Design of distributed compliant micromechanisms with an implicit free boundary representation

In this paper, a parameterization approach is presented for structural shape and topology optimization of compliant mechanisms using a moving boundary representation. A level set model is developed to implicitly describe the structural boundary by embedding into a scalar function of higher dimension as zero level set. The compactly supported radial basis function of favorable smoothness and accuracy is used to interpolate the level set function. Thus, the temporal and spatial initial value problem is now converted into a time-separable parameterization problem. Accordingly, the more difficult shape and topology optimization of the Hamilton–Jacobi equation is then transferred into a relatively easy size optimization with the expansion coefficients as design variables. The design boundary is therefore advanced by applying the optimality criteria method to iteratively evaluate the size optimization so as to update the level set function in accordance with expansion coefficients of the interpolation. The optimization problem of the compliant mechanism is established by including both the mechanical efficiency as the objective function and the prescribed material usage as the constraint. The design sensitivity analysis is performed by utilizing the shape derivative. It is noted that the present method is not only capable of simultaneously addressing shape fidelity and topology changes with a smooth structural boundary but also able to avoid some of the unfavorable numerical issues such as the Courant–Friedrich–Levy condition, the velocity extension algorithm, and the reinitialization procedure in the conventional level set method. In particular, the present method can generate new holes inside the material domain, which makes the final design less insensitive to the initial guess. The compliant inverter is applied to demonstrate the availability of the present method in the framework of the implicit free boundary representation.     

An isogeometrical approach to structural topology optimization by optimality criteria

The Isogeometric Analysis (IA) method is applied for structural topology optimization instead of finite elements. For this purpose, a control point based Solid Isotropic Material with Penalization (SIMP) method is employed and the material density is considered as a continuous function throughout the design domain and approximated by the Non-Uniform Rational B-Spline (NURBS) basis functions. To prevent the formation of layouts with porous media, a penalization technique similar to the SIMP method is used. For optimization an optimality criteria is derived and implemented. A few examples are presented to demonstrate the performance of the method. It is shown that, dissimilar to the element based SIMP topology optimization, the resulted layouts by this method are independent of the number of the discretizing control points and checkerboard free.     

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.     

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.     

Simultaneous shape and topology optimization of shell structures

In this research, Method of Moving Asymptotes (MMA) is utilized for simultaneous shape and topology optimization of shell structures. It is shown that this approach is well matched with the large number of topology and shape design variables. The currently practiced technology for optimization is to find the topology first and then to refine the shape of structure. In this paper, the design parameters of shape and topology are optimized simultaneously in one go. In order to model and control the shape of free form shells, the NURBS (Non Uniform Rational B-Spline) technology is used. The optimization problem is considered as the minimization of mean compliance with the total material volume as active constraint and taking the shape and topology parameters as design variables. The material model employed for topology optimization is assumed to be the Solid Isotropic Material with Penalization (SIMP). Since the MMA optimization method requires derivatives of the objective function and the volume constraint with respect to the design variables, a sensitivity analysis is performed. Also, for alleviation of the instabilities such as mesh dependency and checkerboarding the convolution noise cleaning technique is employed. Finally, few examples taken from literature are presented to demonstrate the performance of the method and to study the effect of the proposed concurrent approach on the optimal design in comparison to the sequential topology and shape optimization methods.     

Topology optimization using a reaction–diffusion equation

This paper presents a structural topology optimization method based on a reaction–diffusion equation. In our approach, the design sensitivity for the topology optimization is directly employed as the reaction term of the reaction–diffusion equation. The distribution of material properties in the design domain is interpolated as the density field which is the solution of the reaction–diffusion equation, so free generation of new holes is allowed without the use of the topological gradient method. Our proposed method is intuitive and its implementation is simple compared with optimization methods using the level set method or phase field model. The evolution of the density field is based on the implicit finite element method. As numerical examples, compliance minimization problems of cantilever beams and force maximization problems of magnetic actuators are presented to demonstrate the method’s effectiveness and utility.     

Topology optimization in OpenMDAO

Recently, topology optimization has drawn interest from both industry and academia as the ideal design method for additive manufacturing. Topology optimization, however, has a high entry barrier as it requires substantial expertise and development effort. The typical numerical methods for topology optimization are tightly coupled with the corresponding computational mechanics method such as a finite element method and the algorithms are intrusive, requiring an extensive understanding. This paper presents a modular paradigm for topology optimization using OpenMDAO, an open-source computational framework for multidisciplinary design optimization. This provides more accessible topology optimization algorithms that can be non-intrusively modified and easily understood, making them suitable as educational and research tools. This also opens up further opportunities to explore topology optimization for multidisciplinary design problems. Two widely used topology optimization methods—the density-based and level-set methods—are formulated in this modular paradigm. It is demonstrated that the modular paradigm enhances the flexibility of the architecture, which is essential for extensibility.

     

Minimum length-scale constraints for parameterized implicit function based topology optimization

This paper introduces explicit minimum length-scale constraint functions suitable for parameterized implicit function based topology optimization methods. Length-scale control in topology optimization has many potential benefits, such as removing numerical artifacts, mesh independent solutions, avoiding thin, or single node hinges in compliant mechanism design and meeting manufacturing constraints. Several methods have been developed to control length-scale when using density-based or signed-distance-based level-set methods. In this paper a method is introduced to control length-scale for parameterized implicit function based topology optimization. Explicit constraint functions to control the minimum length in the structure and void regions are proposed and implementation issues explored in detail. Several examples are presented to show the efficacy of the proposed method. The examples demonstrate that the method can simultaneously control minimum structure and void length-scale, design hinge free compliant mechanisms and control minimum length-scale for three dimensional structures.     

Taking a free ride for routing topology inference in peer-to-peer networks

A Peer-to-Peer (P2P) network can boost its performance if peers are provided with underlying network-layer routing topology. The task of inferring the network-layer routing topology and link performance from an end host to a set of other hosts is termed as network tomography, and it normally requires host computers to send probing messages. We design a passive network tomography method that does not require any probing messages and takes a free ride over data flows in P2P networks. It infers routing topology based on end-to-end delay correlation estimation (DCE) without requiring any synchronization or cooperation from the intermediate routers. We implement and test our method in the real world Internet environment and achieved the accuracy of 92 % in topology recovery. We also perform extensive simulation in OMNeT++ to evaluate its performance over large scale networks, showing that its topology recovery accuracy is about 95 % for large networks.     

Topology optimization of continuum structures subjected to pressure loading

This paper presents a generalization of topology optimization of linearly elastic continuum structures to problems involving loadings that depend on the design. Minimum compliance is chosen as the design objective, assuming the boundary conditions and the total volume within the admissible design domain to be given. The topology optimization is based on the usage of a SIMP material model. The type of loading considered in this paper occurs if free structural surface domains are subjected to static pressure, in which case both the direction and location of the loading change with the structural design. The presentation of the material is given in a 2D context, but an extension to 3D is straightforward. The robustness of the optimization method is illustrated by some numerical examples in the end of the paper. Received August 3, 1999     

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.     

A level set method for structural shape and topology optimization using radial basis functions

This paper presents an alternative level set method for shape and topology optimization of continuum structures. An implicit free boundary representation model is established by embedding structural boundary into the zero level set of a higher-dimensional level set function. An explicit parameterization scheme for the level set surface is proposed by using radial basis functions with compact support. In doing so, the originally more difficult shape and topology optimization, driven by the temporal and spatial Hamilton–Jacobi partial differential equation (PDE), is transformed into a relatively easier size optimization of the expansion coefficients of the basis functions. The design optimization is converted to an iterative numerical process that combines the parameterization with a derivation of the shape sensitivity of the design functions, so as to allow using mathematical programming algorithms to solve the level set-based design problem and avoid directly solving the Hamilton–Jacobi PDE. Furthermore, a numerically more stable and efficient volume integration scheme is proposed to implement calculations of the shape derivatives, leading to the creation of new holes which are generated initially along the boundary and then propagated to the interior of the design domain. Two widely studied examples are used to demonstrate the effectiveness of the proposed optimization method.     

Topology optimization of flow domains using the lattice Boltzmann method

We consider the optimal design of two- (2D) and three-dimensional (3D) flow domains using the lattice Boltzmann method (LBM) as an approximation of Navier-Stokes (NS) flows. The problem is solved by a topology optimization approach varying the effective porosity of a fictitious material. The boundaries of the flow domain are represented by potentially discontinuous material distributions. NS flows are traditionally approximated by finite element and finite volume methods. These schemes, while well established as high-fidelity simulation tools using body-fitted meshes, are effected in their accuracy and robustness when regular meshes with zero-velocity constraints along the surface and in the interior of obstacles are used, as is common in topology optimization. Therefore, we study the potential of the LBM for approximating low Mach number incompressible viscous flows for topology optimization. In the LBM the geometry of flow domains is defined in a discontinuous manner, similar to the approach used in material-based topology optimization. In addition, this non-traditional discretization method features parallel scalability and allows for high-resolution, regular fluid meshes. In this paper, we show how the variation of the porosity can be used in conjunction with the LBM for the optimal design of fluid domains, making the LBM an interesting alternative to NS solvers for topology optimization problems. The potential of our topology optimization approach will be illustrated by 2D and 3D numerical examples.     

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.     

轨道扣件弹性垫板结构优化设计

采用OptiStruct对某型铁路扣件轨下垫板进行拓扑优化和自由形状优化,实现垫板结构轻量化设计.根据优化空间最大化原则,对垂向载荷工况进行拓扑优化、对极限载荷工况进行自由形状优化.使用Abaqus极限载荷扣件系统仿真模型对优化后垫板结构性能进行验证评价.结果表明:优化后的垫板性能基本不变,质量减少约10%,刚度满足设计要求.优化结果可为垫板轻量化设计提供参考.     

A level-set-based topology and shape optimization method for continuum structure under geometric constraints

Recent advances in level-set-based shape and topology optimization rely on free-form implicit representations to support boundary deformations and topological changes. In practice, a continuum structure is usually designed to meet parametric shape optimization, which is formulated directly in terms of meaningful geometric design variables, but usually does not support free-form boundary and topological changes. In order to solve the disadvantage of traditional step-type structural optimization, a unified optimization method which can fulfill the structural topology, shape, and sizing optimization at the same time is presented. The unified structural optimization model is described by a parameterized level set function that applies compactly supported radial basis functions (CS-RBFs) with favorable smoothness and accuracy for interpolation. The expansion coefficients of the interpolation function are treated as the design variables, which reflect the structural performance impacts of the topology, shape, and geometric constraints. Accordingly, the original topological shape optimization problem under geometric constraint is fully transformed into a simple parameter optimization problem; in other words, the optimization contains the expansion coefficients of the interpolation function in terms of limited design variables. This parameterization transforms the difficult shape and topology optimization problems with geometric constraints into a relatively straightforward parameterized problem to which many gradient-based optimization techniques can be applied. More specifically, the extended finite element method (XFEM) is adopted to improve the accuracy of boundary resolution. At last, combined with the optimality criteria method, several numerical examples are presented to demonstrate the applicability and potential of the presented method.     

Topology optimization for femto suspension design

The ongoing increase of track density requirements of hard disk drives (HDDs) and decrease of flying height of sliders have brought along formidable challenges to suspension design. Conventional design processes are quite tedious and inefficient. This paper presents a HDD suspension design process by using topology optimization. An efficient structural topology optimization method, based on the second derivatives information, is proposed to generate structures which satisfy multiple design objectives including both compliance and eigenfrequencies. This topology optimization approach is successfully applied in the HDD suspension design. The design begins with a very simple initial draft, and the design objectives are defined to minimize the spring rate and maximize the resonant frequencies of first bending, first torsion and sway modes of a suspension. An optimal design concept can be generated from the topology optimization. Then the design is further tuned by using the shape optimization. Finally, an optimal suspension design for femto sliders with much better dynamic characteristics is presented.     

Topology optimization of steady and unsteady incompressible Navier–Stokes flows driven by body forces

This paper presents the topology optimization method for the steady and unsteady incompressible Navier–Stokes flows driven by body forces, which typically include the constant force (e.g. the gravity) and the centrifugal and Coriolis forces. In the topology optimization problem, the artificial friction force with design variable interpolated porosity is added into the Navier–Stokes equations as the conventional method, and the physical body forces in the Navier–Stokes equations are penalized using the power-law approach. The topology optimization problem is analyzed by the continuous adjoint method, and solved by the finite element method in conjunction with the gradient based approach. In the numerical examples, the topology optimization of the fluidic channel, mass distribution of the flow and local velocity control are presented for the flows driven by body forces. The numerical results demonstrate that the presented method achieves the topology optimization of the flows driven by body forces robustly.     

Topology optimization of a swing arm type actuator using the response surface method

In the frame of topology optimization, the multi-objective ability has to be considered since structural design is usually required to satisfy more than one requirement. A modified topology optimization method based on the response surface method (RSM) is proposed to generate a structure of a small form factor (SFF) swing arm type actuator satisfying maximum compliance and maximum stiffness at the same time using the multi-objective optimization approach. The multi-objective function is defined to maximize the compliance in the direction of focusing as well as the eigen-frequency of the structure. The design of experiments (DOE) is performed to select sensitive variables. Based on DOE results, the response surface functions are formulated to construct the multi-objective function. The weight factors between conflicting objective functions are determined by the Pareto optimum method. By applying the optimal combination of design variables to the design domain, the optimized topology can be obtained.This work was supported by Korea Research Foundation (KRF) Grant KRF-2004-042-D00004.     

Microstructural topology optimization of structural-acoustic coupled systems for minimizing sound pressure level

The aim of this paper is to present a microstructural topology optimization methodology for the structural-acoustic coupled system. In the structural-acoustic system, the structure is considered to be a thin composite plate composed of periodic uniform microstructures. The discrete design variables are used in the microstructural topology optimization, and the constitutive matrix is interpolated by the power-law scheme at the micro scale. The equivalent macro material properties of the microstructure are computed through the homogenization method. The design objective is to minimize the sound pressure level (SPL) in an interior acoustic medium. The sensitivities of the SPL with respect to design variables are derived. The bi-directional evolutionary structural optimization (BESO) method is extended to solve the structural-acoustic coupled optimization problem to find the optimal material distribution of the microstructure. Numerical examples of a hexahedral box and an automobile passenger compartment are given to demonstrate the efficiency of the presented microstructural topology optimization method.