Topology synthesis of thermomechanical compliant mechanisms with geometrical nonlinearities using meshless method

The element-free Galerkin (EFG) method, one of the important meshless methods, is integrated into topology optimization and a new topology optimization method for designing thermomechanical actuated compliant mechanisms with geometrical nonlinearities is presented. The meshless method is employed to discretize the governing equations and the bulk density field. Using meshless method to analyze the thermomechanical model is better consistent with the natural behavior of large-displacement compliant mechanisms than using the standard finite element method (FEM). The optimization formulation is developed using the SIMP and meshless methods. The nonlinear design sensitivity analysis is performed by incorporating the adjoint approach into the meshless method. The filtering of the sensitivity developed corrects the topology including few discontinuous scattered points. The geometrically nonlinear design sensitivity analysis is performed by incorporating the adjoint approach into the meshless method. The availability of the proposed method is demonstrated by designing compliant actuators in which both linear and nonlinear modeling are considered.     

基于无网格径向点插值方法的简谐激励下的连续体结构拓扑优化

针对用有限元法进行连续体结构拓扑优化时需不断重构网格来处理网格畸变和网格移动,且存在数值计算不稳定等问题,基于无网格径向点插值方法(Radial Point Interpolation Method,RPIM)对简谐激励下的连续体结构进行拓扑优化.选取节点的相对密度作为设计变量,以结构动柔度最小化为目标函数,基于带惩罚的各向同性固体微结构(Solid Isotropic Microstructure with Penalization,SIMP)模型建立简谐激励下的优化模型;采用伴随法求解得到目标函数的敏度分析公式;利用优化准则法求解优化模型.经典的二维连续体结构拓扑优化算例证明该方法的可行性和有效性.     

An efficient decoupled sensitivity analysis method for multiscale concurrent topology optimization problems

The conventional coupled sensitivity analysis method for concurrent topology optimization problems is computationally expensive for microscale design variables. This study thus proposes an efficient decoupled sensitivity analysis method for concurrent topology optimization based on the chain differentiation rule. Two numerical studies are performed to demonstrate the effectiveness of the decoupled sensitivity analysis method for concurrent topology optimization problems with single or multiple porous materials. It can be concluded from the results that the decoupled method is computationally much more efficient than the coupled method, while they are mathematically equivalent. The outstanding merits of the decoupled method are two-fold: (1) computational efficiency of sensitivity analysis with respect to the microscale design variables; and (2) applicability to concurrent topology optimization problems with single or multiple porous materials as well as with composite microstructure and multi-phase materials.     

基于无网格局部Petrov-Galekin法的板结构拓扑优化

为将无网格法的优势集成到结构拓扑优化中,基于无网格局部Petrov-Galerkin(Meshless Local Petrov-Galerkin,MLPG)法进行板结构的拓扑优化.基于带惩罚的各向同性固体微结构(Solid Isotropic Microstructure with Penalization,SIMP...     

On the orthogonal similarity transformation (OST)-based sensitivity analysis method for robust topology optimization under loading uncertainty: a mathematical proof and its extension

The main purpose of this work is to provide a mathematical proof of our previously proposed orthogonal similarity transformation (OST)-based sensitivity analysis method (Zhao et al. Struct Multidisc Optim 50(3):517–522 2014a, Comput Methods Appl Mech Engrg 273:204–218 c); the proof is designed to show the method’s computational effectiveness. Theoretical study of computational efficiency for both robust topology optimization and robust concurrent topology optimization problems shows the necessity of the OST-based sensitivity analysis method for practical problems. Numerical studies were conducted to demonstrate the computational accuracy of the OST-based sensitivity analysis method and its efficiency over the conventional method. The research leads us to conclude that the OST-based sensitivity analysis method can bring considerable computational savings when used for large-scale robust topology optimization problems, as well as robust concurrent topology optimization problems.     

Topology optimization with design-dependent pressure loading

In this paper, the layout of structures under design-dependent pressure loading is optimized using a topology optimization approach. In contrast to topology optimization problems with conventional static external loading, the position and direction of pressure loading are changing with topology of structure during optimization iterations. In order to model the changing structural surface boundaries under design-dependent pressure loading, a pseudo equal-potential function is introduced. Design sensitivity analysis is derived from the adjoint method. Three examples solved by the proposed method are presented.     

Evolutionary topology optimization of continuum structures with smooth boundary representation

This paper develops an extended bi-directional evolutionary structural optimization (BESO) method for topology optimization of continuum structures with smoothed boundary representation. In contrast to conventional zigzag BESO designs and removal/addition of elements, the newly proposed evolutionary topology optimization (ETO) method, determines implicitly the smooth structural topology by a level-set function (LSF) constructed by nodal sensitivity numbers. The projection relationship between the design model and the finite element analysis (FEA) model is established. The analysis of the design model is replaced by the FEA model with various elemental volume fractions, which are determined by the auxiliary LSF. The introduction of sensitivity LSF results in intermediate volume elements along the solid-void interface of the FEA model, thus contributing to the better convergence of the optimized topology for the design model. The effectiveness and robustness of the proposed method are verified by a series of 2D and 3D topology optimization design problems including compliance minimization and natural frequency maximization. It has been shown that the developed ETO method is capable of generating a clear and smooth boundary representation; meanwhile the resultant designs are less dependent on the initial guess design and the finite element mesh resolution.     

Isogeometric topology optimization using trimmed spline surfaces

In the present work, a new spline based topology optimization using trimmed spline surfaces and the isogeometric analysis is proposed. In the proposed approach, the trimmed surface analysis which can treat topologically complex spline surfaces using trimming information provided by CAD systems is employed for structural response analysis and sensitivity calculation in the topology optimization. The outer and inner boundaries of design models are represented by a spline surface and trimming curves. Design variables used in this approach are the coordinates of control points of a spline surface and those of trimming curves. New sensitivity formulations for the control points in the trimmed surface analysis are proposed and their efficiency and accuracy are verified. The creation of new inner fronts during optimization is allowed for the topological flexibility. An inner front merging algorithm is also presented. The proposed spline based topology optimization is used to solve some benchmarking problems. Design space dependency which is one of serious shortcomings in conventional topology optimization approaches is naturally eliminated by the proposed spline based optimization. Design dependent load problems which are difficult to treat with conventional grid based topology optimization methods are easily dealt with by the proposed one. It is also shown that post-processing effort for converting to CAD model is eliminated by using the same spline information in numerical analysis and design optimization.     

Parametric shape optimization techniques based on Meshless methods: A review

In the product development process, structural optimization plays vital role because it deals with size, shape and topology of the structures. However, structural performance greatly depends on its geometric shape and hence structural shape optimization has remained one of the most active research areas since early 1970s. Conventional parametric shape optimization technique employs grid-based numerical tools like FEM and BEM for structural analysis, which experiences some innate limitations like mesh distortion and frequent remeshing, element locking and poor approximation while dealing with large shape changes during the optimization process. Meshless Methods (MMs) can alleviate these issues when used as a structural analysis tool in shape optimization. In last two decades, MMs have been explored for structural shape optimization along with various deterministic and stochastic optimization algorithms. The objective of present work is twofold, first is to review advanced parametric shape optimization techniques which are based on MMs like Element Free Galerkin (EFG) method and Reproducing Kernel Particle Method (RKPM) for linear elastic, thermoelastic, hyperelastic, frictional contact and structure dynamics optimization problems and second is to emphasize benefits of meshless techniques in shape optimization. Based on the review, the article presents some critical observations including Design Sensitivity Analysis (DSA) in meshless environment, numerical integration techniques in MMs and benefits of coupled FEM-MM approach in shape optimization. At the end, promising future research directions in shape optimization field based on MMs are presented along with concluding remarks.     

微型柔性机构的多目标计算机辅助拓扑优化设计

提出了基于结构整体柔度最小化和结构输出位移最大化的多目标拓扑优化设计方法,建立了微型柔性机构的多目标拓扑优化设计模型.提出了适用于微型柔性机构多目标拓扑优化设计的伴随矩阵敏度分析方法,并将广义收敛移动渐进算法用于多目标多约束微型柔性机构拓扑优化问题的求解.最后通过数值计算验证了优化模型的有效性.     

Hinge-free topology optimization with embedded translation-invariant differentiable wavelet shrinkage

In topology optimization applications for the design of compliant mechanisms, the formation of hinges is typically encountered. Often such hinges are unphysical artifacts that appear due to the choice of discretization spaces for design and analysis. The objective of this work is to present a new method to find hinge-free designs using multiscale wavelet-based topology optimization formulation. The specific method developed in this work does not require refinement of the analysis model and it consists of a translation-invariant wavelet shrinkage method where a hinge-free condition is imposed in the multiscale design space. To imbed the shrinkage method implicitly in the optimization formulation and thus facilitate sensitivity analysis, the shrinkage method is made differentiable by means of differentiable versions of logical operators. The validity of the present method is confirmed by solving typical two-dimensional compliant mechanism design problems.     

A new multi-objective programming scheme for topology optimization of compliant mechanisms

This paper presents an alternative method in implementing multi-objective optimization of compliant mechanisms in the field of continuum-type topology optimization. The method is designated as “SIMP-PP” and it achieves multi-objective topology optimization by merging what is already a mature topology optimization method—solid isotropic material with penalization (SIMP) with a variation of the robust multi-objective optimization method—physical programming (PP). By taking advantages of both sides, the combination causes minimal variation in computation algorithm and numerical scheme, yet yields improvements in the multi-objective handling capability of topology optimization. The SIMP-PP multi-objective scheme is introduced into the systematic design of compliant mechanisms. The final optimization problem is formulated mathematically using the aggregate objective function which is derived from the original individual design objectives with PP, subjected to the specified constraints. A sequential convex programming method, the method of moving asymptotes (MMA) is then utilized to process the optimization evolvement based on the design sensitivity analysis. The main findings in this study include distinct advantages of the SIMP-PP method in various aspects such as computation efficiency, adaptability in convex and non-convex multi-criteria environment, and flexibility in problem formulation. Observations are made regarding its performance and the effect of multi-objective optimization on the final topologies. In general, the proposed SIMP-PP method is an appealing multi-objective topology optimization scheme suitable for “real world” problems, and it bridges the gap between standard topological design and multi-criteria optimization. The feasibility of the proposed topology optimization method is exhibited by benchmark examples.     

A new level-set based approach to shape and topology optimization under geometric uncertainty

Geometric uncertainty refers to the deviation of the geometric boundary from its ideal position, which may have a non-trivial impact on design performance. Since geometric uncertainty is embedded in the boundary which is dynamic and changes continuously in the optimization process, topology optimization under geometric uncertainty (TOGU) poses extreme difficulty to the already challenging topology optimization problems. This paper aims to solve this cutting-edge problem by integrating the latest developments in level set methods, design under uncertainty, and a newly developed mathematical framework for solving variational problems and partial differential equations that define mappings between different manifolds. There are several contributions of this work. First, geometric uncertainty is quantitatively modeled by combing level set equation with a random normal boundary velocity field characterized with a reduced set of random variables using the Karhunen–Loeve expansion. Multivariate Gauss quadrature is employed to propagate the geometric uncertainty, which also facilitates shape sensitivity analysis by transforming a TOGU problem into a weighted summation of deterministic topology optimization problems. Second, a PDE-based approach is employed to overcome the deficiency of conventional level set model which cannot explicitly maintain the point correspondences between the current and the perturbed boundaries. With the explicit point correspondences, shape sensitivity defined on different perturbed designs can be mapped back to the current design. The proposed method is demonstrated with a bench mark structural design. Robust designs achieved with the proposed TOGU method are compared with their deterministic counterparts.     

Stress-based topology optimization

Previous research on topology optimization focussed primarily on global structural behaviour such as stiffness and frequencies. However, to obtain a true optimum design of a vehicle structure, stresses must be considered. The major difficulties in stress based topology optimization problems are two-fold. First, a large number of constraints must be considered, since unlike stiffness, stress is a local quantity. This problem increases the computational complexity of both the optimization and sensitivity analysis associated with the conventional topology optimization problem. The other difficulty is that since stress is highly nonlinear with respect to design variables, the move limit is essential for convergence in the optimization process. In this research, global stress functions are used to approximate local stresses. The density method is employed for solving the topology optimization problems. Three numerical examples are used for this investigation. The results show that a minimum stress design can be achieved and that a maximum stiffness design is not necessarily equivalent to a minimum stress design.     

Adjoint parameter sensitivity analysis for the hydrodynamic lattice Boltzmann method with applications to design optimization

We present an adjoint parameter sensitivity analysis formulation and solution strategy for the lattice Boltzmann method (LBM). The focus is on design optimization applications, in particular topology optimization. The lattice Boltzmann method is briefly described with an in-depth discussion of solid boundary conditions. We show that a porosity model is ideally suited for topology optimization purposes and models no-slip boundary conditions with sufficient accuracy when compared to interpolation bounce-back conditions. Augmenting the porous boundary condition with a shaping factor, we define a generalized geometry optimization formulation and derive the corresponding sensitivity analysis for the single relaxation LBM for both topology and shape optimization applications. Using numerical examples, we verify the accuracy of the analytical sensitivity analysis through a comparison with finite differences. In addition, we show that for fluidic topology optimization a scaled volume constraint should be used to obtain the desired “0-1” optimal solutions.     

Optimal design of periodic structures using evolutionary topology optimization

This paper presents a method for topology optimization of periodic structures using the bi-directional evolutionary structural optimization (BESO) technique. To satisfy the periodic constraint, the designable domain is divided into a certain number of identical unit cells. The optimal topology of the unit cell is determined by gradually removing and adding material based on a sensitivity analysis. Sensitivity numbers that consider the periodic constraint for the repetitive elements are developed. To demonstrate the capability and effectiveness of the proposed approach, topology design problems of 2D and 3D periodic structures are investigated. The results indicate that the optimal topology depends, to a great extent, on the defined unit cells and on the relative strength of other non-designable part, such as the skins of sandwich structures.     

Topology optimization of beam cross-section considering warping deformation

The beam cross-section optimization problems have been very important as beams are widely used as efficient load-carrying structural components. Most of the earlier investigations focus on the dimension and shape optimization or on the topology optimization along the axial direction. An important problem in beam section design is to find the location and direction of stiffeners, for the introduction of a stiffener in a closed beam section may result in a topologically different configuration from the original; the existing section shape optimization theory cannot be used. The purpose of this paper is to formulate a section topology optimization technique based on an anisotropic beam theory considering warping of sections and coupling among deformations. The formulation and corresponding solving method for the topology optimization of beam cross-sections are proposed. In formulating the topology optimization problem, the minimum averaged compliance of the beam is taken as objective, and the material density of every element is used as design variable. The schemes to determine the rigidity matrix of the cross-sections and the sensitivity analysis are presented. Several kinds of topologies of the cross-section under different load conditions are given, and the effect of load condition on the optimum topology is analyzed.     

Topology optimization in crashworthiness design

Topology optimization has developed rapidly, primarily with application on linear elastic structures subjected to static loadcases. In its basic form, an approximated optimization problem is formulated using analytical or semi-analytical methods to perform the sensitivity analysis. When an explicit finite element method is used to solve contact–impact problems, the sensitivities cannot easily be found. Hence, the engineer is forced to use numerical derivatives or other approaches. Since each finite element simulation of an impact problem may take days of computing time, the sensitivity-based methods are not a useful approach. Therefore, two alternative formulations for topology optimization are investigated in this work. The fundamental approach is to remove elements or, alternatively, change the element thicknesses based on the internal energy density distribution in the model. There is no automatic shift between the two methods within the existing algorithm. Within this formulation, it is possible to treat nonlinear effects, e.g., contact–impact and plasticity. Since no sensitivities are used, the updated design might be a step in the wrong direction for some finite elements. The load paths within the model will change if elements are removed or the element thicknesses are altered. Therefore, care should be taken with this procedure so that small steps are used, i.e., the change of the model should not be too large between two successive iterations and, therefore, the design parameters should not be altered too much. It is shown in this paper that the proposed method for topology optimization of a nonlinear problem gives similar result as a standard topology optimization procedures for the linear elastic case. Furthermore, the proposed procedures allow for topology optimization of nonlinear problems. The major restriction of the method is that responses in the optimization formulation must be coupled to the thickness updating procedure, e.g., constraint on a nodal displacement, acceleration level that is allowed.     

Topology optimization of heat conduction problems using the finite volume method

This note addresses the use of the finite volume method (FVM) for topology optimization of a heat conduction problem. Issues pertaining to the proper choice of cost functions, sensitivity analysis, and example test problems are used to illustrate the effect of applying the FVM as an analysis tool for design optimization. This involves an application of the FVM to problems with nonhomogeneous material distributions, and the arithmetic and harmonic averages have here been used to provide a unique value for the conductivity at element boundaries. It is observed that when using the harmonic average, checkerboards do not form during the topology optimization process. Preliminary results of the work reported here were presented at the WCSMO 6 in Rio de Janeiro 2005, see Gersborg-Hansen et al. (2005b).