Thesis (MEng)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: We investigate the simultaneous analysis and design (SAND) formulation of the topology optimisation
problem. The characteristics of the formulation are presented considering the simple
compliance/weight constrained problem and the more complex local stress constrained case.
The problems are solved in an efficient sparse sequential approximate optimisation (SAO) framework
with the SAND formulation showing an significant reduction in computational requirements
compared to the traditional and inherently expensive nested analysis and design (NAND) approach.
In SAND the state equations are included in the optimisation problem as a set of equality constraints
and not solved exactly in each iteration, as would be the case in NAND. Decision and state
variables are thus independent, resulting in an immensely sparse optimisation problem. The availability
of simple exact analytic expressions for all the constraint functions (via the finite element
method) allows for the construction of accurate approximate subproblems with little computational
effort. Furthermore, material can be removed completely from the design domain with few complications,
resulting in a decrease in subproblem size as the algorithm progresses, further reducing
computation time.
The inclusion of void material in the design domain leads to the formulation of stress constraints as
so-called ‘vanishing’ constraints. Furthermore, the SAND formulation provides a new perspective
on the infamous singularity problem. Amongst other results, we present some test cases that seem
to scale linearly in computational requirements for a specific range of problem sizes. / AFRIKAANSE OPSOMMING: Die formulering van die topologie optimerings probleem as ’n gelyktydige analise en ontwerp
(simultaneous analysis and design (SAND)) formulering word ondersoek. Die eienskappe van die
formulering word bespreek in die konteks van die eenvoudig begrensde styfheid/gewig geval en
die meer komplekse plaaslike spanning begrensde geval.
Die probleme word opgelos in ’n sekwenti¨ele benaderde optimering (SBO; sequential approximate
optimisation (SAO)) raamwerk met die SAND formulering, wat lei tot ’n wesenlike vermindering
in berekenings vereistes benodig in vergelyking met die tradisionele en inherente duur geneste
analise en ontwerp (nested analysis and design (NAND)) geval. In SAND word die vergelykings
wat die respons van die struktuur beskryf met gelykheidsbegrensings in die optimerings probleem
verteenwoordig. Die respons van die struktuur word dus nie presies opgelos in elke iterasie nie,
soos in die geval van NAND wel gebeur. Alle optimerings veranderlikes is dus onafhanklik en lei
tot ’n baie yl optimerings probleem. Deur middel van die eindige element metode is die analitiese
vorm van alle begrensings beskikbaar en kan dit gebruik word om akkurate benaderde subprobleme
op te stel sonder ekstra berekenings koste. Verder kan materiaal heeltemal verwyder uit van die
ontwerpsgebied met weinig komplikasies. Dit lei tot ’n verkleining van subprobleme soos die
algoritme vordering maak wat berekenings tyd nog meer verminder.
Die feit dat materiaal heeltemal verwyder kan word van die ontwerp gebied lei tot die formulering
van spannings begrensings as sogenaamde ‘verdwynende’ begrensings. Verder gee die SAND
formulering ’n nuwe uitsig op die bekende singulariteitsprobleem. Met verskeie ander resultate
word daar ook gewys dat dit voorkom of ’n spesifieke stel toetsprobleme lineˆer skaal in berekenings tyd.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/86679 |
| Date | 04 1900 |
| Creators | Munro, Dirk Pieter |
| Contributors | Groenwold, A. A., Stellenbosch University. Faculty of Engineering. Dept. of Mechanical and Mechatronic Engineering. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | English |
| Type | Thesis |
| Format | xii, 102 p. : ill. |
| Rights | Stellenbosch University |
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