Topology optimisation and simultaneous analysis and design : material penalisation and local stress constraints

Munro, Dirk Pieter (2014-04)

Thesis (MEng)--Stellenbosch University, 2014.

Thesis

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.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/86679
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