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In this research, a new zeroth-order (non-gradient based) topology optimization methodology for compliance minimization was developed. It is called the Element Exchange Method (EEM). The principal operation in this method is to convert the less effective solid elements into void elements and the more effective void elements into solid elements while maintaining the overall volume fraction constant. The methodology can be integrated with existing FEA codes to determine the stiffness or other structural characteristics of each candidate design during the optimization process. This thesis provides details of the EEM algorithm, the element exchange strategy, checkerboard control, and the convergence criteria. The results for several two- and three-dimensional benchmark problems are presented with comparisons to those found using other stochastic and gradient-based approaches. Although EEM is not as efficient as some gradient-based methods, it is found to be significantly more efficient than many other non-gradient methods reported in the literature such as GA and PSO.

In this dissertation, we study the isotopy problem for a certain three-dimensional contactomorphism which is supported in a neighbourhood of an embedded 2-sphere with standard characteristic foliation. The diffeomorphism which underlies it is the Dehn twist on the sphere, and therefore its square becomes smoothly isotopic to the identity. The main result of this dissertation gives conditions under which any iterate of the Dehn twist along a non-trivial sphere is not contact isotopic to the identity. This provides the first examples of exotic contactomorphisms with infinite order in the contact mapping class group, as well as the first examples of exotic contactomorphisms of 3-manifolds with b_1 = 0. The proof crucially relies on the construction of an invariant for families of contact structures in monopole Floer homology which generalises the Kronheimer--Mrowka--Ozsváth--Szabó contact invariant, together with the nice interaction between this families invariant and the U map in Floer homology.

Yes / The aim of this article is to apply and improve the species-conserving genetic algorithm (SCGA) to search multiple solutions of truss topology optimization problems in a single run. A species is defined as a group of individuals with similar characteristics and is dominated by its species seed. The solutions of an optimization problem will be selected from the found species. To improve the accuracy of solutions, a species mutation technique is introduced to improve the fitness of the found species seeds and the combination of a neighbour mutation and a uniform mutation is applied to balance exploitation and exploration. A real vector is used to represent the corresponding cross-sectional areas and a member is thought to be existent if its area is bigger than a critical area. A finite element analysis model was developed to deal with more practical considerations in modelling, such as the existence of members, kinematic stability analysis, and computation of stresses and displacements. Cross-sectional areas and node connections are decision variables and optimized simultaneously to minimize the total weight of trusses. Numerical results demonstrate that some truss topology optimization examples have many global and local solutions, different topologies can be found using the proposed algorithm on a single run and some trusses have smaller weights than the solutions in the literature.

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With the fast development of communication systems, computers and consumer electronics, the power supplies for telecoms, servers, desktops, laptops, flat-panel TVs, LED lighting, etc. are required for more efficient power delivery with smaller spaces. The LLC resonant converter has been widely adopted for these applications due to the advantages in high efficiency, high power density and holdup time operation capability. However, LLC resonant converter meets some issues, especially in high output current applications. Those issues include magnetic design, start-up, short-circuit protection, synchronous rectifier drive, EMI noise and integration, etc. To solve those issues, like start-up and short-circuit protection, SR driving and EMI, etc., a synthesis method is proposed to find the similar resonant topologies like LLC. Based on this method, lots of multi-element resonant converters are found to solve the issues that LLC resonant converter cannot handle. To evaluate the performance of found numerous valuable topologies. Thus, a general evaluation system is required. State-plane analysis with new normalization factors is utilized. Based on it, the voltage stress, current stresses and apparent power of resonant converters are easy to compare. This method can help select suitable circuit topology for certain applications. Meanwhile, it also can help resonant converters' design. The important performance factors, like start-up, short-circuit protection, SR driving, integration and EMI performance, are also taken into account for the whole evaluation system. The high switching frequency is needed recently for high power density requirement. However, LLC resonant converter suffers high transformer loss. Matrix transformer is introduced to reduce winding loss and total volume. Flux cancellation method is utilized to reduce core size and loss. Synchronous Rectifier (SR) devices and output capacitors are integrated into secondary windings to eliminate termination related winding losses, via loss and reduce leakage inductance. The passive integration is necessary for high power density resonant converter, especially for high order system. Based on stress, suitable passive components are chosen for integration. Then, the magnetic integration method is shown based on multi-winding transformer structure. The passive integration principles are discussed. A novel passive integration method is proposed for multi-elements resonant converters. In conclusion, this work is focus on the topology analysis and integration of resonant converters. Searching the suitable topologies for certain application, and evaluate the performance of them. Then, improve the system power density by integration techniques. / Ph. D.

"The Fine Topology" C(X,Y) where (Y,d) is a metric space is referred to, in an exercise in [14], as the topology generated by basic open neighborhoods of the form B(f,E) = {g: d(f(x),g(x)) < E(x)} where E is a positive continuous real valued function. So in the fine topology, a function g is close to f if g(x) is continuously close to f(x); whereas in the uniform topology, g(x) must be uniformly close to f(x), that is, within a constant distance of f(x). So the fine topology is an obvious refinement of the uniform topology. This topology has not been extensively studied before, and it is the purpose of this paper to see how the fine topology fits in with the lattice of other well studied topologies on C(X,Y), and to study some properties of this topology in itself. Furthermore, other results on these well studied topologies will-be examined and compared with the fine topology. / Ph. D.

<p dir="ltr">We develop a cyclic theory for categorical coalgebras and show that, when applied to the categorical coalgebra of singular chains on a space, this provides an algebraic model for its free loop space as an S¹-space. In other words, the natural circle action on loop spaces, given by rotation of loops, is encoded in the algebraic structure. In particular, the cyclic homology of the categorical coalgebra of singular chains on a topological space X is isomorphic to the S¹-equivariant homology of the free loop space. This extends known results relating cyclic theories for the algebra of chains on the based loop space and the equivariant homology of its free loop space. In fact, our statements do not require X to be simply connected, and we work over an arbitrary commutative ring. Along the way, we introduce a family of polytopes, coined as Goodwillie polytopes, that control the combinatorics behind the relationship of the coHochschild complex of a categorical coalgebra and the Hochschild complex of its associated differential graded category.</p>

This thesis covers fundamental properties of metric spaces, as well as completeness, compactness, and separability of metric spaces.