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Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima
... Filter techniques and projection methods are also called regularization techniques. One of the most common is the density variable filter for topology optimization where the relative densities are the design variables being filtered, for example, Bendsøe and Sigmund (2004), Luo et al. (2019), Sigmund and Petersen (1998), Lazarov et al. (2016), Zhou et al. (2015), Lazarov and Wang (2017) and references therein. The filter techniques and the projection methods can be combined for enforcing a length scale and thereby, ensuring a manufacturable structure for member size requirements, for example, for casted manufactured structures, 3D printed structures, milled structures, etc. ...
... The total mass m( ) of the design domain is constrained to fulfill a certain weight target defined by the relative material fraction f and m full being the mass of the design domain having full material. A sensitivity filter is applied for regularization introducing a length scale and for suppressing checkerboards, see Sigmund and Maute (2013), Sigmund and Petersen (1998). The radius of the sensitivity filter R f for all present optimization results is 1.3 of the averaged element size of all elements specified in the design domain. ...
... (1) and (2) to be well-posed. Usually, a filter is introduced for avoiding the so-called checkerboards and secondly, to introduce a length scale Sigmund and Maute (2013), Sigmund and Petersen (1998). In the present approach s * min and the radius R introduces a maximum length scale for one of the directions. ...
The present contribution derives a theoretical framework for constructing novel geometrical constraints in the context of density-based topology optimization. Principally, the predefined geometrical dimensionality is enforced locally on the components of the optimized structures. These constraints are defined using the principal values (singular values) from a singular value decomposition of points clouds represented by elemental centroids and the corresponding relative density design variables. The proposed approach is numerically implemented for demonstrating the designing of lattice or membrane-like structures. Several numerical examples confirm the validity of the derived theoretical framework for geometric dimensionality control.
... Si l'on considère l'expression de la fonction objectif de l'équation (3.20) ainsi que celle de sa dérivée (3.24), le type de solution finale que l'on a tendanceà obtenir comporte des zones de "damier" [52,83,136], comme illustré dans la figure 3.5b. Afin de rendre la solution plus physique, il faut ajouter une méthode de régularisation de la solution. ...
... Plusieurs techniques existent parmi lesquelles l'utilisation de patchs [21], de contrôle de périmètre [13,70], de contrôle de gradient globaux [22], ou de filtrage [135]. Ces différentes méthodes sont comparées dans [136]. Parmi les filtres existants, deux sont présentés dans cette section, le filtrage des densités [28,32] et le filtrage des sensibilités de la fonction coût [135]. ...
... Dans le domaine statique, elle permet d'agirà la fois sur la raideur et la masse du matériau d'un point de vue global en permettant de trouver une répartition de matièreoptimale dans un domaine de conception défini, maximisant la première tout en contrôlant la seconde[5,19,24,144]. Les solutions engendrées par ce type de problème ne sont toutefois pas uniques. Il est précisé une répartition de matière optimale car les solutions engendrées par ce type de problème ne sont généralement pas uniques[4,22,24,166]. De plus, elles nécessitent généralement l'emploi d'une méthode de régularisation afin de leur donner un sens physique[9,136,139].Ce chapitre a pour but de présenter la méthode d'optimisation topologique par pénalisation SIMP 2[14,20,97,137] parmi l'ensemble des méthodes existantes, et d'évoquer les différentes difficultés inhérentesà l'emploi de celle-ci ainsi que les solutions choisies pour les surmonter. Il estégalement présenté une comparaison de certains résultats avec des exemples provenant de la littérature dans le but de valider le code maison, puis la méthode est employée sur un plot amortissant 3D afin de déterminer la topologie optimale sous plusieurs chargements statiques. ...
Grâce à leurs propriétés amortissantes, les matériaux élastomères sont fréquemment utilisés dans l'industrie aéronautique et spatiale afin d'atténuer les vibrations provenant de sources extérieures.Lorsque ce matériau est placé entre des sous-systèmes d'un assemblage mécanique, il permet de protéger l'intégrité d'équipements sensibles tels que des éléments électroniques ou optiques.Afin d'étudier ces phénomènes, cette thèse de doctorat se base sur un modèle éléments finis représentatif d'une application industrielle comprenant une charge utile que l'on souhaite préserver, un support de charge utile par lequel transite les vibrations provenant du lanceur, et des liaisons entre ces sous-systèmes.Le matériau amortissant est incorporé dans les liaisons qui doivent alors répondre à deux objectifs contradictoires : transmettre les charges statiques et amortir les vibrations.Dans un premier temps, le positionnement du matériau ainsi que ses caractéristiques mécaniques sont déterminés en utilisant une stratégie numérique d'optimisation paramétrique.Puis, dans le but d'améliorer la conception des liaisons, un algorithme d'optimisation topologique est implémenté.Cet algorithme est d'abord utilisé dans le cadre de la statique, puis quelques cas académiques sont résolus en dynamique.
... The optimized topology appears with minor checkerboard effect, as filtering [34,35] that is commonly applied to avoid these unwanted patterns appears to be missing. Setoodeh, et al. [36] developed a stress based SIMP approach to solve for minimum compliance, combined with cellular automata. ...
... In other work, Hoglund [38] extended the SIMP method to accommodate material orientation designated as the CFAO. In this approach, design sensitivities with respect to the density and material orientation were derived, and the density sensitivity was also filtered using a linear weight-average filter [34,35]. The optimized model was then printed with desktop 3D printer. ...
... During the optimization process, a checkerboard pattern is likely to occur, resulting in undesirable structure pattern. To mitigate such effect, a linear sensitivity filter with respect to the density variable [34,35] is employed where the filtered sensitivity becomes ∂c ∂ρ ∑ H ρ ∂c ∂ρ ρ ∑ H (9) with H r dist i, j (10) In the above, H is a weight factor and is a linear function of the distance between the center of element i to the center of neighboring element j. Figure 2 shows the flow of the optimization iteration process employed in this work for CFAO of FFF composites. First, the design domain and boundary conditions are defined, and the domain is discretized into hexahedral elements. ...
Products produced with Additive Manufacturing (AM) methods often have anisotropic microstructure that forms as material layers are added during processing. Unfortunately, current AM design methods do not accommodate the inherent non-isotropic behavior of these materials when determining the best structural layout. This paper presents a three dimensional (3D) topology optimization method that computes the best non-isotropic material distribution and principal material direction for minimum compliance of a statically loaded non-isotropic AM structure. The compliance objective function is calculated using the finite element method with eight node 3D isoparametric elements, and design sensitivities with respect to both density and material orientation are calculated with the Adjoint Variable method. We employ a linear weighted sensitivity filter on the density variables to mitigate checker-boarding of the material distribution. The optimization problem is solved with a nonlinear constraint-based Matlab (The Mathworks, Inc., Natick, MA) optimization solver. Topology optimization of a 3D cantilever beam with different print directions is given to demonstrate the applicability of the optimization scheme.
... The optimized topology appears with minor checkerboard effect, as filtering [34,35] that is commonly applied to avoid these unwanted patterns appears to be missing. Setoodeh, et al. [36] developed a stress based SIMP approach to solve for minimum compliance, combined with cellular automata. ...
... In other work, Hoglund [38] extended the SIMP method to accommodate material orientation designated as the CFAO. In this approach, design sensitivities with respect to the density and material orientation were derived, and the density sensitivity was also filtered using a linear weight-average filter [34,35]. The optimized model was then printed with desktop 3D printer. ...
... During the optimization process, a checkerboard pattern is likely to occur, resulting in undesirable structure pattern. To mitigate such effect, a linear sensitivity filter with respect to the density variable [34,35] is employed where the filtered sensitivity becomes ∂c ∂ρ ∑ H ρ ∂c ∂ρ ρ ∑ H (9) with H r dist i, j (10) In the above, H is a weight factor and is a linear function of the distance between the center of element i to the center of neighboring element j. Figure 2 shows the flow of the optimization iteration process employed in this work for CFAO of FFF composites. First, the design domain and boundary conditions are defined, and the domain is discretized into hexahedral elements. ...
Products produced with Additive Manufacturing (AM) methods often have anisotropic microstructure that forms as material layers are added during processing. Unfortunately, current AM design methods do not accommodate the inherent non-isotropic behavior of these materials when determining the best structural layout. This paper presents a three dimensional (3D) topology optimization method that computes the best non-isotropic material distribution and principal material direction for minimum compliance of a statically loaded non-isotropic AM structure. The compliance objective function is calculated using the finite element method with eight node 3D isoparametric elements, and design sensitivities with respect to both density and material orientation are calculated with the Adjoint Variable method. We employ a linear weighted sensitivity filter on the density variables to mitigate checker-boarding of the material distribution. The optimization problem is solved with a nonlinear constraint-based Matlab (The Mathworks, Inc., Natick, MA) optimization solver. Topology optimization of a 3D cantilever beam with different print directions is given to demonstrate the applicability of the optimization scheme.
... Therefore, it is possible to circumvent local minima in the optimization process by progressively including a predefined number of terms of the sine and cosine series in the calculation of 190 T θ m j . This process is similar to the continuation method used in the pseudo-density variable [55]. ...
... The sensitivity analysis stated in Section 3.3 is used to achieve the objective function 335 derivative with respect to design variables, which is used to linearize the original nonlinear optimization problems using the first-order Taylor series. These sensitivities are filtered to prevent some instabilities in TO [55]. Move limits that correspond to the extreme values that the design variables can take at each linear approximation are computed. ...
... That is, after the sensitivity calculation at each element, it is modified based on a weighted average of the neighboring sensitivities. The filter ensures a feasible solution to the optimization problem, avoiding the formation of instabilities known as checkerboarding and mesh dependency [55]. ...
Laminated piezocomposite actuators (LAPA) are structures composed of piezoelectric and non-piezoelectric materials layers. Due to several parameters and the multiphysics domain involved in the design of LAPA, simple forms are commonly found in industrial applications. However, its design can be systematized by using the topology optimization method (TOM) that permits the solution of complex problems. The design of LAPA with TOM has traditionally considered the optimization of piezoelectric materials over an isotropic substrate, yet some previous researches suggest that LAPA including fiber-reinforced composite layers can increase the performance of these transducers. In addition, works dealing with fiber-based composite focus on static or harmonic analysis with sinusoidal excitations, although other signal inputs are used in practice. In fact, the design of fiber-based LAPA in transient regime has not been assessed before. Thus, a methodology is proposed here to design LAPA with TOM. The actuator is electrically excited with a combined waveform: a sine wave treated as a harmonic problem and a step excitation addressed as a transient problem. Both waves have the same frequency, however they are not applied at the same time. This approach allows the development of a multi-entry actuator since it generates the same level of output displacement independently of the type of excitation input. Consequently, the optimization problem is formulated with the purpose of distributing the material in all layers, the polarization sign in piezoelectric layers and the fiber orientation angle in composite layers, in which the objective function simultaneously seeks for the maximization of the vibration amplitude at certain points of the actuator and its response speed. Eight-node shell elements taking into account the piezoelectric effects are used in the finite element method (FEM) and the “layer wise theory is adopted to model the laminated structure. The Generalized-α method is used to solve the transient problem. In order to optimize the material distribution and the polarization sign, the classical SIMP and PEMAP-P models are used respectively, while to optimize the fiber orientation angles in the composite material, a novel self-penalizable interpolation model is proposed. This optimization problem is solved by using the sequential linear programming (SLP) technique with the CVX solver and the sensitivity analysis is performed with the adjoint method. Discrete signal processing concepts are applied to solve the adjoint problem involving specific points of curves obtained by time integration methods in transient analysis. Numerical techniques are implemented to avoid TOM instabilities. Finally, the potential of this approach is demonstrated with two numerical examples.
... Among many extensions and variations, one can find smooth Heaviside projections (Guest et al. 2004); morphological filters (Sigmund 2007); manufacturing-tolerant formulations (Sigmund 2009;Wang et al. 2011); density filters based on geomteric, harmonic and quasi-arithmetic means (Svanberg and Svärd 2013;Wadbro and Hägg 2015); and recently also spatial variation of the length scale (Amir and Lazarov 2018;Schmidt et al. 2019). Several alternatives to the filtering do exist, e.g., the perimeter control method (Haber et al. 1996), the gradient control scheme (Peterson and Sigmund 1998;Borrvall and Petersson 2001), and the MOLE method (Poulsen 2003); see also the surveys (Sigmund and Peterson 1998;Borrvall 2001). Furthermore, other, non density-based, topology optimization approaches use alternate means for controlling the length scale, for example phase-field methods Ristinmaa 2013, 2015) and level-set methods (Allaire et al. 2016;Wang et al. 2016). ...
... Among many extensions and variations, one can find smooth Heaviside projections (Guest et al. 2004); morphological filters (Sigmund 2007); manufacturing-tolerant formulations (Sigmund 2009;Wang et al. 2011); density filters based on geomteric, harmonic and quasi-arithmetic means (Svanberg and Svärd 2013;Wadbro and Hägg 2015); and recently also spatial variation of the length scale (Amir and Lazarov 2018;Schmidt et al. 2019). Several alternatives to the filtering do exist, e.g., the perimeter control method (Haber et al. 1996), the gradient control scheme (Peterson and Sigmund 1998;Borrvall and Petersson 2001), and the MOLE method (Poulsen 2003); see also the surveys (Sigmund and Peterson 1998;Borrvall 2001). Furthermore, other, non density-based, topology optimization approaches use alternate means for controlling the length scale, for example phase-field methods Ristinmaa 2013, 2015) and level-set methods (Allaire et al. 2016;Wang et al. 2016). ...
Design variables in density-based topology optimization are typically regularized using filtering techniques. In many cases, such as stress optimization, where details at the boundaries are crucially important, the filtering in the vicinity of the design domain boundary needs special attention. One well-known technique, often referred to as “padding,” is to extend the design domain with extra layers of elements to mitigate artificial boundary effects. We discuss an alternative to the padding procedure in the context of PDE filtering. To motivate this augmented PDE filter, we make use of the potential form of the PDE filter which allows us to add penalty terms with a clear physical interpretation. The major advantages of the proposed augmentation compared with the conventional padding is the simplicity of the implementation and the possibility to tune the boundary properties using a scalar parameter. Analytical results in 1D and numerical results in 2D and 3D confirm the suitability of this approach for large-scale topology optimization.
... Moreover, a small change in the initial parameters of the problem may lead to a different optimum (cf. [13]). In [14], a continuation method is proposed to improve the convergence of the optimizer towards the global optimum for compliance problems. ...
... The optimization starts from an unpenalized (p = 1) convex problem, then increases p by a small amount at every optimization iteration until reaching the desired SIMP penalization value (e.g., p = 3). Other continuation methods can be found in [13]. Although continuation methods may also work for non-compliance objective problems, their effectiveness cannot be guaranteed [2]. ...
Topology optimization problems are typically non-convex, and as such, multiple local minima exist. Depending on the initial design, the type of optimization algorithm and the optimization parameters, gradient-based optimizers converge to one of those minima. Unfortunately, these minima can be highly suboptimal, particularly when the structural response is very non-linear or when multiple constraints are present. This issue is more pronounced in the topology optimization of geometric primitives, because the design representation is more compact and restricted than in free-form topology optimization. In this paper, we investigate the use of tunneling in topology optimization to move from a poor local minimum to a better one. The tunneling method used in this work is a gradient-based deterministic method that finds a better minimum than the previous one in a sequential manner. We demonstrate this approach via numerical examples and show that the coupling of the tunneling method with topology optimization leads to better designs.
... Topology optimization of structural layouts has a great impact on the performance of structures and has been extensively studied in the past few decades. Several optimization methods, such as the homogenization method ( Bendsøe and Kikuchi 1988), solid isotropic material with penalization (SIMP) method ( Zhou and Rozvany 1991;Rozvany, Zhou and Birker 1992;Sigmund and Petersson 1998;Ritz 2001), evolutionary structural optimization (ESO) method (Xie and Steven 1993) and level set technique (Sethian and Wiegmann 2000; Wang, Wang and Guo 2003), have been developed in this context. The ESO method, first proposed by Xie and Steven (1993), gradually removes inef- ficient materials from the design domain and the remaining structure evolves to an optimum. ...
... Chequerboard and mesh-dependency problems are typical numerical issues in common topology optimization approaches (Sigmund and Petersson 1998). A mesh-independence filter scheme (Huang and Xie 2009) is employed to smooth the element sensitivities using a low-pass filter of radius r min . ...
A method for the topology optimization on the natural frequency of continuum structures with casting constraints is proposed. The objective is to maximize the natural frequency of vibrating continuum structures subject to casting constraints. When the natural frequencies of the considered structures are maximized using the solid isotropic material with penalization (SIMP) model, artificial localized modes may occur in areas where elements are assigned with lower density values. In this article, the topology optimization is performed by the bi-directional evolutionary structural optimization (BESO) method. The effects of different locations of concentrated lump mass, different volume fractions and meshing sizes on the final topologies are compared. Both two and four parting directions are investigated. Several two- and three-dimensional numerical examples show that the proposed BESO method is effective in achieving convergent solid–void optimal solutions for a variety of frequency optimization problems of continuum structures.
... The method developed and demonstrated is a direct extension of a classical treatment of compliance topology optimization (Bendsøe and Sigmund 2002;Christensen and Klarbring 2009), which is a formulation used also for comparison in the test examples. This means that we use a SIMP penalization (Solid Isotropic Material with Penalization) (Rozvany et al. 1992) and regularize the problem by using Sigmund's sensitivity filtering (Sigmund 1994). The standard optimality criteria update formula (Bendsøe and Sigmund 2002;Christensen and Klarbring 2009) is slightly modified since the sensitivity of crack energy release rate, derived in the paper, is not restricted in sign. ...
... The mostly used remedy for this is filtering which could be applied directly to the design variables or to the sensitives. The latter was suggested by Sigmund (1994) and is used in this work for both f c and f g . In the following a filtered sensitivity is denoted by a hat symbol. ...
Fatigue cracked primary aircraft structural parts that cannot be replaced need to be repaired by other means. A structurally efficient repair method is to use adhesively bonded patches as reinforcements. This paper considers optimal design of such patches by minimizing the crack extension energy release rate. A new topology optimization method using this objective is developed as an extension of the standard SIMP compliance optimization method. The method is applied to a cracked test specimen that resembles what could be found in a real fuselage and the results show that an optimized adhesively bonded repair patch effectively reduces the crack energy release rate.
... Many works are related to optimization of composite laminate designs using genetic algorithms, which are based on evolutionary algorithms and concepts 109,110 . Topology optimization groups strive to optimize properties such as structural compliance based on known loads and boundary conditions, showing how an initial starting geometry slowly forms an optimized truss-like structure [111][112][113][114][115] . Researchers have also studied how to optimize composite topologies with three materials for thermal conductivity properties 116 . ...
... a) A model representation of scanning electron microscope (SEM) images of the dorsal and ventral structures of dragonfly wings near the nodus point, where cracks are most likely to initiate 129 . Adapted from Ref [115], Copyright (2014), with permission from Elsevier. b) A representative design obtained from the optimization algorithm with 12.5% volume fraction of soft material, where the grey zones represent the stiffer material, and the black zones represent the softer material. ...
In the 50 years that succeeded Richard Feynman's exposition of the idea that there is 'plenty of room at the bottom' for manipulating individual atoms for the synthesis and manufacturing processing of materials, the materials-by-design paradigm is being developed gradually through synergistic integration of experimental material synthesis and characterization with predictive computational modeling and optimization. This paper reviews how this paradigm creates the possibility to develop materials according to specific, rational designs from the molecular to the macroscopic scale. We discuss promising techniques in experimental small-scale material synthesis and large-scale fabrication methods to manipulate atomistic or macroscale structures, which can be designed by computational modeling. These include recombinant protein technology to produce peptides and proteins with tailored sequences encoded by recombinant DNA, self-assembly processes induced by conformational transition of proteins, additive manufacturing for designing complex structures, and qualitative and quantitative characterization of materials at different length scales. We describe important material characterization techniques using numerous methods of spectroscopy and microscopy. We detail numerous multi-scale computational modeling techniques that complements these experimental techniques: DFT at the atomistic scale; fully atomistic and coarse-grain molecular dynamics at the molecular to mesoscale; continuum modeling at the macroscale. Additionally, we present case studies that utilize experimental and computational approaches in an integrated manner to broaden our understanding of the properties of two-dimensional materials and materials based on silk and silk-elastin-like proteins.
... Density filtering is needed in thermofluid topology optimization to avoid problems with illposedness of the optimization problem [39]. A Helmholtz-type partial differential equation (PDE) 210 filter [40] is used in this study which is defined as follows: ...
... The optimization is conducted within COMSOL's Optimization Module which automatically solves the adjoint problem to provide sensitivities for the objective and constraint functionals and the globally convergent version of the Method of Moving Asymptotes (GCMMA)[44] is used as the optimization method. A continuation approach[39,45] is 235 applied on the convexity parameters of the interpolation functions, b α , b h , b k , and the steepness parameter of the design projection, β. This is done to ensure a more convex optimization problem in the beginning and to consequently gradually increase the penalization of intermediate densities as well as to increase the sharpness of the solid-fluid interface. ...
This paper investigates the application of density-based topology optimization to the design of air-cooled forced convection heat sinks. To reduce the computational burden that is associated with a full 3D optimization, a pseudo 3D optimization model comprising a 2D modeled conducting metal base layer and a thermally coupled 2D modeled thermofluid design layer is used. Symmetry conditions perpendicular to the flow direction are applied to generate periodic heat sink designs. The optimization objective is to minimize the heat sink heat transfer resistance for a fixed pressure drop over the heat sink and a fixed heat production rate in the base plate. Optimized designs are presented and the resulting fin geometry is discussed from a thermal engineering point of view and compared to fin shapes resulting from a pressure drop minimization objective. Parametric studies are conducted to analyze the influence of the pressure drop on the heat sink heat transfer resistance. To quantify the influence of the assumptions made in the pseudo 3D optimization model, validation simulations with a body-fitted mesh in 2D and 3D are conducted. A good agreement between optimization model and validation simulations is found, confirming the physical validity of the utilized optimization model. Two topology optimized designs are exemplarily benchmarked against a size optimized parallel fin heat sink and an up to 13.6% lower thermal resistance is found to be realized by the topology optimization.
... Mesh dependency[24] ...
... Checkerboard pattern[24] ...
In this work we address the problem of Topology Optimization in the context of the stationary heat equation. In particular, we seek the optimum distribution of material inside a design domain subject to heat generation and/or heat external flow that minimizes the norm of the temperature field, while satisfying a prescribed amount of material to be employed. This problem was already studied by several authors over the years. However, some important concepts from the theoretical point of view are frequently omitted. In this context, the main objective of this work is to present, in the simplest manner possible, the relation between theoretical and numerical aspects of the problem. Emphasis is given to sensitivity analysis, where both the variational problem and its Finite Element Method (FEM) approximation are presented. We also describe in details the Adjoint Method. Some theoretical and numerical difficulties commonly encountered are also discussed, such as the problem of intermediate densities, mesh dependency and complexity of the obtained solution (appearance of checkerboard patterns). In this work these difficulties are addressed using the SIMP (Solid Isotropic Material with Penalization) formulation and sensitivity filtering. The approximate optimization problem is solved using a heuristic numerical scheme available in literature. Finally, numerical examples are presented in order to illustrate the main properties of the problem under study.
... Figura 2.1: Representação da dependência de malha [24] 2.4 Padrões de tabuleiro de xadrez ...
... Figura 2.2: Padrão de tabuleiro de xadrez [24] Diversas estratégias podem ser utilizadas para se evitar o aparecimento de padrões de tabuleiro de xadrez. Em alguns casos o problemá e evitado completamente por conta de uma modelagem mais apropriada do problema [19,1]. ...
Neste trabalho é abordado o problema de Otimização Topológica no contexto da equação do calor em regime estacionário. Em particular, busca-se distribuir material condutor em um domínio de projeto sujeito à geração de calor e/ou fluxos externos de calor de forma a minimizar a norma do campo de temperaturas, respeitando uma quantidade de material prescrita a ser utilizado. Este problema já foi estudado por diversos autores ao longo dos anos. Mesmo assim, alguns conceitos importantes do ponto de vista teórico são frequentemente omitidos. Neste contexto, o principal objetivo deste trabalho é apresentar, da maneira mais simples possível, uma transição entre os aspectos teóricos e numéricos do problema. Ênfase é dada para a análise de sensibilidade, onde é apresentado o problema variacional e sua aproximação utilizando o Método do Elementos Finitos (MEF). Além disso, é descrito em detalhes o Método Adjunto. Algumas dificuldades teóricas e numéricas comumente encontradas são também discutidas, como o problema de densidades intermediárias, a dependência da malha e a complexidade da solução obtida (aparecimento de padrões de tabuleiros de xadrez). Neste trabalho estas dificuldades são contornadas utilizando-se a formulação SIMP (Solid Isotropic Material with Penalization) e filtros de sensibilidade. Para a solução do problema de otimização aproximado foi utilizado um método numérico heurístico disponível na literatura. Por fim, são apresentados exemplos numéricos que ilustram as principais características do problema.
... Despite various differences in these existing methods, most of them are common in having the potential to produce undesirable layouts such as discontinuous regions or checkerboard patterns. The conditions and reasons for such numerical instabilities to arise are discussed by Diaz and Sigmund [5], and a survey of related procedures is given by Sigmund [6]. They implied the general feature of numerical methods to introduce artificially higher stiffness than actual, especially for lower order finite elements resulting in a periodic pattern of dense and porous material. ...
... The design variables x ∈ R n ele are mapped to the density variables ∈ R n ele via a density filter, 35,36 that is, x W → , which is used to address mesh dependence and checkerboard issues, 37 and this (linear) filter operation is given by ...
This work focuses on topology optimization formulations with linear buckling constraints wherein eigenvalues of arbitrary multiplicities can be canonically considered. The non‐differentiability of multiple eigenvalues is addressed by a mean value function which is a symmetric polynomial of the repeated eigenvalues in each cluster. This construction offers accurate control over each cluster of eigenvalues as compared to the aggregation functions such as ‐norm and Kreisselmeier–Steinhauser (K–S) function where only approximate maximum/minimum value is available. This also avoids the two‐loop optimization procedure required by the use of directional derivatives (Seyranian et al. Struct Optim . 1994;8(4):207‐227.). The spurious buckling modes issue is handled by two approaches—one with different interpolations on the initial stiffness and geometric stiffness and another with a pseudo‐mass matrix. Using the pseudo‐mass matrix, two new optimization formulations are proposed for incorporating buckling constraints together with the standard approach employing initial stiffness and geometric stiffness as two ingredients within generalized eigenvalue frameworks. Numerical results show that all three formulations can help to improve the stability of the optimized design. In addition, post‐nonlinear stability analysis on the optimized designs reveals that a higher linear buckling threshold might not lead to a higher nonlinear critical load, especially in cases when the pre‐critical response is nonlinear.
... (mesh dependency) 2 (robustness) (density filter) (illposedness) [74] (Helmholtz-type) (PDE) [75] 2-2 2 2 , [76] (tanh( ( )) tanh( )) , (tanh( (1 )) tanh( ) [77] (globally convergent method of asymptotes, GCMMA) [78] (sequential quadratic programming method, SNOPT) [79] Rojas-Labanda Stolpe [80] ( 2D ) [81] (SNOPT) [79] (coefficient form, PDE) ...
高热流密度散热器广泛应用于电子芯片和航天航空等领域,液冷板(单相流体强制对流)由于制造成本低、散热效率高,是应用最为广泛的散热手段之一。液冷板中的液冷通道布置形态是决定液冷板性能的重要因素,为了得到散热性能良好的散热通道布置形态,设计自由度大、灵活性高的结构拓扑优化技术已成为主要的设计手段。目前,基于结构拓扑优化技术的液冷散热通道布置形态设计已取得长足进展,但现有研究多侧重于流道拓扑优化算法的理论研究,对流道形态和流体流动、散热性能之间的关系阐明不足,同时复杂热载荷工况下的优化设计研究较少,使得结构拓扑优化设计技术在液冷板的实际应用中仍不多。针对目前研究中的不足,本文从工程应用角度出发,基于变密度法,对定常层流状态下的强制对流换热问题,研究基于结构拓扑优化技术的液冷板流道设计方法及其关键技术,并进一步制作基于设计结果的液冷板,研究其流动性能和散热性能的评价和验证方法。分别以流体耗散功最小、换热量最大和两者加权函数为目标进行液冷通道分布优化设计,构建液冷板结构拓扑优化设计数学模型;采用霍尔姆兹偏微分方程形式的密度过滤避免拓扑形态出现数值不稳定,采用双曲正切投影方法以得到清晰的流体通道拓扑形态,并采用连续变化投影斜率方法避免陷入局部最优;对不同进出口布置、不同初始解、不同权重比、复杂热载荷工况(单热源/多热源、均布/非均布热源)开展典型结构的优化设计,并探究上述因素对设计结果的影响;将拓扑优化设计得到的流道与传统直通道比较,通过有限元数值模拟,以流动和传热性能参数(温度、热阻、努塞尔数和压降)为评价指标,对比不同形式通道的散热性能;制作典型液冷板,搭建液冷循环实验系统,通过实验验证进一步验证设计结果。研究结果表明:(1)采用结构拓扑优化设计技术,可根据不同的优化目标(流量分配、热交换、加权多目标)、热载荷工况,实现液冷通道最优设计,设计得到的流固边界清晰、可加工性强;(2)不同出入口布置、雷诺数影响设计结果,设计者可根据实际工况进行选择;(3)拓扑优化设计的通道内压力、速度分布更均匀,压降、热阻更小,温度更低,努塞尔数更大,具有良好的流动和散热性能。本研究可为航空航天、微电子等领域高热流密度环境下的核心零件的热设计提供理论指导和解决方案。
... The Method of Moving Asymptotes (MMA) (Svanberg, 1987(Svanberg, , 2002 is chosen in this paper due to its reliability to catch extremum in various settings of TO. Known problems with SIMP are checkerboarding, mesh dependence, and local minima (Sigmund and Petersson, 1998). Checkerboarding and mesh dependence can be prevented by mesh independent filtering methods (Sigmund, 2007). ...
This paper presents an algorithm for reliability-based topology optimization of linear elastic continua under random-field material model. The modelling random field is discretized into a small number of random variables, and then the interested output is estimated by a stochastic response surface. A single-loop inverse-reliability algorithm is applied to reduce computational cost of reliability analysis. Two common benchmark problems in literature are used for demonstration purposes. Different values of target reliability and ranges of Young's modulus are considered to investigate their effects on resulting optimized topologies. Lastly, Monte Carlo simulation tests the proposed algorithm for correctness and accuracy.
... This method can also be easily extended to the topology optimization of continuum structure with conventional manufacturing methods to avoid enclosed void. [22][23] 和密度过滤 [24][25] 两大类,常用来消除棋盘格和网格依赖性等数值不 稳定现象 [26][27][28] 。此外，不同过滤半径也可以实现对 结构几何特征的控制。GUEST 等 [29] 将密度过滤和 阶跃投影相结合，在获得清晰拓扑结构的同时实现 了对最小特征尺寸的控制。随后，GUEST [30] 又提出 了多相投影法，通过引入两组相对独立的变量，分 别通过密度过滤和阶跃投影函数，实现了两相材料 最小特征尺寸的同时控制。传统过滤方法通常采用 卷积积分函数作为过滤函数，求解时需要临近单元 格的信息，当设计域几何结构较为复杂，单元格不 规则时，求解效率极低 [31] 。为了解决上述问题， LAZAROV 等 [32] 将过滤后的密度定义为具有诺依 曼边界条件的亥姆霍兹偏微分方程的解，用隐式过 滤函数代替了显式过滤函数。该方法对设计域的几 何结构及单元格的形状没有要求，同时可以使用有 限元求解器求解，在复杂模型拓扑优化以及并行计 算中具有优势 [32] 。其并不是一种新的过滤方式，而 是采用不同的过滤函数来实现密度过滤 [32] ，因此其 过滤效果与传统卷积函数相同 [24] ，但在效率和性能 上更具优势。 此外，通过各向异性亥姆霍兹偏微分方程作为 过滤函数，能够实现密度场的各向异性过滤，实现 在不同方向上控制结构几何特征的目的 [33] 。WANG 等 [34] ...
针对增材制造加工拓扑优化结构存在的内部空腔难以加工和支撑材料无法去除的问题，提出一种基于各向异性亥姆霍兹方程的拓扑优化方法，能够简单有效地实现考虑空腔连通性约束的结构拓扑优化。首先，根据封闭空腔结构特点，定义了结构连续性作为结构连通性的等效描述，便于在拓扑优化框架内添加连通性约束；然后，采用各向异性亥姆霍兹方程，通过设置各向异性参数保证实体结构在特定方向上的连续性，构建了包含空腔连通性限制的拓扑优化模型；最后，采用MMA算法求解拓扑优化模型，实现了考虑空腔连通性约束的结构拓扑优化。多个优化算例结果表明，相比于传统方法，该方法能够在不添加新的约束条件和中间变量的基础上，即可抑制拓扑优化设计中封闭空腔结构的产生，构成面向增材制造的拓扑优化结构。同时，该方法很容易拓展到考虑传统制造约束的拓扑优化模型中，避免设计中出现封闭空腔结构。
... This spuriousness leads to checkerboards in the optimized topology [54]. We 380 use sensitivity filtering to smoothen the noisy sensitivities and avoid checkerboarding in structural optimization. ...
This paper presents a framework for the discrete design of optimal multimaterial structural topologies using integer design variables and mathematical programming. The structural optimization problems: compliance minimization subject to mass constraint, and mass minimization subject to compliance constraint are used to design the multimaterial topologies in this work. The extended SIMP interpolation is used to interpolate the different materials available for structural design, and the material phases in each element are represented using binary design variables, one variable per available material. The Topology Optimization of Binary Structure (TOBS) method (Sivapuram and Picelli, 2018) is employed, wherein the nonlinear objective/constraint functions of optimization are sequentially approximated (herein, linearized) to obtain a sequence of Integer Linear Programs (ILPs). A novel truncation error-regulating constraint in terms of the Young’s moduli of the elements is introduced to maintain the sequential approximations valid, by restricting large changes in successive structural topologies. A commercial branch-and-bound solver is used to solve the integer subproblems yielding perfectly binary solutions which guarantee discrete structural topologies with clear material interfaces at each iteration. Adjoint sensitivities are computed to generate the integer subproblems, and the sensitivities are filtered using a conventional mesh-independent sensitivity filter. Few examples show the design of multimaterial structures in the presence of design-dependent loads: hydrostatic pressure loads and self-weight loads. This work also demonstrates through few examples, convergence of optimal multimaterial topologies at inactive constraint values when different type of loadings simultaneously act on the structure.
... This phenomenon typically arises in topology optimization. While earlier it was assumed that this phenomenon can be attributed to the existence of solutions whose figure of merit (for mechanical problems often stiffness) is a consequence of a developing micro-structure, it is nowadays accepted that a bad approximation of the solution of the PDE by the employed discretization method is the reason for it [34]. In the presented approach, this so called checkerboarding is avoided by the employed numerical schemes used to discretize the PDE, whose convergence is granted and uniform with respect to the finite dimensional control space. ...
We present a general numerical solution method for control problems with state variables defined by a linear PDE over a finite set of binary or continuous control variables. We show empirically that a naive approach that applies a numerical discretization scheme to the PDEs to derive constraints for a mixed-integer linear program (MILP) leads to systems that are too large to be solved with state-of-the-art solvers for MILPs, especially if we desire an accurate approximation of the state variables. Our framework comprises two techniques to mitigate the rise of computation times with increasing discretization level: First, the linear system is solved for a basis of the control space in a preprocessing step. Second, certain constraints are just imposed on demand via the IBM ILOG CPLEX feature of a lazy constraint callback. These techniques are compared with an approach where the relations obtained by the discretization of the continuous constraints are directly included in the MILP. We demonstrate our approach on two examples: modeling of the spread of wildfire and the mitigation of water contamination. In both examples the computational results demonstrate that the solution time is significantly reduced by our methods. In particular, the dependence of the computation time on the size of the spatial discretization of the PDE is significantly reduced.
... However, topological optimization methods are likely to encounter numerical difficulties, such as mesh dependency, checkerboard patterns, and local minima [23]. To mitigate such issues, researchers have proposed the use of regularization techniques [24]. One of the most common approaches is the use of density filters [25]. ...
This paper deals with an experimental analysis of stress prediction and simulation prior to 3D printing via the selective laser melting (SLM) method and the subsequent separation of a printed sample from a base plate in two software programs, ANSYS Addictive Suite and MSC Simufact Additive. Practical verification of the simulation was performed on a 3Dprinted topologically optimized part made of AISI 316L stainless steel. This paper presents a typical workflow for working with metallic 3D printing technology and the state-of-the-art knowledge in the field of stress analysis and simulation of printed components. The paper emphasizes the role of simulation software for additive production and reflects on their weaknesses and strengths as well, with regard to their use not only in science and research but also in practice.
... The technique is widely used in industry as well as in academia for myriad applications, such as aerospace, mechanical, and biomedical engineering [48]. A review of the papers published since 1989 sheds light on its extensive application to structural mechanics, as seen from both books and journals [31,173,171,183,137,30,172]. [28] first applied it to the optimization of continuum structures, while [191] considered topology optimization with the homogenization method. ...
Large-scale structural topology optimization has always suffered from prohibitively high computa- tional costs that have till date hindered its widespread use in industrial design. The first and major contrib- utor to this problem is the cost of solving the Finite Element equations during each iteration of the opti- mization loop. This is compounded by the frequently very fine 3D models needed to accurately simulate me- chanical or multi-physical performance. The second is- sue stems from the requirement to embed the high- fidelity simulation within the iterative design proce- dure in order to obtain the optimal design. The pro- hibitive number of calculations needed as a result of both these issues, is often beyond the capacities of ex- isting industrial computers and software. To alleviate these issues, the last decade has opened promising path- ways into accelerating the topology optimization pro- cedure for large-scale industrial sized problems, using a variety of techniques, including re-analysis, multi-grid solvers, model reduction, machine learning and high-performance computing, and their combinations. This paper attempts to give a comprehensive review of the research activities in all of these areas, so as to give the engineer both an understanding as well as a critical appreciation for each of these developments.
... The checkerboard formation is another point that has been studied [11,[16][17][18]. 90 Methodologies using different approaches to describe the density using different elements 91 [11] were proposed to minimize the checkerboard formation. ...
Simulation of the bone remodeling process is extremely important because it makes possible the structure forecast of one or several bones when anomalous situations, such as prosthesis installation, occur. Thus, it is necessary that the mathematical model to simulate the bone remodeling process be reliable; that is, the numerical solution must be stable regardless of initial density field for a phenomenological approach to model the process. For several models found in the literature, this characteristic of stability is not observed, largely due to the discontinuities present in the property values of the models (e.g., Young's modulus and Poisson's ratio). In addition, checkerboard formation and the lazy zone prevent the uniqueness of the solution. To correct these difficulties, this study proposes a set of modifications to guarantee the uniqueness and stability of the solutions, when a phenomenological approach is used. The proposed modifications are: (a) change the rate of remodeling curve in the lazy zone region and (b) create transition functions to guarantee the continuity of the expressions used to describe Young's modulus and Poisson's ratio. Moreover, the stress smoothing process controls the checkerboard formation. Numerical analysis is used to simulate the solution behavior from each proposed modification. The results show that, when all proposed modifications are applied to the three-dimensional models simulated here, it is possible to observe the tendency toward a unique solution.
... The above sensitivity is usually modified to solve the mesh-dependent problem [6,7] using a filtering scheme with ...
Due to the potential to generate forms with high efficiency and elegant geometry, topology optimization is widely used in architectural and structural designs. This paper presents a working flow of form-finding and robotic fabrication based BESO (Bi-directional Evolutionary Structure Optimization) optimization method. In case there are some other functional requirements or condition limitations, some useful modifications are also implemented in the process. With this kind of working flow, it is convenient to foreknow or control the structural optimization direction before the optimization process. Furthermore, some fabrication details of the optimized model will be discussed because there are also many notable technical points between computational optimization and robotic fabrication.
... In order to alleviate checkerboard patterns and other numerical instabilities, a density filter may be applied which also provides a necessary regularization for mesh-independence as discussed in Sigmund and Petersson [39]. The filtering scheme proposed in Bruns and Tortorelli [40], based on a gaussian-type weighting function is used in this work. ...
Optimal design of structures for fracture resistance is a challenging subject. This appears to be largely due to the strongly nonlinear governing equations associated with explicitly modeling fracture propagation. We propose a topology optimization formulation in which low weight structures are obtained with significantly increased resistance to brittle fracture in which crack propagation is explicitly modeled with the phase field approach. In contrast to our previous work, several important features are included which greatly assist the optimizer in dealing with the strongly discontinuous brittle fracture process, including a new objective function which provides additional path information to the optimizer. Increased local control of the topology is introduced via a smoothed threshold function in the phase field fracture formulation and a constraint relaxation continuation scheme is proposed to alleviate some difficulty during the initial optimization iterations. The derivation of the analytical, path‐dependent sensitivities for the relevant functions is provided and the results from two benchmark numerical examples are presented which demonstrate the effectiveness of the proposed method.
... where a design variablex for a lattice member k, contained within a unit cell γ is denoted byx k,γ . The hat superscript implies the design variables have passed through a Helmholtz filter [17], to ensure smoothly-varying geometry is obtained and by consequence eliminate the potential of 'checkerboarding' [18]. The objective weighting and compliance functional for the aerodynamic load case is denoted by w A and J A respectively and equivalently w S and J S for the static load case. ...
A robust multiscale concurrent optimization framework, which enables the precise functional grading of mechanical properties within structures over two-scales, is presented within this paper and applied to a practical aerospace application — the mass minimization of a Goose Neck Hinge. The novelty of this framework lies in the concurrent nature of the response surface which enables the efficient calculation of small-scale mechanical properties during large-scale optimization. The efficacy of this approach permits a large number of design variables to be
used in the parameterization of the small-scale without incurring a significant computational expense. The mass minimization of the Goose Neck Hinge constitutes a multi-objective optimization problem, constrained by a single maximum displacement constraint. Optimization of the Goose Neck Hinge was undertaken using both the framework presented within this paper and a density based topology optimization, to understand the relative performance of the multiscale framework to an industry standard method for structural optimization. The optimized
multiscale geometry was able to satisfy the maximum displacement constraint using 20% less material than the density based topology optimization. This indicates that this framework has the potential to deliver a new generation of optimized aerospace structures.
... To smooth the element sensitivity numbers across the entire domain, a filter scheme is used that alleviates the problem of mesh-independency and checker-boarding, which result from the sensitivity numbers becoming discontinuous across the element boundaries [20]. The filter scheme is similar to that presented by Sigmund and Petersson [21]; however, nodal sensitivity numbers are used when calculating the updated element sensitivity numbers based on the surrounding structure. The nodal sensitivity numbers are defined as the average of the element sensitivity numbers connected to the node: ...
Uncertainty quantification (UQ) within topology optimization (TO) is a growing trend with designers recognizing that deterministic analysis does not reflect the natural variabilities found in real world structural analysis. Thus far the majority of works have dealt with uncertain loading and material properties however minimal research has considered uncertain boundary conditions (BCs), which plays a vital role for both static and dynamic analysis involving natural compliance/frequency objectives and/or constraints. This paper will implement BCs uncertainty on a cantilevered flat plate by assuming that a gaussian-distributed number of nodes can be freed from the structure. Two single objective formulations are conducted involving minimizing compliance and maximizing fundamental frequency while enforcing a volume fraction constraint. A robust formulation is developed using Non-intrusive Polynomial Chaos (NIPC) to minimize the variance of the objectives. Results show that the stochastic algorithm produces topologies which indeed out-perform the deterministic in terms of mean/variance and worst case objective values. Further, the outcomes suggest that there is diminishing returns beyond a certain weighting for standard deviation in the objective function.
... where ( , ) t represents the field variable vector including the temperature field, velocity field and pressure field; ( ( , ) During the process of topology optimization, design filtering is necessary in the transient thermofluid model to avoid checkerboard problems [47]. A partial differential equation (PDE) filter is used in the topology optimization procedure, which defined as: ...
With the increasing power density of electronics components, the heat dissipation capacity of heat sinks gradually becomes a bottleneck. Many structural optimization methods, including topology optimization, have been widely used for heat sinks. Due to its high design freedom, topology optimization is suggested for the design of heat sinks using a transient pseudo-3D thermofluid model to acquire better instantaneous thermal performance. The pseudo-3D model is designed to reduce the computational cost and maintain an acceptable accuracy. The model relies on an artificial heat convection coefficient to couple two layers and establish the approximate relationship with the corresponding 3D model. In the model, a constant pressure drop and heat generation rate are treated. The material distribution is optimized to reduce the average temperature of the base plate at the prescribed terminal time. Furthermore, to reduce the intermediate density regions during the density-based topology optimization procedure, a detailed analysis of interpolation functions is made and the penalty factors are chosen on this basis. Finally, considering the engineering application of the model, a practical model with more powerful cooling medium and higher inlet pressure is built. The optimized design shows a better instantaneous thermal performance and provides 66.7% of the pumping power reduction compared with reference design.
... Generally, complex boundary and interface tracking algorithms and re-meshing strategies are employed to tackle with such challenges. However, regenerating the mesh of design domain and interface tracking in each optimization iteration can be very complex and computationally expensive [35]. ...
Finite element method (FEM) is commonly used with topology optimization algorithms to determine optimum topology of load bearing structures. However, it may possess various difficulties and limitations for handling the problems with moving boundaries, large deformations, and cracks/damages. To remove limitations of the mesh-based topology optimization, this study presents a robust and accurate approach based on the innovative coupling of Peridynamics (PD) (a meshless method) and topology optimization (TO), abbreviated as PD-TO. The minimization of compliance, i.e., strain energy, is chosen as the objective function subjected to the volume constraint. The design variable is the relative density defined at each particle employing bi-directional evolutionary optimization approach. A filtering scheme is also adopted to avoid the checkerboard issue and maintain the optimization stability. To present the capability, efficiency and accuracy of the PD-TO approach, various challenging optimization problems with and without defects (cracks) are solved under different boundary conditions. The results are extensively compared and validated with those obtained by element free Galerkin method and FEM. The main advantage of the PD-TO methodology is its ability to handle topology optimization problems of cracked structures without requiring complex treatments for mesh connectivity. Hence, it can be an alternative and powerful tool in finding optimal topologies that can circumvent crack propagation and growth in two and three dimensional structures.
... The filter functional J filt is based on a spherical filter kernel with radius r κ = 10 nm and is weighted by γ filt = 100. Using a continuation strategy [87] for the grayness term, the penalty parameter is successively increased by the factor 1.3 starting with γ gray = 10 −5 to reach a material distribution where only the materials B (1) , B (2) and B (3) are present. ...
Transparent and conductive thin films find broad application in optoelectronic devices such as touchscreens and solar panels, and are intended to satisfy two opposite properties. On the one hand, these films should appear transparent in the visible light, which means that a large amount of light can pass through such a film. On the other hand, electric energy induced by an applied voltage should be transported with a low electric resistance. Besides the transparent and conductive thin films, we want to consider particle monolayers as another class of photonic nanostructures. The particle monolayers are utilized, for instance, to control the diffuse scattering behavior of photodetectors used in solar cells. This optical property is quantified by the haze factor. Experiments show that the design, which includes both the material composition and the overall shape of the photonic nanostructures, has a noteworthy influence on the performance with respect to the intended purpose. The main objective of this thesis is to optimize the design of such photonic nanostructures with respect to transmission, conductivity and haze factor by changing the material, the shape and the geometry using gradient-based algorithms. Before the individual optimization problems are specified, analytical and numerical solution methods for the involved partial differential equations to determine the optical and electrical properties are discussed. The electromagnetic scattering of a single spherical particle and assemblies of spherical particles is formulated in terms of fundamental solutions of Maxwell's equations, i. e. the vector spherical wave functions. In this context, the order of convergence of dedicated errors is numerically studied with respect to various parameters. In particular, the numerical evaluation of the haze factor for particle monolayers consisting of non-spherical particle is challenging and a suitable numerical solution scheme has been developed. For this purpose, the Finite Element Method and a spectral method based on vector spherical wave functions are combined to a two-stage hybrid simulation scheme in which computationally expensive tasks can be computed in a so-called offline stage. Hence, sophisticated algorithms for the optimization of material, shape and geometry accomplish the gradient-based design optimization of photonic nanostructures.
... The general idea is to find the optimal material distribution of a structure with respect to its design and boundary constraints. However, the main challenge of TO is to provide a design parameterization that leads to a physically optimal design too (Sigmund & Petersson, 1998). ...
Additive manufacturing allows us to build almost anything; traditional CAD however restricts us to known geometries and encourages the re-usage of previously designed objects, resulting in robust but nowhere near optimum designs. Generative design and topology optimization promise to close this chasm by introducing evolutionary algorithms and optimization on various target dimensions. The design is optimized using either 'gradient-based' programming techniques, for example the optimality criteria algorithm and the method of moving asymptotes, or 'non gradient-based' such as genetic algorithms SIMP and BESO. Topology optimization contributes in solving the basic engineering problem by finding the limited used material. The common bottlenecks of this technology, address different aspects of the structural design problem. This paper gives an overview over the current principles and approaches of topology optimization. We argue that the identification of the evolutionary probing of the design boundaries is the key missing element of current technologies. Additionally, we discuss the key limitation, i.e. its sensitivity to the spatial placement of the involved components and the configuration of their supporting structure. A case study of a ski binding, is presented in order to support the theory and tie the academic text to a realistic application of topology optimization.
... 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.
... The entries of the interpolation matrix F are computed using trilinear shape functions and each row of the matrix satisfies a partition of unity property. This interpolation approach serves as a regularization technique, similar to the approach of filtering [55,7], which prevents checkerboarding and other numerical difficulties. Figure 2 illustrates the design interpolation scheme, where the blue analysis mesh is coarsened to create the red design mesh, and the design variables x 1 through x 9 are interpolated from the design variables onto the analysis mesh. ...
... Discretization approach considered here can result in numerical instabilities such as mesh dependence and checkerboard patterns [42]. In order to overcome possible problems, author applies the density filtering method [36], meaning that density within all modeling domain is replaced by filtered density value when calculating mechanical characteristics (2). ...
The method to optimize a topology of 3D continuous fiber-reinforced additively manufactured structures is discussed. The proposed method makes it possible to simultaneously search for density distribution and local reinforcement layup in 3D composite structures of transversely isotropic materials. The approach uses a dynamical systems method to find density distribution, combined with the method for rotation of reinforcement direction to align it in the direction of principal stresses with local minimum compliance. The algorithm is implemented as a built-in material model within Abaqus finite element suite. Both the optimal material density distribution and the distribution of fiber orientation vector are determined for three structural elements used as benchmarks: the bending of simply supported 2D beam under central point load, the loading of 3D cube by vertical load, and the bending of 3D cantilever beam.
... To smooth the element sensitivity numbers across the entire domain, a filter scheme is used that alleviates the problem of mesh-independency and checker-boarding, which result from the sensitivity numbers becoming discontinuous across the element boundaries. The filter scheme is similar to that presented by Sigmund and Petersson [56]; ...
... To achieve a consistent design representation across mesh levels, we use a node-based parametrization. In this node-based parametrization, we utilize separate analysis and design meshes, where the design mesh is generated by coarsening the analysis mesh through TMR, as shown in Figure 2. Design variables are assigned to the nodes of the design mesh and interpolated onto the finer analysis mesh [40]. This linear interpolation process is described by ...
... A large amount of literature exists on preventing checkerboards patterns and mesh dependence. Popular approaches are to restrict the design space placing a constraint on the total perimeter of the structure so that solution exists for the original continuum problem (Haber et al. 1996) or to filter the values of sensitivities (Sigmund and Peterson 1998), which have been used extensively in a significant amount of works. In this work we will adopt a Heaviside projection method (Bourdin 2001), since density filters have demonstrated to be more robust than sensitivity filtering schemes. ...
This work falls within the scope of computer-aided optimal design, and aims to integrate the topology optimization procedures and recent additive manufacturing technologies (AM). The elimination of scaffold supports at the topology optimization stage has been recognized and pursued by many authors recently. The present paper focuses on implementing a novel and specific overhang constraint that is introduced inside the topology optimization problem formulation along with the regular volume constraint. The proposed procedure joins the design and manufacturing processes into a integrated workflow where any component can directly be manufactured with no requirement of any sacrificial support material right after the topology optimization process. The overhang constraint presented in this work is defined by the maximum allowable inclination angle, where the inclination of any member is computed by the Smallest Univalue Segment Assimilating Nucleus (SUSAN), an edge detection algorithm developed in the field of image analysis and processing. Numerical results on some benchmark examples, along with the numerical performances of the proposed method, are introduced to demonstrate the capacities of the presented approach.
... A typical numerical instability in topology optimization is the checkerboard phenomenon. Sigmund and Petersson provided a review on checkerboards, meshdependencies, and local minima in 1998 (Sigmund and Petersson, 1998). One of the standard methods to improve numerical stability is the checkerboard-filtering method, e.g., the mesh-independent filter (16): ...
Topology optimization is one of the most effective tools for conducting lightweight design and has been implemented across multiple industries to enhance product development. The typical topology optimization problem statement is to minimize system compliance while constraining the design space to an assumed volume fraction. The traditional single-material compliance problem has been extended to include multiple materials, which allows increased design freedom for potentially better solutions. However, compliance minimization has the limitations for practical lightweight design because compliance lacks useful physical meanings and has never been a design criterion in industry. Additionally, the traditional compliance minimization problem statement requires volume fraction constraints to be selected a priori; however, designers do not know the optimized balance among materials. In this paper, a more practical method of multi-material topology optimization is presented to overcome the limitations. This method seeks the optimized balance among materials by minimizing the total weight while satisfying performance constraints. This paper also compares the weight minimization approach to compliance minimization. Several numerical examples prove the success of weight minimization and demonstrate its benefit over compliance minimization.
... A fundamental project in the use of TO for architectural design is from Arata Isozaki, in collaboration with structural engineer Matsuro Sasaki for a project called Illa de Blanes at the seaside of Blanes (Costa Brava, Spain) developed in the years 1998(Januszkiewicz 2013. This was one of the first attempts to generate forms obtained from Topology Optimisation algorithms as an architectural form. ...
This chapter illustrates the main approach for a generative use of structural optimization in architecture. Structural optimization is very typical of sectors like mechanical, automotive engineering, while in
architecture it is a less used approach that however could give new possibilities to performative design. Topology Optimization, one of its most developed sub-methods, is based on the idea of optimization of
material densities within a given design domain, along with least material used and wasted energy. In the text is provided a description of TO methods and the principles of their utilization. The process of
topology optimization of micro-structures of cellular materials is represented and illustrated, emphasizing the all-important criteria and parameters for structural design. A specific example is given from the
research at ACTLAB, ACB Dept, Politecnico di Milano, of performative design with lattice cellular solid structures for architecture.
... The mesh independent filter is based on the technique proposed by Sigmund and Petersson (1998) and modifies the sensitivity number of each element based on a weighted average of the element sensitivities (11) in a fixed neighbourhood defined by a minimum radius r min : ...
This paper presents the Matlab implementation of the Sequential Element Rejection and Admission (SERA) method for topology optimization of structures and compliant mechanisms. The lines comprising this code include definition of design domain, finite element analysis, sensitivity analysis, mesh-independency filter, optimization algorithm and display of results. Extensions and changes in the algorithm are also included in order to solve multiple load cases, active and passive elements and compliant mechanisms design. The code is intended for educational purposes and introduces an alternative approach to traditional structural topology optimization algorithms. The complete code is provided in the Appendix.
... The multimodality of an optimization problem depends on its physics, objective, and design variables; however, in some cases, designers need to seek an appropriate initial configuration used to achieve the optimal configuration that has good performance. Ad35 ditionally, the adjustment of artificial parameters, such as the threshold of the filtering scheme [3] or time increment for updating design variables [38], is essential for success- fully obtaining optimal configurations. The tasks, through trial and error, for seeking appropriate initial configurations and adjusting artificial parameters are undesirable work and take a great deal of time in multimodal optimization problems. ...
In this paper, we propose a topology optimization method based on the covariance matrix adaptation evolution strategy (CMA-ES) as a method for solving multimodal structural optimization problems. CMA-ES optimizes level set functions as design variables to minimize the fitness value that is regularized to avoid the ill-posedness of topology optimization using a perimeter constraint. Explicit boundaries between the material and void are obtained using the iso-surface of linearly interpolated level set functions. To show the effectiveness of the proposed method for multimodal structural optimization problems, topology optimization for optical and carpet cloaks is numerically demonstrated. The proposed computational strategy is robust to the settings of the initial configurations, even if the topology optimization problems have multimodal distributions of solutions that include many local minima with insufficient performance, and stably improves the regularized fitness value. The obtained optimal configurations have good performance, and we can obtain them without the trial and error of seeking appropriate initial configurations and adapting strategy parameters.
... In order to ensure the existence of solutions to the topology optimization problem in (1), restrictions on the resulting design need to be introduced [30]. The typical treatment utilizes a sensitivity filtering technique. ...
Topology optimization provides optimized solutions with complex geometries which are often not suitable for direct manufacturing without further steps or post-processing by the designer. There has been a recent progression towards linking topology optimization with additive manufacturing, which is less restrictive than traditional manufacturing methods, but the technology is still in its infancy being costly, time-consuming, and energy inefficient. For applications in automotive or aerospace industries, the traditional manufacturing processes are still preferred and utilized to a far greater extent. Adding manufacturing constraints within the topology optimization framework eliminates the additional design steps of interpreting the topology optimization result and converting it to viable manufacturable parts. Furthermore, unintended but inevitable deviations that occur during manual conversion from the topology optimized result can be avoided. In this paper, we review recent advances to integrate (traditional) manufacturing constraints in the topology optimization process. The focus is on the methods that can create manufacturable and well-defined geometries. The survey will discuss the advantages, limitations, and related challenges of manufacturability in topology optimization.
This paper proposes a new approach for the design of thermal actuators using topology optimization. For the designs based on topology optimization, compliant mechanisms are required to have flexible, stiff, and sustained structures. To achieve this, conventional methods usually consider multi‐loading problems or use spring elements. In this work, a method that only considers one load case and does not use spring elements was developed by introducing an objective function that combines evaluation functions that provide necessary features for the actuators. This way, a structure with the same deformation mode of real phenomena could be derived with a low computational cost and without spring elements. In addition, the stress constraints on the thermal stress, which have not been widely discussed so far, were applied to show the structure with the considered strength characteristics. The study subject is topology optimization of an isotropic thermoelastic model considering a thermo‐mechanical problem. A sensitivity analysis method for the evaluation function required to obtain the optimal configuration was presented, and the validity of the formulated sensitivity was verified in comparison with the finite difference method. Finally, numerical examples for multiple conditions were performed to show that the proposed method can derive the desired optimal structure.
So far, the research on topology optimization against noise has mainly addressed frequency‐domain problems, while time‐domain analysis is also widely used in practical engineering. Nonetheless, the topology optimization work on the latter was rarely reported due to its complexity. This paper presents a new topology optimization scheme of bi‐material curved shell structure to reduce the time‐domain noise generated by transient vibration. A finite element formulation for an eight‐node curved shell element is presented. The Newmark integral method is employed to calculate transient responses, and the obtained results are input into the time‐domain boundary element method to predict transient sound pressure. In the optimization model, the volumetric densities of material in a bi‐material interpolation model constructed by the solid isotropic material with penalization (SIMP) model are chosen as the design variables; the time integral of the squared sound pressure on structural surfaces or prescribed reference points in acoustic medium over a specified time interval of interest is taken as the objective function; and the constraint on material volume is considered. A volume‐preserving Heaviside penalization is introduced to suppress gray elements. Furthermore, the calculation of time‐domain sound radiation sensitivity is transformed into the following two processes: (a) the derivation of transient response based on the Newmark integral method; (b) the derivation of transient sound pressure based on the discrete time‐domain boundary integral equation. Numerical examples demonstrate the validity of the approach proposed in this paper. The influences of volume‐preserving Heaviside penalization, volume constraint, loading position and form, and selection of objective function on the optimal design are discussed. This article is protected by copyright. All rights reserved.
Nanophotonic devices have enabled microscopic control of light with an unprecedented spatial resolution by employing subwavelength optical elements that can strongly interact with incident waves. However, to date, most nanophotonic devices have been designed based on fixed-shape optical elements, and a large portion of their design potential has remained unexplored. It is only recently that free-form design schemes have been spotlighted in nanophotonics, offering routes to make a break from conventional design constraints and utilize the full design potential. In this review, we systematically overview the nascent yet rapidly growing field of free-form nanophotonic device design. We attempt to define the term “free-form” in the context of photonic device design, and survey different strategies for free-form optimization of nanophotonic devices spanning from classical methods, adjoint-based methods, to contemporary machine-learning-based approaches.
Cellular structures in nature have attracted great attention in the field of structural optimization. This paper proposes a Kriging-assisted topology optimization method for design of functionally graded cellular structures (FGCS), which are infilled by smoothly-varying lattice unit cells (LUCs). Specifically, LUCs are depicted by level set functions and the shape interpolation method is employed to generate sample LUCs. Then, one Kriging metamodel is constructed to predict the mechanical properties of LUCs within FGCS, so as to reduce the computational expense involved in finite element analysis of LUCs. Meanwhile, the other Kriging metamodel is created to predict the values of shape interpolation function, and a Kriging-assisted morphological post-process method is put forward to achieve the smooth transition between adjacent graded LUCs. In the proposed method, the effective densities, mechanical properties, and geometrical configurations of LUCs are coupled by Kriging metamodels, so that the multiscale design of FGCS can be realized at a low computational burden by optimizing the distribution of elemental densities, and this also paves the way for design of FGCS with irregular geometries by morphological post-process and geometry reconstruction. Numerical examples are presented to validate the accuracy and effectiveness of the proposed method for design of FGCS. What is more, the design of a pillow bracket with a slightly complex geometry is provided to illustrate the engineering application of the proposed method. The results indicate that the proposed method is effective and universal for designing FGCS with smoothly-varying LUCs.
In recent times, plastics are broadly utilized for car outside and insides trims in any case, with the developing significance of diminishing vehicle weight, plastics application to body and chassis parts is ever on the increment. Within the past the insides and outside trim was glued. With the scope to extend the security and for execute a cogging prepare with reference was built these machines. A few of the applications incorporate profound drawing operations, blanking & punching riveting, stamping & squeezing, powder compacting, etc. Separated from accuracy and precision, pressure driven presses have solid outlines, made of ultrasonically tried steel outlines. Outlines are manufactured from prime steel plates of proper thickness for more noteworthy quality & inflexibility. Additionally, there's distinctive kind of directing course of action to suit the requirements of different applications. The machines are completely robotized, administrator is vital as it were for load/unload the bumpers. Machines offer assistance to fulfill clients with amazing Quality & Security guidelines for manufacturing hydraulic presses in a relatively short span of time. Uncommon reason machines are presently broadly utilized in car industry to manage up with fabricating volume requests with quality and exactness. In this paper, Optimized design is done with the assistance of Bi directional Developmental Basic Optimization Method (BESO). Components are analyzed and rebuilt to meet basic, flexural prerequisites. Optimized plan is focused on to fulfill the adaptability and gigantic alter over necessities for different bumper models to gathering with lessening in weight of basic parts.
This study proposes a topology optimization method for dynamic problems based on the finite deformation theory to derive a structure that can reduce deformations for arbitrary dynamic loads that have large deformations. To obtain a structure that can minimize deformations due to dynamic loading for the isotropic hyperelastic model, the square norm of dynamic compliance is set as the objective function. A sensitivity analysis method of the equation of motion, based on Newmark's 𝛽 method for unknown displacements, is presented in the current study. The analysis is carried out by developing a general design sensitivity equation that can accurately account for the response of the structure to dynamic loading and simultaneously display a high affinity for the general constitutive lawby using the adjoint variable method. The accuracy of the obtained sensitivity is verified by using the finite difference method as a benchmark. Numerical examples are then used to demonstrate the validity of the proposed method. The results show that the proposed method is able to derive optimization results according to the magnitude of the applied load.
In this study, we propose a topology optimization method for dynamic problems to control the deformation of the structure. To derive a structure that minimizes the deformation due to transient loads for an isotropic linear elastic model, the strain energy and the squared norm of dynamic compliance are set as objective functions. The topology optimization method applies a density method based on the RAMP method. In the case of the density method, since a optimal structure is obtained by an optimization algorithm based on the gradient method, it is necessary to formulate design sensitivity equations that can appropriately take into account the target optimization problem. A generalized sensitivity analysis method is proposed by introducing the adjoint method and applying Newmark’s β method, which considers the displacement as an unknown quantity , and considering the equations of motion. Furthermore, the accuracy of the sensitivity is verified by using the finite difference method as a benchmark, and it is shown that the proposed design sensitivity has high accuracy. Finally, as a numerical example, we derive optimal structures for several optimization problems and discuss the optimization problem settings to obtain a structure that can control vibrations. The validity of the proposed method is demonstrated by deriving the optimal structure to control the vibration.
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