Integration of topology and shape optimization for design of structural components

This paper presents an integrated approach that supports the topology optimization and CAD-based shape optimization. The main contribution of the paper is using the geometric reconstruction technique that is mathematically sound and error bounded for creating solid models of the topologically optimized structures with smooth geometric boundary. This geometric reconstruction method extends the integration to 3-D applications. In addition, commercial Computer-Aided Design (CAD), finite element analysis (FEA), optimization, and application software tools are incorporated to support the integrated optimization process. The integration is carried out by first converting the geometry of the topologically optimized structure into smooth and parametric B-spline curves and surfaces. The B-spline curves and surfaces are then imported into a parametric CAD environment to build solid models of the structure. The control point movements of the B-spline curves or surfaces are defined as design variables for shape optimization, in which CAD-based design velocity field computations, design sensitivity analysis (DSA), and nonlinear programming are performed. Both 2-D plane stress and 3-D solid examples are presented to demonstrate the proposed approach. Received January 27, 2000 Communicated by J. Sobieski     

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.     

An integrated approach to topology,sizing, and shape optimization

Topology optimization has become very popular in industrial applications, and most FEM codes have implemented certain capabilities of topology optimization. However, most codes do not allow simultaneous treatment of sizing and shape optimization during the topology optimization phase. This poses a limitation on the design space and therefore prevents finding possible better designs since the interaction of sizing and shape variables with topology modification is excluded. In this paper, an integrated approach is developed to provide the user with the freedom of combining sizing, shape, and topology optimization in a single process.     

A variational binary level-set method for elliptic shape optimization problems

We present a variational binary level-set method to solve a class of elliptic problems in shape optimization. By the ‘ersatz material’ approach, which amounts to fill the holes by a weak phase, the original shape optimization model is approximated by a two-phase optimization problem. Under the binary level-set framework, we need to optimize a smooth functional under a binary constraint. We propose an augmented Lagrangian method to solve the constrained optimization problem. Numerical results are presented and compared with those obtained by level-set methods, which demonstrate the robustness and efficiency of our method.     

A web-based topology optimization program

The paper presents a web-based interface for a topology optimization program. The program is accessible over the World Wide Web at the address http://www.topopt.dtu.dk. The paper discusses implementation issues and educational aspects as well as statistics and experience with the program. Received September 29, 2000     

Topological sensitivity derivative and finite topology modifications: application to optimization of plates in bending

The concept of topological sensitivity derivative is introduced and applied to study the problem of optimal design of structures. It is assumed, that virtual topology variation is described by topological parameters. The topological derivative provides the gradients of objective functional and constraints with respect to these parameters. This derivative enables formulation of the conditions of topology transformation. In this paper formulas for the topological sensitivity derivative for bending plates are derived. Next, the topological derivative is used in the optimization process in order to formulate conditions of finite topology modifications and in order to localize positions of the modifications. In the case of plates they are related to introduction of holes and introduction of stiffeners. The theoretical considerations are illustrated by some numerical examples.     

A note on stress-constrained truss topology optimization

The purpose of this brief note is to demonstrate that general-purpose optimization methods and codes should not be discarded when dealing with stress-constrained truss topology optimization. By using a disaggregated formulation of the considered problem, such methods may find also “singular optima”, without using perturbation techniques like the ε-relaxed approach. Received February 19, 2002     

A new solution for topology optimization problems with multiple loads:The guide-weight method

The guide-weight method is introduced to solve two kinds of topology optimization problems with multiple loads in this paper.The guide-weight method and its Lagrange multipliers' solution methods are presented first,and the Lagrange multipliers' soution method of problems with multiple constraints is improved by the dual method.Then the iterative formulas of the guide-weight method for topology optimization problems of minimum compliance and minimum weight are derived and coresponding numerical examples are...     

On topology optimization of linear and nonlinear plate problems

In this paper we propose a new restriction method based on employing C ⁰-continuous fields of density defined on a set of meshes different from the one used for the finite element analysis. The optimization procedure starts with using a coarse density-mesh compared to the finite element one. Once the convergence is obtained in the optimization steps, a finer density-mesh is nominated for the further steps. Linear and nonlinear plate behaviors are considered and formulated by Kirchhoff or Mindlin–Reissner hypothesis. Comparison is made with element/nodal based approaches using filter. The results show excellent and robust performance of the proposed method.     

The particle swarm optimization algorithm in size and shape optimization

Shape and size optimization problems instructural design are addressed using the particle swarm optimization algorithm (PSOA). In our implementation of the PSOA, the social behaviour of birds is mimicked. Individual birds exchange information about their position, velocity and fitness, and the behaviour of the flock is then influenced to increase the probability of migration to regions of high fitness. New operators in the PSOA, namely the elite velocity and the elite particle, are introduced. Standard size and shape design problems selected from literature are used to evaluate the performance of the PSOA. The performance of the PSOA is compared with that of three gradient based methods, as well as the genetic algorithm (GA). In attaining the approximate region of the optimum, our implementation suggests that the PSOA is superior to the GA, and comparable to gradient based algorithms. Received December 18, 2000     

Constrained adaptive topology optimization for vibrating shell structures

This paper describes an algorithm for structural topology optimization entitled Constrained Adaptive Topology Optimization or CATO which is applied here to produce the optimum design of shell structures under free vibration conditions. The algorithm, based on an artificial material model and an updating scheme, combines ideas from the more mathematically rigorous homogenization (h) methods and the more intuitive evolutionary (e) methods. Thus, CATO can be seen as a hybrid h/e method. The optimization problem is defined as maximizing or minimizing a chosen frequency with a constraint on the structural volume/mass by redistributing the material through the structure. The efficiency of the proposed algorithm is illustrated through several numerical examples. Received February 17, 2000     

Issues of commercial optimization software development

Commercial optimization software development has a different set of goals and constraints than the development of academic or industrial research codes. Commercial codes must be all things to all people. They must contain a wide range of analysis options and be able to handle large, real world, industrial analysis models. As most of the users of the software in industry come from analysis, rather than design optimization backgrounds, the codes must perform in a robust manner. Inconsistent input data must be detected. Optimization methods must be automatically chosen by the program. Optimization parameters need to be adjusted automatically by the program. Another very important aspect is ease of use. A very intuitive and easy to use GUI (Graphical User Interface) should be developed. This work describes some of the development objectives and concerns that are essential to the development of commercial optimization software products. Received December 30, 2000     

A 99 line topology optimization code written in Matlab

The paper presents a compact Matlab implementation of a topology optimization code for compliance minimization of statically loaded structures. The total number of Matlab input lines is 99 including optimizer and Finite Element subroutine. The 99 lines are divided into 36 lines for the main program, 12 lines for the Optimality Criteria based optimizer, 16 lines for a mesh-independency filter and 35 lines for the finite element code. In fact, excluding comment lines and lines associated with output and finite element analysis, it is shown that only 49 Matlab input lines are required for solving a well-posed topology optimization problem. By adding three additional lines, the program can solve problems with multiple load cases. The code is intended for educational purposes. The complete Matlab code is given in the Appendix and can be down-loaded from the web-site http://www.topopt.dtu.dk. Received October 22, 1999