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As studied in previous works, dynamic compliance presents widespread antiresonances in its spectrum, which could trap gradient-based topology optimization for minimum dynamic compliance and cause the optimization procedure to converge prematurely. In order to solve this problem, a novel method for predicting the frequencies that correspond to points of antiresonance in dynamic compliance spectrum is presented in this paper. By leveraging this eigenvalue formulation method, a strategy for introducing antiresonant frequency constraint is developed to solve one-material topology design for minimum dynamic compliance under high-frequency (above the first resonance of the initial design) excitations. In order to facilitate the exploitation of antiresonances, accurate track and eigenvalue sensitivity analysis for the prescribed antiresonant frequency are also discussed in detail. Numerical results show that the proposed method can achieve well-defined topologies with excellent dynamic performance. [ABSTRACT FROM AUTHOR]

Beam features of variable cross-sections are developed in this paper for topology optimization based on feature-driven method. As beam structures are extensively used and mostly hold standard cross-sections in engineering practice, this work is aimed at the simultaneous design of beam layout and beam cross-section. The H-shaped cross-section is taken as general design primitive owing to its flexible evolution into T-shaped, C-shaped, and L-shaped cross-sections. Meanwhile, it is represented by means of the level-set functions (LSFs) with design variables consisting of the beam position, rotation angles, cross-section sizes, and cross-section configurations. Through the continuous change of defined cross-section configuration variables, different specifications of standard cross-sections with normed size values can be obtained and selected from the parts library. Fixed-mesh technique is also implemented for structural analysis and sensitivity analysis. Several numerical examples are presented to demonstrate the effectiveness and merits of the proposed method. [ABSTRACT FROM AUTHOR]

This work presents a topology optimization method for prestressed cable-truss structures considering geometric nonlinearity. Based on the large deformation kinematics theory, we establish a multi-node cable element model, which can fully take into account the geometric nonlinearity caused by external loads and prestress. Besides, nonlinear hyperelasticity principles are used to deal with cable materials’ unidirectional stress characteristics. To consider the impact of prestress on the global structural stiffness, we carried out the rebalancing of prestress and construct an objective function of prestress-modified compliance. Then, we equate the cable-truss structure as a particular two-phase material structure and realize the material interpolation based on a discrete material optimization-like method. Finally, we perform the adjoint sensitivity analysis of the objective function and solve the optimization problem through the gradient-based algorithm. Therefore, we establish a topology optimization method for prestressed cable-truss structures with stiffness as the objective, continuous density/size parameters as variables, and mass fraction as constraints. This method’s feasibility and reliability are demonstrated in 2D and 3D numerical examples. [ABSTRACT FROM AUTHOR]

In this work, a novel optimization method is proposed to pursue high-performance 3D-braided composite structure under thermo-mechanical loads, which optimizes structural topology and braiding parameters concurrently. To realize such design under affordable computational cost, we decompose the optimization method into offline and online stages. In the offline stage, a parameterized geometric model controlled by two braiding parameters, i.e., fiber volume fraction and braiding angle, is established to characterize the microstructure of composite. Subsequently Energy-based Homogenization Method is applied to calculate equivalent composite properties including elastic tensor, thermal conductivity tensor, and Coefficient of Thermal Expansion. A surrogate model based on Radial Basis Network is established to map braiding parameters to equivalent material properties. In the online stage, the surrogate model is integrated into Rational Approximation of Material Properties to build a systematic design scheme for structural topology and braiding parameters. Taking manufacturability into account, the proposed method is combined with stiffener layout design to obtain easy-to-manufacture braided composite structures. Finally, several numerical examples are provided to demonstrate the effectiveness of the proposed optimization method, indicating that braiding parameters have essential impacts on the composite structural design and performance. [ABSTRACT FROM AUTHOR]

Shape memory alloys (SMA) are an ideal class of metallic materials for reusable energy dissipation structures because of pseudo-elasticity. This paper presents a density-based topology optimization framework for the design of SMA structures utilizing pseudo-elastic behaviors to dissipate large amounts of energy under prescribed design constraints. A phenomenological constitutive model is adopted to accurately simulate the mechanical behaviors of SMA, and the corresponding material interpolation scheme is developed via SIMP method. Numerical instability caused by excessive distortion of low-density elements is alleviated by super element method. The degree of phase transformation, which is related to the energy dissipation, is characterized by strain energy and end compliance. Sensitivities are derived via adjoint method. A number of optimized simple-supported beam structures and 2D lattice structures with different energy dissipation performance and stiffness capacity are tailored. In addition, the load dependency and initial design dependency for the optimization of SMA energy dissipation structures are discussed. [ABSTRACT FROM AUTHOR]

This paper presents a MATLAB code implementation of the Adaptive Bubble Method (ABM) published by Cai and Zhang (2020) for topology optimization. The ABM has the main feature of inserting deformable holes adaptively into the design domain to reflect the designer’s motto that “The art of structure is where to put the holes.” This method consists of three modules: (i). implicit description of holes with the closed B-spline (CBS) curve possessing high deformability; (ii). determination of optimal insertion points in each iteration using the topological derivative combined with a matrix operation-based searching scheme; (iii). fixed grid-based finite element analysis (FEA) using the ersatz material model. Representative numerical examples are tested to illustrate the implementation of the ABM. A compact 168-line MATLAB code is provided in the Appendix and explained in detail for educational purpose. [ABSTRACT FROM AUTHOR]

This paper is aimed to develop a load allocation topology optimization approach to meet the design requirement of thermo-elastic structures in engineering practice. To formulate the optimization problem, load allocation constraint is proposed by constraining the ratio of considered reaction forces to a proper range. The average displacement at the prescribed region is taken as the objective function. Based on the adjoint method, sensitivity analyses of the average displacement and the load allocation constraint are carried out and used in combination with the gradient-based algorithm. Numerical examples are presented to demonstrate the validity of the proposed approach. Comparisons are made with solutions of standard topology optimization to highlight the coupling effect between the load allocation constraint and the temperature field. [ABSTRACT FROM AUTHOR]

In this paper, a B-spline parameterized density method is proposed to simultaneously optimize the structural topology and the build direction for additive manufacturing (AM). B-spline parameterization is used to represent the structural density field together with Heaviside projection and Boolean operation. Two constraints and a novel filtering scheme relevant to AM are presented: The overhang angle constraint constructed based on the density gradient with build direction included as design variables. The boundary gradient constraint constructed for possible V-shaped cross-section features of 3D problems in various build directions. The gradient filtering scheme adopted to eliminate the V-shaped areas by computing the global overhang angle at all structural boundaries. Representative 2D and 3D problems are dealt with to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Radar cross section (RCS) reduction for cavities is essential in flight vehicle design. As a conventional method to reduce the RCS, coating radar absorbing materials has been widely employed in engineering. Nevertheless, radar absorbing coatings (RAC) will additionally increase the structural weight. In this paper, a topology optimization approach is introduced for the layout design of RAC on the inner cavity walls. The objective of this problem is to minimize the mean value of RCS under the prescribed incident angles. A SIMP-like model is employed to represent the relative impedance of areas of intermediate density. The design variable is iteratively updated during the optimization process using a gradient-based algorithm. The RCS of the cavity is computed by the iterative physical optics method, which is utilized for the subsequent analytical gradient derivation. The validity of the proposed method is demonstrated by optimizing the RAC layout of two different shaped cavities. In both numerical examples, when optimizing the RCS in both planes with a weight of 1:1 and a volume fraction of 50%, the highest RCS loss rate in both horizontal and pitch planes is 18.02% and the lowest is only 6.89%. The optimization results indicate that the proposed method can be employed as a design procedure to consider both weight cost and cavity RCS reduction when coating the absorbing materials, instead of the classical experience-based RAC distributions. [ABSTRACT FROM AUTHOR]

In this work, we develop a topology optimization method for linear elastic contact problems with maximum contact pressure constraint. First, the Kreisselmeier–Steinhauser (KS) function is adopted as an aggregated measure of the maximum contact pressure over specific contact regions. Then, the maximum contact pressure constraint is introduced into the standard volume-constrained compliance minimization problem and formulated in the framework of B-spline parameterization method. Two geometric constraints are further extended to suppress the intermediate densities and control minimum length scale. The adjoint method is employed for deriving design sensitivities analytically. Finally, both frictionless and frictional problems are tested to demonstrate the effectiveness of the proposed method. It is shown that the maximum contact pressure can be effectively controlled using the contact pressure constraint and thus avoiding the concentration of contact pressure. The influence of maximum contact pressure constraint on the optimization result is discussed in comparison with the standard maximum stiffness design. Effects of friction behavior upon optimized results and contact pressure are also highlighted. It concludes that the maximum contact pressure can be reduced at the cost of the structural stiffness. [ABSTRACT FROM AUTHOR]

Stiffener layout optimization of complex surfaces is fulfilled within the framework of topology optimization. A combined parameterization method is developed in two aspects. One is to parameterize the material distribution of the stiffener layout by means of B-spline. The other is to build the mapping relationship from the known 3D surface mesh of the thin-walled structure to its parametric domain by means of mesh parameterization. The influence of mesh parameterization upon the stiffener layout is discussed to reveal the matching issue of the combined parameterization. 3D complex surfaces represented by the triangular mesh can be dealt with even though analytical parametric equations are not available. Some numerical examples are solved to demonstrate the direct advantage and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The electromagnetic performance of flight vehicles has been attracting considerable interest in recent years. As one of the important indicators of this performance, radar cross section (RCS) has been extensively studied. A significant limitation on the RCS of flight vehicles lies in the fact that tiny gaps will transform into steps during structural deformation, where the local RCS will be significantly affected. This study investigates a topology optimization method to minimize the RCS of step structures when subjected to external loads. To ensure that the structure discretized by shell elements in the mechanical finite element analysis can still simulate the wave reflection of step structures, artificial step elements are defined to simulate the generation of steps during structural deformation. Then, the conventional method of moments is utilized to calculate the RCS. To implement the gradient-based optimization, the sensitivity of RCS with respect to the design variables is derived analytically by the adjoint method. An extended compliance-based topology optimization is performed to determine a stiffness constraint. Then, together with a structural volume limitation, the average RCS of target incident angles will be set as the object to obtain the structural topology. The superiority compared with the general structural design method and the validity have been demonstrated with numerical examples. [ABSTRACT FROM AUTHOR]

Structural stability has attracted increasing attention in topology optimization because of the buckling effect under compression load. In this work, the feature-driven optimization method is developed for structural topology optimization involving buckling constraints. First, the finite cell method is extended to linear buckling analysis. A stress relaxation strategy is proposed to effectively remove the pseudo-buckling modes from low-density regions in combination with the adaptive quadtree/octree-based integral scheme for the geometric stiffness matrix. Then, the feature-based topology variation model is constructed to drive topology optimization. The boundary integral scheme developed in our previous work is adapted to the sensitivity analysis of the buckling load factor, which effectively avoids the time-consuming domain integral. Finally, the influences of solid and void feature definition, initial layout, number of design features, and minimum feature size are systematically studied. The advantages of the present method are demonstrated with the help of numerical examples. [ABSTRACT FROM AUTHOR]

In this work, a novel concurrent optimization framework based on multiscale finite element method is proposed to pursue superior-performance hierarchical structures with parameterized lattice. Distinguished with conventional homogenization-based optimization method, this framework gets rid of the assumption of scale separation while optimizing hierarchical structures. Firstly, a multi-parameter description is presented to generate a family of parameterized lattice cells that share similar geometric features. In order to balance structural performance and computational cost, we decompose the optimization framework into offline and online stages. In the offline stage, the equivalent stiffness matrix of lattice is calculated with the help of multiscale finite element method. A surrogate model based on general regression neural network (GRNN) is established to map lattice parameters to equivalent stiffness matrix. Besides, the discrete topological variables are introduced to determine the distribution of lattice. Discrete material optimization (DMO) model is employed to integrate such two kinds of design variables into a material interpolation model. Hence, the concurrent optimization framework can optimize the macroscopic distribution and their spatially varying lattice configurations simultaneously at an affordable computation cost. Finally, numerical examples are presented to illustrate the effectiveness of the proposed concurrent optimization framework, indicating that lattice size, relative density, and configuration have essential impacts on the hierarchical structural design and performance. [ABSTRACT FROM AUTHOR]

High-resolution additive manufacturing (AM) facilitates engineering applications of cellular lattices with complex geometries. The unique layer-by-layer process of AM causes the mechanical performance of as-fabricated lattice structures to deviate from the ideal value. In this study, the AM process-induced material anisotropy and geometric inaccuracy are integrated into the mechanical evaluation and topology optimization of lattice structures. Specifically, effective mechanical properties of typical strut-based and shell-based lattices are evaluated by the homogenization method combined with the process-induced features. The equivalent stiffness parameters of ideal and as-fabricated lattices are integrated into the topology optimization algorithm to adaptively obtain corresponding macrostructures. Based on the knowledgebase of unit cells and optimized results, the threshold of the variation for microstructure is extracted to guide design and process parameter selection. The proposed design guidelines and evaluation-adjustment framework can be widely applied to various lattices fabricated by AM technologies. [ABSTRACT FROM AUTHOR]

Thin-walled shell structures are widely used in aeronautical and aerospace engineering. This paper develops an effective B-spline parameterization method for stiffener layout optimization of shell structures. Height variables are defined by B-spline control parameters to characterize the stiffener layout reinforcing the shell structure. A continuous height field is subsequently generated via B-spline and basis functions. In view of possible curvatures of shell structures, the height field is projected from parametric space onto the shell structure by means of the parametric mapping. In this work, the finite element method is adopted with the solid-shell coupling method used for structural analysis. Pseudo-densities associated with solid elements are determined based on the B-spline parameterization and Heaviside function. Several numerical examples are dealt with to demonstrate the proposed method. Compared with the standard density-based method, the proposed method produces checkerboard-free design results with a clear layout and naturally avoids overhang stiffeners. [ABSTRACT FROM AUTHOR]

Multidisciplinary topology optimization (TO) for additive manufacturing (AM) exhibits unprecedented design and manufacturing freedom as an effective lightweight methodology. In order to fully utilize the AM advantages of materialized arrangement in spatial and voxel-by-voxel digitalized printing, multidisciplinary TO for AM requires a full understanding of the process-structure-property-performance (PSPP) relationship to establish the quantitative correlation of part property and multiple process factors, including laser-, process-, and material-related parameters. This study incorporates the PSPP relationships of laser sintered material with topology optimizer so as to provide a guideline for TO collaborative AM design. The proposed workflow involves mapping functions of PSPP relationships via a data-driven approach and then optimizes the multiple process parameters and structural topology by the gradient-based algorithm. Compared with conventional optimization approaches, three case-dependent optimizations demonstrate the effectiveness of the proposed approach yielding encouraging improvements of lightweight performance. The multidisciplinary TO can be generally applied to types of AM technologies, establishing a strong linkage of physical product and digital design. [ABSTRACT FROM AUTHOR]

This work extends B-spline parameterization method to topology optimization of elastic contact problems. Unlike the traditional density-based method, design variables directly refer to the control parameters of the B-spline. A continuous pseudo-density field representing the material distribution over the concerned design domain is constructed by means of B-spline parameterization and then discretized onto the finite element (FE) mesh. The threshold projection is further introduced to regularize the B-spline pseudo-density field for the reduction of gray areas related to the local support property of B-spline. 2D and 3D frictionless and frictional problems are solved to demonstrate the effectiveness of the proposed method. Results are also compared with those obtained by the traditional density-based method. It is shown that the B-spline parameterization is independent of the FE model and suitable to deal with contact problems of inherent contact nonlinearity. The optimized configuration with refined details and smoothed boundaries can be obtained. [ABSTRACT FROM AUTHOR]

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]

To date, sealing structures have been widely used in aerospace vehicles. This paper presents a normalized and systematic procedure realizing lightweight design for a square bolt-flange sealing structure of a rocket engine. To measure the sealing behavior, the contact force on the sealing interface is obtained by a nonlinear analysis of node-to-node contact. The topology optimization is established to achieve sufficient contact forces in the sealing area by acquiring the Kreisselmeier-Steinhauser (KS) function to condense multiple contact force constraints into a global sealing constraint. A three-field projection scheme is also adopted to obtain a clear result and a semi-analytical perturbation scheme is proposed to realize the sensitivity calculation. Then, the square flange cover is optimized by the proposed optimization framework and a lightweight design with a clear load-carrying path is obtained. The optimized design is reconstructed and further optimized in detail. Compared with several traditional designs, it shows better leakproofness under inner liquid pressure, which can be directly fabricated by additive manufacturing and used in the rocket engine. The optimization procedure is proved effective and practical in designing sealing structures for aerospace vehicles. [ABSTRACT FROM AUTHOR]

The failure probability function (FPF), aiming to decouple the nested-loop reliability-based design optimization solution into a single-loop optimization problem, has attracted a great deal of interest from designers and researchers. It is defined as a function of failure probability with respect to the design parameter. Among the estimation methods for the FPF, the Bayes theorem based on probability distribution function methods is competitive. It transforms the FPF as the estimation of the conditional joint probability density function (PDF) of design parameters on the failure event and the augmented failure probability. The augmented failure probability can be estimated by Monte Carlo simulation, while for the joint multi-dimensional PDF, the existing estimation methods are unavailable. To alleviate this issue, a novel FPF estimation method is proposed by Bayes theorem and copula. In the proposed method, the FPF is derived as a product of the conditional copula density and the augmented failure probability, in which the vine copula is employed to disassemble the multi-dimensional conditional copula density into several bivariate copula density functions, and they can be completed by the existing PDF estimation methods. In contrast to the existing Bayes theorem–based estimation methods for the FPF, the proposed method interprets the FPF as the dependence function between design parameters and the augmented failure probability in terms of copula, which involuntarily breaks the limitation of the multi-dimensional design parameters. In addition, the adaptive Kriging surrogate model is embedded in the proposed method to improve the efficiency of the proposed method. The presented examples demonstrate the efficiency and accuracy of the proposed method. [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]

This paper presents the topology optimization of hierarchical microtextures for wetting behavior in the Cassie-Baxter mode, considering a structural unit of the hierarchical microtexture composed of base and secondary structures. The geometrical configuration of the considered structural unit can be described as a fiber bundle composed of an external surface of the base structure and the pattern of the secondary structures. Thus, two design variables are defined, one for the external surface of the base structure, and the other for the pattern of the secondary structures. The Young-Laplace equation, including a term depending on the mean curvature of the external surface, is used to describe the liquid/vapor interface imposed with a surface tension in the Cassie-Baxter mode. To overcome the difficulty of numerically computing the second-order derivative of the external surface, two partial differential equation filters are sequentially applied to the design variable of the base structure to ensure the numerical accuracy and feasibility of using an efficient linear-element-based finite element method to solve the Young-Laplace equation. To improve the performance of the hierarchical microtextures, the volume of the liquid bulges suspended at the liquid/vapor interface in the Cassie-Baxter mode, before the transition into the Wenzel mode, is minimized to optimize the match between the external surface of the base structure and the pattern of the secondary structures. In the topology optimization process, penalization of the material density of the surface tension is achieved by an artificial Marangoni phenomenon. In numerical examples, solid surfaces are tiled into textures with axial symmetry, radial symmetry, chirality, and quasiperiodicity; and structural units are derived consisting of base structures with peak shapes and dense secondary structures surrounding the crests of the peaks. The optimized performance of the derived structural units has been confirmed by comparisons. [ABSTRACT FROM AUTHOR]

Additive printing allows the "single step" production of virtually any complex mechanical component. However, the manufacturing process involves a layer-by-layer deposition of material, which leads to an anisotropic mechanical behavior of the whole component. This would then entail a very fine 3D model to simulate the mechanical performance accurately. This simulation would also need to be integrated within an iterative design process in order to obtain the most efficient design. Both reasons explain the prohibitive number of calculations needed, which is currently beyond the capacities of existing software and computers. Recent research papers have opened promising pathways for integrating model reduction techniques within the overall topology optimization process. However, these approaches still present challenges such as choosing the minimum number and appropriate selection of the snapshots required to get accurate simulations. In this work, we present a methodology in the combined field of reduced-order modeling and topology optimization. The key idea consists of projecting the higher dimensional system of equations onto a lower dimensional space with the reduced basis vectors constructed using Proper Orthogonal Decomposition (POD). This reduced basis is updated in an incremental "on-the-fly" manner using alternatively costly high-fidelity and more rapid lower fidelity simulation snapshots. The variable-fidelity resolutions of successive approximations to the global system of equations are then integrated into the topology optimization process. The approaches are tested and computational savings and precision are compared, using both minimum compliance and compliant mechanism design benchmark problems. [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]

A B-spline multi-parameterization method (MPM) is presented in this paper for topology optimization of thermoelastic structures. As thermoelastic topology optimization belongs to a kind of design-dependent problems that are complicated to deal with, this method is aimed to solve thermoelastic problems with multiple materials by means of B-spline parameterization that integrates together the recursive multiphase material interpolation (RMMI) and the uniform multiphase material interpolation (UMMI) schemes. The commonly used discrete pseudo-density variables related to the finite element model are thus replaced with continuous pseudo-density fields dominated by control parameters in the B-spline space. In this sense, B-spline multi-parameterization is used for the first time to represent multiple pseudo-density fields and multi-material properties including elasticity matrix and thermal stress coefficient. Compared with traditional pseudo-density method, the current method has the advantage of not only attaining a reduction of design variables in number but also achieving a regularized distribution of pseudo-density fields with a clear material layout over the whole structure domain. Numerical results show that the stable convergences are achieved with the avoidance of gray elements, checkerboards, and multi-material overlapping owing to the high continuity of B-spline preserved for multi-material distributions. Besides, it is found that the RMMI scheme distributes less stiff materials around stiff materials, while the UMMI scheme tends to gather less stiff materials together. [ABSTRACT FROM AUTHOR]

In this work, a novel design and modeling method is proposed to obtain hierarchical structures with non-uniform lattice microstructures based on density-based topology optimization. First of all, a parametric concept is proposed to generate a family of parameterized lattice microstructures that present similar topological features. In order to balance the structural performance and computational efficiency, we construct a Parameterized Interpolation for Lattice Material (PILM) model and the mathematical formulation incorporates two new design variables. At the macroscale, the relative density variable is applied to describe material volume fraction in the design domain, instead of using the pseudo-density in the Solid Isotropic Material with Penalization (SIMP) model. At the microscale, each macroelement is regarded as an individual microstructure controlled by an aspect ratio variable. The equivalent properties of parameterized lattice microstructures can be derived by interpolating the effective elastic matrixes of several typical microstructure unit cells, which avoid expensive iterative homogenization calculations during optimization procedure. Hence, the multiscale concurrent design method can optimize the macroscopic distribution and their spatially varying microstructural configurations simultaneously at an affordable computation cost. Several numerical examples are presented to demonstrate the effectiveness of the proposed approach. Furthermore, the obtained hierarchical structures with non-uniform lattice microstructures show good manufacturability and remarkably improved structural performance by means of the additive manufacturing and experimental testing, compared to the designs with uniform lattice microstructures. [ABSTRACT FROM AUTHOR]

In this paper, a topology optimization method is developed for elastic continuum structures in frictionless contact to improve the uniformity of contact pressures. The variance constraint of contact pressures is introduced into the standard volume-constrained compliance minimization problem and combined with the three-field SIMP (Solid Isotropic Material with Penalization) model. Two geometric constraints are also incorporated to achieve optimization solutions with desired minimum length scales and clear boundaries without intermediate densities. To ensure a correct evaluation of contact pressure variance, a density threshold method is proposed to assign each contact node pair a certain density value according to the physical densities of its adjacent elements and further to exclude the contact node pairs with density values smaller than the threshold value. The effectiveness of the proposed method is validated through a series of numerical examples including elastic-rigid and elastic-elastic frictionless contact problems. The influence of the variance constraint of contact pressures on the optimization result is discussed in comparison with the standard maximum stiffness design. It concludes that the variance of contact pressures can be reduced at the cost of the structural stiffness, which implies that a trade-off exists between the structural stiffness and the uniformity of contact pressures. [ABSTRACT FROM AUTHOR]

In this paper, a side constraint scheme is presented for topology optimization considering structural connectivity. Void features are used as basic design primitives with their movements and shape changes to drive topology optimization. To ensure the structural connectivity, design variables related to the centers of all the void features are bounded outside the design domain by means of side constraints without introducing additional nonlinear constraints. It is shown that the side constraint scheme is effective to eliminate enclosed voids and well adapted to the additive manufacturing (AM) without concern of unmelted powders inside the structure. Several representative examples are tested to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Spatial gradient information of density field in SIMP (solid isotropic material with penalization) topology optimization is very useful for imposing overhang angle and minimum length (size) manufacturing constraints or achieving shell-infill optimization. However, the computation of density gradient is an approximation since the design space is discretized. There are several operators for this purpose, which arise from the image processing field. This note compares different gradient operators in the context of SIMP topology optimization method and suggests a new computation strategy to improve the accuracy of gradient estimation. We take a case study of spatial gradient-based minimum size constraints. New structural indicator functions are proposed to improve the general applicability of previous gradient-based minimum length constraints. This study is carried out in 2D structure examples to validate the methodology. [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]

This add-on discussion addresses a shortcoming of the paper in the title – the authors missed an important reference by Clausen et al. (2015). Differences between the current work and the said reference are clarified in this comment. In addition, some mistakes in the commented paper were corrected. [ABSTRACT FROM AUTHOR]

In this paper, the attainment of uniform reaction forces at the specific fixed boundary is investigated for topology optimization of continuum structures. The variance of the reaction forces at the boundary between the elastic solid and its foundation is firstly introduced as the evaluation criterion of the uniformity of the reaction forces. Then, the standard formulation of optimal topology design is improved by introducing the variance constraint of the reaction forces. Sensitivity analysis of the latter is carried out based on the adjoint method. Numerical examples are dealt with to reveal the effect of the variance constraint in comparison with solutions of standard topology optimization. [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]

This work is focused on the topology optimization related to harmonic responses for large-scale problems. A comparative study is made among mode displacement method (MDM), mode acceleration method (MAM) and full method (FM) to highlight their effectiveness. It is found that the MDM results in the unsatisfactory convergence due to the low accuracy of harmonic responses, while MAM and FM have a good accuracy and evidently favor the optimization convergence. Especially, the FM is of superiority in both accuracy and efficiency under the excitation at one specific frequency; MAM is preferable due to its balance between the computing efficiency and accuracy when multiple excitation frequencies are taken into account. [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]

The so-called bi-value coding parameterization (BCP) method is developed for the simultaneous optimization of layout design and discrete fiber orientations of laminated structures related to the compliance minimization and natural frequency maximization. Both kinds of problems are transformed into a discrete material selection problem that is then solved as a continuous topology optimization problem with multiphase materials. A new form of the volume constraint is introduced in accordance with the BCP to control the material usage and material removal in the corresponding problem formulation. The BCP scheme assigning the integer value of +1 or -1 to each design variable for the unique “coding” is efficiently used to interpolate discrete fiber orientations and to identify the presence and removal of materials. Meanwhile, a general set-up strategy is proposed by assigning “uniform” weight values in BCP to ensure the feasibility of the initial starting point. Numerical tests illustrate that the BCP is efficient in dealing with both kinds of design problems including the volume constraint. [ABSTRACT FROM AUTHOR]

In view of the prohibited computing time and the complexity of design procedure, a superelement formulation (SEF) is proposed to deal with the simultaneous optimization of component placement and the framework topology. In the iterative design process, each component is modelled as a movable superelement so that the sensitivity analysis with respect to the location design variables can be largely simplified by the SEF. Moreover, based on the Kuhn-Tucker optimality condition, two decomposition strategies are developed as variant approaches for the simultaneous design of multi-component system. By means of numerical examples, these approaches are compared to show their capability and efficiency for the system compliance minimization. [ABSTRACT FROM AUTHOR]

We focus here on a Response Surface Methodology adapted to the Reliability-Based Design Optimization (RBDO). The Diffuse Approximation, a version of the Moving Least Squares (MLS) approximation, based on a progressive sampling pattern is used within a variant of the First Order Reliability Method (FORM). The proposed method uses simultaneously the points in the standard normal space ( U-space) and the physical space ( X-space). The two grids form a 'virtual design of experiments' defined by two sets of points in both design spaces, which are evaluated only when needed in order to minimize the number of the 'exact' thus supposed costly, function evaluations. At each new iteration, the pattern of points is updated with the points appropriately selected from the virtual design, in order to perform the approximation. As an original contribution, we introduce the concept of 'Advancing LHS' which extends the idea of Latin Hypercube Sampling (LHS) for the maximal reuse of already computed points while adding at each step a minimal number of new neighboring points, necessary for the approximation in the vicinity of the current design. We propose panning, expanding and shrinking Latin patterns of sampling points and we analyze the influence of this specific kind of patterns on the quality of the approximation. Then we analyze the minimal number of data points required in order to get well-conditioned approximation systems. In the application part of this work, we investigate the case of optimizing the process parameters of numerically controlled (NC) milling of ultrahigh strength steel. [ABSTRACT FROM AUTHOR]

Structural topology optimization of thermo-elastic problems is investigated in this paper. The key issues about the penalty models of the element stiffness and thermal stress load of the finite element model are highlighted. The penalization of thermal stress coefficient (TSC) measured by the product between thermal expansion coefficient and Young’s modulus is proposed for the first time to characterize the dependence of the thermal stress load upon the design variables defined by element pseudo-densities. In such a way, the element stiffness and the thermal stress load can be penalized independently in terms of element pseudo-density. This formulation demonstrates especially its capability of solving problems with multiphase materials. Besides, the comparison study shows that the interpolation model RAMP is more advantageous than the SIMP in our case. Furthermore, sensitivity analysis of the structural mean compliance is developed in the case of steady-state heat conduction. Numerical examples of two-phase and three-phase materials are presented. [ABSTRACT FROM AUTHOR]