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MSc(Eng),School of Civil and Environmental Engineering, Faculty of Engineering and the Built Environment, University of the Witwatersrand, 2010 / This dissertation presents a new method for the automated optimisation of structures. The method has been developed to: (1) select sections to satisfy strength and deflection requirements using minimum material, and (2) efficiently group members. The member selection method is based on the principle of virtual work, and is called the Virtual Work Optimisation (VWO) method. It addresses multiple deflection and load case constraints simultaneously. The method determines which sections provide the highest deflection and strength resistance per unit mass. When compared to several other methods in the literature, and designs from industry, the VWO method produced savings of up to 15.1%. A parametric investigation of ungrouped, multi-storey frames is conducted using the VWO method to determine optimal mass and stiffness distributions. Unusual mass patterns have been found. Diagonal paths of increased stiffness are formed in the frames, which suggests truss behaviour. A grouping algorithm is presented which determines how efficiently to create a specified number of groups in a structure. The VWO method has been incorporated into the automated algorithm to optimise the grouped structures. Members are grouped according to their mass per unit length. In the algorithm an exhaustive search of all feasible grouping permutations is carried out, and the lightest structure selected. Results produced are up to 5.9% lighter than those obtained using ad hoc grouping configurations found in the literature and based on experience.

Evolutionary Structural Optimisation (ESO) is a relatively new design tool used to improve and optimise the design of structures. It is a heuristic method where a few elements of an initial design domain of finite elements are iteratively removed. Such a process is carried out repeatedly until an optimum design is achieved, or until a desired given area or volume is reached. There have been many contributions to the ESO procedure since its conception back in 1992. For example, a provision known as Bi-Directional ESO (BESO) has now been incorporated where elements may not only be removed, but added. Also, rather than deal with elements where they are either present or not, the designer now has the option to change the element's properties in a progressive fashion. This includes the modulus of elasticity, the density of the material and the thickness of plate elements, and is known as Morphing ESO. In addition to the algorithmic aspects of ESO, a large preference exists to optimise a structure based on a selection of criteria for various physical processes. Such examples include stress minimisation, buckling and electromagnetic problems. In a changing world that demands the enhancement of design tools and methods that incorporate optimisation, the development of methods like ESO to accommodate this demand is called for. It is this demand that this thesis seeks to satisfy. This thesis develops and examines the concept of multicriteria optimisation in the ESO process. Taking into account the optimisation of numerous criteria simultaneously, Multicriteria ESO allows a more realistic and accurate approach to optimising a model in any given environment. Two traditional methods � the Weighting method and the Global Criterion (Min-max) method have been used, as has two unconventional methods � the Logical AND method and the Logical OR method. These four methods have been examined for different combinations of Finite Element Analysis (FEA) solver types. This has included linear static FEA solver, the natural frequency FEA solver and a recently developed inertia FE solver. Mean compliance minimisation (stiffness maximisation), frequency maximisation and moment of inertia maximisation are an assortment of the specific objectives incorporated. Such a study has provided a platform to use many other criteria and multiple combinations of criteria. In extending the features of ESO, and hence its practical capabilities as a design tool, the creation of another optimisation method based on ESO has been ushered in. This method concerns the betterment of the bending and rotational performance of cross-sectional areas and is known as Evolutionary Moment of Inertia Optimisation (EMIO). Again founded upon a domain of finite elements, the EMIO method seeks to either minimise or maximise the rectangular, product and polar moments of inertia. This dissertation then goes one step further to include the EMIO method as one of the objectives considered in Multicriteria ESO as mentioned above. Most structures, (if not all) in reality are not homogenous as assumed by many structural optimisation methods. In fact, many structures (particularly biological ones) are composed of different materials or the same material with continually varying properties. In this thesis, a new feature called Constant Width Layer (CWL) ESO is developed, in which a distinct layer of material evolves with the developing boundary. During the optimisation process, the width of the outer surrounding material remains constant and is defined by the user. Finally, in verifying its usefulness to the practical aspect of design, the work presented herein applies the CWL ESO and the ESO methods to two dental case studies. They concern the optimisation of an anterior (front of the mouth) ceramic dental bridge and the optimisation of a posterior (back of the mouth) ceramic dental bridge. Comparisons of these optimised models are then made to those developed by other methods.

While designing jet-engine components information about the loads that the component will be subjected to, is critical. For this, a full system analysis of the engine is often performed with every component put together in a large finite element model, which is called the whole engine model (WEM). This model will mostly be composed of lower order shell elements with a few thousands of elements. At the design level of a component, the FE model is detailed with several hundred thousand higher order solid elements. The detailed model cannot be directly put into the whole engine model due to excessive run times. Therefore there must be a simpler representation of the component -a sub-model- with much fewer elements so that it can be assembled into the whole engine model. This simple model must have the same stiffness (load/displacement) in chosen directions, the same mass and the basic mode shapes and frequencies should also be the same with the detailed model. 4 different structural optimisation schemes were studied to prepare a model: sizing optimisation, optimisation with material properties as design variables, combined sizing and material property optimisation and free-size optimisation. Among these free-size optimisation where each element in the model has its own design variable -thickness- was found to be the most effective method. The stiffness could be matched to the detailed model as close as 5% and so also could the first two fundamental mode shapes and frequencies. Additionally, the initial sub-model prepared was used to do a preliminary study on how variations in casting dimensions would affect the stiffness of the component in a certain direction. This was done by creating a design of experiments (DoE) for the stiffness. A response surface for the stiffness was created in terms of the dimensions that have the most significant effect. This was later used to predict an interval for the stiffness based on variations in the cast dimensions with a confidence level of 99.7%.

Evolutionary Structural Optimisation (ESO) is a relatively new design tool used to improve and optimise the design of structures. It is a heuristic method where a few elements of an initial design domain of finite elements are iteratively removed. Such a process is carried out repeatedly until an optimum design is achieved, or until a desired given area or volume is reached. There have been many contributions to the ESO procedure since its conception back in 1992. For example, a provision known as Bi-Directional ESO (BESO) has now been incorporated where elements may not only be removed, but added. Also, rather than deal with elements where they are either present or not, the designer now has the option to change the element's properties in a progressive fashion. This includes the modulus of elasticity, the density of the material and the thickness of plate elements, and is known as Morphing ESO. In addition to the algorithmic aspects of ESO, a large preference exists to optimise a structure based on a selection of criteria for various physical processes. Such examples include stress minimisation, buckling and electromagnetic problems. In a changing world that demands the enhancement of design tools and methods that incorporate optimisation, the development of methods like ESO to accommodate this demand is called for. It is this demand that this thesis seeks to satisfy. This thesis develops and examines the concept of multicriteria optimisation in the ESO process. Taking into account the optimisation of numerous criteria simultaneously, Multicriteria ESO allows a more realistic and accurate approach to optimising a model in any given environment. Two traditional methods � the Weighting method and the Global Criterion (Min-max) method have been used, as has two unconventional methods � the Logical AND method and the Logical OR method. These four methods have been examined for different combinations of Finite Element Analysis (FEA) solver types. This has included linear static FEA solver, the natural frequency FEA solver and a recently developed inertia FE solver. Mean compliance minimisation (stiffness maximisation), frequency maximisation and moment of inertia maximisation are an assortment of the specific objectives incorporated. Such a study has provided a platform to use many other criteria and multiple combinations of criteria. In extending the features of ESO, and hence its practical capabilities as a design tool, the creation of another optimisation method based on ESO has been ushered in. This method concerns the betterment of the bending and rotational performance of cross-sectional areas and is known as Evolutionary Moment of Inertia Optimisation (EMIO). Again founded upon a domain of finite elements, the EMIO method seeks to either minimise or maximise the rectangular, product and polar moments of inertia. This dissertation then goes one step further to include the EMIO method as one of the objectives considered in Multicriteria ESO as mentioned above. Most structures, (if not all) in reality are not homogenous as assumed by many structural optimisation methods. In fact, many structures (particularly biological ones) are composed of different materials or the same material with continually varying properties. In this thesis, a new feature called Constant Width Layer (CWL) ESO is developed, in which a distinct layer of material evolves with the developing boundary. During the optimisation process, the width of the outer surrounding material remains constant and is defined by the user. Finally, in verifying its usefulness to the practical aspect of design, the work presented herein applies the CWL ESO and the ESO methods to two dental case studies. They concern the optimisation of an anterior (front of the mouth) ceramic dental bridge and the optimisation of a posterior (back of the mouth) ceramic dental bridge. Comparisons of these optimised models are then made to those developed by other methods.

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.

In this thesis we show that the finite element error for the high contrast elliptic interface problem is independent of the contrast in the material coefficient under certain assumptions. The error estimate is proved using a particularly technical proof with construction of a specific function from the finite dimensional space of piecewise linear functions. We review the multiscale finite element method of Chu, Graham and Hou to give clearer insight. We present some generalisations to extend their work on a priori contrast independent local boundary conditions, which are then used to find multiscale basis functions by solving a set of local problems. We make use of their regularity result to prove a new relative error estimate for both the standard finte element method and the multiscale finite element method that is completely coefficient independent. The analytical results we explore in this thesis require a complicated construction. To avoid this we present an adaptive multiscale finite element method as an enhancement to the adaptive local-global method of Durlofsky, Efendiev and Ginting. We show numerically that this adaptive method converges optimally as if the coefficient were smooth even in the presence of singularities as well as in the case of a realisation of a random field. The novel application of this thesis is where the adaptive multiscale finite element method has been applied to the linear elasticity problem arising from the structural optimisation process in mechanical engineering. We show that a much smoother sensitivity profile is achieved along the edges of a structure with the adaptive method and no additional heuristic smoothing techniques are needed. We finally show that the new adaptive method can be efficiently implemented in parallel and the processing time scales well as the number of processors increases. The biggest advantage of the multiscale method is that the basis functions can be repeatedly used for additional problems with the same high contrast material coefficient.

Shape control of a structure with distributed piezoelectric actuators can be achieved through optimally selecting the loci, shapes and sizes of the piezoelectric actuators and choosing the electric fields applied to the actuators. Shape control can be categorised as either static or dynamic shape control. Whether it is a transient or gradual change, static or dynamic shape control, both aim to determine the loci, sizes, and shapes of piezoelectric actuators, and the applied voltages such that a desired structural shape is achieved effectively. This thesis is primarily concerned with establishing a finite element formulation for the general smart laminated composite plate structure, which is capable to analyse static and dynamic deformation using non-rectangular elements. The mechanical deformation of the smart composite plate is modelled using a third order plate theory, while the electric field is simulated based on a layer-wise theory. The finite element formulation for static and dynamics analysis is verified by comparing with available numerical results. Selected experiments have also been conducted to measure structural deformation and the experimental results are used to correlate with those of the finite element formulation for static analysis. In addition, the Linear Least Square (LLS) method is employed to study the effect of different piezoelectric actuator patch pattern on the results of error function, which is the least square error between the calculated and desired structural shapes in static structural shape control. The second issue of this thesis deals with piezoelectric actuator design optimisation (PADO) for quasi-static shape control by finding the applied voltage and the configuration of piezoelectric actuator patch to minimise error function, whereas the piezoelectric actuator configuration is defined based on the optimisation technique of altering nodal coordinates (size/shape optimisation) or eliminating inefficient elements in a structural mesh (topology optimisation). Several shape control algorithms are developed to improve the structural shape control by reducing the error function. Further development of the GA-based voltage and piezoelectric actuator design optimisation method includes the constraint handling, where the error function can be optimised subjected to energy consumption or other way around. The numerical examples are presented in order to verify that the proposed algorithms are applicable to quasi-static shape control based on voltage and piezoelectric actuator design optimisation (PADO) in terms of minimising the error function. The third issue is to use the present finite element formulation for a modal shape control and for controlling resonant vibration of smart composite plate structures. The controlled resonant vibration formulation is developed. Modal analysis and LLS methods are also employed to optimise the applied voltage to piezoelectric actuators for achieving the modal shapes. The Newmark direct time integration method is used to study harmonic excitation of smart structures. Numerical results are presented to induce harmonic vibration of structure with controlled magnitude via adjusting the damping and to verify the controlled resonant vibration formulation.

Shape control of a structure with distributed piezoelectric actuators can be achieved through optimally selecting the loci, shapes and sizes of the piezoelectric actuators and choosing the electric fields applied to the actuators. Shape control can be categorised as either static or dynamic shape control. Whether it is a transient or gradual change, static or dynamic shape control, both aim to determine the loci, sizes, and shapes of piezoelectric actuators, and the applied voltages such that a desired structural shape is achieved effectively. This thesis is primarily concerned with establishing a finite element formulation for the general smart laminated composite plate structure, which is capable to analyse static and dynamic deformation using non-rectangular elements. The mechanical deformation of the smart composite plate is modelled using a third order plate theory, while the electric field is simulated based on a layer-wise theory. The finite element formulation for static and dynamics analysis is verified by comparing with available numerical results. Selected experiments have also been conducted to measure structural deformation and the experimental results are used to correlate with those of the finite element formulation for static analysis. In addition, the Linear Least Square (LLS) method is employed to study the effect of different piezoelectric actuator patch pattern on the results of error function, which is the least square error between the calculated and desired structural shapes in static structural shape control. The second issue of this thesis deals with piezoelectric actuator design optimisation (PADO) for quasi-static shape control by finding the applied voltage and the configuration of piezoelectric actuator patch to minimise error function, whereas the piezoelectric actuator configuration is defined based on the optimisation technique of altering nodal coordinates (size/shape optimisation) or eliminating inefficient elements in a structural mesh (topology optimisation). Several shape control algorithms are developed to improve the structural shape control by reducing the error function. Further development of the GA-based voltage and piezoelectric actuator design optimisation method includes the constraint handling, where the error function can be optimised subjected to energy consumption or other way around. The numerical examples are presented in order to verify that the proposed algorithms are applicable to quasi-static shape control based on voltage and piezoelectric actuator design optimisation (PADO) in terms of minimising the error function. The third issue is to use the present finite element formulation for a modal shape control and for controlling resonant vibration of smart composite plate structures. The controlled resonant vibration formulation is developed. Modal analysis and LLS methods are also employed to optimise the applied voltage to piezoelectric actuators for achieving the modal shapes. The Newmark direct time integration method is used to study harmonic excitation of smart structures. Numerical results are presented to induce harmonic vibration of structure with controlled magnitude via adjusting the damping and to verify the controlled resonant vibration formulation.

In this paper the parametric design and functional optimisation of thin-walled structures made from plastics for food packaging is considered. These objects are produced in such vast numbers each year that one important task in the design of these objects is to minimise the amount of plastic used, subject to functional constraints, to reduce the costs of production and to conserve raw materials. By means of performing an automated optimisation on the possible shapes of the food containers, where the geometry is parametrised succinctly, a strategy to create the optimal design of the containers subject to a given set of functional constraints is demonstrated.