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Several currently popular methods of topology optimization are closely related to the classical Fully Stressed Design (FSD)/Stress
Ratio (SR) or Minimum Compliance (MC)/Uniform Energy Distribution (UED) methods. The ranges of validity of the above techniques
– and of recent variations on the same themes – are examined critically and possible extensions of their validity considered.
Particular attention is paid to so-called “hard-kill" or Evolutionary Structural Optimization (ESO) or Adaptive Biological
Growth (ABG) methods and to the Generalized Stress Design (GSD) technique.

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... There are two types of compliance methods; the weighted average compliance method and the worst-case compliance. By using very simple examples (Figure 1), Rozvany (2001b) proved that in truss optimization, these compliance methods lead to different optimal layouts when compared to that obtained from stress constrained problem. Moreover, the results are awkward from the aspect of stiffness. ...

... We show that if a truss is statically non-indeterminate, the suggested problem coincides with that of the minimization of the volume under stress constraints. Therefore, this compliance when used as an objective, overcomes the problems raised by the bench-marks of Rozvany (2001b). ...

... For the case of sizing optimization of trusses, it has been proved that at least for statically non-indeterminate trusses, the optimal layout would be similar to the one of the stress constrained problem. Therefore, this compliance overcomes the problems raised by the bench-marks of Rozvany (2001b). ...

There are two popular methods concerning the optimal design of structures. The first is the minimization of the volume of
the structure under stress constraints. The second is the minimization of the compliance for a given volume. For multiple
load cases an arising issue is which energy quantity should be the objective function. Regarding the sizing optimization of
trusses, Rozvany proved that the solution of the established compliance based problems leads to results which are awkward
and not equivalent to the solutions of minimization of the volume under stress constraints, unlike under single loading (the
layouts would be the same if in the compliance problem the volume is set equal to the result of the first problem). In this
paper, we introduce the “envelope strain energy” problem where we minimize the volume integral of the worst case strain energy
of each point of the structure. We also prove that in the case of sizing optimization of statically non-indeterminate (the
term non-indeterminate includes both statically determinate trusses and mechanisms) trusses, this compliance method gives
the same optimal design as the stress based design method.

... Sigmund [12] believed that it is questionable to extend this approach to other cases with non-compliance objectives, multiphysics and multiple constrains. Zhou and Rozvany [13], Rozvany [14] even gave a critical view about the numerical failures of the approach. In fact, neither the stress level nor the sensitivity values can describe the element efficiency exactly. ...

... In this case, u A is approximately invariable with A A R ≈ u u ( 1 3 ) u AR is the assumptive nodal displacement vector of the removed element A calculated at the common nodes of the neighboring elements. The combination of (9) and (13) gives rise to (14) where u T AR K A0 u AR is design independent. The strain energy of the element A is approximated as a directly proportional function of x A , that increases from left to right. ...

The aim of this paper is to enhance the validity of existing evolutionary topology optimization procedures. As this hard-killing scheme related to the element sensitivity values may lead to incorrect predictions of inefficient elements to be removed and the value of the objective function becomes sharply deteriorated during the iterations, a check position (CP) control is proposed to prevent the erroneous topology design generated by the rejection criteria of evolutionary methods. For this purpose, we introduce a sort of orthotropic cellular microstructure (OCM) element with moderate pseudo-density that acts as a compromising element between solid element and void OCM element. In this way, all inefficient elements removed previously are automatically replaced with the moderate OCM elements depending upon the deterioration of the objective function. Erroneously removed elements are then identified in the updated finite element model through a direct sensitivity computing of the moderate OCM elements and will be finally recovered by the bi-directional element replacement. Besides, detailed structures with checkerboard patterns are eliminated by controlling the local structural band-width with the so-called threshold method. Typical optimization examples of structural compliance and natural frequency that were difficult to tackle are solved by the proposed design procedure. Satisfactory numerical results are obtained.

... The chapter III of his book shows that according to his theory, there is a struggle for survival in nature, but the one who survives is not necessarily the strongest, but the one that best adapts to the conditions of the environment in which he lives. Then the following chapter, Darwin [16][17][18][19]2020 selection: directional, disruptive, and stabilizing, Ridley [2]. The first type is a natural selection mode in which a single phenotype is favored, causing the frequency of the allele to change continuously in one direction. ...

... The latter problems were overcome by extending the method called Bidirectional Evolutionary Structural Optimization (BESO) that allowed both material addition and removal, Huang and Xie [15]. However, the solution may worsen in terms of the objective function if the ESO/BESO technique continues with no stop or reach a local optimum, Rozvany [16]. Because the initial development of ESO methods is based on a heuristic concept and lacks theoretical rigor, most of the early work on ESO /BESO neglected significant numerical problems in TO, such as the existence of a solution, checker-board, meshdependency and local optimum, Xia et al. [17]. ...

The present work deals with a new approach for Topology Optimization (TO) of two-dimensional continuum elastic structures through the Progressive Directional Selection (PDS) method. To achieve the best topology of a structure, a typical goal is to define the best material distribution in a domain, considering an objective function and mechanical constraints. In general, most of the studies address the compliance minimization of structures. Numerical methods for TO of continuum structures have been investigated extensively. Most of those methods are based on finite element analysis, where the design domain is discretized into a fine mesh of elements. In such a setting, the optimization procedure is to find a structure's topology by determining for every point in the design domain if there should be material (solid element) or not (void element). To control the process of inserting or removing finite elements without forgotten the continuum representation, standard algorithms as Homogenization, Solid Isotropic Microstructure with Penalization (SIMP), and Evolutionary Structural Optimization (ESO) are applied in many studies. The latter approach is based on the simple concept that the optimal design can be achieved by gradually removing inefficient material (elements) from the design space. The ESO algorithms are easy to understand and implement. However, ESO is heuristic, and there is no proof that an optimum solution can be achieved by element elimination and admission. The original scheme is inefficient once it needs to find the best solution by comparing several intuitively generated solutions. To avoid this problem but taking advantage of ESO's simplicity, the PDS method, which is inspired by natural selection observed in biology, applies a strategy to minimize the strain energy of a discretized analyzed domain with a volume restriction. Based on the performance criteria adopted for the problem, the selected population is reached through an iterative process that converges when the optimal topology does not evolve anymore, i.e., there is no change in the final set of selected elements. An instructional problem that shows the essence of PDS for a rigid body problem and one example of topology optimization of a classical problem found in the literature are investigated.

... Recently the topology optimization is one of the most "popular" topic in the expanding field of optimal design. A great number of papers indicates the importance of the topic [1,[3][4][5]7,9,11,23,25,27,28]. The popularity comes one part from the needs of the industry (car, airplane, etc.) and other part from the complexity of the problem which is a great challenging for the researchers. ...

... In 1988 a new generation of problem formulation were created by Bendsoe and Kikuchi [3] using homogenization. A detailed description of continuum-type optimality criteria method was reported in Rozvany's books [20,23] and several reports [21,22,25,30]. Their work opened a new "road" in research named topology optimization, but the common thing is that all of them use the basic idea of the classical optimality criteria. ...

... Sigmund [12] believed that it is questionable to extend this approach to other cases with non-compliance objectives, multiphysics and multiple constrains. Zhou and Rozvany [13], Rozvany [14] even gave a critical view about the numerical failures of the approach. In fact, neither the stress level nor the sensitivity values can describe the element efficiency exactly. ...

... In this case, u A is approximately invariable with A A R ≈ u u ( 1 3 ) u AR is the assumptive nodal displacement vector of the removed element A calculated at the common nodes of the neighboring elements. The combination of (9) and (13) gives rise to (14) where u T AR K A0 u AR is design independent. The strain energy of the element A is approximated as a directly proportional function of x A , that increases from left to right. ...

... Despite its conceptual simplicity, ESO/BESO methods are subjected to criticism [38][39][40]. These include, intuitive nature of the procedure, obtaining non optimal designs at the end of the process and difficulties encountered with multiple constraints [39]. ...

... Despite its conceptual simplicity, ESO/BESO methods are subjected to criticism [38][39][40]. These include, intuitive nature of the procedure, obtaining non optimal designs at the end of the process and difficulties encountered with multiple constraints [39]. As a consequence of these, a new Derleme An Overview On Topology Optimization Methods Employed In Structural Engineering 170 BESO algorithm is suggested recently [41, 42]. ...

Any mechanical performance measure of a structure is strongly related with its topology. Size and shape optimization cannot give the best structural performance, since these methods cannot change the structure's topology. Hence, topology optimization should be employed to obtain the best performance. In this paper, a review of topology optimization is provided. At first, the general topology optimization problem is defined. Then, modern topology optimization methods are presented and discussed.

... Sigmund [12] believed that it is questionable to extend this approach to other cases with non-compliance objectives, multiphysics and multiple constrains. Zhou and Rozvany [13], Rozvany [14] even gave a critical view about the numerical failures of the approach. In fact, neither the stress level nor the sensitivity values can describe the element efficiency exactly. ...

... In this case, u A is approximately invariable with A A R ≈ u u ( 1 3 ) u AR is the assumptive nodal displacement vector of the removed element A calculated at the common nodes of the neighboring elements. The combination of (9) and (13) gives rise to (14) where u T AR K A0 u AR is design independent. The strain energy of the element A is approximated as a directly proportional function of x A , that increases from left to right. ...

The aim of this paper is to enhance the validity of
existing evolutionary topology optimization procedures. As this
hard-killing scheme related to the element sensitivity values may
lead to incorrect predictions of inefficient elements to be removed
and the value of the objective function becomes sharply deteriorated
during the iterations, a check position (CP) control is proposed to
prevent the erroneous topology design generated by the rejection
criteria of evolutionary methods. For this purpose, we introduce a
sort of orthotropic cellular microstructure (OCM) element with
moderate pseudo-density that acts as a compromising element between
solid element and void OCM element. In this way, all inefficient
elements removed previously are automatically replaced with the
moderate OCM elements depending upon the deterioration of the
objective function. Erroneously removed elements are then identified
in the updated finite element model through a direct sensitivity
computing of the moderate OCM elements and will be finally recovered
by the bi-directional element replacement. Besides, detailed
structures with checkerboard patterns are eliminated by controlling
the local structural bandwidth with the so-called threshold method.
Typical optimization examples of structural compliance and natural
frequency that were difficult to tackle are solved by the proposed
design procedure. Satisfactory numerical results are obtained.

... It is well known that the stress/strain constraint problem may lead to serious numerical difficulties due to the effect of so-called vanishing constraints, see Achtziger and Kanzow (2007): The constraints may be active even in regions where the design variable tends to zero and which are then effectively void. This, in effect, leads to so-called singularity problem 3 , extensively studied in the structural optimization literature; see, e.g., Kirsch (1990); Cheng and Jiang (1992); Rozvany (2001b). Whether or not we get this effect depends on the formulation of the constraint, the topology optimization model and, in particular, interpretation of the design variable. ...

This article is a continuation of the paper Kočvara and Stingl (Struct Multidisc Optim 33(4–5):323–335, 2007). The aim is to describe numerical techniques for the solution of topology and material optimization problems with local stress constraints. In particular, we consider the topology optimization (variable thickness sheet or “free sizing”) and the free material optimization problems. We will present an efficient algorithm for solving large scale instances of these problems. Examples will demonstrate the efficiency of the algorithm and the importance of the local stress constraints. In particular, we will argue that in certain topology optimization problems, the addition of stress constraints must necessarily lead not only to the change of optimal topology but also optimal geometry. Contrary to that, in material optimization problems the stress singularity is treated by the change in the optimal material properties.

... This is generally true for stress-constrained problems under a single load case with the same stress limits in tension and compression. Since for this class of optimization problems, the optimum is a fully stressed design (Rozvany 2001b), and all constraints g in (25) will be active at a minimizer. Consequently, the global constraint value is equal to all local constraint values in that point. ...

In this paper, we propose a unified aggregation and relaxation approach for topology optimization with stress constraints. Following this approach, we first reformulate the original optimization problem with a design-dependent set of constraints into an equivalent optimization problem with a fixed design-independent set of constraints. The next step is to perform constraint aggregation over the reformulated local constraints using a lower bound aggregation function. We demonstrate that this approach concurrently aggregates the constraints and relaxes the feasible domain, thereby making singular optima accessible. The main advantage is that no separate constraint relaxation techniques are necessary, which reduces the parameter dependence of the problem. Furthermore, there is a clear relationship between the original feasible domain and the perturbed feasible domain via this aggregation parameter.

... However, ESO is not an optimization method. Therefore, the results of ESO do not satisfy the objective nor constraint conditions [11,12]. Evaluation optimization algorithms, such as genetic algorithms (GA), simulated annealing (SA), evolution strategy (ES), and evolution programming (EP) to STOPs have been developed. ...

Genetic algorithms (GAs) for structure topology optimization problems (STOPs) have been developed in recently because GAs are flexible and effective to be applied to various complicated engineering problems. A stress-based crossover (SX) operator (1) for continuous STOPs was proposed to suppress the "checkerboard" pattern and disconnection phenomena, which are common for simple GA for STOPs. Here, this SX operator was improved and the details are described. Different generation models were adopted to verify the effectiveness of this operator. For GA to multi-constrained STOPs, how to define fitness function is always an important consideration because the fitness value determines which individuals maybe transmitted to the next generation. The fitness function used in this paper is well defined to compose the objective function and constraints items. These discussions were examined through a number of multi-constrained STOPs. The results demonstrated that improved SX with GA is a global search algorithm with which it is easy to obtain the applicable topology.

... Early versions of the ESO/BESO algorithms have faced criticism beginning with Zhou and Rozvany (2001), who demonstrated a breakdown of the methods on a simple cantilever tie-beam example. Further discussion is offered by Rozvany (2001c) where the ESO method is compared against classical Fully Stressed Design (FSD) and Minimum Compliance (MC) sizing optimization methods. In an effort to explain with technical rigor the workings of ESO, Tanskanen (2002) investigated the theoretical aspects of the method. ...

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.

... Utilizing the concept of ground structure, Gou et al. [2] solved the truss optimization problem by considering the cross-sectional areas of the members as continuous design variables, whereas the nodal locations were fixed, while Wang et al. [3] and Gou et al. [4] solved this problem by considering nodal locations as additional design variables. A critical review of further popular methods can be found in Rozvany [5]. The other more complex problem which has been defined recently is called layout optimization. ...

In this paper a new graph-based evolutionary algorithm, gM-PAES, is proposed in order to solve the complex problem of truss layout multi-objective optimization. In this algorithm a graph-based genotype is employed as a modified version of Memetic Pareto Archive Evolution Strategy (M-PAES), a well-known hybrid multi-objective optimization algorithm, and consequently, new graph-based crossover and mutation operators perform as the solution generation tools in this algorithm. The genetic operators are designed in a way that helps the multi-objective optimizer to cover all parts of the true Pareto front in this specific problem. In the optimization process of the proposed algorithm, the local search part of gM-PAES is controlled adaptively in order to reduce the required computational effort and enhance its performance. In the last part of the paper, four numeric examples are presented to demonstrate the performance of the proposed algorithm. Results show that the proposed algorithm has great ability in producing a set of solutions which cover all parts of the true Pareto front.

... Topology optimization methods play an important role today in structural design. There is significant amount of literature on this discipline; for example, see [1][2][3][4]. Similarly, considerable work has been carried out on multi-objective optimization [5][6][7][8][9]. In this paper, we focus on the intersection of the two disciplines, i.e., on multiobjective topology optimization (MOTO). ...

In multi-objective problems, one is often interested in generating the envelope of the objective-space, where the envelope is, in general, a superset of pareto-optimal solutions. In this paper, we propose a method for tracing the envelope of multi-objective topology optimization problems, and generating the corresponding topologies. The proposed method exploits the concept of topological sensitivity, and is applied to bi-objective optimization, namely eigenvalue-volume, eigenvalue-eigenvalue and compliance-eigenvalue problems. The robustness and efficiency of the method is illustrated through numerical examples.

... The literature has shown significant developments in the ESO procedure since the idea was brought to fruition in 1992 (Xie and Steven 1992). However a lot of the recent developments have come out of the criticism of the algorithms ability to find the optimum solution efficiently and effectively (Rozvany and Querin 2002;Zhou and Rozvany 2001;Sigmund and Petersson 1998;Rozvany 2009Rozvany , 2001a. This lead to the development of a 'soft-kill' ESO/BESO technique (Deaton and Grandhi 2014). ...

Topology optimization has evolved rapidly since the late 1980s. The optimization of the geometry and topology of structures has a great impact on its performance, and the last two decades have seen an exponential increase in publications on structural optimization. This has mainly been due to the success of material distribution methods, originating in 1988, for generating optimal topologies of structural elements. Previous methods suffered from mathematical complexity and a limited scope for applicability, however with the advent of increased computational power and new techniques topology optimization has grown into a design tool used by industry. There are two main fields in structural topology optimization, gradient based, where mathematical models are derived to calculate the sensitivities of the design variables, and non gradient based, where material is removed or included using a sensitivity function. Both fields have been researched in great detail over the last two decades, to the point where structural topology optimization has been applied to real world structures. It is the objective of this review paper to present an overview of the developments in non gradient based structural topology and shape optimization, with a focus on evolutionary algorithms, which began as a non gradient method, but have developed to incorporate gradient based techniques. Starting with the early work and development of the popular algorithms and focusing on the various applications. The sensitivity functions for various optimization tasks are presented and real world applications are analyzed. The article concludes with new applications of topology optimization and applications in various engineering fields.

... In the former, one arrives at a feasible but non-optimal solution. In the latter, the weights are subjective and difficult to establish a priori; the final topology will depend on the weights [20], [38], [39]. Additionally, due to convergence issues, application-specific methods have also been developed [40], [41]. ...

The objective of this paper is to introduce and demonstrate a robust method for multi-constrained topology optimization. The method is derived by combining the topological sensitivity with the classic augmented Lagrangian formulation. The primary advantages of the proposed method are: (1) it rests on well-established augmented Lagrangian formulation for constrained optimization, (2) the augmented topological level-set can be derived systematically for an arbitrary set of loads and constraints, and (3) the level-set can be updated efficiently. The method is illustrated through numerical experiments.

... Das bekannteste Verfahren ist das "fully stressed design" (FSD) [Patnaik und Hopkins 1998], welches auch als Spannungsquotientenverfahren bezeichnet werden kann und zum Ziel hat, Strukturen zu generieren, bei denen jedes Element entsprechend der zulässigen Spannungen ausgenutzt wird. Diese Vorgehensweise, abgeleitet aus physikalischen Überlegungen, ist jedoch nicht generell geeignet, um Optimalstrukturen zu generieren [Rozvany 2001;Hörnlein 2004]. ...

Im Rahmen der Arbeit wird eine Methode entwickelt, die es erlaubt, optimale Stabwerkstopologien für adaptive Tragwerke unter beliebigen, multiplen, statischen Belastungen zu entwerfen.
Adaptive Tragwerke, die eine Sonderform des Leichtbaus darstellen, sind durch ihre Anpassungsfähigkeit in der Lage, sehr große unplanmäßige Einwirkungen durch eine interne Systemreaktion auf mehrere Elemente umzulagern und somit sicher abzutragen.
Als Grundlage für die Entwicklung des Topologieoptimierungsalgorithmus für adaptive Stabwerke wird der Entwurf dieser Strukturen auf unterschiedlichen Ebenen diskutiert. Ausgehend von der Elastostatik, den verschiedenen Optimierungsverfahren sowie der Topologieoptimierung von Stabwerken werden die notwendigen Grundlagen und Zusammenhänge für die numerische Beschreibbarkeit von adaptiven Stabwerken vorgestellt. Im Rahmen dieser Ausführungen werden auch die notwendigen Begrifflichkeiten definiert und die Methodik der Systemaktivierung vorgestellt.
Um eine wirkungsvolle Systemaktivierung zu gewährleisten, wird im Rahmen dieser Arbeit ein eigens entwickelter Aktuatorenpositionierungsalgorithmus verwendet. Des Weiteren wird eine Methodik vorgestellt, die es ermöglicht, die Anzahl der notwendigen Aktuatoren und des damit einhergehenden Energiebedarfs zu minimieren.
Der Schwerpunkt der Arbeit liegt jedoch auf der Entwicklung eines Topologieoptimierungsalgorithmus für die Generierung von adaptiven Stabwerken. Es wird dargestellt, dass mittels dieser Berechnungsmethodik der Entwurf von sehr leichten, adaptiven Stabwerken möglich wird. Diese Stabwerke zeichnen sich neben ihrer Gewichts- und Verformungsminimalität durch eine Unempfindlichkeit gegenüber zufällig auftretenden Belastungen aus.
Dieser Topologieoptimierungsalgorithmus besteht aus einem passiven und einem adaptiven Teil, die über den Entwurfsraum als Schnittstelle miteinander interagieren. Der passive Teil basiert auf dem Verfahren der Inneren-Punkte-Methode, welches in dem hier beschriebenen Zusammenhang für die Identifikation der Lastpfade verwendet wird. In einem nächsten Schritt erfolgen die Systemaktivierung und die Ermittlung der lokalen Adaptionen. Diese dienen als Entscheidungsparameter für die Entwurfsraummanipulation, die iterativ solange durchgeführt wird, bis das globale Abbruchkriterium erreicht wird. Die Sonderfälle der reinen Kraft- und Verformungsadaption werden ebenfalls diskutiert. Um die Funktionsweise dieses Algorithmus zu überprüfen, wird der adaptive Entwurfsansatz in einen genetischen Algorithmus implementiert.

... Most general problems involve material distribution in a design space with given loads and boundary conditions. The design space can be analysed using the finite element method (Rozvany, 2001) or structure-specific methods with the optimisation process carried out by several techniques such as topological-derivatives, level-sets or genetic algorithm. ...

A simple method is presented to carry out a retrospective analysis to examine the development of load-bearing structures. The idea is to eliminate the differences coming from technological changes (such as joints, profiles, loads) by using relative numbers to express the relation of the structures to the possible theoretical solutions under the same circumstances. The method is demonstrated by investigating the impact of historical changes focusing on metal Pratt trusses spanning about 100 ft, located in Indiana, U.S., erected between 1870 and 1937. Data of 87 structures was collected and compared to the results of a multi-objective optimisation computed using a genetic algorithm. Using the relative numbers acquired by evaluating the objective functions for the historical structures, a large time-scale optimisation process through history can be visualised. Plotting them on the Pareto-front diagram determined by the genetic algorithm and examining the historical background of the state revealed that the economic and industrial changes, in fact, had a considerable impact on the design trends, which manifests in changes of the weights of the objective functions.

... The advent of numerical methods has enabled numerical computer-aided engineering tools to be integrated with TO. Notable TO methods include [11][12][13][14][15][16][17] density methods, such as homogenisation [18], and the ensuing Solid Isotropic Material with Penalisation (SIMP) method [19,20]; evolutionary structural optimisation (ESO), which was initially developed by eliminating low stress elements [21], and has been extended to bidirectional (BESO) to allow material addition [22,23]; and level set methods (LSM) which define the structural boundary by implicit contours of some level-set function [24,25]. ...

Additive manufacturing (AM) enables the direct manufacture of complex geometries with unique engineering properties. In particular, AM is compatible with topology optimisation (TO) and provides a unique opportunity for optimal structural design. Despite the commercial opportunities enabled by AM, technical requirements must be satisfied in order to achieve robust production outcomes. In particular, AM requires support structures to fabricate overhanging geometry and avoid overheating. Support generation tools exist; however, these are generally not directly compatible with the voxel-based representation typical of TO geometries, without additional computational steps. This research proposes the use of voxel-based Cellular automata (CA) as a fundamentally novel method for the generation of AM support structures. A number of CA rules are proposed and applied with the objective of generating robust support structures for an arbitrary TO geometry. Relevant CA parameters are assessed in terms of structure manufacturability, including sequential and random CA, rotation of the cellular array, and alternate CA boundary rules, including permutations not previously reported. From this research, CA with complex cell arrangements that provide robust AM support for TO geometries are identified and demonstrated by manufacture with selective laser melting (SLM) and fused deposition modelling (FDM). These CA may be automatically applied to enable TO geometries to be directly fabricated by AM, thereby providing a unique, and commercially significant, design for AM (DFAM) capability.

... (9)), comes at the cost of sub-optimality. It was shown and mathematically verified that FSD can produce a sub-optimal solution [24]. However, it was also shown that, for statically determinant structures, as far as engineering applications are concerned, the difference between the sub-optimum and the true optimum is not significant when only the yield stress limit is considered [25]. ...

The build orientation is one the most influential factors on material properties in additively manufactured parts. Advanced applications, such as lattice structures optimized for lightweight, often rely on small safety margins and are, hence, particularly affected, but research has not gone far beyond the pure empirical characterization. The focus of this paper is to investigate in detail the influence of anisotropy induced through fabrication on the mechanical performance and build orientation of whole structures when subject to optimization. First, a material property model for both compression and tension states is formulated. Then, the Generalized Optimality Criteria method is extended for fixed topology lattice structures with respect to constraints in displacement, stress, and Euler buckling. The two latter are formulated as local constraints that are handled in combination with Fully-Stressed Design recursion. The results reveal significant safety threads likely leading to premature failure when using properties from one-directional tests, as is so far the case, rather than the full anisotropy model developed herein. If used inversely, the algorithm yields the optimal orientation of a structure on the build platform, allowing further weight reduction while maintaining the mechanical properties.

... However, although mesh refinement can avoid some breakdowns, the user of ESO may never know which mesh density is the most suitable to get an optimal result. (3) Using rigorous resizing formulae [28]: secondorder sensitivities are suggested to prevent the failure. is method seems to be reasonable but its use in practice would increase significantly the computational effort. (4) Global difference method [29]: this improvement will search for "singular elements" during the evolution. ...

Genetic evolutionary structural optimization (GESO) method is an integration of the genetic algorithm (GA) and evolutionary structural optimization (ESO). It has proven to be more powerful in searching for global optimal response and requires less computational efforts than ESO or GA. However, GESO breaks down in the Zhou-Rozvany problem. Furthermore, GESO occasionally misses the optimum layout of a structure in the evolution for its characteristic of probabilistic deletion. This paper proposes an improved strategy that has been realized by MATLAB programming. A penalty gene is introduced into the GESO strategy and the performance index (PI) is monitored during the optimization process. Once the PI is less than the preset value which means that the calculation error of some element’s sensitivity is too big or some important elements are mistakenly removed, the penalty gene becomes active to recover those elements and reduce their selection probability in the next iterations. It should be noted that this improvement strategy is different from “freezing,” and the recovered elements could still be removed, if necessary. The improved GESO performs well in the Zhou-Rozvany problem. In other numerical examples, the results indicate that the improved GESO has inherited the computational efficiency of GESO and more importantly increased the optimizing capacity and stability.

... The latter problems were overcome by extending to the Bidirectional Evolutionary Structural Optimization (BESO) method that allowed both material addition and removal, Huang and Xie [10]. However, the solution may worsen in the objective function if the ESO/BESO technique continues with no stop or reaches a local optimum, Rozvany [11]. Because the initial development of ESO methods is based on a heuristic concept and lacks theoretical rigor, most of the early work on ESO/BESO neglected significant numerical problems in TO, such as the existence of a solution, checker-board, mesh-dependency, and local optimum, Xia et al. [12]. ...

This work presents a study applying the von Mises equivalent stress as a performance parameter for topological optimization of two-dimensional continuous elastic structures employing the Progressive Directional Selection (PDS) method. A typical objective to achieve the ideal topology of a structure is to define the best material distribution of the design domain, considering an objective function and mechanical constraints. In general, most studies deal with minimizing the compliance of structures. Numerical methods for optimizing the topology of continuous structures have been widely investigated. Most of these methods are based on finite element analysis, where the design domain is discretized into a fine mesh of elements. Evolutionary Structural Optimization (ESO) is one of these methods based on the simple concept of gradually removing inefficient finite elements from a structure. This method was formulated from the engineering point of view that the topology of the structures is naturally conservative for safety reasons and contains an excess of material. In such a context, the optimization consists of finding the optimal topology of a structure and determining whether there should be a solid or void element for each point in the design domain. ESO's algorithms are easy to understand and implement. The stress level of each element is determined by comparing the von Mises stress of the element and the maximum von Mises stress of the entire structure. After each finite element analysis, elements that present a stress level below the defined rejection ratio are excluded from the model. However, the ESO is a heuristic method, and there is no mathematical proof that an optimal solution can be achieved by eliminating elements. In addition, the original approach is inefficient because it needs to find the optimal topology comparing several solutions generated intuitively, adjusting the rejection ratio and evolutionary rate. To avoid this problem, but taking advantage of the simplicity of applying ESO, a new approach using the PDS method is proposed, inspired by the natural directional selection observed in biology. In the first work using PDS, the optimization problem was the minimization of the strain energy of a structure analyzed through the Finite-Volume Theory (FVT). This investigation discusses a scheme to minimize the von Mises equivalent stress of a discretized domain with a volume constraint. One example of topological optimization of 2D continuous elastic structure inspired by a classic literature problem is investigated.

... However, they commonly suffer from mesh dependency and checkerboard patterns problems [21,113] . Methods for dealing with these numerical instabilities have been extensively studied [114,115] . Among the established methods, substantial computational experience has demonstrated that filtering is 6. ...

Compliant mechanisms have become an important branch of modern mechanisms. Unlike conventional rigid body mechanisms, compliant mechanisms transform the displacement and force at least partly through the deformation of their structural components, which can offer a great reduction in friction, lubrication and assemblage. Therefore, compliant mechanisms are particularly suitable for applications in microscale/nanoscale manipulation systems. The significant demand of practical applications has also promoted the development of systematic design methods for compliant mechanisms. Several methods have been developed to design compliant mechanisms. In this paper, we focus on the continuum topology optimization methods and present a survey of the state-of-the-art design advances in this research area over the past 20 years. The presented overview can be helpful to those engaged in the topology optimization of compliant mechanisms who desire to be apprised of the field’s recent state and research tendency.

... Topology optimization seeks the material distribution ρ(x) that minimizes an objective function f (Ω, ρ) that is subjected to various constraints g i ≤ 0. A common objective function in structural TO is the global structural compliance [26,27,28], but local objective functions, such as those depending on the von Mises stresses, have been proposed [29,30]. The minimization is usually carried out by solving a finite element analysis. ...

Many machine learning methods have been recently developed to circumvent the high computational cost of the gradient-based topology optimization. These methods typically require extensive and costly datasets for training, have a difficult time generalizing to unseen boundary and loading conditions and to new domains, and do not take into consideration topological constraints of the predictions, which produces predictions with inconsistent topologies. We present a deep learning method based on generative adversarial networks for generative design exploration. The proposed method combines the generative power of conditional GANs with the knowledge transfer capabilities of transfer learning methods to predict optimal topologies for unseen boundary conditions. We also show that the knowledge transfer capabilities embedded in the design of the proposed algorithm significantly reduces the size of the training dataset compared to the traditional deep learning neural or adversarial networks. Moreover, we formulate a topological loss function based on the bottleneck distance obtained from the persistent diagram of the structures and demonstrate a significant improvement in the topological connectivity of the predicted structures. We use numerous examples to explore the efficiency and accuracy of the proposed approach for both seen and unseen boundary conditions in 2D.

This article is devoted to topology optimization of trusses under multiple loading conditions. Compliance minimization with material volume constraint and stress-constrained minimum weight problem are considered. In the case of a single loading condition, it has been shown that the two problems have the same optimal topology. The possibility of extending this result for problems involving multiple loading conditions is examined in the present work. First, the compliance minimization problem is formulated as a multicriterion optimization problem, where the conflicting criteria are the compliances of the different loading conditions. Then, the optimal topologies of the stress-constrained minimum weight problem and the multicriterion compliance minimization problem for a simple test example are compared. The results verify that when multiple loading conditions are involved, the stress-constrained minimum weight topology cannot be obtained in general by solving the compliance minimization problem.

Stress constraint is a hard issue for structural topology optimization, especially for large-scale structures, e.g. railcars. Another technique is proposed to combine a sizing optimizer with metamodelling for topology optimization. At the lower level, for each topology design sampled within the topology design space, a sizing optimizer finds feasible and optimal sol- utions in terms of sizing variables (plate thickness in continuum structures). All performance constraints such as stress, displacement, and stability, are handled only at this level. At the upper level, a metamodel is built to fit all the optimal solutions found at the lower level and is optimized for topology design. The only constraints left at the upper level are topological con- straints and topological variable bounds. Only the objective function (e.g. weight) versus topo- logical variables, and not the constraints, is approximated. The number of topology design variables is much smaller than those used in many other topology optimization approaches. Thus it may be able to handle large-scale structural systems. It was applied to two boxcar design projects, resulting in 18 per cent and 36 per cent weight savings and significant reductions in manufacturing cost and total cost.

Continuum topology optimization with stresses constraints is not only important but also complicated. Because stress is a
local quantity, a large number of constraints must be considered, and the complication of optimization algorithm and sensitivity
analysis is also increased. A global stress constraints method based on ICM (Independent Continuous Mapping) method, which
takes minimum weight and strain energy of structure with multi-load-case as design objective and constraint respectively,
is suggested in this paper. Dual quadratic programming is applied to solve the optimal model for continuum structure established
in this paper. As a result, the number of constraints is reduced and the local optimal solution of the weight is also avoided.
Two numerical examples are discussed, and their results show that the present method is effective and efficient.

Evolutionary structural optimization (ESO) is based on a simple idea that an optimal structure (with maximum stiffness but minimum weight) can be achieved by gradually removing ineffectively used materials from design domain. In general, the results from ESO are likely to be local optimums other than the global optimum desired. In this paper, the genetic algorithm (GA) is integrated with ESO to form a new algorithm called Genetic Evolutionary Structural Optimization (GESO), which takes the advantage of the excellent behavior of the GA in searching for global optimums. For the developed GESO method, each element in finite element analysis is an individual and has its own fitness value according to the magnitude of its sensitivity number. Then, all elements in an initial domain constitute a whole population in GA. After a number of generations, undeleted elements will converge to the optimal result that will be more likely to be a global optimum than that of ESO. To avoid missing the optimum layout of a structure in the evolution, an interim thickness is introduced into GESO and its validity is demonstrated by an example. A stiffness optimization with weight constraints and a weight optimization with displacement constraints are studied as numerical examples to investigate the effectiveness of GESO by comparison with the performance of ESO. It is shown through the examples that the developed GESO method has powerful capacity in searching for global optimal results and requires less computational effort than ESO and other existing methods.

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.

Optimal design with thousands of variables is a great challenge in engineering calculations. In this paper beside the short history of optimality criteria methods, a solution technique is introduced for the topology optimization of elastic disks under single parametric static loading. Different boundary conditions and thousands of design variables are applied. Due to a simple mesh construction technique, the checker-board pattern is avoided. The Michell-type problem is investigated minimizing the weight of the structure subjected to a compliance condition. The numerical procedure is based on an iterative formula that is formed by the use of the. first-order optimality condition of the Lagrangian function. The application is illustrated by numerical examples. The effect of the different loading conditions is studied for the Michell-type topologies as well.

Summary Low noise constructions receive more and more attention in highly industrialized countries. Consequently, decrease of noise
radiation challenges a growing community of engineers. One of the most efficient techniques for finding quiet structures consists
in numerical optimization. Herein, we consider structural-acoustic optimization understood as an (iterative) minimum search
of a specified objective (or cost) function by modifying certain design variables. Obviously, a coupled problem must be solved
to evaluate the objective function. In this paper, we will start with a review of structural and acoustic analysis techniques
using numerical methods like the finite- and/or the boundary-element method. This is followed by a survey of techniques for
structural-acoustic coupling. We will then discuss objective functions. Often, the average sound pressure at one or a few
points in a frequency interval accounts for the objective function for interior problems, wheareas the average sound power
is mostly used for external problems. The analysis part will be completed by review of sensitivity analysis and special techniques.
We will then discuss applications of structural-acoustic optimization. Starting with a review of related work in pure structural
optimization and in pure acoustic optimization, we will categorize the problems of optimization in structural acoustics. A
suitable distinction consists in academic and more applied examples. Academic examples iclude simple structures like beams,
rectangular or circular plates and boxes; real industrial applications consider problems like that of a fuselage, bells, loudspeaker
diaphragms and components of vehicle structures. Various different types of variables are used as design parameters. Quite
often, locally defined plate or shell thickness or discrete point masses are chosen. Furthermore, all kinds of structural
material parameters, beam cross sections, spring characteristics and shell geometry account for suitable design modifications.
This is followed by a listing of constraints that have been applied. After that, we will discuss strategies of optimization.
Starting with a formulation of the optimization problem we review aspects of multiobjective optimization, approximation concepts
and optimization methods in general. In a final chapter, results are categorized and discussed. Very often, quite large decreases
of noise radiation have been reported. However, even small gains should be highly appreciated in some cases of certain support
conditions, complexity of simulation, model and large frequency ranges. Optimization outcomes are categorized with respect
to objective functions, optimization methods, variables and groups of problems, the latter with particular focus on industrial
applications. More specifically, a close-up look at vehicle panel shell geometry optimization is presented. Review of results
is completed with a section on experimental validation of optimization gains. The conclusions bring together a number of open
problems in the field.

Describes the development of a joint continuing education project composed of 20 colleges and universities in northeastern New York which became part of a continuing education consortium. The advantages and limitations of this type of cooperation are outlined. Replicability and viability of the continuing education consortium model are also discussed. (SH)

Functionally graded materials (FGMs) and functionally graded structures (FGSs) are special types of advanced composites with peculiar features and advantages. This article reviews the design criteria of functionally graded additive manufacturing (FGAM), which is capable of fabricating gradient components with versatile functional properties. Conventional geometrical‐based design concepts have limited potential for FGAM and multi‐scale design concepts (from geometrical patterning to microstructural design) are needed to develop gradient components with specific graded properties at different locations. FGMs and FGSs are of great interest to a larger range of industrial sectors and applications including aerospace, automotive, biomedical implants, optoelectronic devices, energy absorbing structures, geological models, and heat exchangers. This review presents an overview of various fabrication ideas and suggestions for future research in terms of design and creation of FGMs and FGSs, benefiting a wide variety of scientific fields. Multi‐scale functionally graded additive manufacturing (FGAM) design is reviewed and the state‐of‐the‐art of FGAM technologies and multifunctionalities and potential applications in biomedical implants, optoelectronic devices, energy absorbing structures, geological models, and heat exchangers is summarized. An overview of various fabrication suggestions for future research in terms of design and creation of FGM/FGS is also presented, benefiting a wide variety of scientific fields.

This paper investigates original local update schemes for Cellular Automata (CA) in structural design. Local problems based on mathematical programming and optimality criteria are tested, allowing the isolation of an original local update scheme. The scheme consists of repeating analysis and optimality-based design rules locally. Several systematic experiments on various problem sizes are performed to show the efficiency and robustness of the approach. For comparison, the experiments are also run using a more traditional CA implementation.

In situations outside those identified with routine elastic structural analysis, there is often a need for formulation in mixed form. Small-deformation elastostatics, expressed in terms of stress, strain, and displacement, is described here in the form of either of two complementary constrained-extremum problems. The governing equations and boundary conditions of elastostatics are obtained by an interpretation of the generalized ‘necessary conditions’ for each of these fully mixed variational formulations. While the objectives in the problem statements are bilinear and therefore nonconvex, a simple proof is available to confirm that the solution to these conditions is an extremizer. Extensions of the basic formulation, obtained by the introduction of constraints or optimal relaxations, simulate constitutively nonlinear systems. The mixed formulations also provide a convenient representation of the mechanics requirements in connection with structural optimization.

With the design freedoms afforded by additive manufacturing (AM) processes, an increasing interest in shape synthesis methods has led to a variety of advances in topology optimisation methods and associated synthesis technologies. In this paper, we identify research issues related to the application of AM to shape synthesis methods, review recent advances in topology optimisation, and outline a vision for future synthesis capabilities.

Inertial ampliﬁcation is a novel phononic band gap generation method in which wide vibration stop bands can be obtained at low frequency regions. The engineering importance for this novelty comes from the fact that the phononic band gap structures can be utilized as passive vibration isolators for the low frequency range. In this thesis, primarily the research is focused on the improvements achieved on stop band widths and depths via employment of structural optimization tools. To that end, size, shape and topology optimization studies are conducted on a compliant inertial ampliﬁcation mechanism, then with these compliant unit cell mechanisms, one and two dimensional periodic structures are formed. Consequently, by means of these periodic structures, it is demonstrated that the vibration transmission is inhibited for wide ranges at low frequencies. The work comprises analytical and numerical studies and more importantly experimental validation of the results. Moreover, topology optimization studies performed during the thesis lead to the development of a new fast topology optimization algorithm to obtain structures with maximized fundamental frequency, though this was not originally among the research objectives. Finally, explicit problem formulations and a comprehensive review on topology optimization are also presented.

In this paper, we consider topology optimization of continuum structures subject to multiple loads. The objective is to minimize mass, while satisfying maximum displacement constraints at the points of load-application.
The classic method of including displacement constraints is through penalization of the objective function. Here, we propose an alternate method that relies on the concept of topological sensitivity. The proposed method generalizes the recently proposed single-load PareTO method by the first author. For multi-load problems, the topological sensitivity fields for each of the load-cases are computed, and then combined through weighting functions that continuously adapt to satisfy the constraints. Different weighting functions are considered, and compared through numerical experiments.

Polytechnic institute and state university, Blacksburg, VA 24061 Cellular Automata (CA) is an emerging paradigm for the combined analysis and de-sign of complex systems using local update rules. An implementation of the paradigm has recently been demonstrated successfully for the design of truss and beam structures. In the present paper, CA is applied to two-dimensional continuum topology optimiza-tion problems. The topology problem is regularized using the popular SIMP approach. The design rules are derived based on continuous optimality criteria interpreted as local Kuhn-Tucker conditions. Both CA based and conventional finite element analyses are considered. Numerical experiments with the proposed algorithm indicate that the ap-proach is quite robust and does not suffer from checkerboard patterns, mesh-dependent topologies, or numerical instabilities.

For aeronautical applications of topology optimization, it is of importance to develop topology optimization techniques, that can handle stress constraints in an efficient and accurate manner. The development of such topology optimization techniques is a challeng- ing task due to the local nature of the stress constraints, their highly non-linear behaviour with respect to the design variables and the so-called singularity phenomenon. An accurate sensitivity analysis is essential for these type of problems with multiple constraints. In this paper, we propose a methodology of dealing with stress constraints in a level set based framework. In this framework, the level set function nodal values are related to element densities by an exact Heaviside projection. Stress relaxation and constraint aggregation techniques are used to deal with the singularity phenomenon and the local nature of the stress, respectively. A constrained optimization problem is then solved, in which the design variables (the level set nodal values) are updated in the projected steepest-descent direction, which is determined using a consistent sensitivity analysis.We demonstrate the effectiveness of this technique on two numerical examples. The results show that the level set method with a consistent sensitivity analysis allows for the treatment of multiple constraints by using constrained optimization techniques.

The objective of this paper is to introduce and demonstrate a robust method for multi-constrained topology optimization. The method is derived by combining the topological sensitivity with the classic augmented Lagrangian formulation.
The primary advantages of the proposed method are: (1) it rests on well-established augmented Lagrangian formulation for constrained optimization, (2) the augmented topological level-set can be derived systematically for an arbitrary set of loads and constraints, and (3) the level-set can be updated efficiently. The method is illustrated through numerical experiments.

“Compliant mechanisms” often refers to monolithic or jointless structures that transfer an input force or displacement to another point through elastic deformation. The intrinsic advantages associated with compliant mechanisms include their lack of a need for lubrication and assembly and their high accuracy. These advantages make compliant mechanisms extremely suitable for precision engineering applications that require nanometer or even subnanometer positioning accuracy. This introductory chapter briefly presents compliant mechanisms, their applications and the associated design approaches.

This thesis contains contributions to the development of topology optimization techniques capable of handling stress constraints. The research that led to these contributions was motivated by the need for topology optimization techniques more suitable for industrial applications. Currently, topology optimization is mainly used in the initial design phase, and local failure criteria such as stress constraints are considered in additional post-processing steps. Consequently, there is often a large gap between the topology optimized design and the final design for manufacturing. Taking into account stress constraints directly into the topology optimization process would reduce this gap.
Several difficulties arise in topology optimization with local stress constraints which complicate solving the optimization problem directly. Chapter 2 discusses these difficulties, and reviews solutions that have been applied. Two fundamental difficulties are: (i) the presence of singular optima, which are true optima inaccessible to standard nonlinear programming techniques, and (ii) the fact that the stress is a local state variable, which typically leads to a large number of constraints.
Currently, the conventional strategy to circumvent these difficulties is to apply (i) constraint relaxation, which perturbs the feasible domain to make singular optima accessible, followed by (ii) constraint aggregation to transform the typically large number of relaxed constraints into a single or few global constraints thereby reducing the order of the problem.
Although there is no consensus on the exact choice of aggregation and relaxation functions and their numerical implementation, in general, this approach introduces two additional parameters to the problem: an aggregation and a relaxation parameter. Following this approach, one solves an alternative optimization problem with the aim of finding a solution to the original stress-constrained topology optimization. The feasible domain of this alternative optimization problem is related to the original feasible domain via these parameters.
In Chapter 2, we investigated the parameter dependence of this alternative optimization problem on an elementary two-bar truss problem. It was found that the location of the global optimum of this alternative optimization problem with respect to the true optimum depends in a non-trivial way on these problem parameters (in their range of application); i.e., for a given parameter set, it is difficult to predict the influence of changing one of the parameter values, and if this change will result in a feasible domain in which the global optimum is closer to the true optimum. This complicates determining optimal parameter values \emph{a priori} which, in addition, are problem-dependent.
In Chapter 3, we investigated the effect of design parameterization, and relaxation techniques in stress-constrained topology optimization. An elementary numerical example was considered, representing a situation as might occur in density-based topology optimization. As previously observed in truss optimization, we found that a global optimum of the relaxed optimization problem may not converge to the true optimum as the relaxation parameter is decreased to zero.
In this thesis, we present two novel approaches: a unified aggregation and relaxation approach in Chapter 4, and the damage approach in Chapter 5. In the unified aggregation and relaxation approach, we applied constraint aggregation such that it simultaneously perturbs the feasible domain, and makes singular optima accessible. Consequently, conventional relaxation techniques become unnecessary when applying constraint aggregation following this approach. The main advantage is that the problem only depends on a single parameter, which reduces the parameter dependency of the problem.
The damage approach is presented as a viable alternative for conventional methodologies. Following the damage approach stress constraint violation is penalized by degrading material where the stress exceeds the allowable stress. Material degradation affects the overall performance of the structure, and therefore, the optimizer promotes a design without stress constraint violation. Similar to conventional constraint aggregation techniques a large number of local constraints can be controlled by imposing a single or a few global constraints.
Both novel approaches are validated on elementary truss examples and tested on numerical examples in density-based topology optimization. In contrast to the conventional strategy of relaxation followed by aggregation, there exists a clear relationship between the perturbed feasible domain and the original unperturbed feasible domain in terms of a single problem parameter.

The work-weight-distribution criterion is derived from the first-order extremum conditions of minimal truss weight under the work constraint with self-weight loads. It indicates that structural weight can be optimally distributed according to the difference of external force work and self-weight work. The work-ratio-extremum method of the truss topology optimization is derived from the Kuhn-Tucker conditions with inequality constraints and the regular influences of a ratio step on the first order derivations of the work function. The method includes three steps, i.e., formulate the optimal ratio step and the multiplier, and solve the work-criterion equations. Using the regular influences of a scale of all design variables on the fixed-point iterative solution and its Jacobi matrix, it is proved that the algorithm is globally convergent. Based on the incompatility of the work constraint and the stress constraints, the stress ratio method can be used to optimize the structure next to the topology optimization. Numerical examples of a three-bar truss and a ten-bar truss for multiple loadings show that the methods are effective.

This paper describes a preliminary study of the application of structural optimization techniques to the design of additively manufactured components, using load testing to failure to establish true load carrying capacity. The cantilever component specimens fabricated were designed to resist a tip load and comprised one conventional benchmark design and two designs developed using layout optimization (LO) techniques. The designs were fabricated from Titanium Ti-6Al-4V and then scanned for internal defects using X-Ray Computed Tomography (XCT). All three specimens failed below the design load during testing. Several issues were identified in both the design optimization and fabrication phases of the work, contributing to the premature failure of the specimens. Various recommendations to improve the optimization phase are presented in the paper.

According to the extremum condition of Kuhn-Tucker function, it is proved that the weight minimization problem, under displacement constraints of multiple nodes and stress constraints of multiple bars, can be transformed into several optimization problems under single constraint. And, the displacement weight and the stress weight distribution criterion can be turned into the directional work distribution criterion. On the basis of external work being equal to internal energy, the displacement and stress constraints can be changed into uniform constraints of directional work. Thus, the work method of directional allocation is proposed: the nodal displacements of carrying nodal loads are used as controlled points, and are made up of the displacement constraints which constitute a positive definite set. Or, a combination with displacement and stress constraints also constitutes a positive definite set. The optimal solution is obtained by the displacement extremum scale method, or a method of combinating with it and fully stress design. At last, two numerical examples of statically determinate 2-bar and 6-bar trusses are used to verify the mentioned theories and algorithmic rules.

Little dose the structural topology simulated annealing algorithm (STSA) consider structural mechanical property and more dose it pay attention to the geometric configuration while optimizing the topology of a truss structure. Moreover, fully stressed design (FSD) specializes in the section optimization of truss structures with fixed configuration. Therefore, FSD was introduced into heuristic searching process of STSA to improve the intelligent algorithm. In addition, an approach was proposed to identify the relative configuration stability of a truss structure, and incorporated FSD with STSA to establish a hybrid algorithm. A benchmark optimizing problem was performed based on such modified intelligent algorithm, and the results exhibit that the searching process of hybrid algorithm is more stable than that of STSA in terms of efficiency, robustness and optimal solutions.

Chapter 3 gives a brief overview of structural shape and topology optimization. Then several common optimization methods, such as SQP and genetic algorithms, are investigated in more detail. This is followed by a discussion of sensitivity analysis techniques with an in-depth look at the FDM and SAM.

Describes development work to combine the basic ESO with the additive evolutionary structural optimisation (AESO) to produce bidirectional ESO whereby material can be added and can be removed. It will be shown that this provides the same results as the traditional ESO. This has two benefits, it validates the whole ESO concept and there is a significant time saving since the structure grows from a small initial one rather than contracting from a sometimes huge initial one where 90 per cent of the material gets removed over many hundreds of finite element analysis (FEA) evolutionary cycles. Presents a brief background to the current state of Structural Optimisation research. This is followed by a discussion of the strategies for the bidirectional ESO (BESO) algorithm and two examples are presented.

This paper is concerned with two-dimensional, linearly elastic, composite materials made by mixing two isotropic components. For given volume fractions and average strain, we establish explicit optimal upper and lower bounds on the effective energy quadratic form. There are two different approaches to this problem, one based on the “Hashin-Shtrikman variational principle” and the other on the “translation method". We implement both. The Hashin-Shtrikman principle applies only when the component materials are “well-ordered", i.e., when the smaller shear and bulk moduli belong to the same material. The translation method, however, requires no such hypothesis. As a consequence, our optimal bounds are valid even when the component materials are not well-ordered . Analogous results have previously been obtained by Gibianski and Cherkaev in the context of the plate equation.

The paper is concerned with the following problem: two alternative loads with the same point of application, A, are to be transmitted to a rigid foundation by a plane truss of minimum weight whose load factors for plastic collapse under one or the other load are not to exceed a given value. A necessary and sufficient condition for global optimality is established and used to determine the optimal layout of the truss.

Whereas minimum-weight design undoubtedly serves the useful function indicated above, the main aim of this Introduction is to convince the reader that optimality criteria have become an indispensable tool in the computational repetoire of the structural designer.

A modification of Michell's problem1 is formulated. This concerns the minimum-weight design of a truss T that transmits a given load to a given rigid foundation with the requirement that the axial stresses in the bars of the truss stay within an allowable range. It is then shown that the truss T cannot be less stiff in the elastic range or in stationary creep than any other truss that uses the same amount of material and respects the same allowable range of axial stress. Finally, it is shown that a truss of minimum weight that supports given point masses from the given rigid foundation and has a given fundamental natural frequency has the same layout as the truss T but, in general, different cross-sectional areas.

Consideration is given to aims, problems, and methods of structural
optimization; continuum-based optimality criteria (COC) methods; optimal
layout theory; layout optimization using the iterative COC algorithm;
simple solutions for optimal layout of trusses; CAD-integrated
structural topology and design optimization; structural optimization of
linearly elastic structures using a homogenization method; and mixed
elements in shape optimal design of structures based on global criteria.
Attention is also given to shape optimal design of axisymmetric shell
structures, applications of artificial neural nets in structural
mechanics, mathematical programming techniques for shape optimization of
skeleton structures; exact and approximate static structural reanalysis;
shape optimization with FEM; sensitivity analysis with BEM; and the
theorems of structural and geometric variation for engineering
structures. (No individual items are abstracted in this volume)

Significant performance improvements can be obtained if the topology of an elastic structure is allowed to vary in shape optimization problems. We study the optimal shape design of a two-dimensional elastic continuum for minimum compliance subject to a constraint on the total volume of material. The macroscopic version of this problem is not well-posed if no restrictions are placed on the structure topology; relaxation of the optimization problem via quasiconvexification or homogenization methods is required. The effect of relaxation is to introduce a perforated microstructure that must be optimized simultaneously with the macroscopic distribution of material. A combined analytical-computational approach is proposed to solve the relaxed optimization problem. Both stress and displacement analysis methods are presented. Since rank-2-layered composites are known to achieve optimal energy bounds, we restrict the design space to this class of microstructures whose effective properties can easily be determined in explicit form. We develop a series of reduced problems by sequentially interchanging extremization operators and analytically optimizing the microstructural design fields.

It is shown on a simple test example that ESO’s rejection criteria may result in a highly nonoptimal design. Reasons for this
failure are also discussed.

The theoretical basis for a new optimality criteria method (DCOC) was presented in Part I of this study (Zhou and Rozvany 1992). It was shown in that paper that a considerable gain in efficiency can be achieved by the proposed method, since the Lagrangian multipliers associated with stress constraints can be, in general, evaluated explicitly and hence the size of the dualtype problem is greatly reduced. The superior efficiency of the proposed method, for problems involving a large number of active stress constraints, was also demonstrated through simple numerical examples. In Part II, the computational algorithm of DCOC is presented in detail and several standard test examples are used for verifying the correctness and efficiency of the proposed method.

A highly efficient new method for the sizing optimization of large structural systems is introduced in this paper. The proposed technique uses new rigorous optimality criteria derived on the basis of the general methodology of the analytical school of structural optimization. The results represent a breakthrough in structural optimization in so far as the capability of OC and dual methods is increased by several orders of magnitude. This is because the Lagrange multipliers associated with the stress constraints are evaluated explicitly at the element level, and therefore, the size of the dual-type problem is determined only by the number of active displacement constraints which is usually small. The new optimaliy criteria method, termed DCOC, will be discussed in two parts. Part I gives the derivation of the relevant optimality criteria, the validity and efficiency of which are verified by simple test examples. A detailed description of the computational algorithm for structures subject to multiple displacement and stress constraints as well as several loading conditions is presented in Part II.

The problem of optimum truss topology design based on the ground structure approach is considered. It is known that any minimum weight truss design (computed subject to equilibrium of forces and stress constraints with the same yield stresses for tension and compression) is—up to a scaling—the same as a minimum compliance truss design (subject to static equilibrium and a weight constraint). This relation is generalized to the case when different properties of the bars for tension and for compression additionally are taken into account. This situation particularly covers the case when a structure is optimized which consists of rigid (heavy) elements for bars under compression, and of (light) elements which are hardly/not able to carry compression (e.g. ropes). Analogously to the case when tension and compression is handled equally, an equivalence is established and proved which relates minimum weight trusses to minimum compliance structures. It is shown how properties different for tension and compression pop up in a modified global stiffness matrix now depending on tension and compression. A numerical example is included which shows optimal truss designs for different scenarios, and which proves (once more) the big influence of bar properties (different for tension and for compression) on the optimal design.

It is shown that Michell's problem of least-weight truss layouts can be obtained by asymptotic expansion of the optimized strain and complementary energies derived for minimum compliance topology of perforated plates in plane stress.

Truss topology design for minimum external work (compliance) can be expressed in a number of equivalent potential or complementary energy problem formulations in terms of member forces, displacements and bar areas. Using duality principles and non-smooth analysis we show how displacements only as well as stresses only formulations can be obtained and discuss the implications these formulations have for the construction and implementation of efficient algorithms for large-scale truss topology design. The analysis covers min-max and weighted average multiple load designs with external as well as self-weight loads and extends to the topology design of reinforcement and the topology design of variable thickness sheets and sandwich plates. On the basis of topology design as an inner problem in a hierarchical procedure, the combined geometry and topology design of truss structures is also considered. Numerical results and illustrative examples are presented.

Topology optimization of structures and composite continua has two main subfields: Layout Optimization (LO) deals with grid-like structures having very low volume fractions and Generalized Shape Optimization (GSO) is concerned with higher volume fractions, optimizing simultaneously the topology and shape of internal boundaries of porous or composite continua. The solutions for both problem classes can be exact/analytical or discretized/FE-based.
This review article discusses FE-based generalized shape optimization, which can be classified with respect to the types of topologies involved, namely Isotropic-Solid/Empty (ISE), Anisotropic-Solid/Empty (ASE), and Isotropic-Solid/Empty/Porous (ISEP) topologies.
Considering in detail the most important class of (i.e. ISE) topologies, the computational efficiency of various solution strategies, such as SIMP (Solid Isotropic Microstructure with Penalization), OMP (Optimal Microstructure with Penalization) and NOM (NonOptimal Microstructures) are compared.
The SIMP method was proposed under the terms “direct approach" or “artificial density approach" by Bendsoe over a decade ago; it was derived independently, used extensively and promoted by the author’s research group since 1990. The term “SIMP" was introducted by the author in 1992. After being out of favour with most other research schools until recently, SIMP is becoming generally accepted in topology optimization as a technique of considerable advantages. It seems, therefore, useful to review in greater detail the origins, theoretical background, history, range of validity and major advantages of this method.

This paper presents a simple method for structural optimization with frequency constraints. The structure is modelled by a fine mesh of finite elements. At the end of each eigenvalue analysis, part of the material is removed from the structure so that the frequencies of the resulting structure will be shifted towards a desired direction. A sensitivity number indicating the optimum locations for such material elimination is derived. This sensitivity number can be easily calculated for each element using the information of the eigenvalue solution. The significance of such an evolutionary structural optimization (ESO) method lies in its simplicity in achieving shape and topology optimization for both static and dynamic problems. In this paper, the ESO method is applied to a wide range of frequency optimization problems, which include maximizing or minimizing a chosen frequency of a structure, keeping a chosen frequency constant, maximizing the gap of arbitrarily given two frequencies, as well as considerations of multiple frequency constraints. The proposed ESO method is verified through several examples whose solutions may be obtained by other methods.

The Michell layout problem as a low volume fraction limit of the perforated plate topology optimization problem: an asymptotic study Use of optimality criteria methods for large scale systems

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Stress ratio type methods and con-ditional constraints – a critical review

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