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Three‐dimensional shape optimization
Author(s) -
Imam M. Hasan
Publication year - 1982
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620180504
Subject(s) - finite element method , shape optimization , superposition principle , cantilever , representation (politics) , constraint (computer aided design) , nonlinear programming , shell (structure) , bending , minimum weight , mathematics , optimization problem , beam (structure) , mathematical optimization , nonlinear system , structural engineering , geometry , mathematical analysis , engineering , mechanical engineering , physics , quantum mechanics , politics , political science , law
Optimal structural desingn 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 class 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 desingn 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‐sectinal 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.