A Study On Some Essential Problems In Topology Optimization Of Frame Structures Considering Dynamic Responses

With the fast development of modern society, people need to consider more factors in the design of engineering structures. Not only the service performance and the structural safety, but also the economy measures should also be taken into consideration,which make the structural desigh become increasingly difficult.Structural optimization especially the structural topology optimization design, provides an advanced tool for the design of innovative engineering structures with some optimal characteristics.Frame structure is a kind of commonly used load-carrying enginering structures. For its design, its dynamic characteristics is usually one of the main concerns. A large amount of studies indicate that the layout of structural components in a frame structure (i.e., structural topology) is the decisive factor for determining its dynamic characteristics. Therefore optimizing the layout of a frame structure with use of topology optimization methods is a effective way to improve its dynamic properties and has great potential in engineering applications.In the past, most of the research of topology optimization with dynamic characteristics of frame structure are based on ground strcture, and focus on the numericl method to sovle the large scale problems,lose sight of the mechanical and mathematical characteristics of the problems,which make the numerical method don’t obtain the optimal result or fails to converge. We think one of the most reason is lack of the deep theoretical analysis of this type of prolems,such as the shape of feasible region and optimal result characteristics. In fact, it’s different between topology optimization and size optimization when consider dynamic characteristics of frame strctures.And use the size optimization method to solve the issue problem through the desigh variable degradation strategy are not effective.In the present thesis, emphases are placed on some challenging issues in topology optimization of dynamic characteristics of frame structures, such as the connectivity of feasible region,the singularity/degenerative of low-dimensional sub-feasible region,the symmetry of optimal result,etc. The following results are obtained:(1) Reveals the property of topology optimization of frame strctures with frequency band constraints. Based on the exact dynamic stiffness method and the fundamental theory of W-W algorithm, we investigated the properties of the feasible domain of the corresponding optimization problem in a quantitative way. It was found that feasible domain may be disconnected and include several isolated sub-domains (even degenerated subdomains in topology optimization problems). This renders it very difficult for locating feasible solutions and finding singular optimal solution with use of gradient-based numerical optimization algorithms. Special treatments are needed to deal with this kind of problem.(2) Based on the idea of continuity restoration and constraint relaxation, a so-called PLMP(PoLynomial Material interPolation) method was proposed to deal with the corresponding strongly singular phenomenon.(3) A theoretical explanation is provided for a rather anti-intuition phenomenon in topology optimization of frame structures with frequency constraints, that is, the global optimal solution may be highly asymmetric even for a symmetric problem. Furthermore, a set of sufficient conditions guaranteeing the existence of symmtric global optimal solutions are proposed for a general class of topology optimization problems. This partially solved an open problem raised by Prof. Rozvany in2011.