A CAD-based design parameterization for shape optimization of elastic solids

In this paper a CAD-based design sensitivity analysis (DSA) and optimization method using Pro/ENGINEER for shape design of structural components is presented. The CAD-based design model is critically important for multidisciplinary shape design optimization. Only when each discipline can compute the design sensitivity coefficients of the CAD-based design model, can a true multidisciplinary what-if study, trade-off analysis, and design optimization be carried out. The proposed method will allow the design engineer to compute design sensitivity coefficients of structural performance measures such. as stress and displacement, evaluated using existing finite element analysis (FEA) tools, both h- and p-versions, with respect to design variables defined in the parameterized CAD model. The proposed method consists of (i) a CAD-based design parameterization technique that ties the structural DSA and optimization to a CAD tool; (ii) a design velocity field computation that defines material point movement due to design change in CAD geometry, satisfies linearity and regularity requirements, and supports both hand p-version FEA meshed using existing mesh generators; and (iii) a design optimization method that supports structural geometric and finite element model updates in Pro/ENGINEER during the optimization process.     

Robust aerodynamic shape design based on an adaptive stochastic optimization framework

Optimization techniques combined with uncertainty quantification are computationally expensive for robust aerodynamic optimization due to expensive CFD costs. Surrogate model technology can be used to improve the efficiency of robust optimization. In this paper, non-intrusive polynomial chaos method and Kriging model are used to construct a surrogate model that associate stochastic aerodynamic statistics with airfoil shapes. Then, global search algorithm is used to optimize the model to obtain optimal airfoil fast. However, optimization results always depend on the approximation accuracy of the surrogate model. Actually, it is difficult to achieve a high accuracy of the model in the whole design space. Therefore, we introduce the idea of adaptive strategy to robust aerodynamic optimization and propose an adaptive stochastic optimization framework. The surrogate model is updated adaptively by increasing training airfoils according to historical optimization results to guarantee the accuracy near the optimal design point, which can greatly reduce the number of training airfoils. The proposed method is applied to a robust aerodynamic shape optimization for drag minimization considering uncertainty of Mach number in transonic region. It can be concluded that the proposed method can obtain better optimal results more efficiently than the traditional robust optimization method and global surrogate model method.     

Simultaneous shape and topology optimization of shell structures

In this research, Method of Moving Asymptotes (MMA) is utilized for simultaneous shape and topology optimization of shell structures. It is shown that this approach is well matched with the large number of topology and shape design variables. The currently practiced technology for optimization is to find the topology first and then to refine the shape of structure. In this paper, the design parameters of shape and topology are optimized simultaneously in one go. In order to model and control the shape of free form shells, the NURBS (Non Uniform Rational B-Spline) technology is used. The optimization problem is considered as the minimization of mean compliance with the total material volume as active constraint and taking the shape and topology parameters as design variables. The material model employed for topology optimization is assumed to be the Solid Isotropic Material with Penalization (SIMP). Since the MMA optimization method requires derivatives of the objective function and the volume constraint with respect to the design variables, a sensitivity analysis is performed. Also, for alleviation of the instabilities such as mesh dependency and checkerboarding the convolution noise cleaning technique is employed. Finally, few examples taken from literature are presented to demonstrate the performance of the method and to study the effect of the proposed concurrent approach on the optimal design in comparison to the sequential topology and shape optimization methods.     

A virtual punching method for shape optimization of openings on curved panels using CAD-based Boolean operations

This paper aims at developing new methodologies for shape optimization of openings on three-dimensional curved panels that are used widely in aeronautical and aerospace engineering. To circumvent the difficulties associated with the hole boundary shape parameterization, a virtual punching method that exploits Boolean operations of the CAD modeler is proposed for the definition of shape design variables. Compared with the parametric mapping method developed previously, the virtual punching method is shown to be an implicit boundary representation for this specific kind of structure. Instead, the parametric mapping method is based on the explicit boundary representation.A zero-order genetic algorithm (GA) is correspondingly implemented into the design procedure of the virtual punching method to execute the optimization process for two reasons. First, it makes it possible to avoid sensitivity analysis that is relatively difficult due to the implicit boundary representation formulation and the use of an unstructured mesh. Second, the computing cost of the GA is practically affordable in shape optimization because often only a small number of design variables are involved. Numerical tests are carried out for typical examples of the stress concentration minimization around openings on the curved panels.     

面向分级设计优化的飞行器参数化建模方法

针对飞行器气动隐身外形综合设计优化问题,提出合适的面向分级设计优化流程,建立适应该流程的渐进分层参数化建模方法;用基于敏度分析的参数影响程度分析方法筛选复杂设计变量;采用多学科设计优化（Multidisplinary Design Optimization,MDO）理论和差分进化算法进行飞行器气动隐身外形的综合设计优化.将该方法用于某飞行器外形设计优化,结果表明：该方法合理可行,可为飞行器外形多学科设计优化提供一定参考.     

A gradient-based,parameter-free approach to shape optimization

In shape optimization, the independent node movement approach, wherein finite element node coordinates are used directly as design variables, allows the most freedom for shape change and avoids the time-consuming parameterization preprocess. However, this approach lacks a length scale control that is necessary to ensure a well-posed shape optimization problem and avoid numerical instability. Motivated by the success of filtering techniques that impose minimum length scales in topology optimization, we propose a scheme with consistent filtering to introduce a length scale and thereby ensure smoothness in shape optimization while preserving the advantages of the independent node movement approach.     

A graph-based parameterization concept for global laminate optimization

A new graph-based parameterization concept aimed at the global optimization of laminated structures by the means of evolutionary algorithms and finite element analysis is introduced. The motivation to develop this novel parameterization concept is twofold. First, the entire design space is accessible to optimi zation down to the smallest entity, which is a single finite element, and secondly, this concept guarantees greatest flexibility in terms of laminate layer shape and placement. The finite element mesh of a structure is represented as a mathematical graph. Substructures of this graph form fiber reinforced and possibly overlapping patches and are affiliated to virtual graph vertices representing their properties. Adapted genetic variation operators are directly applied on this graph. The method allows for concurrent optimization of number, size, shape, and position of the patches and an arbitrary number of material related properties for each of them. The novel concept overcomes the limits of traditional geometry-based approaches, as it is able to represent almost arbitrary patch shapes even on curved surfaces. Two numerical examples demonstrate the efficiency of the method.     

平面线弹性问题的等几何形状优化方法

介绍具有等几何分析功能的GeoPDEs平台的数据结构和分析流程,针对二维平面形状优化问题,以控制顶点为设计变量,在推导出等几何分析的灵敏度计算公式后,提出基于GeoPDEs平台的灵敏度分析的高效实现方法,并采用移动渐近线法(Method of Moving Asymptotes,MMA)算法进行等几何形状优化.形状优化实例表明该方法收敛速度快,优化结果较理想.     

Sizing optimization of truss structures using flower pollination algorithm

The recently developed flower pollination algorithm is used to minimize the weight of truss structures, including sizing design variables. The new algorithm can efficiently combine local and global searches, inspired by cross-pollination and self-pollination of flowering plants, respectively. Furthermore, it implements an iterative constraint handling strategy where trial designs are accepted or rejected based on the allowed amount of constraint violation that is progressively reduced as the search process approaches the optimum. This strategy aims to obtain always feasible optimized designs. The new algorithm is tested using three classical sizing optimization problems of 2D and 3D truss structures. Optimization results show that the proposed method is competitive with other state-of-the-art metaheuristic algorithms presented in the literature.     

Numerical shape optimization based on meshless method and stochastic optimization technique

This paper puts forward a newer approach for structural shape optimization by combining a meshless method (MM), i.e. element-free Galerkin (EFG) method, with swarm intelligence (SI)-based stochastic ‘zero-order’ search technique, i.e. artificial bee colony (ABC), for 2D linear elastic problems. The proposed combination is extremely beneficial in structural shape optimization because MM, when used for structural analysis in shape optimization, eliminates inherent issues of well-known grid-based numerical techniques (i.e. FEM) such as mesh distortion and subsequent remeshing while handling large shape changes, poor accuracy due to discontinuous secondary field variables across element boundaries needing costly post-processing techniques and grid optimization to minimize computational errors. Population-based stochastic optimization technique such as ABC eliminates computational burden, complexity and errors associated with design sensitivity analysis. For design boundary representation, Akima spline interpolation has been used in the present work owing to its enhanced stability and smoothness over cubic spline. The effectiveness, validity and performance of the proposed technique are established through numerical examples of cantilever beam and fillet geometry in 2D linear elasticity for shape optimization with behavior constraints on displacement and von Mises stress. For both these problems, influence of a number of design variables in shape optimization has also been investigated.     

Computational procedure for optimum shape design based on chained Bezier surfaces parameterization

Optimum design introduces strong emphasis on compact geometry parameterization in order to reduce the dimensionality of the search space and consequently optimization run-time. This paper develops a decision support system for optimum shape which integrates geometric knowledge acquisition using 3D scanning and evolutionary shape re-engineering by applying genetic-algorithm based optimum search within a distributed computing workflow.A shape knowledge representation and compaction method is developed by creating 2D and 3D parameterizations based on adaptive chaining of piecewise Bezier curves and surfaces. Low-degree patches are used with adaptive subdivision of the target domain, thereby preserving locality. C¹ inter-segment continuity is accomplished by generating additional control points without increasing the number of design variables. The control points positions are redistributed and compressed towards the sharp edges contained in the data-set for better representation of areas with sharp change in slopes and curvatures. The optimal decomposition of the points cloud or target surface into patches is based on the requested modeling accuracy, which works as lossy geometric data-set compression. The proposed method has advantages in non-recursive evaluation, possibility of chaining patches of different degrees, options of prescribing fixed values at selected intermediate points while maintaining C¹ continuity, and uncoupled processing of individual patches.The developed procedure executes external application nodes using mutual communication via native data files and data mining. This adaptive interdisciplinary workflow integrates different algorithms and programs (3D shape acquisition, representation of geometry with data-set compaction using parametric surfaces, geometric modeling, distributed evolutionary optimization) such that optimized shape solutions are synthesized. 2D and 3D test cases encompassing holes and sharp edges are provided to prove the capacity and respective performance of the developed parameterizations, and the resulting optimized shapes for different load cases demonstrate the functionality of the overall distributed workflow.     

A hierarchical design concept for shape optimization based on the interaction of CAGD and FEM

The numerical treatment of shape optimization problems requires sophisticated software tools such as Computer Aided Design (CAD), the Finite Element Method (FEM) and a suitable Mathematical Programming (MP) algorithm. Efficiency of the overall procedure is guaranteed if these tools interact optimally. The theoretical and numerical effort for sensitivity analysis reflect the complexity of this engineering problem.In this paper we outline a general modelling concept for shape optimization problems. Hierarchical design models within Computer Aided Geometrical Design (CAGD) and the interaction of geometry and FEM lead to an efficient overall optimization procedure. Our concept has been derived, implemented and tested for shell structures but it is seen to be generally applicable.After a short introduction containing the state of the art we give an overview of the numerical tools used and outline the interaction of CAGD and FEM within the overall optimization procedure.The paper is mainly devoted to the hierarchical design space based on a hierarchical geometrical modelling. The major part of computational effort is consumed by sensitivity analysis related to the number of design variables. Therefore, this number should be limited and only few powerful design variables corresponding to the special interests of the considered problem should be defined. This procedure may lead to a considerable limitation of the design space. Based on a hierarchy in the geometrical model different types of design variables are introduced: design variables with global, regional and local influence. The new method is based on successive activation of these types of design variables. This procedure leads to a considerable reduction of computational time for the sensitivity analysis without loss of geometrical flexibility.A new method of geometrical refinement and a successive adaptively driven expansion and reduction of the design space is described. It is based on the degree elevation or degree reduction of parametric curves and surfaces, respectively.A numerical example illustrates the new method and the efficiency of the overall optimization procedure.     

最小描述长度优化下的医学图像统计形状建模

统计形状模型(SSM)是有效的图像处理与分析方法。为了建立模型,需要从形状样本集中提取出具有对应关系的轮廓采样点集合,这是决定模型性能的关键。传统的手动标定这些点集来确保对应关系枯燥耗时且带有主观性,更难以向高维拓展。对形状建立逐层的多尺度参数表示,基于最小描述长度(MDL),在粗尺度上建立反映点对应程度的目标函数并最小化,提出首先确保粗尺度上具有最优意义的点对应,同时在精尺度上使用最便捷的弧长参数函数来确定特征点,完成感兴趣目标的快速统计形状建模,进而统计分析以验证模型性能,为后续图像分割或定量分析打下基础。实验对肌肉骨骼核磁共振成像(MRI)中椎骨、椎间盘以及半月板等具有临床意义的结构建立了统计形状模型,验证了本文方法与手动取点相比具有客观可重复性且更加简洁,与单一尺度下的MDL方法相比时间效率更高。基于此模型的图像分割与基于手动建模的分割相比,误差相当或有所降低。     

New approximate-iterative technique in shape optimization of continuum structures

An efficient technique that examines the optimum shape of elastic continuum structures is presented. The total number of stiffness matrix inversions required to obtain the optimal shape was significantly reduced as a result of using a newly developed iterative and approximate algorithm. The new technique is capable of approximating both displacements and stresses, and allows the user to have control over the degree of approximation. The efficiency of the proposed iterative approach was verified through various numerical examples and proved to be a viable alternative to the direct approach (which requires matrix inversion), especially in large-scale problems. The proposed approach is not limited, however, to shape optimization, but can be applied to other branches of structural optimization.     

Combined shape and reinforcement layout optimization of shell structures

This paper presents a combined shape and reinforcement layout optimization method of shell structures. The approach described in this work is applied to optimize simultaneously the geometry of the shell mid-plane as well as the layout of surface stiffeners on the shell. This formulation involves a variable ground structure, since the shape of the shell surface is modified in the course of the process. Here we shall consider a global structural design criterion, namely the compliance of the structure, following basically the classical problem of distributing a limited amount of material in the most favourable way.The solution to the problem is based on a finite element discretization of the design domain. The material within each of the elements is modelled by a second-rank layered Mindlin plate microstructure. By a simple modification, this type of microstructure can be used to find the optimum distribution of stiffeners on shell structures. The effective stiffness properties are computed analytically through a smear-out procedure. The proposed method has been implemented into a general optimization software called Odessy and satisfactorily applied to the solution of some numerical examples, which are illustrated at the end of the paper.     

A level set method for structural shape and topology optimization using radial basis functions

This paper presents an alternative level set method for shape and topology optimization of continuum structures. An implicit free boundary representation model is established by embedding structural boundary into the zero level set of a higher-dimensional level set function. An explicit parameterization scheme for the level set surface is proposed by using radial basis functions with compact support. In doing so, the originally more difficult shape and topology optimization, driven by the temporal and spatial Hamilton–Jacobi partial differential equation (PDE), is transformed into a relatively easier size optimization of the expansion coefficients of the basis functions. The design optimization is converted to an iterative numerical process that combines the parameterization with a derivation of the shape sensitivity of the design functions, so as to allow using mathematical programming algorithms to solve the level set-based design problem and avoid directly solving the Hamilton–Jacobi PDE. Furthermore, a numerically more stable and efficient volume integration scheme is proposed to implement calculations of the shape derivatives, leading to the creation of new holes which are generated initially along the boundary and then propagated to the interior of the design domain. Two widely studied examples are used to demonstrate the effectiveness of the proposed optimization method.     

Full analytical sensitivities in NURBS based isogeometric shape optimization

Non-uniform rational B-spline (NURBS) has been widely used as an effective shape parameterization technique for structural optimization due to its compact and powerful shape representation capability and its popularity among CAD systems. The advent of NURBS based isogeometric analysis has made it even more advantageous to use NURBS in shape optimization since it can potentially avoid the inaccuracy and labor-tediousness in geometric model conversion from the design model to the analysis model.Although both positions and weights of NURBS control points affect the shape, until very recently, usually only control point positions are used as design variables in shape optimization, thus restricting the design space and limiting the shape representation flexibility.This paper presents an approach for analytically computing the full sensitivities of both the positions and weights of NURBS control points in structural shape optimization. Such analytical formulation allows accurate calculation of sensitivity and has been successfully used in gradient-based shape optimization.The analytical sensitivity for both positions and weights of NURBS control points is especially beneficial for recovering optimal shapes that are conical e.g. ellipses and circles in 2D, cylinders, ellipsoids and spheres in 3D that are otherwise not possible without the weights as design variables.     

基于NURBS方法的气动外形优化设计

采用NURBS曲线曲面,对钝锥弹头和钝双锥弹体建立参数化曲面模型,取NURBS曲线控制点作为设计参数,应用高超声速面元法求解气动力特性,在给定设计约束下,采用遗传算法进行气动外形优化设计,并对优化结果进行了比较分析。结果表明,采用NURBS方法构造参数化外形,并结合优化技术可方便快速地获得所需最优外形;与应用二次曲线构造参数化外形相比,该方法对弹体形状控制更加灵活,并可局部修改弹头曲线形状。因此,基于NURBS方法发展整套的系统优化设计算法很有现实意义和应用价值。