Optimum shape design of rotating disks

This paper deals with optimum shape design of the rotating disks by nonlinear programming method. The shape of the cross section is defined by 5th degree polynomial which is completely determined by the boundary conditions and four design variables. The stress analysis of the disk is carried out by finite element method using isoparametric elements. The optimization technique used is with improved movelimit method of sequential linear programming. Progress of optimization is investigated with three different objective functions. After preliminary studies a weighted objective function is selected for detailed investigation. Optimum shapes are obtained for different speeds and for different fit pressures from hub. The results are presented in non-dimensionalised form.     

Sensitivity analysis and optimal design for fiber reinforced composite disks

For a linearly elastic fiber reinforced composite disk, the first variation of an arbitrary stress, strain and displacement functional corresponding to variation of material parameters is derived by using the direct and adjoint approaches to sensitivity analysis. The results are particularized to the case of total potential and complementary energies. The relevant optimality conditions for optimal design and identification problems are then derived.     

An approximation-concepts approach to shape optimal design

In the past, much of the work done in structural optimization consisted in resizing the members of fixed configuration models. In that case, a powerful design procedure has now emerged, which is based on the coordinate use of explicit high-quality approximations of the behavior constraints and dual methods of mathematical programming.There is, however, a large class of problems for which the main degrees of freedom for the designer correspond to the shape of the structure itself.The main objectives of this paper are to recall briefly a convenient geometric representation, in which the boundaries of the structure are represented by Bezier or B-spline curves, and then to discuss the choice of optimization algorithm. It is shown that cost-efficient methods for structural sizing may be advantageously extended to shape optimal design problems. Different approximation schemes are tested and a new general optimization algorithm is presented that combines mixed approximations and dual methods. Many large-scale applications are treated to demonstrate the generality and the efficiency of the new formulation. Finally, considerations are given about an integrated approach including CAD computer codes and finite element optimization software.     

Sensitivity analysis for optimal shape design problems

Various methods for performing the sensitivity analysis in solving optimal shape design problems are outlined. The methods are illustrated in detail in the finite setting of a unilateral boundary value problem of the Dirichlet-Signorini type. The methods are compared in several numerical examples.     

Dynamic problems of shape optimal design for shallow curved plates

The problem of optimal structural design of shallow thin-walled elements such as curved rectangular plates are formulated and solved for dynamic conditions. The distribution of the initial curvature of shallow plates in a nonstrained state is taken as the control function. Dynamic compliance is considered as the minimized performance functional. Optimality conditions are derived for the distributed parameter system considered and applied for the construction of the analytical solution. The rigorous analysis of extremum conditions and behavioural equations shows that the initial optimization problem is decomposed into several problems of classical structural analysis, which can be successfully solved analytically. Some optimal designs obtained for rectangular plates under stretching and bending, and a plate lying on an elastic foundation and subjected to lateral forces are presented. Received: November 27, 1998     

Integrated optimal topology design and shape optimization using neural networks

In this paper, neural network- and feature-based approaches are introduced to overcome current shortcomings in the automated integration of topology design and shape optimization. The topology optimization results are reconstructed in terms of features, which consist of attributes required for automation and integration in subsequent applications. Features are defined as cost-efficient simple shapes for manufacturing. A neural network-based image-processing technique is presented to match the arbitrarily shaped holes inside the structure with predefined features. The effectiveness of the proposed approach in integrating topology design and shape optimization is demonstrated with several experimental examples.     

桁架形状优化的一种改进模拟退火算法研究

通过设计一种产生可行解的状态发生器,由该状态发生器产生的新状态均满足所有的约束条件,从而方便地处理约束条件,并提出一种求解桁架形状优化设计问题的改进的模拟退火算法。算例表明该方法能获得较高质量的解,具有现实的工程意义,同时指出改进的SA算法用于桁架形状优化问题得不到全局最优解。     

Multiparametric optimization for multidisciplinary engineering design

The area of Multiparametric Optimization (MPO) solves problems that contain unknown problem data represented by parameters. The solutions map parameter values to optimal design and objective function values. In this paper, for the first time, MPO techniques are applied to improve and advance Multidisciplinary Design Optimization (MDO) to solve engineering problems with parameters. A multiparametric subgradient algorithm is proposed and applied to two MDO methods: Analytical Target Cascading (ATC) and Network Target Coordination (NTC). Numerical results on test problems show the proposed parametric ATC and NTC methods effectively solve parametric MDO problems and provide useful insights to designers. In addition, a novel Two-Stage ATC method is proposed to solve nonparametric MDO problems. In this new approach elements of the subproblems are treated as parameters and optimal design functions are constructed for each one. When the ATC loop is engaged, steps involving the lengthy optimization of subproblems are replaced with simple function evaluations.     

On the optimal design of the shape of a buckled elastic beam

Analysis of stability, post-buckling bending and vibrations is performed for a beam (a spring element) having an optimal shape. A buckled pin-jointed spring element of a constant thickness and variable width is considered. The optimal shape of this beam is suggested to provide a uniform distribution of maximum bending stresses in its buckled equilibrium configuration for a given value of a supercritical axial force. Sensitivities of a critical force and a buckling mode to variations of the shape of a beam are calculated. A dependence of the static lateral deflection upon an axial force is analysed. Nonlinear equations of large-amplitude oscillations are derived by a use of the Hamilton principle. The natural frequencies of a spring element, compressed by a supercritical force are calculated. Received April 29, 1999     

On time optimal supernode shape

With the objective of minimizing the total execution time of a parallel program on a distributed memory parallel computer, this paper discusses the selection of an optimal supernode shape of a supernode transformation (also known as tiling). We identify three parameters of a supernode transformation: supernode size, relative side lengths, and cutting hyperplane directions. For supernode transformations on algorithms with perfectly nested loops and uniform dependencies, we prove the optimality of a constant linear schedule vector and give a necessary and sufficient condition for optimal relative side lengths. We also prove that the total running time is minimized by a cutting hyperplane direction matrix from a particular subset of all valid directions and we discuss the cases where this subset is unique. The results are derived in continuous space and should be considered approximate. Our model does not include cache effects and assumes an unbounded number of available processors, the communication cost approximated by a constant, uniform dependences, and loop bounds known at compile time. A comprehensive example is discussed with an application of the results to the Jacobi algorithm.     

A sequential coupling of optimal topology and multilevel shape design applied to two-dimensional nonlinear magnetostatics

In this paper, a sequential coupling of two-dimensional (2D) optimal topology and shape design is proposed so that a coarsely discretized and optimized topology is the initial guess for the following shape optimization. In between, we approximate the optimized topology by piecewise Bézier shapes via least square fitting. For the topology optimization, we use the steepest descent method. The state problem is a nonlinear Poisson equation discretized by the finite element method and eliminated within Newton iterations, while the particular linear systems are solved using a multigrid preconditioned conjugate gradients method. The shape optimization is also solved in a multilevel fashion, where at each level the sequential quadratic programming is employed. We further propose an adjoint sensitivity analysis method for the nested nonlinear state system. At the end, the machinery is applied to optimal design of a direct electric current electromagnet. The results correspond to physical experiments. This research has been supported by the Austrian Science Fund FWF within the SFB “Numerical and Symbolic Scientific Computing” under the grant SFB F013, subprojects F1309 and F1315, by the Czech Ministry of Education under the grant AVČR 1ET400300415, by the Czech Grant Agency under the grant GAČR 201/05/P008 and by the Slovak Grant Agency under the project VEGA 1/0262/03.     

Optimal shape of prestressed disks against brittle-rupture under unsteady creep conditions

The effect of thickness distribution of circumferentially prestressed rotating disks on the localization of brittle creep rupture is analysed. For an optimal disk with arbitrary prestressing the first macrocracks appear simultaneously atN-points (N is the numbers of elements of the jump-like variable thickness of the disk) and, hence, the lifetime is maximized. The time to rupture of an optimal disk of jump-like thickness is compared with the lifetime of a disk of uniform elastic equivalent stress and of the same number of thickness elements. The time hardening theory associated with Kachanov's brittle rupture equation is applied as the constitutive equation describing unsteady creep. The optimization procedure used to determine thickness distribution is based on the iterative corrections of the element thicknesses. Apart from disks optimized without any constraints, the optimal shapes of disks under additional geometric constraints (minimal thickness) are considered.     

Efficient morphological shape representation by varying overlapping levels among representative disks

The morphological skeleton transform, the morphological shape decomposition, and the overlapped morphological shape decomposition are three basic morphological shape representation schemes. In this paper, we propose a new way of generalizing these basic representation algorithms to improve representational efficiency. In all three basic algorithms, a fixed overlapping policy is used to control the overlapping relationships among representative disks of different sizes. In our new algorithm, different overlapping policies are used to generate shape components that have different overlapping relationships among themselves. The overlapping policy is selected dynamically according to local shape features. Experiments show that compared to the three basic algorithms, our algorithm produces more efficient representations with lower numbers of representative points.     

Building optimal 2D statistical shape models

Statistical shape models are used widely as a basis for segmenting and interpreting images. A major drawback of the approach is the need, during training, to establish a dense correspondence across a training set of segmented shapes. We show that model construction can be treated as an optimisation problem, automating the process and guaranteeing the effectiveness of the resulting models. This is achieved by optimising an objective function with respect to the correspondence. We use an information theoretic objective function that directly promotes desirable features of the model. This is coupled with an effective method of manipulating correspondence, based on re-parameterising each training shape, to build optimal statistical shape models. The method is evaluated on several training sets of shapes, showing that it constructs better models than alternative approaches.     

Global optimum shape design

An optimum shape design problem can be formulated as a minimization problem of a functional subject to certain constraints. Usually, it is nonlinear and nonconvex. Conventional optimization techniques are gradient-based, they highly depend on the initial design, and are difficult to be applied to find a global solution. Integral global optimization algorithm is proposed to solve optimum shape design problems. Three design examples are given to illustrate the effectiveness of the algorithm.     

Multi-objective optimal fixture layout design

This paper addresses a major issue in fixture layout design to determine and evaluate the acceptable fixture designs based on multiple quality criteria and to select an optimal fixture with appropriate trade-offs among multiple performance requirements. The performance objectives considered are related to the fundamental requirements of kinematic localization and total fixturing (form-closure). Three performance objectives are defined as the workpiece localization accuracy and the norm and dispersion of the locator contact forces. The paper focuses on multi-criteria optimal design with a hierarchical approach. An efficient interchange algorithm is extended and used for different practical cases, leading to proper trade-off strategies for performing fixture synthesis. Examples are given to illustrate empirical observations with respect to the proposed approach and its effectiveness.     

Computer aided optimal plastic design

An interactive method for optimal plastic design of structures that obey the square yield criterion is presented. Sections of fixed dimensions such as standard sections can be used for design and realistic cost estimates can be used in assessment. The design is developed interactively by upgrading in steps, using the rate of increase of strength per unit incremental cost as a guide. The entire method is explained by establishing the physical meaning of the expressions derived. The method is suitable for implementation in microcomputers.     

Shape optimal design using B-splines

Shape optimal design of an elastic structure is formulated using a design element technique. It is shown that Bezier and B-spline curves, typical of the CAD philosophy, are well suited to the definition of design elements. Complex geometries can be described in a very compact way by a small set of design variables and a few design elements. Because of the B-splines flexibility, it is no longer necessary to piece design elements together in order to agree with the shape complexity, nor to restrict the shape variations. Moreover, the additional optimization constraints that are most often needed to avoid unrealistic designs when the shape variables are the nodal coordinates of a finite element mesh, are automatically taken into account in the new formulation. An analytical derivation of the sensitivity analysis will be established, giving rise to numerical efficiency. It will be seen that the resulting optimization problem does not involve highly nonlinear functions with respect to the shape variables, so that simple mathematical programming algorithms can be applied to solve it. Some numerical examples are offered to demonstrate the power and generality of the new approach presented in this paper.     

Optimal design of disks with respect to ductile creep rupture time

The problem of optimal design with respect to ductile creep rupture time for rotating disks is solved. The finite strain theory is applied, the material is described by the Norton-Bailey law generalized for true stresses and logarithmic strains. The set of four partial differential equations describing the problem is derived. The optimal shape of the disk is found using parametric optimization with one or two free parameters. The results are compared with disks of uniform thickness.