Three‐dimensional shape optimization

Abstract

Optimal structural design generally deals with frame or shell structures where the optimization is limited to resizing of structural members to obtain optimum cross-sections or thicknesses. Shape optimization solves another classs of problems involving continuous structural components where the optimum shape (the shape of the boundaries and the surfaces of the components) is determined. This report describes shape optimization of three-dimensional structural components. The finite element method of analysis is used employing the 20-noded isoparametric element. The objective function, mass, is minimized by the direct use of nonlinear mathematical programming, specifically the feasible direction method. Numerical shape representation and the selection of design variables are the most important aspects of the problem. The problem is addressed from a practical standpoint and techniques are presented to minimize the number of design variables. Isoparametric representation of the surfaces and the numerical superposition of shapes are discussed. These techniques are compared and demonstrated on simple cantilever beams and their minimum mass designs are obtained. In one example, the optimum shape of a non-uniform cross-sectional beam is obtained under stress constraint. The final design has a crosssectional shape varying over the length of the beam which could not be predicted using the bending theory of beams. One major difficulty encountered in some problems was that the shape changes during the optimization process may require a change in the finite element mesh because the initial configuration of the finite element mesh may result in very distorted elements for the new shape. The analyis with a mesh containing distorted elements may not be possible at all or the results of the analysis may be inaccurate.

Isogeometric shape optimisation of shell structures using multiresolution subdivision surfaces

“…In the second group of techniques the design variables are the vertex positions of the finite element mesh [15][16][17]. Yet in another group of parameterisation techniques, like the ones based on radial basis functions [18] or free-form deformations [19,20], the design variables are only Figure 1: Shape optimised thin shell roof structure emanating from a flat plate with stiffeners. The structure is loaded with a uniformly distributed vertical load and is supported at its four corners.…”

A Smoothing Method for Shape Optimization: Traction Method Using the Robin Condition

“…The shape gradient with respect to domain variation can be evaluated by the adjoint variable method [Haug et al (1986); Choi and Kim (2005)]. However, direct application of the gradient method often results in oscillating shapes [Imam (1982)]. …”

Multi-Objective Shape Optimization in Forced Heat-Convection Fields

“…(11) − (13) (14) Morimoto (10) (15) (16) − (19) (20) − (22) (5) (18)(23) (24) mesh Jameson (23) Mohammadi (24) (5) (25)(26) (25) (a)…”