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Any structural analysis which gives stresses and displacements for some predefined structure is governed by some physical domain of loading, geometry and boundary conditions. Let this domain be called the structures "problem space." In applying finite element analysis, the solution to any one problem space may be one of many admissible solutions all of which satisfy some given set of boundary conditions. Admissibility is determined by the stated problem with its boundary conditions along with computer storage capacity considerations. Obtaining the most exact approximate solutions is one of major concern to insure adequate results. This problem has been approached from a number of viewpoints [4-9] all of which employ some version of minimum potential energy [5, 10]. This report is a study of current approaches to this problem and their effect on finite element grid optimizations. Selected optimizations [4-9] are shown to be effective in producing better solutions but it is noted that the implementation of these optimizations may be difficult. To survey the situation two fixed problem spaces of a tapered beam and a cantilever beam are chosen for investigation. Conclusions based on this study display that optimizations methods applied to a finite element model give an optimum space arrangement that is a function of the selected element geometry and displacement function. When changes in the element geometry are introduced a new optimum results. Comparing test problem results leads to some speculation employing uniform strain energy as a better guide to "first guess" grid arrangement and a recommendation for further investigation in this direction.

<p> The thesis proposes a novel method for the computation of the design sensitivity of microwave network parameters. The approach is based on the finite-element method. When combined with the iterative update method (the Broyden method) during the gradient-based optimization process, the approach requires practically no overhead for the computation of the response Jacobian, thus accelerating the optimization. </p> <p> The efficiency and accuracy of the gradient-based optimization and the tolerance analysis greatly depend on the computation of the design sensitivity. However, common commercial full-wave electromagnetic solvers do not provide sensitivity information. With them, the design sensitivities are computed from the response themselves using finite-difference or higher-order approximations at the response level. Consequently, for each design parameter of interest, at least one additional full-wave analysis is performed. </p> <p> The proposed self-adjoint sensitivity analysis (SASA) is so far the most efficient way to extract the sensitivity information for the network parameters with the finite-element method. As an improvement of the adjoint-variable method (AVM), it eliminates the additional system analyses. With one single full-wave analysis, the sensitivities with respect to all design parameters are computed. This significantly improves the efficiency of the sensitivity computations. </p> <p> When employed in gradient-based optimization, the computational overhead of the SASA can be further reduced. Instead of the finite-difference approximation, the system matrix derivatives are updated iteratively using the Broyden update. This reduces the computational overhead of the sensitivity analysis to practically zero. Further, several switching criteria between the Broyden update and the finite-difference approximation of the system matrix derivatives is proposed to guarantee the robust convergence of the optimization algorithm. This leads to our Broyden/finite-difference SASA (B/FD-SASA). </p> <p> The efficiency in terms of CPU time as well as the accuracy of the SASA is verified by several numerical examples, where the reference results are provided through the traditional finite-difference approximations. Also, the efficiency of the B/FD-SASA is validated by a filter design example and a microwave imaging example, with implementations exploiting different gradientbased optimization algorithms. </p> / Thesis / Master of Applied Science (MASc)