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Solving topology optimization problems by the Guide-Weight method

Xinjun LIU, Zhidong LI, Liping WANG, Jinsong WANG

《机械工程前沿（英文）》 2011年 第6卷 第1期   页码 136-150 doi: 10.1007/s11465-010-0126-6

摘要：

Finding a good solution method for topology optimization problems is always paid attention to by the research field because they are subject to the large number of the design variables and to the complexity that occurs because the objective and constraint functions are usually implicit with respect to design variables. Guide-Weight method, proposed first by Chen in 1980s, was effectively and successfully used in antenna structures’ optimization. This paper makes some improvement to it so that it possesses the characteristics of both the optimality criteria methods and the mathematical programming methods. When the Guide-Weight method is applied into topology optimization, it works very well with unified and simple form, wide availability and fast convergence. The algorithm of the Guide-Weight method and the improvement on it are described; two formulations of topology optimization solved by the Guide-Weight method combining with SIMP method are presented; subsequently, three numerical examples are provided, and comparison of the Guide-Weight method with other methods is made.

关键词： Guide-Weight method     topology optimization     SIMP method    

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Modular design of typical rigid links in parallel kinematic machines: Classification and topology optimization

Xinjun LIU, Xiang CHEN, Zhidong LI

《机械工程前沿（英文）》 2012年 第7卷 第2期   页码 199-209 doi: 10.1007/s11465-012-0315-6

摘要：

Due to the demand of reconfigurable system in parallel kinematic machines (PKMs), modular design technology is significant and necessary. However, in earlier research, the core joint modules have been concerned about rather than the customized link modules. The modular design to the typical customized links from the point of seeking optimal structures with best mechanical performances is analyzed and processed in two steps: classification and optimization. Firstly, a brief introduction to the current research status and the aims of this paper are outlined. And then, how the typical customized links classified is proposed. Next, the technology method and the iterative formula derivation process of topology optimization are described in detail. Finally, calculation models for each group of classified ones are set up and their optimal structures are achieved through topology optimization technique. The results provide useful references for reconfigurable and modular design in engineering cases.

关键词： parallel kinematic machines (PKMs)     modular design     classification     topology optimization and improved Guide-Weight method    

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Level set band method: A combination of density-based and level set methods for the topology optimization

Peng WEI, Wenwen WANG, Yang YANG, Michael Yu WANG

《机械工程前沿（英文）》 2020年 第15卷 第3期   页码 390-405 doi: 10.1007/s11465-020-0588-0

摘要： The level set method (LSM), which is transplanted from the computer graphics field, has been successfully introduced into the structural topology optimization field for about two decades, but it still has not been widely applied to practical engineering problems as density-based methods do. One of the reasons is that it acts as a boundary evolution algorithm, which is not as flexible as density-based methods at controlling topology changes. In this study, a level set band method is proposed to overcome this drawback in handling topology changes in the level set framework. This scheme is proposed to improve the continuity of objective and constraint functions by incorporating one parameter, namely, level set band, to seamlessly combine LSM and density-based method to utilize their advantages. The proposed method demonstrates a flexible topology change by applying a certain size of the level set band and can converge to a clear boundary representation methodology. The method is easy to implement for improving existing LSMs and does not require the introduction of penalization or filtering factors that are prone to numerical issues. Several 2D and 3D numerical examples of compliance minimization problems are studied to illustrate the effects of the proposed method.

关键词： level set method     topology optimization     density-based method     level set band    

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A MATLAB code for the material-field series-expansion topology optimization method

《机械工程前沿（英文）》 2021年 第16卷 第3期   页码 607-622 doi: 10.1007/s11465-021-0637-3

摘要： This paper presents a MATLAB implementation of the material-field series-expansion (MFSE) topo-logy optimization method. The MFSE method uses a bounded material field with specified spatial correlation to represent the structural topology. With the series-expansion method for bounded fields, this material field is described with the characteristic base functions and the corresponding coefficients. Compared with the conventional density-based method, the MFSE method decouples the topological description and the finite element discretization, and greatly reduces the number of design variables after dimensionality reduction. Other features of this method include inherent control on structural topological complexity, crisp structural boundary description, mesh independence, and being free from the checkerboard pattern. With the focus on the implementation of the MFSE method, the present MATLAB code uses the maximum stiffness optimization problems solved with a gradient-based optimizer as examples. The MATLAB code consists of three parts, namely, the main program and two subroutines (one for aggregating the optimization constraints and the other about the method of moving asymptotes optimizer). The implementation of the code and its extensions to topology optimization problems with multiple load cases and passive elements are discussed in detail. The code is intended for researchers who are interested in this method and want to get started with it quickly. It can also be used as a basis for handling complex engineering optimization problems by combining the MFSE topology optimization method with non-gradient optimization algorithms without sensitivity information because only a few design variables are required to describe relatively complex structural topology and smooth structural boundaries using the MFSE method.

关键词： MATLAB implementation     topology optimization     material-field series-expansion method     bounded material field     dimensionality reduction    

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Efficient, high-resolution topology optimization method based on convolutional neural networks

Liang XUE, Jie LIU, Guilin WEN, Hongxin WANG

《机械工程前沿（英文）》 2021年 第16卷 第1期   页码 80-96 doi: 10.1007/s11465-020-0614-2

摘要： Topology optimization is a pioneer design method that can provide various candidates with high mechanical properties. However, high resolution is desired for optimum structures, but it normally leads to a computationally intractable puzzle, especially for the solid isotropic material with penalization (SIMP) method. In this study, an efficient, high-resolution topology optimization method is developed based on the super-resolution convolutional neural network (SRCNN) technique in the framework of SIMP. SRCNN involves four processes, namely, refinement, path extraction and representation, nonlinear mapping, and image reconstruction. High computational efficiency is achieved with a pooling strategy that can balance the number of finite element analyses and the output mesh in the optimization process. A combined treatment method that uses 2D SRCNN is built as another speed-up strategy to reduce the high computational cost and memory requirements for 3D topology optimization problems. Typical examples show that the high-resolution topology optimization method using SRCNN demonstrates excellent applicability and high efficiency when used for 2D and 3D problems with arbitrary boundary conditions, any design domain shape, and varied load.

关键词： topology optimization     convolutional neural network     high resolution     density-based    

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Independent continuous and mapping method of structural topology optimization based on the global stress

Yunkang SUI, Jili FENG, Hongling YE, Xirong PENG,

《机械工程前沿（英文）》 2010年 第5卷 第2期   页码 130-142 doi: 10.1007/s11465-010-0011-3

摘要： There are three difficulties in topology optimization of continuum structures. 1) The topology under multiple load case is more difficult to be optimized than under single load case, because the former becomes a multiple objective based on compliance objective functions. 2) With local constraints, such as an elemental stress limit, the topology is more difficult to be solved than with global constraints, such as the displacement or frequency limits, because the sensitivity analysis of the former has very expensive computation. 3) With the phenomenon of load illness, which is similar with stiffness illness in the structural analysis, it is not easy to get the reasonable final topological structure, because it is difficult to consider different influences between the loads with small forces and big forces, and some topology paths of transferring small forces may disappear during the iteration process. To overcome difficulties above, four measures are adopted. 1) Topology optimization model is established by independent continuous mapping (ICM) method. 2) Based on the von Mises strength theory, all elements’ stress constraints are transformed into a structural energy constraint. 3) The phenomenon of load illness is divided to classify into three cases. 4) A strategy based on strain energy is proposed to adopt ICM method with stress globalization, and the problems of the above mentioned three cases of load illness are solved in terms of different complementary approaches. Several numerical examples show that the topology path of transferring forces can be obtained more easily by substituting global strain energy constraints for local stresses constraints, and the problem of load illness can be solved well by the weighting method that takes the structural energy as a weighting coefficient.

关键词： independent continuous mapping (ICM) method     global stresses constraints     topology optimization     continuum structure     load illness    

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拓扑优化中采用增材制造填充构件的结构屈曲荷载提升设计 Artical

Anders Clausen, Niels Aage, Ole Sigmund

《工程（英文）》 2016年 第2卷 第2期   页码 250-257 doi: 10.1016/J.ENG.2016.02.006

摘要：

增材制造可实现优质多功能构件所具有的高度复杂几何构型的制备。可以直接制备内含多孔填充的结构部件是其独有特征的一个例证。现有的设计方法还难以充分利用这一设计自由度，直接获得类似结构的设计。本文将展示涂层方法 (coating approach) 的拓扑优化方法来作为多孔填充构件的设计方法，所设计的构件具有显著改进的临界屈曲载荷，从而使得整体结构部件的稳定性增强。传统的柔顺性拓扑优化方法极少在数学模型中考虑构件的屈曲约束，稳定性要求通常要经过后续的校核与改进过程满足。这一后续过程往往只能获得次优设计。本文所展示的方法弥补了传统柔顺性拓扑优化模型中难以考虑构件屈曲约束的缺陷。利用涂层拓扑优化方法与传统柔顺性拓扑优化同时对经典的MBB 梁进行设计，并采用熔丝增材制造技术对设计结果进行了制备。实验结果验证了涂层方法的数学模型的正确性。由于填充材料的性质，在相同条件下，涂层优化得到的多孔填充结构的屈曲载荷比传统优化得到的实体结构高四倍以上。

关键词： 增材制造     填充构件     拓扑优化     屈曲    

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Level set-based isogeometric topology optimization for maximizing fundamental eigenfrequency

Manman XU, Shuting WANG, Xianda XIE

《机械工程前沿（英文）》 2019年 第14卷 第2期   页码 222-234 doi: 10.1007/s11465-019-0534-1

摘要： Maximizing the fundamental eigenfrequency is an efficient means for vibrating structures to avoid resonance and noises. In this study, we develop an isogeometric analysis (IGA)-based level set model for the formulation and solution of topology optimization in cases with maximum eigenfrequency. The proposed method is based on a combination of level set method and IGA technique, which uses the non-uniform rational B-spline (NURBS), description of geometry, to perform analysis. The same NURBS is used for geometry representation, but also for IGA-based dynamic analysis and parameterization of the level set surface, that is, the level set function. The method is applied to topology optimization problems of maximizing the fundamental eigenfrequency for a given amount of material. A modal track method, that monitors a single target eigenmode is employed to prevent the exchange of eigenmode order number in eigenfrequency optimization. The validity and efficiency of the proposed method are illustrated by benchmark examples.

关键词： topology optimization     level set method     isogeometric analysis     eigenfrequency    

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Geometrically constrained isogeometric parameterized level-set based topology optimization via trimmed

Yingjun WANG,David J. BENSON

《机械工程前沿（英文）》 2016年 第11卷 第4期   页码 328-343 doi: 10.1007/s11465-016-0403-0

摘要：

In this paper, an approach based on the fast point-in-polygon (PIP) algorithm and trimmed elements is proposed for isogeometric topology optimization (TO) with arbitrary geometric constraints. The isogeometric parameterized level-set-based TO method, which directly uses the non-uniform rational basis splines (NURBS) for both level set function (LSF) parameterization and objective function calculation, provides higher accuracy and efficiency than previous methods. The integration of trimmed elements is completed by the efficient quadrature rule that can design the quadrature points and weights for arbitrary geometric shape. Numerical examples demonstrate the efficiency and flexibility of the method.

关键词： isogeometric analysis     topology optimization     level set method     arbitrary geometric constraint     trimmed element    

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Isogeometric topology optimization based on energy penalization for symmetric structure

Xianda XIE, Shuting WANG, Ming YE, Zhaohui XIA, Wei ZHAO, Ning JIANG, Manman XU

《机械工程前沿（英文）》 2020年 第15卷 第1期   页码 100-122 doi: 10.1007/s11465-019-0568-4

摘要： We present an energy penalization method for isogeometric topology optimization using moving morphable components (ITO–MMC), propose an ITO–MMC with an additional bilateral or periodic symmetric constraint for symmetric structures, and then extend the proposed energy penalization method to an ITO–MMC with a symmetric constraint. The energy penalization method can solve the problems of numerical instability and convergence for the ITO–MMC and the ITO–MMC subjected to the structural symmetric constraint with asymmetric loads. Topology optimization problems of asymmetric, bilateral symmetric, and periodic symmetric structures are discussed to validate the effectiveness of the proposed energy penalization approach. Compared with the conventional ITO–MMC, the energy penalization method for the ITO–MMC can improve the convergence rate from 18.6% to 44.5% for the optimization of the asymmetric structure. For the ITO–MMC under a bilateral symmetric constraint, the proposed method can reduce the objective value by 5.6% and obtain a final optimized topology that has a clear boundary with decreased iterations. For the ITO–MMC under a periodic symmetric constraint, the proposed energy penalization method can dramatically reduce the number of iterations and obtain a speedup of more than 2.

关键词： topology optimization     moving morphable component     isogeometric analysis     energy penalization method     symmetric constraint    

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Robust isogeometric topology optimization for piezoelectric actuators with uniform manufacturability

《机械工程前沿（英文）》 2022年 第17卷 第2期 doi: 10.1007/s11465-022-0683-5

摘要： Piezoelectric actuators have received substantial attention among the industry and academia due to quick responses, such as high output force, high stiffness, high accuracy, and precision. However, the design of piezoelectric actuators always suffers from the emergence of several localized hinges with only one-node connection, which have difficulty satisfying manufacturing and machining requirements (from the over- or under-etching devices). The main purpose of the current paper is to propose a robust isogeometric topology optimization (RITO) method for the design of piezoelectric actuators, which can effectively remove the critical issue induced by one-node connected hinges and simultaneously maintain uniform manufacturability in the optimized topologies. In RITO, the isogeometric analysis replacing the conventional finite element method is applied to compute the unknown electro elastic fields in piezoelectric materials, which can improve numerical accuracy and then enhance iterative stability. The erode–dilate operator is introduced in topology representation to construct the eroded, intermediate, and dilated density distribution functions by non-uniform rational B-splines. Finally, the RITO formulation for the design of piezoelectric materials is developed, and several numerical examples are performed to test the effectiveness and efficiency of the proposed RITO method.

关键词： piezoelectric actuator     isogeometric topology optimization     uniform manufacturability     robust formulation     density distribution function    

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Topology optimization of piezoelectric bi-material actuators with velocity feedback control

Mariana MORETTI, Emílio C. N. SILVA

《机械工程前沿（英文）》 2019年 第14卷 第2期   页码 190-200 doi: 10.1007/s11465-019-0537-y

摘要： In recent years, the new technologies and discoveries on manufacturing materials have encouraged researchers to investigate the appearance of material properties that are not naturally available. Materials featuring a specific stiffness, or structures that combine non-structural and structural functions are applied in the aerospace, electronics and medical industry fields. Particularly, structures designed for dynamic actuation with reduced vibration response are the focus of this work. The bi-material and multifunctional concepts are considered for the design of a controlled piezoelectric actuator with vibration suppression by means of the topology optimization method (TOM). The bi-material piezoelectric actuator (BPEA) has its metallic host layer designed by the TOM, which defines the structural function, and the electric function is given by two piezo-ceramic layers that act as a sensor and an actuator coupled with a constant gain active velocity feedback control (AVFC). The AVFC, provided by the piezoelectric layers, affects the structural damping of the system through the velocity state variables readings in time domain. The dynamic equation analyzed throughout the optimization procedure is fully elaborated and implemented. The dynamic response for the rectangular four-noded FE analysis is obtained by the Newmark’s time-integration method, which is applied to the physical and the adjoint systems, given that the adjoint formulation is needed for the sensitivity analysis. A gradient-based optimization method is applied to minimize the displacement energy output measured at a predefined degree-of-freedom of the BPEA when a transient mechanical load is applied. Results are obtained for different control gain values to evaluate their influence on the final topology.

关键词： topology optimization method     bi-material piezoactuator     active velocity feedback control     time-domain transient analysis     host structure design     vibration suppression    

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Massively efficient filter for topology optimization based on the splitting of tensor product structure

《机械工程前沿（英文）》 2022年 第17卷 第4期 doi: 10.1007/s11465-022-0710-6

摘要： In this work, we put forward a massively efficient filter for topology optimization (TO) utilizing the splitting of tensor product structure. With the aid of splitting technique, the traditional weight matrices of B-splines and non-uniform rational B-spline implicit filters are decomposed equivalently into two or three submatrices, by which the sensitivity analysis is reformulated for the nodal design variables without altering the optimization process. Afterwards, an explicit sensitivity filter, which is decomposed by the splitting pipeline as that applied to implicit filter, is established in terms of the tensor product of the axial distances between adjacent element centroids, and the corresponding sensitivity analysis is derived for elemental design variables. According to the numerical results, the average updating time for the design variables is accelerated by two-order-of-magnitude for the decomposed filter compared with the traditional filter. In addition, the memory burden and computing time of the weight matrix are decreased by six- and three-order-of-magnitude for the decomposed filter. Therefore, the proposed filter is massively improved by the splitting of tensor product structure and delivers a much more efficient way of solving TO problems in the frameworks of isogeometric analysis and finite element analysis.

关键词： topology optimization     isogeometric analysis     finite element analysis     tensor product structure     sensitivity analysis    

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Concurrent optimization of structural topology and infill properties with a CBF-based level set method

Long JIANG, Yang GUO, Shikui CHEN, Peng WEI, Na LEI, Xianfeng David GU

《机械工程前沿（英文）》 2019年 第14卷 第2期   页码 171-189 doi: 10.1007/s11465-019-0530-5

摘要： In this paper, a parametric level-set-based topology optimization framework is proposed to concurrently optimize the structural topology at the macroscale and the effective infill properties at the micro/meso scale. The concurrent optimization is achieved by a computational framework combining a new parametric level set approach with mathematical programming. Within the proposed framework, both the structural boundary evolution and the effective infill property optimization can be driven by mathematical programming, which is more advantageous compared with the conventional partial differential equation-driven level set approach. Moreover, the proposed approach will be more efficient in handling nonlinear problems with multiple constraints. Instead of using radial basis functions (RBF), in this paper, we propose to construct a new type of cardinal basis functions (CBF) for the level set function parameterization. The proposed CBF parameterization ensures an explicit impose of the lower and upper bounds of the design variables. This overcomes the intrinsic disadvantage of the conventional RBF-based parametric level set method, where the lower and upper bounds of the design variables oftentimes have to be set by trial and error. A variational distance regularization method is utilized in this research to regularize the level set function to be a desired distance-regularized shape. With the distance information embedded in the level set model, the wrapping boundary layer and the interior infill region can be naturally defined. The isotropic infill achieved via the mesoscale topology optimization is conformally fit into the wrapping boundary layer using the shape-preserving conformal mapping method, which leads to a hierarchical physical structure with optimized overall topology and effective infill properties. The proposed method is expected to provide a timely solution to the increasing demand for multiscale and multifunctional structure design.

关键词： concurrent topology optimization     parametric level set method     cardinal basis function     shell-infill structure design     conformal mapping    

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Multiresolution and multimaterial topology optimization of fail-safe structures under B-spline spaces

《机械工程前沿（英文）》 2023年 第18卷 第4期 doi: 10.1007/s11465-023-0768-9

摘要： This study proposes a B-spline-based multiresolution and multimaterial topology optimization (TO) design method for fail-safe structures (FSSs), aiming to achieve efficient and lightweight structural design while ensuring safety and facilitating the postprocessing of topological structures. The approach involves constructing a multimaterial interpolation model based on an ordered solid isotropic material with penalization (ordered-SIMP) that incorporates fail-safe considerations. To reduce the computational burden of finite element analysis, we adopt a much coarser analysis mesh and finer density mesh to discretize the design domain, in which the density field is described by the B-spline function. The B-spline can efficiently and accurately convert optimized FSSs into computer-aided design models. The 2D and 3D numerical examples demonstrate the significantly enhanced computational efficiency of the proposed method compared with the traditional SIMP approach, and the multimaterial TO provides a superior structural design scheme for FSSs. Furthermore, the postprocessing procedures are significantly streamlined.

关键词： multiresolution     multimaterial     topology optimization     fail-safe structure     B-spline    

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标题 作者 时间 类型 操作

Solving topology optimization problems by the Guide-Weight method

Xinjun LIU, Zhidong LI, Liping WANG, Jinsong WANG

期刊论文

Modular design of typical rigid links in parallel kinematic machines: Classification and topology optimization

Xinjun LIU, Xiang CHEN, Zhidong LI

期刊论文

Level set band method: A combination of density-based and level set methods for the topology optimization

Peng WEI, Wenwen WANG, Yang YANG, Michael Yu WANG

期刊论文

A MATLAB code for the material-field series-expansion topology optimization method

期刊论文

Efficient, high-resolution topology optimization method based on convolutional neural networks

Liang XUE, Jie LIU, Guilin WEN, Hongxin WANG

期刊论文

Independent continuous and mapping method of structural topology optimization based on the global stress

Yunkang SUI, Jili FENG, Hongling YE, Xirong PENG,

期刊论文

拓扑优化中采用增材制造填充构件的结构屈曲荷载提升设计

Anders Clausen, Niels Aage, Ole Sigmund

期刊论文

Level set-based isogeometric topology optimization for maximizing fundamental eigenfrequency

Manman XU, Shuting WANG, Xianda XIE

期刊论文

Geometrically constrained isogeometric parameterized level-set based topology optimization via trimmed

Yingjun WANG,David J. BENSON

期刊论文

Isogeometric topology optimization based on energy penalization for symmetric structure

Xianda XIE, Shuting WANG, Ming YE, Zhaohui XIA, Wei ZHAO, Ning JIANG, Manman XU

期刊论文

Robust isogeometric topology optimization for piezoelectric actuators with uniform manufacturability

期刊论文

Topology optimization of piezoelectric bi-material actuators with velocity feedback control

Mariana MORETTI, Emílio C. N. SILVA

期刊论文

Massively efficient filter for topology optimization based on the splitting of tensor product structure

期刊论文

Concurrent optimization of structural topology and infill properties with a CBF-based level set method

Long JIANG, Yang GUO, Shikui CHEN, Peng WEI, Na LEI, Xianfeng David GU

期刊论文

Multiresolution and multimaterial topology optimization of fail-safe structures under B-spline spaces

期刊论文