Dimension reduction and surrogate based topology optimization of periodic structures

This paper will give a short survey about topology optimization of structures. It is particularly focused on topological shape optimization of structures using level-set methods, including the level-set based standard methods and the level-set based alternative methods. The former often directly solve the Hamilton-Jacobi partial differential equation (H-J PDE) to obtain the boundary velocity field using Finite Differential Methods (FDM), and the later commonly employ parametric or equivalent methods to evaluate the velocity field without directly solving the H-J PDE. The unique characteristics of the level-set based topology optimization methods are discussed, and a future perspective and prospects in this research area is also included. A benchmark numerical example is used to showcase the effectiveness of the level-set based methods.

During the last 20 years, structural optimization has become one of the most important topics of engineering applications. Design optimization of structure has been an interesting area of research in the field of engineering design for its ability to short the design cycle and to enhance product quality. Significant research activity has occurred in the area of structural optimization in the last decade. Especially for topology optimization of structure, many new theoretical, algorithmic, and computational contributions have resulted by researchers and engineers. Topology optimization is a powerful tool for global and multi-scale design of macrostructures, microstructures, and the cell of prescribed composite materials. The population based evolutionary algorithms have emerged as powerful mechanism for finding optimum solutions of complex optimization problems in engineering during the last two decades. Evolutionary computation is the study of computational systems which use ideas and get inspiration from natural evolution and adaptation [1]. The thinking has wide application in various engineering fields, such as computer science, artificial intelligence, operations research. Genetic algorithm is another kind of bio-inspired optimization method and it is playing an increasingly important role in studies of complex adaptive systems. Its application ranges from adaptive agents in economic theory to the use of machine learning techniques in the design of complex devices and structures, such as aircraft turbines and integrated circuits [2]. Optimization of structures can be classified into three categories: sizing, shaping, and topology optimization. In the topology optimization, it is concerned with the structure members and connectivity between members. In general, it is easily represented by discrete variables rather than by those used for continuous optimization problems. Topology optimization is the most difficult and complex among three categories and it is special useful in developing innovative conceptual designs. Structural optimization, in particular the topology optimization, has been identified as one of the most challenging tasks in structural design. Various techniques and approaches have been established during the last two decades. Topology optimization usually referred to as layout optimization or general shape optimization [3]. It lets engineers get the optimal topology of structure or new configurations during product design phase, as they are implementing the design of the size

This article proposes an efficient parameterized level set method (PLSM) for structural topology optimization design with minimum compliance as the objective function and volume fraction as the constraint condition. In the stage of establishing the optimization model, the level set function (LSF) is interpolated using compactly-supported radial basis functions (CS-RBFs) to transform the Hamilton-Jacobi partial differential equation (PDE) into ordinary differential equations (ODEs), thereby making the evolution of the LSF more convenient and efficient, and ensuring the smoothness of the optimization result boundary. The method of moving asymptotes (MMA) is used for solving the established optimization model, and meanwhile the shape sensitivity constraint factor is added to improve computational efficiency. During the evolution of the LSF, an approximate re-initialization scheme is employed to prevent the gradient of the LSF boundary from being too large or too small, thereby improving the numerical stability and the convergence speed of structural topology optimization process. Furthermore, the proposed method is also extensible and applicable to topology optimization of multi-material structures. The feasibility and effectiveness of this method have been verified through several typical numerical examples involving topology optimization of single-material structures and multi-material structures within the framework of minimum compliance design. HIGHLIGHTS An efficient parameterized level set method using the approximate re-initialization scheme is proposed for structural topology optimization. The approximate re-initialization scheme is employed when solving the parameterized level set function via the MMA algorithm. This scheme can prevents the gradient of the level set function boundary from being too large or too small, making the level set function update more stable and accelerating the convergence speed of structural topology optimization. The effectiveness and feasibility of the proposed method are demonstrated through examples of single-material and multi-material structural topology optimization.

Structural topology optimization seeks to identify the most efficient way to allocate material within a given domain to achieve a specific design objective. This research area has broad applications in various fields, including mechanical design and other physical disciplines. Over the past two decades, a great variety of applications and diverse research papers on structural topology optimization have been undertaken and published in the last two decades to highlight its acceptance and advantages. This paper presents a scientometric analysis of 5713 papers published between 2000 and 2022 to provide an overview and visualization of the development of structural topology optimization. While there are numerous approaches to topology optimization, this study focuses on presenting a quantitative analysis of the most popular and influential methods. The method used in this paper involved data collection and analysis through CiteSpace, one of the most popular software tools in this field. This paper provides a comprehensive overview of the latest developments of research in structural topology optimization and identifies key directions for future research. The analysis of the top 10 co-cited articles revealed four main approaches that have been developed for structural topology optimization.

Structural topology optimization is a crucial approach for achieving lightweight design. An effective topology optimization algorithm must strike a balance between the objective functions, constraints, and design variables, which essentially reflects the symmetry and tradeoff between the objective and constraints. In this study, a topology optimization method grounded in load path theory is proposed. Element bearing capacity is quantified using the element birth and death method, with an explicit formulation derived via finite element theory. The effectiveness in evaluating structural performance is assessed through comparisons with stress distributions and topology optimization density maps. In addition, a novel evaluation index for element bearing capacity is proposed as the objective function in the topology optimization model, which is validated through thin plate optimization. Subsequently, sensitivity redistribution mitigates checkerboard patterns, while mesh filtering suppresses multi-branch structures and prevents local optima. The method is applied for the lightweight design of a triangular arm, with results benchmarked against the variable density method, demonstrating the feasibility and effectiveness of the proposed method. The element bearing capacity seeks to homogenize the load distribution of each element; the technique in this study can be extended to the optimization of symmetric structures.

Isogeometric analysis (IGA) connects CAE with standard CAD design tools and has been utilized for a decade in structural optimization. FEA has a number of drawbacks, including a lack of linkage between analytic and geometric models, a shortage of continuity between components, and a lack of accuracy for high-order elements. Isogeometric analysis effectively overcomes these obstacles. This paper proposes the isogeometric topological optimization mathematical model depending on the asymmetric constitutive relationship of tension and compression microscopic materials. We provide a new density-based topology optimization formulation in which the design domain is limited to the B-spline space. A rectangular domain with a tensor-product is embedded in a design domain that is arbitrarily shaped. The density field is represented using B-splines. The intrinsic filter provided by the B-spline representation for topology optimization is governed by the degrees of B-spline and the knot spans. This B-spline effectively removes numerical artifacts and controls the minimum feature-length in; when the B-spline basis functions span many analysis elements. Benchmark problems in structural topology optimization are computed through Evolutionary Structural Optimization (ESO) algorithm. Comparisons of the optimal structure topologies using finite element analysis and isogeometric collocation methods are analyzed. The influence of different tension-compression stiffness ratios on the final structure topologies is listed and analyzed. The numerical results show the reliability and performance of the proposed mathematical formulation and asymmetry of natural structure.

Structural topology constraints in topology optimization are an important research topic. The structural topology is characterized by the topological invariance of the number of holes. The holes of a structure in 3D space can be classified as internally enclosed holes and external through-holes (or tunnels). The genus is the number of tunnels. This article proposes the quotient set design variable method (QSDV) to implement the inequality constraint on the maximum genus allowed in an optimized structure for 3D structural topology optimization. The principle of the QSDV is to classify the changing design variables according to the connectivity of the elements in a structure to obtain the quotient set and update the corresponding elements in the quotient set to meet the topological constraint. Based on the standard relaxation algorithm discrete variable topology optimization method (DVTOCRA), the effectiveness of the QSDV is illustrated in numerical examples of a 3D cantilever beam.

According to the requirements of the aerospace industry for high strength, high stiffness, and lightweight structural parts, topology optimization has been proved to be an effective product design method. As one of the most conceptual and prospective structural optimization design methods, topology optimization intends to seek the optimal layout of materials in an allowed design region under a given load and boundary conditions. Thus, the object of study in the article is the method of topological optimization of aircraft structures. The goal of this article is to analyze the existing approaches, algorithms, as well as application of the method of topological optimization in the aerospace field in applied problems. The tasks are to describe the existing various approaches methods, features, and research directions of topological optimization as well as to study the possibility of application in the manufacturing process of composite structures. The following results were obtained. The optimization methods are briefly explained and compared, and the advantages and limitations of each approach are discussed. The various ways of simultaneous optimization of fiber orientation and structural topology were described and analyzed. The features of different methods of continuous fiber orientation optimization method were reviewed. The discrete fiber orientation optimization methods were represented. The possibility of multi-scale concurrent topological optimization was described. The combination of topology optimization and additive manufacturing was considered. Finally, the topology optimization of FRC structures which have been resolved in literature are reviewed and the potential research fields requiring more investigation are pointed out. Conclusions. In the article, a comprehensive review of the topology optimization design of FRC structures was presented. The promising way is to combine topology optimization with additive manufacturing techniques. However, these proposed methods may not suitable for other more complex problems, such as bucking stability and natural frequency. Hence, the topology optimization design of complex FRC components under complicated conditions is the main challenge in the future. This can be a new trend in the topology design of FRC structures.

This thesis aims at understanding and improving topology optimization techniques focusing on density-based level-set methods and geometrical nonlinearities. Central in this work are the numerical modeling of the mechanical response of a design and the consistency of the optimization process itself. Concerning the first topic, we investigate different means to improve the robustness of density-based numerical models including geometrical nonlinearities. The conventional approach (scaling the local material properties) can result in convergence problems due to excessive deformation in low-stiffness finite elements. To avoid excessive deformation, we combine the element connectivity parameterization method (adapting the connectivity between finite elements) with level-set-based topology optimization. Furthermore, we achieve greater robustness of analysis method using a second and improved approach called element deformation scaling. This approach eliminates the need for solving internal equilibrium equations (needed for the element connectivity parameterization method) via an explicit relation between the local internal and global external displacement field. The second focus of this thesis is on the optimization process of level-set-based topology optimization, and in particular its numerical consistency. We observe that signed-distance reinitialization of the level-set function affects the shape of a design in the optimization process. To minimize this effect, we propose a discrete level-set method that is based on an approximate Heaviside function and focusing on the implementation. Furthermore, we also propose a level-set-based topology optimization method using an exact Heaviside function and mathematical programming that effectively eliminates the need for reinitialization. We demonstrate that our density-based level-set method is closely related to conventional density-based topology optimization methods, while offering the advantage of more control over the geometrical complexity. On the other hand, we confirm that the dependence of the final result on the initial design which remains one of the big challenges for level-set-based topology optimization. The potential of the proposed level-set method is shown by applying it on problems with stress constraints and geometrical nonlinearities and performing manufacturing tolerant topology optimization. Finally, this thesis offers a review of level-set methods for structural topology optimization to identify and discuss the different approaches that are available in literature. We can distinguish between level-set methods by examination of their design parameterization, sensitivities, update procedures and regularization techniques. A level-set-based design parameterization offers the advantage of a crisp distinction between subdomains. For this reason, X-FEM approaches and conforming discretizations are an interesting option to retain the crisp nature of the level-set-based description of the design. Many level-set methods are combined with density-based numerical models and are, therefore, closely related to conventional density-based topology optimization methods. In particular, recently proposed projection methods have much in common with a level-set-based design description. The results of level-set-based TO methods often rely heavily on regularization techniques that introduce inconsistencies in the optimization process. Numerical consistency does not necessarily lead to the best search direction, but is essential to find a Karush-Kuhn-Tucker point of the discretized optimization problem.

Conventional topology optimization is concerned with the structures modeled by classical theory of mechanics. Since it does not consider the effects of the microstructures of materials, the classical theory can not reveal the size effect due to material’s heterogeneity. Couple-stress theory, which takes account of the microscopic properties of the material, is capable of describing the size effect in deformations. The purpose of this paper is to investigate the formulation for topology optimization of couple-stress material structures. The artificial material density of each element is chosen as design variable. Based on the basic idea of SIMP (Solid Isotropic Material with Penalization) method, the effective material stiffness matrix of couple-stress material is related to the artificial density by power law with penalty. The structural analysis is implemented by finite element method for couple-stress materials, and a 4-noded quadrilateral couple-stress element is formulated in which C1 continuity requirement is relaxed. Some typical problems are solved and the optimal results based on the couple-stress theory are compared with the conventional ones. It is found that the optimal topologies of couple-stress continuum show remarkable size effect.

Design optimization of beam structures is a significant topic as beams are an efficient load-carrying component in engineering applications. Most of the earlier researches concentrate on the structural optimization of beams with invariant cross-section topology, and design optimization of periodic beam structures, which cross-section topology varies along the axial direction, has not been extensively investigated. This work presents a novel approach based on the relaxed Saint-Venant solution to conduct microstructural topology optimization of periodic beam structures for minimum structural compliance. One benefit of adopting Saint-Venant solution based compliance formulation is that the strain energy induced by transverse shear loading is incorporated in the structural compliance, which is not reflected in that based on first-order homogenization of the asymptotic homogenization (AH) theory, so that a more reasonable objective function is adopted for the optimization problem. In addition, a material connectivity constraint, which is constructed by restraining the ratio of the strain energy calculated from relaxed Saint-Venant solution to that obtained from first-order homogenization of the AH theory, is further proposed to prevent material separation or to strengthen material connection through the thickness direction. The detailed sensitivity analysis of the objective function and the constraints are carried out with the adjoint method, and several numerical examples are given to show the validity of the relaxed Saint-Venant solution based compliance formulation and the effectiveness of the proposed material connectivity constraint.

A method for structural topology optimization of frame structures with flexible joints is presented. A typical frame structure is a set of beams and joints assembled to carry an applied load. The problem considered in this paper is to find the stiffest frame for a given mass. By introducing design variables for beams and joints, a mass distribution for optimal structural stiffness can be found. Each beam can have several design variables connected to its cross section. One of these is an area-type design variable which is used to represent the global size of the beam. The other design variables are of length ratio type, controlling the cross section of the beam. Joints are flexible elements connecting the beams in the structure. Each joint has stiffness properties and a mass. A framework for modelling these stiffnesses is presented and design variables for joints are introduced. We prove a theorem which can be interpreted as the fact that the removal of structural elements, e.g. joints or beams, can be modelled by a small strictly positive material amount assigned to the element. This is needed for the computations of sensitivities used in the applied gradient based iterative method. Both two and three dimensional problems, as well as multiple load cases and multiple mass constraints, are treated.

Topology optimization of continuum structures is a recent field in structural optimization. However, an increasing research activity in this area has been developed since the statement of the very first formulations. These formulations try to obtain the most adequate material distribution that satisfies the imposed structural limitations. The existence or absence of material in each part of the domain is usually defined by using a continuum variable (the relative density) in order to avoid dealing with a discrete optimization problem. This continuum approach of the material properties present important advantages since conventionaloptimization algorithms can be used. However, numerical models must be considered in order to develop the structural analysis for intermediate values of the relative densities. In this paper, we present some improvements in a minimum weight approach of the structural topology optimization problem. The main goal of this paper is to present an improved formulation that tries to reach binary 0-1 material distributions by using a continuum approach of the design variables. Furthermore, a perimeter penalization is included in the objective function to simplify the solutions obtained. In addition, some computational aspects are considered in order to reduce the computational effort. Finally, we compare the solutions obtained by using these formulations in two application examples.

Level-set approaches are a family of domain classification techniques that rely on defining a scalar level-set function (LSF), then carrying out the classification based on the value of the function relative to one or more thresholds. Most continuum topology optimization formulations are at heart, a classification problem of the design domain into structural materials and void. As such, level-set approaches are gaining acceptance and popularity in structural topology optimization. In conventional level set approaches, finding an optimum LSF involves solution of a Hamilton-Jacobi system of partial differential equations with a large number of degrees of freedom, which in turn, cannot be accomplished without gradients information of the objective being optimized. A new approach is proposed in this paper where design variables are defined as the values of the LSF at knot points, then a Kriging model is used sto interpolate the LSF values within the rest of the domain so that classification into material or void can be performed. Perceived advantages of the Kriging-interpolated level-set (KLS) approach include alleviating the need for gradients of objectives and constraints, while maintaining a reasonable number of design variables that is independent from the mesh size. A hybrid genetic algorithm (GA) is then used for solving the optimization problem(s). An example problem of a short cantilever is studied under various settings of the KLS parameters in order to infer the best practice recommendations for tuning the approach. Capabilities of the approach are then further demonstrated by exploring its performance on several test problems.

Structural topology optimization.- On basic properties of Michell's structures.- Validation of numerical method by analytical benchmarks and verification of exact solutions by numerical methods.- Introduction to shape and topology optimization.- Homogenization method for shape and topology optimization.- Level set method for shape and topology optimization.- Compliance minimization of two-material elastic structures.- Some notes on topology optimization of vibrating continuum structures.- Topology optimization of vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps.- On optimum shape design and periodicity of band-gape beam structures.- Topological design for minimum dynamic compliance of continuum structures subjected to forced vibration.- Topological design for minimum sound emission from bi-material structures subjected to forced vibration.- Discrete material optimization of vibrating laminated composite plates for minimum sound emission.- Optimization of diffusive transport problems.- Fluid topology optimization: stokes and Navier-stokes models.- Topology optimization of coupled multi-physics problems.- Topology optimization based on the extended finite element method.- Topology optimization of meso- and nano-scale problems.- Topology optimization under uncertainty.- A brief review of numerical methods of structural topology optimization.- The free material design in planar elasticity.