On structural shape optimization using an embedding domain discretization technique

This contribution presents a novel approach to structural shape optimization that relies on an embedding domain discretization technique. The evolving shape design is embedded within a uniform finite element background mesh which is then used for the solution of the physical state problem throughout the course of the optimization. We consider a boundary tracking procedure based on adaptive mesh refinement to separate between interior elements, exterior elements, and elements intersected by the physical domain boundary. A selective domain integration procedure is employed to account for the geometric mismatch between the uniform embedding domain discretization and the evolving structural component. Thereby, we avoid the need to provide a finite element mesh that conforms to the structural component for every design iteration, as it is the case for a standard Lagrangian approach to structural shape optimization. Still, we adopt an explicit shape parametrization that allows for a direct manipulation of boundary vertices for the design evolution process. In order to avoid irregular and impracticable design updates, we consider a geometric regularization technique to render feasible descent directions for the course of the optimization. Copyright © 2016 John Wiley & Sons, Ltd.     

Multi-Disciplinary Optimization for Multi-Objective Uncertainty Design of Thin Walled Beams

The focus of this paper is concentrated on multi-disciplinary and multi-objective optimization for thin walled beam systems considering safety, normal mode, static loading-bearing and weight, in which the uncertainties of the parameters are described via intervals. The size and shape of the cross-section are treated as design parameters during optimization. Considering the lightweight and safety, the design problem is formulated with two individual objectives to measure structural weight and maximum energy absorption, respectively, constrained by the average force, normal mode and maximum stress. The optimization problem with uncertainties is further transformed into a deterministic optimization based on interval number programming. The approximation models, coupled with the design of experiment (DOE) technique, are employed to construct objective functions and constraints. The uncertain optimization problem characterized with these approximation models is performed and applied to a practical thin walled beam design problems.     

三维约束下的形状记忆合金相变动力学模拟

依据马氏体含量－温度－应力关系,建立了三维约束下形状记忆合金颗粒相变动力学模型。利用这一模型,根据假设的复合材料中的形状记忆合金颗粒所受的应力状态,可对其相变DSC曲线进行数值模拟,为记忆合金所处的实际应力状态分析提供参考依据。     

浮动元件壁面剪切应力传感器研究进展

介绍了浮动元件的类型和浮动元件壁面剪切应力测量的方法,阐述了基于浮动元件的电容式、压阻式和光学式壁面剪切应力传感器的基本原理和研究现状,分析了上述3种类型的浮动元件壁面剪切应力传感器的优缺点,指出可通过优化浮动元件与传感器封装件之间的移动间隙以及浮动元件与被测面间的平齐度来提升浮动元件壁面剪切应力传感器的性能。展望了浮动元件壁面剪切应力传感器在湍流测量、判断边界层转捩、维护飞行器安全和优化飞行器结构等领域的发展方向,提出未来可通过发展MEMS技术、优化传感器后端处理电路和温度补偿方式、采用一体化设计加工方式等,进一步提升浮动元件壁面剪切应力传感器的小型化程度、检测壁面剪切应力极低值时的灵敏度和精度、测量的可靠性和准确性。     

Parametric Optimization of Hybrid Car Engines

We consider the problem of optimal design of hybrid car engines which combine thermal and electric power. The optimal configuration of the different motors composing the hybrid system involves the choice of certain design parameters. For a given configuration, the goal is to minimize the fuel consumption along a trajectory. This is an optimal control problem with one state variable.The simultaneous optimization of design parameters and trajectories can be formulated as a bilevel optimization problem. The lower level computes the optimal control for a given architecture. The higher level seeks for the optimal design parameters by solving a nonconvex nonsmooth optimization problem with a bundle method.     

Shape optimization using the boundary element method with substructuring

Applications of the boundary element method for two- and three-dimensional structural shape optimization are presented. The displacements and stresses are computed using the boundary element method. Sub-structuring is used to isolate the portion of the structure undergoing geometric change. The corresponding non-linear programming problem for the optimization is solved by the generalized reduced gradient method. B-spline curves and surfaces are introduced to describe the shape of the design. The control points on these curves or surfaces are selected as design variables. The design objective may be either to minimize the weight or a peak stress of the component by determining the optimum shape subject to geometrical and stress constraints. The use of substructuring allows for problem solution without requiring traditional simplifications such as linearization of the constraints. The method has been successfully applied to the structural shape optimization of plane stress, plane strain and three-dimensional elasticity problems.     

The inverse problem in creeping film flows

We consider the inverse problem for gravity-driven free surface flows at vanishing Reynolds numbers. In contrast to the direct problem, where information about the underlying topographic structure is given and the steady free surface shape and the flow field are unknown, the inverse problem deals with the flow along unknown topographies. The bottom shape and the corresponding flow field are reconstructed from information at the steady free surface only. We discuss two different configurations for the inverse problem. In the first case, we assume a given free surface shape, and by simplifying the field equations, we find an analytical solution for the corresponding bottom topography, velocity field, wall shear stress, and pressure distribution. The analytical results are successfully compared with experimental data from the literature and with numerical data of the Navier–Stokes equations. In the second inverse problem, we prescribe a free surface velocity and then solve numerically for the full flow domain, i.e. the free surface shape, the topography and simultaneously the wall shear stress and the pressure field. The results are validated with the numerical solution of the corresponding direct problem.     

Use of the Constrained Optimization Algorithms in Some Problems of Fracture Mechanics

In this paper we consider an algorithm of constrained optimization which arises from boundary variational principles of elastodynamics for bodies with cracks and unilateral constraints on the cracks edges. Variational formulation of unilateral contact problems with friction was considered, boundary variational functionals used with boundary integral equations were obtained and algorithm for solution of the unilateral contact problem with friction was developed. Some numerical results for 3-D elastodynamic unilateral contact problem for bodies with cracks are presented.     

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.     

导管架海洋平台结构模糊优化设计

考虑约束条件边界的模糊性,建立了导管架海洋平台结构模糊优化设计模型。对模糊优化模型中的设计变量、目标函数和约束条件进行了模糊处理。针对导管架海洋平台的特点,用模糊优选法确定约束条件边界容差系数,由界限搜索法求解模糊约束集和模糊目标集之交集的最优水平截集*l,进而求得模糊优化问题的最优解。以胜利油田埕北11#井采油平台为例进行了模糊优化设计,并与确定性优化设计相比较,分析了两种优化设计中设计变量的走向及原因,算例结果还显示目标函数值比确定性优化设计值有较大幅度下降,说明考虑模糊因素进行优化设计的可行性和科学性。     

内置碟簧自复位混凝土剪力墙基于性能的截面设计方法

内置碟簧自复位混凝土剪力墙主要由墙体及墙脚两侧的碟簧装置组成,碟簧装置具有较高的抗压能力,卸载后能恢复到变形前的状态,为墙体提供恢复力,减小结构的震后残余变形。为了较好地设计内置碟簧自复位混凝土剪力墙,该文提出基于性能的截面设计方法。定义了四水准下结构的性能目标和损伤状态,直接基于第三水准下的位移目标设计剪力墙截面尺寸,碟簧装置几何尺寸、承载能力和变形能力。根据自复位剪力墙截面的受力分析,推导其承载力理论计算公式,并对设计的内置碟簧自复位混凝土剪力墙进行弹塑性分析。结果表明,按该方法设计的自复位剪力墙具有较好的复位能力,墙体的损伤也得到有效控制,在位移角分别为0.5%和1%时,残余位移角仅分别为0.012%和0.022%,损伤指标分别为0.12和0.21,符合基于性能的设计目标。推导的理论计算公式能很好地评估剪力墙的承载能力,计算结果与有限元模拟结果吻合较好。     

A numerical study on the elements of shape optimum design

This paper discusses the main elements of shape optimization. The material derivative of a stress function using the continuum approach is derived by introducing an adjoint problem, which is then transformed into shape design sensitivity by replacing the velocity field with the change of the design variables. The difficulty related with the appearance of the concentrated adjoint loads is discussed, with two proposals for the modelling of the adjoint problem. A numerical example is used to demonstrate the accuracy of the proposed formulation for different adjoint loads.

Two shape optimization examples are used to investigate the numerical characteristics of the optimization process. Two kinds of design boundary modelling are employed, namely the linear and cubic spline boundary representation. The difference of the final design shapes under different design variables and mesh distributions are also studied.     

SIMULTANEOUS OPTIMIZATION OF SHAPE AND FLOW PARAMETERS FOR COMBINED FREE AND FORCED CONVECTION IN VERTICAL CHANNELS

Simultaneous optimization of shape and flow parameters is performed for a combined free and forced convection flow through vertical rectangular channels with moving walls. The laminar flow is assumed to be fully developed in the axial direction. The wall velocity, the axial pressure gradient and the channel height in the transverse plane are taken as the optimization parameters. The sensitivity expressions of both the objective function and the flow rate constraint of optimization are obtained in terms of the relevant physical variables, as well as adjoint variables which satisfy additional p.d.e.'s. All equations are discretized using the finite element method. Numerical results are provided for the present constrained optimization problem for various values of the problem parameters which include the moving wall segment size and the Rayleigh number. The results indicate that with increased Rayleigh number the optimal values of the wall velocity and the axial pressure gradient are increased, while the optimal value of the channel height is decreased. General sensitivity expressions are also presented in the appendix which might be utilized for arbitrary boundary variations along with arbitrary optimization objectives in other investigations. © 1997 by John Wiley & Sons, Ltd.     

Application of geometric programming to transformer design

This paper considers the transformer design optimization problem. In its most general form, the design problem requires minimizing the total mass (or cost) of the core and wire material while ensuring the satisfaction of the transformer ratings and a number of design constraints. The constraints include appropriate limits on efficiency, voltage regulation, temperature rise, no-load current, and winding fill factor. The design optimization seeks a constrained minimum mass (or cost) solution by optimally setting the transformer geometry parameters and the relevant electrical and magnetic quantities. In cases where the core dimensions are fixed, the optimization problem calls for a constrained maximum volt-ampere or minimum loss solution. This paper shows that the above design problems can be formulated in geometric programming (GP) format. The importance of the GP format stems from two main features. First, GP provides an efficient and reliable solution for the design optimization problem with several variables. Second, it guarantees that the obtained solution is the global optimum. The paper includes a demonstration of the application of the GP technique to transformer design. It also includes a comparative study to emphasize the advantage of including the transformer core dimensions as variables in the design problem.     

OPTIMUM SHAPE DESIGN AND POSITIONING OF FEATURES USING THE BOUNDARY INTEGRAL EQUATION METHOD

A design optimization procedure is developed using the boundary integral equation (BIE) method for linear elastostatic two-dimensional domains. Optimal shape design problems are treated where design variables are geometric parameters such as the positions and sizing dimensions of entire features on a component or structure. A fully analytical approach is adopted for the design sensitivity analysis where the BIE is implicitly differentiated. The ability to evaluate response sensitivity derivatives with respect to design variables such as feature positions is achieved through the definition of appropriate design velocity fields for these variables. How the advantages of the BIE method are amplified when extended to sensitivity analysis for this category of shape design problems is also highlighted. A mathematical programming approach with the penalty function method is used for solving the overall optimization problem. The procedure is applied to three example problems to demonstrate the optimum positioning of holes and optimization of radial dimensions of circular arcs on structures.     

Somigliana dislocations and internal stresses; with application to second phase hardening

A general integral theory for an important class of internal stress problems in anisotropic elastic media has been obtained. The basic problem that was solved was that of the stress field which arises when a certain region within the interior of a crystal undergoes a change in shape while constrained by the bulk of the crystal. The problem assumes that the shape change, whatever its physical origin, can be represented by a polynomial of arbitrary degree in the three spatial variables. When the transformed region is ellipsoidal in shape the following useful theorem has been indicated: If, in the absence of the matrix, the shape change were given by a polynomial of degree N in the spatial variables, then the state of internal strain, within the region, caused by the constraint of the matrix is given by a polynomial of degree N?1. This fact enables us to recast this above internal strain problem to include those cases where the ellipsoidal region is different elastically than the matrix. The theory is applied to a model of dispersion hardening.     

Topology optimization in Bernoulli free boundary problems

In this work we consider the topology optimization of systems governed by the external Bernoulli free boundary problem arising, for example, from the mathematical modelling of electro-chemical machining. In this work we combine, for the first time, the so-called pseudo-solid approach to the solution of governing free boundary problems and the level set method, which is used to define the design domain. Previous studies of the problem showed a tendency towards topological changes in the design, which can now automatically take place thanks to level set parametrization. The scalar function used in the level set method is parametrized using radial basis functions, converting the problem into a parametric optimization problem, which is solved using a gradient-based method.     

内嵌钢板及边缘框架相互作用对带连梁低屈服点钢板剪力墙结构受力性能的影响

为满足快速发展的高层建筑结构对抗震性能及空间灵活性的要求,将高耗能能力、高延性的低屈服点钢材与带连梁钢板剪力墙组合成新型带连梁低屈服点钢板剪力墙结构体系。采用有限元软件ABAQUS建立带连梁钢板剪力墙结构模型,结合国内外已有的典型试验结果验证数值方法的有效性。在此基础上,设计5个不同耦合度的低屈服点钢板剪力墙结构模型进行单调和循环加载,对比分析其损伤机制、承载性能及滞回耗能能力,探讨内嵌钢板与边缘框架的相互作用对结构及构件受力性能的影响,给出设计建议。结果表明:带连梁低屈服点钢板剪力墙结构内嵌钢板与边缘框架相互作用能够有效提高整体结构承载力、承载效率以及耗能能力。综合考虑材料利用率、承载能力及耗能能力,建议连梁耦合度控制在0.45以内。随着连梁耦合度的提高,边缘框架分担剪力多至60%,内部框架柱的轴力显著减小,连梁转角不断减小。因此,在带连梁低屈服点钢板剪力墙结构设计过程中应充分考虑内嵌钢板与边缘框架的相互作用,适当减小内嵌钢板设计厚度及边缘框架截面尺寸,提高材料利用率及设计经济性。同时,与纯框架抗侧性能相比,内嵌钢板与边缘框架的相互作用有效提高了边缘框架的初始抗侧刚度及承载力。     

Boundary layer in shape optimization of convective fins using a meshfree approach

We show how shape optimization of fin arrays for increased heat flux through the base under area constraint leads to non‐existence of optimal solutions. An additional constraint in terms of the boundary layer eliminates the apparent paradox. We consider a variable heat transfer coefficient and we use a fixed‐point iteration scheme to solve the resulting non‐linear boundary value problem for the steady‐state heat operator with temperature, flux, and convection boundary conditions. We propose a simple yet effective algorithm for evaluating the boundary layer constraint and eliminating the constraint violation. There are large shape changes between the initial and final design but no remeshing is required because we use a meshfree method that is not sensitive to shape distortion of integration cells as long as they remain convex. The resulting optimal unit cell is repeated by periodicity to produce the optimal fin array. The obtained shapes display similarities to shapes seen in natural systems governed by diffusion/convection and conduction processes. A length‐scale for the unit cell is naturally introduced by the non‐overlap condition imposed on the thermal boundary layer in the cooling ambient fluid. Copyright © 2004 John Wiley & Sons, Ltd.     

Topology design with optimized,self-adaptive materials

Significant performance improvements can be obtained if the topology of an elastic structure is allowed to vary in shape optimization problems. We study the optimal shape design of a two-dimensional elastic continuum for minimum compliance subject to a constraint on the total volume of material. The macroscopic version of this problem is not well-posed if no restrictions are placed on the structure topoiogy; relaxation of the optimization problem via quasiconvexification or homogenization methods is required. The effect of relaxation is to introduce a perforated microstructure that must be optimized simultaneously with the macroscopic distribution of material. A combined analytical-computational approach is proposed to solve the relaxed optimization problem. Both stress and displacement analysis methods are presented. Since rank-2 layered composites are known to achieve optimal energy bounds, we restrict the design space to this class of microstructures whose effective properties can easily be determined in explicit form. We develop a series of reduced problems by sequentially interchanging extremization operators and analytically optimizing the microstructural design fields. This results in optimization problems involving the distribution of an adaptive material that continuously optimizes its microstructure in response to the current state of stress or strain. A further reduced problem, involving only the response field, can be obtained in the stress-based approach, but the requisite interchange of extremization operators is not valid in the case of the displacement-based model. Finite element optimization procedures based on the reduced displacement formulation are developed and numerical solutions are presented. Care must be taken in selecting the discrete function spaces for the design density and displacement response, since the reduced problem is a two-field, mixed variational problem. An improper choice for the solution space leads to instabilities in the optimal design similar to those encountered in mixed formulations of the Stokes problem.