Dual sequential approximation methods in structural optimisation

[摘要] ENGLISH ABSTRACT:This dissertation addresses a number of topics that arise from the use of a dual method of sequentialapproximate optimisation (SAO) to solve structural optimisation problems. Said approach iswidely used because it allows relatively large problems to be solved efficiently by minimising thenumber of expensive structural analyses required. Some extensions to traditional implementationsare suggested that can serve to increase the efficacy of such algorithms. The work presented hereinis concerned primarily with three topics: the use of nonconvex functions in the definition of SAOsubproblems, the global convergence of the method, and the application of the dual SAO approachto large-scale problems. Additionally, a chapter is presented that focuses on the interpretation ofSigmund's mesh independence sensitivity filter in topology optimisation.It is standard practice to formulate the approximate subproblems as strictly convex, since strictconvexity is a sufficient condition to ensure that the solution of the dual problem correspondswith the unique stationary point of the primal. The incorporation of nonconvex functions in thedefinition of the subproblems is rarely attempted. However, many problems exhibit nonconvexbehaviour that is easily represented by simple nonconvex functions. It is demonstrated herein that,under certain conditions, such functions can be fruitfully incorporated into the definition of theapproximate subproblems without destroying the correspondence or uniqueness of the primal anddual solutions.Global convergence of dual SAO algorithms is examined within the context of the CCSA method,which relies on the use and manipulation of conservative convex and separable approximations.This method currently requires that a given problem and each of its subproblems be relaxed toensure that the sequence of iterates that is produced remains feasible. A novel method, called thebounded dual, is presented as an alternative to relaxation. Infeasibility is catered for in the solutionof the dual, and no relaxation-like modification is required. It is shown that when infeasibility isencountered, maximising the dual subproblem is equivalent to minimising a penalised linear combinationof its constraint infeasibilities. Upon iteration, a restorative series of iterates is producedthat gains feasibility, after which convergence to a feasible local minimum is assured.Two instances of the dual SAO solution of large-scale problems are addressed herein. The firstis a discrete problem regarding the selection of the point-wise optimal fibre orientation in thetwo-dimensional minimum compliance design for fibre-reinforced composite plates. It is solvedby means of the discrete dual approach, and the formulation employed gives rise to a partiallyseparable dual problem. The second instance involves the solution of planar material distributionproblems subject to local stress constraints. These are solved in a continuous sense using a sparsesolver. The complexity and dimensionality of the dual is controlled by employing a constraintselection strategy in tandem with a mechanism by which inconsequential elements of the Jacobian of the active constraints are omitted. In this way, both the size of the dual and the amount ofinformation that needs to be stored in order to define the dual are reduced.

[发布机构] Stellenbosch University