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In this paper, chiral metamaterials (CMM) were optimized from conceptual design to fine design with the effective elastic constants unchanged under finite strain. First, through calculation and comparison of examples, the unit cell method was selected to compute the effective elastic properties of the periodic chiral metamaterials under finite strain. Secondly, the conceptual design of chiral metamaterials with prescribed Poisson's ratios under finite strain was realized through density-based and feature-driven topology optimization. Then, the method of moving asymptotes (MMA) was used to solve the optimization problems. Based on the optimal configuration, chiral metamaterials with prescribed Poisson's ratios and Young's moduli under finite strain were carefully designed through shape optimization. Genetic algorithm was used to solve the optimization problem. Finally, the optimal models were fabricated by 3D printing. The optimal design was validated by tensile test results, i.e., the designed chiral metamaterials can maintain effective elastic properties under large deformation, and the invariance of the effective elastic properties depends on the nonlinearity of the flexible chiral metamaterials. [ABSTRACT FROM AUTHOR]

Despite a solid theoretical foundation and straightforward application to structural design problems, 3D topology optimization still suffers from a prohibitively high computational effort that hinders its widespread use in industrial design. One major contributor to this problem is the cost of solving the finite element equations during each iteration of the optimization loop. To alleviate this cost in large-scale topology optimization, the authors propose a projection-based reduced-order modeling approach using proper orthogonal decomposition for the construction of a reduced basis for the FE solution during the optimization, using a small number of previously obtained and stored solutions. This basis is then adaptively enriched and updated on-the-fly according to an error residual, until convergence of the main optimization loop. The method of moving asymptotes is used for the optimization. The techniques are validated using established 3D benchmark problems. The numerical results demonstrate the advantages and the improved performance of our proposed approach. [ABSTRACT FROM AUTHOR]

In this work, an adaptive T-spline finite cell method is developed for structural shape optimization based on the finite cell method (FCM). This method has the advantage of using local T-mesh refinement on the uniform B-mesh to ensure a good computing accuracy in the shape optimization process. To do this, we first carry out a pre-analysis of the structure using a uniform coarse mesh with relatively low computational effort. Then, cells for local refinements are identified by adopting the stress gradient criterion. Subsequently, an adaptive quadtree-based refinement scheme is employed to generate the adaptive T-mesh which preserves the linear independence of the corresponding T-spline shape functions. Finally, the shape optimization framework is correspondingly established, and the proposed adaptive refinement scheme is validated by numerical examples. Its efficiency is fully demonstrated by shape optimization procedures of mechanical structures. [ABSTRACT FROM AUTHOR]

Thermal stress is an important design factor that will influence structural responses and cause local warping deformations. In this paper, such temperature variation effect is considered in the structural optimization and the shape preserving design approach is extended into thermo-elastic problems to prevent local thermal damages. Based on the weak-coupled thermo-elastic system, nonlinear structural responses at large thermo-mechanical loads are accurately analyzed. The complementary elastic work is minimized to obtain a reasonable rigid structure. Corresponding integrated deformation energy is utilized to calculate the warping deformation accumulated in the incremental thermo-mechanical loading process. Shape preserving effect is then achieved by an additional constraint on the local deformation energy. Through the adjoint method, sensitivity analysis is derived in the coupled field with design-dependent heat conduction and geometrical nonlinearity. In the numerical implementation, an energy interpolation scheme is applied to circumvent numerical instability in low stiffness regions and further modified for multi-material design. Optimization results show that local distortions in thermo-elastic structures are effectively eliminated by the proposed shape preserving design approach. [ABSTRACT FROM AUTHOR]

Summary This work develops a concurrent shape and topology optimization method to solve the design problems of which the pressure loads are applied on designable boundaries. Implicit B‐spline curves including open B‐spline and closed B‐spline are introduced in the form of level‐set functions to act as common design primitives for the description of both moving pressure boundary and inner topology boundaries of a structure. Boolean operations are applied to describe mathematically the shape and topology variations. An artificial nondesignable offset domain of small bandwidth is introduced along the moving pressure boundary to ensure reasonably the imposition of pressure loads on solid materials. The pressure loads are numerically discretized onto the control nodes of finite cell method used for structural analysis with fixed mesh. Sensitivities of design‐dependent pressure loads with respect to both shape and topology design variables are explicitly derived. Numerical problems are finally solved to illustrate the presented method. [ABSTRACT FROM AUTHOR]

Subparts of load carrying structures like airplane windows or doors must be isolated from distortions and hence structural optimization needs to take such shape preserving constraints into account. The paper extends the shape preserving topology optimization approach from simple linear load cases into geometrically nonlinear problems with practical significance. Based on an integrated deformation energy function, an improved warpage formulation is proposed to measure the geometrical distortion during large deformations. Structural complementary elastic work is assigned as the objective function. The average distortion calculated as the integrated deformation energy accumulated in the incremental loading process is accordingly constrained to obtain warpage control. In the numerical implementation, an energy interpolation scheme is utilized to alleviate numerical instability in low stiffness regions. An additional loading case avoids isolation phenomena. Optimization results show that shape preserving design is successfully implemented in geometrically nonlinear structures by effectively suppressing local warping deformations. [ABSTRACT FROM AUTHOR]

In this paper, the inherent problem of the so-called parasitic effect of low density region caused by design-dependent loads is investigated for topology optimization. A CBS (closed B-splines)-based method is developed to solve efficiently the problem in the way of shape optimization. Compared to the standard density method, design variables are unattached to the finite element model and defined by control parameters dominating the parametric equation of the CBS. As a result, design-dependent loads associated with the material layout of a structure are made change indirectly by the boundary variation of the structure. To favour structural reanalyses and sensitivity analysis, the implicit form of the CBS, i.e., level-set function (LSF) is used in conjunction with the fixed computing grid of finite cell method (FCM). A variety of design-dependent body loads is considered for the mathematical formulation of the CBS-based method. Typical examples are given to illustrate that the CBS-based method is free of low density regions in the optimized topology owing to the proper definitions of design variables. [ABSTRACT FROM AUTHOR]

A moving bounds strategy is proposed to implement simultaneous shape optimization of curved shell structures and openings. Design variables related to the hole shape are constrained in a planar reference domain by the moving bounds whose values are adaptively updated as functions of design variables related to the surface by an arc-length rule. It is shown that this strategy is essential not only to ensure the geometric consistence in the simultaneous design process but also to hold the shape-preserving of the mapped FE mesh from reference domains. Numerical results are presented to validate the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a feature-driven topology optimization method is developed. This is the first study on layout design of multiple engineering features using level-set functions (LSFs) and Boolean operations. The novelty of this work is threefold. First, multiple engineering features of arbitrary shape are considered as basic design primitives and topology variation is achieved via the layout and shape optimization of the features. Kreisselmeier–Steinhauser (KS) function constructed by means of Boolean operations is adopted as the LSF, which uses an implicit function to ensure a smooth description and topological changes of basic features and the whole structure. Second, using a modified Heaviside function to smooth the void–solid material transition over a fixed computing mesh, a narrow-band domain integral scheme is developed for the efficient sensitivity analysis. Third, the gray material distribution regions at the feature-connecting portions are analyzed and the underlying reason for that is traced to the non-equidistant distribution of level-set contours of specific features. To avoid the gray regions, an approximated signed distance function is proposed to regularize the LSF and KS function. The bounded normalization property of the KS function is highlighted for its construction with the signed distance functions or normalized first-order approximations. Numerical examples are finally tested to demonstrate the validity and merits of the proposed feature-driven topology optimization for complicated design problems. [ABSTRACT FROM AUTHOR]

In this work, attempts are made for the first time to carry out structural topology optimization under rotating load caused by the centrifugal force of mass unbalance in rotating machinery. As is known, a rotating load can be decomposed into two harmonic excitations in two orthogonal directions with phase difference, the considered topology optimization problem is handled under two associated harmonic excitations. Due to the presence of phase difference, it is however more complicated than existing works related to harmonic excitations in phase. As the structural response is time-dependent, two topology optimization formulations are thus established to minimize the dynamic compliance and the maximum displacement amplitude of the loaded node over a period, respectively. Meanwhile, harmonic analysis of the structure is carried out by means of the full method and the related sensitivity analysis is presented for both optimization formulations as well. Finally, validities and effects of the proposed optimization formulations are illustrated and compared through typical numerical tests. It is found that both formulations have the ability to reduce the structure vibration caused by the rotating load and similar configurations are obtained after optimization in most tests. [ABSTRACT FROM AUTHOR]

Structural topology optimization related to dynamic responses under stationary random force excitation is investigated in this paper. It is shown that the commonly used Complete Quadratic Combination method (CQC) in previous optimization work is not only computationally expensive but also results in non-convergent design pattern due to the low computing accuracy of random responses for large-scale problems. To circumvent these difficulties, an efficient and accurate optimization procedure integrating the Pseudo Excitation Method (PEM) and Mode Acceleration Method (MAM) is introduced into the dynamic topology optimization. In this framework, random responses are calculated using the PEM to ascertain a high efficiency over the CQC. More importantly, the accuracy of random responses is improved indirectly by solving the pseudo harmonic responses involved in the PEM with the help of the MAM. Numerical examples fully demonstrate the validity of the developed optimization procedure and its potential applications in practical designs. [ABSTRACT FROM AUTHOR]

This paper addresses extended shape optimization problems where structural supports, i.e., Dirichlet boundaries and free boundaries are simultaneously optimized. Unlike traditional FEM, the weighted B-spline finite cell method (FCM) is applied as structural analysis tool and combined with the level-set function (LSF) to take into account Dirichlet boundary condition (DBC) automatically through penalization of the displacement field. The proposed shape optimization method achieves a comprehensive integration of fixed grid, B-spline shape function and level-set function. As both the structure and Dirichlet boundaries are described in the form of LSF, any modification of Dirichlet boundaries can be made in a straightforward way as easily as free boundaries by changing continuous design variables. Meanwhile, the computing accuracy is ensured within the framework of fixed grid owing to the quadtree refinements of boundary cells. Stress related shape optimization problems are finally solved to demonstrate the merit and validity of the proposed optimization method in dealing with shape optimization of Dirichlet boundaries. [ABSTRACT FROM AUTHOR]

In this work, two different formulations that minimize the mean compliance and elastic strain energy of a structure as the objective function are comparatively investigated for topology optimization of thermoelastic problems. As thermal load is a design-dependent load compared with mechanical load, the particularity of the problem and the difference between both formulations are highlighted in terms of sensitivity analysis and related optimization results based on numerical tests. The concept of load sensitivity is used to interpret quantitatively the underlying influences of thermal and mechanical loads upon optimized results in both formulations. [ABSTRACT FROM AUTHOR]

Highlights: [•] A material perturbation method using fixed mesh is developed for shape optimization. [•] The method has the advantage of avoiding sophisticated velocity field calculation. [•] Material properties are related to the shape perturbation via the domain function. [•] Stress corrections are carried out for sensible elements in local coordinate systems. [•] It is shown that stress corrections have a remarkable effect on the stress accuracy. [ABSTRACT FROM AUTHOR]

• A CAD-oriented topology optimization method has been developed. • Two schemes have been proposed for implicit representation of CAD model. • Numerical examples have fully demonstrated the validity of the method. • The performance of the method has been investigated. In this paper, we develop a CAD-oriented topology optimization method incorporating directly the CAD model into the feature-driven optimization (FDO) framework. Practically, any free-form design domain model (FDDM), defined by the CAD model in stereolithography (STL) file format, is first represented by a level set function employing a radial basis function (RBF) interpolation technique. Feature-based topology variation model (TVM) consisting of design and non-design features is then constructed, with the former affording design flexibility whilst the latter preserving the components for engineering requirements. By coupling the FDDM with the TVM via Boolean intersection operation, the topology optimization framework is constructed, and each analysis problem is solved by the finite cell method (FCM). A boundary integral scheme is implemented for efficient sensitivity analysis. It is shown that the developed method shall pose almost no restriction on the expertise of modeling and meshing related to the conventional finite element method. Several examples of increasing complexities are dealt with to demonstrate the effectiveness and the generality of the proposed method. [ABSTRACT FROM AUTHOR]

An adaptive bubble method is presented in this paper for shape and topology optimization design of stiffest structures and compliant mechanisms. With the aim of overcoming the shortcomings of the traditional bubble method and retaining all its advantages, the proposed method achieves a comprehensive integration of topological derivative, implicit model and fixed grid. The topological derivative is devoted to the iterative positioning of new holes inside the design domain. Each of the holes is described by smoothly deformable implicit curve (SDIC) characterized by very few parameters and high deformation capacity. Since the structural model could be implicitly constructed using the SDICs, the finite cell method (FCM) is applied to facilitate the mechanical analysis of the implicit model with fixed Eulerian mesh. Two new concepts, i.e., the topological derivative threshold and the total influence region of inserted holes, are further introduced to adaptively control the numbers and positions of new holes so that the numerical stability of topology optimization could be well guaranteed. It is shown that the adaptive bubble method no longer needs the mesh updating process and can perform the merging and separating operations of holes straightforwardly. Numerical examples concerning the stiffest structure design and the compliant mechanism synthesis are tested to demonstrate the universality of the present method, especially its capability of being integrated within CAD systems directly without post-processing. • A novel bubble method is presented to overcome shortcomings of the traditional one. • Two new concepts are proposed to adjust the inserting number and positions of holes. • The SDIC with high deformation capacity is formulated to describe the inserted holes. [ABSTRACT FROM AUTHOR]

In this work, a novel design approach is developed to carry out shape and topology optimization of shell structures simultaneously. This approach effectively integrates the IsoGeometric Analysis (IGA) method, the Adaptive Bubble Method (ABM) and the Finite Cell Method (FCM) to make full use of their respective advantages in shell structure optimization. To be specific, the IGA method using Non-Uniform Rational B-Splines (NURBS) elements is adopted to analyze the shell structure and realize shape optimization with merits of exact shell surface representation geometrically and rotation-free shell formulation physically; the ABM is utilized to insert deformable holes adaptively in the parametric domain of the NURBS surface for topology optimization of the shell structure; the FCM is employed to facilitate the optimization process by simplifying significantly the discrimination and the numerical integration processes of trimmed elements involved in fixed-mesh analysis of the holed shell structure. Based on this integrated design approach of IGA/ABM/FCM, shape optimization can be implemented by directly optimizing the control-point coordinates of the NURBS surface in the physical space, while topology optimization could be carried out simultaneously in the parametric domain as easily as for 2D planar structures. Finally, representative examples are investigated to demonstrate the effectiveness and advantages of the proposed design approach. • An integrated design approach is developed for shell shape and topology optimization. • Tedious operations involved in the merging of intersecting holes and TSA are avoided. • The high-precision boundary integral for analytical sensitivity analysis is realized. • The influences of shell curvature and thickness on optimized topologies are studied. [ABSTRACT FROM AUTHOR]

There are renewed interests in grid-stiffened composite structures due to the high structural stability and damage tolerance enabled by multiple intersected stiffeners. The limited design space of repeated straight grid patterns has been remarkably expanded by introducing non-uniform curved grid-stiffened composite structures (NCGCs). In this work, curved stiffener layout optimization for NCGCs with a central cutout is investigated to maximize the critical buckling load under a given weight constraint. The design framework is developed by calling ABAQUS for simulating stiffened models and using numerical sensitivities in order to adopt gradient-based optimization algorithm, where the computational time depends not only on simulation but also the number of design variables. To increase the efficiency, an improvement is proposed based on globally convergent method of moving asymptotes (GCM), as one of the most popularly used structural optimization algorithms, by relaxing the conservativeness (CR-GCM) with an additional relaxation factor and an adaptive limit of the iteration number. Numerical examples show that optimal curved stiffeners around the cutout have the ability to remarkably increase the structural buckling load and the proposed CR-GCM demonstrates the ability of better balancing the computational efficiency and convergence in the optimization process. [ABSTRACT FROM AUTHOR]

• A new data-driven topology optimization (DDTO) framework is proposed. • Sensitivity analysis for discrete noisy material data sets is developed. • Verification of the new DDTO framework by truss structure designs. The application of structural topology optimization with complex engineering materials is largely hindered due to the complexity in phenomenological or physical constitutive modeling from experimental or computational material data sets. In this paper, we propose a new data-driven topology optimization (DDTO) framework to break through the limitation with the direct usage of discrete material data sets in lieu of constitutive models to describe the material behaviors. This new DDTO framework employs the recently developed data-driven computational mechanics for structural analysis which integrates prescribed material data sets into the computational formulations. Sensitivity analysis is formulated by applying the adjoint method where the tangent modulus of prescribed uniaxial stress-strain data is evaluated by means of moving least square approximation. The validity of the proposed framework is well demonstrated by the truss topology optimization examples. The proposed DDTO framework will provide a great flexibility in structural design for real applications. [ABSTRACT FROM AUTHOR]

Shape optimization involving finite element analysis in engineering design is frequently hindered by the prohibitive cost of function evaluations. Reduced-order models based on proper orthogonal decomposition (POD) constitute an economical alternative. However the truncation of the POD basis implies an error in the calculation of the global values used as objectives and constraints which in turn affects the optimization results. In our former contribution (Xiao and Breitkopf, 2013), we have introduced a constrained POD projector allowing for exact linear constraint verification for a reduced order model. Nevertheless, this approach was limited a to relatively low numbers constraints. Therefore, in the present paper, we propose an approach for a high number of constraints. The main idea is to extend the snapshot POD by introducing a new constrained projector in order to reduce both the physical field and the constraint space. This allows us to search for the Pareto set of best compromises between the projection and the constraint verification errors thereby enabling fine-tuning of the reduced model for a particular purpose. We illustrate the proposed approach with the reduced order model of the flow around an airfoil parameterized with shape variables. [ABSTRACT FROM AUTHOR]

Abstract: While performing numerical simulations for design optimization, one of the major issues is reaching a good compromise between accuracy and computational effort. Simulation methods such as finite elements, finite volumes, etc. provide ‘high fidelity’ numerical solutions at a cost which may be prohibitive in optimization problems requiring frequent calls to the discretized equations’ solver. The Proper Orthogonal Decomposition (POD) constitutes an economical and efficient option to decrease the cost of the solution; however, the truncation of the POD basis implies an error in the calculation of the global quantities used as objectives and optimization constraints, which in turn might bias the optimization results. Our idea is thus to improve the snapshot POD by means of a reduced basis constructed to provide an exact interpolation of the quantities of interest obtained by integration of the ‘physical’ fields. To this end, we reformulate the POD as a minimization problem where the desired properties are expressed as a set of constraints impacting the calculation of both the modes and the coefficients. The main contribution of this paper is to provide a detailed mathematical justification for our constrained variant of the POD, including a graphical interpretation of the proposed approach. The constrained POD method is then applied to the problem of representing the pressure field around a 2D wing, and is compared with the traditional POD. [Copyright &y& Elsevier]

In this paper, we propose a B-Spline Offset Feature (BSOF) for feature-driven topology optimization. The feature is inspired from the geometric morphology of worms characterized by an elongated cylindrical, bilaterally symmetric, and well-segmented body along a highly deformable directrix. B-spline is used to define the directrix with its outward and inward offsets outlining the BSOF. This definition endows the BSOF with high deformability such as self-folding and self-intersection. To construct the implicit level-set function (LSF), the BSOF is decomposed into a sequence of segments whose LSFs can easily be constructed by Kreisselmeier–Steinhauser (KS) function. Regular or irregular shaped segments are identified by the deformation gradient. Irregular segments involving geometrical singularities, e.g. distorted and concave segments, are replaced with convex hulls. The stiffest structure design and compliant mechanism synthesis are studied to illustrate the effectiveness and advantages of the proposed BSOF for topology optimization. • Bio-inspired B-Spline Offset Feature (BSOF) is proposed for topology optimization. • The BSOF retains high deformability of self-folding and self-intersection. • The feature segmentation method is developed to construct the LSF of BSOF. • Convex hull replacement of irregular segments can stable the optimization process. [ABSTRACT FROM AUTHOR]

This paper presents an efficient structural optimization approach for shape memory alloys (SMAs) with respect to fatigue. In the proposed method, a nonparametric shape optimization approach based on optimality criteria is applied to change the structural configuration. The design optimization framework is accomplished by relying on commercially available software and tools. In addition, both low- and high-cycle fatigue criteria are used to compute the fatigue factor at each material point, indicating its degree of safeness with respect to fatigue. Meanwhile, a 3D constitutive model is utilized to predict, with good accuracy, the stabilized thermomechanical stress state of a SMA structure subjected to multiaxial nonproportional cyclic loading. Finally, numerical examples are tested to demonstrate the effectiveness of the proposed method. Unlabelled Image • A non-parametric shape optimization approach based on optimality criteria is applied to change the structural configuration. • Both low- and high-cycle fatigue criteria are used to compute the fatigue factor. • The design optimization framework is accomplished by relying on commercially available software and tools. [ABSTRACT FROM AUTHOR]