Topology optimization of continuum structures with displacement constraints based on meshless method

In this paper, the element free Galerkin method (EFG) is applied to carry out the topology optimization of continuum structures with displacement constraints. In the EFG method, the matrices in the discretized system equations are assembled based on the quadrature points. In the sense, the relative density at Gauss quadrature point is employed as design variable. Considering the minimization of weight as an objective function, the mathematical formulation of the topology optimization subjected to displacement constraints is developed using the solid isotropic microstructures with penalization interpolation scheme. Moreover, the approximate explicit function expression between topological variables and displacement constraints are derived. Sensitivity of the objective function is derived based on the adjoint method. Three numerical examples are used to demonstrate the feasibility and effectiveness of the proposed method.     

The topology optimization design for continuum structures based on the element free Galerkin method

In this paper, the element free Galerkin method (EFG), combined with evolutionary structural optimization method (ESO), is applied to carry out the topology optimization of the continuum structures. Considering the deletion criterion based on the stresses, the mathematical formulation of the topology optimization is developed. The objective function of this model is the minimized weight. Several numerical examples are used to prove the feasibility of the approach adopted in this paper. And the examples show the simplicity and fast convergence of the proposed method.     

Topology optimization of continuum structures for natural frequencies considering casting constraints

A method for the topology optimization on the natural frequency of continuum structures with casting constraints is proposed. The objective is to maximize the natural frequency of vibrating continuum structures subject to casting constraints. When the natural frequencies of the considered structures are maximized using the solid isotropic material with penalization (SIMP) model, artificial localized modes may occur in areas where elements are assigned with lower density values. In this article, the topology optimization is performed by the bi-directional evolutionary structural optimization (BESO) method. The effects of different locations of concentrated lump mass, different volume fractions and meshing sizes on the final topologies are compared. Both two and four parting directions are investigated. Several two- and three-dimensional numerical examples show that the proposed BESO method is effective in achieving convergent solid–void optimal solutions for a variety of frequency optimization problems of continuum structures.     

Structural shape and topology optimization using a meshless Galerkin level set method

This paper aims to propose a meshless Galerkin level set method for shape and topology optimization of continuum structures. To take advantage of the implicit free boundary representation scheme, the design boundary is represented as the zero level set of a scalar level set function, to flexibly handle complex shape fidelity and topology changes by maintaining concise and smooth interface. Compactly supported radial basis functions (CSRBFs) are used to parameterize the level set function and construct the shape functions for meshfree approximations based on a set of unstructured field nodes. The meshless Galerkin method with global weak form is used to implement the discretization of the state equations. This provides a pathway to unify the two different numerical stages in most conventional level set methods: (1) the propagation of discrete level set function on a set of Eulerian grid and (2) the approximation of discrete equations on a set of Lagrangian mesh. The original more difficult shape and topology optimization based on the level set equation is transformed into a relatively easier size optimization, to which many efficient optimization algorithms can be applied. The proposed level set method can describe the moving boundaries without remeshing for discontinuities. The motion of the free boundary is just a question of advancing the discrete level set function in time by solving the size optimization. Several benchmark examples are used to demonstrate the effectiveness of the proposed method. The numerical results show that the proposed method can simplify numerical process and avoid numerical difficulties involved in most conventional level set methods. It is straightforward to apply the proposed method to more advanced shape and topology optimization problems. Copyright © 2011 John Wiley & Sons, Ltd.     

A meshless Pareto-optimal method for topology optimization

A meshless Galerkin Pareto-optimal method is proposed for topology optimization of continuum structures in this paper. The compactly supported radial basis function (CSRBF) is used to create shape functions. The shape function is constructed by meshfree approximations based on a set of unstructured field nodes. Considering the Pareto-optimality theory, the initial single objective topology optimization problem is transformed into multi-objective problem. The optimum solution is traced via the Pareto-optimal frontier in a computationally effective manner. The optimal problem does not need to be solved directly. Finally, several examples are used to prove the validity and effectiveness of the proposed approach.     

周期性多孔结构特征值拓扑优化

为了实现尺度关联周期性多孔结构的隔振性能优化,提出一种周期性多孔结构特征值拓扑优化方法.基于子结构动态凝聚方法对多孔结构的刚度和质量矩阵进行缩减,采用局部水平集函数(LLSF)对多孔结构进行几何隐式描述,以最大化前6阶特征值为目标函数,以结构体积分数为约束条件,建立周期性多孔结构特征值拓扑优化模型,采用优化准则法对拓扑...     

连续体结构的变密度拓扑优化方法研究

为了实现使连续体结构的体积约束和柔顺度最小的拓扑优化及解决采用经典变密度法引起的结构优化结果存在如灰度单元、棋盘格等数值不稳定问题，提出了一种新的拓扑优化方法。首先，采用改进的固体各向同性材料惩罚法作为材料插值方案，建立结构拓扑优化模型；其次，通过引入基于高斯权重函数的敏度过滤法和设计新灰度单元抑制算子来解决数值不稳定问题；最后，借助优化准则法求解优化模型。通过算例分析可知：新策略可以改进拓扑优化方法；新的拓扑优化方法具有收敛速度较快、能更好地获取柔顺度小且拓扑构型好的优化结构和抑制灰度单元产生等优势。研究结果为其他连续体结构的拓扑优化研究提供了新思路。     

A novel reliability-based topology optimization framework for the concurrent design of solid and truss-like material structures with unknown-but-bounded uncertainties

A new nonprobabilistic reliability-based topology optimization method for continuum structures with displacement constraints is proposed in this paper, in which the optimal layout consists of solid material and truss-like microstructure material simultaneously. The unknown-but-bounded uncertainties that exist in material properties, external loads, and safety displacements are considered. By utilizing the representative volume element analysis, rules of macro-micro stiffness performance equivalence can be confirmed. A solid material and truss-like microstructure material structure integrated design interpolation model is firstly constructed, in which design domain elements can be conducted to select solid material or truss-like microstructure material by a combination of the finite element method in the topology optimization process. Moreover, a new nonprobabilistic reliability measuring index, namely, the optimization feature distance is defined by making use of the area-ratio ideas. Furthermore, the adjoint vector method is employed to obtain the sensitivity information between the reliability measure and design variables. By utilizing the method of moving asymptotes, the investigated optimization problem can be iteratively solved. The effectiveness of the developed methodology is eventually demonstrated by two examples.     

模糊参数平面连续体结构的拓扑优化设计

研究了具有模糊参数的连续体结构在模糊载荷作用下的拓扑优化设计问题。利用信息熵将模糊变量转换为随机变量,构建了随机载荷作用下的随机参数的连续体结构的拓扑优化设计数学模型,以结构的形状拓扑信息为设计变量,结构总质量均值极小化为目标函数,满足单元应力可靠性为约束条件,利用分布函数法对应力可靠性约束进行了等价显式化处理。基于随机因子法,利用代数综合法导出了应力响应的数字特征的计算表达式。采用双方向渐进结构优化(BESO)方法求解。通过两个算例验证了该文模型及求解方法的合理性和有效性。     

A unified stochastic framework for robust topology optimization of continuum and truss-like structures

In this article, a unified framework is introduced for robust structural topology optimization for 2D and 3D continuum and truss problems. The uncertain material parameters are modelled using a spatially correlated random field which is discretized using the Karhunen–Loève expansion. The spectral stochastic finite element method is used, with a polynomial chaos expansion to propagate uncertainties in the material characteristics to the response quantities. In continuum structures, either 2D or 3D random fields are modelled across the structural domain, while representation of the material uncertainties in linear truss elements is achieved by expanding 1D random fields along the length of the elements. Several examples demonstrate the method on both 2D and 3D continuum and truss structures, showing that this common framework provides an interesting insight into robustness versus optimality for the test problems considered.     

结构拓扑优化中敏度过滤技术研究

为了抑制连续体结构拓扑优化结果中的棋盘格和灰度单元问题,借鉴粒子群优化算法中粒子状态的更新方法,提出一种改进的敏度更新技术.以结构的柔度最小为优化目标,构建了基于固体各项同性微惩罚结构的结构拓扑优化模型,根据结构的力学响应分析,采用优化准则法进行设计变量更新,进行载荷作用下二维连续体结构的拓扑优化设计,得到了材料在设计域内的最优分布.通过与已有敏度过滤技术的对比分析,验证了文中方法的正确性和有效性．     

Optimal topology design of continuum structures with stress concentration alleviation via level set method

Although the phenomenon of stress concentration is of paramount importance to engineers when they are designing load‐carrying structures, stiffness is often used as the solely concerned objective or constraint functional in the studies of optimal topology design of continuum structures. Sometimes this will lead to optimal designs with severe stress concentrations that may be highly responsible for the fracture, creep, and fatigue of structures. The aim of the present work is to develop some effective numerical techniques for designing stiff structures with less stress concentrations. This is achieved by introducing some specific stress measures, which are sensitive to the existence of high local stresses, in the problem formulation and resolving the corresponding optimization problem numerically in a level set framework. Our study indicates that with use of the proposed numerical schemes, some intrinsic difficulties in stress‐related topology optimization of continuum structures can be overcome in a natural way. Copyright © 2012 John Wiley & Sons, Ltd.     

Topology optimization of continuum structures using element free Galerkin method on irregular nodal distribution

International Journal of Mechanics and Materials in Design - In this paper, topology optimization of structures is achieved through integrating the element free Galerkin (EFG) method and the solid...     

Radial basis functions and level set method for structural topology optimization

Level set methods have become an attractive design tool in shape and topology optimization for obtaining lighter and more efficient structures. In this paper, the popular radial basis functions (RBFs) in scattered data fitting and function approximation are incorporated into the conventional level set methods to construct a more efficient approach for structural topology optimization. RBF implicit modelling with multiquadric (MQ) splines is developed to define the implicit level set function with a high level of accuracy and smoothness. A RBF–level set optimization method is proposed to transform the Hamilton–Jacobi partial differential equation (PDE) into a system of ordinary differential equations (ODEs) over the entire design domain by using a collocation formulation of the method of lines. With the mathematical convenience, the original time dependent initial value problem is changed to an interpolation problem for the initial values of the generalized expansion coefficients. A physically meaningful and efficient extension velocity method is presented to avoid possible problems without reinitialization in the level set methods. The proposed method is implemented in the framework of minimum compliance design that has been extensively studied in topology optimization and its efficiency and accuracy over the conventional level set methods are highlighted. Numerical examples show the success of the present RBF–level set method in the accuracy, convergence speed and insensitivity to initial designs in topology optimization of two‐dimensional (2D) structures. It is suggested that the introduction of the radial basis functions to the level set methods can be promising in structural topology optimization. Copyright © 2005 John Wiley & Sons, Ltd.     

Temperature-constrained topology optimization of thermo-mechanical coupled problems

Temperature-constrained topology optimization for thermo-mechanical coupled problems under a design-dependent temperature field considering the thermal expansion effect remains an open problem. A temperature-constrained topology optimization method is proposed for thermo-mechanical coupled problems. In this article, the temperature values at the heat sources are constrained. The numerical results reveal that the temperature constraints play an important role in topology optimization of thermo-mechanical coupled problems. The optimized structure obtained by the presented method not only has certain strength but also decreases the temperature significantly compared with structures obtained by other methods without considering temperature constraints. The proposed method is applied to the design of the cooling system of a battery package. Numerical examples verify the efficiency of the presented method.     

Isogeometric topology optimization for continuum structures using density distribution function

This paper will propose a more effective and efficient topology optimization method based on isogeometric analysis, termed as isogeometric topology optimization (ITO), for continuum structures using an enhanced density distribution function (DDF). The construction of the DDF involves two steps. (1)  Smoothness: the Shepard function is firstly utilized to improve the overall smoothness of nodal densities. Each nodal density is assigned to a control point of the geometry. (2) Continuity: the high-order NURBS basis functions are linearly combined with the smoothed nodal densities to construct the DDF for the design domain. The nonnegativity, partition of unity, and restricted bounds [0, 1] of both the Shepard function and NURBS basis functions can guarantee the physical meaning of material densities in the design. A topology optimization formulation to minimize the structural mean compliance is developed based on the DDF and isogeometric analysis to solve structural responses. An integration of the geometry parameterization and numerical analysis can offer the unique benefits for the optimization. Several 2D and 3D numerical examples are performed to demonstrate the effectiveness and efficiency of the proposed ITO method, and the optimized 3D designs are prototyped using the Selective Laser Sintering technique.     

Truss layout design under nonprobabilistic reliability-based topology optimization framework with interval uncertainties

A nonprobabilistic reliability-based topology optimization (NRBTO) method for truss structures with interval uncertainties (or unknown-but-bounded uncertainties) is proposed in this paper. The cross-sectional areas of levers are defined as design variables, while the material properties and external loads are regard as interval parameters. A modified perturbation method is applied to calculate structural response bounds, which are the prerequisite to obtain structural reliability. A deviation distance between the current limit state plane and the objective limit state plane, of which the expression is explicit, is defined as the nonprobabilistic reliability index, which serves as a constraint function in the optimization model. Compared with the deterministic topology optimization problem, the proposed NRBTO formulation is still a single-loop optimization problem, as the reliability index is explicit. The sensitivity results are obtained from an analytical approach as well as a direct difference method. Eventually, the NRBTO problem is solved by a sequential quadratic programming method. Two numerical examples are used to testify the validity and effectiveness of the proposed method. The results show significant effects of uncertainties to the topology configuration of truss structures.     

Voigt–Reuss topology optimization for structures with nonlinear material behaviors

This work is directed toward optimizing concept designs of structures featuring inelastic material behaviours by using topology optimization. In the proposed framework, alternative structural designs are described with the aid of spatial distributions of volume fraction design variables throughout a prescribed design domain. Since two or more materials are permitted to simultaneously occupy local regions of the design domain, small-strain integration algorithms for general two-material mixtures of solids are developed for the Voigt (isostrain) and Reuss (isostress) assumptions, and hybrid combinations thereof. Structural topology optimization problems involving non-linear material behaviours are formulated and algorithms for incremental topology design sensitivity analysis (DSA) of energy type functionals are presented. The consistency between the structural topology design formulation and the developed sensitivity analysis algorithms is established on three small structural topology problems separately involving linear elastic materials, elastoplastic materials, and viscoelastic materials. The good performance of the proposed framework is demonstrated by solving two topology optimization problems to maximize the limit strength of elastoplastic structures. It is demonstrated through the second example that structures optimized for maximal strength can be significantly different than those optimized for minimal elastic compliance. © 1997 John Wiley & Sons, Ltd.