Thesis (PhD)--Stellenbosch University, 2012 / ENGLISH ABSTRACT: This dissertation addresses a number of topics that arise from the use of a dual method of sequential
approximate optimisation (SAO) to solve structural optimisation problems. Said approach is
widely used because it allows relatively large problems to be solved efficiently by minimising the
number of expensive structural analyses required. Some extensions to traditional implementations
are suggested that can serve to increase the efficacy of such algorithms. The work presented herein
is concerned primarily with three topics: the use of nonconvex functions in the definition of SAO
subproblems, the global convergence of the method, and the application of the dual SAO approach
to large-scale problems. Additionally, a chapter is presented that focuses on the interpretation of
Sigmund’s mesh independence sensitivity filter in topology optimisation.
It is standard practice to formulate the approximate subproblems as strictly convex, since strict
convexity is a sufficient condition to ensure that the solution of the dual problem corresponds
with the unique stationary point of the primal. The incorporation of nonconvex functions in the
definition of the subproblems is rarely attempted. However, many problems exhibit nonconvex
behaviour 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 the
approximate subproblems without destroying the correspondence or uniqueness of the primal and
dual 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 to
ensure that the sequence of iterates that is produced remains feasible. A novel method, called the
bounded dual, is presented as an alternative to relaxation. Infeasibility is catered for in the solution
of the dual, and no relaxation-like modification is required. It is shown that when infeasibility is
encountered, maximising the dual subproblem is equivalent to minimising a penalised linear combination
of its constraint infeasibilities. Upon iteration, a restorative series of iterates is produced
that 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 first
is a discrete problem regarding the selection of the point-wise optimal fibre orientation in the
two-dimensional minimum compliance design for fibre-reinforced composite plates. It is solved
by means of the discrete dual approach, and the formulation employed gives rise to a partially
separable dual problem. The second instance involves the solution of planar material distribution
problems subject to local stress constraints. These are solved in a continuous sense using a sparse
solver. The complexity and dimensionality of the dual is controlled by employing a constraint
selection 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 of
information that needs to be stored in order to define the dual are reduced. / AFRIKAANSE OPSOMMING: Hierdie proefskrif spreek ’n aantal onderwerpe aan wat spruit uit die gebruik van ’n duale metode
van sekwensi¨ele benaderde optimering (SBO; sequential approximate optimisation (SAO)) om
strukturele optimeringsprobleme op te los. Hierdie benadering word breedvoerig gebruik omdat
dit die moontlikheid skep dat relatief groot probleme doeltreffend opgelos kan word deur die aantal
duur strukturele analises wat vereis word, te minimeer. Sommige uitbreidings op tradisionele
implementerings word voorgestel wat kan dien om die doeltreffendheid van sulke algoritmes te
verhoog. Die werk wat hierin aangebied word, het hoofsaaklik betrekking op drie onderwerpe: die
gebruik van nie-konvekse funksies in die defini¨ering van SBO-subprobleme, die globale konvergensie
van die metode, en die toepassing van die duale SBO-benadering op grootskaalse probleme.
Daarbenewens word ’n hoofstuk aangebied wat fokus op die interpretasie van Sigmund se maasonafhanklike
sensitiwiteitsfilter (mesh independence sensitivity filter) in topologie-optimering.
Dit is standaard praktyk om die benaderde subprobleme as streng konveks te formuleer, aangesien
streng konveksiteit ’n voldoende voorwaarde is om te verseker dat die oplossing van die duale
probleem ooreenstem met die unieke stasionˆere punt van die primaal. Die insluiting van niekonvekse
funksies in die definisie van die subprobleme word selde gepoog. Baie probleme toon
egter nie-konvekse gedrag wat maklik deur eenvoudige nie-konvekse funksies voorgestel kan word.
In hierdie werk word daar gedemonstreer dat sulke funksies onder sekere voorwaardes met vrug in
die definisie van die benaderde subprobleme inkorporeer kan word sonder om die korrespondensie
of uniekheid van die primale en duale oplossings te vernietig.
Globale konvergensie van duale SBO-algoritmes word ondersoek binne die konteks van die CCSAmetode,
wat afhanklik is van die gebruik en manipulering van konserwatiewe konvekse en skeibare
benaderings. Hierdie metode vereis tans dat ’n gegewe probleem en elk van sy subprobleme verslap
word om te verseker dat die sekwensie van iterasies wat geproduseer word, toelaatbaar bly. ’n
Nuwe metode, wat die begrensde duaal genoem word, word aangebied as ’n alternatief tot verslapping.
Daar word vir ontoelaatbaarheid voorsiening gemaak in die oplossing van die duaal, en geen
verslappings-tipe wysiging word benodig nie. Daar word gewys dat wanneer ontoelaatbaarheid
te¨engekom word, maksimering van die duaal-subprobleem ekwivalent is aan minimering van sy
begrensingsontoelaatbaarhede (constraint infeasibilities). Met iterasie word ’n herstellende reeks
iterasies geproduseer wat toelaatbaarheid bereik, waarna konvergensie tot ’n plaaslike KKT-punt
verseker word.
Twee gevalle van die duale SBO-oplossing van grootskaalse probleme word hierin aangespreek.
Die eerste geval is ’n diskrete probleem betreffende die seleksie van die puntsgewyse optimale
veselori¨entasie in die tweedimensionele minimum meegeefbaarheidsontwerp vir veselversterkte
saamgestelde plate. Dit word opgelos deur middel van die diskrete duale benadering, en die formulering wat gebruik word, gee aanleiding tot ’n gedeeltelik skeibare duale probleem. Die tweede
geval behels die oplossing van in-vlak materiaalverspredingsprobleme onderworpe aan plaaslike
spanningsbegrensings. Hulle word in ’n kontinue sin opgelos met die gebruik van ’n yl oplosser.
Die kompleksiteit en dimensionaliteit van die duaal word beheer deur gebruik te maak van ’n
strategie om begrensings te selekteer tesame met ’n meganisme waardeur onbelangrike elemente
van die Jacobiaan van die aktiewe begrensings uitgelaat word. Op hierdie wyse word beide die
grootte van die duaal en die hoeveelheid inligting wat gestoor moet word om die duaal te definieer,
verminder.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/20033 |
| Date | 03 1900 |
| Creators | Wood, Derren Wesley |
| Contributors | Groenwold, A. A., Stellenbosch University. Faculty of Engineering. Dept.of Mechanical & Mechatronic Engineering. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | English |
| Type | Thesis |
| Format | 220 p. : ill. |
| Rights | Stellenbosch University |
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