Multiphase topology optimization with a single variable using the phase-field design method

In this study, a multimaterial topology optimization method using a single variable is proposed by combining the solid isotropic material with penalization method and the reaction-diffusion equation. Unlike ordinary multimaterial optimization, which requires several variables depending on the number of material types, this method intends to represent various materials as one variable. The proposed method combines two special functions in the sensitivity analysis of the objective function to converge the design variable into prespecified density values defined for each of the multimaterials. The composition constraint based on a normal distribution function is also introduced to estimate the distribution of each target density value in a single variable. It enables density exchange between multiple materials by increasing or decreasing the amount of a specific material. The proposed method is applied to structural and electromagnetic problems to verify its effectiveness, and its usefulness is also confirmed from the viewpoint of cost and computation time.     

Multimaterial topology design for optimal elastic and thermal response with material-specific temperature constraints

We present an original method for multimaterial topology optimization with elastic and thermal response considerations. The material distribution is represented parametrically using a formulation in which finite element–style shape functions are used to determine the local material properties within each finite element. We optimize a multifunctional structure that is designed for a combination of structural stiffness and thermal insulation. We conduct parallel uncoupled finite element analyses to simulate the elastic and thermal response of the structure by solving the two-dimensional Poisson problem. We explore multiple optimization problem formulations, including structural design for minimum compliance subject to local temperature constraints so that the optimized design serves as both a support structure and a thermal insulator. We also derive and implement an original multimaterial aggregation function that allows the designer to simultaneously enforce separate maximum temperature thresholds based upon the melting point of the various design materials. The nonlinear programming problem is solved using gradient-based optimization with adjoint sensitivity analysis. We present results for a series of two-dimensional example problems. The results demonstrate that the proposed algorithm consistently converges to feasible multimaterial designs with the desired elastic and thermal performance.     

A level set–based method for stress-constrained multimaterial topology optimization of minimizing a global measure of stress

In multimaterial topology optimization of minimizing a global measure of stress, the maximum stresses in different materials may not satisfy the strength design requirements simultaneously if stress constraints for different materials are not considered. In this paper, a level set–based method is presented to handle the stress-constrained multimaterial topology optimization of minimizing a global stress measure. Specifically, a multimaterial level set model is adopted to describe the structural topology, and a stress interpolation scheme is introduced for stress evaluation. Then, a stress penalty-based topology optimization model is presented. Meanwhile, an adaptive adjusting scheme of the stress penalty factor is employed to improve the control of the local stress level. To solve the stress-constrained multimaterial topology optimization problem minimizing the global measure of stress, the parametric level set method is employed, and the sensitivity analysis is carried out. Numerical examples are provided to demonstrate the effectiveness of the presented method. Results indicate that multimaterial structures with optimized global stress can be gained, and stress constraints for different materials can be satisfied simultaneously.     

Multimaterial multijoint topology optimization

In this paper, a methodology that solves multimaterial topology optimization problems while also optimizing the quantity and type of joints between dissimilar materials is proposed. Multimaterial topology optimization has become a popular design optimization technique since the enhanced design freedom typically leads to superior solutions; however, the conventional assumption that all elements are perfectly fused together as a single piece limits the usefulness of the approach since the mutual dependency between optimal multimaterial geometry and optimal joint design is not properly accounted for. The proposed methodology uses an effective decomposition approach to both determine the optimal topology of a structure using multiple materials and the optimal joint design using multiple joint types. By decomposing the problem into two smaller subproblems, gradient‐based optimization techniques can be used and large models that cannot be solved with nongradient approaches can be solved. Moreover, since the joining interfaces are interpreted directly from multimaterial topology optimization results, the shape of the joining interfaces and the quantity of joints connecting dissimilar materials do not need to be defined a priori. Three numerical examples, which demonstrate how the methodology optimizes the geometry of a multimaterial structure for both compliance and cost of joining, are presented.     

Composite material design of two‐dimensional structures using the homogenization design method

Composite materials of two‐dimensional structures are designed using the homogenization design method. The composite material is made of two or three different material phases. Designing the composite material consists of finding a distribution of material phases that minimizes the mean compliance of the macrostructure subject to volume fraction constraints of the constituent phases, within a unit cell of periodic microstructures. At the start of the computational solution, the material distribution of the microstructure is represented as a pure mixture of the constituent phases. As the iteration procedure unfolds, the component phases separate themselves out to form distinctive interfaces. The effective material properties of the artificially mixed materials are defined by the interpolation of the constituents. The optimization problem is solved using the sequential linear programming method. Both the macrostructure and the microstructures are analysed using the finite element method in each iteration step. Several examples of optimal topology design of composite material are presented to demonstrate the validity of the present numerical algorithm. Copyright © 2001 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.     

Infill sampling criteria for surrogate-based optimization with constraint handling

This article discusses the benefits of different infill sampling criteria used in surrogate-based constrained global optimization. A new method which selects multiple updates based on Pareto optimal solutions is introduced showing improvements over a number of existing methods. The construction of surrogates (also known as meta-models or response surface models) involves the selection of a limited number of designs which are analysed using the original expensive functions. A typical approach involves two stages. First the surrogate is built using an initial sampling plan; the second stage updates the model using an infill sampling criterion to select further designs that offer improvement. Selecting multiple update points at each iteration, allowing distribution of the expensive function evaluations on several processors offers large potential for accelerating the overall optimization process. This article provides a comparison between different infill sampling criteria suitable for selecting multiple update points in the presence of constraints.     

航材包装单元多隔层集装箱装载优化研究

目的 通过对装载模型开展研究,提高航材包装单元在组合式多隔层集装箱中的装载率,为部队进行集装箱装载工作提供技术方法。方法 依据待装载航材包装单元和集装箱尺寸,分析集装箱内部的分割方案,考虑装载容积、载质量、底面积和装载重心等约束条件,综合空间利用率和载质量为目标函数,建立数学模型,并利用遗传算法对每种分割方案进行优化计算,确定最佳的分割与装载方案。结果 以26个航材包装单元待装入B9型组合式集装箱为例,最优的装载方案是将集装箱内部分割成3层,16个航材包装单元被装入,空间利用率达到86.25%,装载后集装箱的稳定性良好。结论 该模型能有效优化航材包装单元的装载方案,提升组合式集装箱的装载效率,对加快军用物资的输送具有重要意义。     

Path planning of functionally graded material objects for layered manufacturing

Path planning is a fundamental process-planning task in layered manufacturing. However, currently, most researches focus on path planning of homogeneous objects and few approaches for path planning of functionally graded material objects have been reported in the literature. Unlike homogeneous objects, functionally graded materials objects contain continuously varying material information as well as geometry information, which makes their path planning more complex and difficult than those of homogeneous objects. This paper presents an approach for path planning of functionally graded material objects. First, the continuous material distribution in each layer is changed into discrete step-wise gradings by subdividing the functionally graded materials slice into multimaterial subregions, which can be regarded as homogeneous material regions so that the functionally graded material layer can be built by mixing materials into desired volume ratio. Then the tool paths and process parameters for each multimaterial subregion in the slice are generated separately. An algorithm that summarized the procedure is described and an example is also presented.     

危化品事故连续消耗型应急物资调度模型

应急物资调度是危化品事故应急救援工作的关键内容,危化品事故会对周边区域造成环境污染,其应急物资调度问题应该考虑多个目标。针对危化品事故应急物资调度特征,建立了连续消耗应急物资调度多目标模型。模型以缺货损失最小、调运时间最短和调运总环境风险最小为优化目标,考虑应急中心物资数量限制和救援时间限制等约束条件,采用NSGAⅡ遗传算法求解。对于求解结果,采用逼近理想解的排序方法进行选择。算例计算结果表明,多目标模型优化方案比单目标模型优化方案有较大的优势,同时说明算法合理可行。     

Geometry update driven by material forces for simulation of brittle crack growth in functionally graded materials

Functionally graded materials (FGMs) are advanced materials that possess continuously graded properties, such that the growth of cracks is strongly dependent on the gradation of the material. In this work a thermodynamic consistent framework for crack propagation in FGMs is presented, by applying a dissipation inequality to a time‐dependent migrating control volume. The direction of crack growth is obtained in terms of material forces as a result of the principle of maximum dissipation. In the numerical implementation a staggered algorithm—deformation update for fixed geometry followed by geometry update for fixed deformation—is employed within each time increment. The geometry update is a result of the incremental crack propagation, which is driven by material forces. The corresponding mesh is generated by combining Delaunay triangulation with local mesh refinement. Furthermore a Newton algorithm is proposed, taking into account mesh transfer of displacements for crack propagation in incremental elasticity. In two numerical examples brittle crack propagation in FGMs is investigated for various directions of strength gradation within the structures. Copyright © 2008 John Wiley & Sons, Ltd.     

Level-set topology optimization for maximizing fracture resistance of brittle materials using phase-field fracture model

Fracture is one of the most common failure modes in brittle materials. It can drastically decrease material integrity and structural strength. To address this issue, we propose a level-set (LS) based topology optimization procedure to optimize the distribution of reinforced inclusions within matrix materials subject to the volume constraint for maximizing structural resistance to fracture. A phase-field fracture model is formulated herein to simulate crack initiation and propagation, in which a staggered algorithm is developed to solve such time-dependent crack propagation problems. In line with diffusive damage of the phase-field approach for fracture; topological derivatives, which provide gradient information for the topology optimization in a LS framework, are derived for fracture mechanics problems. A reaction-diffusion equation is adopted to update the LS function within a finite element framework. This avoids the reinitialization by overcoming the limitation to time step with the Courant-Friedrichs-Lewy condition. In this article, three numerical examples, namely, a L-shaped section, a rectangular slab with predefined cracks, and an all-ceramic onlay dental bridge (namely, fixed partial denture), are presented to demonstrate the effectiveness of the proposed LS based topology optimization for enhancing fracture resistance of multimaterial composite structures in a phase-field fracture context.     

基于双向渐进结构拓扑优化算法的创新设计研究

目的 在数字化设计的背景下,探索基于结构性能化的算法找形方法,运用双向渐进结构拓扑优化算法(Bi-Directional Evolutionary Structural Optimization,BESO)展开创新设计实践研究。方法 在理解双向渐进结构拓扑优化算法的基本内涵、相关理论、历史发展和现状应用的基础上,分析其算法生成的优势及可行性,并以算法的组织模式与生形原理为前提,对其进行几何划分、约束条件、优化技术、结构模拟、材料设定、迭代生形等内容协同一体的生成策略研究,提供了多元选择的设计机会。结果 得到了运用双向渐进结构拓扑优化算法进行的基于初始形态设计、拓扑优化设计和后处理与制造三步骤创新设计实践结果。结论 此设计实践方案验证了该算法生成方法的设计应用可行性,同时也为多领域应用该算法提供了新思路和新方向。     

Topology optimization with incompressible materials under small and finite deformations using mixed u/p elements

This paper focuses on topology optimization utilizing incompressible materials under both small‐ and finite‐deformation kinematics. To avoid the volumetric locking that accompanies incompressibility, linear and nonlinear mixed displacement/pressure (u/p) elements are utilized. A number of material interpolation schemes are compared, and a new scheme interpolating both Young's modulus and Poisson's ratio (E ‐ν interpolation) is proposed. The efficacy of this proposed scheme is demonstrated on a number of examples under both small‐ and finite‐deformation kinematics. Excessive mesh distortions that may occur under finite deformations are dealt with by extending a linear energy interpolation approach to the nonlinear u/p formulation and utilizing an adaptive update strategy. The proposed optimization framework is demonstrated to be effective through a number of representative examples.     

Continuum structural topological optimizations with stress constraints based on an active constraint technique

Stress‐related problems have not been given the same attention as the minimum compliance topological optimization problem in the literature. Continuum structural topological optimization with stress constraints is of wide engineering application prospect, in which there still are many problems to solve, such as the stress concentration, an equivalent approximate optimization model and etc. A new and effective topological optimization method of continuum structures with the stress constraints and the objective function being the structural volume has been presented in this paper. To solve the stress concentration issue, an approximate stress gradient evaluation for any element is introduced, and a total aggregation normalized stress gradient constraint is constructed for the optimized structure under the r?th load case. To obtain stable convergent series solutions and enhance the control on the stress level, two p‐norm global stress constraint functions with different indexes are adopted, and some weighting p‐norm global stress constraint functions are introduced for any load case. And an equivalent topological optimization model with reduced stress constraints is constructed,being incorporated with the rational approximation for material properties, an active constraint technique, a trust region scheme, and an effective local stress approach like the qp approach to resolve the stress singularity phenomenon. Hence, a set of stress quadratic explicit approximations are constructed, based on stress sensitivities and the method of moving asymptotes. A set of algorithm for the one level optimization problem with artificial variables and many possible non‐active design variables is proposed by adopting an inequality constrained nonlinear programming method with simple trust regions, based on the primal‐dual theory, in which the non‐smooth expressions of the design variable solutions are reformulated as smoothing functions of the Lagrange multipliers by using a novel smoothing function. Finally, a two‐level optimization design scheme with active constraint technique, i.e. varied constraint limits, is proposed to deal with the aggregation constraints that always are of loose constraint (non active constraint) features in the conventional structural optimization method. A novel structural topological optimization method with stress constraints and its algorithm are formed, and examples are provided to demonstrate that the proposed method is feasible and very effective. © 2016 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.     

An asymptotically concentrated method for structural topology optimization based on the SIMLF interpolation

In this work, an asymptotically concentrated topology optimization method based on the solid isotropic material with logistic function interpolation is proposed. The asymptotically concentrated method is introduced into the process of optimization cycle after updating the design variables. At the same time, with the use of the solid isotropic material with logistic function interpolation, all the candidate densities are reasonably polarized, relying on the characteristic of the interpolation curve itself. The asymptotically concentrated method can effectively suppress the generation of intermediate density and speed up the process of updating the design variables, hence improving the optimization efficiency. Moreover, the above polarization can weaken the influence of low‐related‐density elements and enhance the influence of high‐related‐density elements. For the single‐material topology optimization problem, gray‐scale elements can be effectively eliminated, and clear boundary and smaller compliance can be obtained by this method. For the multimaterial topology optimization problem, minimum compliance with high efficiency can be achieved by this method. The proposed method mainly includes the following advantages: concentrated density variables, reasonable interpolation, high computational efficiency, and good topological results.     

基于储运单元的航材携行组配研究

目的 为有效提高航材保障应急准备和战时快速保障能力,探索基于储运单元的航材携行组配新模式,解决组配航材需求大与库存条件有限的矛盾问题。方法 科学设计模块化航材携行组配方案,以组配航材需求最少为目标,分别构建了绝对约束和宽松约束条件下的组配模块配置优化模型,并通过Matlab软件实现模型求解自动化。同时,引入模块配置允许偏差约束条件,以灵敏度分析为方法手段,合理调整组配方案中与库存条件不相适应的部分。结果 算例分析表明,模块化航材携行组配方案能够使组配规模和器材需求降低58.9%,可行性大幅提高,文中构建的模块配置优化模型既能够严格按照航材携行标准精确组配,又能够根据现实状况灵活应变,具有较强的实用性、适应性。结论 航材携行组配模块配置优化模型能够支持组配方案的实施,对航材携行组配模式研究的推进、提高航材保障机动性,具有重要的现实意义。     

A pegging algorithm for separable continuous nonlinear knapsack problems with box constraints

This article proposes an efficient pegging algorithm for solving separable continuous nonlinear knapsack problems with box constraints. A well-known pegging algorithm for solving this problem is the Bitran–Hax algorithm, a preferred choice for large-scale problems. However, at each iteration, it must calculate an optimal dual variable and update all free primal variables, which is time consuming. The proposed algorithm checks the box constraints implicitly using the bounds on the Lagrange multiplier without explicitly calculating primal variables at each iteration as well as updating the dual solution in a more efficient manner. Results of computational experiments have shown that the proposed algorithm consistently outperforms the Bitran–Hax in all baseline testing and two real-time application models. The proposed algorithm shows significant potential for many other mathematical models in real-world applications with straightforward extensions.