A LEVEL SET METHOD FOR MICROSTRUCTURE DESIGN OF COMPOSITE MATERIALS

I. INTRODUCTION Material design refers to the generation of composite materials with prescribed or improved propertiesthat cannot be found in the usual materials. This can be achieved by modifying the microstructureof the composite material. Now a syste…     

考虑界面力学性能的组件及结构的协同优化

包含多个内嵌功能组件及支撑结构的多组件结构因其轻量化、多功能等优良特性被广泛应用于航空航天等领域.已有的多组件结构拓扑优化研究大多基于理想界面假设,忽略了材料连接界面可能发生的破坏. 本文针对包含多个内嵌式功能性组件的结构,考虑连接界面的力学性能, 对组件的形状、布局及支撑材料的拓扑进行协同优化,以实现多组件结构的优良承载性能. 首先,基于超椭圆模型对内嵌组件的形状及布局进行显式的参数化描述,并构造其水平集函数表达; 进而,结合组件及支撑材料的水平集拓扑描述、内聚力模型及扩展有限元方法(extended finite element method, XFEM), 在固定网格下对随优化迭代不断演化的结构拓扑及连接界面的力学性能进行准确描述;进一步, 建立水平集法框架下考虑界面力学性能的多组件结构拓扑优化列式,基于伴随变量法推导解析的灵敏度并采用梯度优化算法求解优化问题.本文采用该优化框架分别对内嵌组件的悬臂梁和MBB梁进行协同优化, 在优化过程中,发现组件的初始布局对最终设计有很大的影响, 并且可能导致不良结构.为了避免此情况, 本文提出了两个阶段的优化策略,即首先对组件布局和形状进行优化, 再进行结构和内嵌组件的协同优化.数值结果显示, 在优化结果中功能性组件及界面通常分布于结构受压应力作用的区域,且连接界面最优形状呈现为曲率较小的光滑曲线,该设计避免界面发生拉伸及剪切破坏, 有效提高了结构的承载力,同时也表明了本文所提出考虑连接界面力学性能拓扑优化方法的有效性.      

An explicit level set approach for generalized shape optimization of fluids with the lattice Boltzmann method

This study is concerned with a generalized shape optimization approach for finding the geometry of fluidic devices and obstacles immersed in flows. Our approach is based on a level set representation of the fluid–solid interface and a hydrodynamic lattice Boltzmann method to predict the flow field. We present an explicit level set method that does not involve the solution of the Hamilton–Jacobi equation and allows using standard nonlinear programming methods. In contrast to previous works, the boundary conditions along the fluid–structure interface are enforced by second‐order accurate interpolation schemes, overcoming shortcomings of flow penalization methods and Brinkman formulations frequently used in topology optimization. To ensure smooth boundaries and mesh‐independent results, we introduce a simple, computationally inexpensive filtering method to regularize the level set field. Furthermore, we define box constraints for the design variables that guarantee a continuous evolution of the boundaries. The features of the proposed method are studied by two numeric examples of two‐dimensional steady‐state flow problems. Copyright © 2009 John Wiley & Sons, Ltd.     

Topological shape optimization of power flow problems at high frequencies using level set approach

Using a level set method we develop a topological shape optimization method applied to power flow problems in steady state. Necessary design gradients are computed using an efficient adjoint sensitivity analysis method. The boundaries are implicitly represented by the level set function obtainable from the “Hamilton–Jacobi type” equation with the “Up-wind scheme.” The implicit function is embedded into a fixed initial domain to obtain the finite element response and sensitivity. The developed method defines a Lagrangian function for the constrained optimization. It minimizes a generalized compliance, satisfying the constraint of allowable volume through the variations of implicit boundary. During the optimization, the boundary velocity to integrate the Hamilton–Jacobi equation is obtained from the optimality condition for the Lagrangian function. Compared with the established topology optimization method, the developed one has no numerical instability such as checkerboard problems and easy representation of topological shape variations.     

基于双向演化的局部水平集组合算法用于拓扑优化

本文基于双向渐进结构优化(BESO)方法和局部水平集方法(LLSM),提出局部水平集组合算法,在拓扑优化中实现双向演化。为改进LLSM的孔洞成核能力,新算法以所提离散水平集函数为节点设计变量,拓扑导数为灵敏度,按照BESO优化准则进行双向演化得到稳定拓扑解。然后通过迭代求解距离正则化方程(DRE)来组合LLSM获得最终拓扑。在LLSM中,DRE用来代替重生成步骤,并构造条件稳定差分格式求解DRE。最终给出典型实例验证所提算法的收敛性和数值稳定性。     

A method for the analysis of dynamic response of structure containing non-smooth contactable interfaces

A novel single-step method is proposed for the analysis of dynamic response of visco-elastic structures containing non-smooth contactable interfaces. In the method, a two-level algorithm is employed for dealing with a nonlinear boundary condition caused by the dynamic contact of interfaces. At the first level, and explicit method is adopted to calculate nodal displacements of global viscoelastic system without considering the effect of dynamic contact of interfaces and at the second level, by introducing contact conditions of interfaces, a group of equations of lower order is derived to calculate dynamic contact normal and shear forces on the interfaces. The method is convenient and efficient for the analysis of problems of dynamic contact. The accuracy of the method is of the second order and the numerical stability condition is wider than that of other explicit methods. The project supported by the National Natural Science Foundation of China (59578032) and the Key Project of the Ninth Five-Year Plan (96221030202)     

Optimization of unsteady incompressible Navier–Stokes flows using variational level set method

This paper presents the optimization of unsteady Navier–Stokes flows using the variational level set method. The solid–liquid interface is expressed by the level set function implicitly, and the fluid velocity is constrained to be zero in the solid domain. An optimization problem, which is constrained by the Navier–Stokes equations and a fluid volume constraint, is analyzed by the Lagrangian multiplier based adjoint approach. The corresponding continuous adjoint equations and the shape sensitivity are derived. The level set function is evolved by solving the Hamilton–Jacobian equation with the upwind finite difference method. The optimization method can be used to design channels for flows with or without body forces. The numerical examples demonstrate the feasibility and robustness of this optimization method for unsteady Navier–Stokes flows.Copyright © 2012 John Wiley & Sons, Ltd.     

基于应力及其灵敏度的结构拓扑渐进优化方法

在导出应力灵敏度的基础上，建立了一种改进的基于应力及其灵敏度的结构拓扑双方向渐进优化算法．该方法是对传统ESO和BESO方法的改进和完善．算例表明该方法能较大程度减少解的振荡状态数，获得更佳的拓扑结构．具有概念上的简洁性和应用上的有效性。     

基于功能度量法的桁架结构非概率可靠性拓扑优化方法研究

在结构优化中,拓扑优化相比于尺寸优化和形状优化,设计空间更加广泛,因而能够取得更大的效益.近年来,结构拓扑优化逐渐成为人们研究的热点和难点.随着科学技术的发展,工程结构越来越复杂,材料本身和外部环境的不确定性影响加剧,因此在拓扑优化中需要考虑不确定性的影响.本文研究了桁架结构的非概率可靠性拓扑优化问题,用区间模型来量化不确定性,并利用参数顶点组合法来完成不确定性的传播分析,利用基于面积比的非概率可靠性指标构建可靠性拓扑优化模型,提出了功能度量法对原可靠性约束进行等价转化,从而克服了收敛性问题.采用移动渐近方法(MMA)对优化问题进行了求解.数值算例表明,本文提出的功能度量法能够很好地适用于桁架结构的非概率可靠性拓扑优化问题.     

A hybrid Lagrangian–Eulerian particle‐level set method for numerical simulations of two‐fluid turbulent flows

A coupled Lagrangian interface‐tracking and Eulerian level set (LS) method is developed and implemented for numerical simulations of two‐fluid flows. In this method, the interface is identified based on the locations of notional particles and the geometrical information concerning the interface and fluid properties, such as density and viscosity, are obtained from the LS function. The LS function maintains a signed distance function without an auxiliary equation via the particle‐based Lagrangian re‐initialization technique. To assess the new hybrid method, numerical simulations of several ‘standard interface‐moving’ problems and two‐fluid laminar and turbulent flows are conducted. The numerical results are evaluated by monitoring the mass conservation, the turbulence energy spectral density function and the consistency between Eulerian and Lagrangian components. The results of our analysis indicate that the hybrid particle‐level set method can handle interfaces with complex shape change, and can accurately predict the interface values without any significant (unphysical) mass loss or gain, even in a turbulent flow. The results obtained for isotropic turbulence by the new particle‐level set method are validated by comparison with those obtained by the ‘zero Mach number’, variable‐density method. For the cases with small thermal/mass diffusivity, both methods are found to generate similar results. Analysis of the vorticity and energy equations indicates that the destabilization effect of turbulence and the stability effect of surface tension on the interface motion are strongly dependent on the density and viscosity ratios of the fluids. Copyright © 2007 John Wiley & Sons, Ltd.     

基于拓扑描述函数的连续体结构拓扑优化方法

提出了一种利用拓扑描述函数(TDF)作为拓扑设计变量求解连续体结构拓扑优化问题 的新方法. 优化问题的目标函数是结构的整体柔顺性，约束条件为对于可利用材料的体积限 制. 这种方法不仅可以消除拓扑优化中经常出现的棋盘格式等数值不稳定现象，而且能够有 效地抑制传统算法处理此类优化问题时所引发的边界扩散效应. 与其它的基于水平集描述函 数的拓扑优化方法相比，所提出的算法不仅无需求解控制水平集函数演化的双曲守恒方 程，而且合理地考虑了目标函数的拓扑导数信息，因而使得算法的计算效率有了显著的提高.     

一种基于应力的双方向结构拓扑优化算法

基于传统的渐进结构优化方法，提出了一种基于应力的双方向渐进拓扑优化算法。该方法是对传统方法和目前的双方向法的改进和完善。算例表明该方法能避免目前的双方向法具有的解的振荡问题，具有更高的可靠性，能获得更佳的拓扑结构。     

Finite‐element/level‐set/operator‐splitting (FELSOS) approach for computing two‐fluid unsteady flows with free moving interfaces

The present work is devoted to the study on unsteady flows of two immiscible viscous fluids separated by free moving interface. Our goal is to elaborate a unified strategy for numerical modelling of two‐fluid interfacial flows, having in mind possible interface topology changes (like merger or break‐up) and realistically wide ranges for physical parameters of the problem. The proposed computational approach essentially relies on three basic components: the finite element method for spatial approximation, the operator‐splitting for temporal discretization and the level‐set method for interface representation. We show that the finite element implementation of the level‐set approach brings some additional benefits as compared to the standard, finite difference level‐set realizations. In particular, the use of finite elements permits to localize the interface precisely, without introducing any artificial parameters like the interface thickness; it also allows to maintain the second‐order accuracy of the interface normal, curvature and mass conservation. The operator‐splitting makes it possible to separate all major difficulties of the problem and enables us to implement the equal‐order interpolation for the velocity and pressure. Diverse numerical examples including simulations of bubble dynamics, bifurcating jet flow and Rayleigh–Taylor instability are presented to validate the computational method. Copyright © 2004 John Wiley & Sons, Ltd.     

多工况下结构拓扑优化设计

考虑单元删除和增加对结构应力约束的影响，提出了一种新的多工况下结构的双方向渐进优化方法．首先，基于在结构孔洞或边界周围附加人工材料的思路，建立了结构优化模型和应力灵敏度公式．然后，结合结构应力和应力灵敏度，给出了多工况静力载荷下考虑静应力约束的结构优化准则和算法．开展了结构仿真设计，结果表明给出的方法是正确和有效的，并具有较好的工程应用价值．     

结构拓扑优化ICM方法的改善

对结构拓扑优化的ICM(独立、连续、映射)方法进行了深入探讨，通 过选取不同的过滤函数可以不进行每步删除而得到清晰的拓扑图形. 以位移约束为例阐述了 ICM方法建模及求解过程. 对位移约束、频率约束、位移及频率约束、简谐载荷激励下动位 移幅值约束等拓扑优化进行了研究，计算算例表明ICM方法在处理静力问题及动力问题的拓 扑优化都是可行的. 程序算法都在MSC.Nastran及MSC.Patran的二次开发环境下实现，与 原软件有机结合在一起.     

STUDY ON PARAMETERS FOR TOPOLOGICAL VARIABLES FIELD INTERPOLATED BY MOVING LEAST SQUARE APPROXIMATION

This paper presents a new approach to the structural topology optimization of continuum structures. Material-point independent variables are presented to illustrate the existence condition,or inexistence of the material points and their vicinity instead of elements or nodes in popular topology optimization methods. Topological variables field is constructed by moving least square approximation which is used as a shape function in the meshless method. Combined with finite element analyses,not only checkerboard patterns and mesh-dependence phenomena are overcome by this continuous and smooth topological variables field,but also the locations and numbers of topological variables can be arbitrary. Parameters including the number of quadrature points,scaling parameter,weight function and so on upon optimum topological configurations are discussed. Two classic topology optimization problems are solved successfully by the proposed method. The method is found robust and no numerical instabilities are found with proper parameters.     

Numerical performance of two formulations of truss topology optimization

The present paper studies topology optimization of truss structures in multiple loading cases and with stress constraints. It is pointed out in the paper that the special difficulty of adding bars and/or deleting bars from structure in the numerical algorithm of truss topology optimization is caused by the discontinuity of stress functions at the zero cross sectional area in the conventional formulation. In a new formulation, we replace the stress constraints by new constraints. The new constraints retain the same feasibility of the stress constraints, but are continuous in the closed interval up to zero cross sectional area. The new formulation enables us to solve topology optimization problem in the frame of the existing FEM software and mathematical programming techniques. Powell constrained variable metric method is applied to a number of examples of truss topology optimization. Numerical performances of the two formulations are compared. It is shown that in the conventional formulation the iteration of numerical algorithm may be blocked by discontinuity of the stress constraint and often stops at a nonoptimum solution. And in the new formulation the bar adding and bar deleting is done rationally and a local optimum, even the global optimum can be obtained by iteration. The project supported by the National Natural Science Foundation of China     

A projection scheme for incompressible multiphase flow using adaptive Eulerian grid

This paper presents a finite element method for incompressible multiphase flows with capillary interfaces based on a (formally) second‐order projection scheme. The discretization is on a fixed Eulerian grid. The fluid phases are identified and advected using a level set function. The grid is temporarily adapted around the interfaces in order to maintain optimal interpolations accounting for the pressure jump and the discontinuity of the normal velocity derivatives. The least‐squares method for computing the curvature is used, combined with piecewise linear approximation to the interface. The time integration is based on a formally second order splitting scheme. The convection substep is integrated over an Eulerian grid using an explicit scheme. The remaining generalized Stokes problem is solved by means of a formally second order pressure‐stabilized projection scheme. The pressure boundary condition on the free interface is imposed in a strong form (pointwise) at the pressure‐computation substep. This allows capturing significant pressure jumps across the interface without creating spurious instabilities. This method is simple and efficient, as demonstrated by the numerical experiments on a wide range of free‐surface problems. Copyright © 2004 John Wiley & Sons, Ltd.