An improved multi-objective topology optimization model based on SIMP method for continuum structures including self-weight

Topology optimization is the process of determining the optimal layout of material and connectivity inside a design domain. This paper surveys topology optimization of continuum structures from the year 2000 to 2012. It focuses on new developments, improvements, and applications of finite element-based topology optimization, which include a maturation of classical methods, a broadening in the scope of the field, and the introduction of new methods for multiphysics problems. Four different types of topology optimization are reviewed: (1) density-based methods, which include the popular Solid Isotropic Material with Penalization (SIMP) technique, (2) hard-kill methods, including Evolutionary Structural Optimization (ESO), (3) boundary variation methods (level set and phase field), and (4) a new biologically inspired method based on cellular division rules. We hope that this survey will provide an update of the recent advances and novel applications of popular methods, provide exposure to lesser known, yet promising, techniques, and serve as a resource for those new to the field. The presentation of each method's focuses on new developments and novel applications.

This paper presents an adaptive continuation method for buckling topology optimization of continuum structures using the Solid Isotropic Material with Penalization (SIMP) model. For optimization problems of minimizing structural compliance subject to constraints on material volume and buckling load factors, it has been found that the conflict between the requirements for structural stiffness and stability may have an adverse impact on the performance of existing optimization algorithms. An automatic scheme for adjusting the penalization parameter is introduced to deal with this conflict and thus achieves better designs. Based on an investigation on the effect of the penalization parameter on design evolution during the optimization process, a rule is established to determine the appropriate penalization parameter values. Using this rule, an effective scheme is developed for determining the penalization parameter values such that the buckling constraints are appropriately considered throughout the optimization process. Numerical examples are presented to illustrate the effectiveness of the proposed method.

Topology optimization is the procedure of determining the connectivity, shape and location of voids inside a given design domain. This chapter proposes a quantum-inspired evolutionary algorithm (QEA) for structural topology optimization. It reviews the formulation of the continuum structural topology optimization model and the characteristics of quantum computing. The chapter also proposes the detailed implementation of a QEA-based topology optimization. It discusses two issues: the density-based continuum structural topology optimization formulation and the characteristics of quantum computing, which may be conducive to reducing the computational expense of topology optimization. The entire theoretical procedure of quantum-inspired topology optimization, which mainly combines FEM analysis, design sensitivity solution and quantum-inspired search, is accomplished for continuum structures. As the experimental implementation of quantum computation needs initialization, coherent manipulation, control and read of the fragile quantum system, practically building quantum computers has proved extremely difficult.

Topology optimization has been successful in generating optimal topologies of various structures arising in real-world applications. Since these applications can have complex and large domains, topology optimization suffers from a high computational cost because of the use of unstructured meshes for discretization of these domains and their finite element analysis (FEA). This article addresses this challenge by developing three graphics processing unit (GPU)-based element-by-element strategies targeting unstructured all-hexahedral mesh for the matrix-free precondition conjugate gradient (PCG) finite element solver. These strategies mainly perform sparse matrix multiplication (SpMV) arising with the FEA solver by allocating more compute threads of GPU per element. Moreover, the strategies are developed to use shared memory of GPU for efficient memory transactions. The proposed strategies are tested with solid isotropic material with penalization (SIMP) method on four examples of 3D structural topology optimization. Results demonstrate that the proposed strategies achieve speedup up to 8.2 × over the standard GPU-based SpMV strategies from the literature.

Topology optimization (TO) is currently a focal point for researchers in the field of structural optimization, with most studies concentrating on single-loading conditions. However, real engineering structures often have to work under various loading conditions. Approaches addressing multiple-loading conditions often necessitate subjective input in order to determine the importance of each loading condition, aiming for a compromise between them. This paper proposes a so-called bisection constraint method (BCM), offering a unique, user-preference-independent solution for TO problems amidst multiple-loading conditions. It is well-known that minimizing the system’s compliance is commonly used in TO as the objective. Generally, compliance is not as sufficient as stress to be used as a response to evaluate the performance of structures. However, formulations focusing on minimizing stress levels usually pose significant difficulties and instabilities. On the other hand, the compliance approach is generally simpler and more capable of providing relatively sturdy designs. Hence, the formulation of min–max compliance is used as the target problem formulation of the proposed method. This method attempts to minimize compliance under only one loading condition while compliances under the remaining loading conditions are constrained. During the optimization process, the optimization problem is automatically reformulated with a new objective function and a new set of constraint functions. The role of compliance under different loading conditions, i.e., whether it is to be treated as an objective or constraint function, might be changed throughout the optimization process until convergence. Several examples based on the solid isotropic material with penalization (SIMP) approach were conducted to illustrate the validity of the proposed method. Furthermore, the general effectiveness of the compliance approach in terms of stress levels is also discussed. The calculation results demonstrated that while the compliance approach is effective in several cases, it proves ineffective in certain scenarios.

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

By assuming that the random parameters of a continuum structure obey the normal distribution, this paper utilized “normrnd” function in Matlab software to generate pseudo random numbers, analyzed structural stress by using finite element method, conducted topology optimization by using k nearest neighbor (KNN) method, and as such, the continuum structure has been stochastic topology optimized. The paper put forward the concept of “stochastic topology optimization”. Numerical examples are given. The results show that the stochastic topology optimization of continuum structure with random parameters is necessary.

In order to solve the problem that the weight of large scale theodolite's casting or steel plate welded structure is too heavy, lightweight design of large scale turntable based on topology optimization of continuum structure is investigated. Topology optimization could have better optimization efficiency because it can provide the optimized distribution of material and gain optimized structure in conceptual design phase of the structural design. In order to obtain lightweight and high stiffness structure, a truss turntable based on topology optimization of continuum structure is presented, and the validity of this lightweight structure is examined by finite element analysis. Comparing to the traditional welding steel turntable, the lightweight turntable, without changing the static displacement, can lose 386.48 kg(lightweight ratio is 26.7%), and its first-order natural frequency raises from 167.02Hz to 173.71Hz. Meanwhile, in other aspects this turntable also has better performance than traditional structure.

A topology optimization approach based on the boundary element method (BEM) and the optimality criteria (OC) method is proposed for the optimal design of sound absorbing material distribution within sound barrier structures. The acoustical effect of the absorbing material is simplified as the acoustical impedance boundary condition. Based on the solid isotropic material with penalization (SIMP) method, a topology optimization model is established by selecting the densities of absorbing material elements as design variables, volumes of absorbing material as constraints, and the minimization of sound pressure at reference surface as design objective. A smoothed Heaviside-like function is proposed to help the SIMP method to obtain a clear 0---1 distribution. The BEM is applied for acoustic analysis and the sensitivities with respect to design variables are obtained by the direct differentiation method. The Burton---Miller formulation is used to overcome the fictitious eigen-frequency problem for exterior boundary-value problems. A relaxed form of OC is used for solving the optimization problem to find the optimal absorbing material distribution. Numerical tests are provided to illustrate the application of the optimization procedure for 2D sound barriers. Results show that the optimal distribution of the sound absorbing material is strongly frequency dependent, and performing an optimization in a frequency band is generally needed.

It is of great importance for the development of new products to find the best possible topology or layout for given design objectives and constraints at a very early stage of the design process (the conceptual and project definition phase). Thus, over the last decade, substantial efforts of fundamental research have been devoted to the development of efficient and reliable procedures for solution of such problems. During this period, the researchers have been mainly occupied with two different kinds of topology design processes; the Material or Microstructure Technique and the Geometrical or Macrostructure Technique. It is the objective of this review paper to present an overview of the developments within these two types of techniques with special emphasis on optimum topology and layout design of linearly elastic 2D and 3D continuum structures. Starting from the mathematical-physical concepts of topology and layout optimization, several methods are presented and the applicability is illustrated by a number of examples. New areas of application of topology optimization are discussed at the end of the article. This review article includes 425 references.

The composite structures in the aerospace industry for in recent decades are widely applied however, at the beginning of the 21st century composites are growing rapidly. The largest companies in the aerospace industry are increasing the volume of composites application of in the structures, and nowadays the volume of composites reaches 50%. The different elements of aircraft and even highly loaded structures such as spars, ribs, skin, etc., are currently made from composites. First of all, this is due to the possibility of a significant reduction in the weight of the structure, as well as a decreasing in production costs. The advanced technologies in the engineering software allows to solute different complex problems. One of the main direct of research in the composites is optimization of composite structure due to improving the relative strength and relative stiffness of the composite structure, and improving the efficiency of manufacturing processes. There are a lot of methods of optimizations but currently the topological optimization is the most conceptual and forward-looking method. The main goal of the article is to analyze and estimate the approach for designing wing rib with symmetric laminated plates with the different fiber orientation based on the topology optimization. The following tasks were solved for this: firstly, a topological optimization model was determined. This model was based on maximum stiffness with a specified volume constraint is established. The next step was optimization by the solid isotropic material with penalization (SIMP) model and sensitivity filtering technique; as a result of optimization the topological structures of wing rib with different fibre orientations were obtained. The topological structure and stiffness of the wing rib depend on the fibre orientation. Finally, the corresponding morphing analysis of wing rib with laminated plates is implemented by adopting ANSYS, which verified the anti-deforming capability of topology structure and illustrated the feasibility for designing the wing rib. The result shows that the maximum deformation of optimized structure is 1.57mm, whereas the maximum deformation of the un-optimized structure is 2.02 mm. Under the condition of the same material removals, the optimized structure can decrease by more than 20% deformations.

SUMMARYA method for topology optimization of continuum structures based on nodal density variables and density field mapping technique is investigated. The original discrete‐valued topology optimization problem is stated as an optimization problem with continuous design variables by introducing a material density field into the design domain. With the use of the Shepard family of interpolants, this density field is mapped onto the design space defined by a finite number of nodal density variables. The employed interpolation scheme has an explicit form and satisfies range‐restricted properties that makes it applicable for physically meaningful density interpolation. Its ability to resolve more complex spatial distribution of the material density within an individual element, as compared with the conventional elementwise design variable approach, actually provides certain regularization to the topology optimization problem. Numerical examples demonstrate the validity and applicability of the proposed formulation and numerical techniques. Copyright © 2011 John Wiley & Sons, Ltd.

A several cycles of design-verification-modification are needed during traditional design, which is very time-consuming. Topology optimization is rapidly expanding in the recent years in the field of structural mechanics. Currently it is widely applied to structural design. The combination of topology optimization and sizing and/or shape optimization has been adopted for flight vehicle structural design, which results possible better designs. In the paper, continuum structural topology optimizations as well as size optimization are combined for a folding wing structural design. A structure optimization problem is established using Solid Isotropic Material with Penalization (SIMP) method, it consists of the thickness variable of the skin and the relative element density variables of the tip part of the folding wing. The design objective is to minimize the compliance of the whole structure, and the constraints are structural mass constraints and stress constraints. The problem was solved based on commercial software package HyperWorks™. The resulting design shows that a new tip structure which satisfies the requirements of stiffness and strength was reconstructed.

Topology optimization with meshless density variable approximations and BESO method

This paper presents a zeroth-order (non-gradient based) methodology for topology optimization of continuum structures. The solid and void elements in the design domain are represented by binary design variables. As in the SIMP method, design variables are assumed to represent the non-dimensional density of the individual finite elements and their contribution to structural stiffness in the model. The optimization procedure begins by specifying a desired volume fraction over a discretized domain and associated boundary conditions. Based on a randomized distribution of solid and void elements, a finite element analysis is performed to identify the strain energy of individual elements. While keeping the volume fraction fixed, the solid elements with low strain energy are converted into void elements whereas the void elements with high strain energy are converted into solid elements. The strain energy threshold for element exchange is specified at the beginning and can be adjusted in the course of optimization. Through subsequent repetition of analysis and exchange steps, the optimal topology corresponding to minimum compliance emerges with no gray elements to filter out. The general characteristics of the Element Exchange Method are demonstrated through solutions of several benchmark problems.