system, it is assumed that the macro structure is composed of a periodic material whose effective properties are obtained using the homogenization theory. The designs of the macro and micro structures are conducted simultaneously. The homogenized elasticity matrix used in the finite element analysis of the macro structure is obtained from considering the layout of the micro structure. A series of numerical examples are presented to validate the optimization procedure and to demonstrate the effectiveness and the efficiency of the proposed method.

Topological design of all‐ceramic dental bridges for enhancing fracture resistance

Layered all-ceramic systems have been increasingly adopted in major dental prostheses. However, ceramics are inherently brittle, and they often subject to premature failure under high occlusion forces especially in the posterior region. This study aimed to develop mechanically sound novel topological designs for all-ceramic dental bridges by minimizing the fracture incidence under given loading conditions. A bi-directional evolutionary structural optimization (BESO) technique is implemented within the extended finite element method (XFEM) framework. Extended finite element method allows modeling crack initiation and propagation inside all-ceramic restoration systems. Following this, BESO searches the optimum distribution of two different ceramic materials, namely porcelain and zirconia, for minimizing fracture incidence. A performance index, as per a ratio of peak tensile stress to material strength, is used as a design objective. In this study, the novel XFEM based BESO topology optimization significantly improved structural strength by minimizing performance index for suppressing fracture incidence in the structures. As expected, the fracture resistance and factor of safety of fixed partial dentures structure increased upon redistributing zirconia and porcelain in the optimal topological configuration. Dental CAD/CAM systems and the emerging 3D printing technology were commercially available to facilitate implementation of such a computational design, exhibiting considerable potential for clinical application in the future. Copyright © 2015 John Wiley & Sons, Ltd.

Layout optimization of continuum structures considering the probabilistic and fuzzy directional uncertainty of applied loads based on the cloud model

This paper reports an efficient approach for uncertain topology optimization in which the uncertain optimization problem is equivalent to that of solving a deterministic topology optimization problem with multiple load cases. Probabilistic and fuzzy property of the directional uncertainty of the applied loads is considered in the topology optimization; the cloud model is employed to describe that property which can also take the correlations of the probability and fuzziness into account. Convergent and mesh-independent bi-directional evolutionary structural optimization (BESO) algorithms are utilized to obtain the final optimal solution. The proposed method is suitable for linear elastic problems with uncertain applied loads, subject to volume constraint. Several numerical examples are presented to demonstrate the capability and effectiveness of the proposed approach. In-depth discussions are also given on the effects of considering the probability and fuzziness of the directions of the applied loads on the final layout.

Bi-directional evolutionary optimization for photonic band gap structures

Toward an efficient and easy-implement optimization for photonic band gap structures, this paper extends the bi-directional evolutionary structural optimization (BESO) method for maximizing photonic band gaps. Photonic crystals are assumed to be periodically composed of two dielectric materials with the different permittivity. Based on the finite element analysis and sensitivity analysis, BESO starts from a simple initial design without any band gap and gradually re-distributes dielectric materials within the unit cell so that the resulting photonic crystal possesses a maximum band gap between two specified adjacent bands. Numerical examples demonstrated the proposed optimization algorithm can successfully obtain the band gaps from the first to the tenth band for both transverse magnetic and electric polarizations. Some optimized photonic crystals exhibit novel patterns markedly different from traditional designs of photonic crystals.