Bi-directional Evolutionary Structural Optimization Algorithm Research Articles

Voxel-based evolutionary topological optimization of connected structures for natural frequency optimization

AbstractThe topology optimization methodology is widely utilized in industrial engineering for designing lightweight and efficient components. In this framework, considering natural frequencies is crucial for adequately designing components and structures exposed to dynamic loads, as in aerospace or automotive applications. The scientific community has shown the efficiency of Bi-directional Evolutionary Structural Optimization (BESO), showcasing its ability to converge towards optimal solid-void or bi-material solutions for a wide range of frequency optimization problems in continuum structures. However, these methods show limits when the complexity of the domain volume increases; thus, they are well-suited for academic case studies but may fail when dealing with industrial applications that require more complex shapes. The connectivity of the structures resulting from the optimization also plays a fundamental role in choosing the best optimization approach, as some available commercial and open-source codes nowadays return unfeasible sparse structures. An improved voxel-based BESO algorithm has been developed in this work to cope with current limits in lightweight structure optimization. A significant case study has been developed to evaluate the performances of the new methodology and compare it with existing algorithms. In contrast to previous studies, the method we developed guarantees that the final structure respects constraints on the initial design volume and that the structure’s connection is preserved, thus enabling the manufacturing of the component with Additive Manufacturing technologies. The proposed approach can be complemented by smoothing algorithms to obtain a structure with externally appealing surfaces.

Phase field modelling combined with optimization algorithm for maximizing the resistance in two-phase composites

Recently, the phase field modelling is widely used to model and simulate material damage. In present work, we present a framework of the topology optimization combined with the phase-field modelling with/ without interfacial damage for optimizing the damage resistance of the inclusion-matrix composites. The first phase field method with the interfacial damage described by the phase field variable d(x) and an interfacial phase field variable β(x), thus the crack occurs in the interaction between the bulk fracture and the interfacial one; the second phase field method without the interfacial damage describes the crack by using only a damage variable d(x), thus the crack initiates at the points where stress concentration occurs (damage only occurring in the phases). Extended bi-directional evolutionary structural optimization algorithm (BESO) is used to optimize the inclusion distribution in order to reduce its volume while keeping the fracture resistance value of the initial design unchanged. Moreover, the strain tensor orthogonal decompositions are implemented into the phase field methods to improve the inaccuracy in the mechanical behavior of the materials. We compare the inclusion optimal distribution, the crack propagation and the fracture resistance between the proposed models through several numerical examples.

An evolutionary structural optimization algorithm for the analysis of light automobile parts using a meshless technique

PurposeThe purpose of the present work is to develop the combination of the radial point interpolation method (RPIM) with a bi-directional evolutionary structural optimization (BESO) algorithm and extend it to the analysis of benchmark examples and automotive industry applications.Design/methodology/approachA BESO algorithm capable of detecting variations in the stress level of the structure, and thus respond to those changes by reinforcing the solid material, is developed. A meshless method, the RPIM, is used to iteratively obtain the stress field. The obtained optimal topologies are then recreated and numerically analyzed to validate its proficiency.FindingsThe proposed algorithm is capable to achieve accurate benchmark material distributions. Implementation of the BESO algorithm combined with the RPIM allows developing innovative lightweight automotive structures with increased performance.Research limitations/implicationsComputational cost of the topology optimization analysis is constrained by the nodal density discretizing the problem domain. Topology optimization solutions are usually complex, whereby they must be fabricated by additive manufacturing techniques and experimentally validated.Practical implicationsIn automotive industry, fuel consumption, carbon emissions and vehicle performance is influenced by structure weight. Therefore, implementation of accurate topology optimization algorithms to design lightweight (cost-efficient) components will be an asset in industry.Originality/valueMeshless methods applications in topology optimization are not as widespread as the finite element method (FEM). Therefore, this work enhances the state-of-the-art of meshless methods and demonstrates the suitability of the RPIM to solve topology optimization problems. Innovative lightweight automotive structures are developed using the proposed methodology.

Topological optimization of the stiffness of an irregular structure based on an element size independent filter

Because of the overly averaged element sensitivity in the topological optimization of an irregular structure with a grid independent filter, a topological optimization model was built for the structural domain. The maximization of stiffness was first taken as the goal for the topological optimization of irregular structure stiffness. Subsequently, an element size filter was proposed to address the overly averaged local element sensitivity with the grid independent filter when the designed domain element size varied dramatically. Finally, the element sensitivity of the objective function was derived under the given constraints. A case study was then conducted on a naval gun mount with the maximization of structural flexibility as the objective function and the volume of structural material as a constraint. A stiffness optimization model based on the bi-directional evolutionary structural optimization algorithm was adopted for the topological optimization of the gun mount. Structural optimization was conducted for the gun mount with different shooting angles to realize its optimal stiffness and strength under the constraint of consistent material volume. The optimization results proved that the element independent filter proposed in this paper could be effectively applied in the topological optimization of an irregular structure and used to explore the topological optimization of the supporting structure under impact.

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.

Crashworthiness topology optimization of thin-walled square tubes, using modified Bidirectional Evolutionary Structural Optimization approach

Abstract In this study, topology optimization of a thin-walled square tube subjected to an axial crushing load was carried out to enhance its crashworthiness performance, utilizing the modified Bidirectional Evolutionary Structural Optimization (BESO) procedure. In this way, by systematic searching, topology optimization finds the best material layout for the structure to satisfy the objective function. In each iteration, a new pattern of material-void through the tube walls can trigger various folding mechanisms. Thus, the energy absorption history trend might not be smooth and the ultimate pattern proposed by the algorithm is less likely to be the optimal one. With an emphasis on the BESO hypothesis regarding direct removal and addition of elements, the modified BESO algorithm excavated the optimal design in two steps. First, material-void patterns were produced, considering specific energy absorption as the element efficiency criterion. Then, the pattern with the least material usage which satisfies an energy absorption constraint was selected as the optimal design. Such an optimum layout also boosts the tube performance respecting peak crushing force, since the embedded voids through the structure serve as a crush initiator. The optimization code was developed in MATLAB, which was automatically integrated with ABAQUS software performing the nonlinear crushing analysis. To validate the finite element model, an experimental investigation was carried out on a simple square tube. Having performed the topology optimization and some fine-tuning process, the topologically optimized tube was constructed and experimentally tested. Results showed 16.21% weight reduction enriched with 36.57% enhancement in the peak crushing force by sacrificing less than 3.5% energy absorption, which represents substantial improvements from a crashworthiness viewpoint. The obtained interesting results showed the high potential of the BESO algorithm for the design of thin-walled structures under large deformation and out-of-plane buckling.

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.

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.

A bidirectional evolutionary structural optimization algorithm for mass minimization with multiple structural constraints

SummaryA bidirectional evolutionary structural optimization algorithm is presented, which employs integer linear programming to compute optimal solutions to topology optimization problems with the objective of mass minimization. The objective and constraint functions are linearized using Taylor's first‐order approximation, thereby allowing the method to handle all types of constraints without using Lagrange multipliers or sensitivity thresholds. A relaxation of the constraint targets is performed such that only small changes in topology are allowed during a single update, thus ensuring the existence of feasible solutions. A variety of problems are solved, demonstrating the ability of the method to easily handle a number of structural constraints, including compliance, stress, buckling, frequency, and displacement. This is followed by an example with multiple structural constraints and, finally, the method is demonstrated on a wing‐box, showing that topology optimization for mass minimization of real‐world structures can be considered using the proposed methodology.

Optimal microstructures of elastoplastic cellular materials under various macroscopic strains

Topology optimization has been applied widely in cellular material design by optimizing the microstructure of periodic base cell (PBC). So far, most works are focused on the design of microstructures with linear material properties using the inverse homogenization method. When accounting elastoplastic constitutive behaviors, there is in usual no overall closed-form constitutive expression for the cellular microstructure as the case of linear elasticity, the design is therefore dependent on the specific applied loading. Due to the importance of topology optimization concerning elastoplastic material, the microstructural topology optimization procedure of elastoplastic cellular material with the objective of maximizing their nonlinear responses in terms of the strain energy density is proposed by using the bi-directional evolutionary structural optimization (BESO) method in this paper. Analytical sensitivity analysis is derived in a clear and rigorous manner using the adjoint method and incorporated into the BESO algorithm. The effectiveness of the proposed approach is validated through a series of numerical examples for various macroscopic strain loadings.

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.

An improved soft-kill BESO algorithm for optimal distribution of single or multiple material phases

Finding the optimum distribution of material phases in a multi-material structure is a frequent and important problem in structural engineering which involves topology optimization. The Bi-directional Evolutionary Structural Optimization (BESO) method is now a well-known topology optimization method. In this paper an improved soft-kill BESO algorithm is introduced which can handle both single and multiple material distribution problems. A new filtering scheme and a gradual procedure inspired by the continuation approach are used in this algorithm. Capabilities of the proposed method are demonstrated using different examples. It is shown that the proposed method can result in considerable improvements compared to the normal BESO algorithm particularly when solving problems involving very soft material or void phase.

A simple and compact Python code for complex 3D topology optimization

This paper presents a 100-line Python code for general 3D topology optimization. The code adopts the Abaqus Scripting Interface that provides convenient access to advanced finite element analysis (FEA). It is developed for the compliance minimization with a volume constraint using the Bi-directional Evolutionary Structural Optimization (BESO) method. The source code is composed of a main program controlling the iterative procedure and five independent functions realizing input model preparation, FEA, mesh-independent filter and BESO algorithm. The code reads the initial design from a model database (.cae file) that can be of arbitrary 3D geometries generated in Abaqus/CAE or converted from various widely used CAD modelling packages. This well-structured code can be conveniently extended to various other topology optimization problems. As examples of easy modifications to the code, extensions to multiple load cases and nonlinearities are presented. This code is useful for researchers in the topology optimization field and for practicing engineers seeking automated conceptual design tools. With further extensions, the code could solve sophisticated 3D conceptual design problems in structural engineering, mechanical engineering and architecture practice. The complete code is given in the appendix section and can also be downloaded from the website: www.rmit.edu.au/research/cism/.

Topology optimization of compliant mechanisms with desired structural stiffness

This paper develops a bi-directional evolutionary structural optimization (BESO) method for topological design of compliant mechanisms. The design problem is reformulated as maximizing the flexibility of the compliant mechanism subject to the mean compliance and volume constraints. Based on the finite element analysis, a new BESO algorithm is established for solving such an optimization problem by gradually updating design variables until a convergent solution is obtained. Several 2D and 3D examples are presented to demonstrate the effectiveness of the proposed BESO method. A series of optimized mechanism designs with or without hinge regions are obtained. Numerical results also indicate that the flexibility and hinge-related property of the optimized compliant mechanisms can be controlled by the desired structural stiffness.

Topology Optimization of Photonic Band Gap Crystals

This paper proposes a new topology optimization algorithm based on the bi-directional evolutionary structural optimization (BESO) method for the design of photonic band gap crystals. The photonic crystals are assumed to be periodically composed of two given dielectric materials. Based on the finite element analysis, the proposed BESO algorithm gradually re-distributes dielectric materials within the unit cell until the resulting photonic crystals possess a maximal band gap at the desirable frequency level. Numerical examples for both transverse magnetic (TM) and transverse electric (TE) polarizations are presented, and the optimized photonic crystals exhibit novel patterns markedly different from traditional designs of photonic crystals.

Evolutionary topology optimization of periodic composites for extremal magnetic permeability and electrical permittivity

This paper presents a bidirectional evolutionary structural optimization (BESO) method for designing periodic microstructures of two-phase composites with extremal electromagnetic permeability and permittivity. The effective permeability and effective permittivity of the composite are obtained by applying the homogenization technique to the representative periodic base cell (PBC). Single or multiple objectives are defined to maximize or minimize the electromagnetic properties separately or simultaneously. The sensitivity analysis of the objective function is conducted using the adjoint method. Based on the established sensitivity number, BESO gradually evolves the topology of the PBC to an optimum. Numerical examples demonstrate that the electromagnetic properties of the resulting 2D and 3D microstructures are very close to the theoretical Hashin-Shtrikman (HS) bounds. The proposed BESO algorithm is computationally efficient as the solution usually converges in less than 50 iterations. The proposed BESO method can be implemented easily as a post-processor to standard commercial finite element analysis software packages, e.g. ANSYS which has been used in this study. The resulting topologies are clear black-and-white solutions (with no grey areas). Some interesting topological patterns such as Vigdergauz-type structure and Schwarz primitive structure have been found which will be useful for the design of electromagnetic materials.

Shape optimization of metallic yielding devices for passive mitigation of seismic energy

Bi-directional Evolutionary Structural Optimization (BESO) is a well-established topology optimization technique. This method is used in this paper to optimize the shape of a passive energy dissipater designed for earthquake risk mitigation. A previously proposed shape design of a steel slit damper (SSD) device is taken as the initial design and its shape is optimized using a slightly modified BESO algorithm. Some restrictions are imposed to maintain simplicity and to reduce fabrication cost. The optimized shape shows increased energy dissipation capacity and even stress distribution. Experimental verification has been carried out and proved that the optimized shape is more resistant to low-cycle fatigue.

A New Algorithm for Bi-Directional Evolutionary Structural Optimization

In this paper, a new algorithm for bi-directional evolutionary structural optimization (BESO) is proposed. In the new BESO method, the adding and removing of material is controlled by a single parameter, i.e. the removal ratio of volume (or weight). The convergence of the iteration is determined by a performance index of the structure. It is found that the new BESO algorithm has many advantages over existing ESO and BESO methods in terms of efficiency and robustness. Several 2D and 3D examples of stiffness optimization problems are presented and discussed.