Multiresolution topology optimization using isogeometric analysis

The paper introduces a novel multiresolution scheme to topology optimization in the framework of the isogeometric analysis. A new variable parameter space is added to implement multiresolution topology optimization based on the Solid Isotropic Material with Penalization approach. Design density variables defined in the variable space are used to approximate the element analysis density by the bivariate B‐spline basis functions, which are easily obtained using k‐refinement strategy in the isogeometric analysis. While the nonuniform rational B‐spline basis functions are used to exactly describe geometric domains and approximate unknown solutions in finite element analysis. By applying a refined sensitivity filter, optimized designs include highly discrete solutions in terms of solid and void materials without using any black and white projection filters. The Method of Moving Asymptotes is used to solve the optimization problem. Various benchmark test problems including plane stress, compliant mechanism inverter, and 2‐dimensional heat conduction are examined to demonstrate the effectiveness and robustness of the present method.     

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.     

Optimal design of piezoelectric microstructures

Application of piezoelectric materials requires an improvement in their performance characteristics which can be obtained by designing new topologies of microstructures (or unit cells) for these materials. The topology of the unit cell (and the properties of its constituents) determines the effective properties of the piezocomposite. By changing the unit cell topology, better performance characteristics can be obtained in the piezocomposite. Based on this idea, we have proposed in this work an optimal design method of piezocomposite microstructures using topology optimization techniques and homogenization theory. The topology optimization method consists of finding the distribution of material phase and void phase in a periodic unit cell, that optimizes the performance characteristics, subject to constraints such as property symmetry and stiffness. The optimization procedure is implemented using sequential linear programming. In order to calculate the effective properties of a unit cell with complex topology, a general homogenization method applied to piezoelectricity was implemented using the finite element method. This method has no limitations regarding volume fraction or shape of the composite constituents. Although only two-dimensional plane strain topologies of microstructures have been considered to show the implementation of the method, this can be extended to three-dimensional topologies. Microstructures obtained show a large improvement in performance characteristics compared to pure piezoelectric material or simple designs of piezocomposite unit cells.     

考虑混合约束的柔顺机构拓扑优化设计

为了满足制造工艺和静强度要求,提出一种综合考虑最小尺寸控制和应力约束的柔顺机构混合约束拓扑优化设计方法。采用改进的固体各向同性材料插值模型描述材料分布,利用多相映射方法同时控制实相和空相材料结构的最小尺寸,采用最大近似函数P范数求解机构的最大应力,以机构的输出位移最大化作为目标函数,综合考虑最小特征尺寸控制和应力约束建立柔顺机构混合约束拓扑优化数学模型,利用移动渐近算法求解柔顺机构混合约束拓扑优化问题。数值算例结果表明,混合约束拓扑优化获得的柔顺机构能够同时满足最小尺寸制造约束和静强度要求,机构的von Mises等效应力分布更加均匀。     

考虑混合约束的柔顺机构拓扑优化设计

为了满足制造工艺和静强度要求,提出一种综合考虑最小尺寸控制和应力约束的柔顺机构混合约束拓扑优化设计方法。采用改进的固体各向同性材料插值模型描述材料分布,利用多相映射方法同时控制实相和空相材料结构的最小尺寸,采用最大近似函数P范数求解机构的最大应力,以机构的输出位移最大化作为目标函数,综合考虑最小特征尺寸控制和应力约束建立柔顺机构混合约束拓扑优化数学模型,利用移动渐近算法求解柔顺机构混合约束拓扑优化问题。数值算例结果表明,混合约束拓扑优化获得的柔顺机构能够同时满足最小尺寸制造约束和静强度要求,机构的von Mises等效应力分布更加均匀。     

Level-set method for design of multi-phase elastic and thermoelastic materials

The problem of designing composite materials with desired mechanical properties is to specify the materials microstructures in terms of the topology and distribution of their constituent material phases within a unit cell of periodic microstructures. In this paper we present an approach based on a multi-phase level-set model for the geometric and material representation and for numerical solution of a least squares optimization problem. The level-set model precisely specifies the material regions and their sharp boundaries in contrast to a raster discretization of the conventional homogenization-based approaches. Combined with the classical shape derivatives, the level-set method yields a computational system of partial differential equations. In using the Eulerian computation scheme with a fixed rectilinear grid and a fixed mesh in the unit cell, the gradient descent solution of the optimization captures the interfacial boundaries naturally and performs topological changes accurately. The proposed method is illustrated with several 2D examples for the synthesis of heterogeneous microstructures of elastic and/or thermoelastic composites composed of two and three material phases.     

Scale‐related topology optimization of cellular materials and structures

The integrated optimization of lightweight cellular materials and structures are discussed in this paper. By analysing the basic features of such a two‐scale problem, it is shown that the optimal solution strongly depends upon the scale effect modelling of the periodic microstructure of material unit cell (MUC), i.e. the so‐called representative volume element (RVE). However, with the asymptotic homogenization method used widely in actual topology optimization procedure, effective material properties predicted can give rise to limit values depending upon only volume fractions of solid phases, properties and spatial distribution of constituents in the microstructure regardless of scale effect. From this consideration, we propose the design element (DE) concept being able to deal with conventional designs of materials and structures in a unified way. By changing the scale and aspect ratio of the DE, scale‐related effects of materials and structures are well revealed and distinguished in the final results of optimal design patterns. To illustrate the proposed approach, numerical design problems of 2D layered structures with cellular core are investigated. Copyright © 2006 John Wiley & Sons, Ltd.     

Towards the stabilization of the low density elements in topology optimization with large deformation

This work addresses the treatment of lower density regions of structures undergoing large deformations during the design process by the topology optimization method (TOM) based on the finite element method. During the design process the nonlinear elastic behavior of the structure is based on exact kinematics. The material model applied in the TOM is based on the solid isotropic microstructure with penalization approach. No void elements are deleted and all internal forces of the nodes surrounding the void elements are considered during the nonlinear equilibrium solution. The distribution of design variables is solved through the method of moving asymptotes, in which the sensitivity of the objective function is obtained directly. In addition, a continuation function and a nonlinear projection function are invoked to obtain a checkerboard free and mesh independent design. 2D examples with both plane strain and plane stress conditions hypothesis are presented and compared. The problem of instability is overcome by adopting a polyconvex constitutive model in conjunction with a suggested relaxation function to stabilize the excessive distorted elements. The exact tangent stiffness matrix is used. The optimal topology results are compared to the results obtained by using the classical Saint Venant–Kirchhoff constitutive law, and strong differences are found.     

An n‐material thresholding method for improving integerness of solutions in topology optimization

It is common in solving topology optimization problems to replace an integer‐valued characteristic function design field with the material volume fraction field, a real‐valued approximation of the design field that permits ‘fictitious’ mixtures of materials during intermediate iterations in the optimization process. This is reasonable so long as one can interpolate properties for such materials and so long as the final design is integer valued. For this purpose, we present a method for smoothly thresholding the volume fractions of an arbitrary number of material phases which specify the design. This method is trivial for two‐material design problems, for example, the canonical topology design problem of specifying the presence or absence of a single material within a domain, but it becomes more complex when three or more materials are used, as often occurs in material design problems. We take advantage of the similarity in properties between the volume fractions and the barycentric coordinates on a simplex to derive a thresholding, method which is applicable to an arbitrary number of materials. As we show in a sensitivity analysis, this method has smooth derivatives, allowing it to be used in gradient‐based optimization algorithms. We present results, which show synergistic effects when used with Solid Isotropic Material with Penalty and Rational Approximation of Material Properties material interpolation functions, popular methods of ensuring integerness of solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

特定弹性性能材料的细观结构设计优化

针对具有特定弹性性质的两相复合材料, 研究了特定性能材料优化设计问题的数学模型,提出了基于形状优化的材料设计方法。该方法利用形状优化技术, 设计两相复合材料的细观结构形式, 以使复合材料具有特定的弹性性质。材料的宏观性质由均匀化方法确定。最后给出了零泊松比材料的设计过程和结果。      

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.     

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.     

General topology optimization method with continuous and discrete orientation design using isoparametric projection

A general topology optimization method, which is capable of simultaneous design of density and orientation of anisotropic material, is proposed by introducing orientation design variables in addition to the density design variable. In this work, the Cartesian components of the orientation vector are utilized as the orientation design variables. The proposed method supports continuous orientation design, which is out of the scope of discrete material optimization approaches, as well as design using discrete angle sets. The advantage of this approach is that vector element representation is less likely to fail into local optima because it depends less on designs of former steps, especially compared with using the angle as a design variable (Continuous Fiber Angle Optimization) by providing a flexible path from one angle to another with relaxation of orientation design space. An additional advantage is that it is compatible with various projection or filtering methods such as sensitivity filters and density filters because it is free from unphysical bound or discontinuity such as the one at θ = 2π and θ = 0 seen with direct angle representation. One complication of Cartesian component representation is the point‐wise quadratic bound of the design variables; that is, each pair of element values has to reside in a given circular bound. To overcome this issue, we propose an isoparametric projection method, which transforms box bounds into circular bounds by a coordinate transformation with isoparametric shape functions without having the singular point that is seen at the origin with polar coordinate representation. A new topology optimization method is built by taking advantage of the aforementioned features and modern topology optimization techniques. Several numerical examples are provided to demonstrate its capability. Copyright © 2014 John Wiley & Sons, Ltd.     

A new method to generate the ground structure in truss topology optimization

The aim of this article is to provide an effective method to generate the ground structure in truss topology optimization. The core of this method is to place nodal points for the ground structure at the intersection of the first and third principal stress trajectories, which are obtained by solving the equivalent static problem in the design domain with a homogeneous isotropic material property. It is applicable to generate the ground structure for arbitrary regular and irregular geometric design domains. The proposed method is tested on some benchmark examples in truss topology optimization. The optimization model is a standard linear programming problem based on plastic design and solved by the interior point algorithm. Compared with other methods, the proposed method may use a well-defined ground structure with fewer nodes and bars, resulting in faster solution convergence, which shows it to be efficient.     

Topology optimization of continuum structures with bi-modulus materials

Since the elasticity of bi-modulus materials is stress dependent, it is difficult to apply most conventional topology optimization methods to such bi-modulus structures owing to great computational expense. Therefore, this study employs the material-replacement method to improve the computational efficiency for topology optimization of bi-modulus structures. In this method, first, the bi-modulus material is replaced by two isotropic materials which have the same tensile or compressive modulus. Secondly, the isotropic materials for finite elements are determined by the local stress/strain states. The local elemental stiffness can be modified according to the current modulus and stress state of the element. Thirdly, the relative densities of elements, acting as the design variables, are updated using the optimality criterion method. Finally, the distributions of elemental densities and moduli are obtained for further applications. Several typical numerical examples are used to demonstrate the effectiveness of the proposed method.     

Topology optimization of material‐nonlinear continuum structures by the element connectivity parameterization

The application of the element density‐based topology optimization method to nonlinear continuum structures is limited to relatively simple problems such as bilinear elastoplastic material problems. Furthermore, it is very difficult to use analytic sensitivity when a commercial nonlinear finite element code is used. As an alternative to the element density formulation, the element connectivity parameterization (ECP) formulation is developed for the topology optimization of isotropic‐hardening elastoplastic or hyperelastic continua by using commercial software. ECP varies the stiffness of zero‐length linear elastic links that connect design domain‐discretizing finite elements. Unloading was not considered. But the advantages of ECP in material‐nonlinear problems were demonstrated: considerably simple analytic sensitivity calculation using a commercial code and simple link stiffness penalization regardless of nonlinear material behaviour. Copyright © 2006 John Wiley & Sons, Ltd.     

Voigt–Reuss topology optimization for structures with linear elastic material behaviours

The desired results of variable topology material layout computations are stable and discrete material distributions that optimize the performance of structural systems. To achieve such material layout designs a continuous topology design framework based on hybrid combinations of classical Reuss (compliant) and Voigt (stiff) mixing rules is investigated. To avoid checkerboarding instabilities, the continuous topology optimization formulation is coupled with a novel spatial filtering procedure. The issue of obtaining globally optimal discrete layout designs with the proposed formulation is investigated using a continuation method which gradually transitions from the stiff Voigt formulation to the compliant Reuss formulation. The very good performance of the proposed methods is demonstrated on four structural topology design optimization problems from the literature. © 1997 John Wiley & sons, Ltd.     

零膨胀材料设计与模拟验证

零膨胀材料对提高航空航天结构和电子设备等的热几何稳定性有重要意义。采用拓扑优化技术设计各相材料在单胞域的分布形式, 以获得零膨胀材料的微结构形式。给出了由二相实体材料和空心构成的各向同性零膨胀材料的设计方案, 讨论了初始设计依赖性问题, 分析了该依赖性的存在原因。采用有限元技术代替实际测试, 分析了所设计材料的试件在均匀温度变化下的变形, 验证了所设计材料的零膨胀(低膨胀) 性质, 说明通过拓扑优化技术设计材料的微结构是设计零膨胀材料的有效方法。     

A nodal variable method of structural topology optimization based on Shepard interpolant

A method for topology optimization of continuum structures based on nodal density variables and density field mapping technique is investigated. The original discrete‐valued topology optimization problem is stated as an optimization problem with continuous design variables by introducing a material density field into the design domain. With the use of the Shepard family of interpolants, this density field is mapped onto the design space defined by a finite number of nodal density variables. The employed interpolation scheme has an explicit form and satisfies range‐restricted properties that makes it applicable for physically meaningful density interpolation. Its ability to resolve more complex spatial distribution of the material density within an individual element, as compared with the conventional elementwise design variable approach, actually provides certain regularization to the topology optimization problem. Numerical examples demonstrate the validity and applicability of the proposed formulation and numerical techniques. Copyright © 2011 John Wiley & Sons, Ltd.     

Nature‐Inspired Lightweight Cellular Co‐Continuous Composites with Architected Periodic Gyroidal Structures

Shell‐core cellular composites are a unique class of cellular materials, where the base constituent is made of a composite material such that the best distinctive physical and/or mechanical properties of each phase of the composite are employed. In this work, the authors demonstrate the additive manufacturing of a nature inspired cellular three‐dimensional (3D), periodic, co‐continuous, and complex composite materials made of a hard‐shell and soft‐core system. The architecture of these composites is based on the Schoen's single Gyroidal triply periodic minimal surface. Results of mechanical testing show the possibility of having a wide range of mechanical properties by tuning the composition, volume fraction of core, shell thickness, and internal architecture of the cellular composites. Moreover, a change in deformation and failure mechanism is observed when employing a shell‐core composite system, as compared to the pure stiff polymeric standalone cellular material. This shell‐core configuration and Gyroidal topology allowed for accessing toughness values that are not realized by the constituent materials independently, showing the suitability of this cellular composite for mechanical energy absorption applications.