Producing Smart Pareto Sets for Multi-objective Topology Optimisation Problems

This paper presents an efficient and comprehensive MATLAB code to solve two-dimensional structural topology optimization problems, including minimum mean compliance, compliant mechanism synthesis and multi-load compliance problems. The Unsmooth Variational Topology Optimization (UNVARTOP) method, developed by the authors in a previous work, is used in the topology optimization code, based on the finite element method (FEM), to compute the sensitivity and update the topology. The paper also includes instructions to improve the bisection algorithm, modify the computation of the Lagrangian multiplier by using an Augmented Lagrangian to impose the constraint, implement heat conduction problems and extend the code to three-dimensional topology optimization problems. The code, intended for students and newcomers in topology optimization, is included as an appendix (Appendix A) and it can be downloaded from https://github.com/DanielYago together with supplementary material.

A globally convergent method to accelerate topology optimization using on-the-fly model reduction

The center of mass and the inertia tensor are important parameters characterizing the kinetic properties of many structural systems, e.g., the rotor, spacecraft, and aircraft. In these occasions, designs not consistent with the requirements on the center of mass and inertia tensor cannot be accepted. In this paper, mathematical models are established to carry out the topology optimization of continua considering the mass and inertia characteristics. Two numerical examples, one for minimizing the static compliance considering locations of the center of mass and the other for maximizing the fundamental eigenfrequency considering both locations of the center of mass and values of the inertia products, are presented to demonstrate that the requirements on mass and inertia characteristics could have a profound influence on the final optimized layouts in structural topology optimization problems. On the other hand, the mass and inertia characteristics can also be tuned by topology optimization methods. In the second numerical problem, the smooth minimum operator is used to approximate the fundamental eigenfrequency with the first 4 eigenfrequencies so that the possible non-differentiability of the repeated eigenfrequencies can be avoided. Due to its extreme importance in engineering practice, the work of this paper can be useful to carry out designs in accordance with the requirements on the mass and inertia characteristics under the framework of traditional density-based structural topology optimization methods.

A global optimization method for topology optimization using a genetic algorithm is proposed in this paper. The genetic algorithm used in this paper is assisted by the Kriging surrogate model to reduce computational cost required for function evaluation. To validate the global topology optimization method in flow problems, this research works on two single-objective optimization problems, where the objective functions are to minimize pressure loss and to maximize heat transfer of flow channels, and the multi-objective optimization problem, which combines these two problems. The shape of flow channels is represented by the level set function, and the pressure loss and the temperature of the channels are evaluated by the Building-Cube Method (BCM), which is a Cartesian-mesh CFD approach. The proposed method resulted in an agreement with previous study in the single-objective problems in its topology, and achieved global exploration of non-dominated solutions in the multi-objective problem.

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.

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

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.

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.

In recent years, additive manufacturing has been gaining ground among traditional manufacturing methods in many different applications. This is mainly due to its ease of prototyping, material savings and above all the highly complex geometries that can be achieved. Despite its many advantages, the use of additive manufacturing in electrical machines has been limited. The lack of maturity of the manufacturing processes and the increase in losses when applying additive approaches to some parts, mainly stators, have led little implementation in the field of electrical machines. The geometric freedom offered by additive manufacturing paves the way for the use of novel optimisation techniques, such as topology optimisation. Traditionally, topology optimisation has not been widely used because the complex geometries obtained by these algorithms are not easily produced by traditional manufacturing methods. However, thanks to additive manufacturing, topology optimisation has started to be used, mainly for mechanical applications and it has demonstrated its ability to reduce weight without compromising mechanical stability in several cases. Nevertheless, topology optimisation for physics other than mechanical has not been thoroughly studied, let alone for the simultaneous optimisation of different physics. In view of the scenario described, this thesis investigates the benefits that additive manufacturing and topology optimisation can bring to the design of electrical machines. For the additive manufacturing of electrical machines, a thorough literature review is carried out on the application of this manufacturing approach to different parts of electrical machines, active and non-active. Two main case studies are presented: in the first, the rotor, the shaft, the stator and the housing of an aerospace actuator are manufactured by L-PBF in FeCoV and the assembled actuator is tested. In the second case, a rotor is manufactured in FeSi by LP- DED. Finally, not considered an additive manufacturing case study itself in this thesis, but two electrical conductor prototypes are manufactured in CuCr1Zr and an additional geometry is manufactured in AlSi10Mg by L-PBF. With regard to topology optimisation of electrical machines, a description of the main topology optimisation methods used is given and their application to electrical machine components is presented. In a similar way to additive manufacturing, two case studies are analysed. First, a novel multiphysics - mechanical and electromagnetic - topology optimisation method is presented, in which the rotor and the stator of a permanent magnet motor are optimised simultaneously. This method is compared with two methods found in the literature, SIMP and on-off. Secondly, another topology optimisation method is described and applied to an electrical conductor model. The proposed algorithm involves a multiphysics - thermal and electromagnetic - hybrid parametric topology optimisation approach. Two reduced length geometries and one full length prototype are built via additive manufacturing. Finally, conclusions regarding additive manufacturing and topology optimisation of electrical machines are drawn from the work presented in this thesis. Future lines of work for additive manufacturing and topology optimisation of electrical machines are also presented.

Structural design for obtaining desired eigenfrequencies by using the topology and shape optimization method

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.

An efficient combination of multi-objective evolutionary optimization and reliability analysis for reliability-based design optimization of truss structures

The numerical method of stamp topological optimization taking into account fatigue strength is presented in the work. It is proposed to take into account the restrictions on the stress state in accordance with the curve of the dependence of the maximum stresses on the number of loading cycles in the ESO topological optimization method. An approach to the selection of the evolutionary coefficient with a step-by-step increase in the rejection coefficient is proposed when constructing an iterative scheme for the rejection of elements by the method of topological optimization. The calculation of the stamp optimal topology with a decrease in volume due to the removal and redistribution of material was carried out in the study. The new geometric model of the optimal topology stamp is based on the predicted distribution of elements with a minimum stress level. The verification calculation of the stress state of the stamp of optimal topology with an assessment of fatigue strength was carried out in the work. The numerical calculation was carried out using the finite element method in the Ansys software package. The minimized stamp volume decreased by 35% according to the calculation results. The results of the study can be further applied in the development of topological optimization methods and in the design of stamping tools of optimal topology.

With the rapid advancement of computing technology, mathematical modeling and computer simulations have become common and standard methods for design of complex systems. To exercise these models intelligently and eliminate the burden of manual iteration, manipulating inputs and reviewing outputs, optimization strategies are applied in a simulation-based design environment. In reality, the knowledge that engineers have about the design problem is imperfect and incomplete, uncertainty always exists in simulation-based design. The conventional deterministic optimization techniques do not consider the impact of such uncertainties. If the system responses driving the design decisions are very sensitive to these uncertainties, serious problems might confront the designer. Performance of the actual system may differ greatly from expected optimum performance. Again, if the design solution is near one or more constraint boundaries, then even slight uncertainties or changes in the operating environment could produce failed, unsafe designs, and/or result in substantial performance degradation. In recent years, the concept of reliability and robust design has become very popular. Probabilistic design analysis and statistical modeling allow the identification of designs that qualify as not only feasible, but as consistently feasible in the face of uncertainty. Reliability based topology optimization (RBTO) extends the reliability notion to the area of structural topology optimization. Although reliability based design optimization (RBDO) is a well established research area, very few work has been presented in the area of RBTO. In this study, we give an overview of the various works done in the field of RBTO. Ever since Bendsoe and Kikuchi introduced the topology optimization using a homogenization method, many topology optimization methods have been developed for the both linear and nonlinear structures. Traditional topology optimization methods drive the topology of a structure to an optimum design based on the constraint on mass. Kharmanda et al. introduced reliability constraints in deterministic topology optimization problem. They proposed a heuristic strategy that aims to reduce mass while improving the reliability level of the structure without greatly increasing its weight. But the limit state function used by them was not based on failure criteria for the structure. Their formulation considered uncertainty with respect to ∗Design Automation Laboratory †Advisor geometrical dimension and applied load only. Also their reliability analysis seems to be independent of the boundary and loading condition, so their results showed similar values for the uncertain variables for different structures. Tovar et al. have developed the Hybrid Cellular Automaton (HCA) method for structural synthesis of continuum material where the state of each cell is defined by both density and strain energy. In Agarwal, a decoupled RBDO approach is employed such that the topology optimization is separate from the reliability analysis. Patel et al. showed the use of RBTO using the gradient free Hybrid Cellular Automata (HCA) method. Their formulation incorporates uncertainty with respect to material property also. They considered limit state function based on failure modes on the output displacements. This study aims at giving a comparative study on these methods. It points out the advantages and disadvantages of the methods, and their significance in the field of RBTO. As large numbers of design variables are associated with continuum topology optimization problems, RBTO methods are inherently computationally expensive because of additional required system analysis associate with RBDO. Based on the observations, this study presents an efficient method for incorporating reliability analysis in topology optimization.

Aiming at the problems of a low material utilization rate and uneven stress distribution of cast-steel support joints in cable dome structures, topology optimization and additive manufacturing methods are used for optimization design and integrated manufacturing. First, the basic principle and calculation process of topology optimization are briefly introduced. Then, the initial model of the support joint is calculated and analyzed by using the universal software ANSYS Workbench 2020R2 and Altair OptiStruct, and the optimized joint is imported into Discovery Live to smooth the surface. The static behaviors of three types of joints (topology-optimized joints, joints after the smoothing treatment, and joints from practical engineering) are compared and analyzed. Finally, the joints are printed by using fused deposition modeling (FDM) technology and laser-based powder bed fusion (LBPBF) technology in additive manufacturing. The results show that the new support joint in the cable dome structure obtained by the topology optimization method has the advantages of a novel shape, a high material utilization rate, and a uniform stress distribution. Additive manufacturing technology can allow the manufacture of complex shape components with high precision and high speed. The combination of topology optimization and additive manufacturing effectively realizes the advanced design and integrated manufacturing of support joints for cable dome structures.