Linear and nonlinear topology optimization design with projection-based ground structure method (P-GSM)

A new topology optimization scheme called the projection-based ground structure method (P-GSM) is proposed for linear and nonlinear topology optimization designs. For linear design, compared to traditional GSM which are limited to designing slender members, the P-GSM can effectively resolve this limitation and generate functionally graded lattice structures. For additive manufacturing-oriented design, the manufacturing abilities are the key factors to constrain the feasible design space, for example, minimum length and geometry complexity. Conventional density-based method, where each element works as a variable, always results in complex geometry with large number of small intricate features, while these small features are often not manufacturable even by 3D printing and lose its geometric accuracy after postprocessing. The proposed P-GSM is an effective method for controlling geometric complexity and minimum length for optimal design, while it is capable of designing self-supporting structures naturally. In optimization progress, some bars may be disconnected from each other (floating in the air). For buckling-induced design, this issue becomes critical due to severe mesh distortion in the void space caused by disconnection between members, while P-GSM has ability to overcome this issue. To demonstrate the effectiveness of proposed method, three different design problems ranging from compliance optimization to buckling-induced mechanism design are presented and discussed in details.     

A mixed integer programming for robust truss topology optimization with stress constraints

This paper presents a mixed integer programming (MIP) formulation for robust topology optimization of trusses subjected to the stress constraints under the uncertain load. A design‐dependent uncertainty model of the external load is proposed for dealing with the variation of truss topology in the course of optimization. For a truss with the discrete member cross‐sectional areas, it is shown that the robust topology optimization problem can be reduced to an MIP problem, which is solved globally. Numerical examples illustrate that the robust optimal topology of a truss depends on the magnitude of uncertainty. Copyright © 2010 John Wiley & Sons, Ltd.     

Optimization of truss topology using tabu search

A design procedure for integrating topological considerations in the framework of structural optimization is presented. The proposed approach is capable of considering multiple load conditions, stress, displacement and local/global buckling constraints, and multiple objective functions in the problem formulation. Further, since the proposed method permits members to be added to or deleted from an existing topology and the topology is not defined by member areas, the difficulty of not being able to reach singular optima is also avoided. These objectives are accomplished using a discrete optimization procedure which uses 0–1 topological variables to optimize alternate designs. Since the topological variables are discrete in nature and the member cross-sections are assumed to be continuous, the topological optimization problem has mixed discrete-continuous variables. This non-linear programming problem is solved using a memory-based combinatorial optimization technique known as tabu search. Numerical results obtained using tabu search for single and multiobjective topological optimization of truss structures are presented. To model the multiple objective functions in the problem formulation, a cooperative game theoretic approach is used. The results indicate that the optimum topologies obtained using tabu search compare favourably, and in some instances, outperform the results obtained using the ground–structure approach. However, this improvement occurs at the expense of a significant increase in computational burden owing to the fact that the proposed approach necessitates that the geometry of each trial topology be optimized.     

A level set approach for topology optimization with local stress constraints

The purpose of this work is to present a level set‐based approach for the structural topology optimization problem of mass minimization submitted to local stress constraints. The main contributions are threefold. First, the inclusion of local stress constraints by means of an augmented Lagrangian approach within the level set context. Second, the proposition of a constraint procedure that accounts for a continuous activation/deactivation of a finite number of local stress constraints during the optimization sequence. Finally, the proposition of a logarithmic scaling of the level set normal velocity as an additional regularization technique in order to improve the minimization sequence. A set of benchmark tests in two dimensions achieving successful numerical results assesses the good behavior of the proposed method. In these examples, it is verified that the algorithm is able to identify stress concentrations and drive the design to a feasible local minimum. Copyright © 2014 John Wiley & Sons, Ltd.     

Multi-material plastic part design via the level set shape and topology optimization method

This work presents a new multi-material level set topology optimization method which is developed especially for designing plastic parts. Instead of representing the structure using multiple level set functions, this new method employs only one level set function to describe the material/void interface. The injection moulding filling simulation is used to determine the material/material interfaces. Because plastic parts are targeted, domain-specific knowledge is carefully investigated to improve the optimization algorithm. Both homogeneous and heterogeneous fibre-reinforced plastics are considered as potential material phases. For the latter, one extra design freedom, fibre orientation distribution, is introduced. Instead of generating incremental interior voids, which complicates the mould design and part ejection, shape-fixed interior voids could be predefined inside the design domain for functional or assembly purposes. This is represented by an additional level set function. A few numerical examples are studied to demonstrate the effectiveness of the proposed method.     

Material cloud method for topology optimization

Material cloud method (MCM), a new approach for topology optimization, is presented. In MCM, an optimal structure can be obtained by manipulating the sizes and positions of material clouds, which are material patches with finite sizes and constant material densities. The optimal distributions of material clouds can be obtained by MCM using fixed background finite element meshes. In the numerical analysis procedure, only active elements, where more than one material cloud is contained, are treated. Optimal material distribution can be element‐wise extracted from the distribution of material clouds. With MCM, an expansion–reduction procedure of design domain can be naturally realized through movements of material clouds, so that a true optimal solution can be found without any significant increase of computational costs. It is also shown that a clear material distribution with narrow region of intermediate density can be obtained with relatively fast convergence. Several numerical examples are shown. Some of the results are compared with those of the traditional density distribution method (DDM). Copyright © 2005 John Wiley & Sons, Ltd.     

A mixed FEM approach to stress‐constrained topology optimization

We present an alternative topology optimization formulation capable of handling the presence of stress constraints in a straightforward fashion. The main idea is to adopt a mixed finite‐element discretization scheme wherein not only displacements (as usual) but also stresses are the variables entering the formulation. By doing so, any stress constraint may be handled within the optimization procedure without resorting to post‐processing operation typical of displacement‐based techniques that may also cause a loss in accuracy in stress computation if no smoothing of the stress is performed. Two dual variational principles of Hellinger–Reissner type are presented in continuous and discrete form that, which included in a rather general topology optimization problem in the presence of stress constraints that is solved by the method of moving asymptotes (Int. J. Numer. Meth. Engng. 1984; 24 (3):359–373). Extensive numerical simulations are performed and ongoing extensions outlined, including the optimization of elastoplastic and incompressible media. Copyright © 2007 John Wiley & Sons, Ltd.     

A projection-based method for topology optimization of structures with graded surfaces

Graded surfaces widely exist in natural structures and inspire engineers to apply functionally graded (FG) materials to cover structural surfaces for performance improvement, protection, or other special functionalities. However, how to design such structures with FG surfaces by topology optimization is a quite challenging problem due to the difficulty for determining material properties of structural surfaces with prescribed variation rule. This paper presents a novel projection-based method for topology optimization of this class of FG structures. Firstly, a projection process is proposed for ensuring the material properties of the surfaces vary with a prescribed function. A criterion of determining the values of parameters in projection process is given by a strict theoretical derivation, and then, a new interpolation function is established, which is capable of simultaneously obtaining clear substrate topologies and realizable FG surfaces. Though such structures are actually multimaterial gradient structures, only the design variables of single-material topology optimization problem are needed. In the current research, the classical compliance minimization problem with a mass constraint is considered and the robust formulation is used to control the length scale of substrates. Several 2D and 3D numerical examples illustrate the validity and applicability of the proposed method.     

Applying functional principal components to structural topology optimization

Structural topology optimization aims to enhance the mechanical performance of a structure while satisfying some functional constraints. Nearly all approaches proposed in the literature are iterative, and the optimal solution is found by repeatedly solving a finite element analysis (FEA). It is thus clear that the bottleneck is the high computational effort, as these approaches require solving the FEA a large number of times. In this work, we address the need for reducing the computational time by proposing a reduced basis method that relies on functional principal component analysis (FPCA). The methodology has been validated considering a simulated annealing approach for compliance minimization in 2 classical variable thickness problems. Results show the capability of FPCA to provide good results while reducing the computational times, ie, the computational time for an FEA is about one order of magnitude lower in the reduced FPCA space.     

A new hole insertion method for level set based structural topology optimization

Structural shape and topology optimization using level set functions is becoming increasingly popular. However, traditional methods do not naturally allow for new hole creation and solutions can be dependent on the initial design. Various methods have been proposed that enable new hole insertion; however, the link between hole insertion and boundary optimization can be unclear. The new method presented in this paper utilizes a secondary level set function that represents a pseudo third dimension in two‐dimensional problems to facilitate new hole insertion. The update of the secondary function is connected to the primary level set function forming a meaningful link between boundary optimization and hole creation. The performance of the method is investigated to identify suitable parameters that produce good solutions for a range of problems. Copyright © 2012 John Wiley & Sons, Ltd.     

Simultaneous single-loop multimaterial and multijoint topology optimization

As the aerospace and automotive industries continue to strive for efficient lightweight structures, topology optimization (TO) has become an important tool in this design process. However, one ever-present criticism of TO, and especially of multimaterial (MM) optimization, is that neither method can produce structures that are practical to manufacture. Optimal joint design is one of the main requirements for manufacturability. This article proposes a new density-based methodology for performing simultaneous MMTO and multijoint TO. This algorithm can simultaneously determine the optimum selection and placement of structural materials, as well as the optimum selection and placement of joints at material interfaces. In order to achieve this, a new solid isotropic material with penalization-based interpolation scheme is proposed. A process for identifying dissimilar material interfaces based on spatial gradients is also discussed. The capabilities of the algorithm are demonstrated using four case studies. Through these case studies, the coupling between the optimal structural material design and the optimal joint design is investigated. Total joint cost is considered as both an objective and a constraint in the optimization problem statement. Using the biobjective problem statement, the tradeoff between total joint cost and structural compliance is explored. Finally, a method for enforcing tooling accessibility constraints in joint design is presented.     

Hyperstatic and redundancy thresholds in truss topology optimization considering progressive collapse due to aleatory and epistemic uncertainties

Optimization leads to specialized structures which are not robust to disturbance events like unanticipated abnormal loading or human errors. Typical reliability-based and robust optimization mainly address objective aleatory uncertainties. To date, the impact of subjective epistemic uncertainties in optimal design has not been comprehensively investigated. In this paper, we use an independent parameter to investigate the effects of epistemic uncertainties in optimal design: the latent failure probability. Reliability-based and risk-based truss topology optimization are addressed. It is shown that optimal risk-based designs can be divided in three groups: (A) when epistemic uncertainty is small (in comparison to aleatory uncertainty), the optimal design is indifferent to it and yields isostatic structures; (B) when aleatory and epistemic uncertainties are relevant, optimal design is controlled by epistemic uncertainty and yields hyperstatic but nonredundant structures, for which expected costs of direct collapse are controlled; (C) when epistemic uncertainty becomes too large, the optimal design becomes redundant, as a way to control increasing expected costs of collapse. The three regions above are divided by hyperstatic and redundancy thresholds. The redundancy threshold is the point where the structure needs to become redundant so that its reliability becomes larger than the latent reliability of the simplest isostatic system. Simple truss topology optimization is considered herein, but the conclusions have immediate relevance to the optimal design of realistic structures subject to aleatory and epistemic uncertainties.     

A new efficient convergence criterion for reducing computational expense in topology optimization: reducible design variable method

A new efficient convergence criterion, named the reducible design variable method (RDVM), is proposed to save computational expense in topology optimization. There are two types of computational costs: one is to calculate the governing equations, and the other is to update the design variables. In conventional topology optimization, the number of design variables is usually fixed during the optimization procedure. Thus, the computational expense linearly increases with respect to the iteration number. Some design variables, however, quickly converge and some other design variables slowly converge. The idea of the proposed method is to adaptively reduce the number of design variables on the basis of the history of each design variable during optimization. Using the RDVM, those design variables that quickly converge are not considered as design variables for the next iterations. This means that the number of design variables can be reduced to save the computational costs of updating design variables. Then, the iteration will repeat until the number of design variables becomes 0. In addition, the proposed method can lead to faster convergence of the optimization procedure, which indeed is a more significant time saving. It is also revealed that the RDVM gives identical optimal solutions as those by conventional methods. We confirmed the numerical efficiency and solution effectiveness of the RDVM with respect to two types of optimization: static linear elastic minimization, and linear vibration problems with the first eigenvalue as the objective function for maximization. Copyright © 2012 John Wiley & Sons, Ltd.     

Global optimization of minimum weight truss topology problems with stress,displacement, and local buckling constraints using branch‐and‐bound

We present a convergent continuous branch‐and‐bound algorithm for global optimization of minimum weight truss topology problems with displacement, stress, and local buckling constraints. Valid inequalities which strengthen the problem formulation are derived. The inequalities are generated by solving well‐defined convex optimization problems. Computational results are reported on a large collection of problems taken from the literature. Most of these problems are, for the first time, solved with a proof of global optimality. Copyright © 2004 John Wiley & Sons, Ltd.     

A new level set method for topology optimization of distributed compliant mechanisms

A new level set method for topology optimization of distributed compliant mechanism is presented in this study. By taking two types of mean compliance into consideration, several new objective functions are developed and built in the conventional level set method to avoid generating the de facto hinges in the created mechanisms. Aimed at eliminating the costly reinitialization procedure during the evolution of the level set function, an accelerated level set evolution algorithm is developed by adding an extra energy function, which can force the level set function to close to a signed distance function during the evolution. Two widely studied numerical examples in topology optimization of compliant mechanisms are studied to demonstrate the effectiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

基于信息熵改进的粒子群算法在桁架结构

提出了信息熵改进的粒子群优化算法用于解决有应力约束、位移约束的桁架结构杆件截面尺寸优化设计问题.首先介绍了信息熵基本理论和基本粒子群优化算法理论,然后对粒子群优化算法作了合理的参数设置,并将信息熵引入粒子群优化算法的适应函数和停机判别准则中.最后对2个经典的优化问题进行求解并与其他算法进行了比较.数据结果表明信息熵改进后的粒子群优化算法在桁架结构优化设计中优于其他同类算法.     

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。     

Adaptive moving mesh level set method for structure topology optimization

The level set method is a promising approach to provide flexibility in dealing with topological changes during structural optimization. Normally, the level set surface, which depicts a structure's topology by a level contour set of a continuous scalar function embedded in space, is interpolated on a fixed mesh. The accuracy of the boundary positions is therefore largely dependent on the mesh density, a characteristic of any Eulerian expression when using a fixed mesh. This article combines the adaptive moving mesh method with a level set structure topology optimization method. The finite element mesh automatically maintains a high nodal density around the structural boundaries of the material domain, whereas the mesh topology remains unchanged. Numerical experiments demonstrate the effect of the combination of a Lagrangian expression for a moving mesh and a Eulerian expression for capturing the moving boundaries.     

A geometry projection method for the topology optimization of curved plate structures with placement bounds

Single‐curvature plates are commonly encountered in mechanical and civil structures. In this paper, we introduce a topology optimization method for the stiffness‐based design of structures made of curved plates with fixed thickness. The geometry of each curved plate is analytically and explicitly represented by its location, orientation, dimension, and curvature radius, and therefore, our method renders designs that are distinctly made of curved plates. To perform the primal and sensitivity analyses, we use the geometry projection method, which smoothly maps the analytical geometry of the curved plates onto a continuous density field defined over a fixed uniform finite element grid. A size variable is ascribed to each plate and penalized in the spirit of solid isotropic material with penalization, which allows the optimizer to remove a plate from the design. We also introduce in our method a constraint that ensures that no portion of a plate lies outside the design envelope. This prevents designs that would otherwise require cuts to the plates that may be very difficult to manufacture. We present numerical examples to demonstrate the validity and applicability of the proposed method.