Bi-directional Evolutionary Structural Optimization Method Research Articles

Evolutionary topology optimization of continuum structures including design-dependent self-weight loads

The effectiveness and efficiency of the Bi-directional Evolutionary Structural Optimization (BESO) method has been demonstrated on the minimization compliance problem with fixed external loads. This paper considers the minimization of mean compliance for continuum structure subjected to design-dependent self-weight loads. Due to the non-monotonous behaviour for this type of the optimization problems, the extended BESO method using discrete design variables has its difficulty to obtain convergent solutions for such problems. In this paper, a new BESO method is developed based on the sensitivity number computation utilizing an alternative material interpolation scheme. A number of examples are presented to demonstrate the capabilities of the proposed method for achieving convergent optimal solutions for structures including self-weight loads.

Reinventing the Wheel

A rational approach to the mechanical design of wheel rims as a typical periodic structure is presented in the current work. With novel application of the latest bidirectional evolutionary structural optimization method, a procedure is presented for the optimal topological design of wheel rims. Design applications are studied with realistic loads on a general vehicle in various scenarios, where the results not only demonstrate originalities of wheel patterns, but also provide insights into existing wheel designs. The simplicity and generic nature imply the general applicability of the proposed approach to a wide range of wheel designs.

Topology Optimization of Composite Structure Using Bi-Directional Evolutionary Structural Optimization Method

This paper deals with topology optimization of composite structure using Bi-directional Evolutional Structural Optimization (BESO) method. By redefining the criteria of the optimization evolution progress, the proposed method is able to extend current BESO method from isotropic material to anisotropic composite material. The initial modification of BESO method is to allow for inefficient Material Element String (MES) to be removed while efficient MES to be added in the thickness direction of a composite structure. This modification can reduce the chance of high stress concentration in a composite structure and also produce the geometry which is easy to fabricate using fabrication techniques currently available. The results of a cantilever composite laminate under uniform inplane pressure are presented, showing that the proposed method can produce shape and topology for composite structures with optimal structure stiffness.

Natural frequency optimization of structures using a soft-kill BESO method

Frequency optimization is of great importance in the design of machines and structures subjected to dynamic loading. When the natural frequencies of considered structures are maximized using the Solid Isotropic Material with Penalization (SIMP) model, artificial localized modes may occur in areas where elements are assigned with low density values. In this paper, a modified SIMP model is developed to effectively avoid the artificial modes. Based on this model, a new bi-directional evolutionary structural optimization (BESO) method combined with rigorous optimality criteria is developed for topology frequency optimization problems. Numerical results show that the proposed BESO method is efficient and convergent and solid-void or bi-material optimal solutions can be achieved for a variety of frequency optimization problems of continuum structures.

Recent developments in evolutionary structural optimization (ESO) for continuum structures

Evolutionary Structural Optimization (ESO) and its later version bi-directional ESO (BESO) have gained widespread popularity among researchers in structural optimization and practitioners in engineering and architecture. However, there have also been many critical comments on various aspects of ESO/BESO. To address those criticisms, we have carried out extensive work to improve the original ESO/BESO algorithms in recent years. This paper summarizes some of the latest developments in the BESO method for topology optimization of continuum structures. Numerical results show that the ESO/BESO solutions agree well with those of other well-established topology optimization methods. It indicates that the current BESO method has great potential to become a robust and efficient design tool for practical applications in engineering and architecture.

AN IMPROVED BI-DIRECTIONAL EVOLUTIONARY TOPOLOGY OPTIMIZATION METHOD FOR FREQUENCIES

This paper presents a topology optimization method for dynamic problems with an improved bi-directional evolutionary structural optimization (BESO) technique. The sensitivity derivation for the frequency optimization problem in the case of multiple eigenvalues and for the stiffness–frequency optimization problem is proposed. Algorithms for a filter scheme, sensitivity history-averaging, and sensitivity global-ranking are used in the present method. Techniques for adaptively removing alternative elements and eliminating singular and single-hinged elements are proposed. Solution-convergence and localized modes are discussed through numerical examples. Results show that the improved BESO method is capable of solving the frequency optimization problem and the multi-objective optimization problem for stiffness and frequency effectively.

Evolutionary topological optimization of vibrating continuum structures for natural frequencies

Frequency optimization is of great importance in the design of machines and structures subjected to dynamic loading. When the natural frequencies of considered structures are maximized using the solid isotropic material with penalization (SIMP) model, artificial localized modes may occur in areas where elements are assigned with lower density values. In this paper, a modified SIMP model is developed to effectively avoid the artificial modes. Based on this model, a new bi-directional evolutionary structural optimization (BESO) method combining with rigorous optimality criteria is developed for topology frequency optimization problems. Numerical results show that the proposed BESO method is efficient, and convergent solid-void or bi-material optimal solutions can be achieved for a variety of frequency optimization problems of continuum structures.

Evolutionary topology optimization of continuum structures with an additional displacement constraint

Evolutionary structural optimization (ESO) and its later version Bi-directional ESO (BESO) have been successfully applied to optimum material distribution problems for continuum structures. However, the existing ESO/BESO methods are limited to the topology optimization of an objective function such as mean compliance with a single constraint e.g. structural volume. The present work extends the BESO method to the stiffness optimization with a material volume constraint and a local displacement constraint. As a result, one will obtain a structure with the highest stiffness for a given volume while the displacement of a certain node does not exceed a prescribed limit. Several examples are presented to demonstrate the effectiveness of the proposed method.

Topology optimization of heat conduction problem involving design-dependent heat load effect

Topology optimization of steady heat conduction problem under both design-independent and design-dependent heat loads is studied by means of a modified bidirectional evolutionary structural optimization (BESO) method. Two types of problems are distinguished by their physical meanings and particularly design-dependent load effect is highlighted in the following two points. At the stage of sensitivity analysis, both the heat conductivity matrix and the design-dependent heat generation load associated with the void element are penalized in the same manner. The rationality is illustrated based on numerical tests. Furthermore, as the sensitivity of the objective function changes its sign during the iteration, a modified BESO procedure is presented to deal with the non-monotonicity of the objective function defined by the heat potential capacity. Detailed steps of the BESO procedure are presented for the element removal and growth while the inequality volume constraint is imposed. To conclude the work, numerical results and the element sensitivity obtained are discussed to show the effect of design-dependent load.

Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials

There are several well-established techniques for the generation of solid-void optimal topologies such as solid isotropic material with penalization (SIMP) method and evolutionary structural optimization (ESO) and its later version bi-directional ESO (BESO) methods. Utilizing the material interpolation scheme, a new BESO method with a penalization parameter is developed in this paper. A number of examples are presented to demonstrate the capabilities of the proposed method for achieving convergent optimal solutions for structures with one or multiple materials. The results show that the optimal designs from the present BESO method are independent on the degree of penalization. The resulted optimal topologies and values of the objective function compare well with those of SIMP method.

BESO method for topology optimization of structures with high efficiency of heat dissipation

The purpose of this paper is to present a Bi-Directional Evolutionary Structural Optimization (BESO) method for topology optimization of heat conduction structures. In BESO method the elements are allowed to be added as well as removed. Focused on the heat performance of structure, the additive criterion and rejection criterion were proposed respectively. With the limit volume of the high conductive material, the optimal layout of structure with high efficiency of heat dissipation and uniform temperature distribution can be obtained efficiently by BESO procedure.

A study on ranked bidirectional evolutionary structural optimization (R-BESO) method for fully stressed structure design based on displacement sensitivity

Evolutionary Structural Optimization(ESO) method is well known as one of several topology optimization methods and has been applied to a lot of optimization problems. While ESO method evolves the given model into an optimum by subtracting several elements, in AESO method elements are added in a previous step of the evolutionary procedure. And in BESO(Bidirectional ESO) method, some elements are either generated or eliminated from a previous model of evolutionary procedure. In this paper, Ranked Bidirectional Evolutionary Structural Optimization(R-BESO) method is introduced as one of the topology optimization methods using an evolutionary algorithm and is applied to several optimization problems. The method can get optimum topologies of the structures throughout fewer iterations comparing with previous several methods based on ESO. R-BESO method is similar to BESO method except that elements are generated near a candidate element according to the rank calculated by sensitivity analyses. The displacement sensitivity analysis was adopted by the nodal displacements of a candidate element in order to determine a rank on the free edges for two dimensional model or the free surfaces for three dimensional model. In this paper, R-BESO method is proposed as another useful design tool like the previous ESO and BESO method for the two bar frame problem, the Michell type structure problem and the three dimension short cantilever beam problem, which had been used to verify reasonability of ESO method family. For the three dimensions short cantilever beam problem an optimized topology could be obtained with much fewer iterations with respect to the results of other ESO methods.

Evolutionary Structural Topology Optimization for Continuum Structures with Structural Size and Topology Variables

In this paper, a topology optimization method mixed structures with continuum and discrete elements is proposed. First, based on the stress and displacement formulations of the finite element analysis, a set of stress sensitivity numbers for beam and plane-stress structures are derived. Then, relative difference quotients for topology and size variables of mixed continuum structures with several different types of components are formulated. The optimization criteria, which incorporate the element stress level and relative different quotients, are developed for the topology optimization of continuum domains and the sizing optimization of discrete elements. A new optimization procedure based on the bi-directional evolutionary structural optimization method is proposed. Two examples are provided to demonstrate the validity and effectiveness of the proposed method.

Advantages of Bi-Directional Evolutionary Structural Optimization (BESO) over Evolutionary Structural Optimization (ESO)

The evolutionary structural optimization (ESO) method evolves a structure from the full design domain towards an optimum by gradually removing inefficient material. The bi-directional ESO (BESO) may start from any initial design and evolve a structure to an optimum by adding and removing material simultaneously. In this paper, a detailed comparison between ESO and BESO has been carried out for stiffness optimization problems. Both 2D and 3D examples shows that the BESO method possesses many advantages over the ESO method such as computational efficiency, robustness of the method and manufacturability of the final topology.

Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method

This paper presents an improved algorithm for the bi-directional evolutionary structural optimization (BESO) method for topology optimization problems. The elemental sensitivity numbers are calculated from finite element analysis and then converted to the nodal sensitivity numbers in the design domain. A mesh-independency filter using nodal variables is introduced to determine the addition of elements and eliminate unnecessary structural details below a certain length scale in the design. To further enhance the convergence of the optimization process, the accuracy of elemental sensitivity numbers is improved by its historical information. The new approach is demonstrated by solving several compliance minimization problems and compared with the solid isotropic material with penalization (SIMP) method. Results show the effectiveness of the new BESO method in obtaining convergent and mesh-independent solutions.

Numerical stability and parameters study of an improved bi-directional evolutionary structural optimization method

This paper presents a modified and improved bi-directional evolutionary structural optimization (BESO) method for topology optimization. A sensitivity filter which has been used in other optimization methods is introduced into BESO so that the design solutions become mesh-independent. To improve the convergence of the optimization process, the sensitivity number considers its historical information. Numerical examples show the effectiveness of the modified BESO method in obtaining convergent and mesh-independent solutions. A study of the effects of various BESO parameters on the solution is then conducted to determine the appropriate values for these parameters.

A fixed‐grid bidirectional evolutionary structural optimization method and its applications in tunnelling engineering

AbstractTopology optimization has exhibited an exceptional capability of improving structural design. However, several typical topology optimization algorithms are finite element (FE) based, where mesh‐dependent zigzag representation of boundaries is barely avoidable in both intermediate and final results. To tackle the problem, this paper proposes a new fixed‐grid‐based bidirectional evolutionary structural optimization method, namely FG BESO. The adoption of an FG FE framework enables a continuous boundary change in the course of topology optimization, which provides a means of dealing with not only the non‐smooth boundary of the final design but also the interpretation of intermediate densities. As a class of important practical application, it is interesting to make use of the new FG BESO method to the reinforcement design for underground tunnels. To accommodate the FG BESO technique to geological engineering applications, a nodal sensitivity is derived for a two‐phase material model comprising the artificial reinforcement and original rock. In this paper, some innovative topological designs of tunnel reinforcements are presented for minimizing the floor and sidewall heaves under different geological loading conditions. Copyright © 2007 John Wiley & Sons, Ltd.

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.

Topology Optimization for Frequencies Using an Evolutionary Method

This paper presents an evolutionary method for structural topology optimization subject to frequency constraints. The evolutionary structural optimization (ESO) method is based on the idea that by gradually removing inefficient material, the residual shape of structure evolves toward an optimum. The method is further developed by allowing the material to be added as well as removed, and this new approach is called the bidirectional ESO method (BESO). BESO has been successfully used for problems of stress and stiffness/displacement constraints. Its application to frequency optimization is addressed in this paper. Three kinds of optimization objectives, namely, maximizing a single frequency, maximizing multiple frequencies, and designing structures with prescribed frequencies are considered. Four examples are tested by BESO and ESO. The objective functions yielded by the two methods are close, and BESO is computationally more efficient in most cases.

3D and multiple load case bi-directional evolutionary structural optimization (BESO)

Evolutionary Structural Optimization (ESO), is a numerical method of structural optimization that is integrated with finite element analysis (FEA). Bi-directional ESO (BESO) is an extension to this method and can begin with minimal amount of material (only that necessary to support the load and support cases) in contrast to ESO which uses an initially oversized structure. Using BESO the structure is then allowed to grow into the optimum design or shape by both adding elements where the stresses are the highest and taking elements away where stresses are the lowest. In conducting this research, a methodology was developed (and integrated into the ESO program EVOLVE) which produced the optimal 3D finite element models of a structure in a more reliable way than the traditional ESO method. Additionally, the BESO method was successfully extended to multiple load cases for both 2D and 3D. Two different algorithms were used to find the best structure experiencing more than one load case and the results of each are included.