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.     

Design of dissipative multimaterial viscoelastic-hyperelastic systems at finite strains via topology optimization

This study focuses on the topology optimization framework for the design of multimaterial dissipative systems at finite strains. The overall goal is to combine a soft viscoelastic material with a stiff hyperelastic material for realizing optimal structural designs with tailored damping and stiffness characteristics. To this end, several challenges associated with incorporating finite-deformation viscoelastic-hyperelastic materials in a multimaterial design framework are addressed. This includes consideration of a thermodynamically consistent finite-strain viscoelasticity model for simulating energy dissipation together with F-bar finite elements for handling material incompressibility. Moreover, an effective multimaterial interpolation scheme is proposed, which preserves the physics of material mixtures in the context of density-based topology optimization. A numerically accurate analytical design sensitivity calculation is also presented using a path-dependent adjoint method. Furthermore, both prescribed-load and prescribed-displacement boundary conditions are considered in the optimization formulations, together with various strategies for controlling stiffness. As demonstrated by the numerical examples, the use of the stiffer hyperelastic material phase in a design not only improves stiffness but also increases energy dissipation capacity. Moreover, with the finite-deformation theory, the effect of the loading magnitude on the optimized designs can be observed.     

Conceptual design of reinforced concrete structures using topology optimization with elastoplastic material modeling

Design of reinforced concrete structures is governed by the nonlinear behavior of concrete and by its different strengths in tension and compression. The purpose of this article is to present a computational procedure for optimal conceptual design of reinforced concrete structures on the basis of topology optimization with elastoplastic material modeling. Concrete and steel are both considered as elastoplastic materials, including the appropriate yield criteria and post‐yielding response. The same approach can be applied also for topology optimization of other material compositions where nonlinear response must be considered. Optimized distribution of materials is achieved by introducing interpolation rules for both elastic and plastic material properties. Several numerical examples illustrate the capability and potential of the proposed procedure. Copyright © 2012 John Wiley & Sons, Ltd.     

Bidirectional evolutionary structural optimization for nonlinear structures under dynamic loads

This study aims to develop efficient numerical optimization methods for finding the optimal topology of nonlinear structures under dynamic loads. The numerical models are developed using the bidirectional evolutionary structural optimization method for stiffness maximization problems with mass constraints. The mathematical formulation of topology optimization approach is developed based on the element virtual strain energy as the design variable and minimization of compliance as the objective function. The suitability of the proposed method for topology optimization of nonlinear structures is demonstrated through a series of two- and three-dimensional benchmark designs. Several issues relating to the nonlinear structures subjected to dynamic loads such as material, geometric, and contact nonlinearities are addressed in the examples. It is shown that the proposed approach generates more reliable designs for nonlinear structures.     

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．     

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.     

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 fixed‐grid bidirectional evolutionary structural optimization method and its applications in tunnelling engineering

Topology optimization has exhibited an exceptional capability of improving structural design. However, several typical topology optimization algorithms are finite element (FE) based, where mesh‐dependent zigzag representation of boundaries is barely avoidable in both intermediate and final results. To tackle the problem, this paper proposes a new fixed‐grid‐based bidirectional evolutionary structural optimization method, namely FG BESO. The adoption of an FG FE framework enables a continuous boundary change in the course of topology optimization, which provides a means of dealing with not only the non‐smooth boundary of the final design but also the interpretation of intermediate densities. As a class of important practical application, it is interesting to make use of the new FG BESO method to the reinforcement design for underground tunnels. To accommodate the FG BESO technique to geological engineering applications, a nodal sensitivity is derived for a two‐phase material model comprising the artificial reinforcement and original rock. In this paper, some innovative topological designs of tunnel reinforcements are presented for minimizing the floor and sidewall heaves under different geological loading conditions. Copyright © 2007 John Wiley & Sons, Ltd.     

Topology optimization of energy absorbing structures with maximum damage constraint

A novel density‐based topology optimization framework for plastic energy absorbing structural designs with maximum damage constraint is proposed. This framework enables topologies to absorb large amount of energy via plastic work before failure occurs. To account for the plasticity and damage during the energy absorption, a coupled elastoplastic ductile damage model is incorporated with topology optimization. Appropriate material interpolation schemes are proposed to relax the damage in the low‐density regions while still ensuring the convergence of Newton‐Raphson solution process in the nonlinear finite element analyses. An effective method for obtaining path‐dependent sensitivities of the plastic work and maximum damage via adjoint method is presented, and the sensitivities are verified by the central difference method. The effectiveness of the proposed methodology is demonstrated through a series of numerical examples. Copyright © 2017 John Wiley & Sons, Ltd.     

Shape, sizing optimization and material selection based on mixed variables and genetic algorithm

In this work, we explore simultaneous designs of materials selection and structural optimization. As the material selection turns out to be a discrete process that finds the optimal distribution of materials over the design domain, it cannot be performed with common gradient-based optimization methods. In this paper, material selection is considered together with the shape and sizing optimization in a framework of multiobjective optimization of tracking the Pareto curve. The idea of mixed variables is often introduced in the case of mono-objective optimization. However, in the case of multi-objective optimization, we still face some hard key points related to the convexity and the continuity of the Pareto domain, which underline the originality of this work. In addition to the above aspect, there is a lack in the literature concerning the industrial applications that consider the mixed parameters. Continuous variables refer to structural parameters such as thickness, diameter and spring elastic constants while material ID is defined as binary design variable for each material. Both mechanical and thermal loads are considered in this work with the aim of minimizing the maximum stress and structural weight simultaneously. The efficiency of the design procedure is demonstrated through various numerical examples.     

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.     

Bi-Directional Evolutionary Topology Optimization Using Element Replaceable Method

In the present paper, design problems of maximizing the structural stiffness or natural frequency are considered subject to the material volume constraint. A new element replaceable method (ERPM) is proposed for evolutionary topology optimization of structures. Compared with existing versions of evolutionary structural optimization methods, contributions are twofold. On the one hand, a new automatic element deletion/growth procedure is established. The deletion of a finite element means that a solid element is replaced with an orthotropic cellular microstructure (OCM) element. The growth of an element means that an OCM element is replaced with a solid element of full materials. In fact, both operations are interchangeable depending upon how the value of element sensitivity is with respect to the objective function. The OCM design strategy is beneficial in preventing artificial modes for dynamic problems. Besides, the iteration validity is greatly improved with the introduction of a check position (CP) technique. On the other hand, a new checkerboard control algorithm is proposed to work together with the above procedure. After the identification of local checkerboards and detailed structures over the entire design domain, the algorithm will fill or delete elements depending upon the prescribed threshold of sensitivity values. Numerical results show that the ERPM is efficient and a clear and valuable material pattern can be achieved for both static and dynamic problems.     

Topology optimization with multiple materials via moving morphable component (MMC) method

With the fast development of additive manufacturing technology, topology optimization involving multiple materials has received ever increasing attention. Traditionally, this kind of optimization problem is solved within the implicit solution framework by using the Solid Isotropic Material with Penalization or level set method. This treatment, however, will inevitably lead to a large number of design variables especially when many types of materials are involved and 3‐dimensional (3D) problems are considered. This is because for each type of material, a corresponding density field/level function defined on the entire design domain must be introduced to describe its distribution. In the present paper, a novel approach for topology optimization with multiple materials is established based on the Moving Morphable Component framework. With use of this approach, topology optimization problems with multiple materials can be solved with much less numbers of design variables and degrees of freedom. Numerical examples provided demonstrate the effectiveness of the proposed approach.     

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.     

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.     

Conceptual design of efficient heat conductors using multi-material topology optimization

Conductive heat transfer plays an important role in dissipating thermal energy to achieve lower operating temperatures in various devices. Topology optimization has the potential to provide efficient structural solutions for such devices. The traditional topology optimization approach considers a single material. Adding additional materials with unique properties not only can expand the design options but also may improve the structural performance of the final structure. In this work, a multi-resolution topology optimization approach is employed to design multi-material structures for efficient heat dissipation. The implementation blends an efficient multi-resolution approach to obtain high-resolution designs with an alternating active phase algorithm to handle multi-material giving greater design flexibility. It solves the steady-state heat equation using finite element analysis and iteratively minimizes thermal compliance (maximizes conductivity). Several examples are presented to show the efficacy of the numerical implementation, which involves benchmark problems. Results indicate good prospects when quantitatively compared with single-material structures.     

Simultaneous optimal design of structural topology, actuator locations and control parameters for a plate structure

 Simultaneous optimization with respect to the structural topology, actuator locations and control parameters of an actively controlled plate structure is investigated in this paper. The system consists of a clamped-free plate, a H ₂ controller and four surface-bonded piezoelectric actuators utilized for suppressing the bending and torsional vibrations induced by external disturbances. The plate is represented by a rectangular design domain which is discretized by a regular finite element mesh and for each element the parameter indicating the presence or absence of material is used as a topology design variable. Due to the unavailability of large-scale 0–1 optimization algorithms, the binary variables of the original topology design problem are relaxed so that they can take all values between 0 and 1. The popular techniques in the topology optimization area including penalization, filtering and perimeter restriction are also used to suppress numerical problems such as intermediate thickness, checkerboards, and mesh dependence. Moreover, since it is not efficient to treat the structural and control design variables equally within the same framework, a nested solving approach is adopted in which the controller syntheses are considered as sub processes included in the main optimization process dealing with the structural topology and actuator locations. The structural and actuator variables are solved in the main optimization by the method of moving asymptotes, while the control parameters are designed in the sub optimization processes by solving the Ricatti equations. Numerical examples show that the approach used in this paper can produce systems with clear structural topology and high control performance. Received 16 November 2001 / Accepted 26 February 2002     

Shape morphing of laminated composite structures with photostrictive actuators via topology optimization

In this paper, a new design method is presented for achieving remote wireless shape morphing of laminated composite structures using topology optimization methods. A recently emerging family of smart materials, photostrictive materials, is introduced as the actuation discipline to implement the active control of optical structures by utilizing the photostriction mechanism, which arises from the superposition of photovoltaic effect and converse piezoelectric effect when exposed to the illumination of near ultraviolet light. In terms of the Mindlin plate theory of first-order shear deformation, a finite element formulation including multiphysics effects of photovoltaic, pyroelectric and thermal expansion is developed to model composite structures of ferroelectric materials polarized in 0–1 and 0–3 directions, respectively. The design is formulated as a multi-constrained optimization problem with a least square objective function to minimize structural shape errors. The topology optimization method is used as a systematic design approach to seek the optimal topologies of material layouts for both the photostrictive and host layers as well as the actuator light distribution. In terms of design sensitivity analysis, many gradient-based optimization algorithms can be applied to solve the problem effectively. Numerical examples are presented to demonstrate the effectiveness of this method in the field of active photonic control of laminated composite structures.     

考虑位移要求的大型三维连续体结构拓扑优化数值技术研究

大型复杂三维结构拓扑优化设计既具有理论意义,又具有重要的应用价值。基于等效转换的非奇异的结构优化模型,研究结构位移要求的最小结构重量设计问题。首先,介绍了位移约束的三维结构优化准则和公式。而后,为了提高拥有数万个单元以上的三维结构的计算效率,结合结构位移计算的迭代方法,在分析用于结构特性参数计算模型的基础上,建立了一套三维结构拓扑优化的求解策略和算法。最后,给出了几个典型和复杂的三维结构的拓扑优化设计算例。算例表明求解策略和算法是正确和有效的,且具有广泛的工程应用前景。