Bi-directional Evolutionary Structural Optimization Research Articles

Stress-based multi-material structural topology optimization considering graded interfaces

For the multi-material structural topology optimization, most of works are based on the ideal interface hypothesis (the interfaces of material are considered to be ideal connection), which leads to the optimized structures tend to dangerous. Thus, this paper proposes a topology optimization method to minimize the maximum von Mises stress of multi-material structures with graded interfaces. A filter-based method is proposed to search and determine the locations of the multi-material interfaces, and the widths of the interfaces are determined by the filter radius. The mechanical properties of the multi-material interfaces are interpolated by gradient, and the gradient transition between the multi-material interface is realized. An extended Bi-directional Evolutionary Structural Optimization (BESO) method based on discrete variables is adopted to avoid the stress singularity problem. The maximum stress is measured by the global p-norm stress aggregation method. The adjoint method is used to derive the sensitivities of elements. Benchmark numerical examples are investigated to validate the effectiveness of the proposed method. Results show that the multi-material structure with graded interface can be described and optimized effectively. The width of the interfacial zone and the properties of the graded interface can be well controlled and defined, respectively. The topological results, for stress design, indicate that the maximum stress can be effectively reduced compared with stiffness design. The maximum stress of the topological structure considering the graded interface is higher than that without the graded interface, which indicates that the structure designed considering the graded interface is more safe. The proposed approach can achieve a reasonable design that effectively controls the stress level and reduces the stress concentration effect at the critical stress areas of multi-material structures with graded interfaces.

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.

Stress-based topology optimization of continuum structures for the elastic contact problems with friction.

Structural problems have various nonlinearities in the real world and these nonlinearities should be accommodated in structural topology optimization. This work proposes a topology optimization method for minimizing the maximum von Mises stress of elastic continuum structures with frictional contact under material usage constraint, using an extended Bi-directional Evolutionary Structural Optimization (BESO) method. Stresses are treated as global performance (objective) function, the global von Mises stress is measured by the p-norm stress aggregation approach, and the friction behavior is governed by the Coulomb friction law regularized in analogy with the perfect elasto-plastic theory. BESO method based on discrete variables which can avoid the well-known stress singularity and the numerical instability issue in frictional contact problems. The adjoint sensitivity analysis method is adopted to derive the sensitivity numbers. The effectiveness of the proposed method is validated through a series of comparison studies including elastic-rigid and elastic-elastic contact problems. The influence of varying friction coefficient on the optimized results and the stress distributions are investigated in comparison with the maximum stiffness design. The effect of different parameters including p-norm, volume fraction and mesh density on the optimized results are discussed. The optimized results, for elastic-rigid contact, indicate that the maximum stress can be reduced compared with elastic-elastic contact. The optimized stress decreases as the friction coefficient increases because the friction behavior resists the tangential deformation at the contact interface. The results also show that the proposed approach can achieve a reasonable design that effectively controls the stress level and reduces the stress concentration effect at the critical stress areas.

Hierarchical design of material microstructures with thermal insulation properties

Composite materials with multiple properties are important for a range of engineering applications. Hence, this study focuses on topological design of hierarchical materials with multiple performance in both thermal insulation and mechanics. First, a novel multi-objective optimization function is defined to find a solution from the Pareto frontier, where the weight coefficients can be adjusted adaptively, to keep all the individual objective functions and their sensitivities stabilized at the same level during the optimization. Second, a new design strategy is proposed to achieve the hierarchical designs of biphasic material microstructures, they are periodically arranged by the porous base materials that are known in advance and independent of topology optimization. Third, sensitivity information and algorithm implementation are given in detail, and the bi-directional evolutionary structural optimization method is adopted to iteratively update the micro-structural topologies, by combining with the homogenization method. Last, numerical examples are provided to illustrate the benefits of the proposed design method, such as high efficiency, implementation easiness, good connectivity and clear interface between adjacent phases, etc.

Topology Optimization with Aperiodic Load Fatigue Constraints Based on Bidirectional Evolutionary Structural Optimization

Because of descriptive nonlinearity and computational inefficiency, topology optimization with fatigue life under aperiodic loads has developed slowly. A fatigue constraint topology optimization method based on bidirectional evolutionary structural optimization (BESO) under an aperiodic load is proposed in this paper. In view of the severe nonlinearity of fatigue damage with respect to design variables, effective stress cycles are extracted through transient dynamic analysis. Based on the Miner cumulative damage theory and life requirements, a fatigue constraint is first quantified and then transformed into a stress problem. Then, a normalized termination criterion is proposed by approximate maximum stress measured by global stress using a P-norm aggregation function. Finally, optimization examples show that the proposed algorithm can not only meet the requirements of fatigue life but also obtain a reasonable configuration.

Efficient Local Level Set Algorithm Combined with Bidirectional Evolutionary Criterion Using Discrete Variables and its Appliance to Topology and Shape Optimization

Many level set methods (LSMs) have been used for topology and shape optimization. An efficient topology optimization algorithm is proposed by combining a new bi-directional evolutionary criterion (BEC) with the local LSM (LLSM). The BEC is established using multiple discrete variables (DVs) and introduced into the bi-directional evolutionary structural optimization (BESO) method so that the DV-BESO method obtained can be applied to topology optimization. The LLSM is used instead of the global LSM due to its high efficiency. In the LLSM, the computational efficiency can be further improved by solving a distance regularized equation without the re-initialization procedure. Two benchmarks show the high efficiency of this proposed combined algorithm for topology and shape optimization.

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.

Influence of foundation flexibility on the topology optimization of piled structures

AbstractThis article investigates the influence of foundation and soil flexibility on the topology optimization of piled structures. The structure is modeled with classical finite elements, while the embedded pile group foundation and its interaction with the soil are modeled via the stiffness matrix method. Coupling between the structure and its foundation is obtained by establishing direct continuity and equilibrium conditions at their interface. Topology optimization of the discretized piled structure is performed through the Bi‐directional Evolutionary Structural Optimization (BESO) method for various boundary conditions. The objective function of the optimization problem is the minimization of the compliance of the piled structure under prescribed volume restriction. The results show that the flexibility of the soil and piles may have a significant effect on the optimal topology of the structure, as well as on the achievable optimization objective.

Stress-based structural topology optimization for design-dependent self-weight loads problems using the BESO method

This article aims to propose an approach to the stress-based topology optimization of continuous elastic bi-dimensional structures subjected to design-dependent self-weight loads using the Bi-directional Evolutionary Structural Optimization (BESO) method. Topology optimization is developed through the minimization of P-norm von Mises stress while satisfying a volume constraint. To implement the algorithm, a consistent sensitivity analysis including design-dependent loads has been developed by the adjoint method. A series of tests has been performed to explore and validate the method through three numerical examples: an L-bracket; a doubly supported beam with one pre-existing crack notch; and a cantilever beam. Comparison between traditional compliance minimization and stress minimization analyses, including design-dependent self-weight loads, shows that the method is an effective way to reduce the maximum stress.

Detail control strategies for topology optimization in architectural design and development

With the ability to generate forms with high efficiency and elegant geometry, topology optimization has been increasingly used in architectural and structural designs. However, the conventional topology optimization techniques aim at achieving the structurally most efficient solution without any potential for architects or designers to control the design details. This paper introduces three strategies based on Bi-directional Evolutionary Structural Optimization (BESO) method to artificially pre-design the topological optimized structures. These strategies have been successfully applied in the computational morphogenesis of various structures for solving practical design problems. The results demonstrate that the developed methodology can provide the designer with structurally efficient and topologically different solutions according to their proposed designs with multi-filter radii, multi-volume fractions, and multi-weighting coefficients. This work establishes a general approach to integrating objective topology optimization methods with subjective human design preferences, which has great potential for practical applications in architecture and engineering industry.

Simultaneously optimizing supports and topology in structural design

Topology optimization techniques are typically performed on a design domain with pre-determined support conditions to generate efficient structures. Allowing the optimizer to simultaneously design supports and topology offers new design possibilities to achieve improved structural performance and reduce the cost of supports. However, existing simultaneous optimization techniques are limited, with most methods requiring cumbersome procedures to pre-define support conditions, which may not be easy for the end-users. This study presents a new element-based simultaneous optimization method by introducing a layer of elements to the boundaries where supports are allowed, which can be simply implemented in finite element (FE) models. Computational algorithms are developed based on a combination of an optimality criteria (OC) method and the bi-directional evolutionary structural optimization (BESO) technique to determine support locations and the structural topology, respectively. A variety of examples are presented to demonstrate the effectiveness of the new method. It is found that the number, position, and stiffness of supports may significantly influence the structural topology. A support location analysis is used to validate the new method and confirms optimal designs. This study shows that treating element-based support locations as additional design variables can effectively obtain efficient and innovative structural designs. Two 3D examples are presented to demonstrate potential practical applications of the new method.

Parametric Structural Topology Optimization of High-Rise Buildings Considering Wind and Gravity Loads

Topology optimization has recently been investigated as a technique for the conceptual design of efficient structures during the early stage of the design of buildings to tackle the design challenges. The use of this technique, which leads to optimum structures, mostly results in esthetic, lightweight, and best performance from the perspective of engineers or architects. However, the topology optimization results are not usually known for direct realization in practice, and the engineer and architect should be able to choose the best solution among numerous choices in close cooperation. This paper has focused on defining a parametric framework of continuous optimum design of lateral bracing systems for tall buildings considering wind and gravity loads. The bidirectional evolutionary structural optimization (BESO) method was employed, considering the main optimization parameter, loading scenarios, and constraints. In order to show the effectiveness of the suggested topology optimization framework for minimizing compliance (maximizing stiffness) and minimum consumption of materials in the design of lateral bracing systems, 2D and 3D systems have been discussed. According to the obtained results, this framework could employ topology optimization during the conceptual design to seek a new definition of the optimum layout of lateral bracing systems with high structural performance, elegant geometries, and other characteristics considered by architects and engineers.

Topology optimization of adhesive material in a single lap joint using an evolutionary structural optimization method and a cohesive zone model as failure criterion

This paper presents a study of the use of a structural optimization process coupled to a failure model in the adhesive material in single lap bonded joints. The critical point of these joints is in the region of the adhesive material, which performs the function of stress transmission between the structural elements. Owing to the single lap bonded joints shape, the applied loads are eccentric about the joint axis, resulting in a concentration of stress on the overlapping ends. In this study, numerical simulations in two and three dimensions were performed through finite element analysis to model an single lap bonded joints. An optimization procedure based on the bidirectional evolutionary structural optimization method was used to minimize the single lap bonded joints compliance under volume constraints. The design domain considered was restricted to the adhesive region. The cohesive zone model and the bidirectional evolutionary structural optimization method were simultaneously applied. The numerical results for two types of adhesives, with ductile and brittle failure behavior, enabled the establishment of mechanisms for determining the efficient positioning and quantity of adhesive materials.

Topology optimization design related to size effect using the modified couple stress theory

Traditional topology optimization methods based on classical mechanics can provide practical design solutions for macroscale structural design. However, the rapid development of additive manufacturing, with high-resolution and rapid prototyping characteristics, has promoted their wider application in microscale or nanoscale structures. Furthermore, when the characteristic size of the macroscopic deformation field of the structure is reduced to the microscale or nanoscale, the size effect causes the structure’s mechanical response to be inconsistent with the macroscale structure. Owing to the lack of microscopic characteristic parameters in the constitutive model, the traditional topology optimization framework based on the classical mechanics framework cannot accurately describe the size effect. To solve the topology optimization problem related to the size effect, a general framework that combines the modified couple stress theory with the bi-directional evolutionary structural optimization (BESO) method is constructed to explore the size effect on topology optimization.

Robust topology optimization for thermoelastic hierarchical structures with hybrid uncertainty

This paper presents a robust topology optimization (RTO) framework for thermoelastic hierarchical structures with hybrid uncertainty. Firstly, the thermoelastic concurrent optimization model is established and the uncertainties with interval random parameters are integrated into the thermoelastic hierarchical structure. Then, a reliable and cost-effective hybrid uncertainty perturbation analysis method (HUPAM) is derived for a quick estimate of the robust objective function subject to the mechanical and thermal loads. Finally, by calculating the design variables sensitivities of macroscale and microscale, the robust topological design can be generated efficiently. To obtain clear and optimal topologies for both macro- and micro- structures, the bi-directional evolutionary structural optimization (BESO) method is adopted. Some 2D and 3D numerical examples are presented to demonstrate the influences of the hybrid uncertainties on the final designs. The results also show that the proposed method can effectively improve the thermoelastic structural performance when it comes to uncertainties.

Finite variation sensitivity analysis for discrete topology optimization of continuum structures

This paper proposes two novel approaches to perform more suitable sensitivity analyses for discrete topology optimization methods. To properly support them, we introduce a more formal description of the Bi-directional Evolutionary Structural Optimization (BESO) method, in which the sensitivity analysis is based on finite variations of the objective function. The proposed approaches are compared to a naive strategy; to the conventional strategy, referred to as First-Order Continuous Interpolation (FOCI) approach; and to a strategy previously developed by other researchers, referred to as High-Order Continuous Interpolation (HOCI) approach. The novel Woodbury approach provides exact sensitivity values and is a better alternative to HOCI. Although HOCI and Woodbury approaches may be computationally prohibitive, they provide useful expressions for a better understanding of the problem. The novel Conjugate Gradient Method (CGM) approach provides sensitivity values with arbitrary precision and is computationally viable for a small number of steps. The CGM approach is a better alternative to FOCI since, for appropriate initial conditions, it is always more accurate than the conventional strategy. The standard compliance minimization problem with volume constraint is considered to illustrate the methodology. Numerical examples are presented together with a broad discussion about BESO-type methods.

Inverse design of second-order photonic topological insulators in C3-symmetric lattices

Second-order photonic topological insulators (SPTIs) featured with topological edge and corner states provide an intriguing way for steering light in integrated optics. However, existing SPTIs have narrow bulk gaps for tightly confining topological edge and corner states. In this paper, we propose a bi-directional evolutionary structural optimization (BESO) method to inversely engineer SPTIs in C3-symmetric lattices. At first, we maximize the first band gap of a photonic crystal (PC) enforced with C3 symmetry. Then, the topological phase transition is achieved by inverting the optimized PC. Highly localized topological edge and corner states are formed at the boundary and corner between the optimized PC and its inversion-symmetry partner. The bandwidth of the topological edge state for TE modes achieves 44.9%, exceeding two times of the current record. The robust of the topological corner states is further demonstrated. Our work provides a new route for designing SPTIs, paving the way towards their practical application.

Design of an aircraft engine bracket using stress-constrained bi-directional evolutionary structural optimization method

Design of an aircraft engine bracket using stress-constrained bi-directional evolutionary structural optimization method

Elasto-Plastic limit analysis of reliability based geometrically nonlinear bi-directional evolutionary topology optimization

This paper presents elasto-plastic limit analysis of reliability-based geometrically nonlinear topology optimization. For this purpose, by reason of uncertainties the volume fraction is considered randomly during optimization. Thus reliability-based design has been considered for solving the problems. To perform reliability-based topology optimization design, the Monte-Carlo simulation method has been applied to calculate the probability of failure, thus the reliability index. Besides, bi-directional evolutionary structural optimization (BESO) method is used to consider the effect of geometrically nonlinear design for elasto-plastic analysis. Plastic behavior is controlled by applying a bound on the plastic limit load multipliers using limit analysis. The adequacy of the proposed method is exhibited by three 2D benchmark problems. 2D models of L-shape beam and U-shaped plate are considered for reliability-based design and geometrically nonlinear analysis topology optimization in case of elastic material. Additionally, 2D and 3D elasto-plastic material models have been considered to demonstrate that the proposed method can find the optimal topology of elasto-plastic models for reliability-based design and geometrically nonlinear analysis.

Numerical and experimental investigation on topology optimization of an elongated dynamic system

Minimizing vibration levels of dynamic components at their operating frequency range has been a widely studied topic in engineering. However, the design of structures that satisfy geometric constraints and technical performances is an ongoing challenge. In this work, a topology optimization procedure based on the Bi-directional Evolutionary Structural Optimization (BESO) algorithm is performed to maximize the natural frequency separation interval of an elongated structure. The issues of disconnected and trivial solutions are solved using a connectivity constraint. It is imposed by a proposed procedure based on the heat flux solution of an auxiliary system. An assessment of the feasibility of the structure is done by verifying its accordance with manufacturing and design constraints. The optimized structure was manufactured and validated experimentally. The implemented process produces topologies that maximize the natural frequency separation and reduce the mass of the structure. The obtained results demonstrate the effectiveness of the proposed procedure at satisfying geometric design constraints and technical performances.