Shape optimization using time evolution equations

In this study, we deal with a numerical solution based on time evolution equations to solve the optimization problem for finding the shape that minimizes the objective function under given constraints. The design variables of the shape optimization problem are defined on a given original domain on which the boundary value problems of partial differential equations are defined. The variations of the domain are obtained by the time integration of the solution to derive the time evolution equations defined in the original domain. The shape gradient with respect to the domain variations are given as the Neumann boundary condition defined on the original domain boundary. When the constraints are satisfied, the decreasing property of the objective function is guaranteed by the proposed method. Furthermore, the proposed method is used to minimize the heat resistance under a total volume constraint and to solve the minimization problem of mean compliance under a total volume constraint.     

Robust topology optimization of optical cloaks under uncertainties in wave number and angle of incident wave

This article presents a robust topology optimization method for optical cloaks under uncertainties in the wave number and angle in the incident wave. We first discuss the governing equation derived from Maxwell's equation, and extend it to the entire domain including the dielectric material and air, based on the level set-based topology optimization method. Next, a robust optimization problem is formulated as a minimization problem of the weighted sum of the scattered wave norm and its standard deviation with respect to the wave number and angle of the incident wave. The standard deviation is mathematically expressed by the Taylor series approximation and the use of the adjoint variable method. The design sensitivity of the objective functional is also derived by the adjoint variable method. An optimization algorithm is then constructed, based on the proposed formulation for robust designs of optical cloaks. Several numerical examples are finally provided to demonstrate the validity and utility of the proposed method.     

Topology optimization by a time‐dependent diffusion equation

Most topology optimization problems are formulated as constrained optimization problems; thus, mathematical programming has been the mainstream. On the other hand, solving topology optimization problems using time evolution equations, seen in the level set‐based and the phase field‐based methods, is yet another approach. One issue is the treatment of multiple constraints, which is difficult to incorporate within time evolution equations. Another issue is the extra re‐initialization steps that interrupt the time integration from time to time. This paper proposes a way to describe, using a Heaviside projection‐based representation, a time‐dependent diffusion equation that addresses these two issues. The constraints are treated using a modified augmented Lagrangian approach in which the Lagrange multipliers are updated by simple ordinary differential equations. The proposed method is easy to implement using a high‐level finite element code. Also, it is very practical in the sense that one can fully utilize the existing framework of the code: GUI, parallelized solvers, animations, data imports/exports, and so on. The effectiveness of the proposed method is demonstrated through numerical examples in both the planar and spatial cases. Copyright © 2012 John Wiley & Sons, Ltd.     

Topology optimization of exterior acoustic-structure interaction systems using the coupled FEM-BEM method

This paper presents an efficient topology optimization procedure for exterior acoustic-structure interaction problems, in which the coupled systems are formulated by the boundary element method (BEM) and the finite element method (FEM). So far, the topology optimization based on the coupled FEM-BEM still faces several issues needed to be addressed, especially the efficient design sensitivity analysis for the coupled systems. In this work, we contribute to these issues in two main aspects. Firstly, the adjoint variable method (AVM) formulations are derived for sensitivity analysis of arbitrary objective function, and the feedback coupling between the structural and acoustic domains are taken into consideration in the sensitivity analysis. Secondly, in addition to the application of fast multipole method (FMM) in the acoustic BEM response analysis, the FMM is now updated to adapt to the arising different multiplications in the AVM equations. These accelerations save considerable computing time and memory. Numerical tests show that the developed approach permits its application to large-scale problems. Finally, some basic observations for the optimized designs are drawn from the numerical investigations.     

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.     

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.     

Thermomechanical topology optimization of shape-memory alloy structures using a transient bilevel adjoint method

We present a novel method for computational design of adaptive shape-memory alloy (SMA) structures via topology optimization. By optimally distributing a SMA within the prescribed design domain, the proposed algorithm seeks to tailor the two-way shape-memory effect (TWSME) and pseudoelasticity response of the SMA materials. Using a phenomenological material model, the thermomechanical response of the SMA structure is solved through inelastic finite element analysis, while assuming a transient but spatially uniform temperature distribution. The material distribution is parameterized via a SIMP formulation, with gradient-based optimization used to perform the optimization search. We derive a transient, bilevel adjoint formulation for analytically computing the design sensitivities. We demonstrate the proposed design framework using a series of two-dimensional thermomechanical benchmark problems. These examples include design for optimal displacement due to the TWSME, and design for maximum mechanical advantage while accounting for pseudoelasticity.     

考虑结构稳定性的变密度拓扑优化方法

将稳定性问题引入传统变密度法中,可实现包含稳定性约束的平面模型结构拓扑优化。以单元相对密度为设计变量,结构柔度最小为目标函数,结构体积和失稳载荷因子为约束条件建立优化问题数学模型,提出了一种考虑结构稳定性的变密度拓扑优化方法。通过分析结构柔度、体积、失稳载荷因子对设计变量的灵敏度,并基于拉格朗日乘子法和Kuhn-Tucker条件,推导了优化问题的迭代准则。同时,利用基于约束条件的泰勒展开式求解优化准则中的拉格朗日乘子。通过推导平面四节点四边形单元几何刚度矩阵的显式表达式,得到了优化准则中的几何应变能。最后,通过算例对提出的方法进行了验证,并与不考虑稳定性的传统变密度拓扑优化方法进行对比,结果表明该方法能显著提高拓扑优化结果的稳定性。研究结果对细长受压结构的优化设计有重要指导意义,对结构的稳定性设计有一定参考价值。     

Robust topology optimization of continuum structures with loading uncertainty using a perturbation method

An approach for the robust topology optimization (RTO) of continuum structures with loading uncertainty is investigated. The loading uncertainties are quantified using the second order Taylor series expansion of uncertain loading magnitudes and directions, and then the response statistic mean and standard deviation of compliance are calculated using the uncertain perturbation propagation method. A robust design Lagrange function considering the compliance objective and finite element constraints is developed, and a sensitivity analysis is performed to calculate the Lagrange coefficients. The Lagrange objective function is optimized using the modified solid isotropic material with penalization (SIMP) algorithm; thus, the optimum material distribution under loading uncertainty is acquired. The proposed methodology is used for the RTO of two examples, revealing its efficiency under both concentrated and distributed uncertain loadings. The accuracy of the results is verified by comparison with similar cases found in the literature where a different modelling approach was used.     

Multi-material topology optimization for the transient heat conduction problem using a sequential quadratic programming algorithm

Transient heat conduction analysis involves extensive computational cost. It becomes more serious for multi-material topology optimization, in which many design variables are involved and hundreds of iterations are usually required for convergence. This article aims to provide an efficient quadratic approximation for multi-material topology optimization of transient heat conduction problems. Reciprocal-type variables, instead of relative densities, are introduced as design variables. The sequential quadratic programming approach with explicit Hessians can be utilized as the optimizer for the computationally demanding optimization problem, by setting up a sequence of quadratic programs, in which the thermal compliance and weight can be explicitly approximated by the first and second order Taylor series expansion in terms of design variables. Numerical examples show clearly that the present approach can achieve better performance in terms of computational efficiency and iteration number than the solid isotropic material with penalization method solved by the commonly used method of moving asymptotes. In addition, a more lightweight design can be achieved by using multi-phase materials for the transient heat conductive problem, which demonstrates the necessity for multi-material topology optimization.     

From a quasi-static fluid-based evolutionary topology optimization to a generalization of BESO

A new algorithm is proposed for topology optimization based on a fluid dynamics analogy. It possesses characteristics similar to most well-known methods, such as the Evolutionary Structural Optimization (ESO)/Bidirectional Evolutionary Structural Optimization (BESO) method due to Xie and Steven (1993 Xie, Y. M., and G.P. Steven. 1993. “A Simple Evolutionary Procedure for Structural Optimisation.” Computers and Structures 49 (5): 885–896. doi: 10.1016/0045-7949(93)90035-C[Crossref], [Web of Science ®] , [Google Scholar], “A Simple Evolutionary Procedure for Structural Optimisation.” Computers and Structures 49 (5): 885–896.), which works with discrete values, and the Solid Isotropic Material with Penalization (SIMP) method due to Bendsøe (1989 Bendsøe, M.P. 1989. “Optimal Shape Design as a Material Distribution Problem.” Structural Optimization 1 (4): 193–202. doi: 10.1007/BF01650949[Crossref] , [Google Scholar], “Optimal Shape Design as aMaterial Distribution Problem.” Structural Optimization 1 (4): 193–202.) and Zhou and Rozvany (1991 Zhou, M., and G.I.N. Rozvany. 1991. “The COC Algorithm—Part II: Topological, Geometry and Generalized Shape Optimization.” Computer Methods in Applied Mechanics and Engineering 89 (1–3): 309–336. doi: 10.1016/0045-7825(91)90046-9[Crossref], [Web of Science ®] , [Google Scholar], “The COCAlgorithm–Part II: Topological, Geometry and Generalized Shape Optimization.” Computer Methods in Applied Mechanics and Engineering 89 (1–3): 309–336.) (using Optimality Criterion (OC) or Method of Moving Asymptotes (MMA)), which works with intermediate values, as it is able to work both with discrete and intermediate densities, but always yields a solution with discrete densities. It can be proven mathematically that the new method is a generalization of the BESO method and using appropriate parameters it will operate exactly as the BESO method. The new method is less sensitive to rounding errors of the matrix solver as compared to the BESO method and is able to give alternative topologies to well-known problems. The article presents the basic idea and the optimization algorithm, and compares the results of three cantilever optimizations to the results of the SIMP and BESO methods.     

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.     

Temperature-constrained topology optimization of thermo-mechanical coupled problems

Temperature-constrained topology optimization for thermo-mechanical coupled problems under a design-dependent temperature field considering the thermal expansion effect remains an open problem. A temperature-constrained topology optimization method is proposed for thermo-mechanical coupled problems. In this article, the temperature values at the heat sources are constrained. The numerical results reveal that the temperature constraints play an important role in topology optimization of thermo-mechanical coupled problems. The optimized structure obtained by the presented method not only has certain strength but also decreases the temperature significantly compared with structures obtained by other methods without considering temperature constraints. The proposed method is applied to the design of the cooling system of a battery package. Numerical examples verify the efficiency of the presented method.     

New approach to identify the initial condition in degenerate hyperbolic equation

This paper deals with the determination of an initial condition in degenerate hyperbolic equation from final observations. With the aim of reducing the execution time, this inverse problem is solved using an approach based on double regularization: a Tikhonov’s regularization and regularization in equation by viscose-elasticity. So, we obtain a sequence of weak solutions of degenerate linear viscose-elastic problems. Firstly, we prove the existence and uniqueness of each term of this sequence. Secondly, we prove the convergence of this sequence to the weak solution of the initial problem. Also we present some numerical experiments to show the performance of this approach.     

基于分段常数水平集方法的声振耦合系统拓扑优化

针对结构与无限大声场的声振耦合系统中结构的双材料拓扑优化问题进行了研究。采用有限元与边界元方法分别对结构和声场进行离散。基于分段常数水平集(piecewise constant level set,PCLS)方法,构造了结构的刚度阵、质量阵与阻尼阵。优化目标选为最小化结构指定位置的振幅平方,采用伴随变量法进行灵敏度分析。引入二次罚函数方法来实现体积约束,基于灵敏度信息对优化参数进行重新定义,克服了参数的问题依赖性。数值结果表明优化设计可以显著降低结构的振幅,证实了优化方法的有效性。不同算例下体积约束在相同优化参数下均得到很好满足,说明了重新定义参数的优越性。     

Eigenvalue topology optimization via efficient multilevel solution of the frequency response

The article presents an efficient solution method for structural topology optimization aimed at maximizing the fundamental frequency of vibration. Nowadays, this is still a challenging problem mainly because of the high computational cost required by spectral analyses. The proposed method relies on replacing the eigenvalue problem with a frequency response one, which can be tuned and efficiently solved by a multilevel procedure. Connections of the method with multigrid eigenvalue solvers are discussed in details. Several applications demonstrating more than 90% savings of the computational time are presented as well.     

Compliant mechanism design with non-linear materials using topology optimization

In this paper, compliant mechanism design with non-linear materials using topology optimization is presented. A general displacement functional with non-linear material model is used in the topology optimization formulation. Sensitivity analysis of this displacement functional is derived from the adjoint method. Optimal compliant mechanism examples for maximizing the mechanical advantage are presented and the effect of non-linear material on the optimal design are considered.     

Higher‐order multi‐resolution topology optimization using the finite cell method

This article presents a detailed study on the potential and limitations of performing higher‐order multi‐resolution topology optimization with the finite cell method. To circumvent stiffness overestimation in high‐contrast topologies, a length‐scale is applied on the solution using filter methods. The relations between stiffness overestimation, the analysis system, and the applied length‐scale are examined, while a high‐resolution topology is maintained. The computational cost associated with nested topology optimization is reduced significantly compared with the use of first‐order finite elements. This reduction is caused by exploiting the decoupling of density and analysis mesh, and by condensing the higher‐order modes out of the stiffness matrix. Copyright © 2016 John Wiley & Sons, Ltd.     

Simultaneous partial topology and size optimization of a wing structure using ant colony and gradient based methods

This article presents a methodology and process for a combined wing configuration partial topology and structure size optimization. It is aimed at achieving a minimum structural weight by optimizing the structure layout and structural component size simultaneously. This design optimization process contains two types of design variables and hence was divided into two sub-problems. One is structure layout topology to obtain an optimal number and location of spars with discrete integer design variables. Another is component size optimization with continuous design variables in the structure FE model. A multi city-layer ant colony optimization (MCLACO) method is proposed and applied to the topology sub-problem. A gradient based optimization method (GBOM) built in the MSC.NASTRAN SOL-200 module was employed in the component size optimization sub-problem. For each selected layout of the wing structure, a size optimization process is performed to obtain the optimum result and feedback to the layout topology process. The numerical example shows that the proposed MCLACO method and a combination with the GBOM are effective for solving such a wing structure optimization problem. The results also indicate that significant structural weight saving can be achieved.