Sampling-based approach for design optimization in the presence of interval variables

This paper proposes a methodology for sampling-based design optimization in the presence of interval variables. Assuming that an accurate surrogate model is available, the proposed method first searches the worst combination of interval variables for constraints when only interval variables are present or for probabilistic constraints when both interval and random variables are present. Due to the fact that the worst combination of interval variables for probability of failure does not always coincide with that for a performance function, the proposed method directly uses the probability of failure to obtain the worst combination of interval variables when both interval and random variables are present. To calculate sensitivities of the constraints and probabilistic constraints with respect to interval variables by the sampling-based method, behavior of interval variables at the worst case is defined by the Dirac delta function. Then, Monte Carlo simulation is applied to calculate the constraints and probabilistic constraints with the worst combination of interval variables, and their sensitivities. A merit of using an MCS-based approach in the X-space is that it does not require gradients of performance functions and transformation from X-space to U-space for reliability analysis, thus there is no approximation or restriction in calculating sensitivities of constraints or probabilistic constraints. Numerical results indicate that the proposed method can search the worst case probability of failure with both efficiency and accuracy and that it can perform design optimization with mixture of random and interval variables by utilizing the worst case probability of failure search.     

A review of optimization of structures subjected to transient loads

Various aspects of structural optimization techniques under transient loads are extensively reviewed. The main themes of the paper are treatment of time-dependent constraints, calculation of design sensitivity, and approximation. Each subject is reviewed with corresponding papers that have been published since the 1970s. The treatment of time-dependent constraints in both the direct method and the transformation method is discussed. Two ways of calculating design sensitivity of a structure under transient loads are discussed—direct differentiation method and adjoint variable method. The approximation concept mainly focuses on the response surface method in crashworthiness and local approximation with the intermediate variables. Especially, a method using the equivalent static load is discussed as an approximation method. It takes advantage of the well-established static response optimization. The structural optimization in flexible multibody dynamic systems is reviewed in the viewpoint of the above three themes.     

Interval optimization of dynamic response for structures with interval parameters

This paper presents an interval optimization method for the dynamic response of structures with interval parameters. The matrices of structures with interval parameters are given. Combining the interval extension of function with the perturbation theory of dynamic response, the method for interval dynamic response analysis is derived. The interval optimization problem is transformed into a corresponding deterministic one. Because the mean values and the uncertainties of the interval parameters can be elected as the design variables, more information of the optimization results can be obtained by the present method than that obtained by the deterministic one. The present method is implemented for a truss structure and a frame structure. The numerical results show that the method is effective.     

基于网格实现的汽轮机基础优化设计

工程优化设计往往需要进行大规模的数值计算,拥有大量闲置资源的网格环境为建立这种高性能计算平台提供了可能.但是网格资源的动态性、异构性和分布性的本质特征,阻碍了网格技术在工程应用上的普及.为了利用网格环境中大量的闲置资源来协同解决实际工程中复杂的优化设计问题,建立了一个4层结构的高性能网格计算平台,并利用Kriging近似模型,在该平台上开发了以减轻基础重量和降低基础振幅为目的的多目标汽轮机优化设计的网格算法.使用该算法,在网格平台上对两个汽轮机基础进行了优化设计,与序列线性规划方法的结果比较表明所开发的优化算法有较高的计算精度.还分析了当使用不同数量的计算节点时网格的加速情况,说明所发展的优化方法能够在网格环境中高效地运行,搭建的网格平台也适合于工程优化设计.     

Sensitivity Analysis of Rigid-Flexible Multibody Systems

An important step in the application of automated design techniques to rigid-flexible multibody systems is the calculation of the sensitivities with respect to design variables. Thispaper presents a general formulation for thecalculation of the first order analytical designsensitivities based on the direct differentiationmethod. The analytical sensitivities are comparedwith the numerical results obtained by the finitedifferences method and the accuracy and validity ofboth methods is discussed. Cartesian co-ordinates areused for the dynamic analysis of rigid-flexiblemultibody systems. To reduce the number ofco-ordinates associated with the flexible bodies, thecomponent mode synthesis method is used. Theequations of the sensitivities are obtainedsymbolically and integrated in time simultaneouslywith the dynamic equations. Examples of 2Dsensitivity analysis of the transient response of aslider-crank and of a vehicle with a flexible chassisare presented, and the accuracy and characteristics ofthe sensitivities are analyzed and discussed.     

Computational strategies for reliability-based structural optimization of aeroelastic limit cycle oscillations

Gradient-based optimization, via the adjoint method, is needed to realistically enable the reliability-based design of a nonlinear unsteady aeroelastic system with many random and/or deterministic design variables. The adjoint derivatives of a time-marched system entail a cumbersome reverse-time integration, and so a time-periodic spectral element scheme is used here to efficiently capture the gradients of the limit cycle oscillations. Further reductions in the computational cost of the monolithic-time adjoint vector are obtained with proper orthogonal decomposition, which projects the large system onto a reduced basis. Design reliability is computed with the first order reliability method, which provides an estimate of the failure probability without resorting to sampling-based approaches (infeasible for large systems). Analytical gradients are needed to obtain the most probable point (in the random variable space), as well as the reliability design derivatives. These computational strategies are utilized to locate the optimal thickness distribution of a cantilevered wing operating beyond its flutter point in supersonic flow (via piston theory). Specifically, the wing mass is minimized under both deterministic and non-deterministic limit cycle oscillation amplitude constraints, with both structural and flow uncertainties considered in the latter.     

Optimal topology design of structures under dynamic loads

When elastic structures are subjected to dynamic loads, a propagation problem is considered to predict structural transient response. To achieve better dynamic performance, it is important to establish an optimum structural design method. Previous work focused on minimizing the structural weight subject to dynamic constraints on displacement, stress, frequency, and member size. Even though these methods made it possible to obtain the optimal size and shape of a structure, it is necessary to obtain an optimal topology for a truly optimal design. In this paper, the homogenization design method is utilized to generate the optimal topology for structures and an explicit direct integration scheme is employed to solve the linear transient problems. The optimization problem is formulated to find the best configuration of structures that minimizes the dynamic compliance within a specified time interval. Examples demonstrate that the homogenization design method can be extended to the optimal topology design method of structures under impact loads.Presented at WCSMO-2, held in Zakopane, Poland, 1997     

Design of absorbing material distribution for sound barrier using topology optimization

A topology optimization approach based on the boundary element method (BEM) and the optimality criteria (OC) method is proposed for the optimal design of sound absorbing material distribution within sound barrier structures. The acoustical effect of the absorbing material is simplified as the acoustical impedance boundary condition. Based on the solid isotropic material with penalization (SIMP) method, a topology optimization model is established by selecting the densities of absorbing material elements as design variables, volumes of absorbing material as constraints, and the minimization of sound pressure at reference surface as design objective. A smoothed Heaviside-like function is proposed to help the SIMP method to obtain a clear 0–1 distribution. The BEM is applied for acoustic analysis and the sensitivities with respect to design variables are obtained by the direct differentiation method. The Burton–Miller formulation is used to overcome the fictitious eigen-frequency problem for exterior boundary-value problems. A relaxed form of OC is used for solving the optimization problem to find the optimal absorbing material distribution. Numerical tests are provided to illustrate the application of the optimization procedure for 2D sound barriers. Results show that the optimal distribution of the sound absorbing material is strongly frequency dependent, and performing an optimization in a frequency band is generally needed.     

Topology optimization of frame structures with flexible joints

A method for structural topology optimization of frame structures with flexible joints is presented. A typical frame structure is a set of beams and joints assembled to carry an applied load. The problem considered in this paper is to find the stiffest frame for a given mass. By introducing design variables for beams and joints, a mass distribution for optimal structural stiffness can be found. Each beam can have several design variables connected to its cross section. One of these is an area-type design variable which is used to represent the global size of the beam. The other design variables are of length ratio type, controlling the cross section of the beam. Joints are flexible elements connecting the beams in the structure. Each joint has stiffness properties and a mass. A framework for modelling these stiffnesses is presented and design variables for joints are introduced. We prove a theorem which can be interpreted as the fact that the removal of structural elements, e.g. joints or beams, can be modelled by a small strictly positive material amount assigned to the element. This is needed for the computations of sensitivities used in the applied gradient based iterative method. Both two and three dimensional problems, as well as multiple load cases and multiple mass constraints, are treated.     

Topology optimization for coupled acoustic-structural systems under random excitation

Topology optimization for coupled acoustic-structural systems subjected to stationary random excitations is investigated. Bi-material elastic continuum structures without damping are considered. The finite element method is utilized to deal with coupled acoustic-structural problems. An accurate and highly efficient algorithm series for stationary random analysis techniques, pseudo excitation method, is extended to calculate the acoustic random response generated by the vibrating structures. Minimization of the auto power spectral density of sound pressure is taken as design objective and its sensitivities with respect to topological variables are derived by adjoint method. A penalty term is proposed to suppress the intermediate volumetric densities. Numerical examples are given to demonstrate the validity of the presented methods.     

Application of gradient-based optimization methods for a rotor system with static stress, natural frequency,and harmonic response constraints

This paper demonstrates the application of gradient-based optimization methods to the minimal weight design optimization of rotor systems. A nonlinear constrained optimization problem is considered. Design variables are inner radii and wall thicknesses of shaft sections. Constraints are imposed on torsional and equivalent stresses, natural frequencies, and unbalance response amplitudes. The sizing optimization problem is solved using a gradient projection method and a sequential quadratic programming technique. A typical turbine rotor system is considered. An in-house beam-based finite element method code is used for the prediction of static and dynamic characteristics of the rotor system. Analytical sensitivity analysis is performed for the static and harmonic equations using the adjoint method. Sensitivity coefficients for the natural frequencies are obtained directly from the quadratic eigenvalue problem. Results of several optimization runs with different constraint sets show a significant shaft weight reduction in comparison with the baseline configuration with all constraints being satisfied. The two optimization methods are compared and discussed in regard to their performance.     

Optimal cross-section and configuration design of elastic-plastic structures subject to dynamic cyclic loading

This paper shows an optimal design problem with continuum variational formulation, applied to nonlinear elasticplastic structures subject to dynamic loading. The total Lagrangian procedure is used to describe the response of the structure. The direct differentiation method is used to obtain the sensitivities of the structural response that are needed to solve the optimization problem. Since unloading and reloading of the structure are allowed, the structural response is path-dependent and an additional step is needed to integrate the constitutive equations. It can be shown, consequently, that design sensitivity analysis is also path-dependent. A finite element method with implicit time integration is used to discretize the state and sensitivity equations.A mathematical programming approach is used for the optimization process. Numerical applications are performed on a 3-D truss structure, where cross-sectional areas and nodal point coordinates are treated as design variables. Optimal designs have been obtained and compared by using two different strategies: a twolevel strategy where the levels are defined according to the type of design variables, cross sectional areas or node coordinates, and optimizing simultaneously with respect to both types of design variables. Comparisons have also been made between optimal designs obtained by considering or not considering the inertial term of the structural equilibrium.     

Adhesive surface design using topology optimization

Tailoring adhesive properties between surfaces is of great importance for micro-scale systems, ranging from managing stiction in MEMS devices to designing wall-scaling gecko-like robots. A methodology is introduced for designing adhesive interfaces between structures using topology optimization. Structures subjected to external loads that lead to delamination are studied for situations where displacements and deformations are small. Only the effects of adhesive forces acting normal to the surfaces are considered. An interface finite element is presented that couples a penalty contact formulation and a Lennard–Jones model of van der Waals adhesive forces. Two- and three dimensional design optimization problems are presented in which adhesive force distributions are designed such that load-displacement curves of delaminating structures match target responses. The design variables describe the adhesive energy per area of the interface between the surfaces, as well as the geometry of the delaminating structure. A built-in length scale in the formulation of the adhesion forces eliminates the need for filtering to achieve comparable optimal adhesive designs over a range of mesh densities. The resulting design problem is solved by gradient based optimization algorithms evaluating the design sensitivities by the adjoint method. Results show that the delamination response can be effectively manipulated by the method presented. Varying simultaneously both adhesive and geometric parameters yields a wider range of reachable target load-displacement curves than in the case varying adhesive energy alone.     

Design optimization and sensitivity analysis on buckling stability of piezoelectric truss structures

The design optimization of buckling behavior is studied for piezoelectric intelligent truss structures. First, on the basis of mechanical–electric coupling equation and considering electric load and mechanical loads together, the finite element model of piezoelectric trusses has been built up. Then, the computational formula has been derived for the design sensitivities of critical buckling load factor of the structure with respect to size and shape design variables. The electric voltage is taken as a new kind of design variable and the calculation method of critical load buckling factor with respect to the electric voltage variables is proposed. Particularly, the variations of the loads and pre-buckling inner forces with design variables have been accounted for. Finally, the sequential linear programming algorithm is employed to solve the optimization problem, and a new method of controlling structural buckling stability by optimizing the voltages of piezoelectric active bars is proposed. Numerical examples given in the paper have demonstrated the effectiveness of the methods presented.     

区间非线性路径约束的动态优化问题求解

本文研究路径约束中含有区间参数形式的动态优化问题,提出了一种新的非线性路径约束的确定化描述形式,和采用惩罚函数法的求解算法。对于转化后的极大极小优化命题,论文提出采用Lagrangian多项式加权和的方法得到有限维的确定性优化求解形式。该算法可有效地描述和求解不确定参数动态优化问题。     

直接radau配置的多阶非线性模型预测聚合反应控制

针对半间歇聚合反应强非线性,模型参数的不确定性以及过程存在路径约束和终端约束等特点,将多阶非线性模型预测控制方法应用于半间歇聚合反应控制器的设计。该方法使用离散的场景树表示不确定量对系统的影响,使得未来的控制输入取决于先前的不确定量实现,从而构成闭环的鲁棒性控制方法。为了提高求解效率,使用基于有限元正交配置的联立法对问题求解,通过选取合适的radau配置点对控制变量和状态变量同时离散化,将动态优化问题转化为NLP问题,并使用内点法求解器IPOPT求解。在BASF SE公司提供的半间歇聚合反应模型上验证了提出方法的有效性。     

BLISS/S: a new method for two-level structural optimization

The paper describes a two-level method for structural optimization for a minimum weight under the local strength and displacement constraints. The method divides the optimization task into separate optimizations of the individual substructures (in the extreme, the individual components) coordinated by the assembled structure optimization. The substructure optimizations use local cross-sections as design variables and satisfy the highly nonlinear local constraints of strength and buckling. The design variables in the assembled structure optimization govern the structure overall shape and handle the displacement constraints. The assembled structure objective function is the objective in each of the above optimizations. The substructure optimizations are linked to the assembled structure optimization by the sensitivity derivatives. The method was derived from a previously reported two-level optimization method for engineering systems, e.g. aerospace vehicles, that comprise interacting modules to be optimized independently, coordination provided by a system-level optimization. This scheme was adapted to structural optimization by treating each substructure as a module in a system, and using the standard finite element analysis as the system analysis. A numerical example, a hub structure framework, is provided to show the new method agreement with a standard, no-decomposition optimization. The new method advantage lies primarily in the autonomy of the individual substructure optimization that enables concurrency of execution to compress the overall task elapsed time. The advantage increases with the magnitude of that task. Received December 5, 1999?Revised mansucript received April 26, 2000     

Simulation based optimization of stochastic systems with integer design variables by sequential multipoint linear approximation

Optimization problems are considered for which objective function and constraints are defined as expected values of stochastic functions that can only be evaluated at integer design variable levels via a computationally expensive computer simulation. Design sensitivities are assumed not to be available. An optimization approach is proposed based on a sequence of linear approximate optimization subproblems. Within each search subregion a linear approximate optimization subproblem is built using response surface model building. To this end, N simulation experiments are carried out in the search subregion according to a D-optimal experimental design. The linear approximate optimization problem is solved by integer linear programming using corrected constraint bounds to account for any uncertainty due to the stochasticity. Each approximate optimum is evaluated on the basis of M simulation replications with respect to objective function change and feasibility of the design. The performance of the optimization approach and the influence of parameters N and M is illustrated via two analytical test problems. A third example shows the application to a production flow line simulation model. Received April 28, 2000     

Nonlinear dynamic response structural optimization using equivalent static loads

It is well known that nonlinear dynamic response optimization using a conventional optimization algorithm is fairly difficult and expensive for the gradient or non-gradient based optimization methods because many nonlinear dynamic analyses are required. Therefore, it is quite difficult to find practical large scale examples with many design variables and constraints for nonlinear dynamic response structural optimization. The equivalent static loads (ESLs) method is newly proposed and applied to nonlinear dynamic response optimization. The equivalent static loads are defined as the linear static load sets which generate the same response field in linear static analysis as that from nonlinear dynamic analysis. The ESLs are made from the results of nonlinear dynamic analysis and used as external forces in linear static response optimization. Then the same response from nonlinear dynamic analysis can be considered throughout linear static response optimization. The updated design from linear response optimization is used again in nonlinear dynamic analysis and the process proceeds in a cyclic manner until the convergence criteria are satisfied. Several examples are solved to validate the method. The results are compared to those of the conventional method with sensitivity analysis using the finite difference method.