Bi-directional Evolutionary Structural Optimization Research Articles

Bi-directional evolutionary topology optimization of geometrically nonlinear continuum structures with stress constraints

This paper proposes a design method to maximize the stiffness of geometrically nonlinear continuum structures subject to volume fraction and maximum von Mises stress constraints. An extended bi-directional evolutionary structural optimization (BESO) method is adopted in this paper. 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. By introducing one Lagrange multiplier, the objective of the traditional stiffness design is augmented with p-norm stress. The stiffness and p-norm stress are considered simultaneously by the Lagrange multiplier method. A heuristic method for determining the Lagrange multiplier is proposed in order to effectively constrain the structural maximum von Mises 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 topology 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 benchmark design problems.

Concurrent optimization of macrostructures and material microstructures and orientations for maximizing natural frequency

In this paper, an efficient concurrent optimization method of macrostructures, and material microstructures and orientations is proposed for maximizing natural frequency. It is assumed that the macrostructure is composed of uniform material with the same microstructure but with various orientation. The bi-directional evolutionary structural optimization (BESO) method is applied to optimize the macrostructure and its material microstructure under a given weight constraint. Meanwhile, the optimality condition with respect to local material orientation is derived and embedded in the two-scale design of macrostructures and material microstructures. Numerical examples are presented to demonstrate the capability and effectiveness of the proposed optimization algorithm. The results show that the current design of macrostructures, material microstructures, and local material orientation greatly improves structural dynamic performance.

Determination of strut-and-tie models for structural concrete under dynamic loads

This proposed study aims to develop reliable and efficient numerical optimization methods for generating optimal strut-and-tie models (STMs) in structural concrete members under dynamic loads. The numerical models are developed based on the bidirectional evolutionary structural optimization (BESO) method for the stiffness maximization problems. In this method, a controlling index based on the minimum weight and maximum stiffness is defined as the optimization criterion function and the element virtual strain energy is taken as the element removal and addition criterion. By the dynamical analysis, optimal strut-and-tie models are established based on the BESO method. Several examples are presented to show the efficiency of the proposed approach in finding optimal STMs under dynamic loads. It is shown that optimal STMs and reinforcement layouts under dynamic loads generally differ from those obtained under static loads. The developed numerical models based on dynamic responses can be used by practicing design engineers for the analysis and design of STMs in concrete structures.

Fluid-structure coupling topology optimization based on added mass method

A wet model topology optimization method based on added mass method and bidirectional evolutionary structure optimization algorithm is proposed to maximize the natural frequency of fluid-structure coupling system. The fluid-structure coupling equation of the finite element method and the sensitivity formula of the element are analyzed simultaneously, and then the coupling optimization model is established. Under the constraints of structure volume, the weighted natural frequencies of single-objective and multi-objective of the coupled system are maximized to optimize the underwater coupled structure. The numerical results show that the evolutionary methods can be applied to this kind of problem effectively and efficiently. Besides, the optimization results of cases have smooth boundaries and the numerical stability of the optimization algorithm is good. Under constant volume constraints, the single-order natural frequency is increased by more than 380%. The results of multi-objective weighted optimization show that the weighted coefficients are final. The larger the weighting coefficient is, the closer the topological form is to the optimal result of the model.

Optimal Design of Soft Pneumatic Bending Actuators Subjected to Design-Dependent Pressure Loads

Soft actuators, mainly composed of soft materials, have made a great impact on applications in unstructured or unknown environments due to their high flexibility and customizability. The design methods of soft actuators can be divided into two groups: the bionic design method and the topology optimization method. Compared to the bionic design method that requires numerous experiments, the topology optimization method can generate innovative structures according to the design requirements. However, the existing topology optimization method cannot be applied to soft pneumatic bending actuator (SPBA) designs because SPBAs are usually subjected to design-dependent pressure loads of which the position depends on the structure. In this paper, we propose an optimal design framework of SPBAs to resolve the design-dependent load problem using an adaptive bi-directional evolutionary structural optimization method. Herein, SPBAs are considered as compliant mechanisms and our goal is to achieve maximum bending deformation as well as structural stiffness. In finite element analysis, each element in the design domain is set to solid or void according to sensitivity number, which is approximated by the objective function derivative with respect to the design variables. During the iterative optimization procedure, we explicitly define the movable solid-void boundary surfaces on which the pressure will act. A precision prototype actuator is fabricated, and its performance is evaluated in terms of free travel experiment. Some extensions are supplied to validate the optimality and reliability of the proposed method. This framework paves a way for the diversity of soft actuators.

Bidirectional evolutionary structural optimization for nonlinear structures under dynamic loads

SummaryThis study aims to develop efficient numerical optimization methods for finding the optimal topology of nonlinear structures under dynamic loads. The numerical models are developed using the bidirectional evolutionary structural optimization method for stiffness maximization problems with mass constraints. The mathematical formulation of topology optimization approach is developed based on the element virtual strain energy as the design variable and minimization of compliance as the objective function. The suitability of the proposed method for topology optimization of nonlinear structures is demonstrated through a series of two‐ and three‐dimensional benchmark designs. Several issues relating to the nonlinear structures subjected to dynamic loads such as material, geometric, and contact nonlinearities are addressed in the examples. It is shown that the proposed approach generates more reliable designs for nonlinear structures.

A Modified Big Bang–Big Crunch Algorithm for Structural Topology Optimization

The purpose of this study is to develop a topology optimization scheme based on big bang–big crunch (BB–BC) algorithm, inspired from the evolution of the universe called big bang and big crunch theory. In order to apply the BB–BC algorithm to topology optimization for static and dynamic stiffness problems, the parameters of the algorithm were transformed to those of topology optimization scheme. In addition, some parameters such as big bang (BB) range, BB search, population and non-exchange limit were newly introduced to topology optimization scheme. Also, a parametric study for the parameters involved in the topology optimization scheme was performed to reduce the number of parameters, and find the appropriate ranges for topology optimization. Some examples were provided to examine the effectiveness of the developed topology optimization scheme for both static and dynamic stiffness problems throughout comparing with other metaheuristic topology optimization algorithms and the BESO (bi-directional evolutionary structural optimization) method. It was verified that the suggested algorithm shows superior to the compared typical metaheuristic topology optimization algorithms in the viewpoints of stability, robustness, accuracy and the convergence rate.

Multi-solution nature of topology optimization and its application in design for additive manufacturing

Purpose The purpose of this paper is to introduce the multi-solution nature of topology optimization (TO) as a design tool for additive manufacturing (AM). The sensitivity of topologically optimized parts and manufacturing constraints to the initial starting point of the optimization process leading to structures with equivalent performance is explored. Design/methodology/approach A modified bi-directional evolutionary structural optimization (BESO) code was used as the numerical approach to optimize a cantilever beam problem and reduce the mass by 50 per cent. Several optimized structures with relatively equivalent mechanical performance were generated by changing the initial starting point of the TO algorithm. These optimized structures were manufactured using fused deposition modeling (FDM). The equivalence of strain distribution in FDM parts was tested with the digital image correlation (DIC) technique and compared with that from the modified BESO code. Findings The results confirm that TO could lead to a wide variety of non-unique solutions based on loading and manufacturability constraints. The modified BESO code was able to reduce the support structure needed to build the simple two-dimensional cantilever beam by 15 per cent while keeping the mechanical performance at the same level. Originality/value The originality of this paper lies in introduction and application of the multi-solution nature of TO for AM as a design tool for optimizing structures with minimized features in the overhang condition and the need for support structures.

On the combination of topology optimization and multiple mass dampers

This paper presents a vibration reducing mechanism combining the bi-directional evolutionary structural optimization (BESO) method and the multiple mass damper (MMD) method. The BESO method changes the mass and stiffness distribution of a structure, in order to push the most significant natural frequency away from the driving frequency. The removal of less stressed material by the BESO method creates hollow spaces in the structure. The MMD will be installed in the hollow spaces, and reduces the resonance vibration amplitude of the most significant mode.

Three-dimensional structural optimization of a cementless hip stem using a bi-directional evolutionary method

A correct choice of stem geometry can increase the lifetime of hip implant in a total hip arthroplasty. This study presents a numerical methodology for structural optimization of stem geometry using a bi-directional evolutionary structural optimization method. The optimization problem was formulated with the objective of minimizing the stresses in the bone–stem interface. Finite element analysis was used to obtain stress distributions by three-dimensional simulation of the implant and the surrounding bone under normal walking conditions. To compare the initial and the optimal stems, the von Mises stress distribution in the bone–implant interface was investigated. Results showed that the optimization procedure leads to a decrease in the stress concentration in the implant and a reduction in stress shielding of the surrounding bone. Furthermore, periprosthetic bone adaptation was analyzed numerically using an adaptive bone remodeling procedure. The remodeling results showed that the bone mass loss could be reduced by 16% in the optimal implant compared to the initial one.

Concurrent topology optimization of structures and orientation of anisotropic materials

Both the distribution and orientation of anisotropic materials have great effects on the performance of civil and mechanical structures. Taking the maximum structural stiffness as the design objective, a concurrent method for determining the optimal distribution and orientation of orthotropic materials is proposed in this article. The method for determining material orientation theoretically overcomes the ‘repeated global minimum’ phenomenon, which often occurs in strain-based and stress-based methods, when the material is shear ‘strong’. Then, the concurrent optimization methodology for material distribution and orientation is developed by introducing an independent discrete topology variable and continuous material orientation angle. The optimized designs of material distribution and orientation are obtained concurrently by the gradient-based bi-directional evolutionary structural optimization method and the proposed material orientation optimization method. Numerical two- and three-dimensional examples are presented to demonstrate the effectiveness of the proposed concurrent optimization algorithm.

Sound Radiation Analysis of Constrained Layer Damping Structures Based on Two-Level Optimization.

Constrained layer damping (CLD) is an effective method for suppressing the vibration and sound radiation of lightweight structures. In this article, a two-level optimization approach is presented as a systematic methodology to design position layouts and thickness configurations of CLD materials for suppressing the sound power of vibrating structures. A two-level optimization model for the CLD structure is developed, considering sound radiation power as the objective function and different additional mass fractions as constraints. The proposed approach applies a modified bi-directional evolutionary structural optimization (BESO) method to obtain several optimal position layouts of CLD materials pasted on the base structure, and sound power sensitivity analysis is formulated based on sound radiation modes for the position optimization of CLD materials. Two strategies based on the distributions of average normalized elemental kinetic energy and strain energy of the base plate are proposed to divide optimal position layouts of CLD materials into several subareas, and a genetic algorithm (GA) is employed to optimally reconfigure the thicknesses of CLD materials in the subareas. Numerical examples are provided to illustrate the validity and efficiency of this approach. The sound radiation power radiated from the vibrating plate, which is treated with multiple position layouts and thickness reconfigurations of CLD materials, is emphatically discussed.

Combination of the phase field method and BESO method for topology optimization

In this paper, the phase field method and the BESO (bidirectional evolutionary structural optimization) are combined to solve the topology optimization problems. The phase field function is used to represent the structure, and a time-dependent reaction diffusion equation called the Allen–Cahn equation is used to update the phase field function. In the sensitivity of Lagrange function, the Lagrange multiplier is replaced by the product of the Lagrange multiplier and the phase field function for fairing. The material removal scheme of the BESO which is easy to implement is employed to nucleate holes in the phase field method–based topology optimization. For a given target volume in each iterative step, a threshold of sensitivity is used to determine which elements should be removed; then, the structure is updated to match the target volume. Several numerical examples based on a two-dimensional minimum compliance problem are studied to demonstrate the effectiveness of this method.

Topology optimization of bimorph piezoelectric energy harvesters considering variable electrode location

With the advances on additive manufacturing techniques for piezoelectric materials, it is becoming steadily more viable to produce piezoelectric structures with complex geometries. Since such devices can have more sophisticated shapes, the application of topology optimization techniques can be used to explore optimized topologies for piezoelectric energy harvesters with varying thickness. In this context, challenges arise in the application of the electrode equipotential constraints during the optimization process. In this work, an algebraic formulation was proposed to tackle this issue when optimizing a harvester’s topology. A bimorph piezoelectric energy harvester with series connection of the piezoceramic layers was modeled under plane strain hypothesis in the thickness profile, first not applying and then considering periodicity constraints in the optimization process, while concurrently improving the electrodes. This paper shows how the open-circuit harvesting device was modeled and its finite element formulation. Lastly, an adaptation of the bidirectional evolutionary structural optimization method is proposed, in order to approach this problem.

Ultrawide band gap design of phononic crystals based on topological optimization

In this paper, based on the combined finite element method and bidirectional evolutionary structural optimization algorithm, we perform the topological optimization of phononic crystals to obtain ultrawide band gap between two special adjacent bands for both the in-plane and out-of-plane wave modes. The influences of matrix/scatter materials, material volume fraction, and initial topological design on the unit cell optimization are discussed in detail. The results show that the strategy proposed in this paper is effective and efficient to obtain better optimization results under the similar optimization condition, solutions with ultrawide band gaps can be easily obtained within 30 iterations. Several new patterns for phononic band gap crystals with optimized band gaps are presented. This work provides useful guidance in topological optimization design of phononic crystals.

Topology optimization of continuum structures under hybrid additive-subtractive manufacturing constraints

Additive manufacturing (AM) makes it possible to fabricate complicated parts that are otherwise difficult to manufacture by subtractive machining. However, such parts often require temporary support material to prevent the component from collapsing or warping during fabrication. The support material results in increased material consumption, manufacturing time, and clean-up costs. The surface precision and dimensional accuracy of the workpieces from AM are far from the engineering requirement due to layer upon layer manufacturing. Subtractive machining (SM), by contrast, can fabricate parts to satisfy the requirements of surface precision and dimensional accuracy. Nevertheless, the components need to be relatively uncomplicated for subtractive manufacturing. Thus, hybrid additive-subtractive manufacturing (HASM) is gaining increasing attention in order to take advantages of both processes. There is little research on the topological design methodology for this hybrid manufacturing technology. To address this issue, a method based on geometry approach for topology optimization of continuum structure is proposed in this paper. Both additive manufacturing and subtractive machining constraints are simultaneously considered in each topology optimization iteration. The topology optimization is performed by the bi-directional evolutionary structural optimization (BESO) method. The effectiveness of the proposed method is demonstrated by several 3D compliance minimization problems.

Fracture strength topology optimization of structural specific position using a bi-directional evolutionary structural optimization method

ABSTRACTA new topology optimization procedure based on the extended finite element method (XFEM) for enhancing the fracture strength of structural specific position is proposed. The XFEM is used as an approach for the description of discontinuity states in the design domain, such as cracks, voids and inclusions, by employing the enrichment shape function with discontinuous properties. Based on linear elastic fracture mechanics and the bi-directional evolutionary structural optimization (BESO) method, the J integral, as a criterion for predicting crack growth, is considered to prevent crack propagation and enhance the fracture strength of the structure. Several typical examples are provided to verify the effectiveness of the present method by considering different situations of length and location of the crack, and the interactions between crack, void and inclusion. The results suggest that combining the XFEM with BESO is a promising approach for optimizing the anti-fracturing mechanical properties of structures.

Topology optimization for vibrating structures with the BESO method

There is a lot of literature concerning the topology optimization of structures under static loads with the bi-directional evolutionary structural optimization (BESO), but only few approaches has focus on the dynamics problems with the BESO. This paper presents the von Mises stress as a sensitivity number and a new filter scheme for the BESO method focusing on dynamics problems. Based on the proposed technique and the BESO method, we discuss two measures to reduce vibrations of structures subjected to single external harmonic loads at a user-defined point. The natural frequency-based measure (NFBM) shifts the natural frequency of the most significant mode away from the driving frequency, while the steady state dynamics-based measure (SSDBM) considers several modes and natural frequencies at the same time using modal superposition.

Optimal design, analysis and additive manufacturing for two-level stochastic honeycomb structure

ABSTRACTRecent advances in the aviation, aerospace, and energy industries have created a huge demand for multifunctional structures with lightweight, load-carrying capacity, and heat resistance. In order to further explore the potential of honeycomb structure, a design method for hierarchically architected structures constructed by replacing side walls of pores with Voronoi lattices is put forward. A digital representation model of hierarchical honeycomb structures is first proposed. Then based on centroidal Voronoi tessellation (CVT) algorithm, the first level honeycomb structures are generated by decomposition of design domain as a set of Voronoi unit. At last, based on modified bidirectional evolutionary structural optimization and combined with the control function of the CVT algorithm, the internal pore size and distribution of cell walls are controlled to optimize the distribution of materials. The proposed approach is verified through three case studies, which provides an important basis for the wide application of additive manufacturing technology of the two-level stochastic honeycomb structure.

Follow-up optimized phononic band structure design based on reserved inclusions

For functional demands, inclusions are always reserved beforehand in a structural design. In this paper, a follow-up optimized phononic band structure design is performed based on a 2D structure with reserved inclusions but no absolute band gaps for the purpose to simultaneously achieve acoustic abatement except for functional demands. Firstly, the strategy to optimize the band structure based on bidirectional evolutionary structural optimization algorithm is proposed. Then the effects of geometrical or physical parameters on the final optimization results are discussed. Numerical results show that there is a material gravitation phenomenon around the inclusions in the topological optimization. Factors affecting the material gravitation are discussed in details. This work will lead an important guidance in the multifunctional design of multi-phase phononic crystals.