Shape optimization for beam structural design
Authors:   , Lee Hsiender 
Published: December 2024

"Optimization is concerned with achieving the best outcome of a given objective while satisfying certain restrictions. The central purpose of structural analysis is to predict the behavior of known structural configurations which are subject to known distributions of specified loads, displacements, and temperatures. As the engineers were able to perform structural design, it was natural to introduce optimization schemes into structural analysis. Design optimization provides not only savings in structural weight and cost but also allows sensitivity studies which provide valuable information about the design. Engineering design optimization is a fundamental objective of virtually every engineer who strives to create a component, device, or system to meet a need. To fulfill this objective, engineers must have organized methods tailored to the special features and needs of these fields of engineering design. Thus, an applicable method of optimal design for finite dimensional optimization and a nonlinear mathematical programming method will be presented in this dissertation. Also because the finite element method is the principal modeling method, it is incorporated as an integral part of optimal design method. This is difficult to accomplish with a commercially available finite element code. It is very important to realize that engineering design optimization and engineering analysis are fundamentally different in nature. In analysis, one is generally assured that a solution exists and the numerical methods used are stable. In optimal design, on the other hand, existence of even a nominal design satisfying the constraints is not assured, much less the existence of an optimum design. Even when an optimum design exists, numerical methods for its solution are often quite sensitive to initial estimates and require considerable computational art to achieve iterative convergence. These properties will be observed again and again in this dissertation. In the structural optimization of discrete structures, much of the work has been concerned with truss problems [1,2,3,4,5,6,7,8,9]. Some work has also been reported for frame st ructures[10,11,12]. However, the vast majority of this effort was almost exclusively focused on member sizing(fixed-shape) problems where the design variables are the cross sectional area of the truss members. Relatively little effort has been devoted to the configuration problem where the node positions in the finite element structure are treated as design variables, and virtually no effort has been devoted to the topological problem of choosing the number of members and their connectivity in the structure. The beam member is the most common and an important structural system for a variety of industrial applications. In this dissertation, applications of the finite element method and nonlinear mathematical programming techniques for the optimum design of beam frame structures subject to sizing changes, configurational changes and topological changes is presented. These features are required in order to produce a "true" structural design/optimization tool. Anything less will simply allow the user to "fine tune" an existing design. In the context of shape optimization, the geometric design variables in discrete structures are the coordinates of node locations(joints) in the structure. The shape geometry can be optimized through the coordinate change or use of parametric cubic curves. For beam-type frames, this means node locations are allowed to vary. Commercial general purpose finite element codes do not appear to be particular useful in configuration design because of rigid input and output formats and no way to access intermediate information. This creates a need for a special purpose code for beam finite elements to calculate the static response. The developed beam elements will meet the particular requirements in configuration design and do not have the tradeoffs associated with a commercial general purpose finite element package. For this research project, it was decided to develop a finite element code including tubular circular, tubular rectangular and symmetric open channel cross sections for sizing, configurational and topological design for three dimensional beam frame structures; and to also develop an efficient algorithm based on combining a finite element analysis method and a nonlinear mathematical programming approach subject to multiple loading conditions and displacement and stress constraints. In Chapter 2, an overview of structural optimization will be presented as well as a literature review covering the development and application of structural optimization. The potential for growth in the area of structural optimization is also discussed. In Chapter 3 some general topics related to the optimization of beam frame structural systems are discussed. Approaches for finite element analysis and optimization algorithms are presented. In Chapter 4 the gradient computations required for the development of structural optimization are presented. The calculation of sensitivities with respect to static response is reviewed. Chapter 5 introduces the topic of geometric modeling for structural optimization and topological design, and presents the shape optimization for beam frame design. Chapter 6 contains the research results regarding the application to beam structures subject to sizing, shape geometry and topological changes. Chapter 7 presents the conclusions of the research and recommendations for future studies."--Introduction.