Showing papers on "Shape optimization published in 1988"

A shape optimization approach based on natural design variables and shape functions

Abstract: The general problem of concern is to find the optimum shape of an elastic body, which requires minimizing an objective function subject to stress, displacement, frequency, and manufacturing constraints. The basic approach so far has been to choose a set of geometric design variables that define the shape of the structure. Typically the design variables have been chosen as coefficients of splines and polynomials, coordinates of ‘control’ nodes, and other geometric parameters. An automatic finite element discretization scheme that uses geometric entities such as lines, arcs, splines, and blending functions, is then used to relate changes in position of interior grid points in the finite element mesh to changes in the design variables. In this paper, a set of natural design variables is chosen as the design variables defining the shape. Specifically, the design variables are the magnitudes of a set of fictitious loads applied on the structure. The displacements produced by these fictitious loads, or natural shape functions, are added onto the initial mesh to obtain the new shape. Consequently, a linear relationship is established between changes in grid point locations and design variables through a finite element analysis. Plane elasticity problems are solved using the new approach. The quality of the finite element meshes produced and other salient features of the shape optimal design problem are discussed.

Design‐Sensitivity analysis of Solids Using BEM

Abstract: This paper describes an effective formulation for computing design sensitivities required in the shape optimization of solid objects using the boundary element method (BEM). Implicit differentiation of the discretized boundary integral equations is performed, resulting in a general and efficient analysis technique for design sensitivities of all structural quantities. The numerical integration of kernels is performed, which involves the products of shape functions, fundamental solutions, and their derivatives required for sensitivity calculations. The sensitivities of all components of the boundary stress tensor are obtained without additional numerical integrations. High‐order elements with curved sides are employed for stress and sensitivity analysis. A multizone analysis is implemented and its computational advantages are studied. An approximate method for design sensitivity calculations is also suggested and its performance and computational economy relative to the exact procedure are presented. Compa...

Boundary Integral Equations for Recovery of Design Sensitivities in Shape Optimization

Abstract: A new formulation for obtaining design sensitivities in shape optimization has been developed. The formulation is based on a direct application of the material derivative concept to the appropriate boundary integral equations for displacements and stresses in an elastic solid. As a check on accuracy, the approach was applied to a uniformly loaded infinite plate containing an elliptical hole. In this case, the availability of an analytical solution made it possible to calculate errors exactly. Furthermore, a wide range of stress states could be considered simply by varying the aspect ratio of the elliptical hole. For convenience, the design sensitivities were calculated with respect to changes in the major axis. Numerical convergence was established by comparing results based on four successive boundary meshes. In general, the predictions for both displacement and stress sensitivities were remarkably accurate.

Boundary integral equation method for shape optimization of elastic structures

Abstract: A general method for shape design sensitivity analysis as applied to plane elasticity problems is developed with a direct boundary integral equation formulation, using the material derivative concept and adjoint variable method. The problem formulation is very general and a complete consideration is given to describing the boundary variation by including the tangential component of the velocity field. The method is then applied to obtain the sensitivity formula for a general stress constraint imposed over a small part of the boundary. The accuracy of the design sensitivity analysis is studied with a fillet and an elastic ring design problem. Among the several numerical implementations tested, the second order boundary elements with a cubic spline representation of the moving boundary have shown the best accuracy. A smooth characteristic function is found to be better than a plateau function for localization of the stress constraint. Optimal shapes for the two problems are presented to show numerical applications.

Shape sensitivity analysis of boundary optimal control problems for parabolic systems

Abstract: This paper is concerned with the shape sensitivity analysis of a class of boundary control, constrained, optimal control problems for parabolic systems. The notion of Euler and Lagrange derivatives of a boundary optimal control in the direction of a vector field is introduced. The derivatives are obtained in the form of optimal solutions of auxiliary optimal control problems. The method of sensitivity analysis used in this paper is based on related results on the differential stability of metric projections in Hilbert space onto a convex, closed subset, combined with the material derivative method of shape sensitivity analysis. Parabolic initial-boundary value problems with Dirichlet and Neumann boundary conditions are considered in this paper.

Shape optimal design of elastic bodies using a mixed variational formulation

Abstract: This study is concerned with the development of a variational formulation and a procedure for the computational solution for the shape optimal design of a two-dimensional linear elastic body. The objective is to minimize the maximum value of the Von Mises equivalent stress in the body, subject to an isoperimetric constraint on the area. The optimality conditions for this problem are derived using a mixed variational formulation, where the equations defining the elastostatics problem are dealt with as additional constraints for the optimization. The results of the analysis are implemented via a finite element discretization. The discretized model is tested in two numerical examples, the shape optimization of a hole in a biaxially loaded sheet, and of the design of a fillet.

Shape optimization and identification of solid geometries considering discontinuities

Abstract: Shape sensitivity analysis of heat-conducting bodies is performed in general terms incorporating interface conditions and boundary singularities. Adjoint variables and the material derivative concept are utilized to obtain the material derivaives of volume and surface integrals of temperature and heat flux. Two illustrative examples are then analyzed by iterative numerical techniques incorporating the boundary element method of discretization. In this first problem, the interface position in a nonhomogeneous material is optimized for a minimum of total surface heat flow. The second problem involves the determination of the solidification interface shape in the so-called steady-state one-phase Stefan problem. Numerical results, checked by exact solutions, where available, indicate that the proposed solution procedure is suitable for free boundary problems in heat transfer.

Optimal shape design of thin axisymmetric shells

Abstract: This paper is concerned with the optimal shape design of thin axisymmetric shells loaded symmetrically or non-symmetrically. The objective is to male a reference stress uniform along a part of the boundary to minimize the stress concentrations. For technological reasons, the boundary of the structure is composed of straight or circular segments defined by input data of master point coordinates and radius values. The design variables are easily deduced from the data. The analysis of the structure is performed by the finite element method using two or three node thin shell elements. Some test examples and an application of the program to the shape optimization of a bottle demonstrate the efficiency of the process.

Shape optimization of thermoelastic solids

Abstract: Variation of a general integral functional, defined over a three-dimensional domain and its bounding surface, is obtained with respect to domain variations in a thermoelastic solid body by employing the adjoint variable method and the material derivative concept, as in isothermal structural optimization problems. Since no variational principles are utilized, it is asserted that the present procedure may be applied to other physical systems. For illustration purposes, an axisymmetric problem involving a hollow cylinder is taken, for which analytical solutions are possible for the field variables. The optimal outer radius is calculated, however, through an iterative minimization algorithm utilizing the shape sensitivity expressions. For the numerical calculations, two different objective junctions are adopted leading to two independent optimal radii.

Stochastic form optimization

Abstract: Engineering decision-making processes are the subject of design methodology, engineering optimization, and heuristics. However, a large class of engineering problems exists which cannot be easily classified as suitable for the application of methods belonging to any of these three groups. In particular, engineering problems requiring innovative, patentable solutions are in a gray area between the engineering branches of heuristics, innovative design, and shape optimization, which is considered a subfield of optimization. These problems can be also dealt with using the techniques of artificial intelligence. This paper describes a unique computer-oriented innovative design method. Stochastic Form Optimization, which can also be classified as a nondeterministic optimization method. The proposed model of the generation of innovative solutions is based on a stochastic simulation of morphological analysis. It has also been used in BRZDYL1, a learning expert system for engineering design currently under...

Shape optimization of 3D continuum structures via force approximation techniques

Abstract: The existing need to develop methods whereby the shape design efficiency can be improved through the use of high quality approximation methods is addressed. An efficient approximation method for stress constraints in 3D shape design problems is proposed based on expanding the nodal forces in Taylor series with respect to shape variations. The significance of this new method is shown through elementary beam theory calculations and via numerical computations using 3D solid finite elements. Numerical examples including the classical cantilever beam structure and realistic automotive parts like the engine connecting rod are designed for optimum shape using the proposed method. The numerical results obtained from these methods are compared with other published results, to assess the efficiency and the convergence rate of the proposed method.

Structural Shape Optimization Using OASIS

Abstract: OASIS is a code for structural optimization. This paper demonstrates the capibilities in shape optimization. The CAD system ALADDIN is used for shape description and generation of FE-meshes.

Structural optimization : proceedings of the IUTAM Symposium on Structural Optimization, Melbourne, Australia, 9-13 February 1988

Abstract: Contributed Papers:.- Practical Design of Shear and Compression Loaded Stiffened Panels.- Shape Optimization of Holes in Composite Shear Panels.- Development of a Knowledge-Based System for Structural Optimization.- Modern Trend in Elastic-Plastic Design. Shape and Internal Structure Optimization.- Composite Materials as a Basis for Generating Optimal Topologies in Shape Design.- Performance Characteristics of Optimality Criteria Methods.- Optimal Shape of Cable Structures.- Shape Optimization of Continuum with Crack.- Linear Complementarity Problems: A Cutting Plane Method Based on the Convex Hull of Polyhedra.- Optimization of Vibrating Thin-Walled Structures.- Shape Optimization of Intersecting Pressure Vessels.- Optimization Procedure SAPOP Applied to Optimal Layouts of Complex Structures.- Structural Shape Optimization Using OASIS.- Optimal Design of R.C. Frames Based on Improved Inelastic Analysis Method.- Boundary Element Methods in Optimal Shape Design - An Integrated Approach.- A Michell Type Criterion for Shells.- Shape Optimization of the Cross-Sections of Thinwalled Beams Subjected to Bending and Shear.- Optimization of Structures Using the Finite Element Method.- Comparison of NLP Techniques in Optimum Structural Frame Design.- Structural and Control Optimization with Weight and Frobenius Norm as Performance Functions.- Multicriterion Plate Optimization.- Optimization of Syatems in Bending - Conjectures, Bounds and Estimates Relating to Moment Volume and Shape.- On the Optimal Design of Columns Subjected to Circulatory Loads.- Structural Optimization in a Non-Deterministic Setting.- On the Shape Optimization of Truss Structure Based on Reliability Concept.- On Optimality in Structural and Material Composition of Bamboo.- Optimization of a Hollow Beam-Shaft with Prescribed Inner Contour.- Dynamic Optimization of Machine Systems Configuration.- Design for Minimum Stress Concentration - Some Practical Aspects.- Optimality Conditions for Multiple Loaded Structures - Integrating Control and Finite Element Method.- A Variational Principle Useful in Optimizing Rectangular-Base Shallow Shells.- Minimax Algorithms for Structural Optimization.- Discrete-Continuous Structural Optimization.- Optimality Criteria and Layout Theory in Structural Design: Recent Developments and Applications.- Shape Optimization: Creating a Useful Design Tool.- Optimal Shape of Pendulum Links.- An Integrated Knowledge-Based Problem Solving System for Structural Optimization.- A Method of Feasible Direction with FEM for Shape Optimization.- Experimental Design and Structural Optimization.- On the Shape Determination of Non-Conservative System: A Case of Column under Follower Force.- A Mathematical Programming Approach for Finding the Stochastically Most Relevant Failure Mechanism.- Structural Shape Optimization in Multidisciplinary System Synthesis.- Boundary Element Approach to Optimal Structural Design Based on the Inverse Variational principle.- Optimal Shape of Least Weight Arches.- Optimazation of Conical Shells for Static and Dynamic Loads.- Optimum Designs of Rotating Shaft Systems for Nonlinear Dynamic Responses.- Minimum Compliance Stiffener Layout of a Plate.- Recent Investigations of Structural Optimization by Analytic Methods.- Detailed Machine Structural Shapes Generated from Simplified Models.- Abstracts of papers presented at the Symposium which are likely to appear in an early issue of the Journal "Structural Optimization", (Springer-Verlag)..- On Shape Optimization of Satelite Tanks.- Divergence Instability Conditions in the Optimum Design of Nonlinear Elastic Systems under Follower Loads.- Solution of Max-Min Problems via Bound Formulation and Mathematical Prograrmming.- Natural Structural Shapes for Shells of Revolution in the Membrane Theory of Shells.- Optimal Design of Structures under Creep Conditions.

Shape optimization of continuum with crack

Abstract: The present paper studies the problem of shape optimization from the viewpoint of structural design philosophy based on durability and damage tolerance. Initial cracks are assumed to exist or to occur at an early stage of fatigue life. The objective is to minimize the crack propagation rate, or the stress intensity factor range. Quadratic boundary elements are applied to discretize the continuum to be optimized. To obtain the stress intensity factor range, quarter-point singular elements are placed at the tip of the crack. The sensitivity of the stress intensity factor with respect to the structural shape is derived. A numerical example is presented and dicussed.

A Method of Feasible Direction with FEM for Shape Optimization

Abstract: The avoidance of cracks in zones of stress concentrations by minimizing the maximal von Mises stress is very important in practical problems for the industry. In many applications it is necessary to allow for the change of traction vectors in large time intervals. So, one gets as result a multiple loading problem with a finite number of traction vectors, which will be formulated here as a discrete dynamic programming problem. The proposed algorithm is based on the method of feasible directions. It means, in effect, that one has a non-gradient method, combined with FEM, for which the search direction vector is known from physical reasons, see SCHNACK 1979 and SPORL 1986 [1, 2].

Application of optimization methods in structural systems reliability theory

Abstract: In this paper it is demonstrated that optimization methods are useful tools for structural engineering. Estimation of the reliability of structures can be formulated as optimization problems, but more interesting applications are related to optimal design of structures and dorivation of optimal maintenance strategies.

Shuttle solid rocket booster bolted field joint shape optimization

Abstract: A structural optimization procedure is used to determine the shape of an alternate design for the Shuttle's solid rocket booster field joint. In contrast to the tang and clevis design of the existing joint, this design consists of two flanges bolted together. Configurations with 150 studs of l| in. diam and 135 studs of 1^ in. diam are considered. Using a nonlinear programming procedure, the joint weight is minimized under constraints on either von Mises or maximum normal stresses, joint opening, and geometry. The procedure solves the design problem by replacing it with a sequence of approximate (convex) subproblems; the pattern of contact between the joint halves is determined every few cycles by a nonlinear displacement analysis. The minimum weight design has 135 studs of 1 ^ in. diam and is designed under constraints on normal stresses. It weighs 1144 Ib per joint more than the current tang and clevis design.

Approximation of Contact Problems. Shape Optimization in Contact Problems

Abstract: The main part of this contribution deals with the approximation of variational inequalities, with special emphasize to contact problems. Finite element technique is used, starting from different variational formulations (primal, mixed). More details, concerning the approximation and numerical realization of contact problems can be found in [9]. Second part of this contribution is devoted to the optimization of the shape of contact zone of an elastic body, unilaterally supported by a rigid foundation, in order to obtain an even distribution of normal forces along contact part (for more details see [10]).

Shape Optimal Design Using Natural Shape Functions

Abstract: The problem of concern is to find the optimum shape of an elastic body such that the weight is minimized subject to limits on the Von-Mises stress within each element However, the basic approach and computational aspects apply to other types of shape optimization problems [1,2] Further, the optimized shape should satisfy requirements such as a straight edge remains straight, or that a portion of the boundary remains unchanged

Structural shape optimization in multidisciplinary system synthesis

Abstract: Structural shape optimization couples with other discipline optimization in the design of complex engineering systems. For instance, the wing structural weight and elastic deformations couple to aerodynamic loads and aircraft performance through drag. This coupling makes structural shape optimization a subtask in the overall vehicle synthesis. Decomposition methods for optimization and sensitivity analysis allow the specialized disciplinary methods to be used while the disciplines are temporarily decoupled, after which the interdisciplinary couplings are restored at the system level. Application of decomposition methods to structures-aerodynamics coupling in aircraft is outlined and illustrated with a numerical example of a transport aircraft. It is concluded that these methods may integrate structural and aerodynamic shape optimizations with the unified objective of the maximum aircraft performance.