Efficient shape parameterization method for multidisciplinary global optimization and application to integrated ship hull shape optimization workflow

Multidisciplinary global shape optimization requires a geometric parameterization method that keeps the shape generality while lowering the number of free variables. This paper presents a reduced parameter set parameterization method based on integral B-spline surface capable of both shape and topology variations and suitable for global multidisciplinary optimization. The objective of the paper is to illustrate the advantages of the proposed method in comparison to standard parameterization and to prove that the proposed method can be used in an integrated multidisciplinary workflow. Non-linear fitting is used to test the proposed parameterization performance before the actual optimization. The parameterization method can in this way be tested and pre-selected based on previously existing geometries. Fitting tests were conducted on three shapes with dissimilar geometrical features, and great improvement in shape generality while reducing the number of shape parameters was achieved. The best results are obtained for a small number (up to 50) of optimization variables, where a classical applying of parameterization method requires about two times as many optimization variables to obtain the same fitting capacity.The proposed shape parameterization method was tested in a multidisciplinary ship hull optimization workflow to confirm that it can actually be used in multiobjective optimization problems. The workflow integrates shape parameterization with hydrodynamic, structural and geometry analysis tools. In comparison to classical local and global optimization methods, the evolutionary algorithm allows for fully autonomous design with an ability to generate a wide Pareto front without a need for an initial solution.     

Topology optimization with design-dependent pressure loading

In this paper, the layout of structures under design-dependent pressure loading is optimized using a topology optimization approach. In contrast to topology optimization problems with conventional static external loading, the position and direction of pressure loading are changing with topology of structure during optimization iterations. In order to model the changing structural surface boundaries under design-dependent pressure loading, a pseudo equal-potential function is introduced. Design sensitivity analysis is derived from the adjoint method. Three examples solved by the proposed method are presented.     

Shape design optimization of stationary fluid-structure interaction problems with large displacements and turbulence

This paper presents general and efficient methods for analysis and gradient based shape optimization of systems characterized as strongly coupled stationary fluid-structure interaction (FSI) problems. The incompressible fluid flow can be laminar or turbulent and is described using the Reynolds-averaged Navier-Stokes equations (RANS) together with the algebraic Baldwin–Lomax turbulence model. The structure may exhibit large displacements due to the interaction with the fluid domain, resulting in geometrically nonlinear structural behaviour and nonlinear interface coupling conditions. The problem is discretized using Galerkin and Streamline-Upwind/Petrov–Galerkin finite element methods, and the resulting nonlinear equations are solved using Newtons method. Due to the large displacements of the structure, an efficient update algorithm for the fluid mesh must be applied, leading to the use of an approximate Jacobian matrix in the solution routine. Expressions for Design Sensitivity Analysis (DSA) are derived using the direct differentiation approach, and the use of an inexact Jacobian matrix in the analysis leads to an iterative but very efficient scheme for DSA. The potential of gradient based shape optimization of fluid flow and FSI problems is illustrated by several examples.     

A phase-field method for shape optimization of incompressible flows

In this paper, we present a phase-field method applied to the fluid-based shape optimization. The fluid flow is governed by the incompressible Navier–Stokes equations. A phase field variable is used to represent material distributions and the optimized shape of the fluid is obtained by minimizing the certain objective functional regularized. The shape sensitivity analysis is presented in terms of phase field variable, which is the main contribution of this paper. It saves considerable amount of computational expense when the meshes are locally refined near the interfaces compared to the case of fixed meshes. Numerical results on some benchmark problems are reported, and it is shown that the phase-field approach for fluid shape optimization is efficient and robust.     

Multidisciplinary optimization of a transport aircraft wing using particle swarm optimization

The purpose of this paper is to demonstrate the application of particle swarm optimization to a realistic multidisciplinary optimization test problem. The papers new contributions to multidisciplinary optimization are the application of a new algorithm for dealing with the unique challenges associated with multidisciplinary optimization problems, and recommendations for the utilization of the algorithm in future multidisciplinary optimization applications. The selected example is a bi-level optimization problem that demonstrates severe numerical noise and has a combination of continuous and discrete design variables. The use of traditional gradient-based optimization algorithms is thus not practical. The numerical results presented indicate that the particle swarm optimization algorithm is able to reliably find the optimum design for the problem presented. The algorithm is capable of dealing with the unique challenges posed by multidisciplinary optimization, as well as the numerical noise and discrete variables present in the current example problem.     

Solution to boundary shape optimization problem of linear elastic continua with prescribed natural vibration mode shapes

This paper presents a practical method of numerical analysis for boundary shape optimization problems of linear elastic continua in which natural vibration modes approach prescribed modes on specified sub-boundaries. The shape gradient for the boundary shape optimization problem is evaluated with optimality conditions obtained by the adjoint variable method, the Lagrange multiplier method, and the formula for the material derivative. Reshaping is accomplished by the traction method, which has been proposed as a solution to boundary shape optimization problems of domains in which boundary value problems of partial differential equations are defined. The validity of the presented method is confirmed by numerical results of three-dimensional beam-like and plate-like continua.     

An efficient strategy for reliability-based multidisciplinary design optimization using BLISS

This paper presents an efficient reliability-based multidisciplinary design optimization (RBMDO) strategy. The conventional RBMDO has tri-level loops: the first level is an optimization in the deterministic space, the second one is a reliability analysis in the probabilistic space, and the third one is the multidisciplinary analysis. Since it is computationally inefficient when high-fidelity simulation methods are involved, an efficient strategy is proposed. The strategy [named probabilistic bi-level integrated system synthesis (ProBLISS)] utilizes a single-level reliability-based design optimization (RBDO) approach, in which the reliability analysis and optimization are conducted in a sequential manner by approximating limit state functions. The single-level RBDO is associated with the BLISS formulation to solve RBMDO problems. Since both the single-level RBDO and BLISS are mainly driven by approximate models, the accuracy of models can be a critical issue for convergence. The convergence of the strategy is guaranteed by employing the trust region–sequential quadratic programming framework, which validates approximation models in the trust region radius. Two multidisciplinary problems are tested to verify the strategy. ProBLISS significantly reduces the computational cost and shows stable convergence while maintaining accuracy.     

Two-stage size-layout optimization of axially compressed stiffened panels

In this study, a two-stage optimization framework is proposed for cylindrical or flat stiffened panels under uniform or non-uniform axial compression, which are extensively used in the aerospace industry. In the first stage, traditional sizing optimization is performed. Based on the buckling or collapse-like deformed shape evaluated for the optimized design, the panel can be divided in sub-regions each of which shows characteristic deformations along axial and circumferential directions. Layout optimization is then performed using a stiffener spacing distribution function to represent the location of each stiffener. A layout coefficient is assigned to each sub-region and the overall layout of the panel is optimized. Three test problems are solved in order to demonstrate the validity of the proposed optimization framework: remarkably, the load-carrying capacity improves by 17.4 %, 66.2 % and 102.2 % with respect to the initial design.     

基于目标优化的高光谱图像亚像元定位

目的 高光谱图像混合像元的普遍存在使得传统的分类技术难以准确确定地物空间分布,亚像元定位技术是解决该问题的有效手段。针对连通区域存在孤立点或孤立两点等特例时,通过链码长度求周长最小无法保证最优结果及优化过程计算量大的问题,提出了一种改进的高光谱图像亚像元定位方法。方法 以光谱解混结合二进制粒子群优化构建算法框架,根据光谱解混结果近似估计每个像元对应的亚像元组成,通过分析连通区域存在特例时基于链码长度求周长最小无法保证结果最优的原因,提出修改孤立区域的周长并考虑连通区域个数构造代价函数,最后利用二进制粒子群优化实现亚像元定位。为了减少算法的时间复杂度,根据地物空间分布特点,采用局部分析代替全局分析,提出了新的迭代优化策略。结果 相比直接基于链码长度求周长最小的优化结果,基于改进的目标函数优化后,大部分区域边界更明显,并且没有孤立1点和孤立两点的区域,识别率可以提高2%以上,Kappa系数增加0.05以上,新的优化策略可以使算法运算时间减少近一半。结论 实验结果表明,本文方法能有效提高亚像元定位精度,同时降低时间复杂度。因为高光谱图像中均匀混合区域不同地物的分布空间相关性不强,因此本文方法适用于非均匀混合的高光谱图像的亚像元定位。     

Nonparametric gradient-less shape optimization for real-world applications

A nonparametric gradient-less shape optimization approach for finite element stress minimization problems is presented. The shape optimization algorithm is based on optimality criteria, which leads to a robust and fast convergence independent of the number of design variables. Sensitivity information of the objective function and constraints are not required, which results in superior performance and offers the possibility to solve the structural analysis task using fast and reliable industry standard finite element solvers such as ABAQUS, ANSYS, I-DEAS, MARC, NASTRAN or PERMAS. The approach has been successfully extended to complex nonlinear problems including material, boundary and geometric nonlinear behavior. The nonparametric geometry representation creates a complete design space for the optimization problem, which includes all possible solutions for the finite element discretization. The approach is available within the optimization system TOSCA and has been used successfully for real-world optimization problems in industry for several years. The approach is compared to other approaches and the benefits and restrictions are highlighted. Several academic and real-world examples are presented.     

On simultaneous shape and material layout optimization of shell structures

This work presents a computational method for integrated shape and topology optimization of shell structures. Most research in the last decades considered both optimization techniques separately, seeking an initial optimal topology and refining the shape of the solution later. The method implemented in this work uses a combined approach, were the shape of the shell structure and material distribution are optimized simultaneously. This formulation involves a variable ground structure for topology optimization, since the shape of the shell mid-plane is modified in the course of the process. It was considered a simple type of design problem, where the optimization goal is to minimize the compliance with respect to the variables that control the shape, material fraction and orientation, subjected to a constraint on the total volume of material. The topology design problem has been formulated introducing a second rank layered microestructure, where material properties are computed by a “smear-out” procedure. The method has been implemented into a general optimization software called ODESSY, developed at the Institute of Mechanical Engineering in Aalborg. The computational model was tested in several numerical applications to illustrate and validate the approach.     

A direct search variant of the simulated annealing algorithm for optimization involving continuous variables

A memory-based simulated annealing algorithm is proposed which fundamentally differs from the previously developed simulated annealing algorithms for continuous variables by the fact that a set of points rather than a single working point is used. The implementation of the new method does not need differentiability properties of the function being optimized. The method is well tested on a range of problems classified as easy, moderately difficult and difficult. The new algorithm is compared with other simulated annealing methods on both test problems and practical problems. Results showing an improved performance in finding the global minimum are given.Scope and purposeThe inherent difficulty of global optimization problems lies in finding the very best optimum (maximum or minimum) from a multitude of local optima. Many practical global optimization problems of continuous variables are non-differentiable and noisy and even the function evaluation may involve simulation of some process. For such optimization problems direct search approaches are the methods of choice. Simulated annealing is a stochastic global optimization algorithm, initially designed for combinatorial (discrete) optimization problems. The algorithm that we propose here is a simulated annealing algorithm for optimization problems involving continuous variables. It is a direct search method. The strengths of the new algorithm are: it does not require differentiability or any other properties of the function being optimized and it is memory-based. Therefore, the algorithm can be applied to noisy and/or not exactly known functions. Although the algorithm is stochastic in nature, it can memorise the best solution. The new simulated annealing algorithm has been shown to be reliable, fast, general purpose and efficient for solving some difficult global optimization problems.     

An Efficient Optimization Procedure for Tetrahedral Meshes by Chaos Search Algorithm

A simple and efficient local optimization-based procedure for node reposition-ing/smoothing of three-dimensional tetrahedral meshes is presented.The initial tetrahedral mesh is optimized with respect to a specified element shape measure by chaos search algorithm,which is very effective for the optimization problems with only a few design variables.Examples show that the presented smoothing procedure can provide favorable conditions for local transformation approach and the quality of mesh can be significantly improved by the combination of these two procedures with respect to a specified element shape measure.Meanwhile,several commonly used shape measures for tetrahedral element,which are considered to be equivalent in some weak sense over a long period of time,are briefly re-examined in this paper.Preliminary study indicates that using different measures to evaluate the change of element shape will probably lead to inconsistent result for both well shaped and poorly shaped elements.The proposed smoothing approach can be utilized as an appropriate and effective tool for evaluating element shape measures and their influence on mesh optimization process and optimal solution.     

Efficient use of iterative solvers in nested topology optimization

In the nested approach to structural optimization, most of the computational effort is invested in the solution of the analysis equations. In this study, it is suggested to reduce this computational cost by using an approximation to the solution of the analysis problem, generated by a Krylov subspace iterative solver. By choosing convergence criteria for the iterative solver that are strongly related to the optimization objective and to the design sensitivities, it is possible to terminate the iterative solution of the nested equations earlier compared to traditional convergence measures. The approximation is computationally shown to be sufficiently accurate for the purpose of optimization though the nested equation system is not necessarily solved accurately. The approach is tested on several large-scale topology optimization problems, including minimum compliance problems and compliant mechanism design problems. The optimized designs are practically identical while the time spent on the analysis is reduced significantly.     

On sensitivity analysis of nonsmooth multidisciplinary optimization problems in engineering process line applications

In this paper, we study multidisciplinary optimization problems where the objective functions and the vector-valued mathematical models are not necessarily differentiable in a classical sense. Moreover, we assume that the state equation consists of several submodels, that is, unit-process models. This study is based on the nonsmooth analysis introduced by Clarke and the theory of H-differentiable functions by Gowda. It concentrates on two mathematical tools needed in sensitivity analysis of the multidisciplinary optimization problems, namely, the generalized chain rule for vector-valued functions and the implicit subdifferentiation formula. We employ these tools to obtain subgradient information of the problems considered. Finally, we present a numerical example and compare the obtained results with finite differences by means of accuracy and computational efficiency.     

Microstructural design of composite materials using fixed-grid modeling and noise-resistant smoothed Kriging-based approximate optimization

This paper discusses a microstructural optimization of composites using a fixed-grid modeling technique and an approximate optimization approach. In particular, we design a microscopic structure of composites to improve its reliability. As the response surface becomes nonlinear and inaccuracies may be included in the sampling results in using the fixed-grid model, applicability of several approximation methods such as a polynomial-based approach, neural network, and Kriging method are investigated. Especially, the inaccuracy is regarded as a noise in sampling data, and applicability of the noise-resistant smoothed Kriging (ns-Kriging) is investigated. As an example, cross-sectional shape of fiber in a unidirectional fiber-reinforced plastics is optimized. By applying several approximate optimization methods to the problem, applicability of those methods is investigated. Next, cross-sectional shape of fibers in a composite plate subject to bending and compression is optimized using the ns-Kriging-based method. Numerical results illustrate applicability of the proposed approach.     

Review of formulations for structural and mechanical system optimization

Alternative formulations for optimization and simulation of structural and mechanical systems and other related fields are reviewed. The material is divided roughly into two parts. Part 1 focuses on the developments in structural and mechanical systems, including configuration and topology optimization. Here the formulations are classified into three broad categories: (i) the conventional formulation where only the structural design variables are treated as optimization variables, (ii) simultaneous analysis and design (SAND) formulations where design and some of the state variables are treated as optimization variables, and (iii) a displacement-based two-phase approach where the displacements are treated as unknowns in the outer loop and the design variables as the unknowns in the inner loop. Part 2 covers more general formulations that are applicable to diverse fields, such as economics, optimal control, multidisciplinary problems and other engineering disciplines. In these fields, SAND-type formulations have been called mathematical programs with equilibrium constraints (MPEC), and partial differential equations (PDE)-constrained optimization problems. These formulations are viewed as generalizations of the SAND formulations developed in the structural optimization field. Based on the review, it is concluded that the basic ideas of the formulations presented in diverse fields can be integrated to conduct further research and develop alternative formulations and solution procedures for practical engineering applications. The paper lists 187 references on the subject.     

Numerical method for shape optimization using T-spline based isogeometric method

Numerical methods for shape design sensitivity analysis and optimization have been developed for several decades. However, the finite-element-based shape design sensitivity analysis and optimization have experienced some bottleneck problems such as design parameterization and design remodeling during optimization. In this paper, as a remedy for these problems, an isogeometric-based shape design sensitivity analysis and optimization methods are developed incorporating with T-spline basis. In the shape design sensitivity analysis and optimization procedure using a standard finite element approach, the design boundary should be parameterized for the smooth variation of the boundary using a separate geometric modeler, such as a CAD system. Otherwise, the optimal design usually tends to fall into an undesirable irregular shape. In an isogeometric approach, the NURBS basis function that is used in representing the geometric model in the CAD system is directly used in the response analysis, and the design boundary is expressed by the same NURBS function as used in the analysis. Moreover, the smoothness of the NURBS can allow the large perturbation of the design boundary without a severe mesh distortion. Thus, the isogeometric shape design sensitivity analysis is free from remeshing during the optimization process. In addition, the use of T-spline basis instead of NURBS can reduce the number of degrees of freedom, so that the optimal solution can be obtained more efficiently while yielding the same optimum design shape.     

Multidisciplinary design optimization of mechanisms

The optimization of mechanisms is usually done in the field of mechanism itself. As a result the structural safety of the mechanism is neglected. To ensure the safety and improve the dynamic characteristics of mechanisms, a multidisciplinary design optimization procedure is proposed in this paper to synthesize optimum mechanisms. Two disciplines are involved in the multidisciplinary design optimization. They are the mechanism and the structure. The multi-level decomposition approach is chosen to generate optimum mechanisms. The optimized mechanisms not only satisfy mechanism and structural constraints but also have optimum objective function values in the two disciplines. In order to solve general mechanism design problems two widely used commercial softwares MSC/NASTRAN and MSC/ADAMS are integrated in the procedure to do the structural and the mechanism analysis, respectively. When the structural optimization is performed, a compromised approach is introduced to treat multiple configurations of mechanisms during operation. Two mechanism design problems are given to test the proposed method.