Bi-directional Evolutionary Structural Optimization Research Articles

A hybrid method for fast and robust topology optimization

To accelerate the convergence rate and obtain robust optimal results with clear profiles of structural topologies, this paper proposes a hybrid multi-population genetic algorithm (MPGA) and bi-directional evolutionary structural optimization (BESO) method for structural topology optimization. Each element in the design domain is treated as an individual and the elemental sensitivity is taken as the fitness function of one individual. Based on these treatments, MPGA operators, including crossover, mutation, migration and selection, are modified to adapt to compliance minimization problems. Additionally, some key parameters are controlled to guarantee a convergent solution and to solve the structural unconnectivity problem. A case is used to verify the effectiveness and efficiency of the proposed method. The numerical results show that the proposed method is efficient, and compared with the BESO method and combined simple genetic algorithm and BESO method, the proposed method provides a powerful ability in searching for better robust solutions and improving convergence speed.

Topology Optimization of Aircraft Components for Increased Sustainability

The aim of this work is to explore a topology optimization approach for the design of load-bearing aircraft components. The main objective of the approach is to develop a design methodology that reduces the maximum stress in a component for a given applied load and weight constraint. Consequently, the load carrying capacity and fatigue life of the component are maximized, resulting in less downtime and increased sustainability of the aircraft. A representative component geometry has been selected that is a potential candidate for additive manufacturing in order to enable the enhanced design freedom required of topology optimization approaches. Both linear and nonlinear-static finite element analyses are performed to validate the improved performance of the topologically optimized component, as well as contact analyses to demonstrate that the resulting design cannot be further improved without redesigning the complete assembly. The optimization problem is formulated and solved using a bidirectional evolutionary structural optimization method. The method demonstrates large reductions in maximum stress, leading to significant increases in fatigue life, potentially compensating for the degradation in fatigue properties inherent in additively manufactured metallic structures. The added design flexibility of additive manufacturing combined with topology optimization offer the potential for an anywhere, anytime manufacturing capability that would greatly benefit aircraft sustainability.

Topology Optimization Methods for 3D Structural Problems: A Comparative Study

The work provides an exhaustive comparison of some representative families of topology optimization methods for 3D structural optimization, such as the Solid Isotropic Material with Penalization (SIMP), the Level-set, the Bidirectional Evolutionary Structural Optimization (BESO), and the Variational Topology Optimization (VARTOP) methods. The main differences and similarities of these approaches are then highlighted from an algorithmic standpoint. The comparison is carried out via the study of a set of numerical benchmark cases using industrial-like fine-discretization meshes (around 1 million finite elements), and Matlab as the common computational platform, to ensure fair comparisons. Then, the results obtained for every benchmark case with the different methods are compared in terms of computational cost, topology quality, achieved minimum value of the objective function, and robustness of the computations (convergence in objective function and topology). Finally, some quantitative and qualitative results are presented, from which, an attempt of qualification of the methods, in terms of their relative performance, is done.

Topology optimization of continuum structures under geometric uncertainty using a new extended finite element method

In this article, robust topology optimization under geometric uncertainty is proposed. The design domain is discretized by an extended finite element method. A bi-directional evolutionary structural optimization carries out the optimization process. The performance of the proposed method is compared with the Monte Carlo, solid isotropic material with penalization, perturbation and non-intrusive polynomial chaos expansion methods. The novelty of the present method lies in the following three aspects: (1) this article is among the first to use the extended finite element method in studying the topology optimization under uncertainty; (2) by adopting the extended finite element method for boundary elements in the finite element framework, there is no need for any remeshing techniques; and (3) the numerical results show that the present method has a smoother boundary region and minimum value of the mean and standard deviation of compliance than the other methods, in particular mesh size.

A Novel Method for Structural Lightweight Design with Topology Optimization

Topological optimization is an innovative method to realize the lightweight design. This paper proposes a hybrid topology optimization method that combines the SIMP (solid isotropic material with penalization) method and genetic algorithm (GA), called the SIMP-GA method. In the method, SIMP is used to update the chromosomes, which can accelerate convergence. The filtering scheme in the SIMP method can filter unconnected elements to ensure the connectivity of the structure. We studied the influence of varying the filtering radius on the optimized structure. Simultaneously, in the SIMP-GA method, each element is regarded as a gene, which controls the population number to a certain extent, reduces the amount of calculation, and improves the calculation efficiency. The calculation of some typical examples proves that the SIMP-GA method can obtain a better solution than the gradient-based method. Compared with the conventional genetic algorithm and GA-BESO (Bi-directional Evolutionary Structural Optimization) method, the calculation efficiency of the proposed method is higher and similar results are obtained. The innovative topology optimization method could be an effective way for structural lightweight design.

Topology Optimization Considering Porosity Defects in Metal Additive Manufacturing

In this paper, considering the porosity defects of Additive Manufacturing (AM), a level set topology optimization method for AM with porosity constraints is proposed. The concept of topological sensitivity is used to formulate a global porosity constraint function in the proposed method, and a level set topology optimization model considering porosity defects is obtained. To improve the robustness of the algorithm, the topology optimization model is solved in two phases. At first, the classical level set method without the porosity constraint is used to initially optimize the structure. During this process, the hole nucleation method combining bi-directional evolutionary structural optimization (BESO) and the topological sensitivity is used. Secondly, the topology optimization considering the effects of porosity is implemented on the preliminary optimization results. After performing the two-step optimization, a robust structure that alleviates the harmful impact of porosity defects is obtained. Finally, the robustness and effectiveness of the proposed method are validated by several two-dimensional numerical examples.

Topological designs of mechanical-electromagnetic integrated laminate metastructure for broadband microwave absorption based on bi-directional evolutionary optimization

Microwave absorption plays an important role on many aspects such as stealth technologies and electromagnetic compatibility. In this work, topological optimization is applied for the proposed laminate metastructure (LM) to achieve broadband and wide incident-angle microwave absorption. A new and efficient version of genetic algorithms (GA), namely the large mutation genetic algorithms (LMGA) is introduced in the bi-directional evolutionary optimization (BEO) methodology which is distinguishing from the traditional bi-directional evolutionary structural optimization (BESO). Topological optimization for patterned resistive films and size optimization for spacer thickness are integrated in the program. The optimized two-dimensional patterns of the metasurface are given. The thickness optimization range is limited below 4 mm to reduce the total thickness of LM. The optimized and fabricated specimen achieves −10dB absorption bandwidth in 2.0–22.9 GHz with total thickness of 16.05 mm, small areal density of 4.19 kg/m2 and equivalent flexural strength of 23.12 MPa. A three-stage nonlinear model on the bending and buckling of LM is given, and the experimental and theoretical deflection-load curves match well. The proposed LM achieves multifunctional features of broadband microwave absorption and effective mechanical resistance.

Evolutionary Topology Optimization of Spatial Steel-Concrete Structures

Topology optimization techniques based on finite element analysis have been widely used in many fields, but most of the research and applications are based on single-material structures. Extended from the bi-directional evolutionary structural optimization (BESO) method, a new topology optimization technique for 3D structures made of multiple materials is presented in this paper. According to the sum of each element's principal stresses in the design domain, a material more suitable for this element would be assigned. Numerical examples of a steel- concrete cantilever, two different bridges and four floor systems are provided to demonstrate the effectiveness and practical value of the proposed method for the conceptual design of composite structures made of steel and concrete.

Design, fabrication and verification of a novel auxetic microstructure using topology optimization

This research presents a numerical method to design a two-dimensional auxetic microstructure with negative Poisson’s ratios. The method is established by combining the finite element method (FEM) with two different optimization procedures called bi-directional evolutionary structural optimization (BESO) and solid isotropic material with penalization (SIMP), respectively. The results show that by choosing any of the two methods of optimization (FEM and BESO or FEM and SIMP), each with a different objective function, it is possible to create microstructures that show negative Poisson’s ratio. In addition, there is a significant resemblance between the results of both optimization methodologies. The final part of this study is to verify the outcome of optimization, both numerically and experimentally using three-dimensional (3D) printing techniques.

An approach for the design and validation of high frequency vibration energy harvesting devices

The aim of this work is to explore a topology optimisation approach for the design of high frequency vibration energy harvesting devices capable of operating in the range of 1800–2600 Hz. The main objective of the approach is to develop an optimisation algorithm for tailoring the resonant behaviour of the cantilever energy harvesters to the dominant frequencies of the available vibrations present in a Bell 206B-1 Kiowa helicopter main rotor gearbox. Hence, the devices designed are to be operated in the high frequency, highly loaded, extreme environments commonly found on aircraft structures. A multi-physics model capable of predicting the performance of the developed cantilever energy harvesters is presented and the design optimisation problem is formulated and then solved with the bi-directional evolutionary structural optimisation method. Two designs were experimentally validated; one that is sensitive to a single mode in the range 1800–2600 Hz equipped with a single transducer having a 6.35 mm diameter and 0.5 mm thickness and the other sensitive to three modes in the same range equipped with two transducers. Output powers of 0.3–3.1 mW were measured, demonstrating the approach’s potential to design harvesters capable of powering many useful sensor types.

A Hole Nucleation Method Combining BESO and Topological Sensitivity for Level Set Topology Optimization.

As is known to all, the incapacity to nucleate holes automatically in the design domain is one of the main issues of the classical level set topology optimization method. To solve the issue of hole nucleation, this paper employs the bi-directional evolutionary structural optimization (BESO) method based on the material removal scheme and the frequently used topological sensitivity and proposes the combining BESO and topological sensitivity (CBT) method for level set topology optimization. This method can replace the existing hole nucleation method for level set topology optimization. First, the topological sensitivity is combined with BESO, and the BESO method based on topological sensitivity is proposed. Second, the method is integrated into level set topology optimization to solve the issue of hole nucleation. Two sensitivity thresholds are defined depending on the evolutionary volume ratio and boundary topological sensitivity, respectively, and the smaller one is used as the sensitivity threshold for hole nucleation. The material is removed from the design domain to nucleate holes based on this threshold. Three classical two-dimensional numerical examples are used to validate the proposed hole nucleation method.

A Sequential Approach for Aerodynamic Shape Optimization with Topology Optimization of Airfoils

The objective of this work is to study the coupling of two efficient optimization techniques, Aerodynamic Shape Optimization (ASO) and Topology Optimization (TO), in 2D airfoils. To achieve such goal two open-source codes, SU2 and Calculix, are employed for ASO and TO, respectively, using the Sequential Least SQuares Programming (SLSQP) and the Bi-directional Evolutionary Structural Optimization (BESO) algorithms; the latter is well-known for allowing the addition of material in the TO which constitutes, as far as our knowledge, a novelty for this kind of application. These codes are linked by means of a script capable of reading the geometry and pressure distribution obtained from the ASO and defining the boundary conditions to be applied in the TO. The Free-Form Deformation technique is chosen for the definition of the design variables to be used in the ASO, while the densities of the inner elements are defined as design variables of the TO. As a test case, a widely used benchmark transonic airfoil, the RAE2822, is chosen here with an internal geometric constraint to simulate the wing-box of a transonic wing. First, the two optimization procedures are tested separately to gain insight and then are run in a sequential way for two test cases with available experimental data: (i) Mach 0.729 at α=2.31°; and (ii) Mach 0.730 at α=2.79°. In the ASO problem, the lift is fixed and the drag is minimized; while in the TO problem, compliance minimization is set as the objective for a prescribed volume fraction. Improvements in both aerodynamic and structural performance are found, as expected: the ASO reduced the total pressure on the airfoil surface in order to minimize drag, which resulted in lower stress values experienced by the structure.

Form Finding and Aesthetic Design for Pylons of Cable-supported Bridges

Pylons of cable-supported bridges are especially important components. They are generally large in scale and bear the main loads of the bridge, involving requirements both in structural performance and aesthetics. In this paper, a new form finding method for the pylon is proposed, with the use of bi-directional evolutionary structural optimization (BESO) method and the application of aesthetic rules. The principle of BESO method and the process for the form finding of the pylon are discussed. Pylons of the Hangrui Dongting Lake Bridge and the 3rd Panama Canal Bridge are used as cases to illustrate the proposed method. These two bridges are a suspension and a cable-stayed bridge with a main span of 1480 m and 530 m, respectively. The Golden-section and the Rectangle-division methods, together with the morphologic feature from local culture, are used to achieve harmony between the bridge and the environment. It is shown that the requirements mentioned above can be more easily achieved with the proposed method as well as some aesthetic rules.

Stability analysis of anti-dip bedding rock slopes using a limit equilibrium model combined with bi-directional evolutionary structural optimization (BESO) method

Anti-dip bedding rock slope (ABRS) is a very common type of rock slopes in nature and engineering construction. However, the stability assessment of ABRSs is still a hard problem for geotechnical engineers due to the difficulty in finding the critical failure surface. In this work, a new method called LEM-BESO was proposed to tackle this problem based on limit equilibrium theory and bi-directional evolutionary structural optimization method. Two model tests reported in previous studies were employed to verify the correctness of the LEM-BESO method. Moreover, a comparative study of the LEM-BESO method and the genetic algorithm was also conducted. The highly consistent results demonstrate that using the LEM-BESO method can accurately predict the failure surface of the slope. Then, the LEM-BESO method was extended to study the influence of two key parameters (i.e. friction angle of the joints and tensile strength of the rock layers) on the failure mechanism of anti-dip bedding rock slopes. The results show that these two parameters have a significant influence on the stability of the slope. However, the failure surface is insensitive to the tensile strength of the rock layers. The LEM-BESO method provides an effective means for the stability evaluation and engineering design of ABRSs.

Bi-directional evolutionary topology optimization of continuum structures subjected to inertial loads

This paper proposes to investigate topology optimization of continuum structures with density-dependent inertial loads which including self-weight, centrifugal forces, the magnitude and direction of the acceleration of inertial loads in order to make topology optimization more realistic. By using an extended bi-directional evolutionary structural optimization (BESO) method, this work presents an extension of this procedure to deal with maximum stiffness (minimum compliance) topology optimization of continuum structures when different combinations of inertial loads and fixed forces are applied. Notably, the magnitude and direction of the acceleration of inertial loads are considered. The computation of the element sensitivity numbers is deduced in detail in order to achieve the optimum design. MATLAB programming of proposed mathematical approach is done and compared with the conventional structural optimization problems with fixed external force. The procedure has been tested in several 2D and 3D benchmark examples to illustrate and validate the approach. The inertial loads have great influence on the topological structure, especially when the fixed external force is relatively small. The inertial loads change the magnitude and variation trend of objective function. The iterative curves converge to constant values stably, and the convergence rate is fast.

Topology optimization of material nonlinear continuum structures under stress constraints

This paper proposes a methodology to extend the bi-directional evolutionary structural optimization (BESO) method for compliance minimization (stiffness maximization) design of material nonlinear continuum structures subject to both constraints on volume fraction and maximum von Mises stress. BESO method based on discrete variables can effectively avoid the well-known stress singularity problem in density-based methods with low density elements. The aggregated p-norm global stress measure is adopted to approximate the maximum von Mises stress. The conventional compliance design objective is augmented with p-norm stress measures by introducing one Lagrange multiplier. The Lagrange multiplier is employed to yield compromised designs of compliance and the p-norm stress. A scheme is developed for the determination of the Lagrange multiplier so that the maximum von Mises stress could be effectively constrained. The update of the binary topological design variables lies in the sensitivity numbers derived using the adjoint method. As for the highly nonlinear stress behavior, both topological variables and sensitivity numbers are filtered to stabilize the optimization procedure. The filtered sensitivity numbers are further stabilized with their historical information. A series of comparison studies has been conducted to validate the effectiveness of the method on several benchmark design problems.

Material microstructure topology optimization of piezoelectric composite beam under initial disturbance for vibration suppression

This study proposes a material microstructure topology optimization model for a piezoelectric composite beam subject to initial disturbance. The optimization model is built through the Solid Isotropic Material with Penalization interpolation subject to the material volume fraction on the microlevel. And the objective function is to minimize the time integral about local structural response. By introducing the Lyapunov second equation, this time integral is simplified and does not need any numerical integration technique. It can also be used to measure the global vibration response instead of short time gap. Besides, different objectives such as a regional structural response can also be obtained by only changing the weight matrix form. And the sensitivity of the objective function with respect to the design variables are also deduced by solving the Lyapunov equation. A bidirectional evolutionary structural optimization algorithm is used to acquire the optimal results. Four numerical examples including multiphase material microstructure topology optimization verify the developed method that can effectively suppress local structural vibration level and obtain lighter structure.

An efficient 137-line MATLAB code for geometrically nonlinear topology optimization using bi-directional evolutionary structural optimization method

Topology optimization, as a powerful conceptual design method, has been widely adopted in both academic research and industrial applications. To further promote the development of topology optimization, many computer programs have been published for educational purposes over the past decades. However, most of the computer programs are constructed based on a linear assumption. On the basis of bi-directional evolutionary structural optimization (BESO) method, the paper presents a MATLAB implementation of the geometrically nonlinear topology optimization code for compliance minimization of statically loaded structures. Excluding 19 lines which are used for explanation, only 118 lines are needed for the initialization of the design parameters, nonlinear finite element analysis, sensitivity calculation, sensitivity filtration, and topological design variables update. Different design problems can be solved by modifying several lines in the proposed program. The complete 137-line code is included as an Appendix and is intended for educational purposes only.

A normalization strategy for BESO-based structural optimization and its application to frequency response suppression

The bi-directional evolutionary structural optimization (BESO) method has been widely studied and applied due to its efficient iteration and clear boundaries. However, due to the use of the discrete design variable, numerical difficulties are more likely to occur with this method, especially in cases with strong nonlinearity. This limits the application of the BESO method in certain cases, such as the suppression of structural dynamic frequency response under high-frequency excitation. In this work, a normalization strategy is proposed for the BESO-based topology optimization, by which the magnitude of the sensitivities can be efficiently unified to the same order to avoid the possible numerical instabilities caused by the nonlinearity. To validate its merit in applications, the normalization-based BESO (NBESO) method is proposed for minimizing the structural frequency response. By means of the weighted sum method, a normalized weighted sum method is also proposed for multi-frequency involved problems. A series of 2D and 3D numerical examples is presented to illustrate the advantages of the NBESO. The effectiveness of the NBESO for multi-frequency response suppression is also demonstrated, in which the frequency ranges below and above the eigenfrequency are involved, respectively.

High natural frequency gap topology optimization of bi-material elastic structures and band gap analysis

This work aims to perform the topology optimizationof frequency separation interval of continuous elastic bi-dimensional structures in the high-frequency domain. The studied structures are composed of two materials. The proposed algorithm is an adaptation of the Bidirectional Evolutionary Structural Optimization (BESO). As the modal density is high in this frequency domain, the objective function, based on the weighted natural frequency, is formulated to consider an important number of modes. To implement the algorithm, a mode tracking method is necessary to avoid problems stemming from mode-shifting and local modes. As the obtained results by using structural dynamics analysis present quasi-periodic topology, further calculations are done to compare the results with and without imposed periodicity. A dispersion analysis based on wave propagation theory is performed by using the unit cell previously obtained from the structural optimization to investigate the band gap phenomenon. The resulting band gaps from the dispersion analysis are compared with respect to the dynamic behavior of the structure. The topology optimization methodology and the wave propagation analysis are assessed for different boundary conditions and geometries. Comparison between both analyses shows that the influence of the boundary conditions on the frequency separation interval is small. However, the influence from the geometry is more pronounced. The optimization procedure does not present significant numerical instability. The obtained topologies are well-defined and easily manufacturable, and the obtained natural frequency separation intervals are satisfactory.