Reliability-based robust multi-objective optimization applied to engineering system design

ABSTRACT

Probabilistic and non-probabilistic methods have been proposed to deal with design problems under uncertainties. Reliability-based design and robust design are probabilistic strategies traditionally used for this purpose. In the present contribution, reliability-based robust design optimization (RBRDO) is formulated as a multi-objective problem considering the interaction of both approaches. The proposed methodology is based on the differential evolution algorithm associated with two strategies to deal with reliability and robustness, respectively, namely inverse reliability analysis and the effective mean concept. This multi-objective optimization problem considers the maximization of reliability and robustness coefficients as additional objective functions. The effectiveness of the methodology is illustrated by two classical test cases and a rotor-dynamics application. The results demonstrate that the proposed methodology is an alternative method to solve RBRDO problems.     

Parallel computational strategies for structural optimization

The objective of this paper is to investigate the efficiency of various computational algorithms implemented in the framework of structural optimization methods based on evolutionary algorithms. In particular, the efficiency of parallel computational strategies is examined with reference to evolution strategies (ES) and genetic algorithms (GA). Parallel strategies are implemented both at the level of the optimization algorithm, by exploiting the natural parallelization features of the evolutionary algorithms, as well as at the level of the repeated structural analysis problems that are required by ES and GA. In the latter case the finite element solutions are performed by the FETI domain decomposition method specially tailored to the particular type of problems at hand. The proposed methodology is generic and can be applied to all types of optimization problems as long as they involve large‐scale finite element simulations. The numerical tests of the present study are performed on sizing optimization of skeletal structures. The numerical tests demonstrate the computational advantages of the proposed parallel strategies, which become more pronounced in large‐scale optimization problems. Copyright © 2003 John Wiley & Sons, Ltd.     

STRUCTURAL SHAPE OPTIMIZATION USING EVOLUTION STRATEGIES

The objective of this paper is to investigate the efficiency of combinatorial optimization methods and in particular algorithms based on evolution strategies, when incorporated into shape optimization problems. Evolution strategy algorithms are used either on a stand-alone basis, or combined with a conventional mathematical programming technique. The numerical tests presented demonstrate the computational advantages of the proposed approach which become more pronounced in large-scale optimization problems and/or parallel computing environment.     

Efficient optimization processes using kriging approximation models in electrical impedance tomography

A reduced model technique based on kriging approximations is developed in order to increase the rate of convergence of an evolution strategy (ES) when solving a non‐destructive evaluation problem. The inverse problem investigated consists of identifying the geometry of discontinuities in a conductive material from Cauchy data measurements taken on the boundary. In this study, we use kriging approximation models in order to increase the rate of convergence of the optimization algorithm and to efficiently detect, from a computational time point of view, a subsurface cavity, such as a circle. The algorithm developed by combining evolution strategies and kriging approximations is found to be a robust, fast and efficient method for detecting the size and location of subsurface cavities. Copyright © 2006 John Wiley & Sons, Ltd.     

Calibration of elastoplastic constitutive model parameters from full-field data with automatic differentiation-based sensitivities

We present a framework for calibration of parameters in elastoplastic constitutive models that is based on the use of automatic differentiation (AD). The model calibration problem is posed as a partial differential equation-constrained optimization problem where a finite element (FE) model of the coupled equilibrium equation and constitutive model evolution equations serves as the constraint. The objective function quantifies the mismatch between the displacement predicted by the FE model and full-field digital image correlation data, and the optimization problem is solved using gradient-based optimization algorithms. Forward and adjoint sensitivities are used to compute the gradient at considerably less cost than its calculation from finite difference approximations. Through the use of AD, we need only to write the constraints in terms of AD objects, where all of the derivatives required for the forward and inverse problems are obtained by appropriately seeding and evaluating these quantities. We present three numerical examples that verify the correctness of the gradient, demonstrate the AD approach's parallel computation capabilities via application to a large-scale FE model, and highlight the formulation's ease of extensibility to other classes of constitutive models.     

Design and Optimization of airfoils in non-stalling incompressible flow with a prescribed range of the angle of attack

A simple and reliable method of aerodynamic design of airfoils is proposed for a given range of the angle of attack. The method is based on the solution of a new inverse boundary-value problem of aerohydrodynamics. In the problem the initial velocity distributions are given along the contour of the required airfoil as a function of the are co-ordinate for extreme values of the angle of attack from a fixed range. We justify a method of choice of these distributions which guarantees a non-stalling flow around the airfoil. The problem of airfoil optimization is also considered. Results of numerical calculations are presented.     

EVOLUTION STRATEGIES IN ENGINEERING OPTIMIZATION

The available methods used for the solution of optimization problems in engineering are commonly classified as zero, first and second order methods according to the order of information which is required. In this paper a zero order method based on evolution strategies is described. Evolution strategies are stochastic search methods for solving optimization problems and have their philosophical basis in processes found in nature. The basic evolution strategy is presented first1 followed by generalizations for parallel computing and for the solution of discrete and mixed discrete-continuous problems. Some applications to test problems from the literature and to realistic problems of structural optimization are given.     

Comparison of the born iterative method and tarantola's method for an electromagnetic time-domain inverse problem

Two methods of solving the nonlinear two-dimensional electromagnetic inverse scattering problem in the time domain are considered. These are the Born iterative method and the method originally proposed by Tarantola for the seismic reflection inverse problems. The former is based on Born-type iterations on an integral equation, whereby at each iteration the problem is linearized, and its solution is found via a regularized optimization. The latter also uses an iterative method to solve the nonlinear system of equations. Although it linearizes the problem at each stage as well, no optimization is carried out at each iteration; rather the problem as a whole is posed as a (regularized) optimization. Each method is described briefly and its computational complexity is analyzed. Tarantola's method is shown to have a lower numerical complexity compared to the Born iterative method for each iteration, but in the examples considered, required more iterations to converge. Both methods perform well when inverting a smooth profile; however, the Born iterative method gave better results in resolving localized point scatterers.     

The benefits of parallel multibody simulation

To exploit the benefits of parallel computer architectures for multibody system simulation, an interdisciplinary approach has been pursued, combining knowledge of the three disciplines of dynamics, numerical mathematics and computer science. An analysis of the options available for the formulation and numerical solution of the dynamical system equations yielded a surprising result. A method initially proposed to solve the inverse problem of dynamics is the best choice to generate the system equations required for solving the simulation problem, when relying on implicit integration routines. Such routines have the particular advantage of handling stiff systems, too. The new O(N)-residual formalism, generating the system equations in a form required for implicit numerical integration, has a high potential to benefit from parallel computer architectures. Two strategies of medium and coarse grain parallelization have been implemented on a Transputer network to obtain a package for parallel multibody simulation. An analysis of the performance of this package demonstrates for typical multibody simulation problems that the new code is five times faster than existing codes when implemented on a serial computer. An additional speed-up by the same order of magnitude is obtained when the code is implemented on a Transputer network.     

An inverse method for determining elastic material properties and a material interface

A numerical procedure which integrates optimization, finite element analysis and automatic finite element mesh generation is developed for solving a two-dimensional inverse/parameter estimation problem in solid mechanics. The problem consists of determining the location and size of a circular inclusion in a finite matrix and the elastic material properties of the inclusion and the matrix. Traction and displacement boundary conditions sufficient for solving a direct problem are applied to the boundary of the domain. In addition, displacements are measured at discrete points on the part of the boundary where the tractions are prescribed. The inverse problem is solved using a modified Levenberg-Marquardt method to match the measured displacements to a finite element model solution which depends on the unknown parameters. Numerical experiments are presented to show how different factors in the problem and the solution procedure influence the accuracy of the estimated parameters.     

Scheduling method synchronizing two lines of activities and minimizing waiting and pass-time

In this paper, methods are suggested for solving a common class of job-shop scheduling problem. The type of job-shops considered consists of two parallel production lines ended by a group work station. The methods use a domination principle at first, of one line over the other, defining a master—slave hierarchy in the workshops. The solutions obtained from the optimal scheduling of the master line form the constraint domain of the slave line, the optimal schedules of which are worked out in turn. Parametrization of economical functions permits the dynamic modification of the optimization criteria of the slave line.     

基于同伦延拓的冗余机械臂逆运动学优化算法研究

目的 针对偏置冗余机械臂的逆运动学,采用传统数值法存在依赖初始值、奇异位姿收敛性差等问题,提出一种改进数值法。方法 首先将非线性方程组转化为同伦方程组,引入同伦延拓算法能够有效避免依赖初始值的问题,同时能够获取逆运动学解空间。然后考虑奇异位姿,将同伦方程组转化为最小二乘问题,采用Levenberg Marquardt算法对同伦方程组进行路径追踪,以获取逆运动学解空间。最后将关节极限避免问题映射为解空间优化问题,引入二进制改进粒子群优化算法,获得最优逆运动学解。结果 实验结果表明,相较于传统数值法,文中所提数值法针对逆运动学求解具有更高的收敛率、更快的收敛速度,同时二进制改进粒子群算法能够有效避免关节极限问题。结论 采用文中所提数值法求解逆运动学的精度较高,能够满足实时性要求,对于机械臂用于包装作业具有一定的理论意义和工程应用价值。     

Simultaneous reconstruction of the time-dependent Robin coefficient and heat flux in heat conduction problems

This paper aims to solve an inverse heat conduction problem in two-dimensional space under transient regime, which consists of the estimation of multiple time-dependent heat sources placed at the boundaries. Robin boundary condition (third type boundary condition) is considered at the working domain boundary. The simultaneous identification problem is formulated as a constrained minimization problem using the output least squares method with Tikhonov regularization. The properties of the continuous and discrete optimization problem are studied. Differentiability results and the adjoint problems are established. The numerical estimation is investigated using a modified conjugate gradient method. Furthermore, to verify the performance of the proposed algorithm, obtained results are compared with results obtained from the well-known finite-element software COMSOL Multiphysics under the same conditions. The numerical results show that the proposed algorithm is accurate, robust and capable of simultaneously representing the time effects on reconstructing the time-dependent Robin coefficient and heat flux.     

3-CRS-S并联机构的运动学分析及仿真

目的 对3-CRS-S并联机构进行运动学分析及仿真,验证该机构是否具有优良的运动学性能.方法 运用修正的Kutzbach-Grübler公式对机构进行自由度计算,并分别采用D-H法和数值算法中的粒子群优化算法(PSO)对该3-CRS-S并联机构的位置正逆解进行分析,运用Adams软件对3-CRS-S并联机构进行角度和角速度分析.结果 得出该机构的位置逆解和正解,以及运动学仿真后的角度、角速度图像,该图像曲线均呈现为有规律、周期性的变化,且曲线没有出现有任何断点和突变点,运动范围相对稳定,说明该机构在运行过程中运行平稳.结论 该机构在运动过程中运行平稳,具有良好的运动学性能,在自动化包装机械领域具有广阔的应用前景.     

Solution of Inverse Boundary Optimization Problem by Trefftz Method and Exponentially Convergent Scalar Homotopy Algorithm

The inverse boundary optimization problem, governed by the Helmholtz equation, is analyzed by the Trefftz method (TM) and the exponentially convergent scalar homotopy algorithm (ECSHA). In the inverse boundary optimization problem, the position for part of boundary with given boundary condition is unknown, and the position for the rest of boundary with additionally specified boundary conditions is given. Therefore, it is very difficult to handle the boundary optimization problem by any numerical scheme. In order to stably solve the boundary optimization problem, the TM, one kind of boundary-type meshless methods, is adopted in this study, since it can avoid the generation of mesh grid and numerical integration. In the boundary optimization problem governed by the Helmholtz equation, the numerical solution of TM is expressed as linear combination of the T-complete functions. When this problem is considered by TM, a system of nonlinear algebraic equations will be formed and solved by ECSHA which will converge exponentially. The evolutionary process of ECSHA can acquire the unknown coefficients in TM and the spatial position of the unknown boundary simultaneously. Some numerical examples will be provided to demonstrate the ability and accuracy of the proposed scheme. Besides, the stability of the proposed meshless method will be validated by adding some noise into the boundary conditions.     

Performance-based multi-objective optimization of large steel structures

In recent years, the importance of economical considerations in the field of structures has motivated many researchers to propose new methods for minimizing the initial and life cycle cost of the structures subjected to seismic loading. In this paper, a new framework is presented to solve the performance-based multi-objective optimization problem considering the initial and life cycle cost of large structures. In order to solve this problem, a non-dominated sorting genetic algorithm (NSGA-II) using differential evolution operators is employed to solve the optimization problem, while a specific meta-model is utilized for reducing the number of fitness function evaluations. The required computational time for pushover analysis is decreased by a simple numerical method. The constraints of the optimization problem are based on the FEMA codes. The presented results for application of the proposed framework demonstrate its capability in solving the present complex multi-objective optimization problem.     

Regularization of the inverse heat conduction problem by the discrete Fourier transform

Two methods of solving the inverse heat conduction problem with employment of the discrete Fournier transform are presented in this article. The first one operates similarly to the SVD algorithm and consists in reducing the number of components of the discrete Fournier transform which are taken into account to determine the solution to the inverse problem. The second method is related to the regularization of the solution to the inverse problem in the discrete Fournier transform domain. Those methods were illustrated by numerical examples. In the first example, an influence of the boundary conditions disturbance by a random error on the solution to the inverse problem (its stability) was examined. In the second example, the temperature distribution on the inner boundary of the multiply connected domain was determined. Results of calculations made in both ways brought very good outcomes and confirm the usefulness of applying the discrete Fournier transform to solving inverse problems.     

Optimization under adaptive error control for finite element based simulations

Optimization problems constrained by complex dynamics can lead to computationally challenging problems especially when high accuracy and efficiency are required. We present an approach to adaptively control numerical errors in optimization problems approximated using the finite element method. The discrete adjoint equation serves as a key tool to efficiently compute both parameter sensitivities and goal-oriented error estimates at the same discretized levels. By using a recovery method for the error estimators, we avoid expensive higher order adjoint calculations. We nest the adaptivity of the mesh within the optimization algorithm, which is responsible for converging both the state and optimization algorithms and thereby allowing the reuse of state, parameters, and reduced Hessian in subsequent optimization iterations. Our approach is demonstrated on a parameter estimation problem for contamination transport in a contact tank reactor. Significant efficiency and accuracy improvements are realized in comparison to uniform grid refinement strategies and black-box optimization methods. A flexible and maintainable software interface was developed to provide access between the underlying linear algebra of a production simulator and advanced numerical algorithms such as optimization and error estimation.     

Evolutionary modelling and optimization of grinding processes

The objective of this study is to develop a framework of modelling the complex grinding processes and finding optimal process conditions to meet the general class of process requirements. In order to achieve the above goal, novel modelling schemes and optimization methods based on evolutionary algorithms (EA) are developed. The optimization problem of grinding processes can be formulated as a constrained non-linear programming problem with mixed-discrete variables. The adaptive least-squares (ALS) algorithm proposed by Lee and Shin's 1998 study is extended for modelling multi-input-multi-output (MIMO) complex grinding processes using fuzzy basis function networks (FBFN), while the modified evolution strategies (ES) is proposed for successful optimization of grinding processes. Two grinding optimization problems demonstrate the superior performance of the proposed scheme.     

New Finite Difference Methods Based on IIM for Inextensible Interfaces in Incompressible Flows

In this paper, new finite difference methods based on the augmented immersed interface method (IIM) are proposed for simulating an inextensible moving interface in an incompressible two-dimensional flow. The mathematical models arise from studying the deformation of red blood cells in mathematical biology. The governing equations are incompressible Stokes or Navier-Stokes equations with an unknown surface tension, which should be determined in such a way that the surface divergence of the velocity is zero along the interface. Thus, the area enclosed by the interface and the total length of the interface should be conserved during the evolution process. Because of the nonlinear and coupling nature of the problem, direct discretization by applying the immersed boundary or immersed interface method yields complex nonlinear systems to be solved. In our new methods, we treat the unknown surface tension as an augmented variable so that the augmented IIM can be applied. Since finding the unknown surface tension is essentially an inverse problem that is sensitive to perturbations, our regularization strategy is to introduce a controlled tangential force along the interface, which leads to a least squares problem. For Stokes equations, the forward solver at one time level involves solving three Poisson equations with an interface. For Navier-Stokes equations, we propose a modified projection method that can enforce the pressure jump condition corresponding directly to the unknown surface tension. Several numerical experiments show good agreement with other results in the literature and reveal some interesting phenomena.