Abstract: Biological structures consist of mechanical load carriers, which are highly optimized in terms of mechanical strength and minimum weight. It is demonstrated on some selected examples that a constant Mises-stress at the surface of the biological component can be accepted as significant biological design rule. However, a general proof of this hypothesis seems to be impossible. It is discussed how ready-grown biological designs can be transferred to engineering applications. A new method of structural shape optimization was developed because biological “components” do not always exist exactly in a shape ready to be copied for engineering use. The method is based on the computer-simulation of tree growth which is performed by use of the “volumetric swelling” option or alternatively by stress-controlled thermal expansion in the FEM-code ABAQUS. A number of examples show that the growth of biological structures can be computer-simulated very well and incidentally the “natural” loading case can be defined precisely. Technical applications show that the method is very efficient in structural shape optimization of 2 D and 3 D engineering structures. It is compared with other methods of structural optimization found in the literature.

Abstract: We study an optimal design problem for the domain of an elliptic equation with Dirichlet boundary conditions. We introduce a relaxed formulation of the problem which always admits a solution, and we prove some necessary conditions for optimality both for the relaxed and for the original problem.

Abstract: The objective of this paper is to provide a method of optimizing areas of the members as well as the shape of both two-hinged and fixed arches. The design process includes satisfaction of combined stress constraints under the assumption that the arch ribs can be approximated by a finite number of straight members. In order to reduce the number of detailed finite element analyses, the Force Approximization Method is used. A finite element analysis of the initial structure is performed and the gradients of the member end forces (axial, bending moment) are calculated with respect to the areas and nodal coordinates. The gradients are used to form an approximate structural analysis based on first order Taylor series expansions of the member end forces. Using move limits, a numerical optimizer minimizes the volume of the arch with information from the approximate structural analysis. Numerical examples are presented to demonstrate the efficiency and reliablity of the proposed method for shape optimization. It is shown that the number of finite element analysis is minimal and the procedure provides a highly efficient method of arch shape optimization.

Abstract: CAOS is a system for structural shape optimization. It is closely integrated in a Computer Aided Design environment and controlled entirely from the CAD-system AutoCAD. The mathematical foundation of the system is briefly presented and a description of the CAD-integration strategy is given together with an example.

Abstract: A procedure is presented for the shape optimization of truss structures based on the reliability concept. Nodal coordinates are taken as the shape design variables together with the sizing design variables such as the cross-sectional areas of the members. These variables are determined to minimize the structural volume under the constraint on the structural failure probability.

Abstract: The design sensitivities are obtained economically by implicit differentiation of the boundary integral equations. The mesh generation and regeneration is done using a parametric and auxiliary geometry concept that allows the original mesh to remain adequate for a wide range of subsequent evolving geometries as the optimization proceeds

Abstract: The aim of this paper is to discuss some qualitative properties of the relaxed solutions of a shape optimization problem for the domain of resolution of an elliptic equation with Dirichlet boundary conditions.

Abstract: The research work presented here is based on the concept of the integration of optimization techniques and numerical analysis with the finite element method (FEM) and computer-aided design (CAD). A microcomputer aided optimum design system, MCADS, has been developed for general structures. Certain techniques to be discussed in the paper, e.g. the semi-analytical method for design sensitivity analysis, optimization analysis modelling for shape design, application oriented user interfaces and the coupling of automated optimization and user intervention have rendered MCADS pratical and versatile in applications for engineering structures. The above techniques and an application are presented in this paper.

Abstract: Two procedures, the feasible direction method and sequential linear programming, for shape optimization of gas turbine disks are presented in this paper. The objective of these procedures is to obtain optimal designs of turbine disks with geometric and stress constraints. The coordinates of the selected points on the disk contours are used as the design variables. Structural weight, stress and their derivatives with respect to the design variables are calculated by an efficient finite element method for design sensitivity analysis. Numerical examples of the optimal designs of a disk subjected to thermo-mechanical loadings are presented to illustrate and compare the effectiveness of these two procedures.Copyright © 1990 by ASME

Abstract: In this paper, first- and second-order necessary conditions for optimality are studied for a domain optimization problem. The optimization problem considered is the minimization of an objective function defined on the domain boundary through the solution of a boundary value problem. In order to derive the first and second variations of the objective function due to boundary variation, the first and second variations of the solution of the boundary value problem are calculated using a perturbation technique. An iterative shape optimization algorithm for potential flow problems in R2 with Dirichlet boundary conditions is presented. In the algorithm a boundary element method (BEM) is employed to solve the Laplace equation numerically. The validity and accuracy of the algorithm have been verified on a problem where the final solution is known. Finally, the problem of designing a 90° bend for two-dimensional potential flow is solved.

Abstract: The concept of structural optimization was introduced in the early sixties. It was at that time that Professor Schmit, from Los Angeles, California University, suggested a combination of structural analysis by finite element and optimization methods (Schmit 1960). At present, the use of optimization methods is generally well understood. The capabilities (and limitations) of these methods are known and some industrial softwares exist. One of these is SAMCEF which can be applied to design of thin-walled structures or to shape optimization. The techniques used for the solution of large scale problems are presented in this paper. A summary of the main difficulties involved in shape optimization is also given. Several industrial problems are solved to illustrate the proposed concepts.

Abstract: This paper is concerned with a second-order numerical method for shape optimization problems. The first variation and the second variation of the objective functional are derived. These variations are discretized by introducing a set of boundary-value problems in order to derive the second-order numerical method. The boundary-value problems are solved by the conventional finite-element method.

Abstract: This paper describes the principal methods used within the computer program STARS for the computer-aided design of optimum structures subject to a variety of constraints. In particular, the development of a Newton method for size optimization and a hierarchical approach to shape optimization are outlined. A test problem connected with the latter is presented. Practical examples ere given that show how research that originated at the Royal Aerospace Establishment (RAE) has been continued and applied at Deutsche Airbus GmbH (DA) [formerly MesserschmittBolkow-Blohm GmbH (MSB)]. This includes the design of various components that are typical in aircraft construction and also a description of the manner in which flutter optimization is being accomplished with STARS and DA in combination with the in-house aeroelastic program.

Abstract: Concepts associated with the use of the implicit differentiation technique to derive Boundary Element theoretical formulations for Design Sensitivity Analysis (DSA) of coupled problems are presented. Within the context of a coupled thermal/structural problem, it is shown that the thermoelastic response DSA can be formulated even for transient temperature distributions without the need for explicit domain integrations and without the requirement to factor perturbed matrices. For the case of a coupled structural/fluid problem, a coupled shell structural/fluid acoustic DSA formulation is presented. In this second problem the coupling is twofold. First, there is the obvius coupling of the two physical phenomena governed by different differential equations. Secondly, the shell structural behavior is numerically simulated using the Finite Element Method (FEM) while the fluid acoustic behavior is treated by the employment of the BEM. It is shown that implicit differentiation of the coupled, discretized equations can lead to the formulation of a two step procedure for shape or property DSA for all response quantities involved in the coupled analysis. In both steps of this procedure, it is shown that the DSA approach allows for the reuse of matrix factorizations performed in the earlier coupled analysis step to once again produce a DSA methodology that obviates the need to factor perturbed matrices. These two applications can be considered typical of a much wider class of coupled problems for which effective DSA formulations can be derived via the implicit differentiation approach.

Abstract: The capability to handle multiple, static loading conditions has been added to two-dimensional-shape optimization with adaptive mesh refinement. The strain energy density variations for each mesh were normalized, and then maximum values from all loading conditions were used for refinement. Several sample test cases were considered: a two-dimensional bracket with two in-plane loading cases and a folded, stamped-sheet metal part representing an upper suspension control arm and a folded, stamped-sheet metal part representing a brake pedal support bracket

Abstract: A continuous approach is adopted in order to find the total variation of a general performance criterion. The present analysis may find important physical applications in the simultaneous shape optimization and control of large space structures in time by applied thermal and/or mechanical loads. The adjoint variable method and the material derivative concept are used to find the sensitivity expressions

Abstract: Structural optimization has attracted the attention since the days of Galileo. Olhoff and Taylor have produced an excellent overview of the classical research within this field. However, the interest in structural optimization has increased greatly during the last decade due to the advent of reliable general numerical analysis methods and the computer power necessary to use them efficiently. This has created the possibility of developing general numerical systems for shape optimization. Several authors, eg., Esping; Braibant & Fleury; Bennet & Botkin; Botkin, Yang, and Bennet; and Stanton have published practical and successful applications of general optimization systems. Ding and Homlein have produced extensive overviews of available systems. Furthermore, a number of commercial optimization systems based on well-established finite element codes have been introduced. Systems like ANSYS, IDEAS, OASIS, and NISAOPT are widely known examples. In parallel to this development, the technology of computer aided design (CAD) has gained a large influence on the design process of mechanical engineering. The CAD technology has already lived through a rapid development driven by the drastically growing capabilities of digital computers. However, the systems of today are still considered as being only the first generation of a long row of computer integrated manufacturing (CIM) systems. These systems to come will offer an integrated environment for design, analysis, and fabrication of products of almost any character. Thus, the CAD system could be regarded as simply a database for geometrical information equipped with a number of tools with the purpose of helping the user in the design process. Among these tools are facilities for structural analysis and optimization as well as present standard CAD features like drawing, modeling, and visualization tools. The state of the art of structural optimization is that a large amount of mathematical and mechanical techniques are available for the solution of single problems. By implementing collections of the available techniques into general software systems, operational environments for structural optimization have been created. The forthcoming years must bring solutions to the problem of integrating such systems into more general design environments. The result of this work should be CAD systems for rational design in which structural optimization is one important design tool among many others.

Abstract: An optimal and robust shape is determined for a pipe conveying a fluid. The problem is formulated as a structural optimization problem with the radius of the circular cross section as design variable. The critical velocity of the fluid flow in the pipe serves as the first criterion. The critical velocity is related to the upper limit of the nonconservative fluid force. It is furthermore desired that the critical velocity be insensitive to perturbations in the pipe shape and we thus consider the design’s robustness with respect to such perturbations as the second criterion. The two criteria are ordered in accordance with their importance with the maximum of the follower force as the primary criterion. The optimal design for the force is considered first and this design is then modified to improve the robustness. The results are illustrated for a silicon rubber pipe conveying water at constant velocity.