A boundary density evolutionary topology optimization of continuum structures with smooth boundaries

Based on the variable density method, this article proposes a boundary density evolutionary topology optimization method. The method uses a material interpolation model without penalization. Combined with the density grading filtering method, an optimal topology with only 0/1 cells can be obtained. Compared with the solid isotropic microstructures with penalization method (SIMP), no penalty factor is required in the material interpolation model; compared with the evolutionary structural optimization method (ESO), intermediate-density elements are allowed in the optimization process, but the concept of gradually removing the low-utilization materials near the boundary in the ESO method is retained. After the optimal result is obtained, the structural boundary element is processed by the level set of nodal strain energy, and the optimization result with smooth boundaries similar to the level set method (LSM) can be obtained. The proposed method has the superiority of the variable density method, and it also combines the advantages of the evolutionary method and the level set method, so which is named as boundary density evolution (BDE) method. The four static and one dynamic optimization examples illustrate the stability and efficiency of the proposed method.     

Topology optimization of continuum structures with local stress constraints

We introduce an extension of current technologies for topology optimization of continuum structures which allows for treating local stress criteria. We first consider relevant stress criteria for porous composite materials, initially by studying the stress states of the so-called rank 2 layered materials. Then, on the basis of the theoretical study of the rank 2 microstructures, we propose an empirical model that extends the power penalized stiffness model (also called SIMP for Solid Isotropic Microstructure with Penalization for inter-mediate densities). In a second part, solution aspects of topology problems are considered. To deal with the so-called ‘singularity’ phenomenon of stress constraints in topology design, an ϵ-constraint relaxation of the stress constraints is used. We describe the mathematical programming approach that is used to solve the numerical optimization problems, and show results for a number of example applications. © 1998 John Wiley & Sons, Ltd.     

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.     

Topology optimization of continuum structures with stress constraints and uncertainties in loading

Topology optimization using stress constraints and considering uncertainties is a serious challenge, since a reliability problem has to be solved for each stress constraint, for each element in the mesh. In this paper, an alternative way of solving this problem is used, where uncertainty quantification is performed through the first‐order perturbation approach, with proper validation by Monte Carlo simulation. Uncertainties are considered in the loading magnitude and direction. The minimum volume problem subjected to local stress constraints is formulated as a robust problem, where the stress constraints are written as a weighted average between their expected value and standard deviation. The augmented Lagrangian method is used for handling the large set of local stress constraints, whereas a gradient‐based algorithm is used for handling the bounding constraints. It is shown that even in the presence of small uncertainties in loading direction, different topologies are obtained when compared to a deterministic approach. The effect of correlation between uncertainties in loading magnitude and direction on optimal topologies is also studied, where the main observed result is loss of symmetry in optimal topologies.     

Topology design of structures using a dual algorithm and a constraint on the perimeter

Dual optimization algorithms are well suited for the topology design of continuum structures in discrete variables, since in these problems the number of constraints is small in comparison to the number of design variables. The ‘raw’ dual algorithm, which was originally proposed for the minimum compliance design problem, worked well when a perimeter constraint was added in addition to the volume constraint. However, if the perimeter constraint was gradually relaxed by increasing the upper bound on the allowable perimeter, the algorithm tended to behave erratically. Recently, a simple strategy has been suggested which modifies the raw dual algorithm to make it more robust in the absence of the perimeter constraint; in particular the problem of checkerboarding which is frequently observed with the use of lower‐order finite elements is eliminated. In this work, we show how the perimeter constraint can be incorporated in this improved algorithm, so that it not only provides a designer with a control over the topology, but also generates good topologies irrespective of the value of the upper bound on the perimeter. Copyright © 2002 John Wiley & Sons, Ltd.     

Topology optimization of continuum structures subjected to uncertainties in material properties

This work addresses the use of the topology optimization approach to the design of robust continuum structures under the hypothesis of uncertainties with known second‐order statistics. To this end, the second‐order perturbation approach is used to model the response of the structure, and the midpoint discretization technique is used to discretize the random field. The objective function is a weighted sum of the expected compliance and its standard deviation. The optimization problem is solved using a traditional optimality criteria method. It is shown that the correlation length plays an important role in the obtained topology and statistical moments when only the minimization of the standard deviation is considered, resulting in more and thinner reinforcements as the correlation length decreases. It is also shown that the minimization of the expected value is close to the minimization of the deterministic compliance for small variations of Young's modulus. Copyright © 2016 John Wiley & Sons, Ltd.     

基于变频率区间约束的结构材料优化设计

针对频率约束的结构材料优化问题,基于结构拓扑优化思想,提出变频率区间约束的结构材料优化方法。借鉴均匀化及ICM(独立、连续、映射)方法,以微观单元拓扑变量倒数为设计变量,导出宏观单元等效质量矩阵及导数,进而获得频率一阶近似展开式。结合变频率区间约束思想,获得以结构质量为目标函数、频率为约束条件的连续体微结构拓扑优化近似模型;采用对偶方法求解。通过算例验证该方法的有效性及可行性,表明考虑质量矩阵变化影响所得优化结果更合理。     

Topology and sizing optimization of discrete structures using a cooperative coevolutionary genetic algorithm with independent ground structures

This article proposes a method called the cooperative coevolutionary genetic algorithm with independent ground structures (CCGA-IGS) for the simultaneous topology and sizing optimization of discrete structures. An IGS strategy is proposed to enhance the flexibility of the optimization by offering two separate design spaces and to improve the efficiency of the algorithm by reducing the search space. The CCGA is introduced to divide a complex problem into two smaller subspaces: the topological and sizing variables are assigned into two subpopulations which evolve in isolation but collaborate in fitness evaluations. Five different methods were implemented on 2D and 3D numeric examples to test the performance of the algorithms. The results demonstrate that the performance of the algorithms is improved in terms of accuracy and convergence speed with the IGS strategy, and the CCGA converges faster than the traditional GA without loss of accuracy.     

Topology optimization for tunnel support in layered geological structures

Tunnel construction commonly proceeds in an environment of layered geological formation. Design for tunnel support relies on the tunnel location and the mismatch of different layers. The present paper proposes a topology optimization method for the design of tunnel support. The design domain is discretized by finite elements. An element is composed of the original rock (hard or soft) and the reinforcing material (rock reinforced by grouting or bolting). The design issue involves the distribution of reinforcements. We model the reinforced host ground by a power‐weighted mixture law. The ratios of two phases in various elements are optimized to minimize the deformation of the tunnel. The method enables a computer‐aided design for the supports of underground tunnels embedded within layered geological structures. The reinforced areas for the tunnel are displayed under the passive geological stress and the active external loading. Four kinds of rock formations are examined. They are homogeneous rock, hard–soft–hard (HSH) sandwich structure, two‐layer structure with the soft rock at the top and the hard rock at the bottom (SH), and the one with the hard rock at the top and the soft rock at the bottom (HS). The simulation reveals the high efficiency of tunnel support by optimizing its topologies. Copyright © 2000 John Wiley & Sons, Ltd.     

Topology optimization of structures under variable loading using a damage superposition approach

We present an original algorithm and accompanying mathematical formulation for topology optimization of structures that can sustain material damage and are subject to multiple load cases with varying configurations. Damage accumulation is simulated using a coupled, non‐linear brittle damage model. The structures are optimized for minimum mass subject to stiffness constraints defined as the compliance evaluated at the end of each loading sequence. To achieve robustness of the optimized structures, the respective damage fields caused by each individual load case are computed and combined using superposition to simulate a worst‐case damage field. All load cases are then run a second time using the worst‐case damage distribution as a starting point. In this way, one effectively accounts for the spectrum of possible load sequences to which the structure may be subjected. Results from this method are compared with an exhaustive, brute‐force approach in which all non‐repeating load sequences are analyzed individually. For each method, the corresponding sensitivities are derived and implemented analytically using a path‐dependent adjoint method. The two approaches are implemented on a series of numerical examples, which demonstrate that the superposition method produces structures that are as robust as those obtained using the exhaustive method but require significantly less computational effort. Copyright © 2014 John Wiley & Sons, Ltd.     

Topology optimization of plane structures using smoothed particle hydrodynamics method

This paper presents an alternative topology optimization method based on an efficient meshless smoothed particle hydrodynamics (SPH) algorithm. To currently calculate the objective compliance, the deficiencies in standard SPH method are eliminated by introducing corrective smoothed particle method and total Lagrangian formulation. The compliance is established relative to a designed density variable at each SPH particle which is updated by optimality criteria method. Topology optimization is realized by minimizing the compliance using a modified solid isotropic material with penalization approach. Some numerical examples of plane elastic structure are carried out and the results demonstrate the suitability and effectiveness of the proposed SPH method in the topology optimization problem. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Topology optimization of resonating structures using SIMP method

The present paper is concerned with the layout optimization of resonating actuators using topology optimization techniques. The goal of the optimization is a maximization of the magnitude of steady‐state vibrations for a given excitation frequency. The problem formulation includes an external viscous damper at the output port which models a working load on the structure. Copyright © 2002 John Wiley & Sons, Ltd.     

A robust dual algorithm for topology design of structures in discrete variables

Dual optimization algorithms for the topology optimization of continuum structures in discrete variables are gaining popularity in recent times since, in topology design problems, the number of constraints is small in comparison to the number of design variables. Good topologies can be obtained for the minimum compliance design problem when the perimeter constraint is imposed in addition to the volume constraint. However, when the perimeter constraint is relaxed, the dual algorithm tends to give bad results, even with the use of higher‐order finite element models as we demonstrate in this work. Since, a priori, one does not know what a good value of the perimeter to be specified is, it is essential to have an algorithm which generates good topologies even in the absence of the perimeter constraint. We show how the dual algorithm can be made more robust so that it yields good designs consistently in the absence of the perimeter constraint. In particular, we show that the problem of checkerboarding which is frequently observed with the use of lower‐order finite elements is eliminated. Copyright © 2001 John Wiley & Sons, Ltd.     

Discrete material optimization of general composite shell structures

A novel method for doing material optimization of general composite laminate shell structures is presented and its capabilities are illustrated with three examples. The method is labelled Discrete Material Optimization (DMO) but uses gradient information combined with mathematical programming to solve a discrete optimization problem. The method can be used to solve the orientation problem of orthotropic materials and the material selection problem as well as problems involving both. The method relies on ideas from multiphase topology optimization to achieve a parametrization which is very general and reduces the risk of obtaining a local optimum solution for the tested configurations. The applicability of the DMO method is demonstrated for fibre angle optimization of a cantilever beam and combined fibre angle and material selection optimization of a four‐point beam bending problem and a doubly curved laminated shell. Copyright © 2005 John Wiley & Sons, Ltd.     

Topology optimization for submerged buoyant structures

This paper presents an evolutionary structural topology optimization method for the design of completely submerged buoyant modules with design-dependent fluid pressure loading. This type of structure is used to support offshore rig installation and pipeline transportation at all water depths. The proposed optimization method seeks to identify the buoy design that has the highest stiffness, allowing it to withstand deepwater pressure, uses the least material and has a minimum prescribed buoyancy. Laplace's equation is used to simulate underwater fluid pressure, and a polymer buoyancy module is considered to be linearly elastic. Both domains are solved with the finite element method. Using an extended bi-directional evolutionary structural optimization (BESO) method, the design-dependent pressure loads are modelled in a straightforward manner without any need for pressure surface parametrization. A new buoyancy inequality constraint sets a minimum required buoyancy effect, measured by the joint volume of the structure and its interior voids. Solid elements with low strain energy are iteratively removed from the initial design domain until a certain prescribed volume fraction. A test case is described to validate the optimization problem, and a buoy design problem is used to explore the features of the proposed method.     

Generalized shape optimization of three-dimensional structures using materials with optimum microstructures

This paper deals with generalized shape optimization of linearly elastic, three-dimensional continuum structures, i.e. we consider the problem of determining the structural topology (or layout) such that the shape of external as well as internal boundaries and the number of inner holes are optimized simultaneously. For prescribed static loading and given boundary conditions, the optimum solution is sought from the condition of maximum integral stiffness (minimum elastic compliance) subject to a specified amount of structural material within a given three-dimensional design domain. This generalized shape optimization problem requires relaxation which leads to the introduction of microstructures. A class of optimum three-dimensional microstructures and explicit analytical expressions for their optimum effective stiffness properties have been developed by Gibiansky and Cherkaev (1987) [Gibiansky, L.V., Cherkaev, A.V., 1987. Microstructures of composites of extremal rigidity and exact estimates of provided energy density (in Russian). Report (1987) No. 1155. A.F. Ioffe Physical-Technical Institute, Academy of Sciences of the USSR, Leningrad. English translation in: Kohn, R.V., Cherkaev, A.V. (Eds.), Topics in the Mathematical Modelling of Composite Materials. Birkhaüser, New York. 1997]. The present paper gives a brief account of the results in Gibiansky and Cherkaev (1987) which will be utilized for our microlevel problem analysis. It is a characteristic feature that the use of optimum microstructures renders the global problem convex if an appropriate parametrization is applied. Hereby local optima can be avoided and we can construct a simple gradient based numerical method of mathematical programming for solution of the complete optimization problem. Illustrative examples of optimum layout and topology designs of three-dimensional structures are presented at the end of the paper.     

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.     

Topology optimization of double- and triple-layer grids using a hybrid methodology

In this article, a hybrid methodology combining evolutionary structural optimization (ESO) and gravitational particle swarm (GPS) methods is proposed for topology optimization of double- and triple-layer grids. In the present methodology, which is called the ESO-GPS method, the size optimization of double- and triple-layer grids is first performed by ESO. Then, the outcomes of the ESO are used to improve the GPS through four modifications. Structural weight is minimized against constraints on the displacements of nodes, internal stresses and element slenderness ratio. The GPS is used to investigate the optimum topology of large-scale skeletal structures with discrete variables whose agents update their respective positions by the particle swarm optimization velocity and the acceleration of the gravitational search algorithm. The numerical results show that the proposed algorithm, the ESO-GPS, performs better than the GPS and the other methods presented in the literature.